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"Michelson, David George"@en .
"2009-04-14T18:43:59Z"@en .
"1993"@en .
"Doctor of Philosophy - PhD"@en .
"University of British Columbia"@en .
"Recent developments in radar navigation and remote sensing have led to a requirement for\r\nrugged yet inexpensive location markers and calibration targets which present both a very large\r\nscattering cross section and a specified polarization response over a wide angular range. This\r\nstudy considers several problems related to the analysis and design of passive radar targets\r\nderived from corner reflectors.\r\nTransformation of the polarization response of a target between global and local coordinate\r\nframes is shown to correspond to rotation of the polarization basis by a prescribed angle which is\r\na function of both the coordinate transformation matrix and the direction of propagation. Once\r\nthe angle of rotation has been determined using either spherical trigonometry or vector algebra,\r\nany polarization descriptor can be transformed between coordinate frames by application of a\r\nsuitable rotation operator.\r\nThe scattering cross section and angular coverage of a conventional trihedral corner reflector\r\ncan be altered by modifying the size and shape of its reflecting panels. A numerical algorithm\r\nbased on physical optics is used to predict the contribution of triple-bounce reflections to the\r\nresponse of a reflector with polygonal panels of arbitrary shape. If three-fold symmetry is\r\nbroken and the reflector is simply required to present bilateral symmetry, it is found that the\r\nscattering cross section, elevation beamwidth, and azimuthal beamwidth of the reflector can be\r\nchosen independently of each other.\r\nA method for altering the polarization response of a conventional trihedral corner reflector\r\nby adding conducting fins or corrugations to one its interior surfaces is proposed. In calculating\r\ndesign curves for twist-polarizing or circularly polarizing reflectors by mode-matching, optimum accuracy and efficiency are obtained by setting the ratio of free space to groove modes\r\nequal to the ratio of groove width to the period. Methods for obtaining linear and circular\r\npolarization selective responses are considered. The contribution of triple-bounce reflections to\r\nthe response of such reflectors is a function of the direction of incidence, the orientation of the\r\nreflector, the dimensions of the corrugations, and the size and shape of the reflecting panels.\r\nExperimental results show that prototype twist-polarizing and circularly polarizing reflectors\r\nrespond essentially as predicted."@en .
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"RADAR CROSS SECTION ENHANCEMENTFOR RADAR NAVIGATION AND REMOTE SENSINGByDavid George MichelsonB. A. Sc., The University of British Columbia, 1982M. A. Sc., The University of British Columbia, 1986A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF ELECTRICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1993\u00C2\u00AE David George Michelson, 1993In presenting this thesis in partial fulfilment of the requirements for an advanced degree at theUniversity of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarlypurposes may be granted by the head of my department or by his or her representatives. Itis understood that copying or publication of this thesis for financial gain shall not be allowedwithout my written permission.David G. MichelsonThe University of British Columbia,Department of Electrical Engineering,2356 Main Mall,Vancouver, B.C.CanadaV6T 1Z4Date: 29 December 1993AbstractResearch Supervisor: Prof. E.V. JullRecent developments in radar navigation and remote sensing have led to a requirement forrugged yet inexpensive location markers and calibration targets which present both a very largescattering cross section and a specified polarization response over a wide angular range. Thisstudy considers several problems related to the analysis and design of passive radar targetsderived from corner reflectors.Transformation of the polarization response of a target between global and local coordinateframes is shown to correspond to rotation of the polarization basis by a prescribed angle which isa function of both the coordinate transformation matrix and the direction of propagation. Oncethe angle of rotation has been determined using either spherical trigonometry or vector algebra,any polarization descriptor can be transformed between coordinate frames by application of asuitable rotation operator.The scattering cross section and angular coverage of a conventional trihedral corner reflectorcan be altered by modifying the size and shape of its reflecting panels. A numerical algorithmbased on physical optics is used to predict the contribution of triple-bounce reflections to theresponse of a reflector with polygonal panels of arbitrary shape. If three-fold symmetry isbroken and the reflector is simply required to present bilateral symmetry, it is found that thescattering cross section, elevation beamwidth, and azimuthal beamwidth of the reflector can bechosen independently of each other.A method for altering the polarization response of a conventional trihedral corner reflectorby adding conducting fins or corrugations to one its interior surfaces is proposed. In calculatingdesign curves for twist-polarizing or circularly polarizing reflectors by mode-matching, optimum accuracy and efficiency are obtained by setting the ratio of free space to groove modesequal to the ratio of groove width to the period. Methods for obtaining linear and circularIIpolarization selective responses are considered. The contribution of triple-bouhce reflections tothe response of such reflectors is a function of the direction of incidence, the orientation of thereflector, the dimensions of the corrugations, and the size and shape of the reflecting panels.Experimental results show that prototype twist-polarizing and circularly polarizing reflectorsrespond essentially as predicted.InTable of ContentsAbstract jjTable of Contents ivList of Tables viiList of Figures viiiAcknowledgements xviii1 Introduction1.1 Background and Motivation 11.2 Outline 7References 92 Transformation of Polarization Descriptors Between Coordinate Frames2.1 Introduction 112.2 Wave Polarization 132.3 Transformation of Coordinates 162.4 Evaluation of the Coordinate Transformation Matrix 232.5 Rotation of the Basis of Common Polarization Descriptors 262.6 Conclusions 44References 453 Truncation and Compensation of Trihedral Corner Reflectors3.1 Introduction 473.2 Analysis 49iv3.3 Reflectors with Three-Fold Symmetry3.4 Reflectors with Bilateral Symmetry3.5 Effect of Errors in Construction on Reflector Performance3.6 ConclusionsReferences4 Depolarizing Trihedral Corner Reflectors4.1 Introduction4.2 Scattering by a Conducting Grating with Rectangular Grooves4.3 Scattering by a Depolarizing Trihedral Corner Reflector4.4 Numerical and Experimental Results4.4.1 Design and Construction of the Prototype Trihedral Corner4.4.2 Polarization Response4.4.3 Azimuthal Response4.5 ConclusionsReferences5 Summary, Conclusions, and Recommendations5.1 Summary and Conclusions5.2 Recommendations for Further WorkAppendicesA Design Curves for Top Hat ReflectorsA.1 IntroductionA.2 AnalysisA.3 Design CurvesA.4 Design ExampleReferences63718485879092104112Reflectors . . 112117120126128131136140141146152156V\u00E2\u0080\u00A2 157\u00E2\u0080\u00A2 159\u00E2\u0080\u00A2 159\u00E2\u0080\u00A2 165169174187C Circular Polarization Selective ReflectorsC.1 IntroductionC.2 ConceptC.3 Proposed ImplementationsC.4 DiscussionReferencesD Experimental ArrangementD.1 IntroductionD.2 OverviewD.3 CW Radar ApparatusD.4 Digital Pattern Recorder.D.5 Facility EvaluationReferences189189192198199B Scattering by a Conducting Grating with Rectangular GroovesB.1 IntroductionB.2 AnalysisB.2.1 TM PolarizationB.2.2 TE PolarizationB.3 Verification of Numerical ResultsB.4 ImplementationReferences200200202208214221viList of Tables3.1 Response Characteristics of Selected Trihedral Corner Reflectors with Three-foldSymmetry 703.2 Response Characteristics of Selected Trihedral Corner Reflectors with BilateralSymmetry 834.1 Tolerances on the Corner Angles of a Prototype Trihedral Corner Reflector withTriangular Panels 1164.2 Dimensions of the Prototype Reflection Polarizers 116A.1 Response Characteristics of Selected Top Hat Reflectors at f = 10 GHz 154B.1 Definition of Symbols 158D.1 Digital Pattern Recorder Program Modules 209D.2 Response of Trihedral Corner Reflectors with Triangular Panels at 9.445 GHz . . 216ViiList of Figures1.1 Detection of a point target in ground clutter 31.2 Probability distribution functions of clutter and a signal embedded in clutter. . 31.3 NASA/JPL synthetic aperture radar calibration site at Goldstone, California. . 51.4 Radar-assisted positioning with respect to cooperative shore-based targets. . . 51.5 Use of range and azimuth gates to isolate shore station reflectors from surrounding clutter 61.6 Relative size of corner reflectors which present the same maximum radar crosssection (4500 m2) at f = 10 GHz 62.1 Coordinate frames used to define polarization state in radar scattering problems 112.2 Coordinate system for a plane wave propagating in the direction A 142.3 A polarization ellipse with semi-major axis OA, semi-minor axis OB, and tiltangle r 152.4 Mapping of polarization states onto a Poincar sphere 152.5 A polarization ellipse showing the relationship between the tilt angles T and Tin the xyz and x\u00E2\u0080\u0099y\u00E2\u0080\u0099z\u00E2\u0080\u0099 coordinate frames 172.6 A perspective projection of the parallels of two coordinate frames which arerelated by pure rotation 192.7 A Mercator projection of the parallels of two coordinate frames which are relatedby pure rotation 192.8 Poincar\u00C3\u00A9 sphere representation of the polarization state W of a plane wave. . 272.9 Definition of the phase reference for orthogonal circular components 28viii2.10 Coordinate systems and scattering geometry for the forward scattering alignment(FSA) convention 412.11 Coordinate systems and scattering geometry for the backscatter alignment (BSA)convention 413.1 Relative sizes of trihedral corner reflectors used as location markers and calibration targets in radar navigation, radar-assisted positioning, and radar remotesensing 473.2 Scattering by a trihedral corner reflector with triangular panels 503.3 Problem geometry and coordinate system for scattering by a trihedral cornerreflector composed of triangular, elliptical, or rectangular panels with corners ofarbitrary length 503.4 Spencer\u00E2\u0080\u0099s model for the equivalent flat plate area of a trihedra) corner reflector 513.5 An implementation of Robertson\u00E2\u0080\u0099s model for the contribution of triple-bouncereflections to the response of a trihedral corner reflector 543.6 Examples of trihedral corner reflectors for which Spencer\u00E2\u0080\u0099s model gives accuratepredictions of the equivalent flat plate area 553.7 Examples of trihedral corner reflectors for which Spencer\u00E2\u0080\u0099s model gives erroneouspredictions of the equivalent flat plate area 563.8 Transformation of coordinates by pure rotation 583.9 Execution of the Weiler-Atherton polygon clipping algorithm 623.10 Reflector coordinate frame and global coordinate frame 633.11 Angular coverage of a trihedral corner reflector with triangular panels 663.12 Angular coverage of a trihedral corner reflector with circular panels 663.13 Angular coverage of a trihedral corner reflector with square panels 673.14 Boresight view and panel geometry of trihedral corner reflectors with Robertsonpanels 67ix3.15 Angular coverage of a trihedral corner reflector with Robertson panels whered/d = 0.25 683.16 Angular coverage of a trihedral corner reflector with Robertson panels whered/d = 0.50 683.17 Angular coverage of a trihedral corner reflector with Robertson panels whered/d = 0.75 693.18 Cumulative probability distribution of the response of trihedral corner reflectorswith triangular, circular, square, and Robertson panels (where d/d = 0.25)over the quadrant defined by the axes of the trihedral 693.19 Angle of maximum response 6m of bilaterally symmetric trihedral corner reflectors with triangular, elliptical, and rectangular panels vs. the reflector aspectratio c/a 743.20 Angle of maximum response 8m and angle of the normal to the aperture Oof bilaterally symmetric trihedral corner reflectors with triangular panels vs. thereflector aspect ratio c/a 743.21 Maximum response of a bilaterally symmetric trihedral corner reflectors withtriangular, elliptical, and rectangular panels vs. the reflector aspect ratio c/a. . 753.22 Azimuthal and elevation beamwidths of a bilaterally symmetric trihedral cornerreflector with triangular panels vs. the reflector aspect ratio c/a 753.23 Azimuthal and elevation beamwidths of a bilaterally symmetric trihedral cornerreflector with elliptical panels vs. the reflector aspect ratio c/a 763.24 Azimuthal and elevation beamwidths of a bilaterally symmetric trihedral cornerreflector with rectangular panels vs. the reflector aspect ratio c/a 763.25 Angular coverage of a bilaterally symmetric trihedral corner reflector with triangular panels and reflector aspect ratio c/a = 0.25 773.26 Angular coverage of a bilaterally symmetric trihedral corner reflector with triangular panels and reflector aspect ratio c/a = 4.0 77x3.27 Angular coverage of a bilaterally symmetric trihedral corner reflector with triangular side panels and a circular center panel 783.28 Angular coverage of a bilaterally symmetric trihedral corner reflector with triangular side panels and a square center panel 783.29 Angular coverage of a bilaterally symmetric trihedral corner reflector with circular side panels and a triangular center panel 793.30 Angular coverage of a bilaterally symmetric trihedral corner reflector with circular side panels and a square center panel 793.31 Angular coverage of a bilaterally symmetric trihedral corner reflector with squareside panels and a triangular center panel 803.32 Angular coverage of a bilaterally symmetric trihedral corner reflector with squareside panels and a circular center panel 803.33 Evolution of Lanziner\u00E2\u0080\u0099s bilaterally symmetric trihedral corner reflector 813.34 Angular coverage of a bilaterally symmetric trihedral corner reflector with truncated triangular side panels and a triangular center panel 813.35 Angular coverage of a bilaterally symmetric trihedral corner reflector with truncated and compensated triangular side panels and a triangular center panel. . . 823.36 Angular coverage of a bilaterally symmetric trihedral corner reflector with truncated, compensated, and extended side panels and a triangular center panel. . . 823.37 Effect of errors in all three corner angles on the response of a trihedral cornerreflector with triangular panels for incidence along the the symmetry axis 844.1 Methods for altering the polarization response of a conventional trihedral cornerreflector 904.2 Problem geometry for scattering by a conducting grating with rectangular grooves 934.3 Visible diffracted orders as a function of the grating period (in wavelengths) andthe angle of incidence where the plane of incidence is normal to the grating axis 93xi4.4 Normalized response of a regular reflector as a function of the polarization stateof the incident wave 964.5 Normalized response of a twist-polarizing reflector as a function of the polarization state of the incident wave 964.6 Normalized response of a circularly polarizing reflector as a function of the polarization state of the incident wave 974.7 Normalized response of a vertical polarization selective reflector as a function ofthe polarization state of the incident wave 974.8 Twist polarizer design curves for normal incidence 1004.9 Twist polarizer design curves for 45 degree incidence 1004.10 Circular polarizer design curves for normal incidence 1014.11 Circular polarizer design curves for 45 degree incidence 1014.12 Linear polarization selective reflectors derived from corrugated surfaces 1024.13 Replacement of one panel of a trihedral corner reflector by a reflection polarizerderived from a corrugated surface 1054.14 Twist polarizer design curves for incidence along the symmetry axis of a trihedralcorner reflector 1064.15 Circular polarizer design curves for incidence along the symmetry axis of a tnhedral corner reflector 1064.16 Reflector coordinate frame and global coordinate frame 1084.17 Angle of rotation of the projection of the grating axis onto the view plane forincidence along the symmetry axis and 30 degrees off the symmetry axis 1084.18 Profile view of the RCS measurement range 1134.19 Measurement of the response of a prototype trihedral corner reflector 1134.20 Construction details of the prototype depolarizing trihedral corner reflector. . . 1154.21 Photograph of the prototype twist-polarizing trihedral corner reflector mountedon the antenna range model tower 115xii4.22 A single segment of a prototype reflection polarizer 1164.23 Evaluation of the polarization response of a radar target 1184.24 Polarization response of a prototype regular trihedral corner reflector as a function of rotation about the boresight 1184.25 Polarization response of a prototype twist-polarizing trihedral corner reflector asa function of rotation about the boresight 1194.26 Polarization response of a prototype circularly-polarizing trihedral corner reflector as a function of rotation about the boresight 1194.27 Co-polar azimuthal response pattern of a prototype regular trihedral corner reflector for rotation angle a = 00 and vertically polarized transmission 1214.28 Cross-polar azimuthal response patterns of a prototype regular trihedral cornerreflector for rotation angle a 00 and vertically polarized transmission 1214.29 Co-polar azimuthal response pattern of a prototype twist-polarizing trihedralcorner reflector for rotation angle a = 00 and vertically polarized transmission. . 1224.30 Cross-polar azimuthal response patterns of a prototype twist-polarizing trihedralcorner reflector for rotation angle a = 00 and verticaily polarized transmission. . 1224.31 Co-polar azimuthal response pattern of a prototype twist-polarizing trihedralcorner reflector for rotation angle a = 45\u00C2\u00B0 and vertically polarized transmission. 1234.32 Cross-polar azimuthal response patterns of a prototype twist-polarizing trihedralcorner reflector for rotation angle a = 45\u00C2\u00B0 and vertically polarized transmission. 1234.33 Co-polar azimuthal response pattern of a prototype circularly-polarizing trihedralcorner reflector for rotation angle a = 0\u00C2\u00B0 and vertically polarized transmission. . 1244.34 Cross-polar azimuthal response patterns of a prototype circularly-polarizing tnhedral corner reflector for rotation angle a = 00 and vertically polarized transmission 1244.35 Co-polar azimuthal response pattern of a prototype circularly-polarizing trihedralcorner reflector for rotation angle a = 45\u00C2\u00B0 and vertically polarized transmission. 125XIII4.36 Cross-polar azimuthal response patterns of a prototype circularly-polarizing tnhedral corner reflector for rotation angle o \u00E2\u0080\u0094 45\u00C2\u00B0 and vertically polarized transmission 125A.1 Problem geometry for scattering by a top hat reflector 141A.2 A simplified ray-optical model for scattering by a top hat reflector 142A.3 Problem geometry for forward scattering by a cylinder 143A.4 Angle of maximum response of a top hat reflector vs. , the ratio of the annuluswidth to the cylinder height 146A.5 Maximum scattering cross section of a top hat reflector vs. , the ratio of theannulus width to the cylinder height for fixed values of a and c 148A.6 Scale factors for the cylinder radius a and height c of a top hat reflector vs. theratio of the annulus width to the cylinder height 148A.7 Half-power elevation beamwidth of a top hat reflector vs. the ratio of the annuluswidth to the cylinder height 150A.8 1 dB elevation beamwidth of a top hat reflector vs. the ratio of the annulus widthto the cylinder height 151A.9 Angle of maximum response and angles of median response for 1 and 3 dBelevation beamwidths vs. the ratio of the annulus width to the cylinder height. . 152A.10 Relative size of selected top hat reflectors which present the same maximumscattering cross section 154A.11 Elevation response patterns of selected top hat reflectors which present the samemaximum scattering cross section 155B.1 A unit cell of a conducting grating with rectangular grooves 158xivB.2 Convergence of the phase difference between the TE and TM specular reflectioncoefficients with the number of groove modes for a reflection twist polarizer withd = 0.3333A, aid = 0.5000, and h = 0.2302A illuminated by a plane wave incidentat qS 45 degrees 172B.3 Convergence of the phase difference between the TE and TM specular reflectioncoefficients with the number of groove modes for a reflection twist polarizer withd = 0.3333), aid = 0.9999, and h = 0.3172A illuminated by a plane wave incidentat = 45 degrees 172B.4 Convergence of the phase difference between the TE and TM specular reflectioncoefficients with the number of groove modes for a reflection circular polarizerwith d = 0.3333A, a/d = 0.5000, and h = 0.1466A illuminated by a plane waveincident at = 45 degrees 173B.5 Convergence of the phase difference between the TE and TM specular reflectioncoefficients with the number of groove modes for a reflection circular polarizerwith d = 0.3333), a/d = 0.9999, and h = 0.1641A illuminated by plane waveincident at = 45 degrees 173B.6 Hierarchy of subprograms called by subroutines TMREFL and TEREFL 175B.7 Combined execution time of subroutines TMREFL and TEREFL on a Sun 41380workstation vs. the number of groove modes used in the solution 176B.8 Combined execution time of subroutines TMREFL and TEREFL on a Sun 41380workstation vs. the number of free space modes used in the solution 176B.9 Execution profile of subroutines TMREFL and TEREFL 177C.1 Normalized response of a left circular polarization selective reflector as a functionof the polarization state of the incident wave 191C.2 Normalized response of a right circular polarization selective reflector as a function of the polarization state of the incident wave 191xvC.3 A proposed implementation of a circular polarization selective reflector using atransmission circular polarizer and a linear polarization selective reflector.C.4 A parallel plate transmission circular polarizerC.5 A proposed implementation of a circular polarization selective reflector using acircular polarization selective surface and a trihedral twist reflectorC.6 Scattering by a right circular polarization selective trihedral corner reflector.C.7 A linear polarization selective gridded trihedral and its co-polar and cross-polarazimuthal response patternsC.8 A bilaterally symmetric trihedral corner reflector with triangular panelsC.9 Elements of a circular polarization selective surfaceD.1 Block diagram of the radar cross section measurement rangeD.2 Photograph of the radar cross section measurement rangeD.3 Block diagram of the CW radar transmitterD.4 Photograph of the CW radar transmitterD.5 Mounting arrangement for the transmitting and receiving antennasD.6 Proffle view of the RCS measurement rangeD.7 Arrangement for mechanically aligning the transmitting and receiving antennas.D.8 Radar cross section measurement range link budgetD.9 Photograph of the digital pattern recorder, positioner control unit, and portablemicrowave receiverD.10 Digital pattern recorder: synchro test screenD.11 Digital pattern recorder: receiver calibration screenD.12 Equipment configuration for performing relative and absolute calibration of theCW radar apparatusD.13 Digital pattern recorder: parameter entry screenD.14 Digital pattern recorder: data acquisition screenD.15 Error model for radar cross section measurement\u00E2\u0080\u00A2 193\u00E2\u0080\u00A2 194\u00E2\u0080\u00A2 195195196\u00E2\u0080\u00A2 197198201202\u00E2\u0080\u00A2 204204205\u00E2\u0080\u00A2 206206\u00E2\u0080\u00A2 207208211\u00E2\u0080\u00A2 211\u00E2\u0080\u00A2 212\u00E2\u0080\u00A2 213213214xviD.16 The effective aperture of a trihedral corner reflector with triangular panels forincidence along the boresight and at an azimuth angle of 30 degrees 216D.17 Polarization response of the receiving horn at a range of 11 m 218D.18 Boresight response of a conventional trihedral corner reflector vs. range 218D.19 Azimuthal response pattern of a conventional trihedral corner reflector at rangesof 10 and 12 m 219D.20 Azimuthal response pattern of the model tower at a range of 11 m 219xviiAcknowledgementsI wish to express my appreciation to my advisor, Prof. E.V. Jull; the members of my thesissupervisory committee, including Professors H.W. Dommel, W. McCutcheon, and L. Young;and Professors R.W. Donaldson, D.L. Pulfrey, L.M. Wedepohi, and M.M.Z. Kharadly for theirinterest and support during the past few years. Thanks are also due to Helmut H. Lanziner,President of Offshore Systems Ltd., and Prof. Simon Haykin, Director of the CommunicationsResearch Laboratory at McMaster University, for encouraging me to pursue a thesis projectconcerning the design of passive radar targets.I gratefully acknowledge the scholarships and awards which I have received during mycourse of study including a Science Council of British Columbia Graduate Scholarship, a BritishColumbia Advanced Systems Institute (ASI) Graduate Scholarship, an International Union ofRadio Science (URSI) Young Scientist Award, a Province of British Columbia Graduate Scholarship, and a British Columbia Telephone Company Graduate Scholarship. Additional funding and support were provided by the Natural Sciences and Engineering Research Council ofCanada (NSERC) under Operating Grant 5-88571, Transport Canada (Transportation Development Centre, Montreal, P.Q.) under contract T8200-9-9560/O1-XSB, the National ResearchCouncil of Canada (NRC) under Industrial Research Assistance Program (TRAP) Project No.9-8323-L-19, Offshore Systems Ltd. (North Vancouver, B.C.), Motorola Canada Ltd. (MDIDivision, Richmond, B.C.), and the University of British Columbia (Department of ElectricalEngineering).The assistance provided by Jean Liu, Kirk Jong, Robert Laing, David Clarke, and DonMacNeil in setting up the radar cross section measurement range, implementing the digitalpattern recorder, constructing the prototype radar reflectors, and conducting the experimentalmeasurements is greatly appreciated.xviiiChapter 1INTRODUCTION1.1 Background and MotivationRadars function by radiating electromagnetic energy and detecting the presence and characterof the echoes returned by reflecting objects or targets. If these echoes are correlated withthe original transmitted signal, many of the characteristics of the targets can be estimatedincluding their range, bearing, apparent size or reflectivity, and certain aspects of their physicalgeometry. If a series of radar measurements is processed over time, the original estimate of thesecharacteristics can be refined and the future kinematic behaviour of the target can be predicted.Radar has traditionally been associated with the detection and navigation of ships and aircraft.More recently, it has become an important tool for remote sensing of the environment. Thechief problem of radar is to detect targets of interest and estimate their position and physicalcharacteristics in the presence of interference from clutter returns and noise [1], [2].The scattering cross section a of a target is defined as the area intercepting that amountof power which, when scattered isotropically, would produce an echo equal to that actuallyreturned by the target. Thus,E2 HS2a lim 4irr2 lim 4irr2 (1.1)IE\u00E2\u0080\u009912 r\u00E2\u0080\u0094cc\u00E2\u0080\u0099 HI2where E and W are the incident electric and magnetic fields, E8 and Hs are the scatteredelectric and magnetic fields, and r is the range at which the scattered field is measured.Radar cross section or RCS refers to that portion of the scattering cross section which isassociated with a specified polarization component of the scattered wave and is a function ofthe size, shape, composition, and orientation of the target, the frequency of the incident wave,and the polarization state of the radar transmitting and receiving antennas [3], [4].1Chapter 1. Introduction 2The tendency for both natural and man-made objects to depolarize radar echoes in characteristic ways has been recognized since the earliest days of radar. It is convenient to describethe relationship between the polarization states of the incident and scattered fields by a polarization scattering operator expressed in matrix form. Several representations are in commonuse. The polarization scattering matrix relates incident and scattered fields which have beenexpressed as complex polarization vectors while the Mueller matrix relates fields which havebeen expressed as Stokes vectors. Other forms, such as the Stokes scattering operator and thecovariance matrix, are used in certain methods for synthesizing arbitrary polarization responsesfrom sets of experimental data. Measurement of the complete polarization response of a targetrequires a radar which is capable of antenna polarization control or agility during transmissionand polarization diversity on reception. Although such radars are considerably more complexthan their conventional counterparts, polarimetric radar signal processing has become an important tool for target detection and classification in several fields including radar meteorology,geophysical remote sensing, and certain specialized forms of radar navigation [5]\u00E2\u0080\u0094[8j.It is often necessary to enhance the radar cross section of a cooperative target either toincrease the maximum range at which the target can be reliably detected or to provide atarget with a known response which may be used to assist in radar calibration and performanceverification. Although it is sometimes possible to achieve the desired result simply by makingminor modifications to the natural shape of the target or by disturbing the current distributionon the surface of the body with discrete impedance loading, it is usually more convenient to makeuse of auxiliary devices such as corner reflectors, dielectric lenses, and retrodirective antennaarrays which have been designed specifically to present a large radar cross section over wideangular ranges. The characteristics and relative merits of the various types of RCS enhancementdevices have been widely discussed in the literature [9]\u00E2\u0080\u0094{12]. The response characteristics whichare required of such devices are determined by several factors including the distance betweenthe radar and the target, the reflectivity of the surrounding clutter, the resolution of the radarin range and azimuth, the combination of transmit and receive polarizations employed bythe radar, and the nature of the target detection or radar calibration algorithm. BeamwidthChapter 1. Introduction 3and pulse width limited radar resolution cells are shown in Figure 1.1. The probabilities ofdetection PD and false alarm FFA are determined by the value of the detection threshold andthe probability distribution functions of the target and clutter returns as suggested by theexample presented in Figure 1.2.xFigure 1.1: Detection of a point target in ground clutter where the radar resolution cell is eitherbeamwidth limited (entire ellipse) or pulse width limited (shaded portion of ellipse), a is theradar cross section of the target, r is the range to the target, h is the height of the radar, 8Hand 8 are the half-power beamwidths of the radar antenna in azimuth and elevation, t is thepulse duration, C is the speed of light, and c is the depression angle. (after [2, p. 84])P(x)Figure 1.2: Probability distribution functions of clutter C and a signal embedded in clutterS + C and the corresponding probabilities of detection PD and false alarm PFA for a givendetection threshold. (after [2, p. 42])hIcsin a\u00C2\u00B0fAThresholdChapter 1. Introduction 4In recent years, the development of airborne and spaceborne imaging radar systems forgeophysical remote sensing and radar-assisted positioning systems for marine navigation hasled to a requirement for rugged yet inexpensive calibration targets and location markers whichpresent both a very large scattering cross section and a specified polarization response over awide angular range. The NASA/JPL synthetic aperture radar (SAR) calibration site shownin Figure 1.3 is similar to the calibration ranges which have been established by several otherresearch organizations to assist in the geometric, radiometric, and polarimetric calibration ofSAR imagery [13], [14]. The experimental radar-assisted positioning system depicted in Figures1.4 and 1.5 is being developed by Transport Canada as a method for allowing large vesselsnavigating in inland waterways, harbours, and harbour approaches to accurately determinetheir position with respect to cooperative shore-based targets in real time [15]\u00E2\u0080\u0094[19]. While therequirements for targets used in the calibration of airborne radars can be estimated with afair degree of confidence [12]\u00E2\u0080\u0094[14], scattering by terrain at grazing incidence has not been wellcharacterized [20]\u00E2\u0080\u0094[22] and it is not yet possible to give reliable estimates of the size of targetsrequired to achieve specified probabilities of detection and false alarm in applications such asradar-assisted positioning.Although active targets are physically compact and their scattering cross section, angularcoverage, and polarization response can be modified with relative ease, their usefulness is limitedby several factors including their requirement for an external power source, interaction betweentheir transmitting and receiving antennas which may lead to regenerative feedback and distortion of their response patterns, and the limited reliability and stability of active components.Despite their greater physical size and finer mechanical tolerances, passive targets provide abetter and more reliable alternative when it is necessary to install devices in remote locationsfor extended periods of time. In Figure 1.6, the relative sizes of corner reflectors which presentthe same maximum radar cross section (4500 m2) at a frequency of 10 GHz are compared. Inthe face of conflicting requirements for a target which presents a large response and wide angular coverage while retaining mechanical ruggedness and ease of manufacture, trihedral cornerreflectors frequently represent the best compromise.Chapter 1. IntroductionA A 6,T,8 TrlfledralsI...\u00E2\u0080\u0098 0,45 degree Dihedrals\u00E2\u0080\u00A2 0 0,45,90 degree PARC\u00E2\u0080\u0099s (LandTone Generators (L and C)0 Passive Receivers (L only)5Figure 1.3: NASA/JPL synthetic aperture radar(from [14, p. 227])calibration site at Goldstone, California.Figure 1.4: Radar-assisted positioning with respect(from [15, front cover])to cooperative shore-based targets... pathGoldstone SARCaubratlon Site E4Chapter 1. Introduction 6Building/ I.-< Shore Station<- Reflector/// ////Radar AntennaFigure 1.5: Use of range and azimuth gates to isolate shore station reflectors from surroundingclutter.(a)(b) (e)3m.(c) 2m.-Figure 1.6: Relative size of corner reflectors which present the same maximum radar crosssection (4500 m2) at f = 10 GHz. A spherical target which presents an equivalent responsewould have a diameter of over 75 metres. (a) Trihedral corner reflector with triangular panels,(b) Trihedral corner reflector with square panels. (c) Dihedral corner reflector. (d) Bruderhedral(a cylindrical sector attached to a flat plate). (e) Top hat reflector. (f) Biconical reflector.Range Gate//Azimuth Gate/RadarBeamwidth/Shore StationReflector////(d) (t)Chapter 1. Introduction 71.2 OutlineThis study considers several problems related to the analysis, design, and implementation ofpassive targets including transformation of polarization descriptors between coordinate frames,modification of the angular coverage and polarization response of conventional trihedral cornerreflectors, design of top hat reflectors with specified response characteristics, and design ofreflection polarizers derived from conducting gratings with rectangular grooves.In Chapter 2, the problem of transforming representations of polarization state and polarization scattering operators between coordinate frames is considered. It is shown that suchtransformations correspond to rotation of the polarization basis by a prescribed angle which isa function of both the transformation matrix which relates the two coordinate frames and thedirection of propagation. Two methods for determining the angle of rotation are derived forthe case in which the local vertical is defined by the direction in each frame. Algorithms fortransforming common polarization descriptors are presented.In Chapter 3, the problem of predicting the response and angular coverage of a trihedralcorner reflector with panels of completely arbitrary shape is considered. For most purposes,only triple-bounce reflections from the interior of the reflector need be accounted for since theycompletely dominate the response for most directions of incidence. A simple yet robust RCSprediction algorithm which overcomes many of the limitations of previous work is described. Theresponse patterns of three-fold symmetric and bilaterally symmetric trihedral corner reflectorswith panels of various shapes are compared and design curves for realizing bilaterally symmetrictrihedral corners with specified response characteristics are given.In Chapter 4, a method for altering the polarization response of a conventional trihedralcorner reflector by adding conducting fins or corrugations of appropriate dimensions and orientation to one of its interior surfaces is proposed. Design curves for twist-polarizing andcircularly polarizing trihedral corner reflectors are given. Methods for realizing linear polarization selective trihedral corner reflectors using similar techniques are proposed. An algorithmfor predicting the contribution of triple-bounce reflections to the response of a depolarizingtrihedral corner reflector as a function of the direction of incidence and the orientation of theChapter 1. Introduction 8reflector is derived. Experimental results are presented which show that prototype depolarizingreflectors respond essentially as predicted.In Chapter 5, the results of this study are summarized and recommendations for furtherwork are offered.In Appendix A, the problem of designing top hat reflectors with specified response characteristics is considered. Expressions for the elevation response pattern, maximum scatteringcross section, angle of maximum response, 1 and 3 dB beamwidths, and angle of median response for 1 and 3 dB beamwidths are derived and used to generate design curves. The resultsare used to solve a sample design problem.In Appendix B, the problem of scattering by a conducting grating with rectangular groovesis ccrnsidered. Analytical solutions are derived for the cases of TM- and TE-polarized incidentwaves by mode-matching between the free space and groove regions. Procedures for determiningthe validity of numerical results are described and the problem of determining the minimumnumber of modes required to accurately represent the fields in each region is studied. Animplementation of the analytical solutions as a pair of subroutines coded in Fortran 77 ispresented.In Appendix C, the problem of modifying a conventional trihedral corner reflector to presenta circular polarization selective response is considered. It is shown that such a response cannotbe realized simply by using the techniques of Chapter 4 because the corresponding polarizationscattering matrix cannot be diagonalized. Alternative methods for obtaining such a responsebased on the addition of a suitable transmission polarizer to reflectors which present either alinear polarization selective or a twist polarizing response are proposed.In Appendix D, the experimental facility which was developed to measure the responseof prototype trihedral corner reflectors is briefly described. Details of its physical layout, thedesign and implementation of the CW radar apparatus and digital pattern recorder, and theresults of tests used to verify its suitability for use in the measurement program are given.Recommendations for future modifications and improvements are offered.References[1] F.E. Nathanson, Radar Design Principles: Signal Processing and the Environment, 2nd ed.New York: McGraw-Hill, 1991, pp. 1\u00E2\u0080\u009448.[2] N. Levanon, Radar Principles. New York, Wiley, 1988, pp. 1\u00E2\u0080\u009497.[3] E.F. Knott, J.F. Shaeffer, and M.T. Tuley, Radar Cross Section: Its Prediction, Measurement, and Reduction. Norwood, MA: Artech House, 1985, pp. 47\u00E2\u0080\u009483.[4] A.K. Bhattacharyya and D.L. Sengupta, Radar Cross Section Analysis and Control. Nor-wood, MA: Artech House, 1991, pp. 10\u00E2\u0080\u009432.[5] D. Giuli, \u00E2\u0080\u009CPolarization diversity in radars,\u00E2\u0080\u009D Proc. IEEE, vol. 74, pp. 245\u00E2\u0080\u0094269, Feb. 1986.[6] A. Macikunas and S. Haykin, \u00E2\u0080\u009CPolarization as a Radar Discriminant,\u00E2\u0080\u009D in Selected Topics inSignal Processing, S. Haykin, Ed. Englewood Cliffs, NJ: Prentice-Hall, 1989, pp. 251\u00E2\u0080\u0094286.[7] D. Atlas, Ed., Radar in Meteorology. Boston, MA: American Meteorology Society, 1990.[8] F. T. Ulaby and C. Elachi, Eds., Radar Polarimetry for Geoscience Applications. Norwood,MA: Artech House, 1990.[9] D. R. Brown, R. J. Newman, and J. W. Crispin, Jr., \u00E2\u0080\u009CRCS Enhancement Devices,\u00E2\u0080\u009D inMethods of Radar Cross Section Analysis, J. W. Crispin, Jr. and K. M. Siegel, Eds.New York: Academic Press, 1969, pp. 237\u00E2\u0080\u0094280.[10] C. T. Ruck, \u00E2\u0080\u009CRadar Cross Section Enhancement,\u00E2\u0080\u009D in Radar Cross Section Handbook,G.T. Ruck, Ed. New York: Plenum Press, 1970, pp. 585\u00E2\u0080\u0094601.[11] F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive,vol. 2. Norwood, MA: Artech House, 1982, pp. 766\u00E2\u0080\u0094779.9References 10[12] D. R. Brunfeldt and F. T. Ulaby, \u00E2\u0080\u009CActive reflector for radar calibration,\u00E2\u0080\u009D IEEE Trans.Geosci. Remote Sensing, vol. GE-22, pp. 165\u00E2\u0080\u0094169, Mar. 1984.[13] S. H. Yueh, J. A. Kong, R. M. Barnes, and R. T. Shin, \u00E2\u0080\u009CCalibration of polarimetric radarsusing in-scene reflectors,\u00E2\u0080\u009D J. Electrornagn. Waves Appi., vol. 4, pp. 27\u00E2\u0080\u009448, Jan. 1990.[14] A. Freeman, Y. Shen, and C.L. Warner, \u00E2\u0080\u009CPolarimetric SAR. calibration experiment usingactive radar reflectors,\u00E2\u0080\u009D IEEE Trans. Geosci. Remote Sensing, vol. GE-28, pp. 224\u00E2\u0080\u0094240,Mar. 1990.[15] C. Stiles, R.O. Hewitt, and C.O. MdHale, \u00E2\u0080\u009CPRANS Trials: Evaluation of a Precise RadarNavigation System.\u00E2\u0080\u009D Transport Canada Pubi. No. TP 2800E. Montreal: TransportationDevelopment Centre, Aug. 1981.[16] D. Kalnicki and R. Harrs, \u00E2\u0080\u009CPerformance Evaluation and Demonstration of the RANAV(Radar-Assisted Precise Navigation) System in the St. Lawrence River.\u00E2\u0080\u009D Transport CanadaPubi. No. TP 11326E. Montreal: Transportation Development Centre, April 1992.[17] S. Haykin, \u00E2\u0080\u009CPolarimetric radar for accurate navigation,\u00E2\u0080\u009D Can. J. Elect. Comp. Eng., vol. 17,pp. 130\u00E2\u0080\u0094135, July 1992.[18] H. Lanziner, D. Michelson, S. Lachance, and D. Williams, \u00E2\u0080\u009CExperiences with a commercialECDIS,\u00E2\u0080\u009D mt. Hydrogr. Rev., vol. 67, no. 2, pp. 69\u00E2\u0080\u009486, July 1990.[19] D.C. Michelson et al., \u00E2\u0080\u009CUse of circular polarization in a marine radar positioning system,\u00E2\u0080\u009DProc. IGARSS\u00E2\u0080\u009989, (Vancouver, B.C.), July 1989.[20] M.W. Long, Radar Reflectivity of Land and Sea. Norwood, MA: Artech House, 1983.[21] D.K. Barton, \u00E2\u0080\u009CLand clutter models for radar design and analysis,\u00E2\u0080\u009D Proc. IEEE, vol. 73,pp. 198\u00E2\u0080\u0094204, Feb. 1985.[22] F.T. Ulaby and M.C. Dobson, Handbook of Radar Scattering Statistics for Terrain. Norwood, MA: Artech House, 1989.Chapter 2TRANSFORMATION OF POLARIZATION DESCRIPTORSBETWEEN COORDINATE FRAMES2.1 IntroductionThe polarization state of an electromagnetic wave is a vector quantity which refers to thebehaviour with time of the electric field as observed at a fixed point in space. If the wave ismonochromatic, the tip of the electric field vector will trace an ellipse in the plane orthogonalto the direction of propagation. Such a wave is said to be completely polarized. If the wavecoiltains a random component in amplitude or phase, it will occupy a finite bandwidth andthe polarization ellipse will tend to change shape and orientation with time. Such a wave issaid to be partially polarized. In the extreme case, the tip of the electric field vector will traceout a figure which is totally random in shape and the wave is said to be randomly polarized(or unpolarized). The fundamental aspects of wave polarization have been reviewed by severalauthors, e.g., [1]\u00E2\u0080\u0094[6].zAntenna coordinate framex\u00E2\u0080\u009D -\ /Local or body-fixed coordinate frameGlobal coordinate fra:Figure 2.1: Coordinate frames used to define polarization state in radar scattering problems.11Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 12Since polarization is a vector quantity, its description must be referred to a particularcoordinate frame. When problems involving propagation or scattering in the vicinity of theearth\u00E2\u0080\u0099s surface are considered, it is often convenient to describe the polarization state of anantenna or the polarization response of a target with respect to a global coordinate frame inwhich the earth\u00E2\u0080\u0099s surface is coincident with the z-y plane and the local vertical is parallel tothe z axis. If the radiation or scattering characteristics of an object can be determined moreefficiently in a different frame or if the object is free to rotate about one or more axes as inthe case of airborne or spaceborne platforms, it may be preferable to define the polarizationcharacteristics of the object with respect to a local or body-fixed coordinate frame instead,as suggested by Figure 2.1. In turn, the response of the device is measured in yet anothercoordinate frame which is defined by the antenna. However, with the exception of the specialcases considered by Mott [6] and Krichbaum [7], the problem of transforming polarizationdescriptors between coordinate frames has received little attention in the literature.In section 2.2, the concept of polarization state is briefly reviewed. In section 2.3, it isshown that transformation of a polarization descriptor between coordinate frames correspondsto rotation of its basis by a prescribed angle which is a function of both the transformationmatrix which relates the two coordinate frames and the direction of propagation. Two methodsfor determining the angle of rotation for the case in which the local vertical is defined by thedirection in each frame are derived using spherical trigonometry and vector algebra, respectively.In section 2.3, methods for determining the elements of the coordinate transformation matrixare reviewed. Although the matrix can be determined from either the relative directions ofthe three principal axes in each coordinate frame or the Euler angles which define a seriesof rotations which will transform one coordinate frame into the other, in practice it may bedifficult to obtain these parameters. A third method is derived which permits the elements ofthe transformation matrix to be determined from any pair of arbitrary directions which havebeen expressed in terms of both coordinate frames. In section 2.4, algorithms for rotating thebasis of several common polarization descriptors are presented.Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 132.2 Wave PolarizationThe electric field vector E of a plane wave travelling in the direction k may be characterized interms of a horizontally polarized component EHI and a vertically polarized component Ef\u00E2\u0080\u0099.Thus,E = (EH& + EvI) e_jk.r (2.1)where k = 27r/A is the propagation constant in free space and the time dependence ejwt hasbeen suppressed. The amplitudes Ejq and Ev are complex quantities given byEH = aj , (2.2)= av , (2.3)where a and av are the magnitudes of EH and Ev, respectively, and H and 5v are their phaseangles. The coordinate frame (I, , k) can be specified in terms of the triad (i,, ) defined bya spherical coordinate system such thatk . .k = r = cosqsin8x+sinqsinOy+cos8z, (2.4)h = = \u00E2\u0080\u0094sinqx+cosqy, (2.5)xkj= kxh \u00E2\u0080\u0094 = \u00E2\u0080\u0094cosqcos6\u00E2\u0080\u0094sinq5cos0+sin8, (2.6)as depicted in Figure 2.2. The definitions of I and I given by (2.5) and (2.6) are somewhatarbitrary and any pair of orthogonal directions which form a right hand triad with k may besubstituted. In cases where the direction of propagation coincides with the z axis, e.g., theantenna coordinate frame of Figure 2.1, the definitions of Ii and 13 given by (2.5) and (2.6) areambiguous. If(2.7)it is common to define h and 13 such thatI. = = \u00C2\u00B1, (2.8)15 = kxh = +. (2.9)Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 14When propagation occurs mainly in the vicinity of the z axis, it is often more convenient tosimply redefine polarization in terms of a new coordinate frame in which \u00E2\u0080\u0098, \u00E2\u0080\u0098, and \u00E2\u0080\u0098 correspondto , , and , respectively, in the original frame and z\u00E2\u0080\u0099 defines the new local vertical.zIf the wave is monochromatic, the tip of the electric field vector will trace an ellipse in theplane orthogonal to the direction of propagation. The shape, sense of rotation, and orientationof the effipse are sufficient to specify the polarization state of the wave. Consider a polarizationellipse with semi-major axis OA and semi-minor axis OB, as depicted in Figure 2.3. Themagnitude of the axial ratio R is given byIRH=g, (1IRIoo),According to the IEEE convention, the axial ratio is positive for right-hand polarized wavesand negative for left-hand polarized waves while the reverse is true for the ellipticity angle.The tilt angle r is defined as the angle between the horizontal and the semi-major axis of thepolarization ellipse and is valid over the range _90\u00C2\u00B0 r +900. The entire range of possiblepolarization states can be mapped onto the surface of a Poincar\u00C3\u00A9 sphere as shown in Figure 2.4.kryxFigure 2.2: Coordinate system for a plane wave propagating in the direction k.hwhile the ellipticity angle E is defined as=\u00E2\u0080\u0094 cot1 R, (\u00E2\u0080\u009445\u00C2\u00B0 e +45\u00C2\u00B0).(2.10)(2.11)Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 15A polarization state with ellipticity angle E and tilt angle T corresponds to a point havinglongitude 2T and latitude 2c. Linear and circular polarization states map onto the equator andpoles, respectively, while left and right elliptical polarization states map onto the upper andlower hemispheres.Figure 2.3: A polarization ellipse with semi-major axis OA, semi-minor axis OB, and tilt angle r.EQUATORREPRESENTSLOWER HEMISPHERERIGHT-HAND SENSELONGmJOE =2X TILT ANGLEAhPolarizationellipseLATmJDE =2X ELLIPTICITYANGLECIRCULAR POLARIZATIONSUPPER HEMISPHERELEFT-HAND SENSELINEARFigure 2.4: Mapping of polarization states onto a Poincar sphere. (from [8], p. 82)Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 162.3 Transformation of CoordinatesConsider two coordinate frames which are related by a combination of translation and rotationand let the directions and 2 define the local vertical in each frame. In cartesian coordinates,the position vectors r\u00E2\u0080\u0099 and R in the two frames are related by a coordinate transformationmatrix [T] such thatL m1 X \u00E2\u0080\u0094 Xo= 2 m2 n2 Y \u00E2\u0080\u0094 , (2.12)z\u00E2\u0080\u0099 \u00C2\u00A33 m3 n3 Z\u00E2\u0080\u0094Zowhere the origin 0\u00E2\u0080\u0099 of the x\u00E2\u0080\u0099y\u00E2\u0080\u0099z\u00E2\u0080\u0099 coordinate frame is located at (Xo, Y0, Zo) relative to the XYZcoordinate frame and l, m1,n1; 12, m2n2; and 13, m3,n3 are the direction cosines of the x\u00E2\u0080\u0099, y\u00E2\u0080\u0099, z\u00E2\u0080\u0099axes relative to the X, Y, Z axes, respectively. Since the coordinate transformation matrix is aunitary matrix, its inverse is identical to its transpose and the reverse transformation is simplygiven byXI XOy\u00E2\u0080\u0099 + y0 . (2.13)z\u00E2\u0080\u0099 Z0The polarization state of a propagating wave can be transformed from one coordinate frameto another by direct application of either (2.12) or (2.13) to the components of the electricfield vector as described by Mott [6, pp. 212\u00E2\u0080\u0094219). However, this technique is cumbersomeand cannot be easily generalized to the many different\u00E2\u0080\u0099 methods which are used to representthe polarization state of a wave or the polarization response of a scatterer. The limitations areparticularly apparent in cases where polarization state is described with respect to an ellipticallyor circularly polarized basis or where the wave is partially polarized. A more general approachis suggested by considering transformation of the corresponding polarization ellipse betweencoordinate frames. The ellipticity angle c is invariant under either translation or rotation sinceit depends only on the magnitude of the axial ratio and the polarization sense of the wave.Although the tilt angle r is invariant under translation, it will not be preserved under rotationxYZm1 m2 m3ni 2 fl3Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 17cosb sini& 0[Tj=\u00E2\u0080\u0094sin cosb 00 0 1a = T \u00E2\u0080\u0094 T, (\u00E2\u0080\u009490\u00C2\u00B0 a +900), (2.15)where a corresponds to the angle between the unit vectors and \u00E2\u0080\u0098 which define the horizontalplane in each coordinate frame with respect to the direction of propagation, as depicted inFigure 2.5. In general, it can be shown that transformation of any polarization descriptorbetween coordinate frames may be regarded as a change of basis transformation correspondingto rotation of the polarization basis by an angle a about the direction of propagation.Figure 2.5: A polarization ellipse showing the relationship between the tilt angles r and r\u00E2\u0080\u0099 inthe xyz and x\u00E2\u0080\u0099y\u00E2\u0080\u0099z\u00E2\u0080\u0099 coordinate frames.unless the horizontal planes in both coordinate frames are parallel to each other. This conditionwill be satisfied only if the coordinate transformation matrix is of the form(2.14)which corresponds to rotation about the z aids by an angle b. Otherwise, the difference betweenthe tilt angles of the polarization ellipse in the two coordinate frames is given by6Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 18The angle c between the unit vectors and \u00E2\u0080\u0098 is a function of both the direction of propagation and the coordinate transformation matrix which relates the two coordinate frames. Thisvariation is apparent in Figures 2.6 and 2.7 where the parallels of two coordinate frames whichare related by pure rotation are plotted on perspective and Mercator projections, respectively.Since the Mercator projection is conformal, the angles between the parallels are accurately depicted at all points on the grid and the variation in the angle with the direction of propagationcan be easily visualized. In the context of radar cross section measurement, Krichbaum [7] hasderived an expression for this angle 1 for the special case in which the direction of propagation iscoincident with the z axis and the coordinate transformation corresponds to rotation about thex and y axes. Here, two methods for determining the angle cr for any direction of propagationand coordinate transformation are derived using spherical trigonometry and vector algebra,respectively. Since translation between the coordinate frames can be neglected, it is convenientto definex=X\u00E2\u0080\u0094X0, y=Y\u00E2\u0080\u0094Yo, z=Z\u00E2\u0080\u0094Z0, (2.16)and reduce (2.9) and (2.10) toL m1 n1 x\u00C2\u00A32 m2 n2 y , (2.17)z\u00E2\u0080\u0099 \u00C2\u00A33 m3 n3 zandx 4 \u00C2\u00A32 3y m1 m2 m3 y\u00E2\u0080\u0099 , (2.18)Z i 2 713where the origins of the zyz and z\u00E2\u0080\u0099y\u00E2\u0080\u0099z\u00E2\u0080\u0099 coordinate frames are coincident and l, m1,ni; 12, m2n2;and l3 m3,n3 are the direction cosines of the x\u00E2\u0080\u0099, y\u00E2\u0080\u0099, z\u00E2\u0080\u0099 axes relative to the x, y, z axes, respectively. Expressions for transforming direction expressed in terms of the elevation angle 6 andazimuth angle 4 between coordinate frames can be derived from the coordinate transformations1which he refers to as the polarization angle T.Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 19Figure 2.6: A perspectiveby pure rotation.projection of the parallels of two coordinate frames which are related5.00,:2, 90.0175.0-180 -135 -90 -45 o 45 90 135 180\u00E2\u0080\u00A2 (deg)Figure 2.7: A Mercator projection of the parallels of two coordinate frames which are related bypure rotation. Since the projection is conformal, the angles between the parallels are accuratelydepicted at all points on the grid.I II I,I I /_________I______2Z5\u00C2\u00B0-o(7 I\I I I7.50I\u00E2\u0080\u00A2\u00E2\u0080\u00A2_ I IChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 20(2.17) and (2.18) and the relationsr = 1, 8 = cos1z, 4) = tan(y/x), 219x = sin 6 cos 4), y = sin9 sin 4), z = cos 9.The forward transformation is given by9\u00E2\u0080\u0099 = cos1 { sinS cos 4) + m3 sinS sin 4) + n3 cos 8] , (2.20)4)\u00E2\u0080\u0099 \u00E2\u0080\u0094 tan1[2sin9cos4)+msin ) ncos6] 221\u00E2\u0080\u0094 ii sin8 cos 4) + m1 sin 9 sin 4) + n1 cos 9] \u00E2\u0080\u0098while the reverse transformation is given by8 = cos1 [ni sin 0\u00E2\u0080\u0099 cos 4)\u00E2\u0080\u0099 + n2 sin 9\u00E2\u0080\u0099 sin 4)\u00E2\u0080\u0099 + n3 cos 9\u00E2\u0080\u0099] , (2.22)4) \u00E2\u0080\u0094 tan\u00E2\u0080\u0099 m1 sinS\u00E2\u0080\u0099 cos 4)\u00E2\u0080\u0099 + m2 sin 0\u00E2\u0080\u0099 sin 4)\u00E2\u0080\u0099 + m3 cos \u00E2\u0080\u0098 (2 23)L sin 9\u00E2\u0080\u0099 cos 4)\u00E2\u0080\u0099 + \u00C2\u00A32 sin 8\u00E2\u0080\u0099 sin 4)\u00E2\u0080\u0099 + \u00C2\u00A33 COS 6\u00E2\u0080\u0099Method IA general expression for the angle a may be derived using spherical trigonometry. Consider twocoordinate frames which are related by pure rotation as shown in Figure 2.6. Let the point 0 bethe common origin of the coordinate frames and let the points Z, Z\u00E2\u0080\u0099, and P be the intersectionof the z axis, the z\u00E2\u0080\u0099 axis, and the direction of propagation with a unit sphere centered aboutthe origin. The points Z, Z\u00E2\u0080\u0099, and P define the spherical triangle ZZ\u00E2\u0080\u0099P. The great circle anglesdefined by the arcs Z\u00E2\u0080\u0099P, ZP, and ZZ\u00E2\u0080\u0099 are designated by z, z\u00E2\u0080\u0099, and p. The angles defined bythe vertices of the triangle opposite to arcs z, z\u00E2\u0080\u0099, and p, are designated by Z, Z\u00E2\u0080\u0099, and P. Sincethe vectors which are tangential to and 1 at P correspond to the directions O and \u00E2\u0080\u0098 andthe vectors 1, and,are orthogonal in both the x\u00E2\u0080\u0099y\u00E2\u0080\u0099z\u00E2\u0080\u0099 and xyz coordinate frames, it can beshown that the angle P between the unit vectors 8 and \u00E2\u0080\u0098 is congruent to the angle a betweenthe unit vectors andChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 21In terms of spherical coordinates, the great circle angles z, z\u00E2\u0080\u0099, and p are given byz = , (2.24)= Op, (2.25)p = = , (2.26)while the vertex angles Z and Z\u00E2\u0080\u0099 are given byZ= \u00E2\u0080\u0094(2.27)= (2.28)If the point P lies on the great circle defined by the arc ZZ\u00E2\u0080\u0099, the vertex angles Z and Z\u00E2\u0080\u0099 vanishand a is given bya = 00 , 00 < (Op \u00E2\u0080\u0094 O) < 180\u00C2\u00B0,(2.29)= \u00C2\u00B1180\u00C2\u00B0, \u00E2\u0080\u0094180\u00C2\u00B0 < (Op \u00E2\u0080\u0094 8) < 00However, if the point P is coincident with either the point Z or its antipode, the local horizontalin the xyz coordinate frame will be undefined and the angle a cannot be determined. Similarconsiderations apply if the point P is coincident with either the point Z\u00E2\u0080\u0099 or its antipode. Inall other cases, an expression for the angle a is obtained by applying the laws of sines to thespherical triangle ZZ\u00E2\u0080\u0099P to give eithersinpsinZsina , (2.30)sin z\u00E2\u0080\u0094sinO sin(q5p\u00E2\u0080\u0094c5z\u00E2\u0080\u0099)\u00E2\u0080\u0094sinO\u00E2\u0080\u0099porsinp sinZ\u00E2\u0080\u0099ama = , (2.31)sin z\u00E2\u0080\u0099\u00E2\u0080\u0094sin O\u00E2\u0080\u0099 sin(qY\u00E2\u0080\u0094q5)\u00E2\u0080\u0094sinOpChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 22then applying the law of cosines to givecosa = \u00E2\u0080\u0094cosZcosZ\u00E2\u0080\u0099+sinZsinZ\u00E2\u0080\u0099cosp, (2.32)=\u00E2\u0080\u0094 cos(bp\u00E2\u0080\u0094 \u00E2\u0080\u0098) cos(qV \u00E2\u0080\u0094 qYp) + sin@p \u00E2\u0080\u0094 ) sin( \u00E2\u0080\u0094 p) Cos 8.Since both the sine and cosine of the angle are known, it is a simple matter to determine theangle a using a four-quadrant arctangent function, e.g.,/ sinp sin Z/sinz N\u00E2\u0080\u0094 I\_cosZcosZl+sinZsinZlcosp)\u00E2\u0080\u0094 tan\u00E2\u0080\u0099 ( sin8 sin(4p \u00E2\u0080\u0094 c5\u00E2\u0080\u0099z)/ sinO,\u00E2\u0080\u0094 cos(p\u00E2\u0080\u0094 \u00E2\u0080\u0098) cos(4/ \u00E2\u0080\u0094 ,) + sin(qp \u00E2\u0080\u0094 \u00E2\u0080\u0098) sin( \u00E2\u0080\u0094 \u00C3\u00A71,) cosMethod IIAn alternative expression for the angle a may be derived using vector algebra. Let cp definethe horizontal plane of the xyz coordinate frame with respect to the direction of propagationand be given byp=\u00E2\u0080\u0094sin4+cos, (2.34)and let 74 similarly define the horizontal plane of the x\u00E2\u0080\u0099y\u00E2\u0080\u0099z\u00E2\u0080\u0099 coordinate frame and be given by(2.35)Let ip be the outward normal to the unit sphere at the point P in the xyz coordinate frameand be given by=\u00E2\u0080\u0094 sin 8 cos + sin6 sin + cos 0 . (2.36)The expression for the unit vector in the x\u00E2\u0080\u0099y\u00E2\u0080\u0099z\u00E2\u0080\u0099 coordinate frame given by (2.35) must betransformed to the xyz coordinate frame. This can be accomplished by determining qY in termsof 9 and 4> using (2.21) and converting the basis of the vector using (2.18). The scalar tripleproduct of the unit vectors p, ,, and ip givessina=p.(px4), (2.37)Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 23while the dot product of the unit vectors c7p with , givescosa=p.. (2.38)Since both the sine and cosine of the angle are known, a can be determined using a four-quadrant arctangent function, i.e.,a = tan\u00E2\u0080\u0099 ( \u00E2\u0080\u0098 (2.39)If ip is coincident with z axis then the definition of p given by (2.34) is ambiguous. In suchcases, the direction of the horizontal plane with respect to the direction of propagation mustbe defined arbitrarily. If p is coincident with the z\u00E2\u0080\u0099 axis, similar considerations apply to thedefinition of , given by (2.35).2.4 Evaluation of the Coordinate Transformation MatrixIn order to apply the results presented in the previous section, it is necessary to determine theelements of the coordinate transformation matrix which relates the xyz and x\u00E2\u0080\u0099y\u00E2\u0080\u0099z\u00E2\u0080\u0099 coordinateframes according to (2.17) and (2.18). This can be accomplished if either the relative directionsof the basis vectors defined by the three principal axes in each coordinate frame or a series ofEuler angle rotations which will transform one coordinate frame into the other are known [9],[101. In the first case, the coordinate transformation matrix is given by4 m1 n1[T] \u00C2\u00A32 m2 2 = \u00E2\u0080\u00A2 . . , (2.40)\u00C2\u00A33 m3 ri3 \u00E2\u0080\u0098\u00E2\u0080\u00A2 \u00E2\u0080\u0098\u00E2\u0080\u00A2 I\u00E2\u0080\u0099..where the two sets of basis vectors (i\u00E2\u0080\u0099, \u00E2\u0080\u0098, \u00E2\u0080\u0098) and (, , ) have been expressed with respect to acommon coordinate frame. In the second case, the transformation is described in terms a seriesof angles through which the first frame can be rotated in order to bring it into coincidence withthe second. A maximum of three Euler angle rotations is sufficient to bring any two framesChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 24into coincidence. One possibility is to begin the sequence with rotation about the y axis by anangle 4C,cost 0 sine x= 0 1 0 , (2.41)z11 \u00E2\u0080\u0094sine 0 cos zfollow by rotation about the x\u00E2\u0080\u009D axis by an angle ,1 0 0= 0 cos i sin , (2.42)z\u00E2\u0080\u009D 0 \u00E2\u0080\u0094 sin cos z\u00E2\u0080\u009Dand conclude with rotation about the z axis by angle ,cosC sinC 0 x\u00E2\u0080\u009D=\u00E2\u0080\u0094 C o , (2.43)z\u00E2\u0080\u0099 0 0 1 z\u00E2\u0080\u009Das suggested by Krichbaum [7]. The product of the three rotation matrices given in (2.41),(2.42), and (2.43) yields the transformation matrix which relates the xyz and x\u00E2\u0080\u0099y\u00E2\u0080\u0099z\u00E2\u0080\u0099 coordinateframes,cos cos \u00E2\u0080\u0094 sin sin ii sin sin ( cos cos C sin + sin ( sin7 cos[T] =\u00E2\u0080\u0094sinCcos \u00E2\u0080\u0094 cos(sinisin cos(cos \u00E2\u0080\u0094sinCsin+ cosCsiniicos (2.44)\u00E2\u0080\u0094cosqsin \u00E2\u0080\u0094sing coscosSince rotation is not commutative, this formulation is not unique and there are several othercombinations of Euler angle rotations which will yield an equivalent coordinate transformation,e.g., [6], [9], [101.In practice, it may be difficult to obtain the parameters required by the basis vector andEuler angle methods for determining the elements of the coordinate transformation matrix. Analternative method is derived here which allows the matrix to be determined from two arbitrarydirections (8k, qf1) and (02, 2) which have been expressed in terms of both coordinate frames.Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 25The unit vectors which correspond to these two directions can be determined by applying (2.19)to the ordered pairs (O,q4), (62,2), and (6,2) to yield= (xi,yi,zi) (a4,y,z) , (2.45)= (x2,yz) E = (x,y,z) . (2.46)A third direction, r3, can be determined from the normalized cross products,\u00E2\u0080\u0094 1x2 ,______T3\u00E2\u0080\u0094 , r3= , 2.47I r X r21 Ir X rIin order to obtain two sets of vectors (, i, r) and (, , f) which form a linearly independentset in their respective coordinate frames, i.e.,X1 X2 X3 xc xYi Y2 J3 0, 0. (2.48)Z1 Z2 Z3 Zc Z ZFrom (2.17), it can be shown thatx x x \u00C2\u00A3i m1 n1 z1 X2 X3c Y Y!3 \u00C2\u00A32 m2 fl2 Yi Y2 /3 (2.49)zc z z \u00C2\u00A33 m3 fl3 z1 z2 z3Solving for the coordinate transformation matrix gives\u00E2\u0080\u00941L m1 n1 xc x x x1 x2 x3\u00C2\u00A32 m2 n2 c Y2 Y!3 Yi Y2 J3 (2.50)\u00C2\u00A33 m3 n3 zc Z z!3 zi z2 z3Since the coordinate matrices satisfy (2.48), they will always have an inverse and (2.50) willalways have a valid solution.Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 262.5 Rotation of the Basis of Common Polarization DescriptorsIn this section, methods for rotating the basis of polarization coordinates and the complexpolarization vector e.g., [1], [3], [11], are extended to more general cases. Methods for rotatingthe basis of the Stokes vector and the coherency basis are derived. Some of the results presentedhere have recently been confirmed using a different approach by Mott [6].Polarization Coordinates and Complex Polarization RatiosThe entire range of possible polarization states of a completely polarized wave can be mappedonto the surface of a Poincar\u00C3\u00A9 sphere so that each polarization state is represented by itscoordinates in either latitude and longitude or elevation and azimuth. For example, a sphericalcoordinate system can be devised in which a polarization state W with ellipticity angle E andtilt angle r is represented by a point having longitude 2r and latitude 2e, as shown in Figure2.8. If the basis vectors Ii. and \u00C2\u00A3\u00E2\u0080\u0098 of (2.5) and (2.6) are rotated about the propagation vectorby an angle cr, the coordinates of the polarization state W\u00E2\u0080\u0099 in the new frame are given byC\u00E2\u0080\u0099 = C, (2.51)= r\u00E2\u0080\u0094c. (2.52)Alternatively, the coordinates of the polarization state W may be described in terms of thepolarization angle y and phase angle 6 which are derived from the expression of the corresponding plane wave as the weighted sum of orthogonally polarized basis states. For example, theelectric field vector E may be characterized in terms of horizontally and vertically polarizedcomponents withE (EH, + Evi\u00E2\u0080\u0099) e_ulcr, (2.53)where the time dependence \u00E2\u0082\u00ACi has been suppressed and EH and Ev are complex amplitudesgiven byEH = aj- e3SH , (2.54)= av e5\u00E2\u0080\u009D , (2.55)Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 27135\u00C2\u00B0LINEARHORIZONTALLINEARLHCFigure 2.8: Poincar\u00C3\u00A9 sphere representation of the polarization state W of a plane wave. (after[8], p. 81)as described in section 2.2. The quantities aH and av are the magnitudes of EH and Ev,respectively, and 6H and 5v are their phase angles. The polarization angle 7L and phase angleare given by7L = tarf\u00E2\u0080\u0099(av/aH),=(\u00E2\u0080\u009490\u00C2\u00B0 7L +90\u00C2\u00B0)(\u00E2\u0080\u009490\u00C2\u00B0 \u00E2\u0080\u00985L +900).(2.56)(2.57)Although any pair of orthogonally polarized basis states may be employed, the most commonare horizontal and vertical, 45\u00C2\u00B0 and 135\u00C2\u00B0 linear, and left and right circular. In the latter cases,the polarization angle and phase angle are defined in a similar manner to 7L and 6L with7D = tan1(a35/4)613 6135645,-Ic = tan1(aR/aL),\u00E2\u0080\u00985C(\u00E2\u0080\u009490\u00C2\u00B0 +90\u00C2\u00B0),(\u00E2\u0080\u009490\u00C2\u00B0 5J3 +90\u00C2\u00B0),(\u00E2\u0080\u009490\u00C2\u00B0 7c +90\u00C2\u00B0),(\u00E2\u0080\u009490\u00C2\u00B0 6c +90\u00C2\u00B0).(2.58)(2.59)(2.60)(2.61)LINEAR450 LINEARRHCandChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 28In each case, a polarization state W with polarization angle y and phase angle is represented by a point having elevation 27 and azimuth 5. The orientation of the axes from whichand cc are measured is shown in Figure 2.8. The phase reference for orthogonal circularcomponents is defined in Figure 2.9.VERTICAL8CH OR ZON TA LFigure 2.9: Definition of the phase reference for orthogonal circular components. (from [8],p. 82)Rotation of the basis of polarization coordinates is easily accomplished if they are expressedin terms of the ellipticity angle E and tilt angle r of the corresponding polarization ellipse.Expressions for transforming the polarization coordinates (7L, 6L), (7D, 6D), or (7c, 6c) into thepolarization coordinates (e, r) and back can be derived using spherical trigonometry. Considerthe right spherical triangle defined by the points corresponding to horizontal polarization, thepolarization state W, and the polarization state (0, T). For polarization coordinates expressedwith respect to a horizontally aiid vertically polarized basis, the forward transformation is givenby= sin1 (sin 27L SIfl ccL) , (2.62)r = tan\u00E2\u0080\u0099(tan27L cos6L) , (2.63)while the reverse transformation is given by7L = cos1 (cos 2e cos 2T) , (2.64)= tan\u00E2\u0080\u0099(tan2c csc2T) (2.65)Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 29Next, consider the right spherical triangle defined by the points corresponding to 45\u00C2\u00B0 linearpolarization, the polarization state W(\u00E2\u0082\u00AC, r), and the polarization state (0, r). For polarizationcoordinates expressed with respect to a 45\u00C2\u00B0 and 135\u00C2\u00B0 linearly polarized basis, the forwardtransformation is given by= sin (sifl27D SiflD) , (2.66)T = _tan_1 (cot 27D sec6D) , (2.67)while the reverse transformation is given by= cos1 (cos 2\u00E2\u0082\u00AC sin 2r) , (2.68)tan1 (tan2E sec2r) . (2.69)Finally, consider the right spherical triangle defined by the points corresponding to left circularpolarization (45\u00C2\u00B0,0), the polarization state W(\u00E2\u0082\u00AC,r), and the polarization state (\u00E2\u0082\u00AC,0). For polarization coordinates expressed with respect to a left and right circularly polarized basis, theforward transformation is given by= tan1(1)\u00E2\u0080\u0094 -Ic , (2.70)r= t5cj/2 , (2.71)while the reverse transformation is given by7c = tan\u00E2\u0080\u0099(l)\u00E2\u0080\u0094c, (2.72)= 2r. (2.73)The surface of the Poincar sphere can be mapped onto the entire complex plane by astereographic projection. Each polarization state is represented by a complex number which isreferred to as the complex polarization ratio p. Rotation of the basis of the complex polarizationratio is easily accomplished if the ellipticity angle e and tilt angle i- of the correspondingChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 30polarization ellipse are known. The complex polarization ratio is related to the polarizationcoordinates (7L, 6L), (7D, 5I3), and Qyc, c) byPL = PL (2.74)PD = PD , (2.75)pc = ,\u00C2\u00B0c , (2.76)where the polarization ratio p is defined byPL = tan7L av/aH, (2.77)PD = tan\u00E2\u0080\u0099yD a135/a4, (2.78)Pc tan7c aR/aL, (2.79)and the phase angle is defined by (2.57), (2.59), and (2.61). Together with (2.62)\u00E2\u0080\u0094(2.73), thisis sufficient to define the transformation of the complex polarization ratios PL, PD, pc to thepolarization coordinates (\u00E2\u0082\u00AC, r) and back.Complex Polarization VectorsThe pair of complex amplitudes which arise from the representation of a plane wave as theweighted sum of orthogonally polarized basis states may be arranged to yield a complex polarization vector, e.g.,EHE = , (2.80)where EH and Ev are the complex amplitudes which correspond to a horizontally and verticallypolarized basis. Rotation of the polarization basis about the propagation vector k by an anglea is accomplished by application of a rotation operator {R] to the complex polarization vectorE to yield= [R] E. (2.81)Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 31If the basis of the complex polarization vector is linear, the corresponding rotation operator[RU is simply given bycosa \u00E2\u0080\u0094sina[RL] . (2.82)sina cosaThus,E cosa \u00E2\u0080\u0094sina EH, (2.83)sina cosaandE5 cos a \u00E2\u0080\u0094 sin a E45\u00E2\u0080\u00A2 (2.84)E35 sin a cos a E135If the basis of the complex polarization vector is either elliptical or circular, it must be transformed to linear before the rotation operator given by (2.82) is applied. After it has beenrotated, the polarization basis can be restored to its original effipticity by the reverse transformation. This transformation can be performed by an ellipticity operator,coSE jsinc[H] = , (2.85)jsinc coscwhich will modify the ellipticity angle of the polarization basis states which define the upperand lower element in the polarization vector by \u00E2\u0082\u00AC and -, respectively, without affecting theirrespective tilt angles [11]. Let be the ellipticity angle of the polarization basis state whichdefines the first element in the polarization vector. Since the basis states are orthogonal,the ellipticity angle of the polarization basis state which defines the second element in thepolarization vector is given by -e. Thus, the general rotation operator [R] for the complexpolarization vector is given by[R] = [H(c)] [RL(a)] {IJ(\u00E2\u0080\u0094\u00E2\u0082\u00AC)] (2.86)cos e j sin c cos a \u00E2\u0080\u0094 sin a cos e\u00E2\u0080\u0094j sinEj sin c cos E sin a cos a \u00E2\u0080\u0094j sin c cos eChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 32If the basis is circularly polarized, e = ir/4 and the corresponding rotation operator [Rc] isgiven bye 0[Rd = . (2.87)0 e2Thus,E eic 0 EL=. (2.88)E 0 ei ERThe Stokes VectorThe .Stokes vector representation of a completely polarized wave is given bySo I (IEvI2+ IEHI2) 1= S1 = Q = *(IEvI2 \u00E2\u0080\u0094 IEHI2)= 1cos2ecos2T(2.89)S2 U IEvIIEHIcos6L cos2csin2r53 V jIEvIjEHIsin5L sin2cwhere EH, Ev, and 6L are defined in (2.54), (2.55), and (2.57), \u00E2\u0080\u0098i is the impedance of free space,1 is the total power carried by the wave, and c and T are the ellipticity and tilt angles of thecorresponding polarization ellipse. For a completely polarized wave,s=s?+s+s. (2.90)The normalized Stokes vector is given by0 181 cos2Ecos2rs = (2.91)cos2\u00E2\u0082\u00ACsin2r83 sin2ewhere1 = s + s + s . (2.92)Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 33Since all the elements of the Stokes vector are expressed in units of power, polarimetric datawhich are expressed in Stokes format can be spatially and temporally averaged with relativeease. Also, the elements of the Stokes vector are always expressed in real numbers so recourse tocomplex arithmetic is not required. Unlike the complex polarization vector, the Stokes vectorcan also represent the polarization state of quasi-monochromatic or partially polarized wave.In such cases, the Stokes vector can be resolved into a completely polarized component S, andan unpolarized component S, such thatS=S+S. (2.93)Then,S0 l\u00E2\u0080\u0094d 1 10 cos2\u00E2\u0082\u00ACcos2r dcos2fcos2rS= =10 +d =1 , (2.94)S2 0 cos2Esin2T dcos2Esin2TS3 0 sin2\u00E2\u0082\u00AC dsin2cwhere d is the ratio of the power carried by the polarized component of wave S, to the totalpower carried by the wave and is given by+ S + Sd= , 0dl. (2.95)soThe equivalent normalized Stokes vector is given by1\u00E2\u0080\u0094d 1 10 cos2ecos2T dcos2\u00E2\u0082\u00ACcos2rs = , (2.96)0 cos 2E sin 2T d cos 2e sin 2r83 0 sin2\u00E2\u0082\u00AC dsin2Ewhere the depolarization ratio d is given byd=/4+$+s, 0dl. (2.97)Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 34Consider rotation of the polarization basis about the propagation vector by an angle a.Let S represent the Stokes vector in the original coordinate frame and let 5\u00E2\u0080\u0099 represent theStokes vector in the new coordinate frame. From (2.94), S\u00E2\u0080\u0099 is given byS6 1S dcos2cos2r\u00E2\u0080\u0099= Io . (2.98)dcos2sin2r\u00E2\u0080\u0099dsin2\u00E2\u0082\u00ACFrom (2.15), r\u00E2\u0080\u0099 = r \u00E2\u0080\u0094 a, and1S dcos2cos2(T \u00E2\u0080\u0094 a)1 (2.99)S dcos2csin2(r \u00E2\u0080\u0094 a)dsin2c1d cos 2e(cos 2T cos 2a + sin 2r sin 2a)10d cos 2E(sin 2r cos 2a \u00E2\u0080\u0094 cos 2T sin 2u)dsin2cBy inspection, (2.99) can be factored to yield1 0 0 0 10 cos2a sin2a 0 dcos2\u00E2\u0082\u00ACcos2r= 1 (2.100)0 \u00E2\u0080\u0094 sin 2a cos 2a 0 d cos 2e sin 2T0 0 0 1 dsin2cThus, S\u00E2\u0080\u0099 is related to S by1 0 0 0 So0 cos 2a sin 2a 0 (2.101)0 \u00E2\u0080\u0094sin2a cos2a 0 S20 0 0 1 S3Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 35It is convenient to express (2.101) in the form[5\u00E2\u0080\u0099] = [Rs] [5], (2.102)where [Rs] is a rotation operator given by1 0 0 0o cos2c sin2o 0[Rs] = . (2.103)0 \u00E2\u0080\u0094sin2c cos2c 00 0 0 1A modified form of the Stokes vector is sometimes used to simplify the formulation of radiative transfer problems, e.g., [4], [12]. From (2.89), expressions for the vertically and horizontallypolarized intensity,Il-I = _IEvI2 (I+Q)/2, (2.104)Iv = _IEHI2 = (I\u00E2\u0080\u0094Q)/2, (2.105)are obtained. These expressions are substituted for the first two elements of the Stokes vectorto yield the modified Stokes vector Sm,SmO -1H (jEvI2) (1+dcos2ecos2r)Sml \u00E2\u0080\u0094 Iv \u00E2\u0080\u0094 \u00E2\u0080\u0094(IEHI2) (1\u00E2\u0080\u0094dcos2Ecos2r), (2106)5m2 U IEvIIEHlcos6L dcos2Ecos2TSm3 V IEvWEHIsinL dsin2Ewhich is related to the conventional Stokes vector S byC, 1 1 i Cm0 U U \u00E2\u0080\u009C0Smi 0 0 Si (2.107)5m2 0 0 1 0 S25m3 0 0 0 1 S3Consider rotation of the polarization basis about the propagation vector A by an angle . LetSm represent the modified Stokes vector in the original coordinate frame and let S representChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 36the modified Stokes vector in the new coordinate frame. From (2.101) and (2.107), it can beshown that the rotation operator [Rm] is given by00 1 0 0 0 00_i1 _1 0 0 0 cos2a sin2a 0 1 _1 0 0\u00E2\u0080\u00942 2 2 214\u00E2\u0080\u0099mi \u00E2\u0080\u00940 0 1 0 0 \u00E2\u0080\u0094sin2a cos2a 0 0 0 1 00 001 0 0 0 1 0 001cos2 a sin2 a sin 2a 0sin2 a cos2 a _l sin 2a 02 (2.108)\u00E2\u0080\u0094 sin 2a sin 2a cos 2a 00 0 0 1Thus, S is related to S byS0 cos2 a sin2 a sin 2a 0 SmO= sin2a cos2a \u00E2\u0080\u0094sin2a 0 Smi (2.109)\u00E2\u0080\u0094 sin 2a sin 2a cos 2a 0 Sm2C! 1 C\u00E2\u0080\u0098m3 U U U m3A third variant of the Stokes vector is used to define the data format employed by the JPLpolarimetric imager [4]. The JPL Stokes vector S3 is given by*(JEHI2+ lEvi2) 1S31 = j(lEHl2\u00E2\u0080\u0094 lEvi2)= 10\u00E2\u0080\u0094dcos2\u00E2\u0082\u00ACcos2r (2.110)5j2 \u00E2\u0080\u00941IEvilEHicost5L \u00E2\u0080\u0094dcos2Esin2r5j3 EvllEHIsinbL dsmn2Eand is related to the conventional Stokes vector S bySjO 1 0 0 0 SOSi\u00E2\u0080\u0099 0\u00E2\u0080\u00941 0 0 Si= (2.111)5j2 0 0\u00E2\u0080\u009410 S2S33 0 0 0 1 53Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 37Consider rotation of the polarization basis about the propagation vector k by an angle a.Let S3 represent the JPL Stokes vector in the original coordinate frame and let S representthe JPL Stokes vector in the new coordinate frame. From (2.103) and (2.111), it can be shownthat the rotation operator [R3] is given by\u00E2\u0080\u009411 000 1 0 0 0 1 0000 \u00E2\u0080\u00941 0 0 0 cos2a sin2a 0 0 \u00E2\u0080\u00941 0 0[R3]=0 0 \u00E2\u0080\u00941 0 0 \u00E2\u0080\u0094sin2a cos2a 0 0 0 \u00E2\u0080\u00941 00001 0 0 0 1 00011 0 0 00 cos2a sin2a 0V(2.112)0 \u00E2\u0080\u0094sin2a cos2a 00 0 0 1Thus, S is related to S3 byi 0 0 0 s30S\u00E2\u0080\u0099 0 cos 2a sin 2a 0 Si1 (2.113)S\u00E2\u0080\u00992 0 \u00E2\u0080\u0094 sin 2a cos 2a 0 Sj2S\u00E2\u0080\u00993 0 0 0 1The Coherency MatrixThe coherency matrix is another method for representing the polarization state of a partiallypolarized wave which is sometimes used [5], [6], [13]. In terms of the Stokes vector, the elementsof the coherency matrix are given by[J} 11J12 = (So+S1) (S2+jS3). (2.114)J21 J22 (S2\u00E2\u0080\u0094iS3) (SoSi)Substitution of the expressions for S0, S, S2, and S3 given in (2.94) into (2.114) yields anexpression for the coherency matrix in terms of the wave intensity Io, the ellipticity angle e andChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 38tilt angle T of the po\u00E2\u0080\u0099arized component of the wave and the depolarization ratio d,[J] 1(1 + dcos2ecos2r) d(cos2Esin2r + jsin 2e) (2 115)d(cos2esin2r \u00E2\u0080\u0094jsin2c) (1 \u00E2\u0080\u0094 dcos2ecos2r)The coherency matrix can be resolved into polarized and unpolarized components,{J] = {J] + [Jr] , (2.116)to give[J]\u00E2\u0080\u0094Io 01\u00E2\u0080\u0094d+Iod(1+cos2os2T) (cos2Esin2r+jsin2E)1\u00E2\u0080\u0094d 0 (cos2esin2T\u00E2\u0080\u0094jsin2) (1\u00E2\u0080\u0094cos2\u00E2\u0082\u00ACcos2r)(2.117)Consider rotation of the polarization basis about the propagation vector k by an angle o.Let [J] represent the coherency matrix in the original coordinate frame and let [J\u00E2\u0080\u0099] representthe coherency matrix in the new coordinate frame. Since the unpolarized component [J] isinvariant under rotation of the polarization basis,J1l J,\u00E2\u0080\u0099412 = Jul1 Jul2, (2.118)J2l,\u00E2\u0080\u0098L22 J2l J22and only transformation of the polarized component [Jr] need be considered. In the newcoordinate frame, [J] is given byJ71 J72 (1 + cos2\u00E2\u0082\u00ACcos2r\u00E2\u0080\u0099) (cos2Esin2r\u00E2\u0080\u0099 +jsin2E)Jf J= 10d(cos2csin2r\u00E2\u0080\u0099\u00E2\u0080\u0094jsin2E) (1\u00E2\u0080\u0094dcos2ccos2r\u00E2\u0080\u0099). (2.119)From (2.15), r\u00E2\u0080\u0099 = r \u00E2\u0080\u0094 cr andJ1 J12 (1 + cos2ccos2Qr\u00E2\u0080\u0094 cr)) (cos2Esin2(r \u00E2\u0080\u0094 c) +jsin2c)=10dJ21 J22 (cos2esin2(r\u00E2\u0080\u0094a) \u00E2\u0080\u0094jsin2) (1 \u00E2\u0080\u0094 cos2ccos2(T\u00E2\u0080\u0094a))(2.120)With a little effort, (2.120) can be factored to yieldJ1 J12 \u00E2\u0080\u0094 cosa sina J,11 J12 cosa \u00E2\u0080\u0094sina(2121)J,21 J22 \u00E2\u0080\u0094sina cosa J21 422 sina cosaChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 39Thus, from (2.118) and (2.121), it can be shown that [J\u00E2\u0080\u0099j is related to [J] byil \u00E2\u0080\u0098i2 = cosa sina J11 J12 CO5O. (2.122)J1 J2 \u00E2\u0080\u0094sina cosa J21 J22 sina cosaPolarization Scattering OperatorsThe scattering cross section a of a target is defined as the area intercepting that amount of powerwhich, when scattered isotropically, would produce an echo equal to that actually returned bythe target. Thus,1Es12a = urn 4irr2 , (2.123)r\u00E2\u0080\u0094oo IE\u00E2\u0080\u0099Iwhere E is the incident electric field, ES is the scattered electric field, and r is the range atwhich the scattered field is measured. Radar cross section refers to that portion of the scatteringcross section which is associated with a specified polarization component of the scattered waveand is a function of the size, shape, composition, and orientation of the target, the frequency ofthe incident wave, and the polarization state of the radar transmitting and receiving antennas.The relationship between the polarization states of the incident and scattered fields canbe described by a polarization scattering operator expressed in matrix form. Following thedefinition of scattering cross section presented in (2.123), the polarization scattering matrix [S]relates incident and scattered fields which have been expressed as complex polarization vectors,i.e.,Ef S11 S12 El=. (2.124)E S21 522 EWhen defined with respect to a horizontally and vertically polarized basis, the polarizationscattering matrix can also be used to relate incident and scattered fields which have beenexpressed as coherency matrices, i.e.,Jr1 Jf2 = SHH SHy h J2 5IHT(2.125)47rr2SVH Svv J J2 Sjj S17Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 40The Mueller matrix [L] relates incident and scattered fields which have been expressed as Stokesvectors, i.e.,L11 L12 L13 L14 SS= 1 L21 L22 L23 L24 S\u00E2\u0080\u00A2 (2.126)s 4irr L31 L32 L33 L34 SS L41 L42 L43 L44 SThere are three principal variants of the Mueller matrix which correspond to the conventional,modified, and JPL Stokes vectors, respectively. In practice, polarization scattering matrices andMueller matrices are often normalized by factoring out the scattering cross section of the targetand the range dependence of the response. Other polarization scattering operators which arederived from the polarization scattering matrix and the Mueller matrix, such as the covariancematrix and the Stokes scattering operator, are used in computationally efficient methods forsynthesizing arbitrary polarization responses from experimental data [4].it is convenient to describe scattering problems with respect to a coordinate frame which iscentered on the scatterer. The local coordinate systems used to define the polarization state ofthe incident and scattered fields are specified in a manner similar to that presented in section 2.2for the case of a single propagating wave. However, it is also necessary to specify the relationship between the local coordinate systems. According to the forward scatter alignment (FSA)convention, the propagation vectors of the incident and scattered fields are aligned with the direction of propagation while according to backscatter alignment (BSA) convention, they alwayspoint towards the scatterer. While the expressions for the incident field are identical underboth conventions, the expressions for the scattered field and, by extension, the correspondingpolarization scattering operator, are not. The coordinate systems and scattering geometriescorresponding to the FSA and BSA conventions are depicted in Figures 2.10 and 2.11, respectively. The subscripts i and s refer to fields expressed with respect to the FSA convention whilethe subscripts t and r refer to fields expressed with respect to the BSA convention. The unitChapter 2. Transformation of Polarization Descriptors Between Coordinate FramesAxAAy41Figure 2.10: Coordinate systems and scattering geometry for the forward scattering alignment(FSA) convention. (after [4], p. 18])Figure 2.11: Coordinate systems and scattering geometry for the backscatter alignment (BSA)convention. (after [4], p. 18])AvsAht/ kr IVr/ I/ _JChapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 42vectors in the local coordinate system for the incident (or transmitted) field are given by= = E \u00E2\u0080\u0094 = \u00E2\u0080\u0094cosqsin02 \u00E2\u0080\u0094sinsin0 \u00E2\u0080\u0094cos92 ., (2.127)xk= \u00E2\u0080\u0094,. = \u00E2\u0080\u0094sinx+cosqy, (2.128)Izxklfit f kxh \u00E2\u0080\u0094\u00C3\u0094 = \u00E2\u0080\u0094cosq5cos8\u00E2\u0080\u0094sinqcos8+sin8, (2.129)while the corresponding unit vectors for the scattered (or received) field are given by= \u00E2\u0080\u0094\u00E2\u0080\u00A2k = E \u00E2\u0080\u0094 = \u00E2\u0080\u0094cos5sin68\u00E2\u0080\u0094sin4sinO3\u00E2\u0080\u0094cos68, (2.130)\u00E2\u0080\u00A2 xkhr = \u00E2\u0080\u0094h3 = \u00E2\u0080\u0094q = sinq53x\u00E2\u0080\u0094cos4i\u00E2\u0080\u0099y, (2.131)Izxkl= k x h E \u00E2\u0080\u00948 = \u00E2\u0080\u0094cosq5co 6sin4sO+sin68. (2.132)Since the coordinate systems lii, fit) and (kr, \u00E2\u0080\u0098zr, fi,.) are coincident when the transmittingand receiving antennas are collocated, the BSA convention is a particularly convenient choicefor use in radar scattering problems. Unless otherwise stated, the BSA convention will be theconvention used in the remainder of this study.Once the local coordinate systems for the incident and scattered fields have been defined,it is a simple matter to apply the results derived earlier in this section to the problem ofrotating the basis of either the polarization scattering matrix or the Mueller matrix about theradial vector i by an angle c. Since the propagation vector = \u00E2\u0080\u00941, this transformation isequivalent to rotating the polarization basis about k by an angle \u00E2\u0080\u0094tx. Let [S] represent thenormalized polarization scattering matrix in the original coordinate frame and let [S\u00E2\u0080\u0099] representthe normalized polarization scattering matrix in the new coordinate frame. The correspondingscattering equations are given byES = [Sj E , (2.133)= [S\u00E2\u0080\u0099] Et\u00E2\u0080\u0099 . (2.134)Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 43Since the rotation operator [R] is a unitary matrix,= [R(\u00E2\u0080\u0094a)] (2.135)andEs\u00E2\u0080\u0099= [RJ\u00E2\u0080\u0099 ES , (2.136)E\u00E2\u0080\u009D = [RI\u00E2\u0080\u0094\u00E2\u0080\u0099 E\u00E2\u0080\u0099 , (2.137)where expressions for the rotation operator [RI which are appropriate for use with complexpolarization vectors having linearly polarized, circularly polarized, or arbitrarily polarized basisstates are given by (2.82), (2.87), and (2.86), respectively. Multiplying both sides of (2.136)and (2.137) by [RI yields= [RIE\u00E2\u0080\u0099 , (2138)Et = [Rj E\u00E2\u0080\u0099 . (2.139)Substituting (2.138) and 2.(139) into (2.133) gives[RI Es\u00E2\u0080\u0099 = [SI [RI Et\u00E2\u0080\u0099 . (2.140)Multiplying both sides by [R1\u00E2\u0080\u0099 gives[Rj\u00E2\u0080\u0099 [R] Es\u00E2\u0080\u0099 = [R]\u00E2\u0080\u0099 [SI [RI Et\u00E2\u0080\u0099 , (2.141)which simplifies to= [RI\u00E2\u0080\u0099 [SI [RI Et\u00E2\u0080\u0099 . (2.142)Thus, by equating (2.134) and (2.142), it can be shown that [S\u00E2\u0080\u0099I and [SI are related by[S\u00E2\u0080\u0099I = [Rj\u00E2\u0080\u0099 [5] [RI . (2.143)In a similar fashion, it can be be shown that a Mueller matrix [L\u00E2\u0080\u0099j in the new coordinate frameis related to the Mueller matrix [L] in the original coordinate frame by[L\u00E2\u0080\u0099} = [RsI1 [LI [Rs] (2.144)where expressions for the rotation operator [RsI which are appropriate for use with conventional,modified, and JPL Stokes vectors are given by (2.94), (2.106), and (2.110), respectively.Chapter 2. Transformation of Polarization Descriptors Between Coordinate Frames 442.6 ConclusionsIt has been shown that transformation of a polarization descriptor between coordinate framescorresponds to rotation of its polarization basis by a prescribed angle which is a function ofboth the transformation matrix that relates the two coordinate frames and the direction ofpropagation. Two methods for determining the angle of rotation for the case in which thelocal vertical is defined by the direction in each frame have been derived using sphericaltrigonometry and vector algebra, respectively. Both methods are robust and will yield thecorrect result but the method based on vector algebra is more compact and would be easierto implement in software. Although the elements of the coordinate transformation matrix canbe determined from either the relative directions of the three principal axes in each coordinateframe or the Euler angles which define a series of rotations which will transform one coordinateframe into the other, in practice it may be difficult to obtain these parameters. A third methodhas been derived which overcomes this limitation by allowing the elements of the coordinatetransformation matrix to be determined from any pair of directions which have been expressedin terms of both coordinate frames. Algorithms for rotating the basis of several commonly usedpolarization descriptors, including polarization coordinates, the complex polarization ratio, thecomplex polarization vector, the Stokes vector and several of its variants, the coherency matrix,the polarization scattering matrix, and the Mueller matrix have been derived.References[1] R.M.A. Azzam and N.M. Bashara, Ellipsometry and Polarized Light. Amsterdam: North-Holland, 1977.[2] S.R. Cloude, \u00E2\u0080\u009CPolarimetric techniques in radar signal processing,\u00E2\u0080\u009D Microwave J., vol. 26,no. 7, pp. 119\u00E2\u0080\u0094127, July 1983.[3] J.D. Kraus, Antennas, 2nd ed. New York: McGraw-Hill, 1988, pp. 70\u00E2\u0080\u009481.[4] F.T. Ulaby and C. Elachi, Eds., Radar Polarimetry for Geoscience Applications. Norwood,MA: Artech House, 1990, pp. 1\u00E2\u0080\u009452.[5] W.L. Stutzman, Polarization in Electromagnetic Systems. Norwood, MA: Artech House,1992.[6] H.A. Mott, Antennas for Radar and Communications: A Polarimetric Approach. NewYork: Wiley, 1992.[7] C.K. Krichbaum, \u00E2\u0080\u009CRadar cross-section measurements,\u00E2\u0080\u009D in Radar Cross Section Handbook.(G.T. Ruck et al., Eds.) vol. 2, New York: Plenum, 1970, pp. 893\u00E2\u0080\u0094896.[8] IEEE Standard Test Procedums for Antennas. (ANSI/IEEE Std 149\u00E2\u0080\u00941979.) New York:IEEE, 1979.[9] W.T. Thomson, Introduction to Space Dynamics. New York: Wylie, 1961, pp. 33\u00E2\u0080\u009437.[10] R.R. Bate, D.D. Mueller, and J.E. White, Fundamentals of Astrodynamics. New York:Dover, 1971, pp. 74\u00E2\u0080\u009483.[11] S.H. Bickel, \u00E2\u0080\u009CSome invariant properties of the polarization scattering matrix,\u00E2\u0080\u009D Proc. IEEE,vol. 53, pp. 1070\u00E2\u0080\u00941072, Aug. 1965.45References 46[12] S. Chandrasekhar, Radiative Transfer. New York: Dover, 1960, pp. 24\u00E2\u0080\u009435.[13] M. Born and E. Wolf, Principles of Optics. New York: Pergamon, 1965, pp. 545\u00E2\u0080\u0094553.Chapter 33.1 IntroductionTRUNCATION AND COMPENSATION OFTRIHEDRAL CORNER REFLECTORSA trihedral corner reflector is a reentrant structure formed by the intersection of three mutuallyorthogonal reflecting panels. In general, a ray incident upon one of its interior surfaces willundergo reflection from each of the others in succession and will be returned to the source.Although other scattering mechanisms contribute to the response, triple-bounce reflections fromthe interior of the reflector dominate over most directions of incidence. Since trihedral cornerreflectors present a large scattering cross section over a wide angular range, are mechanicallyrugged, and can be manufactured with relative ease, they are widely used in radar navigationand remote sensing as location markers and calibration targets. The relative sizes of trihedralcorner reflectors in common use are compared in Figure 3.1.(a)3m.2m\u00C2\u00B11m0mFigure 3.1: Relative sizes of trihedral corner reflectors used as location markers and calibrationtargets in (a) radar navigation, (b) radar-assisted positioning, and (b/c) radar remote sensing.(b) (c)47Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 48The dependence of the scattering cross section and angular coverage of a trihedral cornerreflector on the size and shape of its reflecting panels has been recognized since the adventof radar. In the 1940\u00E2\u0080\u0099s and 50\u00E2\u0080\u0099s, closed-form expressions for the contribution of triple-bouncereflections to the response of trihedral corner reflectors with triangular, elliptical or rectangularpanels with corners of arbitrary length were derived [1]\u00E2\u0080\u0094[3]. Although it was apparent that awide variety of response characteristics could be obtained by appropriate shaping of the reflecting panels, a procedure which Robertson [4], [5] referred to as truncation and compensation,work in this area was not pursued due to the lack of either suitable methods for determining theresponse of a trihedral corner reflector with panels of completely arbitrary shape or a need forphysically large targets which would benefit from such modifications. In recent years, interestin altering the response of trihedral corner reflectors in this manner has been renewed by arequirement for physically large targets to serve as location markers and calibration targetsin radar navigation and remote sensing [6]\u00E2\u0080\u0094[9]. However, very little design data and relatedmaterial to guide the development of such reflectors are available in the literature.In section 3.2, the problem of predicting the response of a trihedral corner reflector withpanels of completely arbitrary shape is considered and an efficient and robust numerical methodfor solving Robertson\u00E2\u0080\u0099s model for the equivalent flat plate area of a trihedral corner reflectoris proposed. In section 3.3, the response characteristics of a selected set of trihedral cornerreflectors which present three-fold symmetry are compared. In section 3.4, the response characteristics of trihedral corner reflectors which present bilateral symmetry are considered andthe possibility of increasing the beamwidth of the response of such reflectors in one principalplane relative to the beamwidth in the orthogonal plane by modifying the size and shape ofthe reflecting panels in a suitable manner is examined. Design curves for bilaterally symmetricreflectors which are composed solely of triangular, elliptical, or rectangular reflecting panels aregiven. The response characteristics of a selected set of bilaterally symmetric reflectors whichare composed of combinations of panels with various shapes including triangular, circular, andsquare are compared. A related problem, the design of top hat reflectors with specified responsecharacteristics, is considered in Appendix A.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 493.2 AnalysisA rigorous solution for the scattering cross section of a trihedral corner reflector must accountfor the contributions of single, double, and triple-bounce reflections from the interior of thereflector, deviations of the reflecting panels from perfect flatness and mutual orthogonality, anddiffraction by the panel edges. Numerical techniques such as the finite-difference time-domain(FD-TD) and the shooting and bouncing ray (SBR) methods have been successfully applied tothe problem and can account for most contributions to the response. However, calculating theresponse of a large target is extremely demanding and access to some type of supercomputeror massively parallel processor is generally required [10], [11]. In the case of an ideal reflectorwith reflecting panels which are perfectly flat and mutually orthogonal, the problem can besimplified considerably. A reasonably complete solution can be obtained by using physicaloptics (P0) to account for the contribution of reflections from the interior of the reflector whileusing the method of equivalent currents (MEC) to account for first order diffraction from theedges [12]. Alternatively, a hybrid approach which permits application of the Uniform Theoryof Diffraction (UTD) to the problem can be employed [13]. However, neither of these techniquescan be easily applied to reflectors with panels of completely arbitrary shape.For the purposes of designing trihedral corner reflectors with specified response characteristics, it is usually sufficient to account for the contribution of triple-bounce reflections from theinterior since they completely dominate the response for most directions of incidence. If thereflector is ideal, a ray which is incident upon one of the interior surfaces will generally undergoreflection from each of the others in succession and will be returned to the source. However,the reflecting panels are of finite extent and some rays will fail to intercept one or more of thepanels and will be lost. The equivalent flat plate area A of the reflector can be determined bylaunching a set of parallel rays towards the target, tracing each ray as it is reflected by eachof the interior surfaces, and projecting that portion of the reflector which contributes to thebackscatter response onto a view plane which is normal to the direction of incidence, as suggested by Figure 3.2. The scattering cross section a of the reflector is related to its equivalentChapter 3. Truncation and Compensation of Trihedral Corner Reflectors 50(a) (b)Figure 3.2: Scattering by a trihedral corner reflector with triangular panels. (a) Alternativeray paths. Path 1-1\u00E2\u0080\u0099 represents a ray which has undergone triple-bounce reflection back to thesource while path 2-2\u00E2\u0080\u0099 represents a ray which has undergone double-bounce reflection and beenscattered in a different direction. (b) The equivalent flat plat area of the reflector for incidencealong the symmetry axis. (c) The equivalent flat plat area for incidence off the symmetry axis.(from [5], p. 13-11)xyFigure 3.3: Problem geometry and coordinate system for scattering by a trihedral corner reflector composed of triangular, elliptical, or rectangular panels with corners of arbitrary length.equivalent flat plate area(c)zChapter 3. Truncation and Compensation of Trihedral Corner Reflectors 51A\u00E2\u0080\u0099,encepupilFigure 3.4: Spencer\u00E2\u0080\u0099s model for the equivalent flat plate area of a trihedral corner reflector.In this example, the intersection of the entrance pupil ABC and exit pupil A\u00E2\u0080\u0099B\u00E2\u0080\u0099C\u00E2\u0080\u0099 gives theequivalent flat plat area of a trihedral corner reflector with triangular panels for incidence offthe symmetry axis.flat plate area A by the physical optics relation,A2 (3.1)where A is the wavelength of the incident wave.It is convenient to describe scattering by a trihedral corner reflector with respect to thecoordinate frame shown in Figure 3.3. Spencer [1] empirically derived a simple geometricmodel for predicting the equivalent flat plate area of an ideal trihedral corner reflector based onexperiments that he conducted with reflectors fabricated from optical mirrors. In the model,the polygon which defines the outside edges of the reflecting panels and its inverted image areprojected onto a view plane which is normal to the direction of incidence. The inverted imageis obtained by projecting the original polygon through the apex of the reflector. The projectionand its inverted image are referred to as the entrance pupil and exit pupil of the reflector,respectively. An example for the case of a reflector with triangular panels is shown in Figure3.4. According to the model, the equivalent flat plate area of the reflector is the area commonto the two pupils. Using this model, Spencer derived closed-form expressions for the response oftrihedral corner reflectors composed of triangular and square panels with equal corner lengths.exit pupilAChapter 3. Truncation and Compensation of Trihedral Corner Reflectors 52Later, these were extended to the case of trihedral corner reflectors composed of triangular,elliptical, or rectangular panels with unequal corner lengths by Siegel et al. [2], [3]. In theirformulation, the procedure for calculating the equivalent flat plate area of a reflector beginswith determination of the intermediate quantities p, q, and r from the relations= sin6sinqS (3.2)q= sin6cos (3.3)r= sin6:inqS (3.4)where a, b, and c are the corner lengths of the reflector along the x, y, and z axes, respectively,and the direction of incidence is given by the angles 6 and q as depicted in Figure 3.3. Thevalues given to p, q, and r are then reassigned in order of increasing magnitude such thatjpj JqJ jr. For a reflector with triangular panels, the equivalent flat plate area is given byp2 + q2 + r2abcp+q+r\u00E2\u0080\u00942 , p+qT,p+q+r (35)4abc[p++rj , p+qr,while for a reflector with elliptical panels, the area is given by(M N)2 L2 M N2\u00E2\u0080\u0094Labc [tan_1 ( 4L ) + tan\u00E2\u0080\u0099 ( )+vtan_1((ML2)], L+MN,abc [tan_1 (2) +tan\u00E2\u0080\u0099 (LN)] L +M N,(3.6)where L= p2, M = q2, and N = r2. For a reflector with rectangular panels, the equivalent flatplate area is simply given byabcp(4 \u00E2\u0080\u0094 r/q) , q r/2, (3 74abc(pq/r), q r/2.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 53These closed-form expressions permit rapid and efficient computation of the equivalent flatplate area of trihedral corner reflectors with panels having certain specific shapes. In order toapply Spencer\u00E2\u0080\u0099s model to trihedral corner reflectors with panels having more general shapesKeen [14), [15) devised a numerical method for determining the size and shape of the polygondefined by the intersection of the entrance and exit pupils.Although Spencer\u00E2\u0080\u0099s model accurately predicts the equivalent flat plate area of trihedral corner reflectors with many different panel geometries, it may fail without indication when appliedto reflectors with panels of completely arbitrary shape. This was first noticed by Robertson [4]who proposed an alternative geometric model which will always yield the correct solution.Robertson\u00E2\u0080\u0099s model is based on the observation that the absolute values of the coordinates atwhich a ray incident from a given direction intersects the three planes defined by the trihedralaxes are identical to the coordinates which define the points of reflection of the ray when it isincident upon one of the interior surfaces of a trihedral corner reflector. A physical implementation of Robertson\u00E2\u0080\u0099s model is presented in Figure 3.5. Consider the trihedral corner reflectorwith panels of arbitrary shape which is shown in Figure 3.5(a). First, the panels of the reflector are replaced by complementary apertures which are derived from each reflecting panel byreflection about the trihedral axes as shown in Figure 3.5(b). The optical model which resultsis shown in Figure 3.5(c). To the observer, the polygons defined by the three complementaryapertures are projected onto a view plane which is normal to the direction of incidence. Thearea common to all three polygons is the equivalent flat plate area of the reflector.Examples of trihedral corner reflectors for which Spencer\u00E2\u0080\u0099s model gives accurate predictionsof the equivalent flat plate area are shown in Figure 3.6 while examples for which the predictionsare erroneous are shown in Figure 3.7. From these cases, it appears that Spencer\u00E2\u0080\u0099s model willgive the correct result if the projections of all three complementary apertures in Robertson\u00E2\u0080\u0099smodel are convex polygons. However, if one or more of the aperture polygons is concave, it ispossible that Spencer\u00E2\u0080\u0099s model will fail without indication and give an incorrect result which willbe larger than the actual value. Although a more formal study of the conditions under whichSpencer\u00E2\u0080\u0099s model fails was not pursued, it is clear that Keen\u00E2\u0080\u0099s numerical implementation is notsuitable for use with trihedral corner reflectors with panels of completely arbitrary shape.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 54zxz(a)(c)Figure 3.5: A physical implementation of Robertson\u00E2\u0080\u0099s model for the contribution oftriple-bounce reflections to the response of a trihedral corner reflector. (a) A trihedral corner reflector with panels of arbitrary shape. (b) Aperture planes derived from each panel byreflection about the trihedral axes. (c) An optical model for the equivalent flat plate area of atrihedral corner reflector. (from [16, p. 240])(b)Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 55zI projection ofy-z aperturepolygonprqection ofx-y aperturepolygonexit pupil/prection ofz-x aperturepolygonx_____ yzIxprojection of projection ofentrance pupil z-x aperturex-y aperturepolygon polygon(a) (b) (c)Figure 3.6: Examples of trihedral corner reflectors for which Spencer\u00E2\u0080\u0099s model gives accuratepredictions of the equivalent flat plate area. (a) Reflector geometry. (b) Spencer\u00E2\u0080\u0099s model:projection of the entrance and exit pupils of the reflector onto the view plane and determinationof their intersection. (c) Robertson\u00E2\u0080\u0099s model: projection of the aperture planes onto the viewplane and determination of their intersection.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 56(b)Figure 3.7: Examples of trihedral corner reflectors for which Spencer\u00E2\u0080\u0099s model gives erroneouspredictions of the equivalent flat plate area. (a) Reflector geometry. (b) Spencer\u00E2\u0080\u0099s model:projection of the entrance and exit pupils of the reflector onto the view plane and determinationof their intersection. (c) Robertson\u00E2\u0080\u0099s model: projection of the aperture planes onto the viewplane and determination of their intersection.zentrance pupPxexit pupilprojection ofx-y aperturepolygonprojection ofz-x aperturepolygonzIexit pupil(xNy(a)projection ofz-x aperturepolygonprojection ofx-y aperturepolygon(c)Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 57An algorithm for solving Robertson\u00E2\u0080\u0099s model for the contribution of triple-bounce reflectionsto the equivalent flat plate area of an ideal trihedral corner reflector can be devised using anapproach similar to the one followed by Keen [14], [15] in his solution of Spencer\u00E2\u0080\u0099s model.The four geometric primitives used in this algorithm are defined as follows [17]\u00E2\u0080\u0094[19j: A pointis specified by its coordinates P(x, y, z). A line segment is specified by giving its end pointsPi(xi, yi, zi) and P2(x,Y2, z2). A polyline is a chain of connected line segments which isspecified by giving a list of the vertices F1,. . . , F that define the line segments. The firstvertex is called the initial or starting point while the last vertex is called the final or terminalpoint. A polygon is a closed polyline in which the initial and terminal points coincide. The linesegments P1 F2,F2 F3,. . . , FN P1 are called the edges of the polygon. The vertex list for theexterior boundary of the polygon is traversed in a counterclockwise direction and the enclosedregion has a positive vector area. If the polygon contains interior boundaries (or holes), thecorresponding vertex lists are traversed in a clockwise direction and the enclosed regions havea negative vector area. Once the polygons which represent the panels of the reflector and thedirection of incidence have been specified, the prediction algorithm is executed in four steps:1. The polygons which represent the x-y, y-z, and z-x reflecting panels are converted intocorresponding aperture polygons by reflection about the principal axes of the trihedral.2. The x-y, y-z, and z-x aperture polygons are projected onto a view plane which containsthe origin and is normal to the direction of incidence.3. The polygon which represents the region that is common to the projection of all threeaperture polygons is determined. This is accomplished by calculating the intersection ofthe projection of the x-y aperture polygon and the projection of the y-z aperture polygonthen calculating the intersection of the result and the projection of the z-x polygon.4. The area of the polygon which represents the region that is common to the projection ofall three aperture polygons is calculated.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 58The first step in the prediction algorithm, conversion of the polygons which represent thereflecting panels into aperture polygons by reflection about the principal axes of the trihedral,can be performed by inspection. The second step, projection of the aperture polygons ontoa view plane which contains the origin and is normal to the direction of incidence, may beaccomplished by a transformation of coordinates through pure rotation. If the reflector frameis defined by x, y, and z axes of the trihedral corner reflector, let the x1y\u00E2\u0080\u009D plane define theview plane and let the z\u00E2\u0080\u0099 axis be coincident with the direction of propagation of the reflectedwave, as suggested by Figure 3.8. If the direction of propagation is given by the elevation andazimuth angles 8 and 4, the view plane is defined by\u00E2\u0080\u0094++--=O, (3.8)and the vector \u00E2\u0080\u0098 is given by(3.9)where the direction cosines a, 3, and y are given bya = sin8sinq, (3.10)/3 = sinOcos, (3.11)= cosO. (3.12)zx,/4// Z// \u00E2\u0080\u0094\u00E2\u0080\u0094/ \u00E2\u0080\u0094/ \u00E2\u0080\u0094\u00E2\u0080\u0094/ \u00E2\u0080\u0094\u00E2\u0080\u0094/ \u00E2\u0080\u0094\u00E2\u0080\u0094/\u00E2\u0080\u0094Figure 3.8: Transformation of coordinates by pure rotation.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 59From (3.9), it can be shown that the view plane and reflector coordinate systems are relatedby a transformation matrix of the form4 4 4X \u00C2\u00A311 \u00C2\u00A312 \u00C2\u00A313 X= t21 t22 t23 , (3.13)cosc cosf3 cos zwhere t3 corresponds to elements of the coordinate transformation matrix with unknown values.Since the angle of rotation of the x\u00E2\u0080\u0099-y\u00E2\u0080\u0099 plane with respect to the z\u00E2\u0080\u0099 axes can be defined arbitrarily,it is convenient to set 113 to zero so that a vector parallel to the z axis will have only acomponent in the view plane. Then, from the unitary property of the transformation matrix,it can be shown that 123 = SlIt 7. Thus,29 t11 t12 0 x= t21 122 51117 . (3.14)z\u00E2\u0080\u0099 cosa cos/3 cos zIn order to determine the values of the remaining elements, the coordinate transformationmatrix may be compared to a prototype transformation matrix which corresponds to rotationabout the z-axis through angle \u00E2\u0080\u0098\u00C3\u00A7b followed by rotation about the x\u00E2\u0080\u0099-axis through angle \u00C3\u00A7 whichgives29 cosib sin& 0 x=\u00E2\u0080\u0094cosCsint\u00E2\u0080\u0099 \u00E2\u0080\u0094cosCcos& sin( y . (3.15)z\u00E2\u0080\u0099 sin c sin \u00E2\u0080\u0094 sin (cos cos ( zBy inspection of (3.14) and (3.15), it can be shown thatsin( sin,(3.16)cos( E COS7 , (3.17)sin\u00E2\u0080\u0099\u00C3\u00A7b cosa/sin( , (3.18)cos& E \u00E2\u0080\u0094cos/3/sin(. (3.19)Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 60From (3.16)\u00E2\u0080\u0094(3.19), it can be shown that the view plane coordinate frame is related to thereflector coordinate frame by\u00E2\u0080\u0094cos/3/sin7 cosc/sin\u00E2\u0080\u0099y 0 x= \u00E2\u0080\u0094coscr/tan7 \u00E2\u0080\u0094cos/3/tan7 sin7 y . (3.20)cosc cos,6 cos zwhere , 8, and 7 are given by (3.1O)\u00E2\u0080\u0094(3.12). The projection of each of the aperture polygonsonto the view plane can be determined by transforming the coordinates of each vertex from thereflector frame into the view plane frame using (3.20) and setting their z\u00E2\u0080\u0099 coordinates to zero.The third step in the prediction algorithm, determining the region of the view plane whichis common to the projection of all three aperture polygons, is more difficult. A variety ofalgorithms for determining the intersection of overlapping polygons have been developed for usein computer graphics applications and are widely used. Most of these, including the SutherlandHodgman and Liang-Barskey polygon-clipping algorithms, are unsuitable for use in predictionalgorithm derived from Robertson\u00E2\u0080\u0099s model because they require at least one of the polygons tobe convex. However, the Weiler-Atherton polygon-clipping algorithm overcomes this limitationand is capable of clipping a concave polygon with interior holes to the boundaries of anotherconcave polygon with interior holes [18]\u00E2\u0080\u0094[21j.In the Weiler-Atherton polygon-clipping algorithm, the subject and clip polygons are described by circular lists of vertices S1, 52,. . . , SM and C1,C2,. . . , CN, respectively. Before theactual clipping is performed, the points at which the subject and clip polygons intersect aredetermined. The coordinates of the intersection points are inserted into both the subject andclip polygon vertex lists in the appropriate sequence. In order to establish a bidirectional linkbetween the vertex lists, each intersection point in the subject polygon vertex list is given apointer to the location of the same intersection point in the clip polygon list and vice versa. Theactual clipping is performed as follows: The subject polygon is traversed in a counterclockwisedirection until an intersection is reached. If this series of points lies in the interior of the clippolygon, they are added to the result list. If the next vertex of the subject polygon lies insideChapter 3. Truncation and Compensation of Trihedral Corner Reflectors 61the clip polygon, the subject polygon vertex list is followed. Otherwise, the algorithm jumps tothe clip polygon vertex list and follows it to the next intersection. This process continues untilall the intersections have been traversed and the algorithm has returned to the first point inthe result polygon. Methods for implementing the algorithm and enhancing its efficiency androbustness have beendiscussed in the literature [18)\u00E2\u0080\u0094[21].Application of the Weiler-Atherton polygon-clipping algorithm to the problem of determining the equivalent flat plate area of a trihedral corner reflector with triangular panels of equalcorner length for incidence along the symmetry axis is demonstrated in Figure 3.9. The projections of the x-y, y-z, and z-x aperture polygons onto the view plane are shown in Figure 3.9(a).The shaded region represents the area which is common to the projections of all three aperturepolygons. Symbolically, this area may be described byA = ((Z fl X) fl Y), (3.21)where Z, X, and Y are the projections of the x-y, y-z, and z-x aperture polygons, respectively.In Figure 3.9(b), polygon Z is clipped against polygon Y to yield a first result. In Figure 3.9(c),the first result is clipped against polygon X to yield the final result. The polygon which definesthe region common to the projection of all three aperture polygons is shown in Figure 3.9(d).The last step in the prediction algorithm is calculation of the area of the polygon whichdefines the region common to the projection of all three aperture polygons. If the vertices ofthe polygon are given by {S}, the z-directed vector area of the polygon is simply given byA = S x S1. (3.22)Once the equivalent flat plate area has been calculated from (3.22), the scattering cross sectionof the reflector can be determined from (3.1).Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 62projection ofz-x aperturepolygon_____ ,j \\___________1:: /\ / \ ,,/ \- VI\u00E2\u0080\u0099 \u00E2\u0080\u0094projection ofy-z aperture(a) polygonZ polygon list X polygon listS1/1(C) C1/1(S)52 N 12C212C3/I(S)S3/1/(C)54 2 1414 C4Figure 3.9: Execution of the Weiler-Atherton polygon clipping algorithm. (a) The x-y, y-z, andz-x apertures are projected onto the view plane to yield polygons Z, X, and Y, respectively.(b) Polygon Z is clipped against polygon X to yield the first result list. (c) The polygon definedby the first result list is clipped against polygon Y to yield the final result list. (d) The finalresult list defines the polygon A which contains the area common to polygons Z, X, and Y. Ineach case, S refers to points in the subject polygon, C refers to points in the clip polygon,and I, refers to the points at which the polygons intersect.C4projection ofx-y aperturepolygon14polygon V(b)S5S4polygon ZS3/I3/C3polygon Xfinal result(d)Second Clipping Sequence12(c)First Clipping SequenceFirst result list Ypolygon listSi ClIl C252 1212 1353 C313 C4S4 1414 IChapter 3. Truncation and Compensation of Trihedral Corner Reflectors 633.3 Reflectors with Three-Fold SymmetryIn this section, the extent to which the response characteristics of trihedral corner reflectorswhich present three-fold symmetry can be altered by appropriate shaping of their reflectingpanels is considered. In each case, the contribution of triple-bounce reflections to the scatteringcross section of the reflector was calculated over the entire quadrant defined by the axes of thetrihedral. The resulting array of values in 8 and were then converted to contours expressedin decibels with respect to the maximum response of the reflector. Although the reflectorcoordinate frame of Figure 3.10(a) is a convenient choice for analyzing the response of a trihedralcorner reflector, it is not the most natural choice for presenting response patterns. Instead, theresults were transformed to the global coordinate frame of Figure 3.10(b) in which the z\u00E2\u0080\u0099 axisis aligned with the local vertical and the direction of maximum response (or the boresight ofthe reflector) lies in the horizontal plane and is aligned with the x\u00E2\u0080\u0099 axis.boresightboresightz,(b)Figure 3.10: (a) Reflector coordinate frame and (b) global coordinate frame.z(a)z,x,Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 64The boresight of a trihedral corner reflector which presents three-fold symmetry is usuallycoincident with the symmetry axis defined by 8 54.74\u00C2\u00B0 and 4) = 45\u00C2\u00B0. Using the method ofevaluating coordinate transformation matrices that was derived in section 2.4, it can be shownthat the global coordinate frame is related to the reflector coordinate frame by1 1 1X\u00E2\u0080\u00947 x\u00E2\u0080\u0094 1 1 1 (323\u00E2\u0080\u00947 77\u00E2\u0080\u00A2 / .0 ZThus, direction expressed in terms of the angles 9\u00E2\u0080\u0099 and 4)\u00E2\u0080\u0099 in the global coordinate frame isrelated to direction expressed in terms of the angles 0 and 4) in the reflector coordinate frameby0\u00E2\u0080\u0099 = cos sin8 cos 4) + Vhicos ej , (3.24)\u00E2\u0080\u00943=sin8cos4)+ \u00E2\u0080\u0094=sin8sin4)\u00E2\u0080\u0094 -3=cos04)\u00E2\u0080\u0099 = tan V3 V2 V6 (3.25)*smn0cos4)_ *slnosln4)_ *cos8while the reverse transformation is given by0 = cos1 [_*sino\u00E2\u0080\u0099cos4)\u00E2\u0080\u0099 \u00E2\u0080\u0094 \u00C3\u00A7sin0\u00E2\u0080\u0099sin4)\u00E2\u0080\u0099+ Jjcoso\u00E2\u0080\u0099] , (3.26)\u00E2\u0080\u0094+sin9\u00E2\u0080\u0099cos4)\u00E2\u0080\u0099+ \u00C2\u00B1sin0\u00E2\u0080\u0099sin4)\u00E2\u0080\u00994) = tan1 v2 v2 , (3.27)sin 8\u00E2\u0080\u0099 cos 4)\u00E2\u0080\u0099 + sin 8\u00E2\u0080\u0099 sin 4)\u00E2\u0080\u0099 + cos 9\u00E2\u0080\u0099Although the response contours could be plotted on a conventional rectangular grid, distortionof the pattern can be minimized by using a grid derived from an equal area projection of thetype used in geodesy and cartography [22], [23]. Several common projections are suitable, butthe sinusoidal projection is the easiest to implement and was selected for use here. Whilethe horizontal axis of the response patterns is expressed in terms of the azimuth angle 4)\u00E2\u0080\u0099, thevertical axis is expressed in terms of the co-elevation or altitude angle \u00E2\u0080\u0098 given by8\u00E2\u0080\u0099 = 90 \u00E2\u0080\u0094 9 (3.28)in order to simplify interpretation of the results.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 65The angular coverage of symmetrical trihedral corner reflectors with triangular, circular,and square panels were predicted using the algorithm described in the previous section. Thecircular panels were represented by a twenty-sided polygon of equivalent area. The results arepresented in Figures 3.11, 3.12, and 3.13 and summarized in Table 3.1. It is generally foundthat attempts to increase the angular coverage of a symmetrical trihedral corner reflector witha fixed corner length by modifying the shape of its reflecting panels are generally accompaniedby a corresponding decrease in the ratio of the scattering cross section to the physical size ofthe reflector, as previously noted by Robertson [5].Robertson [4] proposed a method for altering the angular coverage of a symmetrical reflectorwith triangular panels by removing notches of prescribed width aild depth from the outside edgeof each panel, as shown in Figure 3.14. The depth of the notch can be expressed by a parameterd/d which is the ratio of the depth of the notch to the length of the median to the outside edgeof the panel. The angular coverage of a Robertson reflector with shallow notches (d/d = 0.25)is depicted in Figure 3.15. After notching the panels, three equal maxima appear at about12 degrees off the symmetry axis. Although the angular coverage of the reflectors increasesdramatically, the maximum response is correspondingly smaller. The angular coverage of areflector with notches of intermediate depth (d/d = 0.50) is depicted in Figure 3.16. The threemaxima are more pronounced and have shifted outward to about 21 degrees off the symmetryaxis. The angular coverage of a reflector with deep notches (d/d = 0.75) is depicted inFigure 3.17. The null in the response which has formed along the symmetry axis is -25 dB withrespect to the maximum response. The results are summarized in Table 3.1.The cumulative probability distribution of the response of symmetrical trihedral corner reflectors with triangular, circular, and square panels, and Robertson panels with shallow notches(where d/d = 0.25) over the quadrant defined by the axes of the trihedral is shown in Figure3.18. Of the four targets, Robertson\u00E2\u0080\u0099s reflector with shallow notches presents the most uniformresponse. Siegel et al. [2] claimed that the cumulative distribution of a trihedral corner reflectorwith circular panels exceeds that of reflectors with either triangular or square panels. In fact,the results presented here show that its cumulative distribution is only intermediate.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 6660453015C)ci)v 0-15-30-45-60dB-60 -30 0 30 60s\u00E2\u0080\u0099 (deg)Figure 3.11: Angular coverage of a trihedral corner reflector with triangular panels.60453015C)ci).0-15-30-45-60-60 -30 0 30 600\u00E2\u0080\u0099 (deg)Figure 3.12: Angular coverage of a trihedral corner reflector with circular panels.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 67jBFigure 3.14: Boresight view and panel geometry of trihedral corner reflectors with Robertsonpanels where the ratio of notch depth to panel depth, d/d, is (a) 0.25, (b) 0.50, and (c) 0.75.The shaded portion indicates the equivalent flat plate area of the reflector for incidence alongthe boresight.ci604530150-15-30-45-60-60 -30 0 30 600\u00E2\u0080\u0099 (deg)Figure 3.13: Angular coverage of a trihedral corner reflector with square panels.BORESIG HTVIEWPANELGEOMETRYd(a) (b) (c)Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 68I L I SFigure 3.15: Angular coverage of a trihedral corner reflector with Robertson panels whered/d = 0.25.60453015G)V 0-15-3045-60qS\u00E2\u0080\u0099 (deg)a)-D604530150-15-3045-60-60 -30 0 30 60\u00C3\u00A7S\u00E2\u0080\u0099 (deg)-60 -30 0 30 60Figure 3.16: Angular coverage of a trihedral corner reflector with Robertson panels whered/d = 0.50.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 69Figure 3.18: Cumulative probability distribution of the response of trihedral corner reflectorswith triangular, circular, square, and Robertson panels (where d/d = 0.25) over the quadrantdefined by the axes of the trihedral.604530150)a)00-15-30-45-60\u00E2\u0080\u0098 (deg)coverage of a trihedral corner reflector with Robertson panels where-60 -30 0 30 60Figure 3.17: Angulard/d2 = 0.75.1.00.810*6O.4EC)0.20.0-10 -8 -6 -4 -2 0O/L (dB)maxChapter 3. Truncation and Compensation of Trihedral Corner Reflectors 70Table 3.1:Response Characteristics of Selected Trihedral Corner Reflectors with Three-fold SymmetryMaximum ResponseFigure Panel Geometry Umax (1) Umax (2)3.11 Triangular (4ir/3)a/A2 0.0 dB3.12 Circular 15.6a 5.7dB3.13 Square 12ira4/) 9.5 dB3.15 Robertson) 0.39a4/A2 -4.1 dB3.16 Robertson) 0.18a4/)i2 -7.0 dB3.17 Robertson(s) 0.12a4/) -9.4 dB1 and 3 dB Elevation and Azimuthal Beamwidths of the Main Response LobeFigure Panel Geometry 0iB 41dB 03dB 3dB3.11 Triangular 24\u00C2\u00B0 24\u00C2\u00B0 39\u00C2\u00B0 39\u00C2\u00B03.12 Circular 18\u00C2\u00B0 17\u00C2\u00B0 31\u00C2\u00B0 30\u00C2\u00B03.13 Square 8\u00C2\u00B0 8\u00C2\u00B0 22\u00C2\u00B0 20\u00C2\u00B03.15 Robertson(s) 41\u00C2\u00B0 40\u00C2\u00B0 52\u00C2\u00B0 50\u00C2\u00B03.16 Robertson() n/a n/a n/a n/a3.17 Robertson) n/a n/a n/a n/a6 and 10 dB Elevation and Azimuthal Beamwidths of the Main Response LobeFigure Panel Geometry O6 0o 41odB3.11 Triangular 52\u00C2\u00B0 51\u00C2\u00B0 63\u00C2\u00B0 61\u00C2\u00B03.12 Circular 44\u00C2\u00B0 43\u00C2\u00B0 57\u00C2\u00B0 55\u00C2\u00B03.13 Square 36\u00C2\u00B0 35\u00C2\u00B0 50\u00C2\u00B0 50\u00C2\u00B03.15 Robertson(s) 61\u00C2\u00B0 59\u00C2\u00B0 70\u00C2\u00B0 66\u00C2\u00B03.16 Robertson) 65\u00C2\u00B0 64\u00C2\u00B0 73\u00C2\u00B0 68\u00C2\u00B03.17 Robertson(5) n/a n/a n/a n/aNotes: (1) where a is the corner length of the reflector.(2) relative to Umax of a trihedral corner reflector with triangularpanels of the same corner length.(3) where the ratio of notch depth to panel depth, d/d, is 0.25.(4) where d/d = 0.50.(5) where d/d = 0.75.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 713.4 Reflectors with Bilateral SymmetryThe response characteristics of bilaterally symmetric trihedral corner reflectors with triangular,elliptical, and rectangular panels can be predicted using the closed-form expressions for theirequivalent flat plate area which are given by (3.5), (3.6), and (3.7), respectively. Although theform of the expressions makes it difficult to derive closed-form solutions for either the size anddirection of the maximum response or the azimuthal and elevation beamwidths of the mainresponse lobe, it is a relatively simple matter to determine these quantities using a numericalapproach. The problem geometry and the reflector coordinate system are shown in Figure 3.3.It is convenient to define bilateral symmetry by a mirror plane which contains the z axis andbisects the x-y plane at an azimuthal angle pf 45 degrees. If the length of the corner along thez axis is given by c and the length of the corners along the x and y axes are given by a, thenthe reflector aspect ratio is defined as c/a.If a reflector presents a single main response lobe and is bilaterally symmetric, the directionof its maximum response must lie in its mirror plane. While the azimuthal angle cmax of thedirection of maximum response is a fixed quantity, the corresponding elevation angle 6m is afunction of the reflector aspect ratio c/a. A golden section search [24] was used to determine theelevation angle of maximum response of bilaterally symmetric trihedral corner reflectors withtriangular, elliptical, and rectangular panels. The results are presented in Figure 3.19. Althoughthe directions of maximum response are coincident when the reflector aspect ratio is unity,they diverge as the reflector becomes increasingly asymmetric. Trihedral corner reflectors withtriangular panels are of special interest since they present a planar aperture which facilitatesthe attachment of either a transmission polarizer or a protective dielectric cover. Since theintercepts of the plane which defines the reflector aperture are given by the corner lengths ofthe reflector, it can be shown that the elevation angle of the normal to the reflector aperture isgiven by= cos1 (2a+ C2) . (3.29)Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 72The angle of maximum response 8m and the angle of the normal aperture 8 of bilaterallysymmetric trihedral corner reflectors with triangular panels are plotted as a function of thereflector aspect ratio c/a in Figure 3.20. Although the direction of maximum response and thenormal to the reflector aperture are coincident if the reflector aspect ratio is unity, they divergeas the reflector becomes increasing asymmetric. If the reflector aspect ratio is less than unity,the normal to the aperture is lower than the direction of maximum response. If the ratio isgreater than unity, the reverse is true.If a trihedral corner reflector has corners of equal length \u00C2\u00A3, it is convenient to express itsscattering cross section in the formumax \u00E2\u0080\u0094 K. (3.30)The parameter K is a figure of merit which can be used to compare the maximum responseof trihedral corner reflectors with similar dimensions but different panel shapes. Although itis desirable to employ a similar scheme to express the scattering cross section of a bilaterallysymmetric reflector, the equivalent corner length of\u00E2\u0080\u0099 such a reflector must be defined. Althoughthe equivalent corner length could be defined in several different \u00E2\u0080\u0098ways, it is convenient to simplytake the arithmetic mean of the three corner lengths, i.e.,a+2c (3.31)Using this definition, the parameter K for bilaterally symmetric trihedral corner reflectors withtriangular, elliptical, and rectangular panels is plotted as a function of the reflector aspect ratioin Figure 3.21.The azimuthal and elevation beamwidths of bilaterally symmetric trihedral corner reflectorswith triangular, elliptical, and rectangular panels were determined by applying a bracketing andbisection algorithm [24] to (3.5), (3.6), and (3.7) in a global coordinate frame similar to thatdefined in Figure 3.10(b) where the z\u00E2\u0080\u0099 axis is aligned with the local vertical and the x\u00E2\u0080\u0099 axisis aligned with the direction of maximum response. The results are presented as a functionof the reflector aspect ratio c/a in Figures 3.22, 3.23, and 3.24, respectively. The increase inChapter 3. Truncation and Compensation of Trihedral Corner Reflectors 73and the converse for large reflector aspect ratios is evident. The angular coverage of bilaterallysymmetric trihedral corner reflectors with triangular panels and reflector aspect ratios of 4.0and 0.25 are presented in Figures 3.25 and 3.26, respectively.In certain cases, it has been found useful to realize bilaterally symmetric trihedral cornerreflectors which are composed of combinations of triangular, elliptical, and rectangular panels.For example, the trihedral corner reflector developed by the European Space Agency for useas a calibration target in the SAR-580 program is composed of triangular side panels and asquare center panel [8]. The angular coverage of the SAR-580 calibration target and five otherreflectors which have been realized in a similar manner are presented in Figures 3.27 through3.32. The results are summarized in Table 3.2.During the course of this study, an alternative approach to the design of bilaterally symmetric trihedral corner reflectors for use in radar-assisted positioning systems was suggested byHelmut Lanziiier of Offshore Systems Ltd. [25]. The primary design objective was to produce aset of reflectors which would support themselves on a horizontal surface with their direction ofmaximum response in the horizontal plane. A secondary objective was to provide most of thereflectors with a planar aperture in order to facilitate the attachment of a protective cover or atransmission polarizer. Three variations were devised as shown in Figure 3.33. The truncatedreflector of Figure 3.33(a) is simply a symmetrical trihedral corner reflector with triangular panels which has been inverted and had its side panels truncated iii such a way that structure is selfsupporting when placed on a horizontal surface but the effective flat plate area for incidencealong the boresight is unaffected. The compensated reflector of Figure 3.33(b) is developedfrom the truncated reflector by increasing the size of the side panels until maximum apertureefficiency is obtained. The extended reflector of Figure 3.33(c) is developed from the compensated reflector by extending the side panels further still. While the aperture of the reflectoris no longer planar, the result is a large increase in the azimuthal beamwidth. The angularcoverage of the truncated, compensated, and extended reflectors is presented in Figures 3.34,3.35, and 3.36, respectively. The results are summarized in Table 3.2.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 749075604530150I0.1 10Reflector aspect ratio - c/aFigure 3.19: Angle of maximum response 6m of bilaterally symmetric trihedral corner reflectors with triangular, elliptical, and rectangular panels vs. the reflector aspect ratio c/a.90756045301500.1 10Reflector aspect ratio - c/aFigure 3.20: Angle of maximum response \u00C2\u00B0m and angle of the normal to the aperture 8 ofbilaterally symmetric trihedral corner reflectors with triangular panels vs. the reflector aspectratio c/a.1Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 75Figure 3.21: Maximum response of a bilaterally symmetric trihedral corner reflectors with triangular, elliptical, and rectangular panels vs. the reflector aspect ratio c/a where = K \u00C2\u00A3 /A2and \u00C2\u00A3 = (a + 2c)/3.Reflector aspect ratio- c/aFigure 3.22: Azimuthal and elevation beamwidths of a bilaterally symmetric trihedral cornerreflector with triangular panels vs. the reflector aspect ratio c/a.2010I10010I0.100.1 1 10Reflector aspect ratio- c/a-10907560.45E301500.1 I 10Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 76Reflector aspect ratio - c/aFigure 3.23: Azimuthal and elevation beamwidths of a bilaterally symmetric trihedral cornerreflector with elliptical panels vs. the reflector aspect ratio c/a.Reflector aspect ratio - c/aFigure 3.24: Azimuthal and elevation beamwidths of a bilaterally symmetric trihedral cornerreflector with rectangular panels vs. the reflector aspect ratio c/a.90750.1 1 10907545301500.1 1 10Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 7760453015D)a)-15-30-45-60qS\u00E2\u0080\u0099 (deg)Figure 3.25: Angular coverage of a bilaterally symmetric trihedrai corner reflector with triangular panels and reflector aspect ratio c/a = 0.25.A..N-60 -30 0 30 60\u00E2\u0080\u0098 (deg)Figure 3.26: Angular coverage of a bilaterally symmetric trihedral corner reflector with triangular panels and reflector aspect ratio c/a = 4.0.-60 -30 0 30 6060453015C). 0I-15-30-45-60Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 7860453015C)a)-15-30-45-60-60 -30 0 30 60\u00E2\u0080\u0098 (deg)Figure 3.27: Angular coverage of a bilaterally symmetric trihedral corner reflector with triangular side panels and a circular center panel.60453015C)a)DOI-15-30-45-60-60 -30 0 30 60q5\u00E2\u0080\u0099 (deg)Figure 3.28: Angular coverage of a bilaterally symmetric trihedral corner reflector with triangular side panels and a square center panel.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 7960453015C)a)DO-15z-60-3003060\u00E2\u0080\u0098 (deg)Figure 3.29: Angular coverage of a bilaterally symmetric trihedral corner reflector with circularside panels and a triangular center panel.60453015C)DO-15-30-45-60qS\u00E2\u0080\u0099 (deg)Figure 3.30: Angular coverage of a bilaterally symmetric trihedral corner reflector with circularside panels and a square center panel.I /VtE.N\-30-45-60-60 -30 0 30 60Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 801Figure 3.31: Angular coverage of a bilaterally symmetric trihedral corner reflector with squareside panels and a triangular center panel.604530Figure 3.32: Angular coverage of a bilaterally symmetric trihedral corner reflector with squareside panels and a circular center panel.60453015C)-15-30-45-60 V.-60 -30 0 30 60\u00E2\u0080\u0098 (deg)0)a,\u00E2\u0080\u00A20150-15-30-45-60 I \u00E2\u0080\u00A2\u00E2\u0080\u00A2.i / 1 ..\u00E2\u0080\u00A2 I..\u00E2\u0080\u00A2\u00E2\u0080\u00A2-60 -30 0 30 600\u00E2\u0080\u0099 (deg)Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 81flyzflBORESIGHTVIEW22a,\u00E2\u0080\u0098 a--\\u00E2\u0080\u0098 /\u00E2\u0080\u0098 /\ /\u00E2\u0080\u009CI(a) (b) (C)SIDE PANELGEOMETRYFigure 3.33: Evolution of Lanziner\u00E2\u0080\u0099s bilaterally symmetric trihedral corner reflector through(a) truncation, (b) compensation, and (c) extension of the triangular side panels. The shadedportion indicates the equivalent flat plate area of the reflector for incidence along the boresight.604530Figure 3.34: Angular coverage of a bilaterally symmetric trihedral corner reflector with truncated triangular side panels and a triangular center panel.i1 15-15-30-45-60-60 -30 0 30 600\u00E2\u0080\u0099 (deg)Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 82r1604530._ 150)a)0\u00E2\u0080\u0098\u00E2\u0080\u00940I-15-30-45-60....1 \u00E2\u0080\u00A2..\u00E2\u0080\u00A2\u00E2\u0080\u0098 1\u00E2\u0080\u00A2.\u00E2\u0080\u00A2\u00E2\u0080\u0099 I\ i/ /-s\u00E2\u0080\u0099-..\ (///1d-60 -30 0 30 60\u00C3\u00A75\u00E2\u0080\u0099 (deg)Figure 3.35: Angular coverage of a bilaterally symmetric trihedral corner reflector with truncated and compensated triangular side panels and a triangular center panel.flifl604530150)a)--15-30-45-60\u00E2\u0080\u00A2...1 ..-60 -30 0 30 60\u00E2\u0080\u0098 (deg)Figure 3.36: Angular coverage of a bilaterally symmetric trihedral corner reflector with truncated, compensated, and extended side panels and a triangular center panel.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 83Table 3.2:Response Characteristics of Selected Trihedral Corner Reflectors with Bilateral SymmetryMaximum ResponseFigure Side Panels Center Panel (1) max(2) 9m3.27 Triangular Circular 6.4a4/A2 1.8 dB 58.5\u00C2\u00B03.28 Triangular Square 8.7a/A 3.2 dB 61.4\u00C2\u00B03.29 Circular Triangular 10.0a4/A2 3.8 dB 51.6\u00C2\u00B03.30 Circular Square 21.3a4/A2 7.1 dB 56.9\u00C2\u00B03.31 Square Triangular 18.8a4/A2 6.5 dB 49.8\u00C2\u00B03.32 Square Circular 28.6a4/A2 8.3 dB 52.8\u00C2\u00B03.34 Truncated Triangular (4K/3)a4/A2 0.0 dB 54.7\u00C2\u00B03.35 Compensated Triangular (167r/3)a4/A2 6.0 dB 54.7\u00C2\u00B03.36 Extended Triangular (l6ir/3)a4/A2 6.0 dB 54.7\u00C2\u00B01 and 3 dB Elevation and Azimuthal Beamwidths of the Main Response LobeFigure Side Panels Center Panel Oi 4\u00E2\u0080\u0099ii 03dB 3d]33.27 Triangular Circular 21\u00C2\u00B0 22\u00C2\u00B0 35\u00C2\u00B0 37\u00C2\u00B03.28 Triangular Square 17\u00C2\u00B0 17\u00C2\u00B0 31\u00C2\u00B0 31\u00C2\u00B03.29 Circular Triangular 21\u00C2\u00B0 19\u00C2\u00B0 35\u00C2\u00B0 32\u00C2\u00B03.30 Circular Square 13\u00C2\u00B0 14\u00C2\u00B0 27\u00C2\u00B0 26\u00C2\u00B03.31 Square Triangular 14\u00C2\u00B0 11\u00C2\u00B0 28\u00C2\u00B0 25\u00C2\u00B03.32 Square Circular 11\u00C2\u00B0 10\u00C2\u00B0 25\u00C2\u00B0 24\u00C2\u00B03.34 Truncated Triangular 16\u00C2\u00B0 23\u00C2\u00B0 30\u00C2\u00B0 38\u00C2\u00B03.35 Compensated Triangular 8\u00C2\u00B0 8\u00C2\u00B0 20\u00C2\u00B0 24\u00C2\u00B03.36 Extended Triangular 8\u00C2\u00B0 29\u00C2\u00B0 20\u00C2\u00B0 41\u00C2\u00B06 and 10 dB Elevation and Azimuthal Beamwidths of the Main Response LobeFigure Side Panels Center Panel O6 4\u00E2\u0080\u00996dB OiodB 4\u00E2\u0080\u0099lOdB3.27 Triangular Circular 48\u00C2\u00B0 49\u00C2\u00B0 60\u00C2\u00B0 60\u00C2\u00B03.28 Triangular Square 45\u00C2\u00B0 45\u00C2\u00B0 58\u00C2\u00B0 57\u00C2\u00B03.29 Circular Triangular 48\u00C2\u00B0 45\u00C2\u00B0 60\u00C2\u00B0 57\u00C2\u00B03.30 Circular Square 41\u00C2\u00B0 39\u00C2\u00B0 55\u00C2\u00B0 53\u00C2\u00B03.31 Square Triangular 41\u00C2\u00B0 40\u00C2\u00B0 54\u00C2\u00B0 53\u00C2\u00B03.32 Square Circular 38\u00C2\u00B0 39\u00C2\u00B0 52\u00C2\u00B0 52\u00C2\u00B03.34 Truncated Triangular 43\u00C2\u00B0 51\u00C2\u00B0 56\u00C2\u00B0 61\u00C2\u00B03.35 Compensated Triangular 33\u00C2\u00B0 40\u00C2\u00B0 45\u00C2\u00B0 55\u00C2\u00B03.36 Extended Triangular 33\u00C2\u00B0 52\u00C2\u00B0 45\u00C2\u00B0 62\u00C2\u00B0Notes: (1) where a is the corner length of the center panel of the reflector.(2) relative to Um of a trihedral corner reflector with triangularpanels of the same corner length.Chapter 3. Truncation and Compensation of Trihedral Corner Reflectors 843.5 Effect of Errors in Construction on Reflector PerformanceA trihedral corner reflector will present an optimum response if its three reflecting panels areperfectly flat and its corner angles are exactly 90 degrees. Approximate methods for determiningthe extent to which the response degrades as the reflecting panels deviate from perfect flatnessand mutual orthogonality have been presented by Spencer [1], Keen [26], and Trebits [27]. Thereduction in the scattering cross section of a symmetrical trihedral corner reflector with triangular panels due to errors in all three corner angles is presented in Figure 3.37. In general, thetolerances on the corner angles generally decrease as the size of the reflector increases. It hasalso been found that the tolerances on the corner angles depend on the shape of the reflectingpanels and the size of the equivalent flat plate area. Although the approximate methods described by Spencer, Keen, and Trebits may be used to determine the effect of panel deviationson the response of trihedral corner reflectors with modified panel geometries, numerical techniques such as the finite-difference time-domain (FD-TD) and the shooting and bouncing ray(SBR) methods could also be used to perform the necessary calculations if sufficient computational resources are available [10], [11]. Alternatively, experimental techniques may be used toassess the effect of construction errors on the response of trihedral corner reflectors [28].0wFigure 3.37: Effect of errors in all three corner angles on the response of a trihedral cornerreflector with triangular panels for incidence along the the symmetry axis. (from [5, p. 13-12])40w3 300z 20-Jw>w-J89 90 91ANGLE B IN DEGREESChapter 3. Truncation and Compensation of Trihedral Corner Reflectors 853.6 ConclusionsMethods for predicting the scattering cross section and angular coverage of a conventional trihedral corner reflector with panels of completely arbitrary shape have been considered. Althoughnumerical techniques such as the finite-difference time-domain (FD-TD) and the shooting andbouncing ray (SBR) methods can predict both the contributions of single, double, and triple-bounce reflections from the interior and the effect of deviations of the reflecting panels fromperfect flatness or mutual orthogonality, they are extremely demanding computationally. If thereflector is sufficiently large, it is usually sufficient to account for the contribution of triple-bounce reflections from the interior since they completely dominate the response for most directions of incidence. Spencer\u00E2\u0080\u0099s model [1] for the contribution of triple-bounce reflections to theequivalent flat plate area of an ideal trihedral corner reflector with either triangular or squarepanels, and the prediction algorithm subsequently derived from it by Keen [14], [15], may failwhen applied to reflectors with panels of completely arbitrary shape but an alternative modelproposed by Robertson [4] will always provide the correct solution. An efficient and robustnumerical method for solving Robertson\u00E2\u0080\u0099s model has been presented.The response patterns of ideal trihedral corner reflectors which present three-fold symmetryhave been plotted on grids derived from equal area projections and compared. If the cornerlength of the reflector is fixed, it is generally found that attempts to increase the beamwidthof the response by modifying the shape of the panels are accompanied by a reduction in theamplitude of the maximum response. If three-fold symmetry is broken so that the reflectorsimply presents bilateral symmetry about a mirror plane which contains one of the trihedralaxes and bisects the opposite panel, the beamwidth of the response in one principal planecan be increased relative to the beamwidth in the orthogonal plane by modifying the shape ofthe reflecting panels in a suitable manner. The additional degree of freedom may also proveuseful when a reflector must be designed subject to a constraint such as a requirement thatthe modified reflector present a planar aperture in order to facilitate the attachment of eithera transmission polarizer or a protective cover. A set of design curves for bilaterally symmetricChapter 3. Truncation and Compensation of Trihedral Corner Reflectors 86reflectors which are composed solely of triangular, elliptical, or rectangular reflecting panels hasbeen presented. The curves present the elevation angle of the direction of maximum response,the amplitude of the maximum response, and the elevation and azimuthal beamwidths as afunction of the reflector aspect ratio. For the case of a bilaterally symmetric trihedral cornerreflector with triangular panels, the elevation angles of the direction of maximum response andthe normal to the reflector aperture have been compared as a function of the reflector aspectratio. The response patterns of selected bilaterally symmetric reflectors which are composed ofcombinations of panels with various shapes including triangular, circular, and square have alsobeen presented and compared. The European Space Agency\u00E2\u0080\u0099s SAR-580 calibration target isincluded in this set. A related problem, the design of top hat reflectors with specified responsecharacteristics, is considered in Appendix A.Degradation of the response of a trihedral corner reflector caused by deviation of its reflecting panels from perfect flatness or orthogonality is an important consideration in the designand fabrication of calibration targets and location markers. Although the approximate methodspresented by Spencer [1], Keen [26], and Trebits [27] may be used to determine the effect ofpanel deviations on the response of trihedral corner reflectors with modified panel geometries,numerical techniques such as the finite-difference time-domain and the shooting and bouncing ray methods mentioned above can also be used to perform such calculations if sufficientcomputing resources are available.References[1] R. C. Spencer, Optical Theory of the Corner Reflector. Cambridge, MA: MIT Rad. Lab.Tech. Rep. 433, 2 Mar. 1944.[2] K.M. Siegel et al., Studies in Radar Cross Sections XVIII - Airborne Passive Measuresand Countermeasures. Ann Arbor, MI: Univ. Michgan, Jan. 1956. (cited in C.G. Bachman,Radar Targets. Lexington, MA: Heath, 1982, p. 71\u00E2\u0080\u0094101.)[3] A.L. Maffett, Topics for a Statistical Description of Radar Cross Section. New York: Wiley,1989, pp. 190\u00E2\u0080\u0094201.[4] 5. D. Robertson, \u00E2\u0080\u009CTargets for microwave radar navigation,\u00E2\u0080\u009D Bell Syst. Tech. J., vol. 26,pp. 852\u00E2\u0080\u0094869, 1947.[5] W. C. Jakes, Jr., and S. D. Robertson, \u00E2\u0080\u009CPassive Reflectors,\u00E2\u0080\u009D in Antenna EngineeringHandbook, H. Jasik, Ed. New York: McGraw-Hill, 1961, chap. 13.[6] P. 0. Gillard and K. B. Whiting, Ground Plane Corner Reflectors for Navigation andRemote Indication. U.S. Patent No. 4,104,634, dated Aug. 1, 1978.[7] S. Haykin, \u00E2\u0080\u009CPolarimetric radar for accurate navigation,\u00E2\u0080\u009D Can. J. Elect. Comp. Eng., vol. 17,pp. 130\u00E2\u0080\u0094135, July 1992.[8] D.G. Corr, A.D. Woode, and S. Bruzzi, \u00E2\u0080\u009CThe SAR-580 calibration site at RAE, Bedford,\u00E2\u0080\u009Dmt. J. Remote Sensing, vol. 3, pp. 223-227, 1982.[9] A. Freeman, Y. Shen, and C. L. Werner, \u00E2\u0080\u009CPolarimetric SAR calibration experiment usingactive radar reflectors,\u00E2\u0080\u009D IEEE Trans. Geosci. Remote Sensing, vol. GE-28, pp. 224\u00E2\u0080\u0094240,Mar. 1990.87References 88[101 A. Taflove and K.R. Umashankar, \u00E2\u0080\u009CReview of FD-TD numerical modeffing of electromagnetic wave scattering and radar cross section,\u00E2\u0080\u009D Proc. IEEE, vol. 77, pp. 682\u00E2\u0080\u0094699, May1989.[11] J. Baldauf, S-W Lee, L. Lin, S-K Jeng, S.M. Scarborough, and C.L. Yu, \u00E2\u0080\u009CHigh frequencyscattering from trihedral corner reflectors and other benchmark targets: SBR versus experiment,\u00E2\u0080\u009D IEEE Trans. Antennas Propagat., vol. 39, pp. 1345\u00E2\u0080\u00941351, Sept. 1991.[12] A.C. Polycarpou, C.A. Balanis, and P.A. Tirkis, \u00E2\u0080\u009CRadar cross section evaluation of thesquare trihedral corner reflector using P0 and MEC,\u00E2\u0080\u009D in IEEE/AP-S Symp. Dig. (AnnArbor, MI), June 1993, pp. 1428\u00E2\u0080\u00941431.[131 Z.0. Al-hekail and W.D. Burnside, \u00E2\u0080\u009CScattering from corner reflectors: a hybrid approach,\u00E2\u0080\u009Din IEEE/A P-S Symp. Dig. (Ann Arbor, MI), June 1993, pp. 1432\u00E2\u0080\u00941435.[14] K. M. Keen, \u00E2\u0080\u009CNew technique for the evaluation of the scattering cross-sections of radarcorner reflectors,\u00E2\u0080\u009D lEE Proc. H, vol. 130, pp. 322\u00E2\u0080\u0094326, Aug. 1983.[15] K. M. Keen, \u00E2\u0080\u009CFast algorithm for the exact determination of the mapped effective areas oftrihedral radar reflectors,\u00E2\u0080\u009D Electron. Lett., vol. 19, pp. 1014\u00E2\u0080\u00941015, Nov. 24, 1983.[16] D.R. Brown, R.J. Newman, and J.W. Crispin, Jr., \u00E2\u0080\u009CRCS Enhancement Devices,\u00E2\u0080\u009D in Methods of Radar Cross Section Analysis, J.W. Crispin, Jr. and K.M. Siegel, Eds. New York:Academic Press, 1969, pp. 237\u00E2\u0080\u0094280.[17) R. Sedgewick, Algorithms, 2nd ed. Reading, MA: Addison-Wesley, 1988, pp. 347\u00E2\u0080\u0094356.[18] D.F. Rogers, Procedural Elements for Computer Graphics. New York: McGraw-Hill, 1985,pp. 111\u00E2\u0080\u0094188.[19] J.D. Foley, A. van Dam, S.K. Feiner, and J.F. Hughes, Computer Graphics: Principles andPractice, 2nd ed. Reading, MA: Addison-Wesley, 1990, pp. 124\u00E2\u0080\u0094127,References 89[20] K. Weiler and P. Atherton, \u00E2\u0080\u009CHidden surface removal using polygon area sorting,\u00E2\u0080\u009D ComputerGraphics, vol. 11, pp. 214\u00E2\u0080\u0094222, Summer 1977.[21] K. Weiler, \u00E2\u0080\u009CPolygon comparison using a graph representation,\u00E2\u0080\u009D Computer Graphics,vol. 14, pp. 10\u00E2\u0080\u009418, Spring 1980.[22] P. Richardus and R.K. Adler, Map Projections for Geodesists, Cartographers, and Geographers. Amsterdam: North-Holland, 1972.[23] J.P. Snyder and P.M. Voxiand, An Album of Map Projections. U.S. Geol. Surv. Prof. Pap.,no. 1453, 1989.[24] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes,2nd ed. Cambridge Univ. Press: Cambridge, 1992.[25] D. G. Michelson and H. H. Lanziner, Radar Reflector to Enhance Detection. U.S. PatentNo. 4,990,918, dated Feb. 5, 1991.[26] K. M. Keen, \u00E2\u0080\u009CPrediction of scattering cross-section reductions due to plate orthogonalityerrors in trihedral radar reflectors,\u00E2\u0080\u009D Electron. Lett., vol. 19, pp. 115\u00E2\u0080\u0094117, Feb. 3, 1983.[27] R.N. Trebits, \u00E2\u0080\u009CRadar Cross Section,\u00E2\u0080\u009D in Radar Reflectivity Measurement: Techniques andApplications, N.C. Currie, Ed. Norwood, MA: Artech House, 1989, pp. 46\u00E2\u0080\u009449.[28] D. K\u00C3\u00A4hny and J. van Zyl, \u00E2\u0080\u009CHow does corner reflector construction affect polarimetric SARcalibration?\u00E2\u0080\u009D in Proc. IGARSS\u00E2\u0080\u009990 (College Park, MD), May 1990, pp. 1093\u00E2\u0080\u00941096.Chapter 4DEPOLARIZING TRIHEDRAL CORNER REFLECTORS4.1 IntroductionA conventional trihedral corner reflector returns linearly polarized incident waves without modification but reverses the sense of elliptically or circularly polarized waves. This is often referredto as a regular polarization response since it is also characteristic of spheres and flat plates.Although this is an ideal response for targets intended for use with conventional radars whichemploy either horizontally or vertically polarized antennas for both transmission and reception,radars which employ same-sense circular polarization to assist in rain clutter suppression or,more recently, various forms of polarization diversity to assist in target classification and identification often require calibration targets and location markers with twist-polarizing, circularlypolarizing, or linear polarization selective responses [1], [2]. Several methods for altering thepolarization response of conventional trihedral corner reflectors have been developed over theyears [3]\u00E2\u0080\u0094[12]. The three basic approaches are shown in Figure 4.1.Figure 4.1: Methods for altering the polarization response of a conventional trihedral cornerreflector. (a) Removal of one reflecting panel and loading of the interior of the reflector witha low-loss dielectric material (2.3 Er 4). (b) Installation of a transmission polarizer acrossthe reflector aperture. (c) Replacement of one of the reflecting panels by a reflection polarizer.(a) (b) (c)90Chapter 4. Depolarizing Trihedral Corner Reflectors 91Since it is often necessary to deploy large numbers of calibration targets and location markersin the field for extended periods of time where they are exposed the effects of sun, wind, andrain, the mechanical ruggedness of trihedral corner reflectors which incorporate depolarizingelements and the ease with which such reflectors can be manufactured are important considerations. Schemes which require that a transmission polarizer be installed across the reflectoraperture become increasingly difficult to implement as the reflector grows larger and cannot beapplied at all if the reflector doesn\u00E2\u0080\u0099t present a planar aperture [4]\u00E2\u0080\u0094[6]. Schemes which involvereplacing or augmenting one of the reflecting panels with a reflection polarizer oriented suchthat the axis of the grating is parallel to one of the principal axes of the trihedral have attractedconsiderable interest, particularly in recent years, because they avoid many of the mechanicalproblems associated with the use of transmission polarizers and can easily be applied to reflectors with panels of arbitrary size and shape [7J\u00E2\u0080\u0094[12]. However, trihedral corner reflectors whichincorporate reflection polarizers that make substantial use of wire grids and dielectric materialsin their construction are relatively fragile [13]\u00E2\u0080\u0094[14j. Although mechanical and environmentaldamage can be prevented through the use of protective covers and weatherproof seals, suchmeasures substantially increase manufacturing costs and are not always effective.Many of the limitations of previous schemes for altering the polarization response of trihedral corner reflectors can be overcome [15j\u00E2\u0080\u0094[16J by utilizing a reflection polarizer derived fromconducting fins or corrugations [17]\u00E2\u0080\u0094{22]. In section 4.2, the scattering properties of conducting gratings with rectangular grooves are reviewed and design curves for twist polarizing andcircularly polarizing trihedral corner reflectors are presented. Methods for realizing conductinggratings with linear polarization selective responses are proposed. In section 4.3, an algorithmfor predicting the contribution of triple-bounce reflections to the polarization scattering matrixof a trihedral corner reflector which has been modified by the addition of conducting fins orcorrugations to one of its interior surfaces as a function of the direction of incidence and orientation of the reflector is described. In section 4.4, the response patterns of prototype reflectorswith regular, twist-polarizing, and circularly polarizing responses are compared to theoreticalpredictions. The results show that the prototype reflectors respond essentially as expected.Chapter 4. Depolarizing Trihedral Corner Reflectors 924.2 Scattering by a Conducting Grating with Rectangular GroovesConsider a plane wave incident on a conducting grating with rectangular grooves as shown inFigure 4.2. Let the direction of the grooves define the grating axis. If the grating is of infiniteextent, the scattered field will consist of a finite number of propagating plane waves or diffractedorders and an infinite number of nonpropagating or evanescent waves. If the plane of incidenceis perpendicular to the grating axis, the direction in which the mth diffracted order propagatesis given by the grating equation,S1flm =sin+m, m ...,\u00E2\u0080\u00941,0,1,2,... , (4.1)where km is the angle of reflection of the mth diffracted order, 4j is the angle of incidence, A isthe wavelength of the incident wave, and d is the period of the grating. If I sin q5 1, the mthdiffracted order will propagate away from the surface at angle \u00C3\u00A7b with respect to the x axis.If sin mI > 1, the mth diffracted order takes the form of an evanescent field which decaysexponentially with increasing distance from the surface of the grating.The number of diffracted orders which are visible is dependent on the period of the gratingand the angle of incidence. The chart shown in Figure 4.3 is derived by solving the gratingequation at each angle of incidence for the point at which each diffracted order becomes visibleas the period of the grating is gradually increased. Diffracted orders with negative indices willappear at q = 90\u00C2\u00B0 while those with positive indices will appear at 4m = 900. If the gratingperiod and the angle of incidence are chosen such thatsin \u00E2\u0080\u0094 < \u00E2\u0080\u00941, (4.2)then only the specularly reflected order will propagate. However, if the grating period and theangle of incidence are chosen such that the following conditions are satisfied simultaneously,sin \u00E2\u0080\u0094 > \u00E2\u0080\u00941 , (4.3)sin\u00E2\u0080\u00942 < \u00E2\u0080\u00941, (4.4)sin+ > +1, (4.5)Chapter 4. Depolarizing Trihedral Corner Reflectors 93V0I-a)00)C0Figure 4.3: Visible diffracted orders as a function of the grating periodincidence j where the plane of incidence is normal to the grating axis.d/\ and the angle ofFigure 4.2: Problem geometry for scattering by a conducting grating with rectangular grooveswhere d, a, and h are the period of the grating and the width and depth of the grooves,respectively, is the angle of incidence, and 1m is the angle of reflection of the mth diffractedorder.2.01.51.00.50.00 15 30 45 60Angle of Incidence-q5, (degrees)75 90Chapter 4. Depolarizing Trihedral Corner Reflectors 94then both the m = 0 and the m = \u00E2\u0080\u00941 diffracted orders will propagate. The Bragg anglecondition corresponds to propagation of the m = \u00E2\u0080\u00941 diffracted order back towards the sourceand is given byd= 2sinq (4.6)It has been found that a grating which satisfies this condition can be made to diffract all thepower in an incident wave into the m = \u00E2\u0080\u00941 diffracted order by appropriate shaping of its profile.Since the grating is uniform in the z-direction, the spatial derivatives associated with theincident and scattered fields vanish in z and Maxwell\u00E2\u0080\u0099s equations divide into two independentsets,\u00C3\u00B4E \u00C3\u00B4E= jwe0E, = \u00E2\u0080\u0094jw\u00C3\u00A7,E9, \u00E2\u0080\u0094 = \u00E2\u0080\u0094jwi0H , (4.7)Oy ox Ox 9yOE . OII OH-h-\u00E2\u0080\u0094 = -jwH, -h-- = jwH, -h-\u00E2\u0080\u0094 \u00E2\u0080\u0094 = jwe0E. (4.8)The first, which consists of lIe, E, and E components, is called a transverse magnetic (TM)or E-polarized field while the second, which consists of E, H, and H components, is called atransverse electric (TE) or H-polarized field. Let [Sm] be the normalized polarization scatteringmatrix which is associated with diffraction into the mth order. Since the TM- and TE-polarizedfields are decoupled, the off diagonal elements of the matrix vanish and [Sm] reduces to0 c nEErc\u00E2\u0080\u0099 1 EE,m EH,m 1m[JmJ = = ,C c n rHHHE,m HH,m mwhere pE represents the complex reflection coefficient for diffraction of a TM-polarized incident wave into the TE-polarized component of the mth diffracted order and [Sm] is expressedaccording to the forward scattering alignment (FSA) convention. If the amplitude of the complex reflection coefficients 1JE = f0HH1 = 1, then the total power in the specular reflectedorder will be independent of the polarization state of the incident wave. Such a response is saidto be polarization-operative. However, if the amplitude JEE1 JIH1, then the total power inthe specular reflected order will be dependent on the polarization state of the incident wave.Such a response is said to be polarization-selective.Chapter 4. Depolarizing Trihedral Corner Reflectors 95The complex reflection coefficients PEmE and are functions of both the dimensions ofthe grating and the direction and wavelength of the incident wave. Since the free space andgroove regions are defined by separable coordinate systems, a rigorous solution to the problemcan be obtained by representing the fields in each region as the weighted sum of orthonormalbasis functions and determining the relative amplitude and phase of the propagating diffractedorders by mode-matching at the boundary between the regions. A complete derivation ofthe solution and procedures for determining the validity of numerical results obtained by thismethod are presented in Appendix B. Although the solution will invariably converge to anessentially constant result as the number of modes used in the field expansions are systematicallyand gradually increased, it is shown that the manner in which the solution converges and thevalue of the final result will depend on both the number of modes used to represent the fields ineach region and their ratio. This phenomenon is commonly referred to as relative convergence.Numerical results are presented which suggest that the optimum ratio of groove modes to freespace modes is similar in value to the aspect ratio of the grating, aid. An implementation ofthe solution as a pair of subroutines coded in Fortran 77 is also presented.For the purpose of defining the polarization states of the incident and reflected waves, letthe z axis define the local vertical. If period of the grating is less than one-half wavelength,only the specular order will propagate. In such cases, the phase difference 6 between the TEand TM reflection coefficients can be exploited to yield a depolarizing response of the form,[S]= SHH SHy= pEE 0 = 1 0 (4.10)SVH Svy 0 pHH 0 e3A phase difference of 0 will yield a regular polarization response while phase differences of 180and 90 degrees will yield twist-polarizing and circularly polarizing responses, respectively. Theco-polar and cross-polar response of regular, twist, and circular polarizers with polarizationresponses defined by (4.10) are plotted as a function of the polarization state of the incidentwave in Figures 4.4, 4.5, and 4.6. If the grating is modified in such a way that pE 0, thelinear polarization selective response of Figure 4.7 will be obtained.Chapter 4. Depolarizing Trihedral Corner Reflectors 96(a)b(b)Figure 4.4: Normalized response of a regular reflector as a function of the polarization state ofthe incident wave. (a) Co-polar response. (b) Cross-polar response.(a) (b)Figure 4.5: Normalized response of a twist-polarizing reflector as a function of the polarizationstate of the incident wave. (a) Co-polar response. (b) Cross-polar response.Chapter 4. Depolarizing Trihedral Corner Reflectors 97(a)Figure 4.6: Normalized response of a circularly polarizing reflector as a function of the poiarizatiori state of the incident wave. (a) Co-polar response. (b) Cross-polar response.(a) (b)Figure 4.7: Normalized response of a vertical polarization selective reflector as a function of thepolarization state of the incident wave. (a) Co-polar response. (b) Cross-polar response.a,(b)Chapter 4. Depolarizing Trihedral Corner Reflectors 98For a grating with a vanishingly small period and arbitrarily thin fins, the groove depthsrequired to realize twist and circularly polarizing responses for normal incidence are simply A/4and .X/8, respectively. In practice, however, account must be taken of both the finite dimensionsof the grating and the angle of incidence. The groove depths required to realize twist andcircularly polarizing responses were determined as a function of the period and aspect ratio ofthe grating and the direction and wavelength of the incident wave by applying a bracketing andbisection algorithm to a subroutine which calculates the phase difference between the TE andTM specular reflection coefficients. The calculations were performed using the mode-matchingformulation presented in Appendix B. Twist polarizer design curves for essentially normalincidence and 45 degree incidence are presented in Figures 4.8 and 4.9, respectively. (Theresponse of the grating could not be calculated for incidence at 4j = 00 due to the nature ofthe analytical formulation so the design curves for normal incidence were calculated at qSj = 10instead.) As the grating period approaches zero, the groove depth required to yield a twist-polarizing response converges to A/4 for all aspect ratios and angles of incidence. As the periodof the grating increases, the required groove depth increases for large aspect ratios and decreasesfor small aspect ratios. Circular polarizer design curves for essentially normal incidence and45 degree incidence are presented in Figures 4.10 and 4.11. In this case, the ideal groove depthis satisfactory only for very thin fins and at normal incidence. The difference between thegroove depths required to realize twist polarizers and circular polarizers with identical aspectratios is nearly constant as a function of the grating period. A large difference between therequired groove depths implies that the grating will present the desired polarization responseover a wide operating bandwidth while a small difference implies that the grating will presentthe desired response over a relatively narrow operating bandwidth. For normal incidence, thedifferences between the required groove depths for gratings with aspect ratios a/d of 0.3333,0.7500, and 0.9999, are approximately 0.05 A, 0.10 A, and 0.13 A, respectively. Similar resultsare obtained for 45 degree incidence. This suggests that twist and circular polarizers derivedfrom corrugated surfaces should be designed with the largest possible aspect ratios in order toobtain optimum performance.Chapter 4. Depolarizing Trihedral Corner Reflectors 99Three methods for realizing a grating which presents a linear polarization selective responseare depicted in Figure 4.12. A grating with a small period and large aspect ratio which has beenloaded with lossy media, as shown in Figure 4.12(a), will present a TE-polarization selectiveresponse since the groove region functions as a resonant absorber for TM-polarized incidentwaves [23], [24]. This approach has several disadvantages including the limitations on the minimum value of the TM reflection coefficient that can be obtained, the frequency selective natureof the response, and the vulnerability of the lossy groove media to mechanical or environmental damage. Alternatively, a conventional grating which satisfies the Bragg condition can bemade to diffract the TM-polarized component of an incident wave into the m = \u00E2\u0080\u00941 diffractedorder while it specularly reflects the TE-polarized component by appropriate shaping of itsprofile [25]. Although such a grating will act as a retro-reflector for TM-polarized waves andcannot be used to realize a linear polarization selective trihedral corner reflector, it has beenshown that gratings can be perfectly blazed for other angles of incidence as well, as suggestedby Figure 4.12(b) [26]. The phenomena of perfect blazing for non-Bragg angle incidence isnot well understood and relatively few examples are known. Also, the direction of the higherdiffracted order is a function of both the wavelength and direction of the incident wave andmust be accounted for when designing an apparatus which incorporates such a polarizer. Athird approach is shown in Figure 4.12(c). A grating with a small period, large aspect ratio,and a sloped profile which subtends an angle 3, will reflect TM-polarized incident from thebottom of the groove region at an angle qo with respect to the grating normal but will reflectTE-polarized incident waves from the top of the corrugations at an angle o \u00E2\u0080\u0094 23. Since theresponse is independent of the wavelength of the incident wave and the structure is inherentlyrugged, this approach overcomes many of the limitations of the other two schemes. Althoughthe angle 3 should be as large as possible so that the directions of the TM and TE polarizedreflected waves are separated by a wide angle, the maximum height of the fins is given byhm = r sin /3 where r is the maximum dimension of the polarizer in the plane perpendicularto the grating axis. Thus, a practical limit on the slope angle which can be accomodated isimposed when a physically large polarizer of this type is required.Chapter 4. Depolarizing Trihedral Corner Reflectors 1000.400.35x.0.300.25\u00E2\u0080\u0094 y0.200.150.0 0.1 0.2 0.3 0.4Period - d/AFigure 4.8: Twist polarizer design curves for normal incidence.0.400.35X 0.300.25\u00E2\u0080\u0094y0.200.150.0 0.1 0.2 0.3 0.4 0.5Period-d/AFigure 4.9: Twist polarizer design curves for 45 degree incidence.0.5Chapter 4. Depolarizing Trihedral Corner Reflectors 1010.250.200.15a,0.050.000.200.150.100.050.000.0 0.2 0.3 0.4Period - d/AFigure 4.11: Circular polarizer design curves for 45 degree incidence.x\u00E2\u0080\u0094 ya/d\u00E2\u0080\u0094 0.9999- 0.90000.75000.50000.3333Circular polarizer= 1\u00C2\u00B00.0 0.1 0.2 0.3 0.4 0.5Period - d/AFigure 4.10: Circular polarizer design curves for normal incidence.0.25x\u00E2\u0080\u0094 ya/d\u00E2\u0080\u0094 0.9999- 0.90000.75000.50000.3333Circular polarizer0 = 4500.1 0.5Chapter 4. Depolarizing Trihedral Corner Reflectors 102TE,TM TM TEFigure 4.12: Linear polarization selective reflectors derived from corrugated surfaces. (a) Specular reflection of TE-polarized incident waves and dissipation of TM-polarized incident wavesin lossy groove media. (b) Specular reflection of TE-polarized incident waves and diffraction ofTM-polarized incident waves into the m = \u00E2\u0080\u00941 order. (c) Specular reflection of TM-polarized incident waves with respect to the base of a polarizer with a sloped profile and specular reflectionof TE-polarized incident waves with respect to the upper surface.TE(a)__J_ __(b)TE,TM TE TM(c)Chapter 4. Depolarizing Trihedral Corner Reflectors 103Thus far, it has been assumed that the plane of incidence is perpendicular to the gratingaxis with the phase of the incident wave given byexp(jk(2)= exp[jk(xcosq5 \u00E2\u0080\u0094 ysinq1)] . (4.11)Since the grating is uniform along its axis, the results are easily extended to the case of obliqueincidence [27]. In this case, the phase of the incident wave is given byexp(jk() = exp[jk(x sin 6 cos\u00E2\u0080\u0094y sin 6 sin \u00C3\u00A7 + z cos 92)J , (4.12)which is obtained from (4.11) by replacing k by k sin 8: and multiplying by exp(jkz cos 8:). Sincethe scattered field also varies with z as exp(jkzcos82),the spatial derivative \u00C3\u00B4/\u00C3\u00B4z in Maxwell\u00E2\u0080\u0099sequations may simply be replaced with jk cos 8. As in the case of perpendicular incidence, theequations divide into TM-polarized and TE-polarized sets. In the TM-polarized case, E 0and the field components are expressed in terms of H where\u00C3\u00942H \u00C3\u00942HZ 2 22 + 2 + k sin 8 H = 0 , (4.13)ax ayZ0 (\u00C3\u0094HZ t9HEx+Ey= )ksin28: \u00E2\u0080\u0094\u00E2\u0080\u0094z\u00E2\u0080\u0094 \u00E2\u0080\u0094WY) (4.14)cos8 (aHZ aHHx+Hy= Jksin281 \u00E2\u0080\u0094-\u00E2\u0080\u0094X+--\u00E2\u0080\u0094--Y) , (4.15)while in the TE-polarized case, li = 0 and the field components are expressed in terms of Ewhere\u00C3\u00942E \u00C3\u00942E+ (4.16)cos 8 (dEZ OE \u00E2\u0080\u0098\ErX+Eyy= Jksin28 j---\u00E2\u0080\u0094x+--\u00E2\u0080\u0094Y) , (4.17)1 (0E2 OEZIJ x + H \u00E2\u0080\u0098 3kZ0sin2 8X \u00E2\u0080\u0094 Y} . (4.18)Since the boundary conditions are identical, the solution to the problem of scattering by aconducting grating at oblique incidence to the grating axis is identical to the solution forperpendicular incidence if k is replaced by k sin 8 and all fields are multiplied by exp(jkz cos 8).Chapter 4. Depolarizing Trihedral Corner Reflectors 1044.3 Scattering by a Depolarizing Trihedral Corner ReflectorThe polarization response of a trihedral corner reflector can be modified by replacing one ofits panels by a reflection polarizer derived from a conducting grating with rectangular grooves,as depicted in Figure 4.13. Projections of a typical ray path in the x-y, z-z, and y-z planesare also shown. The coordinate frame is identical to that used in the previous section todescribe scattering by a free-standing grating. In order for this type of depolarizing trihedralcorner reflector to function correctly, the grating must be oriented in such a way that all of therays which are incident from a given direction undergo identical polarization transformationsregardless of the sequence in which they are reflected by each of the interior surfaces. It can beshown that this condition will be satisfied if the grating axis is parallel to one of the principalaxes of the trihedral. If the incident field is resolved into orthogonal components which areTM- and TE-polarized with respect to the grating axis, the components will not be coupleddue to reflection from either the grating or any of the three dihedral corners which comprise thetrihedral corner reflector. As a result, reflection from each interior surface can be represented bya diagonal polarization scattering matrix and the cumulative transformation due to the threereflections necessary to return a incident ray to the source will be independent of the sequencein which the reflections occur. Furthermore, it can be shown that the angles of incidence of therays with respect to both the grating normal and the grating axis will be equivalent regardless ofthe sequence in which the rays are reflected by the interior surfaces of the reflector. Thus, all ofthe rays which are incident on the reflector will undergo identical polarization transformationson reflection from the grating and the required condition will be met.For the purpose of defining the polarization states of incident and reflected waves, hencethe polarization response of the reflector, let the z axis define the local vertical in the reflectorcoordinate frame. For an arbitrary direction of incidence, it is convenient to specify horizontaland vertical in terms of the triad (, \u00C3\u0094, ) associated with a conventional spherical coordinatez///(a)(b)Chapter 4. Depolarizing Trihedral Corner Reflectors 105z/x yx yx.\u00E2\u0080\u0094x \u00E2\u0080\u0094.y(c) (d)Figure 4.13: Replacement of one panel of a trihedral corner reflector by a reflection polarizerderived from a corrugated surface. (a) View along the symmetry axis. (b) Projection of atypical ray path in the x-y plane. (c) Projection of a typical ray path in the z-x plane. (d)Projection of a typical ray path in the y-z plane. If the direction of incidence is (6, 4), theangle 0 is given by cos\u00E2\u0080\u0099(cos8cos4) and the angle 6 is given by cos\u00E2\u0080\u0099(cos6sin).z zy0.350.30z0.25-caChapter 4. Depolarizing Trihedral Corner Reflectors 1060.50aid0.45\u00E2\u0080\u0094- 0.90000.25x0.20 Twist polarizer0 = 54.74\u00C2\u00B0, = 45\u00C2\u00B00.150.00Figure 4.14: Twist polarizer design curves for incidence along the symmetry axis of a trihedralcorner reflector.0.10 0.20 0.30Period - d/)0.40 0.50:0.10 - 0.90000.7500x . . 0.50000.05 Circular polarizer. 00 = 5474b, = 4500.00 . .1... .1 \u00E2\u0080\u00A2 ...I. \u00E2\u0080\u00A2 ..I. \u00E2\u0080\u00A20.00 0.10 0.20 0.30 0.40 0.50Period-d/AFigure 4.15: Circular polarizer design curves for incidence along the symmetry axis of a trihedralcorner reflector.Chapter 4. Depolarizing Trihedral Corner Reflectors 107system where= E = (4.19)= ZXkE = \u00E2\u0080\u0094sin+cos, (4.20)Izxkl= kxh \u00E2\u0080\u0094 = \u00E2\u0080\u0094cosqcos8i\u00E2\u0080\u0094sinq5cos9+sin6, (4.21)as described in section 2.2. Here, the component of an incident wave which is TM-polarized withrespect to z corresponds to horizontal polarization while the component which is TE-polarizedwith respect to z corresponds to vertical polarization.The polarization scattering matrix of a depolarizing trihedral corner reflector accordingto the backscatter alignment (BSA) convention is identical to that of a free-standing gratingaccording to the forward scatter alignment (FSA) convention for the same direction of incidence.If only the specular reflected order propagates, the polarization response of the reflector is ofthe form[S] = SHH SHy= 0 = 1 0\u00E2\u0080\u00A2 (4.22)SVH Svv 0 pHH 0 e6Since TM- and TE-polarized waves are decoupled upon reflection from the grating even if theplane of incidence makes an oblique angle with the grating axis, the off diagonal elements ofthe polarization scattering matrix given by (4.19) will always vanish in the reflector coordinateframe of Figure 4.13. The polarization scattering matrix of a target relates the scattered fieldES at the receiver to the incident field E at the target according to\u00E2\u0080\u0094 \u00E2\u0080\u0094jkr S11 S12 E4 2\u00E2\u0080\u0094 r S21 S22 E\u00E2\u0080\u00983)where both the fields and the polarization scattering matrix have been expressed with respectto an arbitrarily polarized basis. Often, this expression is normalized by factoring out thescattering cross section of the target and the range dependence of the response. The polarizationmatch factor or polarization efficiency between the scattered field and a receiving antenna ofChapter 4. Depolarizing Trihedral Corner Reflectorsboresightboresightzz,Figure 4.16: (a) Reflector coordinate frame and (b) global coordinate frame.(a) (b)108Figure 4.17: Angle of rotation of the projection of the grating axis onto the view plane forincidence (a) along the symmetry axis and (b) 30 degrees off the symmetry axis.z(a)z,(b)Chapter 4. Depolarizing Trihedral Corner Reflectors 109polarization state h is given byIEs.h12P OP 1. (4.24)A reflection polarizer designed for incorporation into a trihedral corner reflector shouldpresent the desired polarization response along the same direction of incidence that the reflectorpresents its maximum scattering cross section. For reflectors composed of triangular, circular,or square panels with equal corner lengths, this direction falls along the symmetry axis of thereflector. In the reflector coordinate frame of Figure 4.13, this direction is given in sphericalcoordinates by 8 = 54\u00E2\u0080\u00A2740, = 45\u00C2\u00B0. Design curves for twist polarizers and circular polarizersfor incidence along this direction are presented in Figures 4.14 and 4.15, respectively. In theprevious section, it was shown that the response of a grating for oblique incidence to the gratingaxis is identical to its response for perpendicular incidence to the grating axis if k is replaced byk sin 6 and all fields are multiplied by exp(jkz cos 8). In this case, sin 8 is given by and cos 8is given by i//. This transformation may also be applied in the design of linear polarizationselective reflectors which are based on the concepts described in the previous section.The response of the target can be altered to suit a particular application by rotating thetarget about its symmetry axis [28). After the target has been rotated by an angle c from thevertical, as shown in Figure 4.19, its polarization scattering matrix is given by[S\u00E2\u0080\u0099j = [R]\u00E2\u0080\u0099 [S] [R] . (4.25)If the response is expressed with respect to a linearly polarized basis, the rotation operator [R]is given bycosu \u00E2\u0080\u0094sina[RL] = , (4.26)sina cosawhile if the response is expressed with respect to a circularly polarized basis, the operator isgiven byeja 0[Rc] = . (4.27)0 e3Chapter 4. Depolarizing Trihedral Corner Reflectors 110For example, if a twist-polarizing target is rotated about its symmetry axis to c = 00, \u00C2\u00B190\u00C2\u00B0,or 180\u00C2\u00B0 degrees, it will present a maximum co-polar response for horizontally, vertically, andcircularly polarized incident waves. If the target is rotated to c = \u00C2\u00B145\u00C2\u00B0 or \u00C2\u00B1135\u00C2\u00B0, it will stillpresent a maximum co-polar response to circularly polarized incident waves but will present amaximum cross-polar response to horizontally and vertically polarized incident waves.In practice, it is often necessary to describe the polarization response of a depolarizingtrihedral corner reflector with respect to a global coordinate frame in which the horizontalplane contains the boresight of the reflector and the vertical plane is parallel to the axis of thegrating, as suggested by Figure 4.16. By the methods described in section 2.4, it can be shownthat the global coordinate frame xyz is related to the reflector coordinate frame xyz byI 1 1 1x,/ X= * * .\u00E2\u0080\u0094* , (4.28)0 zwhere the x\u00E2\u0080\u0099 axis in the global coordinate frame corresponds to the symmetry axis of the reflectorand the z\u00E2\u0080\u0099 axis is parallel to vertical. Transformation of the polarization scattering matrix ofa depolarizing trihedral corner from the reflector coordinate frame to the global coordinateframe corresponds to rotation of its basis by a prescribed angle which is a function of both thetransformation matrix which relates the two coordinate frames and the direction of propagation,as described in Chapter 2. Let p define the horizontal plane of the xyz coordinate frame withrespect to the direction of propagation and be given byp=\u00E2\u0080\u0094sin&+cosq, (4.29)and let , similarly define the horizontal plane of the x\u00E2\u0080\u0099y\u00E2\u0080\u0099z\u00E2\u0080\u0099 coordinate frame and be given by(4.30)Let p be the outward normal to the unit sphere at the point P in the zyz coordinate frameand be given by= \u00E2\u0080\u0094sin 6cosq +sinsin+ cos9. (4.31)Chapter 4. Depolarizing Trihedral Corner Reflectors 111The expression for the unit vector , in the x\u00E2\u0080\u0099y\u00E2\u0080\u0099z\u00E2\u0080\u0099 coordinate frame given by (4.30) must betransformed to the xyz coordinate frame. This can be accomplished by determining qY in termsof 6 and according to the relation\u00E2\u0080\u00943-sin9cos+ -3-sin6sinq\u00E2\u0080\u0094 \u00E2\u0080\u00943-cos0= tan , (4.32)and converting the basis of the unit vector \u00C3\u00A7Yp according to the relation given by (4.28). Then,the angle of rotation can be determined from= tan\u00E2\u0080\u0099 . (4.33)\ c5p.14 IIf ip is coincident with z axis then the definition of p given by (4.29) is ambiguous. In suchcases, the direction of the horizontal plane with respect to the direction of propagation mustbe defined arbitrarily, as noted in section 2.2. If ip is coincident with the z\u00E2\u0080\u0099 axis, similarconsiderations apply to the definition of given by (4.30). Although several factors causethe polarization response of a depolarizing trihedral corner reflector to degrade asthe directionof incidence shifts away from the boresight, rotation of the projection of the grating axis inthe aperture plane with respect to the local vertical is the most important since it resultsin an effective rotation of the corresponding polarization scattering matrix, as suggested byFigure 4.17.The polarization scattering matrix corresponding to a circular polarization selective response cannot be diagonalized when expressed with respect to a linearly polarized basis. As aresult, it is not possible to modify a conventional trihedral corner reflector to present a circularpolarization selective response simply by adding conducting fins or corrugations of appropriatedimensions and orientation to one of its interior surfaces. Alternative methods for obtainingsuch a response based on the addition of a transmission circular polarizer to a linear polarization selective reflector or a circular polarization selective surface to a twist-polarizing reflectorare proposed in Appendix C.Chapter 4. Depolarizing Trihedral Corner Reflectors 1124.4 Numerical and Experimental ResultsIn this section, the results of a test program that was conducted in order to evaluate theresponse characteristics of depolarizing trihedral corner reflectors which incorporate a reflectionpolarizer derived from a corrugated surface are given. Prototype trihedral corner reflectorswhich present regular, twist-polarizing, and circularly polarizing responses were designed foruse at the standard marine radar frequency of 9.445 GHz and assembled. Their polarizationand azimuthal response patterns were measured and the results were compared to theoreticalpredictions of the contribution of triple-bounce reflections to their response based on the analysispresented in the previous section. Details of the design and construction of the prototypereflectors are presented in section 4.4.1. The polarization and azimuthal response patterns ofthe prototype reflectors are presented in sections 4.4.2 and 4.4.3, respectively.The response patterns of the prototype reflectors were measured using the microwave antenna range located on the roof of the Electrical Engineering building at the University ofBritish Columbia. A side view of the outdoor portion of the range is shown in Figure 4.18.The model tower supports the target at a height of 2.2 metres and travels on a carriage along15 metres of track down the center of the roof of the east wing of the Electrical Engineeringbuilding. The response pattern of the target will be distorted by near field effects if the rangeto the target is too short or by multipath propagation effects if the range is too long. Testswee conducted and it was found that the optimum range at which to measure the response of aprototype reflector is 11 metres. The model tower can be configured to either rotate the targetabout a vertical axis for conventional azimuthal pattern measurements or roll the target abouta horizontal axis for polarization response measurements as shown in Figure 4.19. The CWradar apparatus and the digital pattern recorder which were used to measure and record theco-polar and cross-polar response of the target under test were developed specifically for use inthis project. A description of the design and implementation of the instrumentation, the resultsof tests performed to verify the suitability of the range for use in the measurement program,and recommendations for future modifications and improvements are presented in Appendix D.Chapter 4. Depolarizing Trihedral Corner ReflectorsOm 5mFigure 4.18: Profile view of the RCS measurement range.0\u00E2\u0080\u0099+(a)(b)\u00E2\u0080\u0094 a2mOm113Figure 4.19: Measurement of the response of a prototype trihedral corner reflector. (a) Polarization response. (b) Azimuthal response.Horn AntennasPenthouse- 3 dBlOm 15m0\u00E2\u0080\u0099Chapter 4. Depolarizing Trihedral Corner Reflectors 1144.4.1 Design and Construction of the Prototype Trihedral Corner ReflectorsA prototype trihedrai corner reflector with a regular polarization response was assembled fromtriangular panels with equal corner lengths of 60 cm. At the design frequency of 9.445 GHz, theprototype reflector presents a maximum scattering cross section of approximately 540 squaremetres (27 dBsm) with a half-power beamwidth of approximately 38 degrees in both azimuthand elevation. The reflecting panels were cut from 12 gauge (approximately 2 mm in thickness)utility grade (3003) sheet aluminum and secured to 50 cm lengths of 90\u00C2\u00B0 angle aluminum ofapproximate dimensions 1 in x 1 in x in with rivets spaced approximately 5 cm apart. Theright hand panel of the reflector (as viewed from the front) was secured to the angle aluminumwith #6 flathead machine screws so that it could be removed easily. A mounting flange whichmatches the corresponding adapter on the antenna range model tower was secured to the rearof the reflector in line with the axis of symmetry. Prototype reflectors with twist-polarizing andcircularly polarizing responses were realized by removing the right hand panel and replacingit with a suitable reflection polarizer. Details of the design and construction of the prototypedepolarizing reflectors are shown in Figure 4.20. A photograph of a prototype reflector with atwist polarizer installed in place of the right hand panel is shown in Figure 4.21.The extent to which the response of a trihedral corner reflector degrades as the reflectingpanels deviate from perfect flatness and mutual orthogonality has been discussed by Trebits [29]and others. The tolerances on the corner angles of a trihedral corner reflector with triangularpanels are presented as a function of the corner length of the reflector in Table 4.1. Each timethe right hand panel of the prototype reflector was replaced, a set square with a corner lengthof 30 cm was used to verify that all three panels were mutually orthogonal and, if required,suitable adjustments were performed. Although commercially designed and fabricated trihedralcorner reflectors intended for use under field conditions are typically manufactured from castalumimum tooling plate between 10 and 15 mm thick in order to realize a structure which willmeet the required tolerances [11], [30], [31], the simpler construction technique employed herewas far easier to implement and was deemed adequate for use in the test program.Chapter 4. Depolarizing Trihedral Corner Reflectors 115900 Angle6JuminumShea ,8juminumMounting FlangeFigure 4.20: Construction details of the prototype depolarizing trihedral corner reflector.(a) Front view. (b) Rear View.Figure 4.21: Photograph of the prototype twist-polarizing trihedral corner reflector mountedon the antenna range model tower.Removable Panel(a) (b)Chapter 4. Depolarizing Trihedral Corner Reflectors 116Table 4.1:Tolerances on the Corner Angles of a Trihedral Corner Reflector with Triangular Panels.(1) Error in a Single Corner Angle(2) Error in Three Corner Angles(2)Corner Length (-1 dB) (-3 dB) (-10 dB) (-1 dB) (-3 dB) (-10 dB)0.15 m (4.7).) \u00C2\u00B15.1\u00C2\u00B0 \u00C2\u00B18.4\u00C2\u00B0 \u00C2\u00B114.4\u00C2\u00B0 \u00C2\u00B12.4\u00C2\u00B0 \u00C2\u00B14.2\u00C2\u00B0 \u00C2\u00B17.5\u00C2\u00B00.30 m (9.4).) \u00C2\u00B12.5\u00C2\u00B0 \u00C2\u00B14.2\u00C2\u00B0 \u00C2\u00B17.3\u00C2\u00B0 \u00C2\u00B11.2\u00C2\u00B0 \u00C2\u00B12.1\u00C2\u00B0 \u00C2\u00B13.8\u00C2\u00B00.45 m (14.2).) \u00C2\u00B11.7\u00C2\u00B0 \u00C2\u00B12.8\u00C2\u00B0 \u00C2\u00B14.9\u00C2\u00B0 \u00C2\u00B10.8\u00C2\u00B0 \u00C2\u00B11.4\u00C2\u00B0 \u00C2\u00B12.5\u00C2\u00B00.60 m (18.9).) \u00C2\u00B11.3\u00C2\u00B0 \u00C2\u00B12.1\u00C2\u00B0 \u00C2\u00B13.7\u00C2\u00B0 \u00C2\u00B10.6\u00C2\u00B0 \u00C2\u00B11.1\u00C2\u00B0 \u00C2\u00B11.9\u00C2\u00B01.00 m (31.4).) \u00C2\u00B10.8\u00C2\u00B0 \u00C2\u00B11.3\u00C2\u00B0 \u00C2\u00B12.2\u00C2\u00B0 \u00C2\u00B10.4\u00C2\u00B0 \u00C2\u00B10.6\u00C2\u00B0 \u00C2\u00B11.1\u00C2\u00B03.00 m (94.3).) \u00C2\u00B10.3\u00C2\u00B0 \u00C2\u00B10.4\u00C2\u00B0 \u00C2\u00B10.7\u00C2\u00B0 \u00C2\u00B10.1\u00C2\u00B0 \u00C2\u00B10.2\u00C2\u00B0 \u00C2\u00B10.4\u00C2\u00B0Notes: 1. )\u00E2\u0080\u0098. = 3.18 cm.. 2. which leads to the indicated reduction in the maximum response of the reflector.-d-afl ri[40mmI150mmFigure 4.22: A single segment of a prototype reflection polarizer.Table 4.2: Dimensions of the Prototype Reflection PolarizersType of Polarizer Period d Groove Width a Groove Depth h Aspect Ratio aIdTwist 13.5 mm 12.0 mm 11.5 mm 0.89(0.426).) (0.379).) (0.355).)Circular 13.5 mm 12.0 mm 6.0 mm 0.89(0.426).) (0.379).) (0.186).)2mmNote: A = 3.18 cmChapter 4. Depolarizing Trihedral Corner Reflectors 117Reflecting panels which present twist-polarizing and circularly polarizing responses weredesigned using the methods described in section 4.3 and were assembled from extrusions thatwere custom manufactured for the purpose by Alcan Extrusions (Richmond, B.C.) from marinegrade (6061) aluminum. A view of a single segment in cross section is depicted in Figure 4.22.The closely spaced row of fins along the top of the structure form the reflection polarizer whilethe three longer fins which form the base of the structure merely provide mechanical supportduring manufacture and points of attachment during assembly. The extrusion was designed topresent a twist-polarizing response. In order to realize a circularly polarizing response, it wasnecessary to mill the upper row of fins down to the height specified in Table 4.2. During themilling procedure, the machinist found it difficult to secure the extrusion to the platform ofthe milling machine and problems with vibration were encountered. As a result, the height ofthe fins varied slightly along the length of the circularly polarizing reflecting panel. This mayhave contributed to the discrepancies that were observed between the measured response of thecircularly polarizing trihedral corner reflector and theoretical predictions.4.4.2 Polarization ResponseThe polarization response of each of the prototype reflectors was verified by rotating the targetabout its boresight and measuring the co-polar and cross-polar response of the target to avertically polarized incident wave as a function of the angle of rotation c, as suggested byFigure 4.19(a). The result corresponds to taking a cross section through the normalized copolar and cross-polar response patterns which are presented in Figures 4.4, 4.5, and 4.6 forellipticity angle e = 0\u00C2\u00B0. as suggested by Figure 4.23. The tilt angle r of the incident wave inthe reflector frame is related to the rotation angle c byr a\u00E2\u0080\u0094 90\u00C2\u00B0,where a = 0\u00C2\u00B0 corresponds to vertically polarized incidence. The response patterns of prototypereflectors designed to present regular, twist-polarizing, and circularly polarizing responses arepresented in Figures 4.24, 4.25, and 4.26, respectively. In the case of the regular and twist-Figure 4.23: Evaluation of the polarization response of a radar target. Measuring the co-polarand cross-polar response of the target to a vertically polarized incident wave as a function of theangle of rotation r, as suggested by Figure 4.19(a), corresponds to taking a cross section throughthe normalized co-polar and cross-polar response patterns which are presented in Figures 4.4,4.5, and 4.6 for ellipticity angle c = 00.Response0 dB-10 dB-20 dB\u00E2\u0080\u0094 MeasuredTheoreticalAngle of rotation - a(a) (b)Figure 4.24: Polarization response of a prototype regular trihedral corner reflector as a functionof rotation about the boresight. (a) Co-polar response- vv. (b) Cross-polar response - HV.Chapter 4. Depolarizing Trihedral Corner Reflectors 118Rigflt circular polanzationUwUHV 0.-30dB.08kI\u00E2\u0080\u0099,.08[Chapter 4. Depolarizing Trihedral Corner Reflectors 119\u00E2\u0080\u0094MeasuredTheoreticalAngle of rotation - a(a) (b)Figure 4.25: Polarization response of a prototype twist-polarizing trihedral corner reflector as afunction of rotation about the boresight. (a) Co-polar response- aVV. (b) Cross-polar response- HV\u00E2\u0080\u00A2\u00E2\u0080\u0094MeasuredTheoreticalAngle of rotation - a(a) (b)Figure 4.26: Polarization response of a prototype circularly-polarizing trihedral corner reflectoras a function of rotation about the boresight. (a) Co-polar response- uvv. (b) Cross-polarresponse-5w 0\u00E2\u0080\u00A2UHV 0\u00E2\u0080\u00A2Responseo dB-10dB-20 dB-30dB.08k .08kUw 0\u00E2\u0080\u00A2Response HV0dB-10dB-20 dB-30dB.08kI\u00E2\u0080\u0099).08kChapter 4. Depolarizing Trihedral Corner Reflectors 120polarizing reflectors, the measured responses and the theoretical predictions agree reasonablywell. In the case of the circularly polarizing reflector, the general form of the measured responsesis correct but the discrepancy between measurement and theory is relatively large. As notedabove, this may have been due to the problems encountered in milling the fins of the circularpolarizer to the correct height.4.4.3 Azimuthal ResponseThe co-polar and cross-polar azimuthal response uvv and HV of each of the prototype tnhedral corner reflectors were measured as a function of the azimuthal angle /\u00E2\u0080\u0098 for selectedangles of rotation a with respect to the local vertical axis as suggested by Figure 4.19. Theresults are compared to theoretical predictions of the contribution of triple-bounce reflectionsto their response based on the analysis presented in section 4.3. The co-polar and cross-polarazimuthal response patterns of the prototype regular reflector for a rotation angle of 0 degreesare presented in Figures 4.27 and 4.28. The measured responses and the theoretical predictionsgenerally agree. The co-polar and cross-polar azimuthal response patterns of the prototypetwist-polarizing reflector after it has been rotated to 0 degrees for maximum co-polar responseare presented in Figures 4.29 and 4.30 while the corresponding response patterns after thereflector has been rotated to 45 degrees for maximum cross-polar response are presented inFigures 4.31 and 4.32. Once again, the measured responses and the theoretical predictionsgenerally agree. The rapid degradation in the polarization response for incidence off the bore-sight is apparent in both cases. The co-polar and cross-polar azimuthal response patterns ofa prototype circularly polarizing reflector for rotation angles of 0 degrees and 45 degrees arepresented in Figures 4.33 through 4.36. Although the general form of the measured response iscorrect, the measured co-polar azimuthal beamwidth of the prototype reflector is slightly widerthan predicted in both cases. Also, the measured cross-polar azimuthal response pattern fora rotation angle of 0 degrees is substantially higher than predicted. However, the measuredcross-polar azimuthal response pattern for a rotation angle of 45 degrees agrees very well withthe predicted values.121Chapter 4. Depolarizing Trihedral Corner Reflectors0.0-10.0-20.0-30.0-45.0 -30.0 15.0 30.0 45.0\u00E2\u0080\u0094 MeasuredTheoretical ** contribution oftriple-bouncereflecbonsAzimuth Angle- \u00E2\u0080\u0098(deg)Figure 4.27: Co-polar azimuthal response pattern of a prototype regular trihedral corner reflector for rotation angle a = 00 and vertically polarized transmission.0.0-10.0a,>-20.0-15.0 0.0-15.0 0.0\u00E2\u0080\u0094 MeasuredTheoretical *-30.0 * contribution of-45.0 -30.0 15.0 30.0 45.0 triple-bouncereflectionsAzimuth Angle- \u00E2\u0080\u0098(deg)Figure 4.28: Cross-polar azimuthal response patterns of a prototype regular trihedral cornerreflector for rotation angle a = 00 and vertically polarized transmission.Chapter 4. Depolarizing Trihedral Corner Reflectors 1220.0-10.0Cl)C..).-20.0-30.0-45.0 -30.0 -15.0\u00E2\u0080\u0094 MeasuredTheoretical ** contribution of15.0 30.0 450 triple-bouncereflectionsAzimuth Angle - (deg)Figure 4.29: Co-polar azimuthal response pattern of a prototype twist-polarizing trihedralcorner reflector for rotation angle c \u00E2\u0080\u0094 00 and vertically polarized transmission.0.0-10.0Cl)C)4--20.0-30.0-45.0 -30.0 -15.0 0.0 15.0 30.0 45.0\u00E2\u0080\u0094 MeasuredTheoretical ** contribution oftriple-bouncereflectionsAzimuth Angle - \u00E2\u0080\u0098(deg)Figure 4.30: Cross-polar azimuthal response patterns of a prototype twist-polarizing trihedralcorner reflector for rotation angle c = 0 0 and vertically polarized transmission.0.0Chapter 4. Depolarizing Trihedral Corner Reflectors 1230.0-10.0Cl)C)-I-20.00.0? -10.0Cl)C)a,>-20.0-30.0-45.0 -30.0 -15.0 0.0 15.0 30.0 45.0\u00E2\u0080\u0094 MeasuredTheoretical ** contribution oftriple-bouncereflectionsAzimuth Angle- \u00E2\u0080\u0098(deg)Figure 4.32: Cross-polar azimuthal response patterns of a prototype twist-polarizing trihedralcorner reflector for rotation angle a = 450 and vertically polarized transmission.\u00E2\u0080\u0094 MeasuredTheoretical *-30.0 * contribution or-45.0 -30.0 -15.0 0.0 15.0 30.0 45.0 triple-bouncereflectionsAzimuth Angle- \u00E2\u0080\u0098(deg)Figure 4.31: Co-polar azimuthal response pattern of a prototype twist-polarizing trihedralcorner reflector for rotation angle a = 45\u00C2\u00B0 and vertically polarized transmission.Chapter 4. Depolarizing Trihedral Corner Reflectors 124Cl)C)a,a,* contribution oftriple-bouncereflectionsFigure 4.34: Cross-polar azimuthal response patterns of a prototype circularly-polarizing trihedral corner reflector for rotation angle a = 00 and vertically polarized transmission.0.0-10.0-20.0-30.0-45.0 -30.0 -15.0\u00E2\u0080\u0094 MeasuredTheoretical ** contribution of0.0 150 300 450 triple-bouncereflectionsAzimuth Angle- V(deg)Figure 4.33: Co-polar azimuthal response pattern of a prototype circularly-polarizing trihedralcorner reflector for rotation angle a = 00 and vertically polarized transmission.0.0-10.020.0\u00E2\u0080\u0094MeasuredTheoretical *-30.0-45.0 -30.0 -15.0 0.0 15.0 30.0 45.0Azimuth Angle- \u00E2\u0080\u0098(deg)Chapter 4. Depolarizing Trihedral Corner Reflectors 125Azimuth Angle- \u00E2\u0080\u0098(deg)Figure 4.35: Co-polar azimuthal response pattern of a prototype circularly-polarizing trihedralcorner reflector for rotation angle c = 450 and vertically polarized transmission.-30.0 * contribution of-45.0 -30.0 -15.0 0.0 15.0 30.0 45.0 triple-bouncereflectionsAzimuth Angle- \u00E2\u0080\u0098(deg)Figure 4.36: Cross-polar azimuthal response patterns of a prototype circularly-polarizing trihedral corner reflector for rotation angle o = 45\u00C2\u00B0 and vertically polarized transmission.:9-Cl)00.0-10.0-20.0-30.0-45.0 -30.0 -15.0 0.0 15.0\u00E2\u0080\u0094 MeasuredTheoretical **contribution oftriple-bouncereflections30.0 45.00.0:9-i -10.0-20.0\u00E2\u0080\u0094 MeasuredTheoretical *Chapter 4. Depolarizing Trihedral Corner Reflectors 1264.5 ConclusionsA method for altering the polarization response of a conventional trihedral corner reflector byreplacing one of its reflecting panels by a reflection polarizer derived from a conducting gratingwith rectangular grooves of appropriate dimensions and orientation has been proposed. Sincea depolarizing reflector of this type does not make use of wire grids or dielectric materials in itsconstruction, it is less vulnerable to mechanical or environmental damage than other schemeswhich have been proposed in recent years. The scattering properties of conducting gratings withrectangular grooves have been reviewed and analytical solutions to the problem of scatteringby a conducting grating with rectangular grooves have been derived for the cases of TM- andTE-polarized incident waves by mode-matching between the free space and groove regions. Theresults, including an implementation of the solution as a pair of subroutines coded in Fortran77, are presented in Appendix B.If the axis of the fins or corrugations are aligned with one of the axes of the trihedral thenall rays incident from a given direction will experience the same polarization transformationupon reflection from the grating regardless of the sequence in which each ray is reflected fromeach of the three interior surfaces of the reflector. If the period of the grating is less than one-half wavelength and the reflector is oriented so that the projection of the grating axis into theaperture plane is oriented vertically, the phase difference 5 between the TM and TE reflectioncoefficients of the specular reflected order can be exploited to yield a depolarizing responsealong the reflector boresight of the form,[S]= SHH SHy= 1 0SVH Svv 0 ewhere S is the ratio of the electric field components of the specular reflected order and the incident field. A phase difference of 0 yields a regular polarization response while phase differencesof 180 and 90 degrees yield twist polarizing and circularly polarizing responses, respectively.Since the grating is uniform in one dimension, the TM- and TE-polarized fields are decoupledChapter 4. Depolarizing Trihedral Corner Reflectors 127and SHy =SVH = 0. Design curves for trihedral corner reflectors which present circularly polarizing and twist polarizing responses along their boresight have been given. The polarizationresponse of the reflector can be modified by rotating the reflector about its boresight by anangle c, yielding a response of the form,cos sinc 1 0 coscr \u00E2\u0080\u0094sino\u00E2\u0080\u0094sina cosc 0 e6 sinc cos\u00C3\u00BCMethods for realizing trihedral corner reflectors which present a linear polarization selectiveresponse of the form,[Sj= SHH SHy = 1 0SVH Syy 0 0have been proposed. The problem of realizing trihedral corner reflectors which present a circularpolarization selective response is considered in Appendix C.An algorithm for predicting the contribution of triple-bounce reflections to the polarizationscattering matrix of a modified trihedral corner reflector as a function of the direction of incidence and the orientation of the reflector has been described. Although several factors causethe polarization response to degrade as the direction of incidence shifts away from the reflectorboresight, rotation of the projection of the grating axis in the aperture plane with respect to thelocal vertical is the most important since it results in an effective rotation of the correspondingpolarization scattering matrix. The polarization and azimuthal response patterns of prototypereflectors with regular, twist polarizing, and circularly polarizing responses were measured usingthe experimental facility which is described in Appendix D. 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(Ann Arbor, MI), June 28\u00E2\u0080\u0094July 2, 1993, pp. 238\u00E2\u0080\u0094241.[17] J.D. Hanfling, G. Jerenic, and L.R. Lewis, \u00E2\u0080\u009CTwist reflector design using E-type and H-typemodes,\u00E2\u0080\u009D IEEE Trans. Antennas Propagat., vol. AP-29, pp. 622\u00E2\u0080\u0094629, July 1981.[18] W.B. Offut and L.K. DeSize, \u00E2\u0080\u009CMethods of Polarization Synthesis,\u00E2\u0080\u009D in Antenna EngineeringHandbook, 2nd ed., R. C. Johnson and H. Jasik, Eds. New York: McGraw-Hill, 1984,chap. 23.[19] E.V. Jull, \u00E2\u0080\u009CReflection circular polarisers,\u00E2\u0080\u009D Electron. Lett., vol. 15, pp. 423\u00E2\u0080\u0094424, July 5, 1979.[20] R. Kastner and R. Mittra, \u00E2\u0080\u009CA spectral-iteration technique for analyzing a corrugatedsurface twist polarizer for scanning reflector antennas,\u00E2\u0080\u009D IEEE Trans. Antennas Propagat.,vol. AP-30, pp. 673\u00E2\u0080\u0094676, July 1982.References 130[21] D.G. Michelson, P. Phu, and E.V. Jull, \u00E2\u0080\u009CMillimetre-wave grating polarizers,\u00E2\u0080\u009D 22nd URSIGeneral Assembly, Tel Aviv, Israel, Aug. 1987.[22] A. Hessel, J. Shmoys, and D.Y. Tseng, \u00E2\u0080\u009CBragg-angle blazing of diffraction gratings,\u00E2\u0080\u009DJ. Opt. Soc. Am., vol. 65, pp. 380\u00E2\u0080\u0094384, Apr. 1975.[23] E.F. Knott, J.F. Shaeffer, M.T. Tuley, Radar Cross Section: Its Prediction, Measurement,and Reduction. Norwood, MA: Artech House, 1985, pp. 244\u00E2\u0080\u0094247.[24] A.K. Bhattacharyya and D.L. Sengupta, Radar Cross Section Analysis and Control. Nor-wood, MA: Artech House, 1991, pp. 208\u00E2\u0080\u0094212.[25] E.V. Jull and J.W. Heath, \u00E2\u0080\u009CReflection grating polarizers,\u00E2\u0080\u009D IEEE Trans. Antennas Propagat., vol. AP-28, pp. 586\u00E2\u0080\u0094588, July 1980.[26] E.V. Jull and J.W. Heath, \u00E2\u0080\u009CRadio applications of rectangular groove corrugations,\u00E2\u0080\u009D inIEEE/AP-S Symp. Dig. (Seattle, WA), June 1979, pp. 515-518.[27] E.V. Jull, Aperture Antennas and Diffraction Theory. Stevenage, UK: Peregrinus, 1981,pp. 80\u00E2\u0080\u009481.[28] S.H. Bickel, \u00E2\u0080\u009CSome invariant properties of the polarization scattering matrix,\u00E2\u0080\u009D Proc. IEEE,vol. 53, pp. 1070\u00E2\u0080\u00941072, 1965.[29] R.N. Trebits, \u00E2\u0080\u009CRadar Cross Section,\u00E2\u0080\u009D in Radar Reflectivity Measurement: Techniques andApplications, N.C. Currie, Ed. Norwood, MA: Artech House, 1989, pp. 46\u00E2\u0080\u009449.[30] A. Freeman, Y. Shen, and C. L. Werner, \u00E2\u0080\u009CPolarimetric SAR calibration experiment usingactive radar reflectors,\u00E2\u0080\u009D IEEE Trans. Geosci. Remote Sensing, vol. GE-28, pp. 224\u00E2\u0080\u0094240,Mar. 1990.[31] Precision Radar Corner Reflectors, Product Brochure No. 9187F. Norcross, GA: SpectrumTechnologies International, 1991.Chapter 5SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS5.1 Summary and ConclusionsIt is often necessary to enhance the radar cross section of a cooperative target either to increasethe maximum range at which the target can be reliably detected or to provide a target witha known response which may be used to assist in radar calibration and performance verification. In recent years, a requirement has arisen for rugged yet inexpensive radar reflectorswhich present both a very large scattering cross section and a specified polarization responseover a wide angular range for use as calibration targets in airborne and spaceborne imagingradar systems for geophysical remote sensing and location markers in radar-assisted positioningsystems for marine navigation. Passive targets, such as trihedral corner reflectors, are oftenbetter suited for use in such applications than active targets since they do not require an external source of power and are inherently more reliable. This study considers several problemsrelated to the analysis, design, and implementation of passive targets including transformation of the polarization response of a target or the polarization state of an antenna betweencoordinate frames, modification of the angular coverage of a conventional trihedral corner reflector by appropriate shaping of its reflecting panels, modification of the polarization responseof a conventional trihedral corner reflector by the addition of conducting fins or corrugationsof appropriate dimensions and orientation to one of its interior surfaces, design of top hat reflectors with specified response characteristics, and design of reflection polarizers derived fromconducting gratings with rectangular grooves.131Chapter 5. Summary, Conclusions, and Recommendations 132In Chapter 2, it is shown that transformation of a polarization descriptor between coordinate frames corresponds to rotation of its basis by a prescribed angle which is a functionof both the transformation matrix that relates the two coordinate frames and the direction ofpropagation. Two methods for determining the angle of rotation for the case in which the localvertical is defined by the direction in each frame are derived using spherical trigonometry andvector algebra, respectively. Both methods are robust and will yield the correct result but themethod based on vector algebra is more compact and would be easier to implement in software.Although the elements of the coordinate transformation matrix can be determined from eitherthe relative directions of the three principal axes in each coordinate frame or the Euler angleswhich define a series of rotations which will transform one coordinate frame into the other, itmaybe difficult to obtain these parameters in practice. A third method is derived which overcomes this limitation by allowing the elements of the coordinate transformation matrix to bedetermined from any pair of dirctions which have been expressed in terms of both coordinateframes. Algorithms for rotating the basis of several polarization coordinates in common use,including the complex polarization ratio, the complex polarization vector, the Stokes vectorand several of its variants, the coherency matrix, the polarization scattering matrix, and theMueller matrix and several of its variants, are presented.In Chapter 3, the problem of predicting the scattering cross section and angular coverage ofa conventional trihedral corner reflector with panels of completely arbitrary shape is considered.Although general purpose numerical techniques such as the finite-difference time-domain (FDTD) and the shooting and bouncing ray (SBR) methods can predict both the contributionsof single, double, and triple-bounce reflections from the interior and the effect of deviationsof the panels from perfect flatness and mutual orthogonality, they are extremely demandingcomputationally. For most purposes, it is sufficient to account for the contribution of triplebounce reflections from the interior since they completely dominate the response for mostdirections of incidence. The empirical model originally proposed by Spencer for predicting theequivalent flat plate area of ideal trihedral corner reflectors with either triangular or squareChapter 5. Summary, Conclusions, and Recommendations 133panels (and the numerical implementation of the model that was devised by Keen) may failwithout indication when applied to reflectors with complex panel shapes. A simple yet robustnumerical prediction algorithm which overcomes this limitation is formulated by applying theapproach used by Keen and a polygon-clipping algorithm formulated by Weiler and Atherton toan alternative model originally proposed by Robertson. The response patterns of ideal trihedralcorner reflectors which present three-fold symmetry about the boresight are plotted on equalarea projection grids and compared. If the corner length of the reflector is fixed, it is generallyfound that attempts to increase the beamwidth of the response by modifying the shape ofthe panels are accompanied by a reduction in the maximum response and vice versa. A setof design curves for bilaterally symmetric reflectors which are composed solely of triangular,elliptical, or rectangular reflecting panels are derived and the response patterns of selectedbilaterally symmetric reflectors which are composed of combinations of panels with variousshapes including triangular, circular, and square are presented and compared. It is found thatif three-fold symmetry is broken so that the reflector simply presents bilateral symmetry abouta mirror plane containing the boresight then the beamwidth of the response in one principalplane can be increased relative to the beamwidth in the orthogonal plane by modifying theshape of the reflector panels. The additional degree of freedom is shown to be useful when areflector must be designed subject to a constraint such as a requirement that it present a planaraperture in order to facilitate the attachment of either a transmission polarizer or a protectivecover.In Chapter 4, a method for altering the polarization response of a conventional trihedralcorner reflector by adding conducting fins or corrugations of appropriate dimensions and orientation to one of its interior surfaces is proposed. Since depolarizing reflectors of this type do notmake use of wire grids and dielectric materials in their construction, they are less vulnerable tomechanical or environmental damage than other schemes which have been proposed in recentyears. They also avoid many of the mechanical problems associated with the attachment of atransmission polarizer across the reflector aperture. If the axis of the fins or corrugations areChapter 5. Summary, Conclusions, and Recommendations 134aligned with one of the axes of the trihedral then all rays incident from a given direction willexperience the same polarization transformation upon reflection from the polarizer regardless ofthe sequence in which each ray is reflected from each of the three interior surfaces. If period ofthe grating is less than one-half wavelength and the reflector is oriented so that the projectionof the grating axis into the aperture plane is oriented vertically, the phase difference t betweenthe TM and TE reflection coefficients of the specular reflected order can be exploited to yielda depolarizing response along the boresight of the reflector of the form,[S]=t SHH SHy = 1 , (5.1)SVH Svv 0 e\u00E2\u0080\u00996where S is the ratio of the electric field components of the specular reflected order and theincident field. A phase difference of 180 degrees yields a twist-polarizing response while a phasedifference of 90 degrees yields a circularly polarizing response. Since the grating is uniform inone dimension, the TM and TE polarizations are decoupled and SHy = SVH = 0. Design curvesfor trihedral corner reflectors which present twist-polarizing and circularly polarizing responsesalong their boresight are given. The polarization response of the reflector can be modified byrotating the target about its boresight by an angle a, yielding a response of the form,cosa sina 1 0 cosa \u00E2\u0080\u0094sina[S9= . (5.2)\u00E2\u0080\u0094 sin a cos a 0 e6 sin a cos aMethods for realizing linear polarization selective trihedral corner reflectors using similar techniques are proposed. An algorithm for predicting the contribution of triple-bounce reflectionsto the polarization scattering matrix of a modified trihedral corner reflector as a function ofthe direction of incidence and the orientation of the reflector is derived. Although severalfactors cause the polarization response to degrade as the direction of incidence shifts awayfrom the boresight, rotation of the projection of the grating axis in the aperture plane withrespect to the local vertical is the most important since it results in an effective rotation ofthe corresponding polarization scattering matrix. Prototype trihedral corner reflectors withChapter 5. Summary, Conclusions, and Recommendations 135regular, twist-polarizing, and circularly polarizing responses were designed and assembled. Experimental results are presented which show that the prototype reflectors respond essentiallyas predicted.In Appendix A, the problem of designing a top hat reflector with specified response characteristics is considered. Since the top hat reflector is a body of revolution, its response pattern isuniform in azimuth. Expressions for the elevation response pattern, maximum scattering crosssection, angle of maximum response, and 1 and 3 dB elevation beamwidths of a top hat reflectorare derived and design curves are presented. As the angle of maximum response becomes appreciably greater or less than 45 degrees, the elevation response pattern will become increasinglyasymmetrical. In such cases, it may be preferable to consider the angle of median response fora given elevation beamwidth rather than the angle of maximum response. Expressions for theangle of median response are derived and design curves are presented. The results are used tosolve a sample design problem.In Appendix B, the problem of scattering by a conducting grating with rectangular groovesis considered. Analytical solutions are derived for the cases of TM- and TE-polarized incidentwaves by mode-matching between the free space and groove regions. Procedures for determining the validity of numerical results which are obtained using this formulation are discussed.Although the solution will invariably converge to an essentially constant result as the numberof modes used in the field expansion are systematically and gradually increased, it is shownthat the manner in which the solution converges and the value of the final result will dependon both the number of modes used to represent the fields in each region and their ratio. Thephenomenon is commonly referred to as relative convergence. Numerical results are presentedwhich suggest that the optimum ratio of groove modes to free space modes is similar in valueto the ratio of the groove width to the grating period. An implementation of the analyticalsolutions as a pair of subroutines coded in Fortran 77 is presented.In Appendix C, the problem of modifying a conventional trihedral corner reflector to presenta circular polarization selective response is considered. It is shown that such a response cannotChapter 5. Summary, Conclusions, and Recommendations 136be realized using the techniques described in Chapter 4 because the corresponding polarizationscattering matrix cannot be diagonalized when expressed with respect to a linearly polarizedbasis. Alternative methods for obtaining such a response based on the addition of a transmissioncircular polarizer to a linear polarization selective reflector or a circular polarization selectivesurface to a twist-polarizing reflector are proposed. The performance of such reflectors andtheir usefulness in practice will depend on a number of factors that are not considered hereincluding (1) degradation of the polarization response of either of the polarizers for incidenceoff the reflector boresight, (2) possible degradation of the polarization response due to multiplereflections between the transmission and reflection polarizers, and (3) the mechanical ruggednessof the modified reflector and the ease with which it can be fabricated.In Appendix D, the experimental facility which was set up to measure the response ofprototype trihedral corner reflectors is briefly described. Details of the physical layout and thedesign and implementation of the CW radar apparatus and digital pattern recorder are given.Test results show that the facility is suitable for use in the prototype reflector measurementprogram. Before the facility is used in future measurement programs, however, considerationshould be given to replacing the existing microwave receiver with a newer model which is lessprone to drift, correcting the backlash in the model tower positioner head, and evaluating theuse of either a berm or a clutter fence to reduce multipath effects.5.2 Recommendations for Further WorkAlthough depolarizing trihedral corner reflectors which incorporate conducting fins or corrugations along one of their interior surfaces will find immediate use as location markers andcalibration targets in polarimetric radar systems used for radar-assisted positioning and geophysical remote sensing, consideration should also be given to using reflectors of this type asradar cross section enhancement devices in conventional marine radar navigation. An increasing number of civil marine radars are equipped with provision for transmitting and receivingcircularly polarized waves in order to suppress rain clutter. However, mariners must exerciseChapter 5. Summary, Conclusions, and Recommendations 137caution when using them in this mode because the conventional trihedral corner reflectors whichare used to augment the response of most buoys, channel markers, and navigations hazards areessentially invisible to a circularly polarized radar. Replacement of conventional reflectors bydepolarizing reflectors which respond equally well to horizontally, vertically, and circularly polarized incident waves would significantly enhance the safety of marine navigation. Furtherengineering studies and field trials should be conducted in order to verify the suitability of depolarizing reflectors of the type proposed in Chapter 4 for use in such applications. Since suchreflectors would be required in large numbers, it would also be desirable to study alternativemethods of manufacture with the aim of minimizing their unit cost.Detailed analysis of modified trihedral corner reflectors for engineering purposes would besimplified if the radar cross section prediction algorithms and software tools described in Chapters 3 and 4 were combined into a single integrated software package designed to run on a highperformance UNIX workstation or other platform suitable for numerically intensive computing. A package designed to assist in the evaluation of alternative schemes for truncating andcompensating trihedral corner reflectors should include implementations of the prediction algorithm for ideal trihedral corner reflectors based on Robertson\u00E2\u0080\u0099s model (as described in Chapter3), either the finite-difference time-domain (FD-TD) or the shooting and bouncing ray (SBR)methods for predicting the response of non-ideal trihedral corner reflectors, and suitable toolsfor entering the reflector geometry and displaying the results of the calculations. The FD-TDand SBR methods are sufficiently demanding of computing resources that it will probably benecessary to include provision for running them either as background tasks for extended periodsor remotely on a sufficiently powerful host. A package designed to assist in the design of depolarizing trihedral corner reflectors which incorporate conducting fins or corrugations would alsorequire code for predicting the response of a conducting grating with rectangular grooves. Thepackages should also include a facility for comparing numerical predictions with experimentalresults obtained by the user.Chapter 5. Summary, Conclusions, and Recommendations 138Although recent trials conducted by Transport Canada (Transportation Development Centre, Montreal, P.Q.) have shown that radar-assisted positioning systems based on measurementof the range and bearing to shore-based cooperative targets can provide vessels navigating ininland waterways, harbours, and harbour approaches with very accurate position data underoptimum conditions, the lack of data concerning the scattering statistics of terrain at grazingincidence makes it difficult to give reliable estimates of the size of targets required to achievespecified probabilities of detection and false alarm. Future development of radar-assisted positioning systems for marine navigation should assign a high priority to the collection of suchdata in a variety of representative clutter environments in order to assist in system planning andevaluation. Future trials should include provision for determining the confidence with whichthe cooperative targets are detected against the clutter background in order to provide somemeasure of the reliability of such systems.Three methods for realizing linear polarization selective reflective surfaces derived fromconducting gratings with rectangular grooves are suggested in Chapter 4 including (1) gratingsdesigned to reflect TE-polarized incident waves while diffracting TM-polarized incident wavesinto the m = \u00E2\u0080\u00941 diffracted order, (2) gratings designed to reflect TE-polarized waves whiledissipating TM-polarized incident waves in lossy groove media, and (3) gratings designed toreflect TE- and TM-polarized waves in different directions by utilizing a sloped grating profile.Each type of grating suffers from various limitations. The scattering characteristics of eachscheme should be examined in further detail so that their relative performance and merits canbe assessed in the context of both realizing linear and circular polarization selective trihedralcorner reflectors and their possible use in other quasi-optical systems. The first scheme canbe analyzed simply using the mode-matching formulation presented in Appendix B and someresults have been presented in the literature. Analysis of the second scheme would requirethat the representation for the fields in the groove region be modified in order to accountfor the effects of the lossy groove media. In both cases, the impact of relative convergenceon calculation of the amplitude of the diffracted orders should be assessed. Since the problemChapter 5. Summary, Conclusions, and Recommendations 139geometry of the third scheme is not periodic and the free space and groove regions do not belongto separable coordinate systems, it will necessary to employ a different approach. Although apurely numerical technique could be used in such an investigation, it may be simpler to beginsimply by setting up a design parameter matrix, constructing a set of prototype reflectionpolarizers, then measuring and comparing the response characteristics of the prototypes.The task of coding solutions to antenna or scattering problems which involve the prediction of polarization dependent effects would be simplified if a standard library of functionsand subroutines was available for performing polarization calculations. Such a library wouldinclude routines for transforming polarization descriptors between coordinate frames based onthe results presented in this study and routines for converting polarization descriptors from onetype to another (e.g., polarization scattering matrix to Mueller matrix), converting polarization descriptors between the backscatter and forward scatter alignment conventions, verifyingthe validity of polarization descriptor data, calculating polarization efficiency, and extractingco-polar and cross-polar nulls and related parameters from polarization response descriptorsbased on results generally available in the literature. Since these routines will make extensiveuse of matrices and complex variables, the library should initially be coded in Fortran 77. Consideration should also be given to producing libraries for use with more specialized numericalanalysis and symbolic algebra packages such as Matlab (The Mathworks, Inc., Natick, MA),Maple (Waterloo Maple Software, Inc., Waterloo, Ont.), Macsyma (Macsyma, Inc., Arlington,MA), and Mathematica (Wolfram Research, Inc., Champaign, IL).Appendix ADESIGN CURVES FOR TOP HAT REFLECTORSA.1 IntroductionThe top hat reflector is a variant of the conventional dihedral corner reflector which has recentlyfound use as a calibration target for airborne imaging radars. It may be regarded as a body ofrevolution formed by rotating a dihedral corner about an axis of symmetry which is perpendicular to both the seam of the dihedral and one of its two reflector panels. As such, its response isindependent of azimuth angle. In practice, a top hat reflector is usually realized by attaching aright circular cylinder to a flat circular plate such that the cylinder axis is perpendicular to andpasses through the center of the plate although other ground plane configurations have beenconsidered. The requirements for the cylinder and base to be flat and mutually orthogonal aresimilar to those for the panels of a conventional dihedral corner reflector.Although the top hat reflector was apparently introduced over twenty-five years ago, it wasnot discussed in the literature until the early 1980\u00E2\u0080\u0099s [1]. It was not treated in standard referencesuntil much later still and, even so, the discussion is brief and contains obvious errors [2], [3].Blejer [4] recently extended the simplified model introduced by Johansen [1] to account for thecontribution of single bounce reflections from the cylinder, the cylinder cap, and the annularbase. The design of top hat reflectors is complicated by the dependence of the scattering crosssection and the angle of maximum response on three parameters: the radius and height ofthe cylinder and the width of the annular base. In addition, the main response lobe becomesincreasingly asymmetrical as the angle of maximum response becomes either much greater ormuch less than 45 degrees to the vertical. Design curves and related material are not generallyavailable.140Appendix A. Design Curves for Top Hat Reflectors 141In this appendix, the problem of designing top hat reflectors with specified response characteristics is considered. In section A.2, an approximate expression for the scattering cross sectionof a top hat reflector is derived as a function of the elevation angle of the incident ray usinga simplified physical optics model which accounts only for the contribution of double-bouncereflections from the cylinder and the ground plane. In section A.3, design curves which relatethe physical dimensions of a top hat reflector to the direction of its maximum response, itsmaximum scattering cross section, and its elevation beamwidth are presented. In section A.4,the results are used to solve a sample design problem.A.2 AnalysisConsider a top hat reflector which consists of a right circular cylinder of radius a and heightc attached to a circular ground plane of radius b as shown in Figure A.1. The cylinder axiscoincides with the z-axis and the ground plane lies in the x-y plane. The angle of elevation 8is measured from the positive z-axis. If the plane of incidence contains the cylinder axis andthe opposite panel is of sufficient extent, a top hat reflector will return an incident ray to thesource in the same manner as would a dihedral corner. Since a top hat reflector is a body ofrevolution, its response pattern is uniform in azimuth. The locus of maximum response fordouble-bounce reflections is an inverted circular cone which intersects the cylinder in the x-yplane.Figure A.1: Problem geometry for scattering by a top hat reflector.Appendix A. Design Curves for Top Hat Reflectors 142The contribution of double-bounce reflections to the scattering cross section of a top hatreflector can be determined using a ray-optical model in which the ground plane is treated as animage plane of finite extent as shown in Figure A.2. If an incident ray intercepts the cylinderat height h, radius a, and angle 9, the reflected ray will intercept the x-y plane at a radiush tan 9 + a. If h> (b \u00E2\u0080\u0094 a) cot 9, the point of interception will occur beyond the ground planeradius b and the reflected ray will not be returned to the source. The effective height c\u00E2\u0080\u0099 of theportion of the cylinder which contributes to the response is therefore a function of the angle ofincidence and is given byO<07, (A.1)( c, y 9< ir/2,where(b\u00E2\u0080\u0094a\u00E2\u0080\u00997=tan ) . (A.2)TOP-HATREFLECTORIMAGE2c 2cNNNFigure A.2: A simplified ray-optical model for scattering by a top hat reflector.Appendix A. Design Curves for Top Hat Reflectors 143If a top hat reflector is sufficiently large in terms of wavelength, the contribution of singlebounce reflections from the cylinder, the cylinder cap, and the ground plane of the reflector tothe backscatter response will be extremely small for other than vertical or horizontal incidenceand can be neglected for the purposes of design. This reduces the problem to one of simplypredicting the forward scattering cross section of a cylinder of radius a and height 2c\u00E2\u0080\u0099 where theplane of incidence contains the cylinder axis. The corresponding problem geometry is shownin Figure A.3. The intersection of the plane of incidence with the surface of the cylinder isreferred to as the specular line. The unit vectors i and . give the directions of the incident andscattered rays. The unit vectors \u00C3\u00B1 and \u00C3\u00B10 denote the outward normal to the cylinder at anypoint on its surface and along the specular line, respectively, and are given byn = cos+sinq, (A.3)and\u00C3\u00B10=cos?SI+sinq. (A.4)specular line\EZ JFigure A.3: Problem geometry for forward scattering by a cylinder.Following Knott [5], an approximate expression for the bistatic scattering cross section ofa target can be obtained by evaluating a physical optics integral over the illuminated portionAppendix A. Design Curves for Top Hat Reflectors 144of its surface. In the case of scattering by a cylinder with radius a and height 2c\u00E2\u0080\u0099, the physicaloptics integral can be expressed as the product of axial and circumferential components= _j i i, (A.5)where the axial component I is given by\u00E2\u0080\u0098C,I,= J ejkz((i_)) dz , (A.6)\u00E2\u0080\u0094C,\u00E2\u0080\u0094 2sin[kc\u00E2\u0080\u0099( (i\u00E2\u0080\u0094 A 7C kc\u00E2\u0080\u0099(2. ( \u00E2\u0080\u0094 , ( .and the circumferential component I,!, is given byI,, = I (\u00C3\u00B1 i) d , (A.8)J\u00E2\u0080\u0094ir/2Although (A.8) will yield an exact solution, the resulting expression contains special functionsand is difficult to handle. It is generally more convenient to approximate IqS by the method ofstationary phase where the stationary phase point is the specular line shown in Figure A.4 andthe stationary phase approximation is of the form [6]2ii- 1/2I= J g(q) e)dq g(0)e\u00E2\u0080\u0099\u00C2\u00B0 e\u00E2\u0080\u0099\u00E2\u0080\u0099, (A.9)whereg(q) = (\u00C3\u00B1.\u00C3\u00AE), (A.1O)f() = ka (n. (i\u00E2\u0080\u0094(A.11)andg(40)=(n0 i) (A.12)f(,) = ka (\u00C3\u00B10 (1\u00E2\u0080\u0094, (A.13)= ka (n0.(1\u00E2\u0080\u0094 ..)) . (A.14)Substituting (A.12) \u00E2\u0080\u0094 (A.14) into (A.9) gives1/2(n ) jka(iio.(f-)) [a (n0 .\u00E2\u0080\u0094. (A.15)Appendix A. Design Curves for Top Hat Reflectors 145An expression for the bistatic scattering cross section of the cylinder is obtained by substituting (A.15) and (A.7) into (A.5) which gives= \u00E2\u0080\u0094j2c\u00E2\u0080\u0099 [ 2ka ] 1/2 sin kC(Z .(z s)) . i) ejka(no()) (A.16)n.(i\u00E2\u0080\u0094s) kc (z.(z\u00E2\u0080\u0094s))In the case of forward scattering, the vectors \u00E2\u0082\u00AC, , \u00C3\u00B1, and are related as follows:(1\u00E2\u0080\u0094.) = \u00E2\u0080\u00942sin8 \u00C3\u00B1,, , (A.17)\u00E2\u0080\u00A2 (1\u00E2\u0080\u0094.) = \u00E2\u0080\u00942sin0 , (A.18). (f\u00E2\u0080\u0094= 0, (A.19)= \u00E2\u0080\u0094sin0. (A.20)An expression for the forward scattering cross section of the cylinder is obtained by substituting(A.17) \u00E2\u0080\u0094 (A.20) into (A.16) which gives= \u00E2\u0080\u0094j2c\u00E2\u0080\u0099 v\u00E2\u0080\u009Dka sin8 &32ka sinO \u00E2\u0080\u00A2 (A.21)Since (A.18) is negative, a phase factor ejlr/2 is introduced when its square root is extracted.Taking the amplitude of (A.21) and squaring the result gives= acI2sin6. (A.22)Finally, an approximate expression for the scattering cross section of a top-hat reflector isobtained by substituting (A.1) into (A.22) to giveac2sinO, 0<87,a(0)= 8ir cos2 (A.23)\u00E2\u0080\u0094i- a(b \u00E2\u0080\u0094 a)2-y 8 < ir/2,where a, b, c, and 8 are defined in Figure A.1 and the parameter 7 is given by (A.2). If theground plane of the reflector is of very large extent, i.e., b>> c, the angle-y \u00E2\u0080\u0094* r/2 and (A.23)reduces toa(8) = ac2sin8, 0< 8< ir/2. (A.24)Appendix A. Design Curves for Top Hat Reflectors 146A.3 Design CurvesAngle of Maximum ResponseThe contribution of double-bounce reflections to the response of a top hat reflector will reach itsmaximum value when the angle of incidence is such that any ray incident on the cylinder willintercept the annulus and vice versa. From Figure A.2, this will occur when c = (b \u00E2\u0080\u0094 a) cot 6so the angle of maximum response 6m is given by6max = tan1(b a)= tan\u00E2\u0080\u0099 , (A.25)where is defined as the ratio of the width of the annulus to the height of the cylinder. A graphdepicting 8m as a function of is presented in Figure A.4. If the height of the cylinder andthe width of the base are equal, the angle of maximum response is 45 degrees. Doubling theratio will lower the angle of maximum response to approximately 65 degrees while halving itwill raise the angle to approximately 25 degrees.9075600)45301500.1 10.0Figure A.4: Angle of maximum response of a top hat reflector vs. , the ratio of the annuluswidth to the cylinder height.1.0EAppendix A. Design Curves for Top Hat Reflectors 147Maximum Scattering Cross SectionSubstitution of (A.25) into (A.23) gives an expression for the maximum scattering cross sectionof a top hat reflector8ir a(b \u00E2\u0080\u0094 a)c2Umax= A (b \u00E2\u0080\u0094 a)2\u00E2\u0080\u0099 (A.26)8ir 2_____= \u00E2\u0080\u0094acwhere (sin 8max) is given bySfl umax= +. (A.27)Consider a top hat reflector with a cylindrical component of fixed radius a0 and height c0 whichpresents a maximum scattering cross section of o for = 1. As the ratio is altered toobtain the desired angle of maximum response, the maximum scattering cross section of thereflector will vary as shown in Figure A.5. The response will drop off rapidly for < 1 and willasymptotically approach a value of crc, for > 1.The radius and height of the cylinder can be scaled to compensate for this variation andso yield a family of top hat reflectors with different angles of maximum response but the samemaximum scattering cross section. With the cylinder height c fixed, the scale factor for thecylinder radius a is given by______a/1+2 (A.28)a 2With the cylinder radius a fixed, the scale factor for the cylinder height c (and the correspondingannulus width b \u00E2\u0080\u0094 a) is given by_c 11+c2\u00E2\u0080\u0094=/ . (A.29)c0 2In either case, the corresponding value of b is simply given byb=a+c. (A.30)A graph depicting the scale factors (A.28) and (A.29) as functions of the ratio is presentedin Figure A.6.Appendix A. Design Curves for Top Hat Reflectors 148I04-\u00E2\u0080\u00990201.50.51.0E0.00.1 10.0Figure A.5: Maximum scattering cross section of a top hat reflector vs. , the ratio of theannulus width to the cylinder height for fixed values of a and c.5.04.03.02.01.00.00.1 10.0Figure A.6: Scale factors for the cylinder radius a and height c of a top hat reflector vs. , theratio of the annulus width to the cylinder height.1.0CAppendix A. Design Curves for Top Hat Reflectors 149Half Power BeamwidthThe half-power elevation beamwidth of a top hat reflector is determined by finding the anglesof incidence above and below the angle of maximum response, 0m for which the scatteringcross section fails to half its maximum value and taking their difference. From (A.26),A (I. 2oir a \u00E2\u0080\u0094a1c (A31)2 \u00E2\u0080\u0094 A Vc2+(b_a)Equating (A.23a) and (A.31) then solving for 6 gives the upper half-power angle,= sin\u00E2\u0080\u0099(2 i+ 2). (A.32)The lower half-power angle 3dB is determined in a similar fashion. Equating (A.23b) and(A.28) then cancelling common factors gives a quadratic equation in sin 6,1 1\u00E2\u0080\u0094sin20 A2/1+2 = sineRearranging terms gives an expression in standard form,sin26+ (2V-+2)sin0\u00E2\u0080\u0094 1 = 0, (A.34)with roots given by______\u00E2\u0080\u00A2 _B+v\u00E2\u0080\u0099B2+4sin6= 2(A.35)whereB=1\u00E2\u0080\u00A2 (A.36)2/1+.2Since is always positive, so too are B and the solution to (A.34) given by\u00E2\u0080\u00A2 -B+VB2+4sin 0= 2\u00E2\u0080\u00A2 (A.37)Solving for 6 gives the lower half-power angle,\u00C2\u00B03dB = 2 J \u00E2\u0080\u00A2 (A.38)Appendix A. Design Curves for Top Hat Reflectors 1507560,453015Finally, taking the difference between (A.38) and (A.32) gives the total half-power elevationbeamwidth of the reflectorf\u00E2\u0080\u0094B + /B2 + 4\u00E2\u0080\u0099\ . I03d]3 = sin 2 \u00E2\u0080\u0094 sin 2 s/i. + 2(A.39)Care must be taken in applying (A.39) since the main response lobe of a top hat reflector isnot symmetric about the angle of maximum response, particularly if > 1. In addition to O3,the total half-power beamwidth, it is convenient to define the quantities 0U3dB = 9u3dB \u00E2\u0080\u0094and 0e3dB = \u00E2\u0080\u0094 as the half-power beamwidths above and below the angle of maximumresponse, respectively. All three quantities are graphed in Figure A.7 as a function of the ratio. Although both the total and upper half-power beamwidths increase with , the lower half-power beamwidth reaches its maximum value when 0.55. The upper and lower half-powerbeamwidths are equal and the top hat presents a symmetrical elevation response pattern when0.51.9000.1 1.0 10.0Figure A.7: Half-power elevation beamwidth of a top hat reflector vs. , the ratio of the annuluswidth to the cylinder height.Appendix A. Design Curves for Top Hat Reflectors 151where B\u00E2\u0080\u0099 is given by0)w0(10\u00E2\u0080\u00940.16UIdB = sin1 /1 + 2)B+%/B2+4\u00E2\u0080\u009D\84dB = Sfl1 2Similar relations may be derived for the 1 dB beamwidth of the elevation response. Theupper and lower 1 dB angles are given by(A.40)(A.41)111\u00E2\u0080\u00940.1B\u00E2\u0080\u0099\u00E2\u0080\u0094 \u00E2\u0080\u009C (A42)and the total 1 dB elevation beamwidth is given by 0u1dB or-1 (\u00E2\u0080\u0094B\u00E2\u0080\u0099 + i/B\u00E2\u0080\u00992 + 4 -1 (10-0.1=2 } \u00E2\u0080\u0094 sin + 2} . (A.43)As in the previous case, it is convenient to define the quantities 0uldB = 8uldB \u00E2\u0080\u0094 \u00C2\u00B0m and= 9max\u00E2\u0080\u0094 0eldn as the 1 dB beamwidths above and below the angle of maximum response,respectively. All three quantities are graphed in Figure A.8 as a function of the ratio .907545301500.1 1.0 10.0Figure A.8: 1 dB elevation beamwidth of a top hat reflector vs. , the ratio of the annuluswidth to the cylinder height.Appendix A. Design Curves for Top Hat Reflectors 152Since the elevation response pattern of a top hat reflector is generally asymmetric about thedirection of maximum response, it may be useful to instead specify the median angle of responsegiven by 8m = (9, + 8)/2. The angle of maximum response and angles of median response for1 and 3 dB elevation beamwidths are graphed as a function of the ratio in Figure A.9.9075I30w1500.1 10.0Figure A.9: Angle of maximum response and angles of median response for 1 and 3 dB elevationbeamwidths vs. , the ratio of the annulus width to the cylinder height.A.4 Design ExampleConsider a requirement for three even-bounce targets to be used in the calibration of an airborneradar system which operates at a frequency of 10 GHz. The three targets must each presenta maximum scattering cross section of 30 dBsm at elevation angles of 25, 45, and 65 degrees,respectively, over a wide range in azimuth. Top hat reflectors which meet these requirementscan be designed using the results presented in the previous section.The ratio which gives the desired angle of maximum response can be determined usingeither (A.25) or the graph in Figure A.4. It is found that top hat reflectors which present aratio of annulus width to cylinder height of 0.5, 1.0, and 2.0, respectively, will present their60451.0Appendix A. Design Curves for Top Hat Reflectors 153maximum response at the specified angles. This information is sufficient to obtain the 1 and 3dB beamwidths of the three reflectors using either (A.39) and (A.43) or the graphs in FiguresA.7 and A.8.The reflector dimensions required to realize a maximum scattering cross section of 30 dBsmat 10 GHz are determined using (A.26). Once the radar wavelength A and the specified angleof maximum response have been specified, the maximum scattering cross section depends onlyon the product of the cylinder radius a and the square of the cylinder height c. The design ofa top hat reflector which presents its maximum response at 45 degrees is considered first. It isconvenient to choose the diameter and height of the cylinder to be approximately equal. From(A.26), this gives a cylinder radius a of 0.75 m and a height c of 1.5 m. Since the ratio is1, the annulus width (b \u00E2\u0080\u0094 a) is also 1.5 m which gives a total radius b of 2.25 m. The resultsare then scaled to obtain suitable dimensions for the other two reflectors. Assuming that thecylinder radius is fixed, the dimensions of the other two reflectors can be determined using thescale factors presented in (A.28) and (A.29) and graphed in Figure A.6.The results are summarized in Table A.1. In Figure A.10, the three reflectors are drawnto scale for comparison. In each case, the direction of the maximum response is indicated bya vector. In Figure A.11, the elevation response patterns of the three reflectors are predictedusing Blejer\u00E2\u0080\u0099s [4] formulation which accounts for double-bounce reflections from the interior ofthe reflector and single bounce reflections from cylinder, cylinder cap, and annular base.Appendix A. Design Curves for Top Hat ReflectorsTable A.1: Response Characteristics of Selected Top Hat Reflectors at f = 10 GHz154Fi ure Dimensions Response Characteristicsa b C umax 9max OldBA.10(a) 0.75 m 1.69 m 1.89 m 0.5 30 dBsm 25\u00C2\u00B0 27\u00C2\u00B0 100A.10(b) 0.75 m 2.25 m 1.50 m 1.0 30 dBsm 45\u00C2\u00B0 36\u00C2\u00B0 15\u00C2\u00B0A.10(c) 0.75 m 3.75 m 1.33 m 2.0 30 dBsm 65\u00C2\u00B0 450 21\u00C2\u00B0(c)Figure A.10: Relative size of selected top hat reflectors which present the same maximum scattering cross section. The direction of maximum response is indicated by a vector. (a) 0.5.(b) = 1.0. (c) = 2.0.43N Tb(a) (b)TI.. jAppendix A. Design Curves for Top Hat Reflectors 1550\u00E2\u0080\u00A2(a)0\u00E2\u0080\u00A2(b)(c)a (dB)Figure A.11: Elevation response patterns of selected top hat reflectors which present the samemaximum scattering cross section. (a) = 0.5. (b) = 1.0. (c) = 2.0.-10 -6 -3-10-10-6 -3-100\u00E2\u0080\u00A2-10 -6 -3-10References[1] E.L. Johansen, \u00E2\u0080\u009CTop hat reflectors cap radar calibration,\u00E2\u0080\u009D Microwaves, vol. 20, no. 12,pp. 65\u00E2\u0080\u009466, Dec. 1981.[2] C.H. Currie and N.C. Currie, \u00E2\u0080\u009CMMW Reflectivity Measurement Techniques,\u00E2\u0080\u009D in Principlesand Applications of Millimetre-Wave Radar, N. C. Currie and C.E. Brown, Eds. Norwood,MA: Artech House, 1987, pp. 767\u00E2\u0080\u0094774.[3] R. N. Trebits \u00E2\u0080\u009CRadar Cross Section,\u00E2\u0080\u009D in Radar Reflectivity Measurement: Techniques andApplications, N. C. Currie, Ed. Norwood, MA: Artech House, 1989, pp. 44\u00E2\u0080\u009448.[4] D. Blejer, \u00E2\u0080\u009CPhysical optics polarization scattering matrix for a top hat reflector,\u00E2\u0080\u009D IEEETrans. Antennas Propagat., vol. AP-39, pp. 857\u00E2\u0080\u0094859, June 1991.[5] E.F. Knott, J.F. Shaeffer, and M.T. Tuley, Radar Cross Section: Its Prediction, Measurement, and Reduction. Norwood, MA: Artech House, 1985, pp. 119\u00E2\u0080\u0094130.[6] N. Bleistein and R.A. Handelsman, Asymptotic Expansions of Integrals. New York: bit,Rinehart, and Winston, 1975, pp. 219\u00E2\u0080\u0094220.156Appendix BSCATTERING BY A CONDUCTING GRATINGWITH RECTANGULAR GROOVESB.1 IntroductionIn this appendix, the problem of scattering by a conducting grating with rectangular groovesis solved for the case in which the plane of incidence is perpendicular to the direction of thegrooves [1], [2]. In section B.2, analytical solutions to the scattering problem are derived for thecases of TM- and TE-polarized incidence by mode-matching across the boundary between thefree space and groove regions. Since the grating is periodic in y, the fields in adjacent unit cellsdiffer only by the phase factor exp(jkdsin4) and it is sufficient to consider the fields in a singleunit cell such as the one shown in Figure B.l. Symbols are defined in Table B.l. The fields inthe free space region (x > 0) are represented by an infinite sum of propagating plane waves andevansecent waves while the fields in the groove region (x < 0) are represented by an infinitesum of propagating and evanescent parallel-plate waveguide modes. A doubly infinite set oflinear equations in the complex coefficients of either the free space or groove modes is obtainedby applying the condition of continuity of the tangential electric and magnetic fields over theplanar junction between the free space and groove regions and the property of orthogonalitybetween the normal modes in each set. The set must be truncated or partitioned before it canbe solved numerically. In section B.3, procedures for verifying the correctness and accuracy ofnumerical results obtained by mode-matching are described and the problem of determiningthe minimum number of modes required to accurately represent the fields in the free space andgroove regions and their optimum ratio is considered. In section B.4, an implementation of theanalytical solutions as a pair of subroutines coded in Fortran 77 is presented.157Appendix B. Scattering by a Conducting Grating 158yhFigure B.1: A unit cell of a conducting grating with rectangular grooves.Table B.1: Definition of SymbolsSymbol Definitiond period of the gratinga width of the grooveh height of the grooveaid aspect ratio of the gratingA wavelength\u00C3\u00A7b angle of incidenceq angle of reflection for the mth free space mode or diffracted orderm index for free space modesn index for groove modesM number of free space modes in the truncated set of equationsN number of groove modes in the truncated set of equationsk propagation constant in free space (= 2ir/A)k propagation constant of the nth groove modefm orthonormal basis function for TM and TE free space modesg orthonormal basis function for TM groove modesg orthonormal basis function for TE groove modesA2 complex amplitude of the incident TM free space model3 complex amplitude of the incident TE free space modeAm complex amplitude of the mth scattered TM free space modeC complex amplitude of the nth TM groove mode\u00E2\u0080\u00983m complex amplitude of the mth scattered TE free space modeD complex amplitude of the nth TE groove modei, characteristic impedance of free space (= f)Fm complex reflection coefficient of the mth free space modexK aK d >1Appendix B. Scattering by a Conducting Grating 159B.2 AnalysisB.2.1 TM PolarizationFree Space RegionIf a TM-polarized plane wave is incident on a grating of infinite extent such that the planeof incidence is perpendicular to the direction of the grooves, the tangential components of theelectric and magnetic fields in the free space region are given byH(x, y) = A1 ei 0stxfo + Am e_c0mxfm, (B.1)m -_E(x, y) = \u00E2\u0080\u0094A1 cos j ejkcos + Am o cos m e_jk COS mXf , (B.2)m=- oowhereE(x,y) = (B.3)\u00E2\u0080\u0094 1 jksinmy B4Jm \u00E2\u0080\u0094 e ,k = , (B.5)sinqm = sinq!\u00E2\u0080\u0099j+m, (B.6)cosq = \u00E2\u0080\u0094 sii2 Qm, I SiflmI < 1, (B.7)= _jsin2 m \u00E2\u0080\u0094 1, sin mI > 1.The first term in each of (B.1) and (B.2) represents the incident field while the second termrepresents the infinite sum of propagating and non-propagating waves which comprise the scattered field. The function fm is an orthonormal basis function for both TM- and TE-polarizedmodes in the free space region. Equation (B.6) is usually referred to as the grating equation.The remaining quantities and symbols are defined in Table B.1 and Figure B.1.Appendix B. Scattering by a Conducting Grating 160Groove RegionThe tangential components of the TM-polarized fields in the groove region are given byH(x,y) = (B.8)E(x,y) = (B.9)where the function gn is an orthonormal basis function for TM-polarized modes in the grooveregion given by= i/cos(nir(x + a/2)) (B.10)and,.k = /k2 \u00E2\u0080\u0094 (!E)2 , < k , (B.11)_j(!)2_k2, II>k,e = 1, n=0, (B.12)= 2, n=1,2,3Boundary ConditionsThe y-z plane is the interface between the free space and groove regions. The boundary conditions for the tangential components of the TM-polarized field are given byH(0,y) = H(0,y), I\u00E2\u0080\u0099 a/2, (B.13)Hz(O,Y) = \u00E2\u0080\u0094K11(0,y), a/2 lvi < d/2,andE(0,y) = E(0,y), lvi a/2, (B.14)E(0+,y) = 0, a/2 < vi are given byIT 2ksinqrn (kasinq)m.71/\u00E2\u0080\u0094 2 2 cosj j , n=1,3,5,...v ad (ksincm) \u00E2\u0080\u0094 (n2r/a) 2 j(fm,gn) = (B.26)fT 2ksinqm (kasin\u00E2\u0080\u009DVad (ksinq5m)2\u00E2\u0080\u0094 (nir/a)2 sin 2 ) n = 0,2,4,...andI (fm,gn) , n=1,3,5,...= (B.27)( (fm,gn) , n=0,2,4,...since (f,g) = (g,f)*, by definition [4, p. 26].Substitution and SolutionThe inner product of (B.15) with g yields an expression for the groove mode coefficients C interms of the free space mode coefficients Am,Am(fm,gn) = Cn, (B.28)m= -00while the inner product of (B.16) with fm yields\u00E2\u0080\u00942Acosq + AmCO5bm = Cntan(knh)(gn,fm) . (B.29)The result is then rearranged to give an expression for the free space mode coefficient Am flterms of the groove mode coefficients C,Am =k,tan(kh)n, fm> + (B.30)n=O cos4mAppendix B. Scattering by a Conducting Grating 163A direct solution for the coefficients of the free space modes can be obtained by substituting(B.28) into (B.30) to giveAm,6mmI= > (B.31)\u00E2\u0080\u0094\u00E2\u0080\u0094\u00E2\u0080\u0094j cosqS,m\u00E2\u0080\u0099\u00E2\u0080\u0094\u00E2\u0080\u0094oo fl_0_then rearranging the result to yield an infinite set of linear simultaneous equations in thecoefficients of the free space modes Am,:Am\u00E2\u0080\u0099 (fm\u00E2\u0080\u0099,gn) (g,f) tan(kh) \u00E2\u0080\u0094 S, = \u00E2\u0080\u00942A . (B.32)\u00E2\u0080\u0094 \u00E2\u0080\u0094j cosq5m\u00E2\u0080\u0099\u00E2\u0080\u0094\u00E2\u0080\u0094-oo n OIt is convenient to express (B.32) as a matrix equation in the form([B][C]+[G])A=F, (B.33)where the elements of [B], [C], [G], A, and F are given by13\u00E2\u0080\u0094 , : k tan(kh)\u00E2\u0080\u0094 \gn,JmJ-\u00E2\u0080\u0094- , .34jk COSmCnm\u00E2\u0080\u0099 = (fm\u00E2\u0080\u0099,gn) , (B.35)\u00E2\u0080\u0094\u00E2\u0080\u0094Ami = Am\u00E2\u0080\u0099, (B.37)Fm = \u00E2\u0080\u00942A. (B.38)The number of free space and groove modes included in the solution must be reduced froman infinite number to M and N, respectively, before a numerical solution to (B.33) can beobtained. The resulting truncated set of equations is of order M.Alternatively, the coefficients of the free space modes can be determined in an indirect wayby first solving for the coefficients of the groove modes then multiplying the result by the matrixwhich relates the two. Since this truncated set of equations is the order N, the time requiredto compute a solution can be considerably reduced compared to the direct approach if the ratioof groove modes to free space modes IV/M in the truncated set is much less than unity. First,Appendix B. Scattering by a Conducting Grating 164(B.30) is substituted into (B.28) to yield( (g\u00E2\u0080\u0099,f) + 2A6) (fm,gn) = (B.39)m=\u00E2\u0080\u0094-x n\u00E2\u0080\u0099=O m n\u00E2\u0080\u0099Othen the result is rearranged to yield an infinite set of linear simultaneous equations for thecoefficients of the groove modes Ci:Cstan(kh) _icot(knih)) = \u00E2\u0080\u00942A(f0,g) . (B.40)It is convenient to express (B.40) as a matrix equation in the form([C\u00E2\u0080\u0099][B\u00E2\u0080\u0099] + [G\u00E2\u0080\u0099j)X = F\u00E2\u0080\u0099 , (B.41)where the elements of [C\u00E2\u0080\u0099], [B\u00E2\u0080\u0099], [G\u00E2\u0080\u0099], X, and F\u00E2\u0080\u0099 are given byC,cm = (fm,gn> , (B.42)= jkcosqrn (g\u00E2\u0080\u0099,f) , (B.43)= \u00E2\u0080\u00946,k1cotk1h, (B.44)X1 = Ctankh, (B.45)F, = \u00E2\u0080\u00942A (f0,g) . (B.46)On solution, the vector X contains the groove mode coefficients C multiplied by the factortan(kh). From (B.30), the free space mode coefficients Am are determined by applying therelationA\u00E2\u0080\u0094[C\u00E2\u0080\u0099]X+2A6. (B.47)From (B.17), the specularly reflected component of the scattered field is given byARJ = A0 \u00E2\u0080\u0094 A, (B.48)so the specular reflection coefficient for a TM-polarized incident wave is given bypEE \u00E2\u0080\u0094 ARf (B 490\u00E2\u0080\u0094 A \u00E2\u0080\u0098A0= --;+1.Appendix B. Scattering by a Conducting Grating 165B.2.2 TE PolarizationFree Space RegionIf a TE-polarized plane wave is incident on a grating of infinite extent such that the planeof incidence perpendicular to the direction of the grooves, the tangential components of theelectric and magnetic fields in the free space region are given byE2(x,y) = !3je3kc0txfo+ Bme_c0mxfm, (B.5o)m=-_H(x,y) B2COSc Skcostf \u00E2\u0080\u0094 BmOS4m e_ikc0mxfm , (B.51)m=\u00E2\u0080\u0094oo 1owhere fm, k, \u00E2\u0080\u0098q0, sinq5m, and cos4m are defined in (B.3) through (B.7) andj OE(x,y).5\u00E2\u0080\u009Clo \u00E2\u0080\u0098\u00E2\u0080\u0098YGroove RegionThe tangential components of the TE-polarized fields in the groove region are given byE(x,y) = (B.53)H(x, y) = _jD-!_ cos( (B.54)where k is given by (B.11) and g is the orthonormal basis function for TE-polarized modesin the groove region and is given by(B.55)Boundary ConditionsThe y-z plane is the interface between the free space and groove regions. The boundary conditions for the tangential components of the TE-polarized field are given byH(O,y) = H(O,y), II a/2 (B.56)H(O,y) = K(O,y), a/2 II d/2,Appendix B. Scattering by a Conducting Grating 166where K represents the surface current density along the top surface of the corrugation andE(O,y) = E(O,y), yIa/2, (B57)E(O,y) = 0, a/2 I\u00E2\u0080\u0099I d/2.Mode MatchingThe modal expansions of the tangential electric field, (B.50) and (B.53), and the tangentialelectric field, (B.51) and (B.54), are evaluated at x = 0 then substituted into the boundaryconditions (B.56) and (B.57). Continuity of the magnetic field across the boundary gives2I3 cos\u00C3\u00A7j f0 \u00E2\u0080\u0094 >2 \u00E2\u0080\u00983m CO5m frn = >2 Dcot(kh)g, lxi jkcos (gn\u00E2\u0080\u0099,fm) (fm,g) + 6,kicot(k;h) = 2Bjkcos1(f0,g) . (B.77)n\u00E2\u0080\u00991 m\u00E2\u0080\u0094It is convenient to express (B.77) in the form([B\u00E2\u0080\u0099J[C\u00E2\u0080\u0099] + [G\u00E2\u0080\u0099]) X\u00E2\u0080\u0099 = F\u00E2\u0080\u0099 , (B.78)Appendix B. Scattering by a Conducting Grating 169where the elements of [B\u00E2\u0080\u0099], [C\u00E2\u0080\u0099], [G\u00E2\u0080\u0099], X\u00E2\u0080\u0099, and F\u00E2\u0080\u0099 are given by:Bm = (fm,g) , (B.79)= jkcosq5m (g-1,fm) , (B.80)G1 = 6,kcot(kh) , (B.81)X, = , (B.82)F, = 2Bjkcosc(f0,gj. (B.83)On solution, the vector X contains the groove mode coefficients D. From (B.63), the freespace mode coefficients 13m are given by1 [B\u00E2\u0080\u0099]x, (B.84)jk cosFrom (B.60), the complex amplitude of the specularly reflected component of the scattered fieldis given byl3Ref 13o \u00E2\u0080\u0094 B, (B.85)so the specular reflection coefficient for a TE-polarized incident wave is given by:pHH (B.86)L3 113iB.3 Verification of Numerical ResultsThe analysis presented in section B.2 is exact but truncation of the mode-matching matrix anduse of a finite word length during computation introduce unavoidable approximations. Furthermore, errors in analysis and coding sometimes occur. It is therefore desirable to implementprocedures to assess the correctness and accuracy of the numerical solutions which are obtained.Three numerical checks which are applicable to this problem include conservation of energy,reciprocity, and convergence [5].Appendix B. Scattering by a Conducting Grating 170Conservation of Energy and ReciprocityThe physical plausibility of numerical results obtained by mode-matching may be verified bytesting for conservation of energy and/or reciprocity among the propagating modes but neitheris a sufficient condition for a valid solution. Such tests serve mainly to indicate problemsintroduced by errors in coding or excessive round-off during computation [5], [6]. In the contextof scattering by a reflection grating, conservation of energy among the free space modes maybe expressed asIAiI2cos4i Ii4mt2C0 , (B.87)where A and 4j are the complex amplitude and the direction of the incident wave, Am and 4\u00E2\u0080\u0099mare the complex amplitude and the direction of the mth free space mode, and the summationincludes only those free space modes which are propagating. Reciprocity is a statement thatthe response of a system will be unchanged when the source and the observer are interchangedand can be expressed as= fm(bm), (B.88)where Fm(q5) is the complex reflection coefficient of the mth diffracted order in response to awave incident at angle q. In the case of the specularly reflected free space mode, \u00C3\u00A7bm = qj and(B.88) reduces to= l(\u00E2\u0080\u0094q,). (B.89)Testing for conservation of energy incurs little computational overhead but testing for reciprocity effectively doubles the length of time required to compute a solution and is rarely doneso routinely.ConvergenceIt is necessary to determine whether a sufficient number of modes have been used to approximate the fields in the free space and groove regions. Although the solution will invariablyconverge to an essentially constant result as the number of modes used in the field expansionAppendix B. Scattering by a Conducting Grating 171are systematically and gradually increased, the manner in which the solution converges and thevalue of the final result will depend on both the number of modes used in each region and theirratio, N/M. The phenomenon, which has been dubbed relative convergence, was originallyidentified in connection with the formulation of the boundary value problem associated witha bifurcated waveguide [7]. It has since been shown to occur in a variety of other problemsincluding diffraction by a strip grating on a dielectric slab [8], [9]. In these cases, Mittra andothers have shown that there usually exists a unique choice for the so-called partitioning ratiowhich will yield the correct solution. Although results obtained using an arbitrary partioningratio and a large number of modes are usually accurate to within engineering tolerances, use ofthe optimum ratio provides both greater accuracy and greater computational efficiency [10].In Figures B.3 through B.6, the phase difference between the TE and TM specular reflectioncoefficients of selected gratings are plotted as a function of the number of groove modes N usedin the solution for fixed values of the corresponding number of free space modes M. In allcases, the gratings have a period of 0.3333) and are illuminated perpendicular to the directionof the grooves at an angle of 45 degrees to the normal. Figures B.3 and B.4 present the resultsfor a pair of twist polarizers with aspect ratios aid of 0.5000 and 0.9999, respectively, whileFigures B.5 and B.6 present the results for a corresponding pair of circular polarizers. Thephase difference converges monotonically until the ratio of groove modes to free space modesreaches a value equal to the aspect ratio of the grating. At that point, the slope of the curveincreases abruptly and the phase difference begins to converge to a new value which depends onthe number of free space modes. In the limit as the number of free space modes becomes verylarge, the difference between the final result and the result obtained at the point of inflectionvanishes which suggests that the result obtained at the point of inflection corresponds to thecorrect solution. This is similar to behaviour reported in connection with other formulations inwhich relative convergence has been observed and implies that the optimum number of groovemodes is given by truncating the product of the aspect ratio of the grating and the number offree space modes to be used in the solution, i.e., N0 = Trunc(a/d. M).Appendix B. Scattering by a Conducting Grating 172a,0Ca,1811790 10 20 30 40 50 60Number of groove modesFigure B.2: Convergence of the phase difference between the TE and TM specular reflectioncoefficients with the number of groove modes for a reflection twist polarizer with d = 0.3333),aid = 0.5000, and h 0.2302A illuminated by a plane wave incident at = 45 degrees.181180179Figure B.3: Convergence of the phase difference between the TE and TM specular reflectioncoefficients with the number of groove modes for a reflection twist polarizer with d = 0.3333 A,aid = 0.9999, and h = 0.3172A illuminated by a plane wave incident at q5 = 45 degrees.0 10 20 30 40 50 60Number of groove modesAppendix B. Scattering by a Conducting Grating 173C091890 10 20 30 40 50 60Number of groove modesFigure B.4: Convergence of the phase difference between the TE and TM specular reflectioncoefficients with the number of groove modes for a reflection circular polarizer with d = 0.3333A,aid = 0.5000, and h = 0.1466A illuminated by a plane wave incident at = 45 degrees.9190890 10 20 30 40 50 60Number of groove modesFigure B.5: Convergence of the phase difference between the TE and TM specular reflectioncoefficients with the number of groove modes for a reflection circular polarizer with d = 0.3333),aid 0.9999, and h = 0.1641A illuminated by plane wave incident at = 45 degrees.Appendix B. Scattering by a Conducting Grating 174B.4 ImplementationSubroutines TMREFL and TEREFL implement the analytical solutions given by (B.41) and (B.78)and are presented in Listings B.1 and B.2. Given the period d, groove width a, and groovedepth h of the grating (in metres), the angle of incidence 4 (in degrees) and wavelength A(in metres) of the incident wave, and the number of free space and groove modes M and Nto be used in the solution, they return the complex specular reflection coefficient P0 for TM-and TE-polarized incident waves, respectively, and an error code to the calling program. Errorcode 1 indicates that the solution violates conservation of energy among the free space modeswhile error code 2 indicates that the mode-matching matrix passed to subroutine CDSOLN issingular. Error code 0 indicates that the solution has passed both these tests. If tests for eitherreciprocity or convergence are required, they must be performed by the calling program.Because the solution makes extensive use of complex variables, the subroutines were codedin Fortran 77. Although the use of language extensions was generally avoided in order to keepthe source code portable between different compilers and platforms, two extensions which aresupported by virtually all modern Fortran 77 compilers were allowed. First, double precisioncomplex variables (type COMPLEX*16) and the corresponding intrinsic functions were used sothat all floating point calculations could be performed using double precision. Second, variablesof type DOUBLE PRECISION were identified as REAL*8 in type declaration statements for clarity.Otherwise, the code is fully compliant with the ANSI standard and adheres to the generallyaccepted principles of programming style [11]\u00E2\u0080\u0094[13}.A chart depicting the hierarchy of subprograms which are called by subroutines TMREFL andTEREFL is presented in Figure B.6. Functions TMINNR and TEINNR are presented in ListingsB.3 and B.4. Given the indices of the free space and groove modes rn and n, the propagationconstant k, the grating period d, the groove width a, and a character variable 0 which specifiesthe desired sequence of factors, they return the value of the inner products (f, g) and (g, f)as given by (B.26/27) and (B.61/62), respectively. Functions KN, ODD, SEQ, E, and CDCOT arepresented in Listings B.5 through B.9. Complex function KN returns the propagation constantAppendix B. Scattering by a Conducting Grating 175of the nth groove mode as given by (B.11). Logical function ODD returns true if its argumentis an odd integer. In conjunction with ODD and IF... THEN constructs in functions TMINNRand TEINNR, real function SEQ is used to select the appropriate sign for the inner products asspecified by (B.27) and (B.55). Complex function CDCOT implements the complex cotangentfunction. The remaining subprograms called by subroutines TMREFL and TEREFL were suppliedby University Computing Services and perform operations not directly supported by Fortran77 such as addition and subtraction of matrices, multiplication of matrices with vectors, andsolution of systems of linear equations. They are presented in Listing B.10 and are documentedin the publication UBC Matrix [14].TNREFL TEREFLThIINNR KN TEINNR KNI CDCOT CDCOTCDADD CDADDODD CDMATV ODD CDMATVSEQ CDMULT SEQ CDMULTE CDSET CDSETCDSOLN CDSOLNFigure B.6: Hierarchy of subprograms called by subroutines TMREFL and TEREFL.The subroutines were compiled on a Sun 4/380 workstation (equipped with 32 Megabytesof RAM) under SunOS 4.1.1 (a variant of 4.2 BSD UNIX) using the standard Sun Fortrancompiler, f77, with all options set to their default values. The UNIX operating system provides a variety of timing and profiling tools which can be used to assist in the evaluation andoptimization of Fortran programs [15]. The total length of time required to execute the twosubroutines, including both system and user time, was determined by running the shell utilitytime in conjunction with a test program which simply set the parameters to be passed, calledsubroutines TMREFL and TEREFL in succession, then terminated. The results are presented inFigures B.7 and B.8 as a function of the number of free space and groove modes used in thesolution.Appendix B. Scattering by a Conducting Grating 1760ES.0E80 10 20 30 40 506420Number of groove modesFigure B.7: Combined execution time of subroutines TMREFL and TEREFL on a Sun 4/380workstation vs. the number of groove modes used in the solution.00 10 20 30 40 50Number of free space modesFigure B.8: Combined execution time of subroutines TMREFL and TEREFL on a Sun 4/380workstation vs. the number of free space modes used in the solution.Appendix B. Scattering by a Conducting Grating 177A call graph analysis of the code was conducted using the profiling utility gprof. Theresults are presented in Figure B.9 for numerical solutions using (a) 25 groove modes and 51free space modes and (b) 50 groove modes and 51 free space modes. In each case, calls tosubroutine CDMtJLT (which calculates the product of matrices [B] and [C] from (B.41) and [B\u00E2\u0080\u0099]and [C\u00E2\u0080\u0099] from (B.78)) account for approximately two-thirds of the total execution time. Theloading increases quadratically with the number of groove modes and linearly with the numberof free space modes. In contrast, calls to subroutine CDSOLN (which solves for the amplitudeof the groove modes) account for only one-fifth of the total execution time where the loadingalso increases quadratically with the number of groove modes but is independent of the numberof free space modes. The remainder of the execution time is spent calculating the elementsof the various vectors and matrices in (B.41) and (B.78). Since the elements of matrices [B]and [C] (and [B\u00E2\u0080\u0099] and [C\u00E2\u0080\u0099]) are independent on the groove depth h, these results suggest thatexecution time could be reduced considerably by not recalculating their product if h is the onlyparameter which has changed since the previous call. Such a modification could be implementedby inserting a SAVE statement into each subroutine in order to preserve the values of the localvariables between calls and applying an appropriate branch on entry.Figure B.9: Execution profile of subroutines TMREFL and TEREFL for numerical solutions using(a) 25 groove modes and 51 free space modes and (b) 50 groove modes and 51 free space modes.(a) (b)Appendix B. Scattering by a Conducting Grating 178ListingB.1: SUBROUTINE TMREFL(D, A, H, PHI, L, MFS, NWG, REFL, ERR)SUBROUTINE TJIREFL( D, A, H, PHI, L, NFS, luG, REFL, ERR)* Given:* D - grating period (metres)* A \u00E2\u0080\u0094 groove width (metres)* H - grating depth (metres)* PHI \u00E2\u0080\u0094 angle of incidence (degrees)* L \u00E2\u0080\u0094 wavelength (metres)* KFS\u00E2\u0080\u0094 number of free space modes* lUG \u00E2\u0080\u0094 number of groove modes* Result:* REFL \u00E2\u0080\u0094 complex reflection coefficient of zeroth free space mode* ERR\u00E2\u0080\u0094 error return 0 = no errors,* 1 = energr in free space modes not conserved,* 2 = matrix to be inverted is singular.** All floating point calculations are done in double precisionPARAJIETER (MAXFS=51, MAXUG51)* KAXFS, MAXUG- maximum array dimensions* (see also SUBROUTINE TMIINR)EXTERNAL CDCOTINTEGER IFS, JUG, ERR, I, 3, P1, N, ZEROREAL*8 D, A, H, PHI, L, K, PRRAD, P1, S, SINPH, SUMCOMPLEX*16 REFL, B, C, \u00E2\u0080\u00A2E, F, G, RCOMPLEX*16 CDCOT, CDET, COSPH, KI, ThuRDIMENSION KOLAXFS)DIMENSION COSPH(-MAXFS/2 :IIAXFS/2), SINPH(-MAXFS/2 :IIAXFS/2)DIMENSION B(MAXFS,MAXWG), C(MAXWG,MAXFS)DIMENSION GOIAXUG HAXUG), EOIAXUG,MAXUG)DIMENSION FOIAXUG5COMMON P1, SIIPHZERO = (MFS+1)/2P1 = 4D0*DATAN(1DO)** convert angle to radiansPHRAD = PHI * PI/180D0* calculate free space propagation constantK = 2D0*PI/L* tabulate values of sin[phi(m)] and cos[phi(m)) for later useDO 50, I = -MFS/2, IIFS/2S = DSIN(PHRAD) + DBLE(H)*L/DSINPH(M) = SIF ((S.S).LE.1D0) THENCOSPE(H) = DCMPLX(DSQRT(1DO-S*S), ODO)ELSECOSPH(M) = DCJIPLX(ODO, -DSQRT(S*S-1DO))EIDIFSO COITINUEAppendix B. Scattering by a Conducting Grating 179Listing B.1: (continued)* set up the matrices* matrixBDO 100, I = 1, IFSDO 100, 3 = 1, lUGI = I - ZERO1=3 -1B(I,i) = TIIIINR(M, I, K, D, A, \u00E2\u0080\u0098gf\u00E2\u0080\u0099) *$ Kl(K,1,A)/(DCMPLX(ODO,K)*COSPH(M))100 COITIIUE* matrixCDO 200, 3 = 1, lUGDO 200, I = 1, IFS1=3 -111 = I - ZEROC(3,I) = TNIN1R(N, 1, K, D, A, \u00E2\u0080\u0098fg\u00E2\u0080\u0099)200 CONTIJUE** [C]*[B)\u00E2\u0080\u0094> [E]CALL CDNULT(C,B,E,NWG,IIFS,NWG,MAXUG,MAIFS,MIZWG)* vector FDO 300, 3 = 1, lUG13\u00E2\u0080\u0094110F(3) = -2.0*TNIIIR(M, 1, K, D, A, \u00E2\u0080\u0098fg\u00E2\u0080\u0099)300 COITIIUE* matrixG400 CALL CDSET(G, lUG, lUG, ?IAXVG, (ODO,000))DO 500, 3 1, lUGl3-1G(J,J) = \u00E2\u0080\u0094CDCOT(K1(K,1,A)*H)500 CONTINUE*\u00E2\u0080\u00A2 [E]+[G] \u00E2\u0080\u0094> [E]CALL CDADD(E,G,E,IUG,IUG,NAXUG,KAXUG,MAIWG)\u00E2\u0080\u00A2 [E]*[X] = [F]; solve for [I]; [X]\u00E2\u0080\u0094>[F]CALL CDSOL1(E ,F,1WG,IIAXVG ,CDET)IF (ABS(CDET).LT.1E-20) THENREFL = (000,000)ERR2RETURNEIDIF* [B]*[F] \u00E2\u0080\u0094> [R]CALL CDMATV(B ,F,R,MFS,NWG,JIAXFS)R(ZERO) = R(ZERO) + (100,000)* check for conservation of energy in propagating free space modes* (a necessary but not a sufficient condition for a valid solution)SUM = ODOERR = 0DO 1000, I = 1, IFSI = I - ZEROIF (ABS(SINPHOI)) .LE.1DO)$ SUM = SUM + (CDABs(R(I))**2) COSPH(H)1000 CONTINUEIF ( ABS(SUI-DCOS(PHRAD)).GT.0.000IDO ) ERR = 1* calculation complete!REFL = R(ZERO)*(-1DO,000)RETURNENDAppendix B. Scattering by a Conducting Grating 180ListingB.2: SUBROUTINE TEREFL(D, A, H, PHI, L, MFS, NWG, REFL, ERR)SUBROUTINE TEREFL( D, A, H, PHI, L, PIFS, IVG, REFL, ERR)* Given:* D \u00E2\u0080\u0094 grating period (metres)* A \u00E2\u0080\u0094 groove width (metres)* H \u00E2\u0080\u0094 grating depth (metres)* PHI \u00E2\u0080\u0094 angle of incidence (degrees)* L \u00E2\u0080\u0094 wavelength (metres)* MFS \u00E2\u0080\u0094 number of free space modes\u00E2\u0080\u00A2 JUG \u00E2\u0080\u0094 number of groove modes* Result:* REFL \u00E2\u0080\u0094 complex reflection coefficient of zeroth free space mode* ERR \u00E2\u0080\u0094 error return 0 = no errors,\u00E2\u0080\u00A2 1 energy in free space modes not conserved,* 2 = matrix to be inverted is singular.** All floating point calculations are done in double precisionPARAMETER (MAXFS=51, MAXUG=51)* MAXFS, MAXUG - maximum array dimensions* (see also SUBROUTINE TEINIR)EXTERNAL CDCOTINTEGER MFS, JUG, ERR, I, J, P1, N, ZEROREAL*8 D, A, H, PHI, L, K, PERAD, P1, S, SINPH, SUMCOMPLEX*16 REFL, B, C, E, F, G, RCOMPLEX*16 CDCOT, CDET, COSPH, KR, TEIJJRDIMENSION R(MAXFS)DIMENSION COSPH(\u00E2\u0080\u0094MAXFS/2:JIAXFS/2), SIJPH(-MAXFS/2:MAIFS/2)DIMENSION BOIAXFS,MAIUG), C(MAXVG,MAXFS)DIMENSION G(KAXWG KAXUG), E(MAXUG,MAZUG)DIMENSION F(MAXWGSCOMMON P1, SINPHZERO = (MFS+1)/2P1 = 4D0*DATAN(1DO)** convert angle to radiansPURAD = PHI * PI/180D0* calculate free space propagation constantK = 2D0*PI/L* tabulate values of sin[phi(m)J and cos[phi(m)] for later useDO 50, P1 = \u00E2\u0080\u0094HFS/2, PIFS/2S DSIN(PHRAD) + DBLE(M)*L/DSIJPH(M) SIF ((S.S).LE.1DO) THENCOSPH(M) = DCMPLX(DSQRT(1DO-S*S), ODO)ELSECOSPHOI) = DCMPLX(ODO, \u00E2\u0080\u0094DSQRT(S*S-1DO))ENDIF50 CONTINUE*Appendix B. Scattering by a Conducting Grating 181Listing B.2: (Continued)* set up the matrices* matrixEDO 100, I = 1, MFSDO 100, J 1, lUGN = I - ZERO1=1B(I,3) = TEIIIROI, 1, K, D, A, \u00E2\u0080\u0098gf\u00E2\u0080\u0099) *$ DCMPLI(ODO,K)*COSPE(M)100 CONTIIUE* matrixCDO 200, 3 = 1, JUGDO 200, I = 1, MFS\u00E2\u0080\u0098=3N = I - ZEROC(J,I) = TEIIIR(M, 1, K, D, A, \u00E2\u0080\u0098fg\u00E2\u0080\u0099)200 COITIJUE*\u00E2\u0080\u00A2 [C)*tBJ \u00E2\u0080\u0094> [E]CALL CDMULT(C,B,E,IVG,MFS,IUG,MAXUG,MAIFS,MAIWG)* vectorFDO 300, 3 1, lUG1=3M0F(3) = 2.0*TEIIIR(M, 1, K, D, A, \u00E2\u0080\u0098fg\u00E2\u0080\u0099) \u00E2\u0080\u00A2$ DCMPLZ(ODO,K)* DCIIPLX(DCOS(PERAD))300 COITIJUE* matrix G400 CALL CDSET(G, lUG, JUG, NAXUG, (ODO,ODO))DO 500, 3 = 1, lUG\u00E2\u0080\u0098=3G(J,J) = Kl(K,l,A)*CDCOT(Kl(R,l,A)*E)500 CONTINUE-* [E)+[G] \u00E2\u0080\u0094> [E)CALL CDADD(E,G,E,IVG,IWG,NAIUG,MAXWG,KAIWG)* [E]*[X] = [F]; solve for [I]; [X]\u00E2\u0080\u0094>[F]CALL CDSOLI(E ,F,IWG,NAXUG,CDET)IF (ABS(CDET).LT.1E-20) THENREFL = (000,OD0)ERR=2RETURNEIDIF* [B]*[F) -> [R]CALL CDMATV(B,F,R,MFS,IUG,NAXFS)R(ZERO) R(ZERO)-DCMPLI(ODO,K) * DCIIPLX(DCOS(PHRAD))* check for conservation of energy lit propagating free space modesSUM = ODOERR = 0DO 1000, I = 1, MFSN = I \u00E2\u0080\u0094 ZEROIF (ABS(SIIPH(M)) .LE.1DO)$ SUM = SUN + C CDABS( R(I)/( DCMPLI(ODO,K) \u00E2\u0080\u00A2$ COSPE(M) ) ) )*.2 COSPH(M)1000 CONTINUEIF C ABS(SUM-DCOS(PHRAD)).GT.0.000100 ) ERR = 1* calculation complete!REFL R(ZERO)/(DCMPLI(ODO,K)*DCMPLZ(DCOS(PHRAD)))RETURNENDAppendix B. Scattering by a Conducting Grating 182ListingB.3: COMPLEX*16 FUNCTION TMINNR(M, N, K, D, A, 0)COKPLEX*16 FUNCTION TMIIIRO(, I, K, D, A, 0)PARAMETER (KAIFS=51)LOGICAL ODDINTEGER M, IREAL.8 K, D, A, DEN, E, NUN, P1, SEQ, SINPHCEARACTER*2 0DIMENSION SIJPH(\u00E2\u0080\u0094MAXFS/2 :MAIFS/2)COMMON P1, SIJPEDEN = K.SIIPR(M)*K*SINPH(M)- (DBLE(N)*pI/A).(DBLE(I)*pI/A)IF (ODD(I)) THENNUN = DCOS (A.KeSINPH (N) /2D0) *2D0.K*SINPH (K)TMINNR = DCMPLI(SEQ(O))*DCMPLI(ODO,\u00E2\u0080\u0094IUM/(DEI4\u00E2\u0080\u0099DSQRT(A*D)))ELSENUN = SIN(A*K.SIIPH(M)/2D0) *2D0*K*SIIPROI)TMIIIR = DCMPLI(DSQRT(E(N)/(A.D)) * IUJI/DEN,ODO)EIDIFRETURNENDListing B.4: COMPLEX*16 FUNCTION TEINNR(M, N, K, D, A, 0)COMPLEI*16 FUNCTION TEINNR(M, I, K, D, 1, 0)PARAMETER (JIAXFS=51)LOGICAL ODDINTEGER N, NREAL*8 K, D, A, DEN, NUN, P1, SEQ, SIIPECRARACTER*2 0DIMENSION SIIPH(-KAIFS/2 :KLIFS/2)COMMON P1, SliPliDEN K*SIIPH(M)*K*SIIPHOI) - (DBLE(I)*PI/A)*(DBLE(I)*PI/A)IF (ODD(I)) THENNUN = DCOS(A*K*SIIPH(M)/2D0)*2D0*DBLE(N).PI/ATEIIIR = DCMPLX(-(IUN/DEI)*DSQRT(2D0/(A*D)) ,ODO)ELSENUN = SIN(A*KSSINPH(M)/2D0)*2D0*DBLE(N)*PI/ATEIINR = DCWPLI(SEQ(O))*$ DCMPLI(ODO,-(JUMIDEI)*DSQRT(2D0/(A*D)))EIDIFRETURNENDAppendix B. Scattering by a Conducting Grating 183ListingB.5: COMPLEX*16 FUNCTION KN(M, N, K, D, A, 0)CONPLEI*16 FUNCTION KI(K,I,A)REAL*8 K, A, P1, QINTEGER NP1 = 4D0*DATAN(1DO)Q = DBLE(N)*PI/AIF (ABS(Q).LE.K) THENKN DCNPLX(DSQRT(K*K-Q.Q), 0Db)ELSEIN = DCMPLI(ODO, \u00E2\u0080\u0094DSQRT(Q*Q-K*K))ENDIFRETURIENDListing B.6: LOGICAL FUNCTION ODD(N)LOGICAL FUNCTION ODD(N)* true if N is odd, false if N is evenINTEGER NIF (KOD(N,2).EQ.o) THENODD = .FALSE.ELSEODD = .TRUE.ENDIFRETURNENDListing B.7: REAL*8 FUNCTION SEQ(O)REAL*8 FUNCTION SEQ(O)* if n is odd, = the order is unimportant* if n is even, - the order must be accounted for\u00E2\u0080\u0099CHAR.ACTER*2 0IF (O.EQ.\u00E2\u0080\u0099fg\u00E2\u0080\u0099) THENSEQ = 1DbELSESEQ = -1DbENDIFRETURNENDListing B.8: REAL*8 FUNCTION E(N)REAL*8 FUNCTION E(N)* calculates the Neumann numberINTEGER NIF (N.EQ.b) THENE = 1DbELSEE = 2DbENDIFRETURNENDAppendix B. Scattering by a Conducting Grating 184Listing B.9: COMPLEX*16 FUNCTION CDCOT(ARG)COMPLEX*16 FUICTIOJ CDCOT(LRG)* find the cotangent of a complex numberCWIPLEX*16 CDSIJ, CDCOS, ARGREALs8 DIMAGIF (DIIIAG(ARG).GT.100DO) TEEJCDCOT = (ODO,\u00E2\u0080\u00941DO)ELSE IF (DIIIAG(ARG).LT.\u00E2\u0080\u0094100DO) TBEkCDCOT = (0DO,1DO)ELSE IF (ABs(cDsII(ARG)).LT.1E-1o) TRUICDCOT = (1E1O,O.O)ELSECDCOT = CDCOS(ARG)/CDSIJ(AKG)EWDIFRETURIENDAppendix B. Scattering by a Conducting Grating 185Listing B.1O: UBC Complex Matrix SubroutinesSUBROUTIIE CDADD(1,B,C,M,1,1DIMA,1Dh1,1DIMC)* [C) <\u00E2\u0080\u0094 [A)+[B] Ref: UBC Matrix, p. 15COMPLEX*16 A(NDIMA,*),B(IDIKB,*) ,C(NDIMC,*)DO 1 3=1,1DO 1 I=1,M1 C(I,J) = 1(1,3) + B(I,3)RETURNENDSUBROUTINE CDMATV(A,V,V,M,1,NDINI)** w <- [A].v Ref: USC Matrix, p. 17COMPLEX*16 A(NDIMA,e) ,V(*) ,w(*)DO 1 1=1,1U(I) = (0.00,0.00)DO 1 3=1,11 11(I) = w(I) + 1(1,3) \u00E2\u0080\u00A2 V(i)RETURNENDSUBROUTINE CDMULT(A,B,C,M,J,L,NDIMA,NDIMB,NDIMC)\u00E2\u0080\u00A2 [C] <\u00E2\u0080\u0094 [A].[B] Ref: USC Matrix, p. 18COMPLEX*16 1(IDIMA,*),B(IDIMB,.) ,C(NDIIIC..)DO 1 J=1,LDO 1 11,MC(I,J) = (0.DO,O.D0)DO 1 1=1,11 C(I,J) = C(I,J) + 1(I,K) * B(K,J)RETURNENDSUBROUTIIE CDSET(1,M,1,IDIMA,X)\u00E2\u0080\u00A2 [C] \u00E2\u0080\u0094 [A]+[B] Ref: USC Matrix, p. 21COMPLEX*16 *(IDIMA,*),IDO 1 3=1,1DO 1 I=1,M1 1(1,3) = IRETURNENDSUBROUTINE CDSUB(A,B,C,M,N,NDIMA,NDIMB,NDIMC)* [C] <\u00E2\u0080\u0094 [1]-[B] Ref: UBC Matrix, p. 22COMPLEI*16 A(NDIMA,*) ,B(IDIMB,.),C(IDIMC,*)DO 1 J11DO 1 11,M1 C(I,J) = 1(1,3) \u00E2\u0080\u0094 B(I,J)RETURNENDSUBROUTINE CDVMAT(V,A,W,M,N,NDIMA)** a <- v.[A] Ref: USC Matrix, p. 24COMPLEX*16 1(1DIMI,s) ,V(*) ,V(e)DO 1 3=1,NW(J) = (0.D0.0.D0)DO 2 11,M2 W(J) = V(J) + v(I) \u00E2\u0080\u00A2 1(1,3)1 CONTINUERETURNENDAppendix B. Scattering by a Conducting Grating 186Listing B.1O: (continued)SUBROUTINE CDSOLI (A, B, I ,MM ,DET)** [A).x = b; x\u00E2\u0080\u0094>b Ref: UBC Matrix, p. 55C This routine finds the solution of a system of equations AIB, and takesC paging into consideration; i.e., operations are done by columns,C A\u00E2\u0080\u0094 matrix of coefficintsC B\u00E2\u0080\u0094 vector of right hand sides; on exit array B wiil contain the solution.C I\u00E2\u0080\u0094 order of matrix.C MM- first dimension of a.C DET- determinant of matrix is DET*10**JEXCOMMON /NATEXP/ JEXDIMENSION AOIfI,N),B(*)COMPLEX*16 A,B,DET,T,TB,DCMPLXREAL*8 RMAI , QDET , CDABSJEXODETDCMPLX(1 .D0,0.D0)C decompose a into upper triangular matrixDO 6 K1,IM=KRIIAX=CDABS (A (K , K) )IF(K.EQ.I) GO TO 14KP1K+1DO 1 IKP1,NIF(CDABS(A(I,K)).LE.RMAX) GO TO 1M1RAAXCDABS(A(I ,K))1 CONTINUE14 T=A(M,K)IF(RAAX.LT.1.D-20) GO TO 20IF(K.EQ.N) GO TO 5A (11, K) =A (K , K)A (K , K)TBB (K)BOO =B(K)B(K)=TBDO 2 I=KP1,NA(I X)=-A(I K)/T2 B(I5B(I)+AI,K)*TBDO 4 JKP1,NTBA(M,J)A(M,J)=A(X,J)A(K,J)TBIF(CDABS(TB).EQ.0.D0) GO TO 4DC 3 IKP1,N3 A(I,J)A(I,J)+A(I,K)*TB4 CONTINUEC now get determinant5 DETDET*TQDET=CDABS (DET)IF(QDET.LT.1.D15) GO TO 25DET=DET*1.D\u00E2\u0080\u009415JEXJEX+1525 IF(QDET.GT.1.D-15) GO TO 30DETDET*1 .015JEXJEX-1 530 IF(M.NE.K) DET\u00E2\u0080\u0094DET6 CONTINUEC now do back substitutionIF(N.EQ.1) GO TO 9IM1N-1DO 8 KB1,NM1KM1N-KBKKM1+1B(K)B(K)/A(K,K)T=-B(K)DO 8 1=1 XMl8 B(I)B(I+A(I,K)*T9 B(1)=B(1)/A(1,1)RETURN20 DETDCMPLX(0.D0,0.D0)JEXORETURNENDReferences[1] A. Hessel, J. Shmoys, and D. Y. Tseng, \u00E2\u0080\u009CBragg-angle blazing of diffraction gratings,\u00E2\u0080\u009DJ. Opt. Soc. Am., vol. 65, pp. 380\u00E2\u0080\u0094384, Apr. 1975.[2] J. W. Heath, Scattering by a Conducting Periodic Surface with a Rectangular GrooveProfile, M.A.Sc. Thesis, Univ. of British Columbia, Vancouver, B.C., 1977.[3] J.J. Van Zyl and F.T. Ulaby, \u00E2\u0080\u009CScattering matrix representation for simple targets,\u00E2\u0080\u009D inRadar Polarimetry for Geoscience Applications. F.T. Ulaby and C. Elachi, Eds. Norwood,MA: Artech House, 1990, Pp. 17\u00E2\u0080\u009419.[4] J. W. Dettman, Mathematical Methods in Physics and Engineering. New York: McGraw-Hill, 1969, p. 26.[5] C. P. Wu, \u00E2\u0080\u009CMethods of Solution,\u00E2\u0080\u009D in Theory and Analysis of Phased Array Antennas,N. Amitay, V. Galindo, and C. P. Wu, Eds. New York: Wiley, 1972, pp. 75\u00E2\u0080\u0094119.[6] N. Amitay and V. Galindo, \u00E2\u0080\u009COn energy conservation and the method of moments in scattering problems,\u00E2\u0080\u009D IEEE Trans. Antennas Propagat., vol. AP-17, pp. 747\u00E2\u0080\u0094751, Nov. 1969.[7] R. Mittra, \u00E2\u0080\u009CRelative convergence of the solution of a doubly infinite set of equations,\u00E2\u0080\u009D J.Res. Nat. Bur. Stand., vol. 67D, Mar-Apr. 1963, pp. 245\u00E2\u0080\u0094254.[8] T. Itoh and R. Mittra, \u00E2\u0080\u009CRelative convergence phenomenon arising in the solution of diffraction from strip grating on a dielectric slab,\u00E2\u0080\u009D Proc. IEEE, vol. 59, pp. 1363\u00E2\u0080\u00941365, Sept. 1971.[9] R. Mittra, T. Itoh, and T-S Li, \u00E2\u0080\u009CAnalytical and numerical studies of the relative convergence phenomenon arising in the solution of an integral equation,\u00E2\u0080\u009D IEEE Trans. MicrowaveTheory Tech., vol. MTT-20, pp. 96\u00E2\u0080\u0094104, Feb. 1972.187References 188[101 C.P. Wu, \u00E2\u0080\u009CConvergence test and the relative convergence problem,\u00E2\u0080\u009D in Computer Techniques for Electromagnetics. Oxford: Pergamon, 1973, pp. 300\u00E2\u0080\u0094304.[11] B.W. Kernighan and P.J. Plauger, The Elements of Programming Style. New York:McGraw-Hill, 1974.[12] M. Metcalf, Effective Fortran 77. Oxford: Clarendon Press, 1985.[13] B. Kruger, Efficient Fortran Programming. New York: Wiley, 1990.[14] T. Nicol, Ed. UBC Matrix: A Guide to Solving Matrix Problems. Vancouver, B.C.: Univ.of British Columbia Computing Centre, Mar. 1982.[15] M. Loukides, UNIX for Fortran Programmers. Sebastopol, CA: O\u00E2\u0080\u0099Reilley & Associates,1990.Appendix CCIRCULAR POLARIZATION SELECTIVE REFLECTORSC.1 IntroductionBecause the polarization scattering matrix which corresponds to a circular polarization selectiveresponse cannot be diagonalized when expressed with respect to a linearly polarized basis, atrihedral corner reflector cannot be modified to present such a response simply by placing finsor corrugations of appropriate dimensions and orientation along one of its interior surfaces oracross its aperture. Alternative methods for obtaining such a response based on the additionof a suitable transmission polarizer to either a linear polarization selective or a twist polarizingtrihedral corner reflector are proposed.C.2 ConceptA circular polarization selective reflector will return an incident wave of the chosen sense backto the source but will either absorb an incident wave of the opposite sense or scatter it in adifferent direction. With respect to a circularly polarized basis, the normalized polarizationscattering matrices of left and right circular polarization selective reflectors are given bySLL SLR 1 0[SJ = = (C.1)SRL SRR 0 0andSLL SLR 0 0=. (C.2)SRL SRR 0 1The co-polar and cross-polar response of circular polarization selective reflectors are plottedas a function of the polarization state of the incident wave in Figures C.1 and C.2. The189Appendix C. Circular Polarization Selective Reflectors 190parameters E and T refer to the ellipticity and tilt angles of the corresponding polarization ellipse.Although the amplitudes of the co-polar and cross-polar responses of a circular polarizationreflector are invariant under rotation of the target about the line-of-sight, the relative phase ofthe response varies linearly with the angle of rotation.In Chapter 4, it is shown that a conventional trihedral corner reflector can be modified topresent a given polarization response along its boresight simply by placing conducting fins orcorrugations of appropriate dimensions and orientation along one of its three interior surfacesor across its aperture if the corresponding polarization scattering matrix [S] can be transformedthrough rotation about the line-of-sight into a diagonal matrix [S\u00E2\u0080\u0099] of the form[S\u00E2\u0080\u0099] = SH0, (C.3)0 S(7A polarization scattering matrix expressed with respect to a circularly polarized basis [Sj canbe transformed to the equivalent polarization scattering matrix expressed with respect to alinearly polarized basis [S] by applying the unitary change of basis transformation[SJ = {U*J [S] {U*]_l , (C.4)where [U] is the transformation matrix for change of basis from linear to circular polarizationgiven by1 1 1\u00E2\u0080\u0094j.(C.5)Applying (C.4) to the matrices of (C.1) and (C.2) gives the normalized linear polarizationscattering matrices of left and right circular polarization selective reflectors,[Se] = SHH SHy = \u00E2\u0080\u00941 (C.6)SVH Svv 2 1andSHH SHy = 1 \u00E2\u0080\u0094j (C.7)SVH Svv\u00E2\u0080\u0094j 1Appendix C. Circular Polarization Selective Reflectors 191(b)Figure C.1: Normalized response of a left circular polarization selective reflector as a functionof the polarization state of the incident wave. (a) Co-polar response. (b) Cross-polar response.(a)b.(b)Figure C.2: Normalized response of a right circular polarization selective reflector as a functionof the polarization state of the incident wave. (a) Co-polar response. (b) Cross-polar response.(a)Appendix C. Circular Polarization Selective Reflectors 192If a depolarizing reflector with the polarization scattering matrix of (C.6) or (C.7) is rotatedabout the line-of-sight through an angle a, the off diagonal elements of its new polarizationscattering matrix [S\u00E2\u0080\u0099] are given byS1 = 2SHV cos2 a \u00E2\u0080\u0094 SHy + (Svv \u00E2\u0080\u0094 SHH) sin a cos a, (C.8)where the off diagonal elements SHy and SVH are identical in all coordinate frames for themonostatic case [1]. Since the diagonal elements of the original matrix are real and the offdiagonal elements are imaginary, S, is complex. In order to diagonalize [5\u00E2\u0080\u0099], the angle ofrotation a must be chosen such that both the real and imaginary components of S, are setto zero, i.e.,Re(Sy) = (Svv\u00E2\u0080\u0094SHH)sinacosa = 0, (C.9)Im(Sy) = 2SHV cos2 a\u00E2\u0080\u0094 SHy = 0. (C.10)Since SHH Svv, (C.9) is satisfied only if a = n r/2 where n is an integer. However, thesevalues of a will not satisfy (C.10) unless SHy = 0. Therefore, the polarization scatteringmatrices of (C.6) and (C.7) cannot be diagonalized and it is not possible to realize a trihedralcorner reflector with a circular polarization selective response simply by placing conducting finsor corrugations of appropriate dimensions and orientation along one of its interior surfaces oracross its aperture.C.3 Proposed ImplementationsMethod IIt may be possible to realize a trihedral corner reflector with a circular polarization selectiveresponse by incorporating both transmission and reflection polarizers in the modified reflector.One such scheme is shown in Figure C.3. Here a transmission circular polarizer of the typeshown in Figure C.4 is placed across the aperture of a linear polarization selective reflector whichhas been oriented to return horizontally polarized incident waves but reject vertically polarizedincident waves. Subject to mechanical constraints, any of the several types of transmissioncircular polarizers that have been described in the literature (e.g., [2]) would be satisfactory.Appendix C. Circular Polarization Selective Reflectors 193(a) (b)Figure C.3: A proposed implementation of a circular polarization selective reflector using atransmission circular polarizer and a linear polarization selective reflector. (a) Left circularpolarization selective reflector. (b) Right circular polarization selective reflector.In Figure C.3(a), the axis of the transmission circular polarizer is oriented at -45 degreesto the vertical. Since left circularly polarized incident waves are converted to horizontal polarization by the transmission polarizer, they are returned by the linear polarization selectivereflector then converted back to left circular polarization as they pass through the transmissionpolarizer a second time. However, right circularly polarized incident waves are converted tovertical polarization by the transmission polarizer and are either scattered away from the sourceor absorbed by the linear polarization selective reflector. Thus, a left circular polarization selective reflector has been realized. The polarization scattering matrix of the reflector is givenby[Sc] = [T][S][T},\u00E2\u0080\u0094 1 1+j 1+j \u00E2\u0080\u00941 0 1 1+j 1\u00E2\u0080\u0094j\u00E2\u0080\u00942 1\u00E2\u0080\u0094j 1+j 0 0 2 1\u00E2\u0080\u0094j 1+j (C.11)\u00E2\u0080\u0094 j \u00E2\u0080\u00941 j\u00E2\u0080\u0094 jlwhere [T] and [5] are the polarization matrices of the transmission circular polarizer orientedas shown and a vertical polarization selective reflector, respectively, according to the BSAAppendix C. Circular Polarization Selective Reflectors 194LINEAR POLARIZATION4a 0.671 >>\u00E2\u0080\u0098qfZX0 FREE SCEWAVELENGThCIRCULAR POLARIZATIONFigure C.4: A parallel plate transmission circular polarizer. (from [2, sec. 23-5])convention. If the transmission circular polarizer is oriented with its axis at 45 degrees to thevertical instead, as shown in Figure C.4(b), a right circular polarization selective reflector isrealized. If the response of the polarizer is now given by [T\u00E2\u0080\u0099], the polarization scattering matrixof the reflector will be given by[Sn] = [T\u00E2\u0080\u0099][S][T\u00E2\u0080\u0099j,1 1+j \u00E2\u0080\u00941+j \u00E2\u0080\u00941 0 1 1+j \u00E2\u0080\u00941+j= 2\u00E2\u0080\u00941+j 1+j 0 0 2 1+j \u00E2\u0080\u0098 (C.12)\u00E2\u0080\u0094\u00E2\u0080\u00941\u00E2\u0080\u0094j\u00E2\u0080\u0094 2\u00E2\u0080\u0094j 1Method IIA second scheme for implementing a circular polarization selective trihedral corner reflector isshown in Figure C.5. Here a circular polarization selective surface (CPSS) [4, 5] is placed acrossthe aperture of a trihedral twist reflector. Over most aspects, incident waves of the chosen sensepass through the CPSS and are returned back to the source by the trihedral twist reflector whileincident waves of the opposite sense are specularly reflected away from the source as suggestedby Figure C.6(a).Appendix C. Circular Polarization Selective Reflectors 195Figure C.5: A proposed implementation of a circular polarization selective reflector using a circular polarization selective surface and a trihedral twist reflector. (a) Left circular polarizationselective reflector. (b) Right circular polarization selective reflector.If the angle of incidence is normal to the reflector aperture, however, both incident polarizations will be returned to the source and the response will no longer appear to be polarizationselective, as shown in Figure C.6(b). For a reflector with corners of equal length, this condition occurs along the boresight or direction of maximum response which is clearly undesirable.This problem also limits the performance of the gridded trihedral, a linear polarization selectivereflector which is realized by mounting a closely spaced parallel grid of thin wires across thereflector aperture as shown in Figure C.7.(.R.L(R.L R(a) (b)Figure C.6: Scattering by a right circular polarization selective trihedral corner reflector.(a) Oblique incidence. (b) Normal incidence.(a) (b)Appendix C. Circular Polarization Selective Reflectors 19650Figure C.7: A linear polarization selective gridded trihedral and its co-polar and cross-polarazimuthal response patterns. (from [3])Backscatter returns from either the circular polarization selective surface or the closelyspaced grid of parallel wires can be avoided by mounting the reflector in such a way that itsboresight is pointed away from the radar over most aspects. However, this is undesirable if thecorners of the reflector are of equal length because the scattering cross section of the reflectorwill be substantially reduced compared to its maximum value and will become much moresensitive to small changes in the orientation of the reflector. An alternative solution is to forcethe boresight and the normal to the aperture to point in different directions by incorporatingbilateral symmetry into the reflector geometry. Consider a trihedral corner reflector with onecorner of length c and two of length a as shown in Figure C.8. If the ratio of the center andside corner lengths of the reflector (c/a) is increased slightly from unity, the normal to thereflector aperture will rise in elevation relative to the the reflector boresight angle as describedin Chapter 3. The elevation beamwidth of the reflector will also increase slightly at the expenseof its azimuthal beamwidth. Decreasing the ratio (c/a) instead would have the opposite effectalthough the angle of separation will be less pronounced. In either case, the angle of separation,8 between the normal to the aperture and the reflector boresight will increase and the desiredresult will be achieved as suggested by Figure C.8(b).Trihedral Responseto Cross ComponentGrid Response toParallel ComponentC050 25 25Aspect AngleAppendix C. Circular Polarization Selective Reflectors 197aC(a)a(b)Figure C.8: A bilaterally symmetric trihedral corner reflector with triangular panels. 0 is theangle between the boresight and the normal to the aperture. (a) Front view. (b) Side view.The polarization scattering matrix of a hybrid reflector which combines a left CPSS withpolarization matrix [Tj and a trihedral twist reflector with polarization matrix [S] is given by[Se] = [T] {S} {T]1 \u00E2\u0080\u00941 j \u00E2\u0080\u00941 0 1 \u00E2\u0080\u00941 j=j 1 0 1 \u00E2\u0080\u00A2 j 1\u00E2\u0080\u0094 1 \u00E2\u0080\u00941 j\u00E2\u0080\u0094 jl(C.13)By substituting a right CPSS with polarization transmission matrix [T\u00E2\u0080\u0099] for the original, aright circular polarization selective reflector can be realized. Its polarization scattering matrixis given by[Sj = [T\u00E2\u0080\u0099]{S][T\u00E2\u0080\u0099]\u00E2\u0080\u0094 1 \u00E2\u0080\u00941 .\u00E2\u0080\u0094j \u00E2\u0080\u00941 0 1 \u00E2\u0080\u00941 \u00E2\u0080\u0094j\u00E2\u0080\u0094\u00E2\u0080\u0094j 1 0 1 \u00E2\u0080\u0094j 1\u00E2\u0080\u00941 \u00E2\u0080\u00941 \u00E2\u0080\u0094j\u00E2\u0080\u0094\u00E2\u0080\u0094j 1(C.14)Appendix C. Circular Polarization Selective Reflectors 198A method for realizing a circular polarization selective surface was first described by Tilstonet a!. [4]. An alternative implementation which is easier and less expensive to fabricate thanthe original was subsequently devised by Morin [5]. The improved surface consists of a slab oflow dielectric constant material of thickness A/4 which has been divided into a square patternof cells of dimension A/2 by A/2. A resonant element which consists of a single piece of wirewhich has been bent into three sections is inserted into each cell such that the two end sectionsare flush against the front and back faces of the dielectric slab. Left and right CPSS elementsare shown in Figure C.9 where the z-y plane is the plane of the aperture.(b)Figure C.9: Elements of a circular polarization selective surface (CPSS). (a) Left CPSS element.(b) Right CPSS element.C.4 DiscussionTwo methods for realizing circular polarization selective trihedral corner reflectors have beenproposed in this appendix. The performance of such reflectors and their usefulness in practice will depend on a number of factors which have not been considered here including (1)degradation of the polarization response of either of the polarizers for incidence off the reflector boresight, (2) possible degradation of the polarization response due to multiple reflectionsbetween the transmission and reflection polarizers, and (3) the ease with which the modifiedreflector can be assembled and the mechanical ruggedness of the finished product.x3/8(a)References[1] S.H. Bickel, \u00E2\u0080\u009CSome invariant properties of the polarization scattering matrix,\u00E2\u0080\u009D Proc. IEEE,vol. 53, pp. 1070\u00E2\u0080\u00941072, Aug. 1965.[2] W.B. Offut and L.K. DeSize, \u00E2\u0080\u009CMethods of Polarization Synthesis,\u00E2\u0080\u009D in Antenna EngineeringHandbook, 2nd ed., R. C. Johnson and H. Jasik, Eds. New York: McGraw-Hill, 1984,chap. 23.[3] .J.A. Scheer, \u00E2\u0080\u009CRadar Reflectivity Calibration Procedures,\u00E2\u0080\u009D in Radar Reflectivity Measureinent: Techniques and Applications, N.C. Currie, Ed. Norwood, MA: Artech House, 1989,p. 109.[4] W.V. Tilston, T. Tralman, and S.M. Khanna, \u00E2\u0080\u009CA polarization selective surface for circularpolarization,\u00E2\u0080\u009D in IEEE/A P-S Symp. Dig. (Syracuse, NY), June 1988, pp. 762\u00E2\u0080\u0094765.[5] G.A. Morin, \u00E2\u0080\u009CA simple circular polarization selective surface (CPSS),\u00E2\u0080\u009D in IEEE/A P-SSymp. Dig. (Dallas, TX), May 1990, pp. 100\u00E2\u0080\u0094103.199Appendix DEXPERIMENTAL ARRANGEMENTD.1 IntroductionDuring the course of this study, the microwave antenna range located on the roof of the Electrical Engineering building at the University of British Columbia was upgraded and used tomeasure the response patterns of prototype depolarizing trihedral corner reflectors as describedin Chapter 4. The principles of radar cross section measurement have been widely discussedin the literature, e.g., [1]\u00E2\u0080\u0094[5]. In this appendix, the modifications and improvements that weremade to the UBC antenna range in support of the prototype reflector measurement programare briefly described. In section D.2, the general layout of the antenna range is described. Insections D.3 and D.4, respectively, the design and implementation of the CW radar apparatusand digital pattern recorder that are discussed. In section D.5, the results of tests performed toverify the suitability of the antenna range for use in the measurement program are presented.Recommendations for future modifications and improvements are offered.D.2 OverviewA block diagram of the UBC microwave antenna range as it was configured for the prototypereflector measurement program is shown in Figure D.1. The CW radar apparatus is designedto measure the co-polar and cross-polar response of targets in the range from 8\u00E2\u0080\u009412 GHz. Thedigital pattern recorder is designed to calibrate the CW radar apparatus and record the response patterns of antennas or targets under test. It replaces the Scientific Atlanta series 1520mechanical chart recorder used previously.200\u00E2\u0080\u0094E-B0 0 A A U\u00E2\u0080\u0099 It1ICD CDm<:.:*Lm-...D2,\u00E2\u0080\u00A2:..\u00E2\u0080\u00A2:O0CD\u00E2\u0080\u00A2:\u00E2\u0080\u00A2.:\u00E2\u0080\u00A2:\u00E2\u0080\u00A2\u00E2\u0080\u00A2.IL[I1I4$?..G)S:-ai.0:\u00E2\u0080\u00A2._z\u00E2\u0080\u0094\u00E2\u0080\u0098\u00E2\u0080\u0094I1G)WiTitImI\/_Xr\u00C2\u00B00\u00E2\u0080\u0094IoG)j%.)).3Cl,T.0 IN r\u00E2\u0080\u0099) 3I.Appendix D. Experimental Arrangement 202The outdoor portion of the antenna range is shown in Figure D.2. The Scientific Atlantamodel 5851 model tower travels on a carriage along 15 metres of track down the center of theroof of the east wing of the Electrical Engineering building. The model tower supports theantenna or target under test at a height of 2.2 metres and can be configured either to rotatethe device about a vertical axis for conventional azimuthal pattern measurements or to roll thedevice about a horizontal axis for polarization response measurements. Detailed mechanicalspecifications are presented in the model tower operating manual [6].D.3 CW Radar ApparatusThe CW radar apparatus consists of a CW transmitter, a microwave receiver equipped with anexternal crystal mixer, and two standard gain horns mounted side by side on a custom-builtfeedthrough mounting adapter. A block diagram of the transmitter and a photograph of thetransmitter shelf are shown in Figures D.3 and D.4, respectively. The signal source is a MarconiFigure D.2: Photograph of the radar cross section measurement range.Appendix D. Experimental Arrangement 2036052B microwave signal generator which can provide at least 25 mW (14 dBm) of power to amatched load over the range from 8\u00E2\u0080\u009412 GHz. The transmitted signal is sampled using an HPX752C 20 dB directional coupler and monitored using a Marconi 6593A VSWR meter equippedwith a HP X485B detector mount. The exact frequency of the transmitted signal is determinedby tuning the HP X532B frequency wavemeter until a dip is observed on the VSWR meter.The HP X382A precision attenuator is used to adjust the transmitter output power duringcalibration. Detailed specifications for each component are presented in references [7]\u00E2\u0080\u0094[9].The receiver section of the radar apparatus consists of a Scientific Atlanta model 171OAPportable microwave receiver equipped with a model M8.2 external crystal mixer. Detailedspecifications are given in the receiver operating manual [10]. The receiver local oscillator (LO)is tunable from 0.985\u00E2\u0080\u00942.5 GHz. During operation, the LO output is fed through a RF pad tothe LO arm of a frequency selective tee. This tee couples the LO signal through a coaxial cableto the external mixer where harmonic mixing takes place. The resulting 45 MHz IF signal isconducted back through the same coaxial cable and frequency selective tee to a 45 MHz IFpreamplifier and subsequent stages. The receiver provides a signal to the monitor meter onits front panel which is proportional to the received signal strength. This signal was tappedand passed through a signal conditioning unit containing an op-amp based current-to-voltageconverter, a low pass filter, and an adjustable gain block in order to provide the 0\u00E2\u0080\u009410 VDCoutput signal required by the digital pattern recorder.A pair of Scientific Atlanta model 12-8.2 standard gain horn antennas are used as thetransmitting and receiving antennas. The antennas present a boresight gain of 22.10 \u00C2\u00B1 0.05dB at a wavelength of 3.2 cm with E-plane and H-plane half-power beamwidths of 12.5 and13.5 degrees, respectively [11]. They are mounted side by side on a custom-built feedthroughmounting adapter which is attached to the side of the building penthouse at a height of 2.2metres as shown in Figures D.5 and D.6. The transmitting horn is mounted so that it isvertically polarized. The mounting adapter permits the receiving horn to be rotated by 90degrees in order to permit either the co-polar (0vv) and cross-polar (CHV) response of theAppendix D. Experimental Arrangement 204target to be measured. Arrangements for aligning the horns are shown in Figure D.7. Coarsealignment is performed using bubble levels which are mounted on the horns and waveguideas shown. The alignment is completed by optical sighting an alignment target mounted onthe model tower using cross hairs mounted on the front of the horns and the flanges of thewaveguide.Figure D.3: Block diagram of the CW radar transmitter.Standard Gain HornScientific AtlantaModel 12-8.2FrequencyMeterHPXS32BPrecision 10dBVariable DirectionalAttenuator CouplerHP X382A HP X752CDetectorMountHPX485BFigure D.4: Photograph of the CW radar transmitter.CD0\u00E2\u0080\u00A2\u00E2\u0080\u00A2ti CD CD CD CD+-CDCl)\u00E2\u0080\u0094CD\u00E2\u0080\u0094.CDCD Cl) CIaI.01Appendix D. Experimental Arrangement 206Figure D.6: Profile view of the RCS measurement range.\u00E2\u0080\u0094 a2mOm(a)Standard Gain HornFront viewCross-hairs25cm(b)Figure D.7: Arrangement for mechanically aligning the transmitting and receiving antennas.(a) Alignment aids mounted on the transmitter shelf. (b) Alignment target mounted on themodel tower.Horn Antennas -3dBOm 5m lOm 15mCircular Levels90\u00C2\u00B0 twist section(as required)Side\1iewFeedthrough Mounting AdapterAppendix D. Experimental Arrangement 207A graphical representation of the link budget for the radar cross section measurement rangeas configured for measurement of prototype trihedral corner reflectors with a corner length of60 cm is presented in Figure D.8. The results correspond to the ideal case in which the poweravailable to the radar receiver is given by the radar range equation,p \u00E2\u0080\u0094 PtGtGr(/A2)\u00E2\u0080\u0094 (47r)3R/Awhere G = G,. = 22.1 dB are the gains of the transmitting and receiving antennas, respectively,u = 27.3 dBsm is the radar cross section of the target, ..\ = 3.18 cm is the radar wavelength,and R = 11 m is the range to the target. In order to achieve the required dynamic range, theminimum return from the target must be greater than either the background reflectivity of thefacility or the receiver thermal noise level.Ea -19ci)-J-49ci)Facility Background Level (Crosspolar)Receiver Thermal Noise Level140Transmitted Power LevelMaximum RCS ReturnDynamic RangeMinimum RCS Return0dB-10dB-20dB-30dBFacility ckgroundJ\u00C3\u00A7vel(opplar)Figure D.8: Radar cross section measurement range link budget.Appendix D. Experimental Arrangement 208D.4 Digital Pattern RecorderThe digital pattern recorder consists of an industry standard personal computer equipped witha colour VGA display, an 80386SX-25 CPU with a numerical coprocessor, 2 MB RAM, an80 MB hard disk drive, and custom-designed interface hardware and data acquisition software.A PC-26 analog-to-digital converter (ADC) expansion card (Boston Technologies, Boston, MA)equipped with the custom-designed signal conditioning unit described in the previous section is used to measure the output from the portable microwave receiver. A custom-built12-bit synchro-to-digital converter (SDC) expansion card based on a Control Sciences Inc.(Chatsworth, CA) 168F309 integrated 12-bit SDC module is used to measure the angle ofrotation of the antenna or target under test. Design considerations for PC-based laboratoryinstrumentation have been widely discussed in the literature, e.g., [12], [13]. A photograph ofthe digital pattern recorder, the positioner control unit, and the portable microwave receiver isshown in Figure D.9.Figure D.9: Photograph of the digital pattern recorder, positioner control unit, and portablemicrowave receiver.Appendix D. Experimental Arrangement 209A set of five independent program modules perform calibration, data acquisition, and diagnostic functions. The program modules store calibration data, configuration notes, and responsepattern data in ASCII format and employ a consistent user interface. They were coded usingTurbo Pascal, version 5.5 (Borland International, Scotts Valley, CA) and make use of graphicsroutines from the Science and Engineering Tools for Turbo Pascal subroutine library, version 6.1(Quinn-Curtis, Needham, MA). The function of each module is summarized in Table D.1.Table D.1: Digital Pattern Recorder Program ModulesModule Name FunctionSynchro Test DPRSYN \u00E2\u0080\u00A2 display position of model tower in real time andverify correct operation of the synchro interfaceReceiver Calibration DPR_CAL \u00E2\u0080\u00A2 generate relative calibration curve for receiver andverify correct operation of the receiver interfacePattern Recorder DPR..PAT \u00E2\u0080\u00A2 record pattern data and configuration notesfor analysis and presentationData View DPR_VIEW \u00E2\u0080\u00A2 view and compare previously recorded dataReceiver Stability DPRSTAB \u00E2\u0080\u00A2 sample receiver output signal and generateamplitude distributions and frequency spectraThe synchro test module, DPLSYN, is used to verify that the model tower synchro, the positioner control unit synchro repeater, the synchro-to-digital converter and the interconnectingcables are functioning correctly. It can also be used to calibrate the speed control on the positioner control unit. The program displays the current angle of rotation and rate of rotation ofthe antenna or target under test while it generates a plot of the angle of rotation versus time.A sample screen display is presented in Figure D.10.The receiver calibration module, DPR_CAL, is used to measure the transfer characteristic ofthe microwave receiver. During a measurement sequence, the pattern recorder module uses thisdata to translate the binary code read from the ADC into a relative measure of the receivedsignal strength in decibels. Relative calibration of the CW radar apparatus is performed withthe equipment configured as shown in Figure D.11(a). After the operator has set the receiverinput signal to the highest level likely to be encountered during the measurement, the programAppendix D. Experimental Arrangement 210prompts the operator to decrement the received signal strength in 5 dB steps using either theprecision microwave attenuator in the transmitter or the IF step attenuator in the receiver andsample the receiver output. A sample screen display is presented in Figure D.10. After thecalibration sequence is complete, the operator is prompted to either save the relative calibrationdata to a ifie or begin the calibration sequence again. Absolute calibration of the CW radarapparatus can be performed either directly by measuring the response of a calibration target ofsimilar size to the target under test or indirectly by measuring the effective path loss betweenthe transmitter and receiver.An arrangement for performing absolute calibration using an indirect method is shown inFigure D.12. First, the response of the target is measured using the configuration of Figure D.12(a). Next, the crystal mixer is removed from the receiving horn and mounted on the10 dB directional coupler in the transmitter as shown in Figure D.12(b). Finally, the precisionmicrowave attenuator is adjusted until an identical response is observed at the receiver output.The total insertion loss includes the contributions of both the attenuator and the directionalcoupler. Since the distance to the target, the radar wavelength, and the gain of transmittingand receiving antennas are known, the absolute radar cross section of the target can be determined simply by equating the geometric path loss predicted by the radar equation to the totalinsertion loss. Closure is obtained when the results of direct and indirect calibration agree towithin a prescribed tolerance.The pattern recorder module, DPRPAT, is used to record both the parameters of the testconfiguration and the response pattern of the antenna or target under test. A sample parameter entry screen is shown in Figure D.13. When a new measurement sequence is begun, theparameter entry screen is replaced by a data acquisition screen which displays the responsepattern as it is being measured. The data acquisition screen can be configured to present theresults on either a rectangular or polar chart spanning either + 90 or +180 degrees. A sample\u00C2\u00B1 180 degree polar display with a dynamic range of 30 dB is shown in Figure D.14. After datacollection is complete, the operator is returned to the parameter entry screen and prompted toeither save the response pattern data to a file or begin the measurement sequence again.Appendix D. Experimental Arrangement 211Figure D.10: Digital pattern recorder: syncliro test screen.Use arrow keys for nenu selectionFigure D.11: Digital pattern recorder: receiver calibration screen.Appendix D. Experimental Arrangement 212(a)TransmittingChannel (V)ReceivingChannel (VocH)(b)Figure D.12: Equipment configuration for performing relative and absolute calibration of theCW radar apparatus.90 twist section( required)Digital Pattern RecorderDigital Pattern RecorderAppendix D. Experimental Arrangement 213X Axis Label:X Axis Scale:Sa,ipling Interval:Use arrou ke!Js for eenu eiect ionFigure D.13: Digital pattern recorder: parameter entry screen.Output Folder;Frequency:Polarization:9.4 GUzJert 1Output Filenai,e: stiCalibration Status-Calibration File:Plot TeAveraging:Receiver iplitude: Standard Gain iFix4.... .w3Test Description:V Axis Label:Top Of Scale:Dmanic Range: 3BEGIN EX! T IFigure D.14: Digital pattern recorder: data acquisition screen.Appendix D. Experimental Arrangement 214The last two program modules are test articles used to evaluate functions which will beincorporated into future versions of the receiver calibration and pattern recorder modules. Thereceiver stability module, DPR..STAB, is used to assess the short and long-term stability of theportable microwave receiver while a constant amplitude signal is applied to its input. After theoutput of the receiver has been sampled over a period of time ranging from several minutes toseveral hours, the results are either viewed directly as time series or processed to yield amplitudedistribution functions and/or frequency spectra. The data viewing module, DPR_VIEW, is amodified version of the pattern recorder module which is used to view and compare previouslyrecorded data.D.5 Facility EvaluationRadar cross section measurements are affected by a combination of random and systematicerrors. It is convenient to depict these errors by the signal flow graph presented in FigureD.15 where S represents the actual response of the target, R and T represent errors due to(1) deviation of the incident field from a plane wave and (2) multipath reflections from thesurrounding facility, and I represents the contribution of general background due to (3) returnsfrom the target support structure and surroundings and (4) direct transmission between thetransmitting and receiving antennas. In order to determine the suitability of a RCS facility foruse in a measurement program, a series of tests must be performed to assess the magnitude ofthese errors and their potential impact on the accuracy of the results obtained.TransmitterIReceiverRFigure D.15: Error model for radar cross section measurement.Appendix D. Experimental Arrangement 215Ideally, the target is located sufficiently far away from the transmitting antenna that theincident field presents a planar wavefront. For a small outdoor antenna range of the typeconsidered here, such an arrangement is rarely practical since the contribution of the multipathray to the response increases rapidly with range. It is often suggested that satisfactory resultswill be obtained if the phase deviation over the largest dimension of the target is less than 7r/8radians. This leads to the so-called far-field criterion which requires that the range to the targetbe greater than2 d2where d is the largest dimension of the target and A is the radar wavelength. In practice, physicalconstraints often make it necessary to measure radar cross section of targets at ranges as low asone-quarter of the recommended value. The consequences are similar to those encountered whenconducting antenna measurements under similar circumstances. In particular, the apparentvalue of the radar cross section obtained by direct solution of the radar range equation will beless than the actual value that would have been obtained had the incident field been a planewave.A trihedral corner reflector of the type used in the measurement program has triangularpanels of equal length and presents a maximum radar cross section of4ir \u00C2\u00A3umax =where \u00C2\u00A3 is the corner length of the reflector and A is the radar wavelength. Although the largestphysical dimension of a reflector of this type is the distance from the tip of one corner to themid-point of the opposite panel, the maximum dimension of the effective aperture is muchsmaller and is given byas suggested by Figure D.16. The maximum aperture dimension, maximum radar cross section,and far-field range of trihedral corner reflectors with various corner lengths are summarized inTable D.2.Appendix D. Experimental Arrangement 216(a)Figure D.16: The effective aperture of a trihedral corner reflector with triangular panels forincidence (a) along the boresight and (b) at an azimuth angle of 30 degrees.Table D.2: Response of Trihedral Corner Reflectors with Triangular Panels at 9.445 GHzCorner length Max. aperture Maximum RCS Far-field range(cm) dimension (cm) (m2) (dBsm) (m)15 14 2 3.2 230 28 33 15.3 545 42 169 22.3 1160 57 537 27.3 2075 71 1310 31.2 31Four tests were conducted in order to assess the suitability of the antenna range for use inthe prototype reflector measurement program. All the tests were performed at the standardmarine radar frequency of 9.445 GHz.In the first test, the polarization response of the CW radar apparatus was evaluated. Thereceiving horn was removed from the feedthrough mounting adapter and attached to the modeltower. The response of the receiving horn was measured at a range of 11 metres as the hornwas rotated about its boresight. The results generally agree with the predicted values andare presented in Figure D.17. The isolation between the transmitting and receiving horns isconsiderably greater than 30 dB when the horns are orthogonally polarized.In the next two tests, the contributions of near-field and multipath effects to the responseof a target under test were evaluated. A conventional trihedrai corner reflector with a corner(b)Appendix D. Experimental Arrangement 217length of 60 cm was attached to the model tower. First, the response of the target was measuredas its range was increased from 8 to 16 metres in 0.25 m increments. The results are presentedin Figure D.18. At ranges greater than about 12 metres, rapid variations in the response areobserved which suggest that multipath effects are becoming important. At ranges less thanabout 9 metres, evidence of a reduction in the response which is apparently due to near-fieldeffects is observed. At ranges between 9 and 12 metres, the response varies as hR4 with onlya few perturbations. A second test was performed in order to assess the effect of small changesin range on the fine structure of the response pattern. The azimuthal response pattern of thereflector was measured at ranges of 10 and 12 metres and compared. The results are presentedin Figure D.19 where it can be seen that the response patterns are nearly identical. On this basisof these results, it was concluded that the optimum range at which to measure the prototypereflectors is 11 metres.In the last test, the contributions of direct transmission between the horns and unwantedreturns from the model tower and surroundings were evaluated. The conventional trihedralcorner reflector was removed from the model tower and the positioner head was covered by asmall section of microwave absorber. The co-polar and cross-polar responses of the backgroundwas measured in turn as the model tower was rotated in azimuth at a range of 11 metres.The results are presented in Figure D.20. The co-polar response rises above -30 dB at oniy afew angles and never rises above -28 dB relative to the maximum response of the prototypereflector. The cross-polar response never rises above -30 dB.During the evaluation of the facility, several problems were noted. The gears that rotate thepositioner head about a horizontal axis exhibit a small but noticeable backlash. The effect isparticularly obvious when a large target such as a prototype reflector is mounted on the modeltower. Since procedures for reducing the backlash require the services of a skilled machinist [6],the problem could not be corrected simply by conducting routine maintenance during the courseof the measurement program. It is recommended that the problem be corrected in the nearfuture, however.Appendix D. Experimental Arrangement 2180.0-2.0I&-10.0-12.08.0 10.0 12.0 14.0 16.0Range (m)Figure D.18: Boresight response of a conventional trihedral corner reflector vs. range.0\u00E2\u0080\u00A20dB-10dB-20 dB-30 dB-Measured\u00E2\u0080\u00A2...Theoretical.09kFigure D.17: Polarization response of the receiving horn at a range of 11 m.\u00E2\u0080\u0094 MeasuredIdealAppendix D. Experimental Arrangement 2190.0-10.01-20.0\u00E2\u0080\u0094 MeasuredTheoretical-30.0-45.0 -30.0 -15.0 0.0 15.0 30.0 45.0Azimuth Angle-(deg)Figure D.19: Azimuthal response pattern of a conventional trihedral corner reflector at rangesof 10 and 12 m.0.0_vvU:2-bC;) 10.0a,4-a,a,-20.0-30.0 . . it\. . . . fl ,i. .A.-45.0 -30.0 -15.0 0.0 15.0 30.0 45.0Azimuth Angle-,(deg)Azimuthal response pattern of the model tower at a range of 11 m.Figure D.20:Appendix D. Experimental Arrangement 220The transfer characteristic of the portable microwave receiver tends to drift noticeably afteronly an hour. As a result, it was necessary to recalibrate the receiver each time a new seriesof measurements was conducted. Since the receiver is over twenty years old and is nearing theend of its useful life, it is recommended that consideration be given to replacing it.As noted earlier, multipath reflections were observed when the range to the target wasgreater than about 12 metres. It might be possible to significantly reduce multipath effects(and extend the model tower\u00E2\u0080\u0099s useful range of travel) by employing either a berm or a seriesof radar fences to scatter the multipath ray as described by Knott [2, pp. 369\u00E2\u0080\u0094370]. It isrecommended that an experimental program be conducted to assess the effectiveness of suchmethods in suppressing multipath effects at this antenna rangeReferences[1] C. G. Bachman, Radar Targets. Lexington, MA: D. C. Heath, 1982, pp. 109\u00E2\u0080\u0094158.[2] E.F. Knott, J.F. Shaeffer, and M.T. Tuley, Radar Cross Section: Its Prediction, Measurement, and Reduction. Norwood, MA: Artech House, 1985, pp. 315\u00E2\u0080\u0094382.[3] Radar Cross-Section Measurements with the HP 8510 Network Analyzer. Product NoteNo. 8510-2, Santa Rosa, CA: Hewlett-Packard, Apr. 1985.[4] R.B. Dybdal, \u00E2\u0080\u009CRadar cross section measurements,\u00E2\u0080\u009D Proc. IEEE, vol. 75, pp. 498\u00E2\u0080\u0094516,Apr. 1987.[5] N.C. Currie, Ed., Radar Reflectivity Measurement: Techniques and Applications, Norwood,MA: Artech House, 1989.[6] Series 5800 Model Towers, 3rd ed. Atlanta, GA: Scientific-Atlanta, Dec. 1970.[7] Instruction Manual for Signal Source Type 6058B. Stevenage, UK: Marconi Instruments(Sanders Div.), 1976.[8] Coaxial Waveguide Catalog and Microwave Measurement Handbook. Palo Alto, CA:Hewlett-Packard, 1979.[9] Instruction Manual for VSWR Indicator and Selective Amplifier Type 6593A. Stevenage,UK: Marconi Instruments (Sanders Div.), 1976.[10] Series 1710 Portable Microwave Receiver, 2nd ed. Atlanta, GA: Scientific-Atlanta,May 1970.[11] W. T. Slayton, Design and Calibration of Microwave Antenna Gain Standards. NRL Report. No. 4433, U.S. Naval Research Lab, Washington, DC, 1954.221References 222[12] S.C. Gates and J. Becker, Laboratory Automation using the IBM PC. Englewood Cliffs,NJ: Prentice-Hall, 1989.[13] B.G. Thompson and A.F. Kuckes, IBM-PC in the Laboratory. Cambridge: CambridgeUniv. Press, 1989."@en .
"Thesis/Dissertation"@en .
"1994-05"@en .
"10.14288/1.0065024"@en .
"eng"@en .
"Electrical and Computer Engineering"@en .
"Vancouver : University of British Columbia Library"@en .
"For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en .
"Graduate"@en .
"Radar cross section enhancement for radar navigation and remote sensing"@en .
"Text"@en .
"http://hdl.handle.net/2429/7038"@en .