Evanescent Field Interactions inthe Scattering from Cylinderswith Applications in Super-ResolutionbyPeter Cornelius PawliukB.A.Sc., The University of British Columbia, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate Studies(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2013c? Peter Cornelius Pawliuk 2013AbstractThe diffraction limit defines the maximum resolution of an imaging systemthat collects and focuses waves. This limited resolution arises from the finitelength of the waves used to create the image. Therefore, the only way toincrease the resolution is to use higher frequencies with shorter wavelengths.For situations in which increasing the frequency is not possible or not de-sirable, super-resolution imaging techniques can be applied to overcome thediffraction limit. Super-resolution is possible with the inclusion of evanescentwaves, which exhibit unlimited spatial frequencies.Evanescent waves decay exponentially away from their surface of originso they are difficult to recover. One way to recover evanescent wave informa-tion is to scatter the wave from a small object. This scattering converts partof the evanescent wave into radiation that can propagate into the far-fieldwhere it can be detected. In order to characterize this conversion, the two-dimensional scattering of evanescent fields from a single cylinder and frommultiple cylinders is investigated. The scattering models are derived usingan analytical approach where the electromagnetic fields are broken down intocylindrical waves so that the boundary conditions on the cylinders can beapplied directly. The incident field can be formulated from a vector plane-wave spectrum, which allows for an arbitrary combination of radiative andevanescent waves. Multiple cylinders of various sizes can be used to approx-imate the scattering from many two-dimensional objects. For simulating theimaging of objects buried underneath a surface, or near a planar interface,the model is separated into two dielectric half-spaces.An example of a super-resolution application for these models is the sim-iiulation of apertureless near-field scanning optical microscopy (ANSOM). InANSOM, a probe is placed in the extreme near-field of an object in orderto scatter the evanescent fields that are formed by the illumination of theobject. Images created by ANSOM are fundamentally different from tradi-tional images and are difficult to interpret. The simulations provide insightinto how the images are formed and what information they contain.iiiPrefacePublished WorksBelow is a list of publications arising from work presented in this thesis andthe chapter in which the work is located.? c? 2009 OSA. Reprinted, with permission, from P. Pawliuk and M.Yedlin, ?Gaussian beam scattering from a dielectric cylinder, includingthe evanescent region,? J. Opt. Soc. Amer. A, vol. 26, no. 12, pp.2558-2566, 2009. Chapter 2? c? 2010 OSA. Reprinted, with permission, from P. Pawliuk and M.Yedlin, ?Gaussian beam scattering from a dielectric cylinder, includingthe evanescent region: erratum,? J. Opt. Soc. Amer. A, vol. 27, no.2, pp. 166, 2010. Incorporated into Chapter 2? c? 2010 OSA. Reprinted, with permission, from P. Pawliuk and M.Yedlin, ?Truncating cylindrical wave modes in two-dimensional mul-tiple scattering,? Opt. Lett., vol. 35, no. 23, pp. 3997-3999, 2010.Chapter 3? c? 2011 OSA. Reprinted, with permission, from P. Pawliuk and M.Yedlin, ?Scattering from cylinders using the two-dimensional vectorplane-wave spectrum,? J. Opt. Soc. Amer. A, vol. 28, no. 6, pp.1177-1184, 2011. Chapter 4? c? 2012 OSA. Reprinted, with permission, from P. Pawliuk and M.Yedlin, ?Scattering from cylinders using the two-dimensional vectorivplane-wave spectrum: addendum,? J. Opt. Soc. Amer. A, vol. 29, no.3, pp. 352-353, 2012. Incorporated into Chapter 4? c? 2012 IEEE. Reprinted, with permission, from P. Pawliuk and M.Yedlin, ?Scattering From Cylinders Near a Dielectric Half-Space Usinga General Method of Images,? IEEE Trans. Antennas Propag., vol. 60,no. 11, pp. 5296 - 5304, 2012. Chapter 5? c? 2013 IEEE. Reprinted, with permission, from P. Pawliuk and M.Yedlin, ?Multiple Scattering Between Cylinders in Two Dielectric Half-Spaces,? IEEE Trans. Antennas Propag., vol. 61, no. 8, pp. 4220 -4228, 2013. Chapter 6CollaborationIn the papers above, I collaborated with my supervisor Dr. Matthew Yedlin.I performed the background research and the analytical derivations, I pro-grammed the simulations and interpreted the results, and I wrote and sub-mitted the papers. Dr. Yedlin offered feedback, insight, and direction for mywork, related my work to that of previous authors, and helped to edit thepapers.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation: Super-Resolution . . . . . . . . . . . . . . . . . . 11.2 Scattering Models . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Single Cylinder . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Multiple Cylinders . . . . . . . . . . . . . . . . . . . . 81.2.3 Multiple Cylinders Near a Dielectric Half-Space . . . . 111.3 Sample Application: ANSOM . . . . . . . . . . . . . . . . . . 142 Gaussian Beam Scattering from a Dielectric Cylinder, In-cluding the Evanescent Region . . . . . . . . . . . . . . . . . . 18vi2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Incident Electric Field . . . . . . . . . . . . . . . . . . . . . . 202.2.1 Radiated Field . . . . . . . . . . . . . . . . . . . . . . 242.2.2 Evanescent Field . . . . . . . . . . . . . . . . . . . . . 252.3 Scattered and Transmitted Waves . . . . . . . . . . . . . . . 282.4 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . 292.5 Applying Boundary Conditions . . . . . . . . . . . . . . . . . 302.6 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 312.6.1 Numerical Analysis for the Incident Radiated Field . . 312.6.2 Numerical Analysis of the Evanescent Region . . . . . 352.6.3 Scattering Effect of the Evanescent Field Incident onthe Cylinder . . . . . . . . . . . . . . . . . . . . . . . 372.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Truncating Cylindrical Wave Modes in Two-Dimensional Mul-tiple Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Lower Bound Limit . . . . . . . . . . . . . . . . . . . . . . . 443.3 Upper Bound Limit . . . . . . . . . . . . . . . . . . . . . . . 463.4 Scattering Simulations . . . . . . . . . . . . . . . . . . . . . . 473.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Scattering from Cylinders Using the Two-Dimensional Vec-tor Plane-Wave Spectrum . . . . . . . . . . . . . . . . . . . . . 514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Incident Electric Fields . . . . . . . . . . . . . . . . . . . . . 544.2.1 Incident Radiated Fields . . . . . . . . . . . . . . . . . 584.2.2 Incident Evanescent Fields . . . . . . . . . . . . . . . 584.3 Scattered Fields . . . . . . . . . . . . . . . . . . . . . . . . . 594.4 Transmitted Fields . . . . . . . . . . . . . . . . . . . . . . . . 614.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 61vii4.6 T-Matrix Formulation . . . . . . . . . . . . . . . . . . . . . . 644.7 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 674.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735 Scattering from Cylinders Near a Dielectric Half-Space Us-ing a General Method of Images . . . . . . . . . . . . . . . . 745.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Algorithm Outline . . . . . . . . . . . . . . . . . . . . . . . . 765.3 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . 775.3.1 V i - Incident Field . . . . . . . . . . . . . . . . . . . . 785.3.2 V r - Incident Field Reflection . . . . . . . . . . . . . . 795.3.3 V s - Scattered Fields from Cylinders . . . . . . . . . . 825.3.4 V d - Diffracted Fields from the Planar Interface . . . . 825.3.5 V t Transmitted Fields Inside the Cylinders . . . . . . 865.3.6 Applying the Cylinders? Boundary Conditions . . . . . 865.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 895.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956 Multiple Scattering Between Cylinders in Two DielectricHalf-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.2 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . 1006.2.1 Vp Principal Field . . . . . . . . . . . . . . . . . . . . 1036.2.2 Vs Scattered Field from a Cylinder . . . . . . . . . . . 1056.2.3 Vc Field Inside a Cylinder . . . . . . . . . . . . . . . . 1066.2.4 Vr Reflection from Planar Interface . . . . . . . . . . . 1066.2.5 Vt Transmission through Planar Interface . . . . . . . 1076.2.6 Applying the Cylinders? Boundary Conditions . . . . . 1096.2.7 Stationary Phase Approximation . . . . . . . . . . . . 1126.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 1156.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123viii7 Apertureless Near-Field Scanning Optical Microscopy Sim-ulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247.2 The Reference Images . . . . . . . . . . . . . . . . . . . . . . 1267.3 Interpreting Collected Data . . . . . . . . . . . . . . . . . . . 1297.4 Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.5 Interference of Waves . . . . . . . . . . . . . . . . . . . . . . 1377.6 Tip-Object Interaction . . . . . . . . . . . . . . . . . . . . . . 1407.7 Probe to Object Distance . . . . . . . . . . . . . . . . . . . . 1427.8 Wavelength and Object Parameters . . . . . . . . . . . . . . 1437.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1478 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . 1498.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155AppendicesA Diffraction Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 168B Truncating Cylindrical Wave Modes for Very Small Cylin-ders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171C Convergence of Finite Element Method Comparison . . . . 173ixList of Tables2.1 Numerical Comparison between the Cartesian Formula (2.23)and the Cylindrical Wave Representation (2.22) for the Evanes-cent Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.1 The matrices are linked to their corresponding equations. . . . 88C.1 The mesh size parameters of the finite element method arecompared to the resulting error. . . . . . . . . . . . . . . . . . 173xList of Figures1.1 A basic imaging setup is shown. . . . . . . . . . . . . . . . . . 11.2 The progression of the scattering models is shown. . . . . . . . 41.3 A simple ANSOM imaging setup is shown. . . . . . . . . . . . 152.1 Coordinate system and layout for a Gaussian beam emittedfrom (y0, z0) at any angle ?in. The dielectric cylinder withradius a0 and permittivity ?2 is centred at the origin. Theexterior region has permittivity ?1. The x coordinate is per-pendicular to the y ? z plane. . . . . . . . . . . . . . . . . . . 202.2 The integration contour follows the path 1-2-3-4. The points1 and 4 extend infinitely, parallel to the imaginary axis. . . . . 232.3 Cylindrical wave coefficients for the evanescent field of a Gaus-sian beam centred about the origin with ? = 5 at a frequencyof 1 GHz. The coefficients are purely real. . . . . . . . . . . . 262.4 Spatial convergence of the evanescent field with increasingBessel modes N . The field is taken along the y? axis at z? = 0for a Gaussian beam with ? = 5 at a frequency of 1 GHz. . . . 272.5 Incident radiated part of a Gaussian beam with ? = 2.5 at afrequency of 1 GHz scattering from a dielectric cylinder withradius a0 = 0.3m and relative permittivity ? = 10. Displayedare the (a) incident field, (b) scattered field, (c) transmittedfield inside the cylinder, (d) total outside field (a)+(b), and(e) a scaled version of the transmitted field. (a), (b), (c), and(d) have the same scaling. . . . . . . . . . . . . . . . . . . . . 32xi2.6 Incident radiated part of a Gaussian beam with ? = 2.5 at afrequency of 1 GHz scattering from a dielectric cylinder withradius a0 = 0.3m and relative permittivity ? = 1000. Dis-played are the (a) incident field, (b) scattered field, (c) trans-mitted field inside the cylinder, (d) total outside field (a)+(b),and (e) a scaled version of the transmitted field. (a), (b), (c),and (d) have the same scaling. . . . . . . . . . . . . . . . . . . 332.7 Incident radiated part of a Gaussian beam with ? = 2.5 at afrequency of 1 GHz scattering from a dielectric cylinder withradius a0 = 0.03m and relative permittivity ? = 1000. Dis-played are the (a) incident field, (b) scattered field, (c) trans-mitted field inside the cylinder, (d) total outside field (a)+(b),and (e) a scaled version of the transmitted field. (a), (b), (c),and (d) have the same scaling. . . . . . . . . . . . . . . . . . . 342.8 Evanescent field along the y? axis at z? = 0 for a Gaussianbeam with ? = 5 at a frequency of 1 GHz. . . . . . . . . . . . 352.9 Evanescent field for a Gaussian beam with ? = 5 at a fre-quency of 1 GHz, generated from the Cartesian formula (2.23). 362.10 Evanescent field for a Gaussian beam with ? = 5 at a fre-quency of 1 GHz, generated from the cylindrical wave formula(2.22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.11 Evanescent part of a Gaussian beam with ? = 2.5 at a fre-quency of 1 GHz scattering from a dielectric cylinder with ra-dius a0 = 0.03m and relative permittivity ? = 1000. Displayedare the (a) incident field, (b) scattered field, (c) transmittedfield inside the cylinder, (d) total outside field (a)+(b), and(e) a scaled version of the transmitted field. (a), (b), (c), and(d) have the same scaling. . . . . . . . . . . . . . . . . . . . . 392.12 The scattered field from Figure 2.11 showing three time stamps(a)-(c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40xii2.13 The scattered field from Figure 2.11 displaying real and imag-inary parts. The sectional view is taken along the z? axis onthe far side of the cylinder. . . . . . . . . . . . . . . . . . . . . 402.14 Evanescent part of a Gaussian beam with ? = 2.5 at a fre-quency of 1 GHz scattering from a dielectric cylinder with ra-dius a0 = 0.3m and relative permittivity ? = 1000. Displayedare the (a) incident field, (b) scattered field, (c) transmittedfield inside the cylinder, (d) total outside field (a)+(b), and(e) a scaled version of the transmitted field. (a), (b), (c), and(d) have the same scaling. . . . . . . . . . . . . . . . . . . . . 413.1 The minimal accuracy mode limits Mv are displayed for kavfactors that are (a) small, (b) medium, and (c) large. . . . . . 453.2 Multiple scattering of a Gaussian beam from 11 perfectly con-ducting cylinders with broadside incidence. . . . . . . . . . . . 483.3 Multiple scattering of a Gaussian beam from 11 perfectly con-ducting cylinders with in-line incidence. . . . . . . . . . . . . . 483.4 Matrix condition number compared to the mode limit. Theresults are identical for broadside and in-line incidence. . . . . 494.1 Geometry and coordinate systems for the cylinders and theincident field. The x direction is perpendicular to the page. . . 544.2 Triangle showing the coordinates of transformation in the Grafaddition theorem (4.24) . . . . . . . . . . . . . . . . . . . . . . 604.3 Setup for the numerical simulations. The grating is made upof 41 conducting cylinders with a = 0.005m. The dimensionsare not drawn to scale. . . . . . . . . . . . . . . . . . . . . . . 674.4 Far-field intensity (W/m) plot for scattering of the incidentfield, with aperture distribution Eix(0, y?) = exp(?j1.1ky?),from the grating. . . . . . . . . . . . . . . . . . . . . . . . . . 68xiii4.5 Far-field intensity (W/m) plot for scattering of the incidentfield, with aperture distribution Eix(0, y?) = exp(?j1.4ky?),from the grating. . . . . . . . . . . . . . . . . . . . . . . . . . 694.6 Far-field intensity (W/m) plot for scattering of the incidentfield, with aperture distribution Eix(0, y?) = exp(?j1.7ky?),from the grating. . . . . . . . . . . . . . . . . . . . . . . . . . 694.7 Far-field intensity (W/m) plot for scattering of the incidentfield, with aperture distribution Eix(0, y?) = exp(?j2ky?), fromthe grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.8 Far-field intensity (W/m) polar plot for scattering of the evanes-cent field of a square shaped beam. . . . . . . . . . . . . . . . 714.9 Far-field intensity (W/m) plot for scattering of the evanescentfield of a square beam for ??/2 ? ? ? 0. . . . . . . . . . . . . 724.10 Absolute value of the spatial frequency content of the incidentevanescent field from k to 2k . . . . . . . . . . . . . . . . . . . 735.1 The coordinate systems and geometry for the scattering areshown. The z direction is normal to the page. Image cylinderv is not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 The integration contour C1 follows the path 1-2-3-4. Thecontour C2 includes the paths 1-2 and 3-4 only. The points 1and 4 extend infinitely, parallel to the imaginary axis. . . . . . 835.3 A Gaussian beam is scattered from three identical dielectriccylinders near a dielectric half-space. The incident Gaussianbeam has a waist size of w0 = 1, an amplitude factor of E0 = 1,and a frequency f = 1GHz (? ? 0.3m). . . . . . . . . . . . . . 905.4 The error in ~L between the Fourier series method with evanes-cent field corrections (5.39) and the plane-wave integral method(5.38) is shown for TM and TE polarizations. . . . . . . . . . 91xiv5.5 The error in ~L between the Fourier series method with (5.39)and without (5.40) evanescent field corrections is shown forTM and TE polarizations. . . . . . . . . . . . . . . . . . . . . 925.6 The error in ~L between the Fourier series method with andwithout the true evanescent field interaction term U2 is shownfor TM and TE polarizations. . . . . . . . . . . . . . . . . . . 946.1 The coordinate systems and geometry for the scattering areshown. The z direction is normal to the page. The indices wand i refer to cylinders in the first medium and the indices vand u refer to cylinders in the second medium. . . . . . . . . . 1016.2 The break down of electromagnetic fields is depicted. . . . . . 1026.3 The scattering setup for comparing the presented method withthe finite element method is shown. . . . . . . . . . . . . . . . 1166.4 The electric field norm across the planar interface is shownfor both the presented method and the finite element method.The two methods produce similar results. . . . . . . . . . . . . 1176.5 The scattering setup for two cylinders being moved away fromthe interface is shown. . . . . . . . . . . . . . . . . . . . . . . 1186.6 The error for the approximate matrices and the scattering co-efficients is given for the case when both cylinders are movedaway from the interface. . . . . . . . . . . . . . . . . . . . . . 1196.7 The error for the approximate matrices and the scattering co-efficients is given for the case when only one cylinder is movedaway from the interface. . . . . . . . . . . . . . . . . . . . . . 1206.8 The scattering setup for simulations involving four cylindersis shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.9 The error for the approximate matrices and the scattering co-efficients is given for the case when four cylinders are separatedfrom each other in the y direction. . . . . . . . . . . . . . . . . 121xv7.1 The scattering setup for the ANSOM simulations is shown butthe dimensions are not to scale. The z direction is normal tothe page. The wavelength in free space is ?. . . . . . . . . . . 1267.2 The probe-tip is scanned across the image plane and its scat-tered power is recorded to create simulated ANSOM images. . 1287.3 The electromagnetic intensity is compared to the electric fieldintensity across the image plane. TM polarization is used. . . 1307.4 The electromagnetic intensity is compared to the magneticfield intensity across the image plane. TE polarization is used. 1327.5 The first, second, and third harmonic signals are demodulatedusing homodyne detection. . . . . . . . . . . . . . . . . . . . . 1367.6 Altered ANSOM images are shown where the probe-tip sam-ples only the scattered field from the object. . . . . . . . . . . 1387.7 The ANSOM images for TM polarization with and withouttip-object coupling are shown. . . . . . . . . . . . . . . . . . . 1417.8 ANSOM images are shown for various distances separating thehalf-space and the probe. TE polarization was used. . . . . . . 1437.9 The ANSOM images for TE polarization with different inci-dent wavelengths are shown. . . . . . . . . . . . . . . . . . . . 1447.10 The ANSOM images for TE polarization with different objectcylinder depths are shown. . . . . . . . . . . . . . . . . . . . . 1457.11 The ANSOM images for TE polarization with different objectcylinder radii are shown. . . . . . . . . . . . . . . . . . . . . . 1457.12 The ANSOM images for TE polarization with different objectcylinder separation distances are shown. . . . . . . . . . . . . 146A.1 A basic imaging setup is shown, copied from Figure 1.1. . . . . 168xviList of AbbreviationsAFM atomic force microscopeANSOM apertureless near-field scanning optical microscopyDNG double negative materialsFFT fast Fourier transformGLMT generalized Lorenz-Mie theoryGMI general method of imagesGPR ground penetrating radarMRI magnetic resonance imagingNSOM near-field scanning optical microscopyPEC perfect electric conductorPEMC perfect electromagnetic conductorPMC perfect magnetic conductorTE transverse electricTM transverse magneticVPWS vector plane-wave spectrumxviiAcknowledgementsI would like to acknowledge the mentoring provided by my supervisor Dr.Matthew Yedlin throughout the development of my research. His insight andguidance have been a fundamental source of support for me over the past fewyears. I am also grateful for useful discussions with Daryl Van Vorst.Funding support for this research was provided by the Natural Sciencesand Engineering Research Council of Canada.xviiiTo my father Reg Pawliuk for encouraging me to pursue this research.xixChapter 1Introduction1.1 Motivation: Super-ResolutionFigure 1.1: A basic imaging setup is shown.A basic imaging setup is shown in Figure 1.1. The object to be studied isilluminated with waves by a source. The waves scatter from the object inall directions and some of these scattered waves are collected by the lensand refocused back onto the focal plane. The purpose of the imaging systemis to reproduce, on the focal plane, the scattered fields emanating from thesurface of the object. The wave distribution at the surface of the objectcontains valuable information about the size, shape, and properties of the1object. However, there are limitations on how well the imaging system canreproduce this distribution on the focal plane.All imaging systems that use propagating waves to collect data are subjectto a maximum resolution limit known as the diffraction limit. The diffractionlimit was first described by Ernst Karl Abbe in the nineteenth century [1].The diffraction limit defines the maximum resolution of an imaging systemto bed = ?2NA (1.1)where ? is the wavelength and NA = n sin(?) is the numerical aperture of theimaging system. The numerical aperture depends on n, the refractive indexof the background medium, and the half-angle ? shown in Figure 1.1. In thelimit as the aperture becomes infinitely large or half-encircles the object, thediffraction limit becomes d = ?/2. A derivation of the diffraction limit usingFourier optics is provided in Appendix A.Although the resolution at the focal plane is subject to the diffractionlimit, the resolution of the wave-field at the surface of the object is unlimited.The wave-field must be capable of taking on an arbitrary form with unlimitedspatial resolution in order to satisfy the object?s boundary conditions duringthe scattering. Unlimited resolution cannot be satisfied by radiative wavesalone, which have a maximum spatial frequency that is determined by theirwavelength. However, unlimited higher spatial frequencies are possible withevanescent waves.Evanescent waves are formed during scattering, and travel along the sur-face of objects. Evanescent waves are super-oscillatory along the source orscattering surface from which they originate; this means that their spatialfrequency exceeds the wavenumber k [2]. It is this super-oscillatory propertythat allows evanescent waves to provide super-resolution. Away from thesource or scattering surface, the evanescent waves die away exponentially.Evanescent waves store energy locally and do not transfer energy outwardlike radiation. Thus, the evanescent waves cannot be collected at the lens nor2recovered at the focal plane. Without these evanescent waves, an imagingsystem is said to be diffraction limited and cannot obtain resolution beyondd = ?/2.If the data contained in the evanescent waves could be recovered, theresolution of an imaging system could be extended indefinitely. There arethree primary methods for recovering evanescent field data: near-field de-tection, metamaterial super-lenses, and scattering. Near-field detection issimply attempting to measure the wave fields extremely close to the surfaceof the object where the evanescent waves have a significant presence. Meta-material super-lenses are man-made materials that exhibit negative values ofpermittivity and permeability [3]. These super-lenses are able to convert theexponential decay of evanescent waves into exponential growth. The focus ofthis thesis is on the third method: the conversion of evanescent waves intoradiation through scattering.When an evanescent wave is incident upon a scattering body, part ofits energy can be converted into radiation. This radiation can be collectedin the far-field and used to recover the evanescent wave?s super-resolutiondata. Several scattering models are developed to investigate and charac-terize the scattering of evanescent waves into radiation for applications insuper-resolution imaging and focusing. The models are based on the two-dimensional scattering from cylinders.1.2 Scattering ModelsThe scattering models developed in this thesis are analytical or a combina-tion of analytical and numerical components. Analytical models were chosenbecause they provide more insight into the scattering interactions than purelynumerical models. This is important when we are trying to characterize aspecific interaction like the conversion of evanescent waves into radiation.Also, these analytical models do not require a spatial discretization, which3allows them to easily handle subwavelength phenomenon without resolutionlimits. The far-field data can also be obtained easily from the scatteringsolutions.Figure 1.2: The progression of the scattering models is shown.There are four scattering models that build on each other: the scatter-ing from one cylinder in Chapter 2, the scattering from multiple cylindersin Chapter 4, the scattering from cylinders near a dielectric half-space inChapter 5, and the scattering from cylinders on both sides of a dielectrichalf-space in Chapter 6. The progression of the scattering models is out-lined in Figure 1.2. Finally, the scattering models are applied to ANSOMsuper-resolution imaging simulations in Chapter 7.Building on the work of previous authors, these new scattering modelsintroduce the scattering of evanescent fields and the extension to includecylinders on both sides of a dielectric half-space. In Chapters 2 and 4, inci-4dent field coefficients for evanescent fields are derived for the first time. InChapters 5 and 6, the scattering models are extended to include a perme-able half-space for simulating the imaging of buried objects or objects on asurface. There is a particular emphasis on near-field interactions and themodelling of evanescent waves.1.2.1 Single CylinderAnalytical solutions to Maxwell?s equations for the scattering of electromag-netic waves from simple geometrical objects such as cylinders and sphereshas been considered since the beginning of the twentieth century. The firstanalytical solutions for the scattering of a plane-wave from a sphere were pro-duced by Mie [4] and Lorenz [5]. Mie produced his solution to the scatteringproblem in 1908, deriving his work from Maxwell?s equations. Previously in1890, Lorenz had produced an equivalent solution but had derived his workfrom a mechanical theory of aether [6]. Originally Mie theory, or Lorenz-Mietheory, referred to their analytical solution to the scattering from sphericalparticles. Today referring to Mie theory, or Lorenz-Mie theory, indicates thatthe scattering solution is based on the exact Maxwell?s equations and an ap-plication of boundary conditions that involves separation of the coordinates.The advantages of such a solution include a fast computational implemen-tation, an unbounded spatial domain, and a field representation in terms ofscattering coefficients.The first exact solutions of Maxwell?s equations for the scattering froma single circular cylinder were presented by Lord Raleigh in 1918 [7]. Sincethen authors have considered many variations of this single scattering sce-nario. Cylinders with different electromagnetic properties have been con-sidered, including conductors, homogeneous and inhomogeneous dielectrics,ferrite cylinders, meta-materials, anisotropic cylinders, cylindrical shells, andmultilayered cylinders [8?10]. Variations on the geometrical setup, includ-ing oblique incidence, off-axis beam incidence, and elliptical cylinders, have5also been considered [11?14]. In each case the authors followed the samegeneral approach that was instituted by Mie when he derived the scatteringof a plane-wave from a sphere [4]. The exact Maxwell?s equations are ap-plied to the fields and the boundary conditions are satisfied by separatingthe coordinates tangential to the surface of the cylinders.The simplest case to consider for the scattering problem is that of plane-wave incidence. The incident plane-wave has a simple representation as aweighted sum of Bessel functions in cylindrical coordinates. For cylindricalscattering it is essential that the incident fields be expressed as an infinite sumof cylindrical wave modes using Bessel functions. This allows the boundaryconditions to be applied to each mode separately, leading to a closed formsolution for the scattering coefficients. The case of beam incidence is morecomplicated. For Gaussian beams, there is only an analytical solution for theparaxially approximated beam and its higher order corrections [15,16]. Thisis due to the fact that Maxwell?s equations cannot be solved exactly for such abeam. An exact representation for a Gaussian beam is possible through use ofthe plane-wave spectrum technique which represents an arbitrary electric fieldas an infinite sum of homogeneous and inhomogeneous plane-waves travellingin various directions [17]. The plane-wave spectrum integral, however, mustbe solved numerically as no closed form solution exists.For the two-dimensional scattering from cylinders, a two-dimensionalGaussian beam is used. The two-dimensional beam is uniform in the di-rection of the cylinder axis. This simplification exploits the fact that thecylinder is also taken to be infinite and uniform along its axis. Using a threedimensional Gaussian beam severely complicates the scattering problem byintroducing field components that are incident at an oblique angle to thecylinder.Gaussian beam incidence upon an infinite circular cylinder was first con-sidered by Alexopoulos and Park in 1972 [18]. Early work with Gaussianbeams was difficult because the incident beam coefficients had to be numer-6ically computed as sums or integrals [12, 18]. In 1982 Kozaki [19] found away to calculate the incident beam coefficients for a paraxial Gaussian beamusing a closed form equation. The speed and accuracy of using this closedform expression over the previous integration or summation methods wasimmense. Kozaki [19, 20] also provided closed form equations for a secondorder paraxial approximation, and indicated a method for obtaining higherorder corrections. However, the higher order corrections become tedious andinaccurate as the beam waist approaches the wavelength of radiation. Theparaxial approximation, even at higher orders of correction, still completelyeliminates the evanescent region and replaces it with erroneous results. Foranalysis of the evanescent fields, which become more significant as the beamwaist decreases relative to the wavelength, the closed form solution providedby Kozaki cannot be used. Some authors noted this important limitation intheir work and chose not to use a paraxial approximation [13].In Chapter 2 an analysis is made of the full radiative and evanescentregions of a Gaussian beam scattering from a dielectric cylinder. Previousauthors focused primarily on the scattering of radiative waves from a cylinder,whereas the focus of this analysis is on evanescent wave incidence. Incidentfield coefficients are developed for evanescent fields and their properties areexamined. In the numerical simulations, the conversion of incident evanes-cent waves into radiation through scattering is demonstrated. This conver-sion is the mechanism responsible for spatial super-resolution. The use of aGaussian beam was motivated by its ability to model more complicated fieldpatterns by acting as a basis function [21]. However, if evanescent fields areto be included in the analysis of the Gaussian beam, then the plane-wavespectrum method must be used. The plane-wave spectrum then requires nu-merical integration which eliminates the benefit of using Gaussian beams asbasis functions. Therefore, in our next model we use the vector plane-wavespectrum (VPWS) to create arbitrary radiative and evanescent fields.71.2.2 Multiple CylindersMultiple scattering problems are significantly more involved than the singlescattering case. In the single scattering case there are only two boundarycondition equations and two unknowns, the scattering and transmission co-efficients. Each cylindrical wave mode reacts independently in the scattering,allowing for a closed form solution. In the multiple scattering case we havean infinite matrix solution which accounts for the interaction of radiationcoupling between cylinders. The use of the Graf addition theorem, for shift-ing cylindrical functions from one origin to another, causes the translatedscattered fields to contain two infinite sums. Only one of these sums cancelsin the application of the boundary conditions, leaving the other sum in thesolution. The multiple scattering solution relates the scattering coefficientsof each cylinder to those of every other cylinder.The general solution to the multiple scattering of radiation from circu-lar cylinders was first proposed by Twersky in 1952 [22]. Twersky?s firstmathematical derivation used a simplified approach where all the cylinderswere perfectly conducting, although he mentions that his methods could beextended to include any cylinder type. His solution sought to calculate theexcitation of one cylinder by combining the incident field and the scatteredfields produced by the other cylinders. The boundary conditions are appliedaround each cylinder simultaneously and include the contributions from thescattered fields due to the presence of the other cylinders. The solution tothe set of boundary condition equations is presented as an iterative proce-dure where each successive order of scattering is calculated from the previousorder. The matrix solution was introduced in later work (1965) by Burke,Censor, and Twersky [23]. However, they applied the matrix solution to amore complicated scenario where the cylinders were of arbitrary shape ratherthan circular.In 1970 Olaofe solved the boundary value problem for multiple circularcylinders to yield a solution in terms of an infinite set of linear equations [24,825]. Olaofe states that these equations can be solved using matrix methodsor iterative methods, as was done by Twersky [22,23]. The general equationfor the infinite set of linear equations is presented but it is not broken downinto a matrix solution. Olaofe used his solution to derive equations for theextinction, back-scattering cross section, and total-scattering cross sectionfor the multiple scattering from circular cylinders.Experimental results were compared to numerical simulations by Youngand Bertrand in 1975 [26]. The multiple scattering from two cylinders withplane-wave illumination was investigated using the matrix inversion method,the iterative method and direct experimental measurement. They reporteda good agreement between their calculations and their measurements. Theynoted that the matrix method was more efficient than the iterative method.The efficiency and accuracy of the matrix method in computing multiplescattering problems was confirmed by Elsherbeni [27] who compared it to theiterative method and two other computational electromagnetic techniques.The matrix method was shown to be the most flexible technique, as it canbe applied to any scenario regardless of the cylinder radii and separationdistance.Later, these methods were expanded upon by many authors. The matrixmethod for multiple circular conducting cylinders was explicitly derived andnumerically simulated by Ragheb and Hamid in 1985 [28]. They used thematrix method as a reference to compare other approximate methods to.Bever and Allebach [29] performed numerical simulations for the case of aplanar array of dielectric cylinders. They used their results to investigatethe convergence of the iterative method and the conditioning of the matrixmethod. Polewski [8] extended the scattering of a plane-wave by cylindersto include the effects of conducting, lossy dielectric, ferrite, and pseudochiralcylinders. Another advanced extension was given by Henin et al. [30], whoextended the scattering geometry to include oblique incidence between theplane-wave and the cylinder axes.9To compute the scattering from multiple cylinders, the number of cylin-drical wave modes used to model the scattered fields needs to be truncated.In Chapter 3 the truncation of cylindrical wave modes is considered in termsof model accuracy and matrix conditioning. If the number of modes is trun-cated too soon, the accuracy of the solution will be compromised. In multiplescattering, if the number of modes is truncated too late, the matrix inversionwill become ill-conditioned, producing erroneous results. Therefore, in or-der to maximize the accuracy of the solution, it is necessary to have propermodal truncation. Since, the modal truncation is primarily dependant onthe size of the cylinder relative to the wavelength, it may be necessary tohave separate truncation limits for each cylinder involved in the scattering.This analysis is applicable to all cylindrical scattering models and helps tomaximize the accuracy of the solutions.Gaussian beam illumination in the multiple scattering from circular cylin-ders was first investigated by Kojima et al. [31], and also by Sugiyama andKozaki [32]. Sugiyama and Kozaki used the matrix inversion technique tocalculate the scattering of a Gaussian beam from two cylinders of differentradii. They also performed experiments to verify their results. Kojima et al.investigated higher order beam modes [31]. For a Gaussian beam scatteringfrom dielectric cylinders, Yokota et al. [33] derived the matrix solution andapplied it to a scenario with eight cylinders. Elsherbeni et al. also consideredthis problem but used the iterative procedure [34]. Most authors used theparaxial Gaussian beam representation which contains no evanescent fieldcomponents [31?33]. Only Yang et al. [35] provided a means of including theevanescent components into their multiple scattering solution.Using plane-waves and Gaussian beams, previous authors have primarilyfocused on radiative incidence. In Chapter 4 the vector plane-wave spec-trum (VPWS) is scattered from multiple dielectric and conducting cylinders.The VPWS allows for both polarization states and arbitrary radiative andevanescent components in the incident field. The scattering of evanescent10waves from arrays of cylinders is particularly important for applications insuper-resolution imaging and microwave source location. Fink et al. [36]and Lerosey et al. [37] demonstrated that an array of cylindrical scattererscould partially convert an evanescent field into radiation and vice-versa. Byusing time-reversal techniques, they were able to differentiate signals sentto antennas that were spaced only ?/30 apart. Malyuskin and Fusco [38]also demonstrated subwavelength source resolution using cylindrical near-field scatterers.1.2.3 Multiple Cylinders Near a Dielectric Half-SpaceTo simulate a more realistic imaging setup, a dielectric half-space was incor-porated into the scattering model. This model allows the imaging of objectson top or inside of a substrate to be simulated. The scattering from buriedcylinders is important for applications such as ground penetrating radar(GPR), detection of underground landmines, tunnels, conduits, and pipes,underground communications, and biological imaging [39?41]. In Chapter7, the scattering models we present will be applied to imaging simulationsof apertureless near-field scanning optical microscopy (ANSOM), which isdescribed in Section 1.3.The introduction of a perfectly conducting plane into the multiple scat-tering geometry is easily incorporated through the method of images. Themethod of images allows the multiple scattering between the conductingplane and the cylinders to be modelled by the interaction of the real cylin-ders with image cylinders. Twersky used the method of images to model asingle cylinder above a conducting plane as the scattering from two cylin-ders [42]. Bertrand and Young [43] also used the method of images forcylindrical scattering and compared it to experimental results. The scatter-ing from cylinders partially buried in a conducting plane was investigated byRao and Barakat [44, 45].For dielectric half-spaces, the reflection of waves depends on the angle of11incidence. Vdeen and Ngo [46] used an approximate method of images fordielectric half-spaces, where all the incident waves were assumed to strike atnormal incidence. This approximation is only accurate when the cylinders aresufficiently separated from the planar interface. Coatanhay and Conoir [47,48] developed a more accurate general method of images (GMI) for cylindersin front of a dielectric half-space. The GMI uses a Fourier series to convertthe plane-wave reflection coefficients for the planar interface into a cylindricalform. In this form, a modified image cylinder can be represented usinga cylindrical reflection coefficient matrix. This solution accounts for theangular dependence of the reflections from the planar interface but is limitedfor evanescent waves. Depending on the form of the plane-wave reflectioncoefficients, the Fourier series will only converge for a certain range of thecomplex angles that arise in evanescent waves.A solution that is valid for all evanescent wave interactions is made possi-ble through the use of the plane-wave integral method. The Sommerfeld inte-gral is used to convert cylindrical waves into a sum of plane-waves [49], so thatthe planar reflection coefficients can be applied directly. This method is accu-rate for small distances between the cylinders and the dielectric half-space,however, it requires a significant amount of numerical integration [50, 51].Frezza et al. [52] presented the scattering from multiple cylinders near avacuum-plasma interface using a general approach that could be applied toany reflective planar surface. The scattering of a single conducting cylinderin front of a generally reflective planar surface was presented by Borghi etal. [53] and subsequently extended to multiple cylinders [54].In Chapter 5 the Fourier series method is derived along side the plane-wave integral method in order to compare the two methods. Using a Fourierseries to transform the reflection coefficients is applied in vector electromag-netic scattering for the first time. The Fourier series method is much fasterand more intuitive to implement than the plane-wave integration method,but may cause errors for evanescent waves coupling between the cylinders12and the planar interface. Evanescent field correction terms are derived forthe Fourier series method to correct for these errors. Simulations for bothpolarizations demonstrate the convergence of the Fourier series method asthe cylinders are separated from the interface.The case of a buried cylinder can be considered if the incident field origi-nates in the half-space that does not contain the cylinders. Ciambra et al. [40]considered a cylindrical wave approach for the scattering from a single cylin-der buried in a dielectric half-space. This was extended to multiple cylindersby Di Vico et al. [55]. The scattering of a perfect electromagnetic conductor(PEMC) cylinder buried in a dielectric half-space was considered by Ahmedand Naqvi [39]. Lee and Grzesik [56] derived a solution for the scatteringfrom multilayered cylinders buried in a dielectric half-space with oblique in-cidence. For the buried case, the incident field must be transmitted into thesecond half-space before the scattering is considered. The transmission ofthe cylinders? scattered fields back into the first half-space is also consideredusing the Sommerfeld integral to convert cylindrical waves into plane-waves.The scattering from cylinders embedded in a finite dielectric slab is evenmore complicated because of the infinite number of reflections within the slab.Frezza et al. derived an algorithm for the scattering from multiple cylindersembedded in a dielectric slab, for both conducting [57] and dielectric [41]cylinders. In these algorithms, the number of multiple reflections withinthe dielectric slab must be truncated to a finite amount. Lee [58] derived asimilar solution for the scattering from multilayered cylinders embedded ina dielectric slab with oblique incidence.The scattering from cylinders on both sides of a dielectric half-spaces isintroduced in Chapter 6. In previous work, all the cylinders have been con-tained in one of the two half-spaces or inside of a slab. This new scatteringalgorithm accounts for the multiple scattering interactions between cylinderson opposite sides of a dielectric half-space. To avoid the numerical integrationthat arises from the plane-wave spectrum of cylindrical waves, the method13of stationary phase is employed. This method provides an accurate approxi-mation when the cylinders are sufficiently separated from the interface. Thenumerical simulations demonstrate the accuracy of the approximation in dif-ferent scenarios.The final model presented in Chapter 6 is suitable for simulating ANSOMwith a probe on one side of the dielectric half-space and buried objects on theother side. These simulations are presented in Chapter 7, and are used as anexample of how these models can be useful in super-resolution applications.There are many other applications for these models including the analysis ofmetamaterials and photonic crystals, and ground penetrating radar (GPR)simulations, which are discussed briefly in Section 8.1.1.3 Sample Application: ANSOMIn 1928, Synge [59] proposed a subwavelength imaging system that used asmall hole punched in a metallic plate. A subwavelength portion of an ob-ject could be illuminated by placing the plate in the near-field of the object.These early subwavelength images were produced using near-field scanningtechniques at microwave frequencies. Ash and Nicholls [60] built a near-fieldscanning microscope that used a small hole in a thin diaphragm to illumi-nate a subwavelength portion of an object. The object was vibrated at aspecific frequency in order differentiate the small scattering effect amid thelarge background radiation signal. Their experiments demonstrated resolu-tion capabilities of ?/60. They concluded with the suggestion of adaptingtheir techniques for infrared or optical frequencies.Near-field scanning optical microscopy (NSOM) techniques were devel-oped later using subwavelength illumination or collection methods [61]. Forthe illumination method, a small aperture is fabricated at the end of an op-tical fiber, which allows objects to be illuminated by a subwavelength focalspot. The aperture is scanned across the image and the scattered power from14the object is collected to recreate an image. For the collection method, theobjects are illuminated from the far-field and the small aperture is used tocollect the scattered fields from a subwavelength spot on the objects. Thesize of the subwavelength aperture determines the resolution of the devicebut is limited by the skin depth of the metal aperture. The apertures arescanned across a two-dimensional plane and the data is collected one pixelat a time. Three imaging modes can be used in NSOM: constant height,constant intensity, and constant distance [61, 62]. In the constant heightmode, the probe is scanned across a flat plane above the base of the object.In constant intensity mode, the probe is forced to adjust its height abovethe object such that the measured intensity is kept constant. In constantdistance mode, the probe follows the contours of the object with the separa-tion distance being fixed. The resolution of these methods is in the range of30-100nm [63?65].Figure 1.3: A simple ANSOM imaging setup is shown.Apertureless near-field scanning optical microscopy (ANSOM) is the most15powerful of the subwavelength imaging techniques because the probe-tip doesnot need an aperture, and therefore it can be much finer [63, 66]. A basicimaging setup for ANSOM is shown in Figure 1.3. In ANSOM, the probe isjust a metallic or semiconductor rod with an extremely fine tip. This rod isvibrated in the direction normal to the object?s surface so that the scatteredpower from the probe-tip can be demodulated and distinguished from thebackground signal. An atomic force microscope (AFM) is used to perform themodulation and to calculate the distance from the probe-tip to the surface ofthe object. When used together, the AFM and ANSOM techniques provideboth the topographical and optical properties of the object [66]. Surfaceplasmon waves are often used to obtain a field enhancement at the apex ofthe probe-tip, which greatly improves the signal to noise ratio [67]. Theresolution limits of ANSOM have been reported around 3-10nm [63,64].The net effect of the demodulation in ANSOM is believed to be propor-tional to the scattered power emanated by the probe-tip [68,69]. The probe-tip scatters light from the extreme near-field of the object, which containsstrong evanescent field components. These evanescent fields are convertedinto radiation by the probe, transmitted to the far-field, and recorded by thedetector. The evanescent field conversion is responsible for the increase inspatial resolution of the images. The background signal is much greater thanthe small modulation of the probe-tip?s scattered field, which makes the datacollection difficult. By adding a strong reference signal with a controllablephase, the background signal can be suppressed through homodyne detection.This method of measurement also allows for phase recovery [70, 71]. Usinga florescent tip in order to separate the probe-tip?s scattered field throughfrequency differentiation has also been proposed [65].Subwavelength images are useful in many fields including research, medicine,and manufacturing. Because of their ability to extract electromagnetic fieldswith high precision, they have been useful for characterizing surface plasmonpolaritons in nano-devices [64]. This has been particularly important for16chemical and biochemical sensors that depend on plasmon resonance shiftsand local field enhancements [71].In Chapter 7, the cylindrical scattering models are used to provide simula-tions of ANSOM images. The insight provided by these analytical scatteringmodels is exploited to characterize and interpret the images.17Chapter 2Gaussian Beam Scattering froma Dielectric Cylinder, Includingthe Evanescent Region2.1 IntroductionIn this paper we will investigate the scattering of a two-dimensional Gaussianbeam from a homogeneous dielectric cylinder. The effects of the evanescentfield incident on the cylinder will be a key aspect of our analysis.Much work on the scattering of electromagnetic waves from cylindersand spheres has already been conducted. In 1908, Gustav Mie publisheda famous paper on the interaction of a plane-wave and a sphere [4]. Be-cause of a prior version of similar work by Lorenz [5], the theory becameknown as Lorenz-Mie theory, or in the case of an arbitrary beam, generalizedLorenz-Mie theory (GLMT). An in-depth history of the GLMT has beenproduced by Gouesbet [6]. A general theoretical framework for the scatter-ing of an arbitrarily shaped beam by an infinite cylinder has also been givenby Gouesbet [72]. Practical Gaussian beam models with simulations havebeen produced by Ren et al. [73] and Mees et al. [74]. These authors use theDavis framework for representing the Gaussian beam. The Davis frameworkdoes not satisfy Maxwell?s equations, but it approaches a correct solution toMaxwell?s equations at higher orders [73]. This paper does not utilize theDavis framework.Gaussian beams are most commonly analyzed in their paraxial form. This18form may be derived by approximating a plane-wave spectrum as being fo-cused over a small angular range [75]. Many previous authors [9,10,12,20,76]have ignored the evanescent field by using such a paraxial approximation.This approximation eliminates the evanescent field entirely. Other authors[13,34] used a plane-wave spectrum approach that did not include a descrip-tion of the beam coefficients An in the evanescent region. Yang et al. [35]mentions that the evanescent field scattering from a dielectric cylinder cre-ates propagating harmonics. However, they conclude that the evanescentfield effects are insignificant for their application and can be ignored.In this paper we will fully analyze the cylindrical beam coefficients Anso that we can compute the full scattering effect of the evanescent field.This computation is important when the evanescent field is converted intoa significant propagating field. For completeness we will allow the beam tohave any initial offset (y0, z0) and any angle of propagation ?in. The layoutand coordinate system are shown in Figure 2.1. We will begin by derivingsolutions to the wave equation for the incident, scattered, and transmittedwaves. Then to solve the scattering problem we will equate the electromag-netic boundary conditions at the cylinder surface ? = a0. The final resultswill include numerical simulations of the scattering of the radiated field andof the evanescent field.19Figure 2.1: Coordinate system and layout for a Gaussian beam emitted from(y0, z0) at any angle ?in. The dielectric cylinder with radius a0 and permit-tivity ?2 is centred at the origin. The exterior region has permittivity ?1. Thex coordinate is perpendicular to the y ? z plane.2.2 Incident Electric FieldThe incident electric field can be represented by a plane-wave spectrum witha Gaussian aperture distribution [17]. First we start with a two-dimensionalplane-wave spectrum equation for a wave travelling in the z? direction froman arbitrary point (y0, z0) as shown in Figure 2.1. For now we will use thealtered coordinate system (x, y?, z?). We will be using transverse magnetic(TM) polarization with the electric field in the x direction tangential to thecylinder at all points. In the three-dimensional Gaussian beam case, a planarpolarization cannot satisfy Maxwell?s equations [77]. Since we are using atwo-dimensional Gaussian beam, we are able to use a planar polarization and20still satisfy Maxwell?s equations.Our plane-wave spectrum is given by (2.1), where ? and ? are the y? andz? components of the wave-vector k, respectively:Eix(y?, z?) =exp(j?t)2?? ???F (?) exp[?j(?y? + ?z?)]d?. (2.1)When |?| > k, we have entered the evanescent region where the plane-waves are inhomogeneous and decay exponentially away from the y? axis[78, 79]. This exponential decay in the direction of propagation means thatthe evanescent field will be significant in magnitude only close to the beamorigin. To ensure that the evanescent field decays exponentially we need toadopt the correct sign conventions when we define ? in (2.2). Note that theevanescent region is only valid in one half-space z? ? 0:? =????k2 ? ?2 |?| ? k?j??2 ? k2 |?| > k z? ? 0(2.2)The function F (?) is known as the angular spectrum of the field and canbe computed [78,79] by taking the inverse Fourier transform of the aperturedistribution f(y?):F (?) =? ???f(y?) exp(j?y?)dy?. (2.3)The aperture distribution is the electric field taken across the y? axis at z? = 0,orf(y?) = Eix(y?, 0). (2.4)We will define our Gaussian aperture distribution asf(y?) = E0 exp[?(?y?)2]. (2.5)Here ? is the inverse of the beam width at (y0, z0) and E0 is the amplitude21constant. Using our Gaussian distribution (2.5) to calculate the angularspectrum (2.3) we obtain the following result:F (?) = E0??? exp(??24?2 ). (2.6)Now we substitute (2.6) into our plane-wave spectrum (2.1):Eix(y?, z?) =E0 exp(j?t)2???? ???exp[? ?24?2 ? j(y?? + z??k2 ? ?2)]d?. (2.7)Equation (2.7) contains the full description of a two-dimensional Gaussianbeam in Cartesian coordinates. This representation satisfies Maxwell?s equa-tions, since its divergence is zero and it obeys the Helmholtz equation.Holding to our sign conventions in (2.2) we will now convert our variableof integration in (2.7) into an angular function:sin(?) = ?k , (2.8)withcos(?) =??????1? sin2(?) sin2(?) ? 1?j?sin2(?)? 1 sin2(?) > 1 z? ? 0, (2.9)to obtainEix(y?, z?) =E0k exp(j?t)2??? ? (2.10)? ?/2+j???/2?j?exp{?k2 sin2(?)4?2 ? jk[y? sin(?) + z? cos(?)]} cos(?)d?,where the integration contour is shown in Figure 2.2.22Figure 2.2: The integration contour follows the path 1-2-3-4. The points 1and 4 extend infinitely, parallel to the imaginary axis.Now we will transform the coordinates (x, y?, z?) into (x, ?, ?) using (2.11)and (2.12) [35, 80]:y? = ?[? sin(?)? y0] cos(?in) + [? cos(?)? z0] sin(?in), (2.11)z? = ?[? sin(?)? y0] sin(?in)? [? cos(?)? z0] cos(?in). (2.12)After simplifying the exponentials using trigonometric identities we obtainEix(?, ?) =E0k exp(j?t)2???? ?/2+j???/2?j?exp[?k2 sin2(?)4?2 + jk? cos(? ? ?in ? ?)?jk?0 cos(?0 ? ?in ? ?)] cos(?)d?. (2.13)According to the restrictions in (2.9), the electric field in (2.13) is valid onlyin the region z? ? 0. The equation will be invalid for z? < 0 because theevanescent field will not decay exponentially in that region.23Using Bessel function expansions [81], we can define the following identity:exp[jk? cos(? ? ?in ? ?)] =??n=??jn exp[jn(? ? ?in ? ?)] Jn(k?). (2.14)Now we can represent our Gaussian beam as a sum of cylindrical waves (2.15)with weights (2.16):Eix(?, ?) = exp(j?t)??n=??jn exp(jn?) Jn(k?)An, (2.15)An =E0k exp(?jn?in)2??? ? (2.16)? ?/2+j???/2?j?exp[?k2 sin2(?)4?2 ? jn?? jk?0 cos(?0 ? ?in ? ?)] cos(?)d?.2.2.1 Radiated FieldThe radiated field corresponds to |?| ? k or | sin(?)| ? 1, so we can separatethe radiated portion of the beam shape coefficients (2.16) from the evanescentpart:Aradn =E0k exp(?jn?in)2??? ? (2.17)? ?/2??/2exp[?k2 sin2(?)4?2 ? jn?? jk?0 cos(?0 ? ?in ? ?)] cos(?)d?.Inside the radiated region there are only real angles so all the functionsare bounded. There is no analytical solution to the integral in (2.17) so itmust be computed numerically.242.2.2 Evanescent FieldThe evanescent field coefficients corresponds to |?| > k or | sin(?)| > 1 in(2.16):Aevann =E0k exp(?jn?in)2??? ? (2.18)? ?/2+j??/2exp[?k2 sin2(?)4?2 ? jn?? jk?0 cos(?0 ? ?in ? ?)] cos(?)d?+E0k exp(?jn?in)2??? ?? ??/2??/2?j?exp[?k2 sin2(?)4?2 ? jn?? jk?0 cos(?0 ? ?in ? ?)] cos(?)d?.We can simplify and combine the integrals in (2.18 by transforming thevariables of integration. In the positive region (?/2, ?/2 + j?) we will usetransformation (2.19), and in the negative region (??/2,??/2?j?) we willuse transformation (2.20):? = ?2 + ju, (2.19)? = ??2 ? ju. (2.20)After simplifying (2.18) we obtainAevann =E0k exp(?jn?in)2??? ? (2.21)? ?0exp[?k2 cosh2(u)4?2 ] sinh(u){j?n exp[nu? jk?0 sin(?0 ? ?in ? ju)]+jn exp[?nu+ jk?0 sin(?0 ? ?in + ju)]}du.The coefficients in the evanescent region Aevann increase exponentially forhigher modes n. This behaviour is caused by the exponential growth of thefield in the region z? < 0. However, the field is still valid in the z?0 region.We can see the general form of the coefficients jnAevann , which are purely real,25in Figure 2.3.Figure 2.3: Cylindrical wave coefficients for the evanescent field of a Gaussianbeam centred about the origin with ? = 5 at a frequency of 1 GHz. Thecoefficients are purely real.When numerically evaluating the evanescent field, the number of modesmust be limited to some maximum value N :Eevanx (?, ?) = exp(j?t)N?n=?Njn exp(jn?) Jn(k?)Aevann . (2.22)The evanescent field in the z? ? 0 region converges spatially as the numberof modes N increases. This means that for a given number of modes N ,the evanescent field given by (2.22) will converge within a specific radius? < ?conv. In Figure 2.4 we can see that as N is increased the field convergesspatially outward. This convergence is similar to the convergence of theTaylor series of a sine or cosine. The spatial convergence of the evanescent26field makes it hard to compute the field at large distances from the origin.However, in general, the field in this region will have died off exponentiallyand become insignificant.Figure 2.4: Spatial convergence of the evanescent field with increasing Besselmodes N . The field is taken along the y? axis at z? = 0 for a Gaussian beamwith ? = 5 at a frequency of 1 GHz.As a reference, for accurate modelling of the entire evanescent region, wecan use Cartesian coordinates. To obtain the evanescent field in Cartesiancoordinates from (2.7) we limit the range of integration to |?| > kEevanx (y?, z?) =E0 exp(j?t)???? ?kexp(? ?24?2 ?z???2 ? k2) cos(y??)d?. (2.23)The advantage of using Cartesian coordinates is that the field no longer hasspatial convergence issues. Our equations of coordinate transformation from27(x, y?, z?) to (x, y, z) are given byy? = ?(y ? y0) cos(?in) + (z ? z0) sin(?in), (2.24)z? = ?(y ? y0) sin(?in)? (z ? z0) cos(?in). (2.25)To evaluate (2.23) the integral needs to be integrated numerically, as thereis no analytical solution. For numerical integration, the upper integrationlimit must be truncated to k ? ? ? ?max, where ?max depends on the beamwidth variable ?. Larger values of ? will demand a larger ?max value foraccurate results.2.3 Scattered and Transmitted WavesThe general solution of the wave equation in cylindrical coordinates is givenbyE = exp(?j?t) exp(?jn?)[R Jn(k?) + S Yn(k?)]. (2.26)Both the transmitted and scattered fields must be a weighted sum of cylin-drical waves of this form.For the transmitted field we must have S = 0 because Yn is divergent forarguments approaching zero [82]. Therefore, the transmitted field (2.27) willhave only Jn terms with weights jnCn:Etx = exp(j?t)??n=??jn exp(jn?) Jn(??k?)Cn, (2.27)? = ?2/?1. (2.28).The scattered field is composed entirely of outward travelling waves that28are known to be represented by Hankel functions, where H(2)n (x) = Jn(x) ?j Yn(x). The scattered field is represented as a sum of Hankel functions withweights jnBn:Esx = exp(j?t)??n=??jn exp(jn?) H(2)n (k?)Bn. (2.29)2.4 Magnetic FieldsTo obtain the magnetic fields we use the Maxwell-Faraday law:?? E = ??B?t . (2.30)Since we have only an x component in the electric field, the curl simplifiesto?? E = (1??Ex?? )??+ (??Ex?? )??. (2.31)After computation the resultant magnetic fields are as follows:Hi =?exp(j?t)?????n=??jnn exp(jn?) Jn(k?)An?? (2.32)?exp(j?t)k????n=??jn+1 exp(jn?)Jn?(k?)An??,Hs =?exp(j?t)?????n=??jnn exp(jn?) H(2)n (k?)Bn?? (2.33)?exp(j?t)k????n=??jn+1 exp(jn?)H(2)n?(k?)Bn??,29Ht =?exp(j?t)?????n=??jnn exp(jn?) Jn(??k?)Cn?? (2.34)?exp(j?t)??k????n=??jn+1 exp(jn?)Jn?(??k?)Cn??.The primes on the Hankel and Bessel functions denote a derivative withrespect to the whole argument.2.5 Applying Boundary ConditionsTo find the unknown coefficients Bn and Cn, we need to apply the boundaryconditions at the interface of the cylinder and the surrounding medium ? =a0. At this interface, the tangential components of the electric field andmagnetic field must be continuous:Eix(a0, ?) + Esx(a0, ?) = Etx(a0, ?), (2.35)H i?(a0, ?) +Hs? (a0, ?) = H t?(a0, ?). (2.36)Below are the expressions for the coefficients derived from the boundaryconditions (2.35) and (2.36):Bn =Jn(??ka0)Jn?(ka0)? Jn(ka0)Jn?(??ka0)????Jn?(??ka0) H(2)n (ka0)? Jn(??ka0)H(2)n?(ka0)An, (2.37)Cn =2j/(?ka0)??Jn?(??ka0) H(2)n (ka0)? Jn(??ka0)H(2)n?(ka0)An. (2.38)We now have the full solution to our scattering problem. The results forthe coefficients Bn (2.37) and Cn (2.38) are identical to those of previousauthors [13, 20, 35].302.6 Numerical SimulationsThere are many unique situations in which we can apply the techniques wehave derived. These situations include large/small cylinders, wide/narrowbeam widths, normal/offset incidence, and high/low permittivity contrasts.All these factors will affect the fields that are produced in the scattering.Simulations have been performed to determine the effects of a dielectriccylinder on the electromagnetic fields of a Gaussian beam. For completeness,simulations of incident radiated and evanescent waves have been included. Anumerical analysis comparing the evanescent field in Cartesian coordinateswith the cylindrical wave representation was also performed in order to showthe convergence of the field. In all the subsequent figures an arrow displaysthe direction of incident beam propagation.2.6.1 Numerical Analysis for the Incident RadiatedFieldWithin the radiated region we will analyze three cases. The first case involvesa large cylinder a0 = 0.3m with a relative permittivity of ? = 10. The secondcase uses the same size cylinder but with a relative permittivity of ? = 1000.The third case uses a small cylinder a0 = 0.03m with the same permittivityas the second case. For all three cases the incident beam is identical with? = 2.5 at a frequency of 1GHzThe results of the simulations for the first case are shown in Figure 2.5.From the scattered field we can see that cylindrical waves are generated thatpropagate outward in all directions from the cylinder. The high scatteredamplitude on the far side of the cylinder creates a cancellation of the incidentfield in order that a shadow may be created. Inside the cylinder we can seethat there are six wave peaks [see figure 4e], which is consistent with the sizeof the cylinder and the relative wave speed. From the total outer field wenote that there are high peaks of radiation on the illuminated side of the31cylinder.Figure 2.5: Incident radiated part of a Gaussian beam with ? = 2.5 at afrequency of 1 GHz scattering from a dielectric cylinder with radius a0 = 0.3mand relative permittivity ? = 10. Displayed are the (a) incident field, (b)scattered field, (c) transmitted field inside the cylinder, (d) total outsidefield (a)+(b), and (e) a scaled version of the transmitted field. (a), (b), (c),and (d) have the same scaling.The second case is shown in Figure 2.6. By comparing these results withthose of the first case, we can determine the effects of increasing the dielectricpermittivity. The high permittivity cylinder causes stronger reflection andweaker transmission. In the total outer field we can see that the regionbehind the cylinder is lower in amplitude; it has become a stronger shadowregion. Inside the cylinder there are many waves because the wave speed is3231.6228 times slower. The large reduction in speed inside the cylinder alsocauses a focusing effect. The incident waves are focused toward the centre ofthe cylinder and then dispersed.Figure 2.6: Incident radiated part of a Gaussian beam with ? = 2.5 at afrequency of 1 GHz scattering from a dielectric cylinder with radius a0 = 0.3mand relative permittivity ? = 1000. Displayed are the (a) incident field, (b)scattered field, (c) transmitted field inside the cylinder, (d) total outside field(a)+(b), and (e) a scaled version of the transmitted field. (a), (b), (c), and(d) have the same scaling.Case three is displayed in Figure 2.7. We have now reduced the cylinderradius by 10 times, so that it is much smaller than one wavelength of radiationin the surrounding medium. The scattered field is becoming cylindricallysymmetric. In our previous simulations, the cylinder had multiple radiation33phases incident on its surface at any time. Now the radiation phase incidenton the outside of the cylinder is nearly the same at all points. This causesthe field to permeate the cylinder uniformly as inward travelling cylindricalwaves.Figure 2.7: Incident radiated part of a Gaussian beam with ? = 2.5 at afrequency of 1 GHz scattering from a dielectric cylinder with radius a0 =0.03m and relative permittivity ? = 1000. Displayed are the (a) incidentfield, (b) scattered field, (c) transmitted field inside the cylinder, (d) totaloutside field (a)+(b), and (e) a scaled version of the transmitted field. (a),(b), (c), and (d) have the same scaling.342.6.2 Numerical Analysis of the Evanescent RegionFirst, we analyze the evanescent region in the Cartesian coordinates of (2.23),as it will stand as a reference for us to compare the cylindrical wave case with.All of the following simulations were performed with a frequency of 1GHzand a beam width variable ? = 5.Figure 2.8 and Figure 2.9 show the general shape of a Gaussian beam?sevanescent field. The evanescent field has a Gaussian envelope in the y? di-rection (see Figure 2.8) and decays exponentially in the positive and negativez? directions (see Figure 2.9). The evanescent field has only a real part andtherefore oscillates like a standing wave without propagating. The maximumamplitude of this evanescent field is approximately 0.003 for an amplitudeconstant E0 = 1. The distance between peaks comparable to the wavelengthof the radiation.Figure 2.8: Evanescent field along the y? axis at z? = 0 for a Gaussian beamwith ? = 5 at a frequency of 1 GHz.35Figure 2.9: Evanescent field for a Gaussian beam with ? = 5 at a frequencyof 1 GHz, generated from the Cartesian formula (2.23).Now we will compare the results obtained in the Cartesian analysis withequivalent simulations using our cylindrical wave representation from (2.22).Table 2.1 compares the fields produced by the cylindrical wave formula (2.22)within the convergent region ? < ?conv, with those of the Cartesian formula(2.23). The results from the two formulas within the convergent region areextremely close. The tiny percentage of error is most likely due to the imper-fection of numerical integration. An evanescent field plot from the cylindricalwave equation is shown in Figure 2.10 and can be compared with the equiv-alent Cartesian plot in Figure 2.9.36Table 2.1: Numerical Comparison between the Cartesian Formula (2.23) andthe Cylindrical Wave Representation (2.22) for the Evanescent Fieldy? z? Cartesian Cylindrical Waves N = 1000 0 3.036954670? 10?3 3.036954703? 10?30 0.1 1.420888146? 10?3 1.420888145? 10?30 0.2 7.718096192? 10?4 7.718096156? 10?40 0.3 4.680267318? 10?4 4.680267350? 10?40 0.4 3.080132582? 10?4 3.080132516? 10?40.1 0 ?1.978901439? 10?3 ?1.978901438? 10?30.2 0 ?3.710670998? 10?4 ?3.710670940? 10?40.3 0 2.217569218? 10?3 2.217569234? 10?30.4 0 ?2.250000000? 10?3 ?2.250344039? 10?3Figure 2.10: Evanescent field for a Gaussian beam with ? = 5 at a frequencyof 1 GHz, generated from the cylindrical wave formula (2.22).2.6.3 Scattering Effect of the Evanescent FieldIncident on the CylinderTwo cases were simulated for the evanescent field scattering from a dielectriccylinder. The first case involves a small cylinder a0 = 0.03m with relativepermittivity ? = 1000. The second case has a larger cylinder a0 = 0.3m with37the same permittivity. The Gaussian beam used to model these simulationsis the same as in the radiation simulations above. The distance from thecylinder edge to the beam origin is fixed at 0.7m so that enough evanescentfield amplitude remains to enable us to visualize the scattering.The results of the simulation for the first case are shown in Figure 2.11.The scattered field displayed in three time frames in Figure 2.12 demon-strates that propagating cylindrical waves are formed by the reflections atthe cylinder boundary. Figure 2.13 plots the real and imaginary parts of thescattered field on the far side of the cylinder. The incident evanescent fieldcontains a real part only. From Figure 2.13 we can see that an imaginarycomponent 90? out of phase with the real component has been introduced.These newly formed radiating waves have the same frequency as the originalGaussian beam; there are no harmonics. The amplitude of this radiationapproaches the amplitude of the evanescent field at the cylinder boundaryasymptotically as the relative permittivity of the cylinder is increased. Thephase of the radiation at the cylinder boundary also shifts asymptotically to180? as the permittivity is increased.38Figure 2.11: Evanescent part of a Gaussian beam with ? = 2.5 at a frequencyof 1 GHz scattering from a dielectric cylinder with radius a0 = 0.03m andrelative permittivity ? = 1000. Displayed are the (a) incident field, (b)scattered field, (c) transmitted field inside the cylinder, (d) total outsidefield (a)+(b), and (e) a scaled version of the transmitted field. (a), (b), (c),and (d) have the same scaling.39Figure 2.12: The scattered field from Figure 2.11 showing three time stamps(a)-(c).Figure 2.13: The scattered field from Figure 2.11 displaying real and imag-inary parts. The sectional view is taken along the z? axis on the far side ofthe cylinder.40Figure 2.14: Evanescent part of a Gaussian beam with ? = 2.5 at a frequencyof 1 GHz scattering from a dielectric cylinder with radius a0 = 0.3m andrelative permittivity ? = 1000. Displayed are the (a) incident field, (b)scattered field, (c) transmitted field inside the cylinder, (d) total outsidefield (a)+(b), and (e) a scaled version of the transmitted field. (a), (b), (c),and (d) have the same scaling.The second case is shown in Figure 2.14. The scattered field still containspropagating waves in the case of the large cylinder. The wave interferenceeffects taking place in the scattered and transmitted fields are qualitativelythe same as in the previous case. The only difference is that the scatteredwaves are not as uniformly distributed as in the small cylinder case.412.7 ConclusionA full analysis, including the evanescent region, of a two-dimensional Gaus-sian beam scattering from a homogeneous dielectric cylinder was performed.For the radiated region the results were common to those of other papers[10, 13, 20]. The calculation and analyses performed within the evanescentregion provided a more complete description of the scattered fields. We wereable to obtain expressions for the evanescent field in Cartesian coordinatesand in cylindrical coordinates as a sum of weighted cylindrical waves. Withinthe evanescent region the sum of weighted cylindrical waves was found to con-verge spatially outward as we increased the number of Bessel modes. Theevanescent field scattering from a dielectric cylinder was shown to createpropagating waves in all directions. This scattering takes energy that is nor-mally stored in the near-field of the incident Gaussian beam and transformsit into radiation energy. The strength of these newly formed propagatingwaves depends on the amplitude of the evanescent field incident on the cylin-der surface and on the permittivity of the dielectric cylinder. To maximizethis radiating effect one would place a high permittivity cylinder as close tothe beam origin as possible.42Chapter 3Truncating Cylindrical WaveModes in Two-DimensionalMultiple Scattering3.1 IntroductionThe two-dimensional multiple scattering of plane-waves and Gaussian beamsby infinite circular cylinders has been considered by several authors [24, 26?29, 83]. The technique involves representing the incident, scattered, andtransmitted waves as infinite sums of cylindrical wave modes:E(?v, ?v) = exp(j?t)??n=??j?n exp(jn?v) Zn(k?v) bv n, (3.1)with coefficients bv n. The function Z is a Bessel function of the first kind forincident and transmitted fields and for scattered fields it is a Hankel function.Once in this form, the boundary conditions can be applied at the surface ofeach cylinder and solved in terms of a matrix inversion. However, in orderto calculate the scattering matrix or any of the electromagnetic fields, theinfinite sum of cylindrical wave modes must be truncated to some maximummagnitude limit Mv. A unique Mv may be used for each cylinder v involvedin the multiple scattering. In this chapter we discuss how the backgroundwavenumber k, the cylinder radii av, the inter-cylinder separation d, and thecylinder permittivities ?v define appropriate upper and lower bounds on the43limit Mv.Many previous authors chose the limit Mv by iteratively analyzing theconvergence of a given scattering setup [29,83]. Others chose to make linearestimates, which are good for mid-range kav values [26?28]. For dielectriccylinders, Elsherbeni [27] suggests the use of Mv ? 3kvav, which becomesexcessive for large ?. For conducting cylinders, Ragheb and Hamid [28] usedMv ? 3kav, with the restriction that kav is reasonably large. For two cylin-ders, Young and Bertrand [26] suggest using Mv ? 2kav when the cylindersare sufficiently separated d ? a. They also indicate that for low values ofkav, five to ten extra modes are necessary.3.2 Lower Bound LimitThe lower bound limit of Mv is necessitated by the fact that we need enoughmodes to represent the electromagnetic fields at the cylinder boundary withsufficient precision. This limit is determined solely by the wavenumber k inthe surrounding medium and the radius of the cylinders a. The factor kavdetermines the bounds beyond which further cylindrical wave modes becomeseverely attenuated in the scattering. The lower bound mode limit for accu-racy will be similar for the single and multiple scattering cases because we canconsider the multiple scattering case in terms of its iterative solution. Theiterative solution involves recursively applying the single scattering solutionto obtain a convergent total field. Every time the single scattering solutionis applied, the same range of modes will be attenuated due to the kav fac-tor. By analyzing the single scattering coefficients of conducting (3.2) anddielectric (3.3) cylinders with the incident field coefficient factor removed,fv n = ?Jn(kav)H(2)n (kav), (3.2)44fv n =Jn?(kav) Jn(??rvkav)???rv Jn(kav)Jn?(??rvkav)??rv H(2)n (kav)Jn?(??rvkav)? H(2)n?(kav) Jn(??rvkav), (3.3)we can determine proper lower bound mode limits for various values of kav.The single scattering coefficients (3.2) and (3.3) apply to transverse magnetic(TM) polarization, where the electric field is directed along the cylinder axes.For transverse electric (TE) polarization, the mode limits are similar exceptfor the case of extremely small cylinders which is covered in appendix B.Figure 3.1: The minimal accuracy mode limits Mv are displayed for kavfactors that are (a) small, (b) medium, and (c) large.The coefficient factors (3.2) and (3.3) act like filters on the incident fieldcoefficients, attenuating modes beyond a limit determined by kav. The filterwidths of the conducting, dielectric, and lossy dielectric cylinders are verysimilar, even for very weak permittivity dielectrics. Since the filter widthsnever vary by more than one or two modes between the three cases, we cancharacterize the conducting cylinder case, which always has highest moderequirements, and use it for all three cases. Our lower bound limit on Mvis the first mode that causes the magnitude of the scattering coefficient todrop below 1% of its largest value. Plots of the mode limit Mv for mini-mal accuracy representation under these conditions are shown in Figure 3.1.Using a mode limit below this minimal accuracy representation may resultin completely misrepresented fields in the multiple scattering solution. Twolinear equations (3.4) were fitted to the plots (b) and (c) in Figure 3.1 so45that the minimum mode limits could be easily calculated:Mv = [1.0302kav + 4.5585]ceil 10 ? kav ? 200Mv = [1.2174kav + 2.0578]ceil 0.5 ? kav ? 10. (3.4)For the small kav from (a) in Figure 3.1) a linear approximation is not ap-propriate so we use a discrete definitionMv =???2 0.08125 ? kav ? 0.51 kav < 0.08125. (3.5)The extinction and scattering cross sections will also converge to a min-imal accuracy representation at the minimum mode limit, and increase inaccuracy as Mv is raised. Olaofe [24] calculated the extinction cross sectionfor the scattering of a plane-wave from multiple cylinders:Cext =4k Re{??n=??N?v=1bv n exp[?jkd1v cos(?1v)]}, (3.6)and also outlined the calculation of the scattering cross section. However,when the cylinders are non-absorbing, the scattering cross section is equalto the extinction cross section in (3.6). Since the extinction cross section isproportional to a sum of the scattering coefficients bv n, it will converge as allthe coefficients of significant magnitude are included in the sum. Our minimalaccuracy requirements ensure that all coefficients of significant magnitudewill be included.3.3 Upper Bound LimitThe upper bound limit of Mv is due to matrix conditioning. As we increaseMv, the condition number of our multiple scattering matrix increases. Alarge condition number indicates that the matrix is unstable and may produce46erroneous results when an attempt is made to invert it. The matrix conditionnumber is obtained from a singular value decomposition by taking the ratioof the largest singular value divided by the smallest. There tends to be athreshold after which the condition number begins to increase drasticallyfor every increase in the mode limit Mv. Using a mode limit beyond thethreshold will increase the risk of errors in the matrix inversion. Therefore,we will define our upper-bound limit of Mv to be the threshold where thescattering matrix condition numbers begin to increase exponentially.Aside from the mode limit, the condition number is affected by the back-ground wavenumber k, the cylinder radii av, the separation between cylindersd, and the permittivities of the cylinders ?v. These parameters define the rigidstructure created by the multiple cylinders, but there is no restriction on therotational orientation of the structure. The condition number will also becompletely independent of the incident field, because the scattering matrixthat we invert is independent of the incident field.3.4 Scattering SimulationsThe multiple scattering scenarios of Figure 3.2 and Figure 3.3 were eachsimulated for separation distances d = 1.1? and d = 2.2?. Each simulationconsisted of calculating the condition number of the scattering matrix for agiven mode limit Mv. All the mode limits from 1 to 20 were simulated. Aswe would expect, the condition numbers for the broadside and inline caseswere identical to machine precision because they both involve scattering fromthe same rigid structure of cylinders. Remember that the condition numbersare independent of rotations and incident fields. Hence, only one graph ofresults, Figure 3.4, is necessary for both the broadside and inline cases.47Figure 3.2: Multiple scattering of a Gaussian beam from 11 perfectly con-ducting cylinders with broadside incidence.Figure 3.3: Multiple scattering of a Gaussian beam from 11 perfectly con-ducting cylinders with in-line incidence.48MMatrix Condition Numberminimumthreshold threshold Figure 3.4: Matrix condition number compared to the mode limit. Theresults are identical for broadside and in-line incidence.Since kav ? 3.14 in both scenarios, we can use (3.4) to calculate our min-imum mode limit Mv = 6. For the case when d = 2?, the condition numberthreshold is nearMv = 15, well above the minimal accuracy requirement. Forthe case when d = 1.1?, the coupling between cylinders is much stronger.The threshold is now Mv = 9, which is very close to our minimum modelimit. It will be harder to model the d = 1.1? case because the lower- andupper-bound limits on Mv are closer together, limiting the accuracy of thesolution. In general, as cylinders are placed closer together, the thresholdwill decrease, making it harder to obtain higher accuracy results.The case of multiple scattering from cylinders of various radii brings anew aspect into our analysis. Each cylinder involved in the scattering shouldhave a mode limit Mv that matches its kav factor and the desired level ofaccuracy. Suppose we have a multiple scattering situation in which there isa large cylinder with ka1 = 12 and a small cylinder with ka2 = 3. Using49(3.4) to calculate the minimal accuracy mode limits, we get M1 = 17 forthe large cylinder and M2 = 6 for the small cylinder. Now, if we imposethe limit M = 17 for both cylinders, the small cylinder may have an excessnumber of modes and cause the scattering matrix to become ill-conditioned.If instead we impose M = 6 on both cylinders, the large cylinder?s scatteredfield will not have enough cylindrical wave modes to be properly representedand erroneous results will ensue. By using separate mode limits, for exampleM1 = 20 and M2 = 8, we can accurately represent the multiple scatteringwhile maintaining a properly conditioned scattering matrix.3.5 ConclusionIn conclusion, the cylindrical mode limit Mv should be chosen carefully, con-sidering the minimal accuracy requirements and the conditioning of the mul-tiple scattering matrix. The highest degree of accuracy can be obtained bypushing the mode limits Mv, for each cylinder, above their minimal accu-racy requirements until the scattering-matrix condition number reaches itsthreshold.50Chapter 4Scattering from CylindersUsing the Two-DimensionalVector Plane-Wave Spectrum4.1 IntroductionTwo-dimensional multiple scattering from infinite circular cylinders was firstconsidered by Twersky in 1952 [22]. Twersky considered the scattering ofa plane-wave by an arbitrary number of cylinders, first using an iterativesolution and later developing a matrix inversion solution with Burke andCensor [23]. In 1970, Olaofe broke down the scattering problem into aninfinite set of linear equations, which also could be solved iteratively or bymatrix inversion [24, 25]. The effectiveness of the matrix inversion methodwas emphasized by the experimental results of Young and Bertrand [26] andthe analysis of Elsherbeni [27]. We will use the matrix inversion solutionand apply it to T-matrix formalism which directly relates the incident fieldcoefficients to the multiple scattering coefficients [84].Aside from plane-wave incidence, Gaussian beam incidence and arbitrarybeam incidence have also been considered. The scattering of two-dimensionalGaussian beams by two cylinders was considered by Kojima et al. [31], andSugiyama and Kozaki [32]. This was extended to include an arbitrary numberof cylinders by Yokota et al. [33] and Elsherbeni et al. [34]. Gouesbet [72]considered arbitrary beam incidence under generalized Lorentz-Mie theory(GLMT). He allowed for any arbitrary, three-dimensional, incident field by51using the theory of distributions. We are considering a novel applicationof the two-dimensional vector plane-wave spectrum (VPWS) to produce anarbitrary incident field in the multiple scattering from cylinders.The VPWS breaks down an arbitrary aperture distribution into an infi-nite sum of plane-waves which are the eigenfunctions of the homogeneous,isotropic wave equation in Cartesian coordinates. The VPWS can produceboth homogeneous and inhomogeneous plane-waves, which represent the ra-diative and evanescent regions of the incident field respectively. Because ofthe convergence properties of evanescent fields in cylindrical coordinates, itis essential that our solution separates the two regions [85]. Our numeri-cal simulations will focus on evanescent field incidence, since most previousauthors, using a cylindrical wave approach, only considered radiative inci-dence [24?27,31?34]. Exceptions to this include Yang et al. [35], who includedevanescent components of a Gaussian beam in their multiple scattering solu-tion, and Chapter 2, where the scattering of the evanescent components of aGaussian beam from a single cylinder was presented. The Gaussian beam?sevanescent field is well defined, whereas our VPWS formulation allows forarbitrary evanescent fields.Evanescent fields are composed of inhomogeneous plane-waves, in whichthe equiamplitude and equiphase surfaces do not coincide [78]. These evanes-cent waves are only significant in the near-field because they decay exponen-tially away from their source. Evanescent fields are present in the near-fieldof practically every scattering or radiating source because they contain allthe sub-wavelength details of the electromagnetic field distribution. Thisphenomenon is currently exploited in apertureless near-field scanning opti-cal microscopy (ANSOM), where a small metallic probe is used to scatter anobject?s evanescent field into radiation in order to obtain super-resolution im-ages [64]. Arrays of probes, gratings and larger near-field diffractive elementshave also been proposed for scattering evanescent fields [86]. Sub-wavelengthresolution in microwave source localization has been investigated using sin-52gle and multiple scatterers [38]. An investigation by Gulayaev et al. [87]demonstrated the scattering of incident evanescent fields from a periodicgrating. They defined two methods for recovering evanescent field data fromthe scattering: distant-spatial and interference-spatial spectroscopy. Numer-ical examples presented herein focus on the conversion of evanescent fieldsinto radiating waves through near-field scattering from a similar grating toGulayaev et al. [87]. By using sub-wavelength cylinders and half-wavelengthspacing in the grating, we are able to convert evanescent fields into beams ofradiation with directionality based on the incident evanescent field?s spatialfrequency.By writing generalized equations for the incident, scattered and trans-mitted waves, and solving the boundary conditions on each cylinder simulta-neously, we can obtain a T-matrix solution for the multiple scattering fromcylinders. We extend our formulation of the solution to include any arbitrarycombination of conducting and dielectric cylinders with various radii. Thegeometry and coordinate systems are displayed in Fig. 4.1.53Figure 4.1: Geometry and coordinate systems for the cylinders and the inci-dent field. The x direction is perpendicular to the page.4.2 Incident Electric FieldsUse of the VPWS technique allows us to create an incident electric fieldwith arbitrary distribution and polarization. Here we have derived the two-dimensional VPWS from the three-dimensional version provided by Guo etal. [88]. The VPWS breaks down the electromagnetic field across an apertureinto a sum of homogeneous and inhomogeneous plane-waves, while properlymaintaining the polarization of the field. Thus, the VPWS always satis-fies Maxwell?s equations. For cylindrical scattering, the transverse magnetic(TM) polarization has the magnetic field transverse to the cylinder axes andthe transverse electric (TE) polarization has the electric field transverse tothe cylinder axes.For an electromagnetic wave in cylindrical coordinates there are six com-ponents to the vector fields Ex, E?, E?, Hx, H?, and H?. However, since any54polarization state can be described by a combination of TM and TE modes,all the field information can be contained within Ex and Hx. The other com-ponents can be computed from Ex and Hx using Maxwell?s equations withno charges or currents:?? E = ??B?t (4.1)??H = ?D?t (4.2)To simplify the number of equations, only the Ex and Hx components willbe shown.The two-dimensional VPWS in Cartesian coordinates centred about thecoordinate system (z?, y?, x) can be represented byEix(z?, y?) =12?? ???E?TM(ky) exp[?j(kyy? + kzz?)]dky, (4.3)H ix(z?, y?) =12?? ???H?TE(ky) exp[?j(kyy? + kzz?)]dky, (4.4)E?TM(ky) =? ???Eix(0, y?) exp(jkyy?)dy?, (4.5)H?TE(ky) =? ???H ix(0, y?) exp(jkyy?)dy?. (4.6)The fields are defined from the initial aperture distribution along the planez? = 0. Because of uniformity in the x direction, we have kx = 0, so thatk2 = k2y + k2z . The time dependence exp(j?t) is assumed and suppressedthroughout.The two Fourier transforms, (4.5) and (4.6), represent the spatial fre-quency content across the aperture for the x? components of the electricand magnetic fields respectively. When the spatial frequency ky exceeds thewavenumber k, the z? component becomes imaginary kz = ?j?k2y ? k2 for55z? ? 0. These waves are called evanescent because they decay exponentiallyin the z?? direction.It is expedient to use the fast Fourier transform (FFT) in (4.5) and (4.6)when the distributions Eix(0, y?) and H ix(0, y?) are discretized or when theintegral cannot be evaluated analytically.Now we will convert from Cartesian coordinates (z?, y?, x) to polar coor-dinates (?v, ?v, x) centred about each cylinder v:y? = ?v sin(?v)? ?v0 sin(?v0), (4.7)z? = ?v cos(?v)? ?v0 cos(?v0). (4.8)The unit vectors ??v and ??v are unique for the coordinates of each cylinder v.We will also convert our variable of integration into an angular representation:ky = k sin(?). (4.9)Applying these transformations to equations (4.3) and (4.4) yieldsEix(?v, ?v) =k2?? ?/2+j???/2?j?E?TM [k sin(?)] cos(?) (4.10)exp[?jk?v cos(?v ? ?) + jk?v0 cos(?v0 ? ?)]d?,H ix(?v, ?v) =k2?? ?/2+j???/2?j?H?TE[k sin(?)] cos(?) (4.11)exp[?jk?v cos(?v ? ?) + jk?v0 cos(?v0 ? ?)]d?.Now we manipulate and simplify equations (4.10) and (4.11) using the Bessel56function identity [81]:exp[?jk?v cos(?v ? ?)] =??n=??j?n exp[jn(?v ? ?)] Jn(k?v). (4.12)Finally, we can write our incident electromagnetic fields as sums of TM (4.13)and TE (4.14) cylindrical waves with coefficients (4.15) and (4.16) respec-tively,Eix =??n=??j?n exp(jn?v) Jn(k?v) Av n, (4.13)H ix =??n=??j?n exp(jn?v) Jn(k?v) Qv n, (4.14)Av n =k2?? ?/2+j???/2?j?E?TM [k sin(?)] exp[jk?v0 cos(?v0 ? ?)? jn?] cos(?)d?.(4.15)Qv n =k2?? ?/2+j???/2?j?H?TE[k sin(?)] exp[jk?v0 cos(?v0 ? ?)? jn?] cos(?)d?.(4.16)The coefficients (4.15) and (4.16) contain both the radiative and evanescentcomponents of the incident field. From these representations, the evanescentcomponents will only be valid in the half-space z? ? 0 because when z? <0, the exponential decay of the evanescent fields will switch to exponentialgrowth. In Chapter 2 it was demonstrated that in representing evanescentfields from a planar source as a sum of cylindrical wave modes [ (4.13) or(4.14)], the fields converge spatially outward as more modes n are included.This will make the entire incident field extremely difficult to represent in thefar-field. To overcome this, we split the coefficients into separate radiativeand evanescent components. The evanescent components only need to becalculated for near-field scattering, and they can be ignored in the far-field.574.2.1 Incident Radiated FieldsThe incident radiated fields can be isolated by limiting the range of integra-tion in the coefficients (4.15) and (4.16) to ??/2 ? ? ? ?/2. In this region,the plane-waves are homogeneous and propagate energy outward from thesource. In contrast to evanescent fields, the radiative components will bevalid for all z? and converge quickly in the far-field. In general, the inte-gration will have no analytical solution so that numerical integration will benecessary. If (4.5) and (4.6) were computed using the FFT, then the integralsin (4.15) and (4.16) should be discretized accordingly.4.2.2 Incident Evanescent FieldsThe incident evanescent fields can be isolated by limiting the range of in-tegration in the coefficients (4.15) and (4.16) to the complex regions where| sin(?)| > 1. In order to combine the two halves of this region into oneintegral, we use the following transformations on the negative and positivehalves of the integration range respectively:? = ??/2? ju, (4.17)? = ?/2 + ju, (4.18)to obtainAevanv n =k2?? ?0sinh(u){ S+v n E?TM [k cosh(u)] + S?v n E?TM [?k cosh(u)]}du,(4.19)Qevanv n =k2?? ?0sinh(u){ S+v n H?TE[k cosh(u)] + S?v n H?TE[?k cosh(u)]}du,(4.20)58S?v n = j?n exp[?nu? jk?v0 sin(?v0 ? ju)]. (4.21)Again, the integrals in (4.19) and (4.20) need to be numerically integrated,or discretized appropriately if the FFT has been previously used to computeE?TM and H?TE.4.3 Scattered FieldsThe scattered electric and magnetic field components can all be derived fromthe Ex and Hx components alone. The scattered fields will be represented asa sum of cylindrical waves with coefficients bv n and gv n, representing the TMand TE modes respectively. For a time dependence exp(j?t) we use Hankelfunctions of the second kind H(2)n (k?v) to represent outgoing waves:Esvx =??n=??j?n exp(jn?v) H(2)n (k?v) bv n, (4.22)Hsvx =??n=??j?n exp(jn?v) H(2)n (k?v) gv n. (4.23)We also need to represent the scattered field from each cylinder in thecoordinate system of all the other cylinders. To do this we will apply theGraf addition theorem [89]:Zv(z) exp(jv?) =??m=??Zv+m(x) Jm(y) exp(jm?), (4.24)the coordinates of which are displayed in Fig. 4.2. When Z is a Besselfunction of the second or third kind, we have the restriction| exp(?j?)y| < |x|. (4.25)59Figure 4.2: Triangle showing the coordinates of transformation in the Grafaddition theorem (4.24)By applying the Graf addition theorem (4.24) to our geometry in Fig.4.1, we obtainH(2)n (k?w) exp(jn?w) =??m=??H(2)n?m(kdwv) Jm(k?v) exp[j(n?m)?wv + jm?v].(4.26)Now we apply the results (4.26) to our scattered fields from (4.22) and (4.23):Eswx =??n=??j?n exp(jn?v) Jn(k?v) Bwv n, (4.27)Bwv n =??m=??jn?m exp[j(m? n)?wv] H(2)m?n(kdwv) bw m, (4.28)Hswx =??n=??j?n exp(jn?v) Jn(k?v) Gwv n, (4.29)60Gwv n =??m=??jn?m exp[j(m? n)?wv] H(2)m?n(kdwv) gw m. (4.30)The coefficients Bwv n and Gwv n are the coupling coefficients for the TMand TE polarizations respectively. They translate the scattered field fromcylinder w to an incident field on cylinder v.4.4 Transmitted FieldsSince we want to consider the solution for perfect conducting and dielectriccylinders, we will need two definitions for the transmitted fields. The perfectelectric conductor (PEC) and perfect magnetic conductor (PMC) will bothhave electric and magnetic fields equal to zero inside the cylinders. Thedielectric case defines the transmitted electric and magnetic fields asEtvx =??n=??j?n exp(jn?v) Jn(??rvk?v) dv n, (4.31)H tvx =??n=??j?n exp(jn?v) Jn(??rvk?v) hv n. (4.32)The relative permittivity is the ratio of cylinder permittivity to backgroundpermittivity ?rv = ?v/?b, and k = ???b?b is the background wavenumber.The transmission coefficients for TM and TE scattering are dv n and hv nrespectively, and can be found by applying the boundary conditions.4.5 Boundary ConditionsFor a dielectric cylinder, the tangential electric and magnetic field compo-nents must be continuous across the boundary of the cylinders. Therefore,we equate the corresponding x? and ?? components of the electric and magnetic61fields at the cylinder boundaries ?v = av:Etvx (av, ?v) = Eix(av, ?v) + Esvx (av, ?v) +?w 6=vEswx (av, ?v), (4.33)H tv? (av, ?v) = H i?(av, ?v) +Hsv? (av, ?v) +?w 6=vHsw? (av, ?v), (4.34)Etv? (av, ?v) = Ei?(av, ?v) + Esv? (av, ?v) +?w 6=vEsw? (av, ?v), (4.35)H tvx (av, ?v) = H ix(av, ?v) +Hsvx (av, ?v) +?w 6=vHswx (av, ?v). (4.36)For a PEC cylinder, the tangential electric fields must vanish at theboundary0 = Eix(av, ?v) + Esvx (av, ?v) +?w 6=vEswx (av, ?v), (4.37)0 = Ei?(av, ?v) + Esv? (av, ?v) +?w 6=vEsw? (av, ?v). (4.38)The magnetic fields for a PEC cylinder do not vanish at the boundary butare not necessary to consider when solving for the scattering coefficients. Fora PMC cylinder, the tangential magnetic fields must vanish at the boundary0 = H i?(av, ?v) +Hsv? (av, ?v) +?w 6=vHsw? (av, ?v), (4.39)0 = H ix(av, ?v) +Hsvx (av, ?v) +?w 6=vHswx (av, ?v). (4.40)The electric fields for a PMC cylinder do not vanish at the boundary but arenot necessary to consider when solving for the scattering coefficients [90].The magnetic and electric ?? components can be calculated from the elec-tric and magnetic x? components by using (4.1) and (4.2), respectively. In62the two-dimensional scattering case there is no cross-polarization betweenthe TM and TE polarizations. Therefore, the solution is found by using Exand H? to solve for the TM scattering coefficients, and by using E? and Hxto solve for the TE scattering coefficients:bv n = fv n Av n + fv n?w 6=vBwv n, (4.41)gv n = fv n Qv n + fv n?w 6=vGwv n. (4.42)The TM solution (4.41) and TE solution (4.42) share the same functionalform, the only differences are in the single scattering coefficients fv n. For adielectric cylinder with TM polarization, we havefv n =Jn?(kav) Jn(??rvkav)???rv Jn(kav)Jn?(??rvkav)??rv H(2)n (kav)Jn?(??rvkav)? H(2)n?(kav) Jn(??rvkav), (4.43)and with TE polarization it becomesfv n =??rvJn?(kav) Jn(??rvkav)? Jn(kav)Jn?(??rvkav)H(2)n (kav)Jn?(??rvkav)???rvH(2)n?(kav) Jn(??rvkav). (4.44)For a PEC cylinder with TM polarization, we havefv n = ?Jn(kav)H(2)n (kav), (4.45)and with TE polarization it becomesfv n = ?Jn?(kav)H(2)n?(kav). (4.46)63For a PMC cylinder with TM polarization, we havefv n = ?Jn?(kav)H(2)n?(kav), (4.47)and with TE polarization it becomesfv n = ?Jn(kav)H(2)n (kav). (4.48)The PMC cylinder single scattering coefficients 4.47 and 4.48 are identicalto the PEC case with the opposite polarization. 4.46 and 4.45 respectively.The scattering from a perfect electromagnetic conductor cylinder, as a gen-eralization of the PEC and PMC cylinders, was provided by Ruppin [91].The prime indicates a derivative of the function with respect to the wholeargument. The subscript v indicates that the multiple scattering solutionmay contain cylinders with different radii and different complex permittivi-ties. In terms of the scattering solution, the properties of each cylinder arecompletely confined to the single scattering coefficients fv n.4.6 T-Matrix FormulationThe T-matrix formalism, which is commonly used in spherical scattering[84], represents the multiple scattering coefficients L [(4.49) and (4.50)] by amatrix T, which multiplies the incident field coefficients P [(4.51) and (4.52)].In this case, because the TM and TE modes do not interact, we computetheir scattering separately using (4.53) and (4.54), respectively. The matrixT depends upon the polarization, the cylinder properties, and the locationof the cylinders. The total number of cylinders is M .The number of cylindrical wave modes needs to be truncated to somemaximum magnitude Nv, which can be unique for each cylinder. The choiceof an appropriate mode limit Nv depends mostly on the value of the factor64kav. It is important to note that setting the mode limit too high may causethe matrix inversion in (4.59) to become ill-conditioned. A complete anal-ysis of choosing mode limits Nv in two-dimensional multiple scattering wasprovided in Chapter 3.LTM = [ b1 ?N1 , . . . , b1 N1 , b2 ?N2 , . . . ]T , (4.49)LTE = [ g1 ?N1 , . . . , g1 N1 , g2 ?N2 , . . . ]T , (4.50)PTM = [ A1 ?N1 , . . . , A1 N1 , A2 ?N2 , . . . ]T , (4.51)PTE = [ Q1 ?N1 , . . . , Q1 N1 , Q2 ?N2 , . . . ]T , (4.52)LTM = TTMPTM , (4.53)LTE = TTEPTE. (4.54)Our T matrix for the TM and TE polarizations is similar, except for thesingle scattering coefficients, so we can derive it from (4.41) or (4.42). Bothequations define an infinite set of linear equations which can be truncatedand solved by matrix inversion. We can rewrite (4.41) or (4.42) in matrixform asL = FP+ FCL, (4.55)using the appropriate form of L, P, and F for the TM and TE modes. Thediagonal matrix F contains the single scattering coefficients:F = diag[ f1 ?N1 , . . . , f1 N1 , f2 ?N2 , . . . ]. (4.56)65The matrix C is defined by the second term on the right side of (4.41) or(4.42), and the definition of the coupling coefficients (4.28) or (4.30):C =??????????0 C12 . . . C1MC21 0 . . . C2M... ... . . . ...CM1 CM2 . . . 0??????????. (4.57)The sub-matrices Cvw relate the scattered field of cylinder w to the scatteringcoefficients of cylinder v. The elements cn,m of the sub-matrices Cvw comedirectly from the coupling coefficients, (4.28) and (4.30), and are given bycn,m = jn?m H(2)m?n(kdwv) exp[j(m? n)?wv]. (4.58)The Bessel modes n index the rows of the sub-matrices, while the Besselmodes m index the columns. The mode limits Nv might not be the same forn and m, so that Cvw may not be square. However, the total matrix C willstill be square.The solution to (4.55) is found through matrix inversionL = (F?1 ?C)?1P = TP. (4.59)Our T matrix is given by (F?1 ? C)?1. A great advantage of this form ofsolution (4.59) is the separability of the aspects involved in the scattering.The properties of each cylinder are contained as separate elements in the Fmatrix, which represents the solution of the single scattering problem L =FP. The location of each cylinder relative to the others is accounted for bythe matrix C, which allows for multiple scattering interactions. Therefore,our matrix T is independent of the form of the incident field except for itspolarization. T acts as a matrix operator that takes, as input, the incidentfield coefficients and produces, as output, the scattered field coefficients.664.7 Numerical SimulationsOur simulations will focus on evanescent field scattering which has mostlybeen considered using other methods [38, 86, 87]. Most previous authorswho considered cylindrical scattering using a cylindrical wave method onlyconsidered radiative incidence [24?27, 31?34]. We will consider the effectsof evanescent fields scattering from a grating of sub-wavelength cylinders, asshown in Fig. 4.3. The incident fields have a frequency of f = 1GHz and areseparated from the cylinders by ?/4 in all cases. The incident fields are TMpolarized with the electric fields oriented in the x? direction. The cylindersare spaced d = ?/2 apart, inside a vacuum. They are all perfectly conductingcylinders of radii a = 0.005m. The distance from the incident field to thecylinders is critical when scattering evanescent fields because the decay rateof an evanescent wave depends upon its spatial frequency. Aside from this,evanescent fields from an infinite planar source do not diverge, so that theshape of the field is preserved in the z?? direction.Figure 4.3: Setup for the numerical simulations. The grating is made up of41 conducting cylinders with a = 0.005m. The dimensions are not drawn toscale.67In order to characterize the fundamental effects of the grating on evanes-cent fields, we scattered four evanescent fields composed of single spatialfrequencies ranging from 1.1k to 2k from the grating. Results are shownfor the frequencies which travel in the direction of increasing y?; equivalentfrequencies travelling in the opposite direction are identical except that theyscatter into the right-half plane. The resulting far-field radiation intensitypolar-plots are displayed in Figures 4.4 to 4.7, with ? = 0 along the z?? axis.The far-field radiation intensity in two-dimensions was computed asU = ? |Ex|22? + ??|Hx|22 W/m, (4.60)where ? =??b/?b is the background impedance and ? is the radial distancefrom the scattering centre.Figure 4.4: Far-field intensity (W/m) plot for scattering of the incident field,with aperture distribution Eix(0, y?) = exp(?j1.1ky?), from the grating.68Figure 4.5: Far-field intensity (W/m) plot for scattering of the incident field,with aperture distribution Eix(0, y?) = exp(?j1.4ky?), from the grating.Figure 4.6: Far-field intensity (W/m) plot for scattering of the incident field,with aperture distribution Eix(0, y?) = exp(?j1.7ky?), from the grating.69Figure 4.7: Far-field intensity (W/m) plot for scattering of the incident field,with aperture distribution Eix(0, y?) = exp(?j2ky?), from the grating.It is clearly seen from Figures 4.4 to 4.7 that the grating scatters theincident evanescent field into two beams of radiation which travel in directionsthat depend on the spatial frequency of the incident evanescent field. Thegrating works like a phased array antenna; each cylinder scatters a smallportion of the phase along the incident evanescent field. The scattered beamdirection is where the scattered fields from each cylinder add constructively,and is determined by the phase variation along the grating.Because single spatial frequencies form scattered beams with a specificdirectionality, an incident field with multiple spatial frequencies should formmultiple scattered beams in various directions. To test the effects of multiplespatial frequency components scattering simultaneously, we have scatteredthe evanescent field of a square shaped beam, with an aperture distributiongiven by (4.61), from the grating of Figure 4.3. The results are shown inFigure 4.8 and Figure 4.9. The radiative components of the square beam arenot included in this analysis. We can see several beams travelling in various70directions from the grating, each representing part of the spatial frequencycontent of the incident field. The spatial frequency content of the incidentfield from k to 2k is shown in Figure 4.10 and can be directly compared to thescattered far-field intensity rectangular plot of Figure 4.9. Each peak in thespatial frequency content corresponds to a scattered beam in the far-field.The scattered beams from higher spatial frequencies are more attenuatedbecause they decay faster from the incident field origin. Incident spatialfrequencies near 2k and ?2k both scatter power near ? = 0, resulting inthe slight amplitude increase there. The components beyond 2k only have asmall scattering effect because they decay extremely fast in the z?? direction.Eix(0, y?) =?????1 |y?| ? 10 |y?| > 1(4.61)Figure 4.8: Far-field intensity (W/m) polar plot for scattering of the evanes-cent field of a square shaped beam.71(rad)WmFigure 4.9: Far-field intensity (W/m) plot for scattering of the evanescentfield of a square beam for ??/2 ? ? ? 0.72Spatial Frequency (m )Spectral Density-1Figure 4.10: Absolute value of the spatial frequency content of the incidentevanescent field from k to 2k4.8 ConclusionThe VPWS was applied to the multiple scattering from infinite circular cylin-ders in order to obtain a solution in terms of T-matrix formalism, whichrepresents the multiple scattering coefficients as a matrix operation on theincident field coefficients. The VPWS allowed us to define our incident fieldwith arbitrary radiative and evanescent components.Numerical simulations were performed by imposing incident evanescentwaves upon a grating of cylinders. It was demonstrated that the gratingcould transform the evanescent field into a set of propagating beams. Eachpropagating beam had a direction that corresponded to a spatial frequencycomponent of the incident evanescent field. Such a grating could be used toanalyze the spatial frequency content of evanescent fields.73Chapter 5Scattering from Cylinders Neara Dielectric Half-Space Using aGeneral Method of Images5.1 IntroductionSolutions to the scattering of electromagnetic waves from multiple dielectricand conducting cylinders have been investigated by many authors [8, 24, 25,28, 29, 34, 35, 92]. Decomposing the electromagnetic fields into cylindricalwaves enables the boundary conditions to be solved directly, leading to ananalytic solution of the scattering problem. The inclusion of a dielectric half-space in the multiple scattering setup complicates the solution because theboundary conditions on the planar interface cannot be applied directly usingcylindrical waves. Many authors have used numerical approaches [93?95],but we will focus on cylindrical wave decomposition methods.Borghi et al. [53, 54], Frezza et al. [52, 57], and Lee [56, 58] proposedscattering algorithms that apply the boundary conditions on the cylindersdirectly by equating cylindrical wave modes. The waves scattered from thecylinders are translated, via a plane-wave spectrum, to the planar interfacewhere the plane-wave reflection coefficients are applied. The field is thentranslated back to the cylinders and converted back into a sum of cylindricalwaves. This allows the boundary conditions to be applied around the cylin-ders while completely accounting for all of the multiple interactions betweenthe cylinders and the interface. However, these methods require a significant74amount of numerical integration which can be computationally intensive.For a perfectly conducting plane, the method of images can be applied toobtain an exact solution [43?45]. The method of images can also be appliedat a vacuum-plasma interface where the reflection coefficients are not angle-dependent [52]. However, for a dielectric half-space the reflection coefficientsare angle-dependent, which makes the analysis more complicated. Videenand Ngo [46] and Wang et al. [93] formulated an approximate method ofimages for dielectric half-spaces by assuming that the scattered fields fromthe cylinders strike the planar interface at normal incidence, a solution whichis only accurate if the cylinders are sufficiently separated from the interface.Coatanhay and Conoir [47, 48] later produced a general method of images(GMI) approach where the reflection coefficients of the interface are convertedinto a cylindrical form using a Fourier series. Their GMI accounts for theangular dependence of the reflection coefficients, while avoiding the need fornumerical integration. The only shortfall of this technique is that the Fourierseries can only account for a limited range of evanescent field interactionsbetween the cylinders and the interface.Our formulation applies a GMI to vector electromagnetic scattering, in-cluding multiple cylinders and arbitrary beam incidence. By deriving theFourier series method from the well-known plane-wave integral method, weare able to draw parallels between the two. To make the two methods equiv-alent, we add to the Fourier series method two additional terms that correcterroneous evanescent field interactions between the cylinders and the planarinterface. The significance of these terms should decay exponentially as thecylinders are separated from the interface.Using the plane-wave integral method as a reference solution, we investi-gate the accuracy of the Fourier series method with and without evanescentcorrection terms. The Fourier series method with evanescent correctionsshould produce accurate results even when the cylinders are near the pla-nar interface. Without the corrections, the method should converge as the75cylinders are separated from the interface. The convergence will depend onhow far the Fourier series can extend into the complex domain to modelevanescent field interactions between the cylinders and the interface. Thesignificance of such interactions will also be isolated and examined.5.2 Algorithm OutlineWe approach the scattering problem by representing all the electromagneticfields in terms of cylindrical wave modes, which will allow us to apply theboundary conditions around each cylinder directly. We will break down thescattering into five electromagnetic fields:A. V i, the incident field, is an arbitrary beam of radiation. Evanescent fieldincidence is not considered.B. V r is the reflection of the incident field from the planar interface. Itcan be calculated by numerically evaluating a plane-wave integral or byapplying the Fourier series of the planar reflection coefficients.C. V s, the scattered field emanating from a cylinder, is represented by out-going cylindrical waves with unknown coefficients.D. V d, the diffracted field emanating from an image cylinder, accounts forthe reflection of a cylinder?s scattered field V s from the planar interface.From the scattered field V s there may be both radiative and evanescentwaves incident upon the interface.E. V t, the transmitted field inside of a cylinder, accounts for the field thatpenetrates a dielectric cylinder.For transverse magnetic (TM) polarization V = Ez and for transverseelectric (TE) polarization V = Hz. In isotropic two-dimensional scattering76there can be no cross-coupling between the two polarizations, so their so-lutions can be computed independently. Using these fields, we equate thetangential components of the electric and magnetic fields across the bound-aries of the cylinders to solve the multiple scattering system. This solutionaccurately accounts for all of the multiple scattering interactions between thecylinders and the planar interface.5.3 Scattering TheoryWe will consider the two-dimensional scattering of an arbitrary beam of ra-diation from multiple dielectric cylinders near a dielectric half-space. Thecoordinate systems and geometry are shown in fig. 5.1. The labels v and weach refer to an arbitrary cylinder. The time dependence exp(j?t) is assumedand suppressed throughout.Figure 5.1: The coordinate systems and geometry for the scattering areshown. The z direction is normal to the page. Image cylinder v is notshown.775.3.1 V i - Incident FieldAn arbitrary incident beam is created by using the radiative part of a plane-wave spectrum:V i(x, y) = 12?? k?kF (ky) exp[?jkxx? jkyy]dky. (5.1)The wave vector has components ~k = (kx, ky, 0), and its magnitude in medium1 is k = ???1?0. The spatial frequency content is given byF (ky) =? ???V i(0, y) exp(jkyy)dy, (5.2)where V i(0, y) defines the arbitrary beam shape.We now convert to cylindrical coordinates (?, ?) and change the variableof integration into an angular form ky = k sin(?):V i(?, ?) = k2?? ?/2??/2F [k sin(?)] cos(?) exp[?jk? cos(? ? ?)]d?. (5.3)Using Bessel function identities [81] we can transform (5.3) into a sum ofcylindrical wave modes:V i(?, ?) =??m=??j?m Jm(k?) exp(jm?)Am, (5.4)Am =k2?? ?/2??/2F [k sin(?)] cos(?) exp(?jm?)d?. (5.5)By applying the Graf addition theorem [96], the incident field is translatedto the coordinates of each cylinder v:V i(?v, ?v) =??n=??j?n Jn(k?v) exp(jn?v) Av n, (5.6)78Av n =??m=??jn?m Jm?n(k?v0) exp[j(m? n)?v0]Am, (5.7)where Av n are the coefficients Am translated from the incident field origin tocylinder v.This translational method only works well for radiative fields. In Chapter2 it was shown that when evanescent fields are represented as a sum ofBessel functions of the first kind, they converge spatially outward as themodal truncation is extended. If they are translated using the Graf additiontheorem, they often end up outside of their region of convergence. Therefore,if evanescent wave incidence is desired, the numerical integration schemeproposed in Chapter 4 must be used to translate the evanescent fields.5.3.2 V r - Incident Field ReflectionFor plane-waves incident upon a planar interface between two dielectric half-spaces, we have the well-known reflection coefficients (5.8,5.9), which dependon the angle of incidence ?, the permittivities of the two half-spaces ?1 and ?2,and the polarization. It is important to note that TM and TE polarizationsare not the same with reference to cylinders and planar interfaces; we willmaintain the convention for cylinders throughout. For our TM polarizationthe reflection coefficients areR(?) =cos(?)???2?1 ? sin2(?)cos(?) +??2?1 ? sin2(?), (5.8)and for our TE polarization they areR(?) =?2?1 cos(?)???2?1 ? sin2(?)?2?1 cos(?) +??2?1 ? sin2(?). (5.9)To form the reflected field from the incident field, we propagate the in-79cident field to the planar interface, apply the reflection coefficients (5.8,5.9),and then propagate it to a cylinder v. Taking (5.3) and representing thepropagation to and from the interface using the image coordinates (??, ??), weobtainV r(??, ??) = k2?? ?/2??/2R(?)F [k sin(?)] cos(?) exp[jk?? cos(?? + ?)]d?. (5.10)The tildes ? are used to denote image coordinates and dimensions as shown inFigure 5.1. Converting the exponential to Bessel functions and translatingthe field to the coordinates of cylinder v using the Graf addition theoremyieldsV r(?v, ?v) =??n=??j?n Jn(k?v) exp(jn?v) Qv n, (5.11)Qv n =??m=??jm+n Jm?n(k??v0) exp[j(m? n)??v0]Qm, (5.12)Qm =k2?? ?/2??/2R(?)F [k sin(?)] cos(?) exp(jm?)d?, (5.13)where Qv n are the coefficients Qm translated from the image of the incidentfield to cylinder v. As seen in Figure 5.1, the vector (??v0, ??v0) points fromthe image of the incident field to cylinder v.For the plane-wave integral method, we evaluate (5.13) numerically. Al-ternatively, we can express R(?) as a Fourier series:R(?) =??m=??Rm exp(jm?), (5.14)Rm =12?? ???R(?) exp(?jm?)d?. (5.15)Using this notation is the key to the Fourier series method, as it allows usto avoid numerical integration. Inserting the Fourier series representation80(5.14) into (5.13) yields the convolutional sum:Qm =??l=??R?l?m Al. (5.16)Reflection from the planar interface occurs at real angles ??/2 ? ? ? ?/2for radiative waves and at complex angles ??/2 ? ju, 0 < u < ?, forevanescent waves. As pointed out by Coatanhay and Conoir [47], the Fourierseries (5.14) is equivalent to a Laurent series in complex space:R(?) =??m=??Rmzm, (5.17)where z = exp(j?). The Laurent series (5.17) will converge in the annulusr1 < |z| < r2, extending from the unit circle r1 < 1 < r2, if R(?) is analyticwithin the annulus [97]. Therefore, the Laurent series can only model thereflection of evanescent waves with complex angles ? = ??/2?ju that satisfyr1 < exp(?u) < r2. The limits of the annulus (r1, r2) depend on the functionR(?), where smoother functions tend to have wider limits [48].The reflection coefficients for TE polarization (5.9) contain a singularityon |z| = 1 that makes it impossible to calculate the Laurent series there.The singularity always occurs within the angles of incidence that point awayfrom the interface ?/2 < |?| < ? and do not represent a physical situation.Allowing R(?) to take on arbitrary values in this region will not affect thereal angles of incidence 0 < |?| < ?/2, but it will affect the Laurent series?convergence. To remove the singularity and maintain smoothness, it is ex-pedient to fit a spline to the region ?/2 < |?| < ? in (5.9) before integrating(5.15).815.3.3 V s - Scattered Fields from CylindersThe scattered field from each cylinder v can be represented as a sum ofcylindrical waves:V sv(?v, ?v) =??n=??j?n H(2)n (k?v) exp(jn?v) bv n. (5.18)The scattering coefficients bv n are the unknowns that we are ultimately tryingto solve for.By applying the Graf addition theorem, the scattered field (5.18) fromcylinder w can be represented in the coordinate system of cylinder v:V sw(?v, ?v) =??n=??j?n Jn(k?v) exp(jn?v) Bwv n, (5.19)Bwv n =??m=??jn?m H(2)m?n(kdwv) exp[j(m? n)?wv] bw m. (5.20)5.3.4 V d - Diffracted Fields from the Planar InterfaceThe scattered fields from the cylinders (5.18) will propagate to the planarinterface and partially reflect from the surface. To completely account forthe multiple scattering interactions that occur between the cylinders and theinterface, we need to find a representation for this diffracted field in termsof the scattering coefficients bv n. We begin by transforming a cylindricalwave emanating from cylinder w, into plane-waves using the Sommerfeldintegral [98]:H(2)n (k?w) exp(jn?w) =jn??C1exp[jn??jxwk cos(?)?jywk sin(?)]d?, (5.21)where the contour C1 is taken from ??/2 ? j? to ?/2 + j? as shown inFigure 5.2. Cincotti et al. [49] derived a similar plane-wave expansion for82Hankel functions of the first kind.Figure 5.2: The integration contour C1 follows the path 1-2-3-4. The contourC2 includes the paths 1-2 and 3-4 only. The points 1 and 4 extend infinitely,parallel to the imaginary axis.To calculate the reflection of this wave (5.21), we propagate it to theplanar interface, apply the reflection coefficients (5.8, 5.9), and propagate itto a cylinder v. Again, the propagation to and from the interface is done bytaking the image cylinder?s coordinates (??w, ??w) and translating them to thecoordinates of cylinder v:Cw(?v, ?v) =jn??C1R(?) exp[jk?v cos(?v+?)] exp[jn?+jkd?wv cos(??wv+?)]d?,(5.22)where Cw is the reflection of the cylindrical wave H(2)n (k?w) exp(jn?w) fromthe interface. The exponential is now converted into a sum of Bessel functionsand we substitute the entire expression (5.22) into the scattered field forcylinder w (5.18):V dw(?v, ?v) =??n=??j?n Jn(k?v) exp(jn?v) Dwv n, (5.23)83Dwv n =(?1)n???m=??bw m?C1R(?) exp[j(n+m)? + jkd?wv cos(??wv + ?)]d?.(5.24)The diffracted field V dw represents the reflection of the scattered field V swfrom the interface, using the coordinates of cylinder v.If we use numerical integration in (5.24), we obtain the plane-wave inte-gral method [52?54, 56?58], which is well known. Use of the Fourier seriesmethod will allow us to avoid numerical integration in (5.24), but it mayno longer be an exact solution since the evanescent wave reflections may bepartially misrepresented. We will therefore develop two evanescent field cor-rection terms for our Fourier series approach: the first term U1wv n removesthe erroneous evanescent wave reflections, and the second U2wv n contains thetrue evanescent wave reflections. The coefficients Dwv n can now be brokendown into three components Dwv n = U0wv n ? U1wv n + U2wv n, with U0wv nbeing the term for the uncorrected Fourier series method. Substituting theFourier series (5.14) into (5.24) and converting back to Hankel functions usingthe Sommerfeld integral (5.21) producesU0wv n =??l=??jl+n H(2)l?n(kd?wv) exp[j(l ? n)??wv] U0w l, (5.25)U0w l =??m=??R?l?m bw m. (5.26)where U0wv n are the coefficients U0w l translated from image cylinder w tocylinder v.It is important to note that (5.25) is in the form of the Graf additiontheorem for Hankel functions, which is a translational operation. If we remainin the coordinates of image cylinder w, the uncorrected Fourier series methodproducesV dw(??w, ??w) =??n=??jn H(2)n (k??w) exp(jn??w) U0w n, (5.27)84which represents the diffracted field from the planar interface as scatteredfields emanating from the image cylinders. This notation is useful for mod-elling the electromagnetic fields in the far-field where the evanescent waveshave died away.The evanescent field correction terms are derived by separating out theevanescent regions of the integral in (5.24), which follow the paths in C2 fromFigure 5.2. The first term U1wv n represents the erroneous evanescent fieldsfrom the Fourier series coefficients (5.14), and the second term U2wv n rep-resents the true evanescent fields using R(?). After changing the integrationvariable and simplifying, we obtainU1wv n = (?1)n??l=??U0w l? ?0Wwv n?l(u)du, (5.28)U2wv n = (?1)n??m=??bw m? ?0R(?2 + ju) Wwv n+m(u)du, (5.29)whereWwv p(u) =j2? exp[kd?wv cos(??wv) sinh(u)]? (5.30)cos[p(?2 + ju)? kd?wv sin(??wv) cosh(u)].The numerical evaluation of the true evanescent fields (5.29) is common be-tween the plane-wave integral method (5.24) and the Fourier series method.Including the evanescent field correction terms in the solution is not al-ways necessary. The significance of U1wv n and U2wv n depends on the conver-gence limits of the Laurent series (5.17), and the separation distance betweenthe cylinders and the planar interface. The convergence limits of the Laurentseries will be determined by the permittivity contrast ?2/?1 and the wave po-larization. If the cylinders have different separation distances from the planarinterface, it may only be necessary to compute U1wv n and U2wv n for specificcylinders w and v. The numerical simulations demonstrate the significance85of the evanescent field correction terms under various conditions.5.3.5 V t Transmitted Fields Inside the CylindersThe fields transmitted inside the dielectric cylinders can be represented bycylindrical waves:V tv(?v, ?v) =??n=??j?n Jn(kv?v) exp(jn?v) dv n, (5.31)with unknown coefficients dv n. The wavenumber inside cylinder v is denotedkv = ???v?0.5.3.6 Applying the Cylinders? Boundary ConditionsWe will consider homogeneous dielectric cylinders where ?v 6= ?0 and ?v =?0. The boundary conditions require that the tangential components of theelectric and magnetic fields must be continuous across the boundary of eachcylinder. For both polarizations, at the surface of the cylinders ?v = av, thefirst boundary condition isV i + V r +?vV sv +?vV dv = V tv. (5.32)For the TM case, the magnetic boundary condition is???v[V i + V r +?vV sv +?vV dv] = ???vV tv, (5.33)and for the TE case, the electric boundary condition is1?1???v[V i + V r +?vV sv +?vV dv] = 1?v???vV tv. (5.34)86Solving the set of two equations: (5.32) with (5.33) or (5.32) with (5.34), forthe scattering coefficients, yieldsbv n = fv n[ Av n + Qv n +?w 6=vBwv n +?wDwv n], (5.35)where fv n are the single scattering coefficients for dielectric cylinders, similarto Chapter 4. For TM polarization, the single scattering coefficients arefv n =Jn(kvav)Jn?(kav)?? ?v?1 Jn(kav)Jn?(kvav)? ?v?1 Jn?(kvav) H(2)n (kav)? H(2)n?(kav) Jn(kvav), (5.36)and for TE polarization they arefv n =? ?v?1 Jn(kvav)Jn?(kav)? Jn(kav)Jn?(kvav)Jn?(kvav) H(2)n (kav)?? ?v?1H(2)n?(kav) Jn(kvav), (5.37)where the primes denote a derivative with respect to the function?s argument.Equation (5.35) can be evaluated iteratively or through matrix inversion.If the Bwv n and Dwv n coefficients are set to zero initially then (5.35) can besolved iteratively by updating the values of the scattering coefficients bv n. Wecan also define an exact matrix inverse solution. The well-known plane-waveintegral solution [52?54,56?58] is given by the system~L = (F?1 ?G3?D)?1(G1 ~A+G2 ~Q). (5.38)The equivalent Fourier series solution is given by the system~L = [F?1 ?G3? (G4?U1)R2?U2]?1(G1+G2R1) ~A, (5.39)and the uncorrected Fourier series method [47,48] is given by the system~L = (F?1 ?G3?G4R2)?1(G1+G2R1) ~A. (5.40)87Table 5.1: The matrices are linked to their corresponding equations.Matrix Equation Matrix Equation Matrix EquationG1 (5.7) R1 (5.16) ~A (5.5)G2 (5.12) R2 (5.26) ~Q (5.13)G3 (5.20) U1 (5.28) F (5.36) or (5.37)G4 (U0) (5.25) U2 (5.29) D (5.24)The elements of each matrix are taken directly from the equations referencedby Table 5.1. The vector ~L contains the scattering coefficients bv n, the vector~A contains the incident field coefficients An, and the vector ~Q contains thereflected field coefficients Qn. The R matrices are made up of the Fourierseries coefficients and the G matrices are Graf addition theorem matricesthat translate coefficients from one coordinate system to another. The Umatrices are for evanescent field corrections. Similar to Chapter 4, the di-agonal matrix F contains the single scattering coefficients. The matrix D,or its equivalent (G4 ? U1)R2 + U2, accounts for the multiple scatteringbetween the cylinders and the planar interface.For comparison, we can also calculate the scattering from multiple cylin-ders without a planar interface [8, 24, 25, 28,29,34, 35,92]:~L = (F?1 ?G3)?1G1 ~A, (5.41)and the scattering from a single cylinder [20, 85]:~L = FG1 ~A, (5.42)using the same matrices from Table 5.1.885.4 Numerical SimulationsThree numerical simulations will be performed to demonstrate the accuracyof the Fourier series method and the strength of the evanescent field inter-actions. The first simulation compares the well-known plane-wave integralmethod (5.38) to the Fourier series method with both evanescent field correc-tion terms (5.39). Theoretically, the two methods should produce identicalresults regardless of the separation distance between the cylinders and theplanar interface. The second simulation compares the Fourier series methodwith both evanescent field correction terms (5.39) to the uncorrected Fourierseries method (5.40). As the cylinders are moved closer to the interface,the error produced by the uncorrected Fourier series method (5.40) shouldincrease due to stronger evanescent field interactions. The final simulationcompares the solutions of the Fourier series method (5.39) with and withoutthe true evanescent field interaction term U2. This last simulation demon-strates the significance of evanescent field interactions between the cylindersand the interface. For all three simulations the general setup is shown inFigure 5.3.89Figure 5.3: A Gaussian beam is scattered from three identical dielectriccylinders near a dielectric half-space. The incident Gaussian beam has a waistsize of w0 = 1, an amplitude factor of E0 = 1, and a frequency f = 1GHz(? ? 0.3m).When computing the solution (5.38, 5.39, 5.40) numerically, the infinitesums of cylindrical wave modes need to be truncated to appropriate limits.For the incident field and its reflection from the planar interface, the modelimit will depend on the shape of the incident beam. For the fields scat-tered from the cylinders and image cylinders, the truncation limits will beproportional to the factor kav. Choosing appropriate truncation limits incylindrical scattering was investigated in Chapter 3.900 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210?610?510?410?3Gap from Cylinder Boundary to Interface ?gap (?)Error on L (%)TM Polarization Method Error (d?2a)=?/10(d?2a)=?(d?2a)=10?0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210?310?210?1Gap from Cylinder Boundary to Interface ?gap (?)Error on L (%)TE Polarization Method Error (d?2a)=?/10(d?2a)=?(d?2a)=10?Figure 5.4: The error in ~L between the Fourier series method with evanescentfield corrections (5.39) and the plane-wave integral method (5.38) is shownfor TM and TE polarizations.First, we verify that the Fourier series method with both evanescent fieldcorrection terms (5.39) produces similar results to the plane-wave integralmethod (5.38). The three cylinders are illuminated by a Gaussian beam asshown in Figure 5.3, and the solutions are calculated multiple times withvarious distances between the planar interface and the cylinders ?gap, and91between the cylinders d. The results for TM and TE polarizations are shownin Figure 5.4. The small numerical error between the two methods is due tomodal truncation [99] and numerical integration. Therefore, the two methodsproduce equivalent results regardless of the scattering setup.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110?1010?810?610?410?2100102Gap from Cylinder Boundary to Interface ?gap (?)Error on L (%)TM Polarization Fourier Series Error (d?2a)=?/10(d?2a)=?(d?2a)=10?0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110?210?1100101102103Gap from Cylinder Boundary to Interface ?gap (?)Error on L (%)TE Polarization Fourier Series Error (d?2a)=?/10(d?2a)=?(d?2a)=10?Figure 5.5: The error in ~L between the Fourier series method with (5.39)and without (5.40) evanescent field corrections is shown for TM and TEpolarizations.Now that we have verified the accuracy of the Fourier series method with92evanescent field corrections (5.39), we compare it to the Fourier series methodwithout the two evanescent field correction terms (5.40). The simulationsare performed using the same setup shown in Figure 5.3. The TM and TEpolarization results are shown in Figure 5.5. In the TM case, the error diesaway very quickly as the cylinders are separated from the interface. For a1% error tolerance, the evanescent field correction terms would only becomenecessary when ?gap < ?/10. In the TE case, the error extends further beforedecaying, which indicates that the annulus of convergence of the Laurentseries (5.17) must be narrower than in the TM case. We would expect anarrow convergence because the Laurent series for the TE coefficients (5.9)was calculated using an 8th order spline to remove a singularity. For a 1%error tolerance, the evanescent field correction terms would become necessarywhen ?gap < 3?/4.930 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2100101102Gap from Cylinder Boundary to Interface ?gap (?)Error on L (%)TM Polarization Evanescent Field Error (d?2a)=?/10(d?2a)=?(d?2a)=10?0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2100101102Gap from Cylinder Boundary to Interface ?gap (?)Error on L (%)TE Polarization Evanescent Field Error (d?2a)=?/10(d?2a)=?(d?2a)=10?Figure 5.6: The error in ~L between the Fourier series method with andwithout the true evanescent field interaction term U2 is shown for TM andTE polarizations.Finally, we compare the Fourier series method (5.39) with and withoutthe true evanescent field interaction term U2. These simulations show theeffect of not considering evanescent field interactions between the cylindersand the planar interface. The error plots for TM and TE polarizations areshown in Figure 5.6. The evanescent field interactions between the interface94and the cylinders can remain significant until the separation distance ?gapreaches a few wavelengths. The effect of the true evanescent fields in thescattering between the cylinders and the interface dies away much slower thanthe error due to the uncorrected Fourier series method. This indicates thatthe uncorrected Fourier series method accounts for some of the evanescentfield interactions between the cylinders and the interface, which is what wewould expect if the Laurent series (5.17) converged partially into the complexdomain.Many other configurations of cylinder sizes, separation distances, and per-mittivity contrasts were simulated to confirm that these conclusions hold inthe general case. In the case where ?2/?1 < 1, both coefficients (5.8,5.9) be-come non-smooth and their Laurent series have narrower convergence limits.5.5 ConclusionTwo approaches to the scattering from cylinders near a dielectric half-spacewere derived and compared: the plane-wave integral method [52?54, 56?58]and the Fourier series method [47,48]. The plane-wave integral method pro-vides an accurate solution but requires a significant amount of numerical in-tegration that can be computationally intensive. The Fourier series methodavoids numerical integration but it can only accurately represent the reflec-tion of a limited range of evanescent waves. When the cylinders are veryclose to the dielectric half-space, the evanescent wave interactions may bedistorted. To overcome this deficiency, we introduced evanescent field cor-rection terms: U1 and U2. The cylinder-to-interface interaction matrix Din the plane-wave integral method is exactly equal to the corrected Fourierseries matrix (G4?U1)R2+U2. However, numerical simulations demon-strated that the uncorrected Fourier series method (5.40) converges fasterwith respect to ?gap than the solution (5.39) without the true evanescentfield interactions between the cylinders and the planar interface provided by95U2. This implies that the uncorrected Fourier series method is able to ac-curately model part of the evanescent field interactions. This was attributedto the fact that the Fourier series becomes a Laurent series in complex spaceand converges within an annulus extending from the real angles.A summary of the advantages of using the evanescent corrected Fourierseries method (5.39) is provided below.i. It produces accurate results for cylinders close to the planar interface.ii. For cylinders far from the interface, the evanescent field corrections canbe left out, yielding a fast computational implementation.iii. Numerical integration is not necessary for calculating the reflection ofthe incident beam (5.16).iv. Modelling the diffracted fields V d in the far-field is made efficient throughthe use of image cylinders (5.27).Our GMI solution can also be verified in the limiting case by taking ?2?1 ??, so that the dielectric half-space now acts like a perfect conductor. In thiscase, the Fourier series has only one coefficient R0 = ?1, so it converges insidethe entire complex plane. The evanescent interaction terms U1 and U2disappear and the main term G4R2 becomes the representation of perfectimage cylinders. Thus, our solution converts directly into the method ofimages for a conducting plane.Our evanescent wave corrected GMI was formed in order to simulate aper-tureless near-field scanning optical microscopy (ANSOM) imaging, whereevanescent fields interact between samples, probes, and substrates [61, 86,100]. Such an analytical solution is able to give insight into the role of multi-ple scattering in ANSOM experiments. The evanescent field interactions areparticularly important in ANSOM, as evanescent waves from an illuminatedsample are converted into radiative waves by a probe. As demonstrated96by our simulations, an evanescent field?s multiple scattering interactions caneffect the scattering coefficients significantly.97Chapter 6Multiple Scattering BetweenCylinders in Two DielectricHalf-Spaces6.1 IntroductionThe scattering from multiple cylinders near a dielectric half-space has ap-plications in ground penetrating radar (GPR), remote sensing of the earth,metamaterials, photonic crystals, and optical imaging. Several numericaltechniques have been proposed for calculating the scattering from cylindersnear planar interfaces, including extinction theorem [94,101], pseudospectraltime-domain algorithms [102, 103], and the method of moments [93]. In ourapproach, we use a cylindrical wave decomposition to find an analytical-numerical solution to the scattering problem that is accurate, computesquickly, and provides insight into the multiple scattering behaviour of thesystem.For an analytical approach to the scattering problem, we need to considerboth cylindrical waves and plane-waves. To satisfy the boundary conditionsfor a planar interface separating two dielectric half-spaces, the electromag-netic fields must be broken down into plane-waves. To satisfy the boundaryconditions for a cylinder, cylindrical waves must be used. Therefore, theinclusion of both cylinders and a planar interface will require the transfor-mation of cylindrical waves into plane-waves and vice-versa. Cincotti etal. [49] explicitly derived the plane-wave expansion for a cylindrical wave98with a Hankel function of the first kind. A major difficulty here is thatthe transformation from cylindrical waves to plane-waves leaves a numericalintegral that can be complicated and time consuming to compute [50, 51].For cylinders on one side of a dielectric half-space, Borghi et al. [53, 54],and Lee and Grzesik [56] proposed scattering algorithms that use a transfor-mation of cylindrical waves into plane-waves. Since they make no approx-imations, their scattering algorithms apply to all cases, but the numericalintegrals involved can be difficult to compute.The method of images can be applied for conducting planes [43?45] andother planar interfaces where the reflection from the interface is not angle-dependent [52]. The reflection of a cylinder?s scattered field from the planarinterface can be modelled by an image cylinder on the opposite side of theconducting plane. The method of images provides an exact analytical solu-tion to the multiple scattering without requiring numerical integration.Coatanhay and Conoir [47, 48] introduced a method for modelling thereflection of cylindrical waves from a penetrable, angle-dependent interfacewithout having to convert to plane-waves. Their method involves taking theFourier series of the planar reflection coefficients of the interface. Once inthis form, the reflection from the interface can be modelled by an imagecylinder with scattering coefficients equal to the convolutional sum of theFourier series coefficients and the cylinder?s scattering coefficients. Theirmethod does not require numerical integration but may fail when the cylin-ders are very close to the interface due to erroneous evanescent field reflec-tions. This method was adapted for electromagnetic scattering in Chapter5, where evanescent wave correction terms were also derived.Transmission of cylindrical waves across a planar interface has been con-sidered by Ciambra et al. [40], Ahmed and Naqvi [39] and Di Vico et al. [55].The scattered fields emanating from buried conducting cylinders are trans-lated back into the initial medium using a cylindrical wave to plane-wavetransformation. The resulting numerical integrals are investigated in detail.99Transmission of cylindrical waves through a planar interface was also con-sidered for cylinders embedded in a dielectric slab [41,57,58]. The dielectricslabs have two planar interfaces, which cause the cylindrical waves to reflectmultiple times between them. Each time a wave reflects inside the dielectricslab, part of the wave is transmitted to the outer medium.Our scattering model incorporates cylinders in two dielectric half-spaces,which means that the scattered fields from cylinders in one half-space interactwith the cylinders in the other half-space. To do this, we use a decompo-sition of cylindrical waves into plane-waves. This decomposition introducesintegrals that must be solved numerically. When the cylinders are signifi-cantly separated from the interface, we can approximate the integrals usingthe method of stationary phase [104, 105]. Numerical simulations are usedto compare the accuracy, efficiency, and limitations of the solution computedusing the stationary phase approximation with the solution computed usingdirect numerical integration.6.2 Scattering TheoryWe consider the two-dimensional scattering of a plane-wave from multipledielectric cylinders in two dielectric half-spaces. In our approach, we com-pute the boundary conditions in each dielectric half-space, then link the twoformulations to define the full multiple scattering solution. The coordinatesystem shown in Figure 6.1 was created to have inversion symmetry in or-der to facilitate computation from the perspective of either half-space. Theanalysis that we do from the perspective of the first medium applies directlyto the perspective of the second medium because of the inversion symmetry.The indices w and i refer to arbitrary cylinders in the first medium and theindices v and u refer to arbitrary cylinders in the second medium. The timedependence exp(j?t) is assumed and suppressed throughout.100Figure 6.1: The coordinate systems and geometry for the scattering areshown. The z direction is normal to the page. The indices w and i referto cylinders in the first medium and the indices v and u refer to cylinders inthe second medium.Our main objective is to satisfy the boundary conditions on the surfaceof the cylinders and the planar interface separating the two dielectric half-spaces. The tangential electric and magnetic fields must be continuous acrossall boundaries. For the planar interface, the boundary conditions can be sat-isfied by breaking down the incident fields into plane-waves and applyingthe well-known planar reflection and transmission coefficients. For the cylin-ders, the electromagnetic fields must be broken down into cylindrical wavescentred about each cylinder. As shown in Figure 6.2, we break down the z?directed fields into five categories:A) Vp, the principal field, depends on which side of the dielectric half-spaceis being considered. For the first medium, it incorporates the incidentplane-wave Vinc and its reflection from the planar interface Vref . Forthe second medium, it is the transmission of the incident plane-wavethrough the interface Vtr.101B) Vs, the total scattered field emanating from a cylinder, is representedby outgoing cylindrical waves with unknown coefficients.C) Vc, the total field inside of a cylinder, is represented by traversing cylin-drical waves with unknown coefficients.D) Vr accounts for the reflection of a cylinder?s scattered field Vs from theplanar interface.E) Vt accounts for the transmission of a cylinder?s scattered field Vs throughthe planar interface.Figure 6.2: The break down of electromagnetic fields is depicted.There is a problem defining the polarization states because the transversemagnetic (TM) convention for cylinders is the transverse electric (TE) con-vention for planar interfaces and vice-versa. We will use the convention forcylinders throughout. For the TM case V = Ez, and for the TE case V = Hz.1026.2.1 Vp Principal FieldFor simplicity, we have chosen to use a single plane-wave as the incidentfield. Techniques for incorporating Gaussian beam incidence or arbitrarybeam incidence have been presented for similar scattering problems [85,92].In the first medium, the principal field Vp incorporates the incident plane-wave Vinc and its reflection from the planar interface Vref . In the secondmedium, the principal field Vp is the transmission of the incident plane-wavethrough the interface Vtr. If the incident plane-wave isVinc = exp[?j?y ? j?(x? x0)], (6.1)then its reflection from the interface isVref = R(?) exp[?j?y + j?(x+ x0)], (6.2)and its transmission through the interface isVtr = T (?) exp[?j?y ? j?x+ j?x0], (6.3)where ? = k1 sin(?) and ? = k1 cos(?). The incident plane-wave angle ? isshown in Figure 6.1. The wavenumber is defined in the first medium by k1 =???1?0 and in the second medium by k2 = ???2?0. The perpendicular wave-vector component in the second medium can be calculated ? =?k22 ? ?2.The plane-wave reflection and transmission coefficients for the planar in-terface can be found by applying the boundary conditions across the interface.For TM polarization the reflection coefficients areR(ky) =kx1 ? kx2kx1 + kx2, (6.4)103and for TE polarization they areR(ky) =(?2/?1)kx1 ? kx2(?2/?1)kx1 + kx2. (6.5)For TM polarization the transmission coefficients areT (ky) =2kx1kx1 + kx2, (6.6)and for TE polarization they areT (ky) =(?2/?1)2kx1(?2/?1)kx1 + kx2. (6.7)Since the parallel wave-vector component ky is conserved across the planarinterface, the perpendicular components can be calculated as kx1 =?k21 ? k2yand kx2 =?k22 ? k2y.The fields (6.1), (6.2), and (6.3) are translated to the coordinates of acylinder i or u on their respective sides, and transformed into a sum of cylin-drical waves. The transformation from plane-waves to cylindrical waves canbe achieved by applying the Jacobi-Anger expansion [106]. Now the principalfield in the first medium Vp = Vinc + Vref , from the coordinates of a cylinderi, isVp(?i, ?i) =??n=??j?n Jn(k1?i) exp(jn?i)(Ain +Qin), (6.8)Ain = exp[?j?yi0 ? j?(xi0 ? x0)? jn sin?1(?k1)], (6.9)Qin = (?1)nR(?) exp[?j?yi0 + j?(xi0 + x0) + jn sin?1(?k1)]. (6.10)and the principal field in the second medium Vp = Vtr, from the coordinates104of a cylinder u, isVp(?u, ?u) =??n=??j?n Jn(k2?u) exp(jn?u)P un , (6.11)P un = (?1)nT (?) exp[?j?yu0 ? j?xu0 + j?x0 ? jn sin?1(?k2)]. (6.12)6.2.2 Vs Scattered Field from a CylinderThe scattered field from a cylinder i in the first medium can be representedas a sum of outgoing cylindrical waves:V is (?i, ?i) =??n=??j?n H(2)n (k1?i) exp(jn?i)bin. (6.13)The scattering coefficients bin, along with bun from the cylinders on the oppositeside of the half-space, are the unknowns that we are ultimately trying to solvefor.In the first medium, the scattered field (6.13) from a cylinder w can berepresented in the coordinate system of a cylinder i by applying the Grafaddition theorem [96]:V ws (?i, ?i) =??n=??j?n Jn(k1?i) exp(jn?i)Bwin , (6.14)Bwin =??m=??jn?m H(2)m?n(k1dwi) exp[j(m? n)?wi]bwm, (6.15)dwi =?(xi0 ? xw0)2 + (yi0 ? yw0)2, (6.16)?wi = tan?1(yi0 ? yw0xi0 ? xw0), (6.17)105where the tan?1 function should be defined for (??, ?].6.2.3 Vc Field Inside a CylinderThe fields transmitted inside a dielectric cylinder i in the first medium canbe represented by cylindrical waves:V ic (?i, ?i) =??n=??j?n Jn(ki?i) exp(jn?i)din, (6.18)where din are the unknown coefficients. The wavenumber inside a cylinder iis ki = ???i?0.6.2.4 Vr Reflection from Planar InterfaceThe scattered field from each cylinder (6.13) will travel to the planar inter-face where it will partially reflect and partially transmit. The reflection isconsidered first. Transforming the cylindrical waves into plane-waves usingthe Sommerfeld integral [49, 98] yieldsH(2)n (k1?w) exp(jn?w) =jn? ? (6.19)? ???exp[jn sin?1(kyk1 )? jxwkx1 ? jywky]kx1dky.If this wave (6.19) is propagated to the planar interface, reflected, and thenpropagated to another cylinder i in the first medium, it becomesRWwn =jn? ?? ???R(ky)kx1exp[jn sin?1(kyk1)]? (6.20)exp[jkx1(xi + xi0 + xw0)? jky(yi + yi0 ? yw0)]dky.We now substitute the reflected wave representation (6.20) into the scatteredfield representation (6.13), and apply the Jacobi-Anger expansion to convert106it into a new sum of cylindrical waves. This produces the reflection of thescattered field from a cylinder w in the coordinates of a cylinder i:V wr (?i, ?i) =??n=??j?n Jn(k1?i) exp(jn?i)Dwin , (6.21)Dwin =(?1)n???m=??bwm? ???R(ky)kx1exp[j(m+ n) sin?1(kyk1)]? (6.22)exp[jkx1(xi0 + xw0)? jky(yi0 ? yw0)]dky.The plane-wave reflection coefficients R(ky) were given in Section 6.2.1 (6.4,6.5). The 1/kx1 term causes a singularity in the integral in (6.22) at ky = k1.By substituting a new integration variable ky = k1 sin(?), the singularity canbe removed and the integral can be evaluated numerically. Much work hasbeen done in developing methods for evaluating similar integrals [50, 51].6.2.5 Vt Transmission through Planar InterfaceOur approach to calculating the transmission of scattered fields through theinterface is analogous to Section 6.2.4. The scattered field that emanates froma cylinder u in the second medium will partially transmit through the planarinterface and interact with a cylinder i in the first medium. To calculate thisinteraction, we need to first consider the scattered fields of a cylinder u inthe second medium:V us (?u, ?u) =??n=??j?n H(2)n (k2?u) exp(jn?u)bun. (6.23)107Transforming the cylindrical waves into plane-waves using the Sommerfeldintegral yieldsH(2)n (k2?u) exp(jn?u) =jn?? ???exp[jn sin?1(kyk2 )? jxukx2 ? jyuky]kx2dky,(6.24)which is similar to (6.19). If this wave (6.24) is propagated to the planarinterface, transmitted through, and then propagated to a cylinder i in thefirst medium, it becomesTW un =jn?? ???T (ky)kx2exp[jn sin?1(kyk2)? jkx2xu0]? (6.25)exp[jkx1(xi + xi0) + jky(yi + yi0 ? yu0)]dky.We now substitute the transmitted wave representation (6.25) into the scat-tered field representation (6.23), and apply the Jacobi Anger expansion toconvert it into a new sum of cylindrical waves. This produces the transmis-sion of the scattered field from a cylinder u in the coordinates of a cylinderi:V ut (?i, ?i) =??n=??j?n Jn(k1?i) exp(jn?i)Suin , (6.26)Suin =(?1)n???m=??bum? ???T (ky)kx2exp[jm sin?1(kyk2)? jn sin?1(kyk1)]? (6.27)exp[?jkx2xu0 + jkx1xi0 + jky(yi0 ? yu0)]dky.The plane-wave transmission coefficients T (ky), for a wave transmitting fromthe second medium to the first, can be found by switching the 1 and 2 indicesof (6.6) and (6.7). Although it appears that the 1/kx2 term would cause asingularity in the integral in (6.27), the coefficients T (ky) always contain a kx2term in the numerator to cancel it out. When numerically evaluating (6.27),we can exploit the inversion symmetry property: suim,n = siu?n,?m for TM108polarization and suim,n = ?1?2 siu?n,?m for TE polarization, where the s coefficientsare defined bySuin =??m=??bum ? suim,n. (6.28)6.2.6 Applying the Cylinders? Boundary ConditionsThe electromagnetic boundary conditions require that the tangential com-ponents of the electric and magnetic fields must be continuous across theboundaries of the cylinders and of the planar interface separating the twohalf-spaces. The boundary conditions for the planar interface have alreadybeen satisfied by applying the appropriate reflection (6.4, 6.5) and transmis-sion (6.6, 6.7) coefficients.We will first consider the boundary conditions for cylinders on one side ofthe planar interface, and later consider the other side. For both polarizations,at the surface of a cylinder i (?i = ri), the first boundary condition isVp +?wV ws +?wV wr +?uV ut = V ic . (6.29)For the TM case, the magnetic boundary condition is???i[Vp +?wV ws +?wV wr +?uV ut ] =???iV ic , (6.30)and for the TE case, the electric boundary condition is1?1???i[Vp +?wV ws +?wV wr +?uV ut ] =1?i???iV ic . (6.31)The sums ?w are over all cylinders in the first medium including cylinder i.The sums ?u are over all cylinders in the second medium.Solving the set of two equations: (6.29) with (6.30) or (6.29) with (6.31),109for the scattering coefficients yieldsbin = f in[Ain +Qin +?w 6=iBwin +?wDwin +?uSuin ], (6.32)where f in are the single scattering coefficients for dielectric cylinders, similarto Chapter 4. For TM polarization the single scattering coefficients aref in =Jn(kiri)Jn?(k1ri)?? ?i?1 Jn(k1ri)Jn?(kiri)? ?i?1Jn?(kiri) H(2)n (k1ri)? H(2)n?(k1ri) Jn(kiri), (6.33)and for TE polarization they aref in =? ?i?1 Jn(kiri)Jn?(k1ri)? Jn(k1ri)Jn?(kiri)Jn?(kiri) H(2)n (k1ri)?? ?i?1H(2)n?(k1ri) Jn(kiri), (6.34)where the primes denote a derivative with respect to the function?s argument.If we perform the same analysis from the perspective of the second medium,we obtainbun = fun [P un +?v 6=uBvun +?vDvun +?iSiun ]. (6.35)The principal waves switch from the incident wave Ain and its reflection fromthe planar interface Qin, to its transmission through the interface P un . Thesingle scattering coefficients fun are for cylinders in the second medium, sowe need to switch the background wavenumber k1 ? k2 and permittivity?1 ? ?2 in (6.33) and (6.34).The scattering equations on each side of the dielectric half-space (6.32,6.35) are linked through the transmission coefficients Sn, which depend onthe scattering coefficients bn from the opposite side. We can re-write (6.32)and (6.35) in matrix form respectively:~L1 = F1[ ~A1 +G1 ~L1 +D1 ~L1 + S1 ~L2], (6.36)110~L2 = F2[ ~A2 +G2 ~L2 +D2 ~L2 + S2 ~L1]. (6.37)The vectors ~L contain the cylinders? scattering coefficients bn. The vectors ~Acontain the appropriate principal waves: Ain + Qin for the first medium andP un for the second medium. The matrices G represent the direct interactionbetween cylinders in the same half-space, and the elements come from (6.15).The matrices D represent the reflection of the cylinders? scattered fields fromthe planar interface, and the elements come from (6.22). The matrices Srepresent the transmission of the cylinders? scattered fields from one half-space to the other, and the elements come from (6.27).To combine the two matrix systems (6.36, 6.37), we define~L =???~L1~L2??? , (6.38)~A =???~A1~A2??? , (6.39)F =???F1 00 F2???, (6.40)G =???G1 00 G2??? , (6.41)D =???D1 00 D2??? , (6.42)S =???0 S1S2 0??? . (6.43)111The full multiple scattering solution can now be described by~L = F[ ~A+G~L+D~L+ S~L], (6.44)which can be solved by truncating the cylindrical wave modes m and n tofinite limits and inverting~L = [F?1 ?G?D? S]?1 ~A. (6.45)Criteria for selecting appropriate limits when truncating the cylindrical wavemodes was provided in Chapter 3. These limits depend on the radius of thecylinders compared to the appropriate half-space wavelength.6.2.7 Stationary Phase ApproximationIn order to obtain the Dn and Sn coefficients, it is necessary to apply numer-ical integration to (6.22) and (6.27) respectively. The numerical integrationcan become difficult when the separation distances become large or the modenumbers m and n become large, causing the integrand to oscillate rapidly.However, for these cases we can apply the method of stationary phase toevaluate the integrals [104, 105]. The rapidly oscillating integrands in (6.22)or (6.27) will cause most of the integration path along ky to yield negligibleresults. The only significant integrand contribution to the integrals comesfrom the neighbourhood of the stationary point, determined by setting thederivative of the argument of the exponential to zero. For the reflectionintegral (6.22), the argument isf(ky) = (m+ n) sin?1(kyk1) + kx1(xi0 + xw0)? ky(yi0 ? yw0), (6.46)112and for the transmission integral (6.27), the argument isg(ky) =? n sin?1(kyk1) +m sin?1(kyk2) (6.47)+ kx1xi0 ? kx2xu0 ? ky(yu0 ? yi0).The derivatives ared f(ky)dky= (m+ n)kx1? (xi0 + xw0)kykx1? yi0 + yw0, (6.48)andd g(ky)dky= ?nkx1+ mkx2? xi0kykx1+ xu0kykx2? yu0 + yi0, (6.49)respectively. When the derivatives are set to zero it is only possible to findimplicit solutions for ky. From (6.48) we obtainky =m+ n+ (yw0 ? yi0)kx1xi0 + xw0, (6.50)and from (6.49) we obtainky =?mkx2 +nkx1 ? yi0 + yu0xu0kx2 ?xi0kx1. (6.51)where kx1 and kx2 are dependent on ky. For the implicit equations (6.50) and(6.51), we apply Newton?s Method to solve for the stationary phase pointky = kys. At the stationary point kys, the x components of the wave-vectorin the first and second mediums are kx1s and kx2s respectively. In both thereflection and transmission cases, there will be a unique stationary point kysfor each cylinder pair, and each of the mode numbers m and n.Now we Taylor-expand the square root and arc-sin functions from (6.22)113and (6.27) about the stationary phase point kys:kx1 =?k21 ? k2y ? k2y(?12 )(1kx1s+ k2ysk3x1s) + ky(k3ysk3x1s) + kx1s +k2ys2kx1s? k4ys2k3x1s,(6.52)sin?1(ky/k1) ? k2y(kys2k3x1s)+ky(1kx1s? k2ysk3x1s)+sin?1(kysk1)? kyskx1s+ k3ys2k3x1s, (6.53)The Taylor expansions for kx2 and sin?1(ky/k2) are similar to (6.52) and(6.53) respectively. The functions R(ky)/kx1 and T (ky)/kx2 are approximatedby the constants R(kys)/kx1s and T (kys)/kx2s respectively. With these ap-proximations, the reflection integral in (6.22) can be evaluated to yieldDwin ?(?1)n????m=??bwmR(kys)kx1s?a exp(b24a ? c), (6.54)a = ?j(m+ n) kys2k3x1s+ j (xw0 + xi0)2 (1kx1s+ k2ysk3x1s), (6.55)b = j(m+ n)( k2ysk3x1s? 1kx1s)? j(xw0 + xi0)k3ysk3x1s+ j(yi0 ? yw0), (6.56)c = ?j(m+n)[sin?1(kysk1)? kyskx1s+ k3ys2k3x1s]?j(xw0+xi0)(kx1s+k2ys2kx1s? k4ys2k3x1s),(6.57)and the transmission integral in (6.27) can be evaluated to yieldSuin ?(?1)n????m=??bumT (kys)kx2s?a exp(b24a ? c), (6.58)a = ?jm kys2k3x2s+ jn kys2k3x1s? j xu02 (1kx2s+ k2ysk3x2s) + j xi02 (1kx1s+ k2ysk3x1s), (6.59)114b = ?jm( 1kx2s? k2ysk3x2s)+ jn( 1kx1s? k2ysk3x1s)+ jxu0k3ysk3x2s? jxi0k3ysk3x1s+ j(yu0? yi0),(6.60)c =? jm[sin?1(kysk2)? kyskx2s+ k3ys2k3x2s] + jn[sin?1(kysk1)? kyskx1s+ k3ys2k3x1s]+ jxu0(kx2s +k2ys2kx2s? k4ys2k3x2s)? jxi0(kx1s +k2ys2kx1s? k4ys2k3x1s).In the numerical simulations we investigate the accuracy of these approxi-mations for various cylinder sizes and separation distances.6.3 Numerical SimulationsTo verify the accuracy of the presented method, the first simulation is com-pared to the finite element method. For this simulation, the stationary phaseapproximations will not be used. The finite element method is implementedusing the commercial software Comsol Multiphysics R?. The setup for thesimulations is shown in Figure 6.3. For the finite element method, the do-main was truncated to a 60m?60m square with a 2m embedded perfectlymatched layer (PML). Due to the infinite extent of both the plane-waveand the planar interface, it is very hard to obtain accurate results inside atruncated domain. The absorption of the extent of the plane-wave by thePML causes significant distortions in both the incident and scattered fields.Therefore, it is necessary to use a spatially limited incident field, such as aGaussian beam, to properly compare the two methods. How to incorporate aGaussian beam or an arbitrary beam into cylindrical scattering was providedin Chapters 2 and 4 respectively. The Gaussian beam used in the simulationis TM polarized with a wavelength of 4m. The absolute value of the electricfield across the planar interface, shown in Figure 6.4, demonstrates that thetwo methods produce visually identical results. The L1 norm error between115the two methods was only 1.57%.Thesis Addendum to Section 6.3: The convergence of the finite ele-ment method is described in Appendix C.Figure 6.3: The scattering setup for comparing the presented method withthe finite element method is shown.116?20 ?10 0 10 2000.10.20.30.40.50.60.70.80.91y (m)Electric Field Norm (V/m)Electric Field Norm across the Planar Interface Presented MethodFinite ElementFigure 6.4: The electric field norm across the planar interface is shown forboth the presented method and the finite element method. The two methodsproduce similar results.The advantages of our analytical-numerical approach over purely numer-ical methods include:? Unlimited domain of computation? Unlimited spatial frequency compatibility for evanescent waves? Efficient calculation of scattering from small cylinders? Compact representation in terms of scattering coefficients? Multiple scattering interactions are addressed in separate matricesNow the approximations for the reflection (6.54) and transmission (6.58)of cylindrical waves will be compared to their numerical integral counterparts:(6.22) and (6.27) respectively. Our aim is to gauge the accuracy of theapproximations for various positions of the cylinders with respect to theplanar interface. In our simulations we consider the error in the matrices117D1, D2, S1, and S2, and in the resulting scattering coefficients ~L. Becauseof the symmetry between S1 and S2, they always share the same error. Thepercentage errors are given by the L1 norm of the difference between thematrix computed using the stationary phase approximation and the matrixcomputed using numerical integration, divided by the L1 norm of the matrixcomputed using numerical integration. For example, for the D1 matrix, thepercentage error would be calculatedError = 100 ? |D1int ?D1app|1|D1int|1%. (6.61)All the simulations shown are for TM polarization, where the electric fieldis in the z direction. The TE polarization case has also been simulated andfound to produce similar conclusions.Figure 6.5: The scattering setup for two cylinders being moved away fromthe interface is shown.1180 1 2 3 4 510?210?1100101102103Separation Distance ?1,?2 (m)Error (%)Two Cylinders x Separation LD1D2S1,S2Figure 6.6: The error for the approximate matrices and the scattering coef-ficients is given for the case when both cylinders are moved away from theinterface.Two simulations will be performed using the setup of Figure 6.5. Theincident plane-wave approaches normal to the planar interface ? = 0. Thebackground wavelength in medium one has been normalized to ? = 1m, andin medium two the wavelength is half ? = 0.5m. The dielectric cylindershave a radius of a = 0.5m and a wavenumber k = 50.In the first case both cylinders are moved away from the interface. Theerror is shown in Figure 6.6. As the cylinders are moved away from theinterface, all the approximations to the D and S matrices converge rapidly.The approximated D2 matrix is less accurate than the D1 matrix becausethe cylinder in the second medium appears twice as large when comparedto the surrounding wavelength. This indicates that the size of the cylinderrelative to the wavelength affects the convergence significantly. The largespike in the error for the S matrices is caused by the stationary points inthe approximation becoming complex. However, the stationary points onlybecome complex when the cylinders are very close to each other and thestationary phase approximation is poor regardless. The scattering coefficients119in ~L become accurate to within 1% after the cylinders are moved away byonly ?1 = ?2 = 1m.0 1 2 3 4 510?1100101102103Separation Distance ?1 (m)Error (%)Two Cylinders x Separation LD1D2S1,S2Figure 6.7: The error for the approximate matrices and the scattering coef-ficients is given for the case when only one cylinder is moved away from theinterface.In the second case one cylinder is left near the interface ?2 = 0.1m andthe other cylinder is moved away from the interface. The error is shownin Figure 6.7. The D2 matrix approximation does not converge becausethe cylinder in the second medium remains at a fixed distance from theplanar interface. It is important to note that the approximations to the Smatrices both converge, even when one cylinder remains close to the interface.However, the scattering coefficients ~L do not converge to accurate resultsbecause of the large error produced by the D2 matrix.120Figure 6.8: The scattering setup for simulations involving four cylinders isshown.0 2 4 6 8 1010?210?1100101102Separation Distance ? (m)Error (%)Four Cylinders y Separation LD1D2S1,S2critical angle point for D2Figure 6.9: The error for the approximate matrices and the scattering coeffi-cients is given for the case when four cylinders are separated from each otherin the y direction.121Now we consider the case of four cylinders that are separated from eachother in the y direction, as shown in Figure 6.8. The cylinders, backgroundmediums, and incident wave are identical to the previous two simulations.The error plot is shown in Figure 6.9. The matrix D2 embodies the effectof a wave from a cylinder in the second medium reflecting from the planarinterface. When the ray path from one cylinder to another is close to thecritical angle of 30?, the error in the approximate D2 matrix peaks. This maybe caused by the sudden changes in the magnitude and phase of the reflectionsnear the critical angle. The matrix D1 embodies the effect of a wave from acylinder in the first medium reflecting from the interface. In this case therecannot be critical angle reflections, so the error is not affected strongly byseparating the cylinders in the y direction. The S matrices embody the effectof cylinders interacting through the interface from opposite sides. The raypath between cylinders on opposite sides of the half-space asymptoticallyapproaches the critical angle but never reaches it. This accounts for theasymptotic increase in the error of the S matrices.As a final example, to demonstrate the advantages of our approximations,we consider the computational speed and accuracy of scattering from cylin-ders far from the planar interface. The setup in Figure 6.8 is altered to have? = 30m and ? = 10m for the simulation. The incident plane-wave is alsomoved back to x0 = ?50m. Comparing the two methods, the total L1 normerror (6.61) for the scattering coefficient vector ~L is only 0.017%. However,the computational time for the approximate method is 500 times faster thanthat of the numerical integration method. The integrals were evaluated witha trapezoidal scheme, and with integration limits truncated inside the evanes-cent region at ky ? |6?|. The D matrices required 7000 integration pointsand the S matrices required 50000 integration points. Numerical integrationis clearly not a practical solution for this scattering problem because of thehighly oscillatory integrand. In this case our approximation is necessary toproduce fast and accurate results.1226.4 ConclusionAn analytical-numerical technique was presented for the scattering fromcylinders in two dielectric half-spaces. The accuracy of the presented methodwas verified by comparing its results to the finite element method. An ap-proximation based on the method of stationary phase was introduced to elim-inate the need for numerical integration when the cylinders are sufficientlyseparated from the planar interface. The stationary point was found to bedependent on the position of the cylinders and the cylindrical wave modenumbers m and n. The numerical simulations demonstrated the convergenceof the stationary phase approximation. For the reflection of cylindrical waves(6.22), the approximation converged as the cylinders were separated from theinterface. For the transmission of cylindrical waves (6.27), the approximationconverged if at least one of the two interacting cylinders was separated fromthe interface. In both cases, the approximation improved for smaller cylin-ders and angles of interaction that were close to the normal of the interface.In the case where the cylinders were far from the interface, computing thenumerical integrals became difficult due to their oscillating integrand, so thestationary phase approximation became essential. The stationary phase ap-proximation was not valid when the cylinders were near the planar interface.However, computing the numerical integrals was much easier in that case.123Chapter 7Apertureless Near-FieldScanning Optical MicroscopySimulations7.1 IntroductionThere are many aspects of ANSOM that differentiate it from traditional lensbased imaging methods [61]. Traditional images are a refocusing of the scat-tered fields emanating from the object under consideration, whereas ANSOMimages sample the near-field of the object, which contains both the incidentand scattered fields. The incident field can be removed by using evanescentwave illumination from underneath, but this method puts restrictions on thethickness and properties of the object [64].Traditional images are composed of the refocused radiation intensity fromthe surface of the source or scatterer. In ANSOM images, the field that issampled contains evanescent waves as well as radiation. The intensity ofelectromagnetic fields containing evanescent components differs from that ofradiative fields alone and needs to be considered carefully. In addition, howthe probe samples the near-field will depend strongly on the size and materialproperties of the probe-tip.The image resolution is determined primarily by the distance from theprobe-tip to the object and by the size of the probe-tip. Evanescent fields of ahigher spatial frequency decay faster than those of a lower spatial frequency.Thus, the electromagnetic fields are better defined closer to the surface of the124object. The size of the probe-tip determines the spatial size of the sampling,the signal power, and the multiple scattering effects. A larger tip will samplea larger portion of the near-field at every probe position, leading to a possiblereduction in resolution. The scattered power from a smaller probe-tip isreduced, leading to a weaker signal to noise ratio. When the probe-tip is astrong enough scattering body, there is the possibility that its scattered fieldwill interact with the object and distort the image.ANSOM imaging procedures are often modelled by replacing the vibrat-ing tip with a scattering sphere [63,100]. The size of the sphere is determinedby the radius of curvature of the probe-tip. The full shaft of the probe issimulated when shaft effects such as wave coupling or surface plasmons arebeing investigated [67, 100]. Adapting this for two-dimensional scattering,the probe sphere will be modelled by a small cylinder. Using the previouslyderived scattering models, the objects under consideration can be modelledby other dielectric or metallic cylinders in a homogeneous medium, in frontof a half-space, or buried in a dielectric half-space.To separate the scattered power of the probe-tip from the backgroundscattering, the probe is vibrated normal to the surface of the object. Thiscreates a non-linear modulation of the scattered field from the probe-tip. Inour models, this modulation will be simulated by taking several measure-ments while varying the size of the probe-tip.1257.2 The Reference ImagesFigure 7.1: The scattering setup for the ANSOM simulations is shown butthe dimensions are not to scale. The z direction is normal to the page. Thewavelength in free space is ?.In order to analyze the image collection procedure in ANSOM, we need tofirst produce reference images for comparison. The setup for the simulationsis shown in Figure 7.1. The dimensions are all with respect to the wavelengthin free space ?. The incident field is a Gaussian beam with an amplitude ofE0 = 1 V/m and a beam waist of w0 = 200?. All of the cylinders are perfectelectric conductors (PEC). The probe-tip is modelled by the small cylinderin free space and the object to be analyzed is composed of the two largercylinders buried in the dielectric half-space.The probe-tip is scanned across the image plane at a distance of 0.01?from the planar interface, and the scattered power from the probe-tip ismeasured. Since ANSOM uses far-field power measurements, the scattered126field from the probe-tipVs(?, ?) =??n=??j?n H(2)n (k1?) exp(jn?)bn, (7.1)needs to be converted into a measure of time averaged intensity Sav, in W/m2.The time averaged intensity in the far-field isSav(?, ?) =C|Vs(?, ?)|22 , (7.2)where C = 1/? for TM polarization and C = ? for TE polarization. Theintrinsic impedance of the background medium is ? =??? . TM polarizationis defined as V = Ez and TE polarization is defined as V = Hz. In thefar-field, the Hankel function in (7.1) can be approximated by [96]H(2)n (k1?) ??2?k1?exp(?jk1?+ jn?/2 + j?/4). (7.3)Inserting our far-field approximation (7.3) into the scattered field (7.1) andtaking the square magnitude yields|Vs|2 ?2?k1?|??n=??exp(jn?)bn|2. (7.4)To calculate the total backscattered power, we integrate the time averagedintensity over a semi-circular region? 3?/2?/2Sav(?, ?)?d? =C?k1? 3?/2?/2|??n=??exp(jn?)bn|2d?. (7.5)The total backscattered power is measured in W/m, where the per-meterrefers to the z dimension.127?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 1468101214 x 10?6Probe Location in y Direction (?)Probe?s Backscattered Power (W/m)(a) A simulated ANSOM image is shown for TM illumination.?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 11.51.551.61.651.71.751.81.851.9 x 10?12Probe Location in y Direction (?)Probe?s Backscattered Power (W/m)(b) A simulated ANSOM image is shown for TE illumination.Figure 7.2: The probe-tip is scanned across the image plane and its scatteredpower is recorded to create simulated ANSOM images.128The image results for TM polarization are shown in Figure 7.2a and theresults for TE polarization are shown in Figure 7.2b. The centres of theobject cylinders are indicated with dashed lines. Even though the two objectcylinders are spaced ?/10 apart, their effects are clearly distinguished withsubwavelength resolution in each case. However, it is not obvious how tointerpret the data contained in either image.7.3 Interpreting Collected DataOne of the main tasks in ANSOM is to correctly interpret the collected datacontained in the images. As shown by Kim and Song [64], the images arenot equivalent to a topographical profile of the object, but also contain theeffects of the optical properties of the object. Since our object is made up oftwo cylinders buried in a dielectric half-space, the topological profile wouldbe flat. To find where the ANSOM image data originates, the near-field ofthe object should be investigated.The time averaged electromagnetic field intensity in the near-field of theobject isSav =12?{E?H?}. (7.6)In the presence of evanescent waves, the electromagnetic intensity is nolonger proportional to the square magnitude of the electric or magnetic fieldsSav = C/2|Vs|2. However, we will use C|Vs|2 as a measure of the individualelectric/magnetic field intensity to compare the images to, even though itdoes not represent the electromagnetic field intensity in the near-field.129?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 10.0240.0250.0260.0270.0280.0290.030.0310.032Probe Location in y Direction (?)Time Average Intesity (W/m2 )(a) The electromagnetic field intensity is plotted across the image plane.?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 10.010.01050.0110.01150.0120.01250.0130.01350.014Probe Location in y Direction (?)Electric Field Intesity |V s|2 /?(b) The tangential electric field intensity is plotted across the image plane.Figure 7.3: The electromagnetic intensity is compared to the electric fieldintensity across the image plane. TM polarization is used.130First, we will consider the TM polarization case. Figure 7.3a shows theelectromagnetic field intensity along the image plane. For comparison, wealso provide the intensity of the electric field across the same plane in Fig-ure 7.3b. It is clear from Figure 7.2a that the power scattered by the probe isdirectly proportional to the electric field intensity but not to the electromag-netic field intensity. This phenomenon can be attributed to the scatteringproperties of the probe cylinder. Since the cylinder is a perfect electric con-ductor, the tangential electric field must be zero at the boundary of thecylinder. For TM polarization, the electric field is always completely tangen-tial to the cylinder, causing the scattering coefficients to be proportional tothe electric field.131?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 10.0270.02750.0280.02850.0290.0295Probe Location in y Direction (?)Time Average Intesity (W/m2 )(a) The electromagnetic field intensity is plotted across the image plane.?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 10.2890.290.2910.2920.2930.294Probe Location in y Direction (?)Magnetic Field Intesity ? |V s|2(b) The tangential magnetic field intensity is plotted across the image plane.Figure 7.4: The electromagnetic intensity is compared to the magnetic fieldintensity across the image plane. TE polarization is used.132Now we will consider the TE polarization case. Figure 7.4a shows theelectromagnetic field intensity along the image plane. For TE polarization,we also provide the intensity of the magnetic field across the same plane inFigure 7.4b. Unlike the TM polarization case, the image from Figure 7.2bis closely related to the electromagnetic field intensity. We expect that thePEC probe would scatter power proportional to the tangential electric field.However, in the TE polarization case, the tangential electric field is in the ??direction which is dependant upon the position of the cylinder.7.4 DemodulationIn an actual ANSOM procedure, to recover the scattered power from theprobe-tip alone, the probe shaft is modulated in the x? direction at a fre-quency ?. The distance from the probe-tip to the object is given by x(t) =x0 + A cos(?t), where x0 is the midpoint offset and A is the amplitude ofoscillation. The total scattered field at a point (x?, y?) in the far-field iscomposed of the background scattering Eb exp(j?1) and the probe-tip scat-tering Ep exp(j?2). Both the phase ?2 and amplitude Ep of the scatteredfield from the probe-tip depend on the phase of the spatial modulation (?t).The time dependence of the radiation exp(j?t) has been suppressed. Thetime-averaged (over the period of the radiation 2?/?) scattering intensity atthe point (x?, y?) in the far-field is given bySav =|E|22? =|Eb|2 + |Ep|2 + 2EbEp cos(?1 ? ?2)2? . (7.7)The first term |Eb|2 can easily be removed through demodulation becauseit has no time dependence. The second term |Ep|2 is the power scatteredby the probe-tip, which we would like to recover. The third term is theinterference between the background scattering and the probe-tip scattering.Homodyne detection can be used to remove this interference signal [69]. A133large reference signal Er, with a well defined phase ?3, can be used to overridethe interference between Eb and Ep with a much larger interference betweenEr and Ep. With the addition of the reference signal, the scattered intensitybecomesSav =|Eb|2 + |Ep|2 + |Er|2 + 2EbEp cos(?1 ? ?2)2? (7.8)+2EbEr cos(?1 ? ?3) + 2ErEp cos(?3 ? ?2)2? .After demodulation, all of the constant terms are removed leavingSav =|Ep|2 + 2EbEp cos(?1 ? ?2) + 2ErEp cos(?3 ? ?2)2? ?ErEp cos(?3 ? ?2)? .(7.9)If the amplitude of the reference signal is large compared to the backgroundsignal and the probe-tip signal Er ? Eb ? Ep, then the last term in (7.9)will dominate. By taking two measurements, one with ?3 = 0 and one with?3 = ?/2, the magnitude Ep and phase ?2 of the scattered field from theprobe-tip can be recovered.The only difficulty that still remains is how to perform the demodulation.Due to the spatial modulation, the scattered field from the probe-tip will varyin a non-linear way, creating an infinite set of harmonics. Kim and Leone [71]state that the harmonics n? of the scattered field from the probe-tip areapproximately equal to the x-coordinate partial derivative of the local fieldEn(x0, y) =12?? 2?0E[f(?), y] exp(jn?)d? ? ?nE(x, y)?nx |x=x0 . (7.10)This approximation converges in the limit as the amplitude of modulation Abecomes negligibly smaller than the wavelength ? [69]. To reconstruct a two-dimensional map of the near-field, all of the harmonic components of the fieldwould be necessary. It is common for only one harmonic component to bedemodulated, and this component is assumed to be proportional to the near-134field E(x0, y) ? En(x0, y). This assumption relies on the x coordinate of thenear-field being locally separable E(x, y) = Y (y)X(x), so that the derivativeof the field (7.10) will be directly proportional to the field profile in they direction. Knoll and Keilmann [68] demonstrated that higher harmonicdemodulation is able to reject the background scattering better, resultingin better image resolution and contrast. However, the scattering signal forhigher harmonics decreases in amplitude significantly.135?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 10.20.30.40.50.60.70.80.91Probe Location in y Direction (?)Demodulated Signal (Normalized) n=1n=2n=3(a) Demodulated ANSOM images are shown for TM polarization.?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 10.20.30.40.50.60.70.80.91Probe Location in y Direction (?)Demodulated Signal (Normalized) n=1n=2n=3(b) Demodulated ANSOM images are shown for TE polarization.Figure 7.5: The first, second, and third harmonic signals are demodulatedusing homodyne detection.136Since our ANSOM model does not include the probe-shaft, we will simu-late the modulation by changing the size of the probe-tip. Our probe-tip willmodulate from no cylinder at all a = 0, to the full sized cylinder a = 0.001?.A set of new simulations were created using this modulation scheme with TMand TE polarizations and the setup shown in Figure 7.1. Plots of the firstthree demodulated harmonics are displayed in Figure 7.5. The demodulatedpower was measured by taking a discrete spatial derivative of the far-fieldscattering intensity measurements for different probe-cylinder sizes. The de-modulated signal amplitude is proportional to the scattering intensity of theprobe-tip but the magnitude is irrelevant. Hence, the magnitudes have allbeen scaled to a maximum amplitude of one. Comparing the TM harmonicsignals from Figure 7.5a with the same image produced directly from thescattered power of the probe-tip in Figure 7.2a, we can see that all threeharmonics recover the image well. For TE polarization, the third harmonicfrom Figure 7.5b provides a better recovery of the scattered power profilefrom Figure 7.2b. The difference in performance of the two polarizationsis due to the difference in the non-linear response of the modulation of theprobe cylinder in each case. The rejection of the background signal in thethird harmonic demodulation is not observed here, which may be due to notmodelling the probe shaft.7.5 Interference of WavesOne of the major differences between a traditional image and an imageformed from a scanning probe method is the interference between the in-cident field and the scattered field. In a traditional image, a lens is used torefocus the scattered waves back into their original form at the surface of theobject. In a scanning probe image, the probe samples the near-field distri-bution which contains both the incident and scattered fields. The additionof the incident and scattered fields creates an interference pattern.137?1 ?0.5 0 0.5 10123456 x 10?5Probe Location in y Direction (?)Probe?s Backscattered Power (W/m)(a) An altered ANSOM image is shown for TM polarization.?1 ?0.5 0 0.5 100.511.522.533.5x 10?14Probe Location in y Direction (?)Probe?s Backscattered Power (W/m)(b) An altered ANSOM image is shown for TE polarization.Figure 7.6: Altered ANSOM images are shown where the probe-tip samplesonly the scattered field from the object. 138If we do not allow the incident field to scatter from the probe-tip, thenthe ANSOM images for TM and TE polarizations are shown in Figure 7.6aand Figure 7.6b respectively. Comparing these plots to our original ANSOMimages in Figure 7.2a and Figure 7.2b, it is clear that the interference pat-terns from the addition of the incident field into the image, completely alterthe interpretation of the image. In a real ANSOM image, the presence ofthe object cylinders creates localized dips in the field strength because thescattered field from a PEC cylinder is out of phase with the incident field. Ina traditional image, the scattering from a PEC cylinder would appear as aspike in localized field strength because the incident field has been removed.Comparing the results for the TM (Figure 7.6a) and TE (Figure 7.6b)polarizations, we notice that the resolution of the TE case far surpasses thatof the TM case. This may be due to enhanced evanescent field recovery inthe TE case. This can be seen by analyzing the electric and magnetic fieldsof an evanescent waveV = exp(?jkyy ? ?x)z?, (7.11)where V = Ez for TM polarization and V = Hz for TE polarization. Foran evanescent wave, one component of the wave-vector k = (kx, ky, 0) isimaginary kx = ?j?k2y ? k2 = ?j?, and the other is super-oscillatory ky >?/c. The corresponding field in the x ? y plane can be calculated fromMaxwell?s equations. The magnetic field for TM polarization isH = V??(kyx?+ j?y?), (7.12)and the electric field for TE polarization isE = V??(?kyx?? j?y?). (7.13)As the spatial frequency increases above the wavenumber ky > ?/c, the139fields in the x? y plane grow in magnitude compared to the corresponding z?directed field V. In the TM case, the evanescent wave skews towards strongermagnetic fields; in the TE case, the evanescent wave skews towards strongerelectric fields. However, when the evanescent wave is converted into radiationthrough scattering by the probe-tip, the wave impedance of the radiationmust return to the intrinsic material impedance. For a PEC probe-tip, thescattering will be proportional to the local electric field. If the polarization isTE, the electric component of the evanescent waves will be stronger, leadingto an enhanced conversion of evanescent waves into radiation.7.6 Tip-Object InteractionIn ANSOM, one of the assumptions is that the probe-tip does not distort thenear-field distribution through multiple scattering. If the probe-tip is verysmall then the scattered fields emanating from it will also be very small.If these small fields are negligible compared to the scattered field from theobject, then this approximation will be good.One benefit of our model in simulating these effects is that the multiplescattering between any cylinders can be removed easily. To determine if tip-object coupling is distorting our image, we need to remove the ability of theprobe-tip?s scattered field to affect the object cylinders. This can be doneeasily by removing one of the S matrices in the simulation.140?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 124681012 x 10?6Probe Location in y Direction (?)Probe?s Backscattered Power (W/m) couplingno couplingFigure 7.7: The ANSOM images for TM polarization with and without tip-object coupling are shown.For the TM case, a distortion of the fields due to tip-object interactionscan already be observed at a tip radius of a = 0.001?. The effects of tip-object coupling are displayed in Figure 7.7. It is clear that the tip-objectinteractions distort the image from its original field representation. Theresolution of the distorted image has been reduced. Ideally, the ANSOMimage should follow the distribution of the tangential electric field shownin Figure 7.3b. Reducing the size of the probe-tip will help to reduce thedistortion. The drawback is that a smaller probe-tip also reduces the amountof scattered power and consequently, the signal to noise ratio.1417.7 Probe to Object DistanceEvanescent waves with higher spatial frequencies decay faster with distancefrom their surface of origin. This means that the distance from the probe tothe surface of the object will limit the resolution of the image. The closerthe probe gets to the surface, the stronger and more clear the evanescentcomponents will be. Ideally, the probe should be brought as close to thesurface of the object as possible without touching it.The setup in Figure 7.1 was simulated using TE polarization with severaldistances between the half-space and the probe xg: 0.02?, 0.05?, 0.1?, and0.5?. The images produce at each distance are displayed together in Fig-ure 7.8. It is clear that the resolving power of the scanning probe methoddeteriorates quickly as the probe is separated from the object by only halfa wavelength. The change in power level between the different distances isdue to the partial standing wave produced by the reflection of the incidentbeam from the dielectric half-space.142?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 11.52x 10?12Probe Location in y Direction (?)Probe?s Backscattered Power (W/m) xg=0.02 ?xg=0.05 ?xg=0.1 ?xg=0.5 ?Figure 7.8: ANSOM images are shown for various distances separating thehalf-space and the probe. TE polarization was used.7.8 Wavelength and Object ParametersFinally, we observe how the images are affected by changes in the incidentwavelength and in the object parameters. The changes in the images willindicate how the collected super-resolution data relates to the object?s prop-erties.143?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 10.820.840.860.880.90.920.940.960.981Probe Location in y Direction (?)Normalized Backscattered Power Smaller WavelengthOriginalFigure 7.9: The ANSOM images for TE polarization with different incidentwavelengths are shown.The main advantage of ANSOM is that we can obtain higher resolutionimages without decreasing the incident wavelength. In Figure 7.9, the nor-malized incident wavelength is compared to an incident wavelength of half thesize. The two sets of image data have had their maximum amplitude?s nor-malized in order to compare their contours rather than their overall powerlevels. The resolution of the smaller incident wavelength may be slightlybetter because we can observe the higher peak between the two object cylin-ders. However, the super-resolution achievement at this new wavelength isonly ?/5.144?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 11.51.551.61.651.71.751.81.851.9 x 10?12Probe Location in y Direction (? original)Probe?s Backscattered Power (W/m) Deeper ObjectsOriginalFigure 7.10: The ANSOM images for TE polarization with different objectcylinder depths are shown.?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 11.51.551.61.651.71.751.81.851.9 x 10?12Probe Location in y Direction (?)Probe?s Backscattered Power (W/m) Smaller ObjectsOriginalFigure 7.11: The ANSOM images for TE polarization with different objectcylinder radii are shown.145?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 11.51.551.61.651.71.751.81.851.9 x 10?12Probe Location in y Direction (?)Probe?s Backscattered Power (W/m) Wider SeparationOriginalFigure 7.12: The ANSOM images for TE polarization with different objectcylinder separation distances are shown.In the last three simulations we compare the images produced by vary-ing the object?s properties. In Figure 7.10, the effect of burying the objectcylinders ten times deeper within the dielectric half-space is observed. Afterburying the object cylinders deeper, we can no longer recover their evanes-cent fields at the image plane because evanescent fields are only local to thesurface of a scattering body. Therefore, ANSOM is only useful at recoveringsuper-resolution data for features within a very limited surface depth of theobject. However, ANSOM still detects the effects of buried features even ifit cannot resolve them. In Figure 7.11, the effect of decreasing the objectcylinders? radii by one half is observed. The effect of changing the radii isclearly distinguishable from the effect of burying the cylinders deeper. Theprimary outcome from modifying the cylinder radii is the change in ampli-tude of the object cylinders? scattering and consequently of the recoveredimage contrast. In Figure 7.12, the effect of separating the cylinders withtwice as much distance is observed. The shape of the recovered image follows146the location of the cylinders which may indicate that geometrical informationabout the object cylinders could be recovered using this technique.7.9 ConclusionThe electromagnetic scattering models developed in the previous chapterswere applied to simulations of ANSOM. The power scattered by the probe-tip was shown to be proportional to the square magnitude of the tangentialelectric field for a PEC cylinder. For evanescent waves, the electromagneticfield intensity is not directly proportional to the square magnitude of theelectric field. This implies that the information collected through ANSOMwill depend on the material properties of the probe-tip and the polarizationof the incident electromagnetic field.The effects of vibrating the probe-shaft and demodulating the receivedfield through homodyne detection were analyzed. The vibration of the probe-shaft modulates its scattered field in a non-linear way, leading to a widerange of harmonics. The results of our simulations demonstrated that thethird harmonic demodulation may contain higher resolution than the firsttwo harmonics, but this effect was polarization dependent.The probe-tip scatters both the incident field and the scattered field fromthe object. These two fields add together to create interference patterns thatare not present in traditional images that only contain the scattered fields.These interference patterns must be taken into account when interpretingthe ANSOM images.One of the most important assumptions in ANSOM is that the probe-tipdoes not distort the near-field through multiple scattering. This assumptionwas tested by removing the tip-object coupling matrix S in the scatteringcalculation. This new image displayed noticeable changes in field shape com-pared to the original ANSOM image, proving that tip-object coupling canhave significant effects.147One of the most important aspects of ANSOM imaging was also charac-terized: the distance from the probe-tip to the object. Due to the exponentialdecay of the evanescent fields, the simulated ANSOM images lost resolutiondrastically as the probe-tip was separated from the object.Varying the object properties and observing the changes in the imagesallowed us to better understand the capabilities of ANSOM. The super-resolution data was produced by evanescent fields that were local to thesurface of scattering bodies. Buried bodies needed to be within a fractionof a wavelength from the surface for their evanescent fields to be recoveredfrom the other side of the surface. Our observations from moving the objectcylinders laterally across the surface indicate that it may be possible to re-cover the geometrical and optical properties of surface features using thesetechniques.148Chapter 8Summary and ConclusionsSuper-resolution imaging is possible if evanescent field data can be collected.When an evanescent field scatters from a subwavelength object it partiallyconverts into radiation that is able to transfer energy into the far-field where itcan be detected. To characterize this phenomenon and to simulate the super-resolution imaging process, four electromagnetic scattering models were de-veloped. A cylindrical wave decomposition was used to satisfy the boundaryconditions for the cylinders directly. These analytical solutions have severaladvantages over numerical methods including:A) No spatial discretizationB) No limits to far-field modellingC) More efficient modelling of high spatial resolutionsD) Efficient modelling of small scattering bodiesE) A solution in terms of compact scattering coefficientsF) A specific breakdown of scattering interactions is accessibleHence, for looking at specific evanescent wave scattering phenomena theanalytical solutions are ideal.In Chapter 2, the analytical solution for the scattering from a cylinder wasextended to include evanescent field incidence. The near-field of a Gaussianbeam was scattered from a dielectric cylinder and the conversion of evanes-cent waves into radiation was observed. The conversion was attributed to149the spatial redistribution of the electromagnetic fields during the scattering.In order to compute the scattering, the incident evanescent field had to berepresented by cylindrical waves composed of Bessel functions of the firstkind. When the sum of cylindrical waves was truncated, the evanescent fieldonly converged within a specific radial distance from the origin. This conver-gence is important because it limits the use of the Graf addition theorem intranslating evanescent fields represented in this way. Using a sum of Gaus-sian beams to represent an arbitrary field with evanescent components wasfound to be inefficient because the evanescent components make numericalintegration necessary.The proper truncation of cylindrical wave modes in the two-dimensionalscattering from cylinders was investigated in Chapter 3. Previous authorsgave linear estimates for proper modal truncation [26?28], but one linearapproximation cannot suffice for cylinders of all sizes. Therefore, estimatesfor minimum mode limits were formed for cylinders with small, medium, andlarge radii with respect to the wavelength. The ratio of the cylinder radius tothe wavelength was the primary factor in determining the appropriate modelimit. If too few modes were used in the scattering calculation, the fieldswere misrepresented. If too many modes were used in multiple scatteringcalculations, the matrix inversion became ill-conditioned, leading to numeri-cal errors. To maximize the accuracy of the scattering calculation, the modelimit was chosen above a minimum limit for accuracy and below a maximumlimit for matrix conditioning.In Chapter 4, the vector plane-wave-spectrum (VPWS) was used to in-troduce arbitrary radiative and evanescent field incidence into the multiplescattering from dielectric and conducting cylinders. The solution was formedinto a T-matrix, which multiplies the incident field coefficients to producethe scattering coefficients. A method of evanescent field analysis was pro-posed using a grating of cylinders. The grating of cylinders converted anincident evanescent wave into a radiative beam that propagated at an angle150that depended on the spatial frequency of the incident field. The total an-gular distribution of converted beams could then be used to determine thespatial frequency content of the incident evanescent field.In many imaging scenarios the objects under consideration are placed ontop of a planar surface or buried inside of a dielectric. To accommodatethese cases, the scattering from cylinders was adapted to include a dielec-tric half-space. Two methods for calculating the multiple scattering fromcylinders near a dielectric half-space are the Fourier series method and theplane-wave integral method. In Chapter 5 the application of a Fourier seriesto transform the planar reflection coefficients into an angular form was intro-duced into vector electromagnetic scattering for the first time. The Fourierseries method and the plane-wave integral method were derived alongsideeach other in order to draw parallels between the two methods. The meth-ods would have been exactly equivalent except for the fact that the Fourierseries only converged a limited distance into the complex domain. This con-vergence depended on the form of the reflection coefficients for the planarinterface separating the two dielectric half-spaces. This limited convergenceinto the complex domain meant that some of the evanescent field interactionsbetween the cylinders and the planar interface were misrepresented. To ac-count for these errors, evanescent field correction terms were derived. Theplane-wave integral method was always accurate for evanescent fields, but itrequired a significant amount of numerical integration.The scattering from cylinders on both sides of a dielectric half-space wasconsidered for the first time in Chapter 6. This model allowed us to sim-ulate a scanning probe cylinder on one side of the half-space and objectcylinders on the opposite side. The scanning probe cylinder could be usedto scatter evanescent fields emanating from the object cylinders buried inthe half-space, converting them into radiation. This radiation could then becollected and used to form a super-resolution image of the buried cylindersas demonstrated in Chapter 7. The multiple scattering between cylinders on151opposite sides of a dielectric half-space was analyzed using the Sommerfeldintegral to transform cylindrical waves into plane-waves. To eliminate theneed for numerical integration when applying the transformation, an approx-imation based on the method of stationary phase was introduced. For themultiple scattering between cylinders in opposite half-spaces, the cylindersneeded to be sufficiently separated from each other for the approximation tobe accurate. For the multiple scattering between the cylinders and the planarinterface, the cylinders need to be sufficiently separated from the interface forthe approximation to be accurate. The approximation was necessary becausedirect numerical integration became extremely difficult when the separationdistances were large.Finally, in Chapter 7 the previously derived scattering models were usedto simulate ANSOM images. The collected image data was demonstratedto be dependant on the scattering properties of the probe-tip and the polar-ization. Because of the skewed proportion of electric to magnetic fields thatoccurs in evanescent waves, TE polarization recovered better resolution thanTM polarization, using a PEC cylinder. The modulation and demodulationof the probe-tip was characterized using homodyne detection with a largereference signal. The difference in the results obtained using the first threeharmonics was attributed to the non-linear response of the scattered powerto the modulation. The effects of tip-object interactions were demonstratedexplicitly by removing the matrix that couples the tip to the object whencomputing the scattering. A smaller tip prevented tip-object distortion butalso reduced the signal to noise ratio. A major consideration when interpret-ing ANSOM images was that the probe-tip scatters both the incident fieldand the scattered field from the object. Traditional imaging methods onlyrecover the scattered field from the object. The addition of both the incidentfield and the scattered field caused interference patterns to arise in the image.To demonstrate the underlying super-resolution principles, the relationshipbetween resolution and tip-object distance was observed for distances from152?/100 to ?/2. Finally, simulations varying the object?s properties and theincident wavelength were produced to uncover the limitations of the imagedata. The super-resolution data was shown to contain information primarilyabout the near-surface of the object.8.1 Future WorkThe electromagnetic scattering models derived in chapters 2, 4, 5, and 6 areuseful for many applications that have not been considered in this thesis,including the design and analysis of metamaterials and photonic crystals,and ground penetrating radar (GPR) simulations.Metamaterials are man-made structures that are able to mimic uniquematerial properties such as negative permittivities, negative permeabilitiesand low/zero refractive index [3]. The metamaterial must be made out ofstructures with a subwavelength periodicity so that it appears homogeneousto the wavelength under consideration. The electromagnetic scattering mod-els produced in the preceding chapters can be used to model a metamaterialas an array of cylinders arranged in a repeatable pattern. The analytical so-lutions are particularly helpful in the design process as they provide a lot ofinformation about the coupling between cylinders, which ultimately providesthe effective material properties. These models are particularly well-suitedfor simulating wire media metamaterials as proposed by Pendry et al. [107].These metamaterials are composed of thin conducting wires positioned in agrid.Metamaterials are often used to create double negative materials in whichthe permittivity and the permeability of the material are negative. In thiscase the refractive index becomes negative but the impedance remains pos-itive. These double negative materials (DNG) can be used as a super-lens,capable of providing super-resolution images. The double negative materialenhances evanescent fields that are incident upon its surface by switching the153exponential decay into exponential growth. Other applications for metamate-rials include artificial magnetism for magnetic resonance imaging (MRI) [108]and phase compensation in transmission lines.The models that have been presented are also excellent for simulatingtwo dimensional photonic crystals composed of cylindrical structures. Pho-tonic crystals are periodic structures that contain elements of higher andlower permittivities. Unlike metamaterials, the photonic crystal?s size andperiodicity is on the order of the wavelength. The material effects due todiffraction between elements cannot be described by an effective permittiv-ity and permeability. However, photonic crystals can produce photonic bandgaps, which allow for the control and manipulation of light. 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(A.2)Since the wavenumber is defined by k = ????, it must be constant unlessthe medium changes. Therefore, the x component of the wave-vector can bedefinedkx =?k2 ? k2y |ky| ? k radiative?j?k2y ? k2 |ky| > k evanescent. (A.3)The waves emanating from the object (A.1) disperse at various anglesdefined by ? = arcsin(ky/k). If the collecting lens is in the far-field, wecan assume that these waves behave like rays travelling at an angle ?. Thelens can only collect rays with a maximum angle of ?, or equivalently, amaximum ky value of kmax = k sin(?). When backpropagated or refocused,the reconstruction of the original wave-field V (0, y) will beVrc(0, y) =12?? kmax?kmaxF (ky) exp(?jkyy)dky, (A.4)which now has a spatial frequency limit of kmax. By the Nyquist samplingcriteria for perfect reconstruction, the field Vrc needs to be sampled at afrequency 2kmax2? =2n sin(?)?0 samples/m, where ?0 is the wavelength in freespace. Inverting this spatial sampling gives the diffraction limit d = ?02n sin(?)m/sample. From the sampling criteria, there is no information contained inthe reconstructed image about points spaced closer than d apart.The resolution can be increased by moving the lens closer to the objector by expanding the size of the lens. In the limiting case ? ? ?/2 and theresolution reaches d = ?/2. However, it is clear from the integration extentsof (A.1) that infinite resolution exists at the surface of the object and thehigher spatial frequencies are manifested as evanescent waves that exist onlyin the near-field due to their exponential decay. To recover resolution beyond?/2, near-field scanning methods must be used to collect evanescent wave169information.170Appendix BTruncating Cylindrical WaveModes for Very SmallCylindersAs the cylinder size a becomes very small compared to the surrounding wave-length ?, the minimum mode limits for the two polarizations need to be con-sidered more carefully. Applying Bessel function approximations for smallarguments (ka? 0) [96] to the single scattering coefficients for a PEC cylin-der under TM illumination yields? Jn(ka)H(2)n (ka)?? j?2 ln(ka) n = 0j?(ka/2)2|n||n|!(|n|?1)! n 6= 0, (B.1)and under TE illumination yields? Jn?(ka)H(2)n?(ka)?j?(ka/2)2 n = 0? j?(ka/2)2|n||n|!(|n|?1)! n 6= 0. (B.2)The magnitude of the n 6= 0 modes for the two polarizations is equivalentfor very small cylinders. However, the zeroth mode differs greatly betweenthe two polarizations. Under TM polarization, the PEC cylinders continueto scatter strongly due to the induction of currents along the shafts of thecylinders. Under TE polarization, the zeroth mode approaches the samemagnitude as the n = ?1 modes. Therefore, when modelling the scatteringfrom very small PEC cylinders under TE illumination, it will always be171necessary to include the modes n = ?1, 0, 1, but under TM illumination, themode n = 0 will dominate over the much smaller modes n = ?1.If the cylinder had magnetic properties then the reverse would be true.For a subwavelength PMC cylinder under TE illumination, the mode n = 0will dominate over the much smaller modes n = ?1. Under TM illumination,the three modes n = ?1, 0, 1 will approach the same magnitude.172Appendix CConvergence of Finite ElementMethod ComparisonTable C.1: The mesh size parameters of the finite element method are com-pared to the resulting error.maximumelementsizeminimumelementsizemaximumelementgrowthrateresolutionof curva-tureresolutionof narrowregionserror(%)19.8 3 2 1 0.9 31.657.8 0.36 1.5 0.6 1 1.894.02 0.018 1.3 0.3 1 0.552.22 0.0075 1.25 0.25 1 0.49In Section 6.3, a comparison was made between the presented method and thefinite element method. To demonstrate the convergence of the two methods,several mesh sizes were simulated for the finite element method and comparedto a simulation of the presented method with high precision. A table of themesh parameters and the resulting error between the two methods is shownin Table C.1. The primary sources of error are the discretization error for thedifferential equation and the shape mismatch of fitting triangular elementsto circular scatterers. Simulating the presented method with even higherprecision showed no apparent effect on the resulting error. The decrease inthe convergence rate of the error for very fine mesh grids may be attributedto other possible sources of error such as the finite size of the spatial domainor the imperfection of the PMLs.173
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Evanescent field interactions in the scattering from cylinders with applications in super-resolution Pawliuk, Peter Cornelius 2013
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Title | Evanescent field interactions in the scattering from cylinders with applications in super-resolution |
Creator |
Pawliuk, Peter Cornelius |
Publisher | University of British Columbia |
Date Issued | 2013 |
Description | The diffraction limit defines the maximum resolution of an imaging system that collects and focuses waves. This limited resolution arises from the finite length of the waves used to create the image. Therefore, the only way to increase the resolution is to use higher frequencies with shorter wavelengths. For situations in which increasing the frequency is not possible or not desirable, super-resolution imaging techniques can be applied to overcome the diffraction limit. Super-resolution is possible with the inclusion of evanescent waves, which exhibit unlimited spatial frequencies. Evanescent waves decay exponentially away from their surface of origin so they are difficult to recover. One way to recover evanescent wave information is to scatter the wave from a small object. This scattering converts part of the evanescent wave into radiation that can propagate into the far-field where it can be detected. In order to characterize this conversion, the two-dimensional scattering of evanescent fields from a single cylinder and from multiple cylinders is investigated. The scattering models are derived using an analytical approach where the electromagnetic fields are broken down into cylindrical waves so that the boundary conditions on the cylinders can be applied directly. The incident field can be formulated from a vector plane-wave spectrum, which allows for an arbitrary combination of radiative and evanescent waves. Multiple cylinders of various sizes can be used to approximate the scattering from many two-dimensional objects. For simulating the imaging of objects buried underneath a surface, or near a planar interface, the model is separated into two dielectric half-spaces. An example of a super-resolution application for these models is the simulation of apertureless near-field scanning optical microscopy (ANSOM). In ANSOM, a probe is placed in the extreme near-field of an object in order to scatter the evanescent fields that are formed by the illumination of the object. Images created by ANSOM are fundamentally different from traditional images and are difficult to interpret. The simulations provide insight into how the images are formed and what information they contain. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2013-08-13 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0074071 |
URI | http://hdl.handle.net/2429/44809 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2013-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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