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UBC Theses and Dissertations

The use of fin-corrugated periodic surfaces for the reduction of interference from large reflecting surfaces Ebbeson, Gordon Robert 1974

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THE USE OF FIN-CORRUGATED PERIODIC SURFACES FOR, THE REDUCTION OF INTERFERENCE FROM LARGE REFLECTING SURFACES by GORDON ROBERT EBBESON B.A.Sc. University of Br i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of E l e c t r i c a l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1974 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h C olumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date Abstract The use of periodic structures to reduce interference from large reflecting surfaces i s proposed. Instrument landing system (ILS) interference from large hangars and terminal buildings i s cited as a typical problem. An analytical and numerical investigation of an in f i n i t e fin-corrugated surface composed of i n f i n i t e l y thin fins of spacing \/2<a<\ under TM polarized plane wave illumination i s described. Specu-lar reflection from this surface can be completely converted to back-scatter in a direction opposite to the incident wave when the angle of incidence from the normal to the surface and the f i n height are properly chosen. Experiments were performed at 35 and 37 GHz. using f i n i t e size fin-corrugated surfaces with fins of f i n i t e thickness under non-plane wave illumination and the results indicate that these surfaces behave essentially as predicted. In addition, the experimental surfaces remain completely effective for small oblique angles of incidence and have sufficient bandwidth for ILS applications. i i Table of Contents Page List of Illustrations v Li s t of Tables . . v i i List of Symbols v i i i Acknowledgements x 1. Introduction . 1 2. Theoretical Analysis 5 2.1 Formulation of the Overall Problem 5 2.1.1 Boundary and Edge Conditions 7 2.1.2 General Solutions . 9 2.2 Representation of the Fin-Air Discontinuity by a Scattering Matrix H 2.3 Formulation of the Component Problems 15 2.4 The Transformed Problem . 1 7 2.4.1 Solution of the Transformed Problem 18 2.4.2 Determination of the Final Integral Representation 20 2.5 Solution of the Field Amplitude Coefficients by the Method of Residues 22 2.6 Solution of the Remaining Component Problems 25 2.7 Determination of the Component Reflection and Transmission Coefficients . 28 2.7.1 Reflection and Transmission Coefficients at the Optimum Angle of Incidence . . . 2 8 2.7.2 Attenuation of the n=l Mode in the Parallel Fin Region 33 3. Numerical Results 35 3.1' Introduction 35 3.2 Relative Power as a Function of Fin Height 35 3.3 Relative Power as a Function of Incident Angle . . . . . 39 3 .4 Optimum Fin Height as a Function of Fin Spacing . . . . ^2 3.5 Attenuation of the n=l Mode in the Fin Region 2^ 4 . Experimental Results ^ 4.1 Introduction . 4.2 Experimental Arrangement A5 4.3 Results and Discussion . . . . . . . . . . . . . . . . . $2 4.3.1 Plates 3A and 3B 5 2 4.3.2 Plates IA and IB • 6 1 i i i Page 5. Conclusions 69 Appendix A. Itetermination of the Region of Analyticity . . . . 72 Appendix B. Determination of g(s) 74 Appendix C. Convergence of the Infinite Products in g(s) . . . 78 Appendix D. Asymptotic Behaviour of g(s) 79 Appendix E. Determination of P(s) in g(s) 8 1 Appendix F. Experimental Data 83 References 90 iv List of Illustrations Figure Page 1.1 Periodic Surface Demonstrating Bragg's Law 1 2.1 Fin Corrugated Structure with Incident TEM Plane Wave . . . . 6 2.2 Single Period with Boundary Conditions 3 2.3 Four Port Scattering Junction Representation 12 2.A Component Problems . 16 2.5 Pole Plot of the Integral Representation, Equation (2.54), Problem #3 21 2.6 Pole Plot of the Integral Representation, Equation (2.75), Problem 7/4 26 2.7 Pole Plot of the Integral Representation, Equation (2.76), Problem #1 26 3.1 Relative Power of the n=0 and n=-l Modes vs. Fin Height (no attenuation) 36 °.2 Rcl-^ivs Pc.rcr and Relative ^ has 0 o*~ "=Q p.^ d •"=—1 Modes vs. Fin Height (no attenuation) 38 3.3 Relative Power of the n=0 and n=-l Modes vs. Angle of Incidence (no attenuation) 40 3.4 Relative Power and Relative Phase of the n=0 and n=-l Modes vs. Angle of Incidence (no attenuation) 41 3.5 Optimum Fin Height vs. Fin Spacing . 43 4.1 A TM Polarized Plane Wave Incident on a Fin-Corrugated Surface with t=0.028 cm. 1 0.002 cm 46 4.2 Plates IA and IB . . . . . 48 4.3 Experimental Arrangement 49 4.4 Experimental Range for Plates 3A and 3B. Transmitting Horn in Foreground 50 4.5 Plate 3A on Mounting Platform, a.j=0 51 4.6 Relative Power of the n=0 Mode vs. Fin Height, Plates 3A and 3B (with attenuation) 53 4.7 Relative Power of the n=0 Mode vs. Angle of Incidence, Plate 3A (with attenuation) 54 v Figure Page 4.8 Predicted Relative Power of the n=0 Mode vs. Angle of Incidence, Plate 3A (with attenuation) . . 55 4.9 Relative Power of the n=0 Mode vs. Angle of Incidence, Plate 3B (with attenuation) 57 4.10 Predicted Relative Power of the n=0 Mode vs. Angle of Incidence, Plate 3B (with attenuation) . . . 58 4.11 Relative Power of the n=0 Mode vs. Angle of Rotation, Plate 3A 59 4.12 Relative Power of the n=0 Mode vs. Frequency of the Incident Wave, Plate 3A, (with attenuation) 60 4.13 Relative Power of the n=0 Mode vs. F i n Height, Plates IA and IB (with attenuation) 62 4.14 Relative Power of the n=0 Mode vs. Angle of Incidence, Plate IB (with attenuation) 63 4.15 Predicted Relative Power of the n=0 Mode vs. Angle of Incidence, Plate IB (with attenuation) 64 4.16 Relative Power of the n=0 Mode vs. Angle of Incidence, Plate IA 6 6 4.17 Relative Power of the n=0 Mode vs. Frequency of the Incident Wave, Plate IB (with attenuation) 67 A. l The St r i p of Common A n a l y t i c i t y 73 B. l Zero Plot of [u sin(ua)] and [cos(ha)-cos(ua)] with losses ^6 B.2 Pole P l o t of g( s ) / [ c o s ( u a ) - e ~ ^ h a ] 76 D.l z-Plane Representation Showing Branch Cut 80 v i L i s t of Tables Table Page I Amplitude R e f l e c t i o n and Transmission C o e f f i c i e n t s , Sub-Optimum Case 29 II Amplitude R e f l e c t i o n and Transmission C o e f f i c i e n t s , Sub-Optimum Case . . . . 30 III Amplitude R e f l e c t i o n and Transmission C o e f f i c i e n t s , q-i Optimum Case J X IV Amplitude R e f l e c t i o n and Transmission C o e f f i c i e n t s , Optimum Case 32 V Transmission Distance vs. Angle of Incidence . . . . . . . 83 VI Relative Power of the n=0 Mode vs. Angle of Incidence, Plate 3A, f= 35 GHz 84 VII Relative Power of the n=0 Mode vs. Angle of Incidence, Plate 3B, f=35 GHz 85 VIII Relative Power of the n=0 Mode vs. Frequency of the Incident Wave, Plate 3A, 0^=0°, 61=54.5° . . . 86 IX Relative Power of the n=0 Mode vs. Angle of Incidence, Plate IB, 04=0°, f=37 GHz . . . 87 X Relative Power of the n=0 Mode vs. Angle of Incidence, Plate IA, 04=0°, f=37 GHz 88 XI Relative Power of the n=0 Mode vs. Frequency of the Incident Wave, Plate IB, a.j=0o, 9i=69.0° 89 v i i List of Symbols = f i n period = f i n spacing = incident f i e l d amplitude coefficient in the overall problem = incident f i e l d amplitude coefficients in the component problems = amplitude reflection coefficients, n=0 and n=-l modes = reflected f i e l d amplitude coefficient in the overall problem V Bl» V B l = reflected f i e l d amplitude coefficients in the component problems = transmitted f i e l d amplitude coefficient in the overall problem = constant 0' C-l> V c_ 1 = transmitted f i e l d amplitude coefficients in the component problems = f i n height = frequency of the incident wave = H0,s). = k sinO. o 1 = propagation constant of free space = constant = transmission distance between horn and surface = mode numbers = power reflection coefficients, n=0 and n=-l modes v i i i P(s) = special function Rl» R2' V R4,m' V V R7,m =: residues in Problem #3 R = surface r e s i s t i v i t y of the fins m 1 t = f i n thickness 2 ,2 X 2 u = k + s o x, y, z = space coordinates Z Q = impedance constant of free space a = attenuation coefficient of the n=l mode in the f i n region = angle of rotation Y = propagation coefficient of the n*"*1 mode, n=*0,l,2,. n n l op 0<z<d th T = propagation coefficient of the n mode, n=±0,l,2, z<0 T(z) = gamma function = angle of incidence = optimum angle of incidence \ = free space wavelength <|>(x,z) = magnetic f i e l d component in the y-direction, H(x,z) f(x,s) = bi l a t e r a l Laplace transform of <j>(x,z) oi = angular frequency ix Acknowledgements I would like to thank the National Research Council of Canada for the bursary I received in 1972/73, the Ministry of Transport for their support under Grant MOT 65-8114 in 1973/74, the British Columbia Telephone Company for the scholarship I received in 1973/74 and the University of British Columbia for research assistantships received from 1972 to 1974. I am deeply grateful to my research supervisor Dr. E.V. J u l l for his helpful suggestions and constant encouragement. I would also like to thank Dr. E.V. Bohn and Dr. M. Kharadly for reading the original draft and for their valuable comments. I would like to give a special thanks to my office-mate Gary Brooke for his suggestions and for the many f r u i t f u l discussions we had. I would also like to thank Jack Stuber and Derek Daines for the many patient hours they spent milling the fin-corrugated surfaces, Brian Nuttall for his technical assistance, Herb Black for taking the photos and Miss T i l l y Martens for typing the manuscript. Finally, I would like to thank a l l my very special friends in and out of U.B.C. for a wonderful two years. 1. . Introduction Instrument landing system (ILS) interference from large hangars and terminal buildings.is a problem confronting many of today's . crowded airport installations. Although some problems can be eliminated by placing the offending structures at non-critical angles to the run-ways, this becomes very d i f f i c u l t in a multi-runway system. It is sug-gested that the interference could be eliminated by placing a properly designed periodic surface on the reflecting structure so that a l l of the incident energy i s scattered back in the direction of the ILS transmitter. In this way, no interfering reflections would be received along the runway. It i s known from the theory of diffraction gratings that a maximum amount of energy i s transferred from the specularly reflected mode to one or more backscattered modes when the period of the surface, 'a', satisfies Bragg's Law, k a.sine.=Tr, where k =2ir/A, X i s the free o 1 o space wavelength and 0^ i s the angle of incidence from the normal to the surface as shown in figure 1.1. However, the range of periods over which only one backscattered mode is generated depends entirely on the surface. INCIDENT 2a sin9 i= m> m=0,l,2, ... / Figure 1.1 Periodic Surface Demonstrating Bragg's Law. j 2 A number of authors [1-6] have analysed the problem of plane wave incidence on a periodic surface consisting of an i n f i n i t e set of semi-infinite parallel conducting plates. They found that a single backscattered mode, travelling in a direction opposite to the incident wave, i s generated for some angles of incidence when the plates have a period in the range \/2<a<\. This surface is not particularly useful in i t s e l f , but similar structures, consisting of an i n f i n i t e set of parallel plates terminated by a perfectly-conducting plane, are of interest. These structures, sometimes referred to as comb gratings or fin-corrugated surfaces, have been rigorously analysed by Tseng [7] and Tseng, Hessel and Oliner [8] using a scattering matrix approach and an integral trans-form .technique similar to that used by Collin [6, P.430] for semi-infinite plates. Tseng et a l [8] have shown that, for TE polarization of the incident wave and the proper choice of corrugation depth, there i s com-plete concellation of specular reflection at an angle of incidence 9^=sin~l(X/2a). Later DeSanto [12,13], using a modified calculus of residues technique, confirmed these results and described similar results for TM plarization. The problem of TM polarized plane wave scattering from a f i n -corrugated structure with a modulated corrugation depth and f i n spacing in the range 0<a<X/2 has been analysed rigorously by Hessel and Hochstadt [9] using the above-mentioned scattering matrix and integral transform technique. While no numerical results were given, this structure may also be expected, with the proper choice of corrugation depths, to trans-fer a l l reflected power to the backscattered mode. Perfectly conducting sinusoidal surfaces have been investigated by Zaki and Neureuther [10,11] using a numerical solution of the appro-3 priate integral equations. They found, for both TE and TM polarization, the same scattering properties as observed by Tseng et a l [8] for f i n -corrugated surfaces. However, for TM polarization, the reduction of specular reflection seemed to occur over a much wider range of incident angles. A generalization of the fin-corrugated surface, the rectangular groove or lamellar grating has been studied by Wirgin and Deleuil [14] and Wirgin [15,16] for both polarizations. Numerical results were obtained by truncating and then solving a set of simultaneous linear equations for the spectral order amplitudes. Numerical results at 8^=30° exhibiting complete cancellation of the specularly reflected mode were confirmed by experiment. This structure has also been investigated by Hessel and Schmoys.[17], who were interested in i t s application as a frequency sensitive mirror in a laser cavity. A similar structure to the above, the triangular groove or echelette grating, has been used extensively in spectroscopy. However, this structure is very d i f f i c u l t to analyse rigorously and only a limited number of accurate numerical results have been obtained [18]. From practical considerations, i t is apparent that the f i n -corrugated periodic surface is the most promising in the ILS application as i t is much simpler to construct than the other structures. In addi-tion i t can be analysed rigorously and investigated numerically without the costly inversion of matrices required by the lamellar surfaces. However, i t i s also apparent that the work to date has only been con-cerned with the idealized fin-corrugated surface, that i s , the i n f i n i t e surface composed of perfectly-conducting, i n f i n i t e l y thin, fins under plane wave illumination. No experimental work has been carried out on more realizable surfaces. The purpose of this thesis is to provide such an experimental investigation using numerical results, from the analysis of idealized structures, as a guide in designing the surfaces. In Chapter 2, a rigorous analysis i s presented for the problem of TM polarized plane wave incidence on an in f i n i t e fin-corrugated sur-face composed of perfectly-conducting, i n f i n i t e l y thin fins. The theore-t i c a l approach i s the same as that used by Tseng [7] and Tseng et a l [8] for TE polarization and Hessel and Hochstadt [9] for the modulated f i n -corrugated surface. TM polarization was chosen because this i s the polarization used in ILS installations. An investigation of attenuation in the parallel f i n region concludes the chapter. In Chapter 3, the numerical results for some specific cases are presented and compared with those obtained by other workers. Some interesting relations between the f i n height and f i n spacing for optimum cancellation of specular reflection are included. The chapter i s con-cluded with a discussion of the affects of attenuation. In Chapter 4, the experimental results for four specific f i n i t e sized fin-corrugated surfaces are presented together with a procedure for predicting the behaviour of any fin-corrugated surface composed f i n i t e l y thick fins. Conclusions from the investigation are presented in Chapter 5. 2. Theoretical Analysis 2.1 Formulation of the Overall Problem The fin-corrugated structure which s h a l l be analysed i s shown schematically i n figure 2.1. It consists of an i n f i n i t e set of per-fectly-conducting p a r a l l e l f i n s mounted on an i n f i n i t e , p e r f e c t l y - c o n -ducting plane. The f i n s , which are of height "d" and p e r i o d i c spacing "a", s h a l l be assumed i n f i n t e l y t h i n . The notation and approach here - follows that used by C o l l i n [6, P.430] for a simpler structure. Consider a y - i n v a r i a n t , TM p o l a r i z e d , TEM plane wave with compon-ents H v(x,z), E x(x,z) and E z(x,z) incident at an angle 0^ from the normal to the str u c t u r e , as shown i n figure 2.1. From Maxwell's equations E x ( x , z ) = j|a- 3-<Kx.z) .. (2.1) o Z E z(x,z) = -i|a-3<K*>z) (2.2) k d X . o where <j>(x,z) = Hy(x,z) (2.3) and Z Q and k D are the impedance and propagation constants, r e s p e c t i v e l y , of free space. The magnetic f i e l d component i s • i C x . z ) = a Q e " j h x " F o Z z<o (2.4) where h = k Q sine^ (2.5) r o = J k o c o s 6 i (2.6) V Figure 2.1 Fin-Corrugated Structure with Incident TEM Plane Wave. and the time dependence e J U , u has been omitted. Because of the discon-t i n u i t y at the f i n - a i r i n t e r f a c e , a r e f l e c t e d and transmitted wave w i l l r e s u l t . The r e f l e c t e d wave w i l l be composed of an i n f i n i t e sum of y- i n v a r i a n t , TM p o l a r i z e d TEM modes whereas the transmitted wave w i l l contain one TEM mode plus an i n f i n i t e sum of s i m i l a r , but higher order, TM modes. The transmitted wave, however, w i l l be r e f l e c t e d by the terminating plane back toward the f i n - a i r i n t e r f a c e r e s u l t i n g i n further r e f l e c t i o n and transmission. Thus, the t o t a l r e f l e c t e d f i e l d above' the i n t e r f a c e w i l l be governed by multiple r e f l e c t i o n s within the f i n region. As mentioned i n Chapter 1, a procedure f o r analysing such a problem has been described by Tseng [7] and used s u c c e s s f u l l y by Tseng, Hessel and Oliner [8] and Hessel and Hochstadt [9]. It consists essent-i a l l y of representing the d i s c o n t i n u i t y at the f i n - a i r i n t e r f a c e by a sc a t t e r i n g matrix, S, which r e l a t e s , at the d i s c o n t i n u i t y , the amplitudes of the scattered modes to those of the incident modes. This allows the o v e r a l l problem to be broken up into a f i n i t e number of component prob-lems, the number depending on the number of propagating modes i n the f i n region, which can be analysed one at a time. The only assumption that must be made here i s that the f i n height, d, i s large enough to prevent r e f l e c t i o n of evanescent modes.-at-the terminating plane. •-This procedure w i l l form the basis of the t h e o r e t i c a l analysis to follow. 2.1.1 Boundary and Edge Conditions Because of the p e r i o d i c i t y of the fin-corrugated structure, i t i s necessary to consider only a s i n g l e period as depicted i n figure 2.2. In the region z < o, the e l e c t r i c and magnetic f i e l d s are p e r i o d i c i n x. Thus, from Floquet's Theorem [6, P.368J and equations (.2.1) to (2.3), 3d) (x, z) I 3x 'x=ma -jhma „., x e J 3 e ( x , z ) m= t o ,1,2,... z<o (2.7) O X x = o cf)(ma,z) = e j m a <$>(o,z) m=*0,l,2,... z<o. (2.8) Also, i n the region o < z < d, because the tangential component of the e l e c t r i c f i e l d i s zero at the surface of a perfect-conductor, 9<Hx,z) 3x + = 0 m= -0,1,2, ... o < z < d (2.9) x=ma 3<Kx,z) 3z 0 a l l x z=d (2.10) These equations make up the boundary conditions of the o v e r a l l problem. e J <f> z<0 | 3<t> 3x - j h a 3ij> 6 3x 0<z<d | d± 3x - j h a 34 3x" 3x i t= o 3x *£- o 3z ' • • z Figure 2.2 Single Period with Boundary Conditions. Notice that equation (2.9) allows the region of z in.equation (2.7) to be extended to -°° < z < d. However, because the tangential component of the magnetic f i e l d is discontinuous across the conducting f i n by an amount equal to the current on the f i n , the region of equation (2.8) cannot likewise be extended. The edge condition, or the behaviour of the f i e l d at the edge of the parallel fins, is also of importance. It can be shown as in Collin [6,P. 18] that 1/2 + Kma.z) ^ z as z -*- o m = -0,1,2,... (2.11) asymptotically, and hence the magnetic f i e l d component is f i n i t e i n the neighbourhood of the f i n edge. In addition, the radiation condition, (J)(x,z) -*• o as z -°° (2.12) must also be met in the solution of the overall problem. 2.1.2 General Solutions The expressions for the magnetic fields above and below the f i n - a i r interface of figure 2.1 must satisfy the reduced Helmholtz equation (V 2 + k02)<f>(x,z) = o (2.13) subject to the conditions of section 2.1.1. The incident f i e l d expres-sion, (2.1), and the expression for the total reflected f i e l d in the region z < o, oo 2n7r <j> (x,z) = ^ b n e ~ j ( h + ~a~ ) 3 d" rn z z<o (2.14) s a t i s f y the boundary conditions (2.7) and (2.8). However, for (2.13) to hold, i t i s necessary that T n 2 = (h+ ^ ) 2 - k Q 2 n= ±0,1,2,... (2.15) S i m i l a r i l y , the expression f o r the t o t a l f i e l d i n the region o < z < d, 00 <)>t(x,z) = ]>~j c o s ( ^ ) e " Y n z o<z<d, (2.16) n=o s a t i s f i e s (2.7), (2.8) and (2.13) i f and only i f Y 2 = ( — ) 2 " k 2 n= +0,1,2,... (2.17) n a. u Notice that i f Y n 2 < 0 for some n, Y n i s imaginary and p o s i t i v e ( i f the p o s i t i v e root i s chosen to s a t i s f y the r a d i a t i o n condition, (2.12)) and the corresponding mode i s propagating. However, i f Y n 2^0 for some n, Y n i s r e a l and p o s i t i v e and the corresponding mode i s evanescent. A s i m i l a r set of rules apply to equation (2.15). Thus, the number of propagating modes i n the region z<o i s determined by the values of a and h, whereas the number i n the region o < z < d depends only on a. As stated i n Chapter 1, Tseng, Hessel and O l i n e r [8] analysed the problem of a TE pol a r i z e d plane wave incident on a fin-corrugated str u c t u r e . They found, for th i s p o l a r i z a t i o n , the power i n the specular l y r e f l e c t e d mode can be completely transferred to a backscattered mode t r a v e l l i n g i n a d i r e c t i o n opposite to the incident wave i f two propagat-ing modes are i n the f i n region. More recent works by DeSanto [12,13] have confirmed these results and have shown i d e n t i c a l r e s u l t s f o r TM p o l a r i z a t i o n ^ Also, Hessel and Hochstadt [9] investigated a f i n - , corrugated structure with a modulated corrugation depth. They discovere 11 for TM p o l a r i z a t i o n and the general case of a uniform corrugation depth (as i n figure 2.1), a backscattered mode i s not excited i f only one' propagating mode i s i n the f i n region. I t was therefore decided, f o r the present study, that the n = o and n = 1 modes s h a l l be assumed to propagate In the f i n region. This assumption r e s t r i c t s the f i n spacing to the range A/2 < a < A, where A i s the free space wavelength, and the number of propagating modes i n the a i r region to two; the n = o specularly r e f l e c t e d mode and the n = -1 backscattered mode. I t should be noted, however, that the n = -1 mode may be evanescent for some' angles of incidence. 2.2 Representation of the F i n - A i r Discontinuity by a Scattering Matrix The notation and approach here follows that used by Hessel and Hochstadt [9]. The f i n - a i r d i s c o n t i n u i t y of the fin-corrugated structure may be represented as a four port s c a t t e r i n g junction as shown i n figure 2.3(a). Each port may be considered as a modal waveguide supporting one of the four propagating modes under consideration. Assume that each port i s excited one at a time as shown i n figures 2.3(b) and 2.3(c). The r e s u l t i n g four problems are termed the component problems and the A's, B's and C's of figure 2.3(c) are the i n c i d e n t , r e f l e c t e d and trans-mitted f i e l d amplitude c o e f f i c e i n t s , r e s p e c t i v e l y , corresponding to these problems. The matrix equation of the j u n c t i o n i s of the form b = S . • a (2.18) or s Y 0 Y l z<0 z=0 z>0 (a) a1 K k K a2I 1 • 1 • 1 PROBLEM #1 PROBLEM #2 (b) s1 ft ft J1 | s | s 1 1=1 ft PROBLEM #1 PROBLEM #2 (c) k k k 8 | S | 3l } 4 h 4 PROBLEM #3 PROBLEM #4 W 1 • 3 1 4 ft fc PROBLEM #3 PROBLEM #4 Figure 2.3 Four Port Scattering Junction Representation, (a) Ov e r a l l Problem, (b) and (c) Component Problems showing the F i e l d Amplitude C o e f f i c i e n t s . where a and b are column vectors whose components are the modal amplitudes associated with the component problems and S is a scattering matrix with elements consisting of the appropriate amplitude reflection and trans-mission coefficients. Since the n = o mode i s the only mode incident on the structure from z<0, (2.20) In addition, because of the boundary condition (2.10) , a 3 - e " 2 ^ (2.21) a 4 - (2.22) These conditions, together with the matrix equation (2.19), constitute a system of seven equations which may be solved uniquely in terms of the amplitude of the incident f i e l d , a^. The expressions for the specular amplitude reflection coefficient, — • , and the backscatter amplitude a l ' bo reflection coefficient, —^ , which result are a l b i Bo + co " 2 V V " 2 V A — = 1 e H e A 1 Ao Ao "° A. -1 *>2 B , C_x - 2 y d + C - 1 - 2 V = + e A q + e A 1 A Q A Q A (2.23) (2.24) ^0 B x -2 Y ld 1 e B Q C -2 Y ld + — . — — e A 1 A 0 1 -Bo " 2 V — e A 0 B - 2 Y l d 1 e AoX 6 (2.25) 1 - — e A 0 Bo ~ 2 V 4 B i f i o ~ 2 V + — . — e A 0 A 0 J - l e Bo " 2 V B 2 -2 Y ld 1 e AoXe (2.26) The corresponding power reflection coefficients are b, 2 (2.27) Pi . r D ' a i ' (2.28) where (2.29) due to the conservation of power in a lossless system. 2.3 Formulation of the Component Problems Three of the four component problems represented by figure 2.3(c) are shown schematically in figure 2.4. Problem #2 has been omitted because, as equation (2.20) indicates, i t i s not needed in the solution of the overall problem. The boundary conditions for the re-maining problems are the same as equations (2.7) to (2.9) except that the value d is now replaced by +°°. The general solutions to these component problems, as l i s t e d below, satisfy these boundary conditions and Helm-holtz's equation, (2.13), subject to equations (2.15) and (2.17). Problem # 1 z<o 0(x,z) = A, e ^ h x - r o z + V B n e " ^ h + ^ T ^ n 2 n= -°° (2.30) z>o <f>(x,z) = > cos( ) e 'n n=o a (2.31) Problem # 3 . e x ^ „ - j ( h + — ) x + r n z z<o *(x,z) = > L e J V a n Kx,z) = y n=^-~ (2.32) :>o <Kx,z) = AQ e Y ° Z + j^T B N cos(^) e Y n z iPb ° (2.33) Problem # 4 z<o <j>(x,z) = ^ C n e C . " J C h + ^ x + r n Z —OO (2.34) xu INCIDENT TEM PLANE WAVE y ^ N i * 1 ^o- X PROBLEM #1 i 1 **— a * i (a) PROBLEM #3 INCIDENT TEM MODE IN EACH REGION WITH PHASE SHIFT e ~ j h a BETWEEN REGIONS 4 H G H y cj>e 4 O H -jha x (b) INCIDENT TL^ MODE IN EACH REGION WITH-PHASE SHIFT e ~ j h a BETWEEN REGIONS PROBLEM #4 <f>e jha 6 H i H 4>e A -jha 6 I H (c) Figure 2.4 -Component Problems, (a) Problem #1, TEM Mode Incident in Region z<0. (b) Problem #3, TEM Mode Incident in Region z>0. (c) Problem #4, TM Mode Incident in Region z>0. z>o cb(x,z) = Ai_ c o s ( - ^ ) e Y l Z + >^ 't B n c o s ( - — ) e 'n 3. (2.35.) n=o In the following sections, a formal s o l u t i o n for the amplitude c o e f f i c i e n t s of Problem # 3 w i l l be presented using an i n t e g r a l trans-form construction technique described by C o l l i n 16, P.418] and used by Tseng, Hessel and O l i n e r [8] and Hessel and Hochstadt [9]. I t w i l l be-come apparent i n section 2.6 that not a l l of the component problems need to be solved i n d e t a i l . The i n t e g r a l transform expression obtained for Problem # 3 can be transformed to s u i t the others, thus e l i m i n a t i n g a great deal of unnecessary a n a l y s i s . 2.4 The Transformed Problem suppress the var i a b l e z by taking the b i l a t e r a l Laplace transform of the f i e l d s o l u t i o n <}>(x,z). A s o l u t i o n f or the transformed problem which s a t i s f i e s the transformed boundary conditions and the transform of Helm-holtz's equation i s then constructed. In the f i n a l step, the s o l u t i o n for <j>(x,z) i s found by inverting; the transform and evaluating the i n -version i n t e g r a l i n terms of i t s residues. The desired amplitude coef-f i c i e n t s can then be obtained by matching this s o l u t i o n with the o r i g i n a l f i e l d expansion. Let the b i l a t e r a l Laplace transform of the t o t a l f i e l d <j>(x,z) be [6, P.433] The i n i t i a l step of the i n t e g r a l transform technique i s to oo (2.36) — C O The inverse transform i s then -~-r \ e S Z K x , s ) d s (2.37) where T is a specially chosen contour i n the complex s-plane running parallel to the imaginary s-axis through a region i n which cf>(x,s) i s analytic. It i s shown in Appendix A that the region of analyticity for Problem J 3 is well defined. For convenience, a special function g(s) shall be defined as follows: oo g(s)=Ko,s) = J e"Sz<),(o,z)dz (2.38) — C O 2.4.1 Solution of the Transformed Problem The Laplace transform of the reduced Helmholtz's equation i s [6, P.434] a 2Kx,s) + u 2 ^ ( x > s ) = 0 ( 2 > 3 9 ) 3x 2 where u 2 = k D 2 + s 2 (2.40) The transformed variable, <fi(x,s), of this differential equation must satisfy the following transformed boundary conditions obtained from equations (2.7), (2.8) and (2.9): 3<Kx,s) I _ -jhma 3ji(x,s) I m=-0,l,2,... <=°>z>--° (2.41) ai- x=ma °*- x=o Kma,s)= e~ j h m a^(o,s) m=-0,1,2,.. . z<o (2.42) 3$(x,s)j = 0 m=±0,l,2,... z>o 3.x x=ma C2.43) A general solution for this transformed problem is of the form <Kx,s) = A(s) sin(ux) + B(s) cos(ux) (2.44) Substituting this expression into the boundary condition (2.41) with m=l, yields A ^ = B(s) sin(ua) [cos(ua)-e j h a ] (2.45) However, from equation (2.38) g(s) = Ho-,s) = B(s) (2.46) and hence A ( s ) = g(s) sin(ua) Icos(ua)-e j h a ] (2.47) Therefore, from equation (2.44) it \ g(s)[cos(u(a-x))-e ^ a c o s (ux) J <Kx,s) = -rr-[cos(ua)-e J n a ] (2.48) and the inverse transform, equation (2.37), is [6, P.434] . , , s 1 ( e S Zg(s) [cos (u(a-x) )-e ^ 3 c o s (ux) ]ds <p(x,z) = -z—r ^TT 2 U J JT [cos(ua)-e J h a ] (2.49) 2 2 2 where u = k Q + s . In order to evaluate equation (2.49), a representation for g(s) must be obtained. This can be done as follows: Substituting equation (2.49) into the boundary condition (2.9) with m=o, yields 20 3 2 e_ ^ ( s ) u sin(ua)ds = Q Z > Q r I c o s ( u a ) - e j h a ] . (2.50) S i m i l a r i l y , s u b s t i t u t i n g equation ( 2 . 4 9 ) i n t o the boundary condition ( 2 . 8 ) with m=l, y i e l d s s z e g(s)[cos(ha)-cos(ua)]ds _ z < o T [ c o s ( u a ) - e " j h a ] (2.51) Thus, i t i s apparent that the integrands of (2.50) and (2.51) are-regular (that i s , contain no s i n g u l a r i t i e s ) for z>o and z<o, r e s p e c t i v e l y . Therefore, g(s) i s a meromorphic function (contains only poles) and can be constructed from r a t i o s of e n t i r e functions subject to (2.50) and (2.51). I t i s shown i n Appendix B that a s u i t a b l e construction f o r g(s) would be [6, P.435] P(s) {cos(ua)-e~ j h a} 8 ( S ) = ~ ~ ( a + y ) -sa - ( s _ r ) ( s _ r } sa 1 a ; 1 1 a } , \ / \ f \ 1—r n nTr i — r n -n r (s-y 0)(s+y 0)(s-r 0) | | - ^ - e | | ^ 2 6 (2.52) where P(s) i s an e n t i r e function yet to be determined. It i s shown i n Appendix C that the exponential factors and 2 *™ the terms (—) and (——) i n the denominator of (2.52) are used to en-a a sure the uniform convergence of the i n f i n i t e products. [6, P.435] 2.4.2 Determination of the F i n a l I n tegral Representation In order to f i n d a s u i t a b l e function P ( s ) , i t i s f i r s t neces-sary to determine the asymptotic behaviour of g(s). This i s done i n — 3 / 2 Appendix D where i t i s shown that g(s) i s asymptotic to s as |s|-> « If P(s) is now chosen so that g(s) has algebraic growth at -3/2 i n f i n i t y and is asymptotic to s as j s} °^» i t is assured that £(x,z) has the correct behaviour at the fin edges, equation (2.11).. It is shown in Appendix E that P(s) is of the form -sa P(s) = C e ln2 (2.53) where C i s a constant. Substituting (2.53) into equation (2.52) and the result into equation (2.49) yields the f i n a l integral representation of the total f i e l d , >(x,z) 2nj -sa sz — I n Z e e {cos(u(a-x))-e cds(ux)} dS CO - J c o ( 8 - Y 0 ) ( 8 + Y 0 ) ( 8 - r 0 ) (S+Y) " (s-r )(s-r ) ^ n nn T r n -n mr e — r — e . ( HI) a a (2.54) o 9 o where u = k Q + s . The pole plot for this problem is shown i n figure 2.5. A 3*2 n = +2,3,., z>0 P Y 0 INCIDENT MODE $ ro * ) - ^ i xxx n = +1,±2,3,. z<0 Figure 2.5 Pole Plot of the Integral Representation, Equation (2.54), Problem #3. 22 2.5 Solution of the Field Amplitude Coefficients by the Method of  Residues The fin a l step in the analysis of Problem // 3 is the evalua-tion of equation (2.54) using the method of residues. Once this is done, the result can be matched with the original f i e l d expansion for <Hx,z) to obtain the desired f i e l d amplitude coefficients. For z>o, the integration contour of figure 2.5 is closed in the LHP. Therefore <Kx,z) = C ^ > ' Residues (2.55) If the residues at S=+YQ , S=-Y0, S=~YI and s=-YN are denoted by Rj_, R2> R 3 and R^  m , respectively, where m=+2,3,..., then •ln2 R, = ( l - e ^h a ) e V % % - r o ) -YQa ( W n^ T "(HI) a <Fn-'0>"-n^ 0) j f (-2HI) 2 a (2.56) R. In2 -Y nz -jha,. 0 (1-e J ) e 2 V W Y o a e (HI) a (^HI) 2 (2.57) Y l a R. (In2-1) H / - 1 > ~ J n a \ /Tr\ z 7 1 ^ (1+e J ) (—) cos(—) e a a ( y 1 + r 0 ) (YI2-Y02) TT 2 (Yn~V (HI) a Y-j* nir ( r ^ X r . ^ ) nu (^HI) 2 a i(2.58) (In2 ) • ., -y z e (.cos (rati)-e J j {—) c o s ( ) e ; R ; a a < , 4,m Y a . -y a : CO TTi OD F l 1 o , (Y - Y ) — „ ( r + y ) ( r + y •) — 1 (Y +r n)(Yn ~Y ) e ~^ z e rm 0 '0 ' m l 1 nTr. 1 ,2nir. 2 1 ( a } 1 { a ) (2.59) o nnT ^  9 where = (—) - k Q . Therefore, from equation (2.55), oo q.(x,z) = C(R1+R2+R3+ ^ R 4 ) m) - (2.60) m=2 However, the original f i e l d expansion for <Kx,z) in the region z>o, equation (2.33), may be rewritten in the form (j)(x,z)=A0eYoz + B 0e" Yo z + -&± c o s ( ^ ) e ~ Y l z CO + y B n c o s ( — ) e Y n ^ n=2 Matching terms in this expansion with equation (2.60) yields the follow-ing amplitude coefficients: A_ = CRie - Yo z (2.62) o B Q = CR 2e Yo z (2.63) n - C R 3 e Y l Z B l . ,TTX^  (2.64) cos (—) a Bm = CR4,me m=+2,3,... (2.65) ,m7rx, cos ( ) a For z<o, the integration contour of figure 2.5 is closed i n the RHP and <Hx,z) = -C >^ ^ Residues (2.66) If the residues at s=r Q, s=r_i_ and s=r m are denoted by R5, Rg and Ry^, respectively, where m=+l,-2,3,..., then •In2 R. = sin(ha) e -jhx+r Qz 2 2 <ro V> (Y n+r 0) ^ . ( r _ - r n ) ( r „-r n) e n"'0/ v l-n '0' n-e ,2njT^ 2 a (2.67) " r - l a R, 2TT (ln2+l) • . 2 i r 2 - j ( h - — ) x + r _ 1 z sin(ha) (—) e 3. < r o - r - i ) ( r f r - i ) ( r - n o 2 ) " - P - a r _ i a (Y +r .) -=^- °° (r - r ) (r - r ') — — ' n - 1 niT 1 — r n - 1 - n - 1 n7r e s ?; e a ^2mr^ 2 a C2.68) R. "^!(ln2+ k - j (bf ^ f L)x+ r mz j e sm(ha) ( ) e 3. 7 ,m ( r n - r ) ( r - r J ( y n 2 - r 2) 0 m m -m 0 m -r a r a • (Y +r ) — (r - r ) ( r - r ) — ! n m nfr T — r n m -n m nrr 1 ^ I n r\ e t (BE, 1 ,2mr^ 2 a (2.69) where r 2=(h+ -2-S^-)2-k 2 . Therefore, from equation (2.66) Kx,z) = "C(R5+R6+ ]T R7,m) m= -°° m#),-l (2.70) However, the original f i e l d expansion for cb(x,z) in the region z<o, equation (2.32), may be rewritten in the form 2TT <Kx,z) = C 0 e " j h x + r o z + C_ i e- j ( h- ^ ) x + F - l z + 0 0 • /-u_i_ 2njr. E r,- -J (h+ — ) x + r Z G h e a n n= -°° n#),-l (2.71) Matching terms in this expansion with equation (2.70) yields the remain-der of the amplitude coefficeints. CQ = -CR 5e j h x" ro z (2.72) C-1 = - C R 6 e j ( h - ^ ) x - r - l Z (2.73) • />,+ 2mTT. _ Cm = -CR7iXBe3Kn a ) x 1mz m= +1,±2,3,... (2.74) 2.6 Solution of the Remaining Component Problems As stated in section 2.3, i t is not necessary to analyse a l l of the component problems in detail. The integral transform expression and pole plot obtained for Problem # 3 can be easily transformed to suit the others simply by removing the pole corresponding to the incident TEM mode and adding the pole which corresponds to the incident mode of the problem to be analysed. For example, in Problem # 4, the incident mode in the region z>o is a TMi_ mode. Therefore, the incident pole would be located at S=+YIl on the complex s-plane, as depicted in the pole plot of figure 2.6 and the integral representation corresponding to this problem would be n. z>0 *> Y l V U r _ i INCIDENT MODE XXX n n=+2,3,... W~yl n = +1'±2'3> *T Y0 z<0 26 Figure 2.6 Pole P l o t of the Integral Representation,^Equation (2.75), Problem #4. & ro r - l XXX n = +2,3,... z>0 >> r 0 n n = +1,±2,3,... z<0 INCIDENT MODE Figure 2.7 Pole Plot of the Integral Representation, Equation (2.76), Problem #1. as follows Problem # 4 '-sa. ' . —;ln2 _.. ,, . „ , v. ^ Tl {cos(u(a-x))--?. J' cos (tix)} dS <!>(x,z) - -, , , -sa 0 0 , ., \ /* „ v sa (s+ Y ) ^ (s-ri ) ( s - i ) (s. y ) ( s % ) (s - r 0 ) I I e . I I e 1 1 (1T> (2.75) However, in Problem #1, the incident mode in the region z<o is a TEM mode. In this case, the incident pole would be located at s=-r 0 on the complex s-plane, as shown in the pole plot of figure 2.7, and the inte-gral representation would be as given in equation (2.76). Problem # 1 ( sz .. — l n 2 • _., e c " {c.os(u(a-x))-e cos(ux)} dS - ( S + T ) - ( s - D ( s - r ) M * 1—r n niT 1 — r n -n n-rr ( s + r o ) (S +Y 0 ) (s-r Q) T T e U . 2 ^ 2 6 1 ^ a' 1 ^ a ; (2.76) The f i e l d amplitude coefficients for Problem #4: A x, B 0, B l 5 CQ, and C_i (2.77) and for Problem #1: A Q, B D, B l 5 C D, and C_x (2.78) can now be computed using the method of residues and coefficient matching technique described in section 2.5. Once this i s complete, the f i n a l component reflection and transmission coefficients for a l l three problems may be determined. 2.7 Determination of the Component Reflection and Transmission Coefficients of equation (2.19) are determined by taking the proper ratios of the f i e l d amplitude coefficients obtained in sections 2.5 and 2.6. The results are tabulated in tables I and II. It should be clear from the analysis carried out in Appendix C that the i n f i n i t e products contained in these coefficients are uniformly convergent. into equations (2.25) and (2.26) and the results into equations (2.23) and (2.24) yields the overall amplitude reflection coefficients of the n=o reflected and n=-l backscattered modes. Substituting these coeffi-cients into equations (2.27) and (2.28) yields the f i n a l power reflec-tion coefficeints of the overall problem. 2.7.1 Reflection and Transmission Coefficients at the Optimum Angle of Incidence The condition, relates the angle of incidence giving the maximum reduction in specular reflection (the optimum angle of incidence, 6i Qp) to the periodicity of the corrugated surface (the f i n spacing, a). Substituting this value of sin6^ into the expression for h, equation (2.5), gives The f i n a l component reflection and transmission coefficeints Substituting equations (2.79) to (2.90) of tables I and II k^a sinG.- = TT ° 1op (2.91) h, 7T (2.92) op a -(2r0)fm2 ( Y - r Q ) ( Y n - r 0 ) ( r n + r 0 ) ( r _ n + r 0 ) -e < W V ( V rO ) ( rn- rO ) ( r-n- rO ) -^0+T-l>^2 ( 2 F 0 ) < Vr0> < V V ( P l + r 0 ) T T ( V r 0 ) ( I W ( V V ( V r - l ) ( Y l + r - l } VlT-l> ' 2 ' ( Yn + r-l ) (Tn-T-l> ( r-n- r-l> ( Y o - r 0 ) fm2 (i-e-J h a ) ( 2 r 0 ) ( Y o - r 0 ) _ ( Y n - r 0 ) ( r n + r Q ) ( r _ n + r 0 ) 3 6 h 2sin(ha) V ( V ^ O ^ W ^ ( V V f M d + e - j h a ) (2rQ) ( Y Q-r 0) (r 1 +r Q) (r_1+rQ) ( Y n-rQ) (r n +r Q) (r_n+rQ) J £ sinChaJCrQ+^XyQ-y^Cr^^Cr^+y^ ' I ( Y ^ ) ( y ^ ) ^-^J -e 2(Y ( J)fln2 ( r 0 - Y Q ) (Y n+Y 0) ( 1 ^ ) (r_n-yQ) <W ( V V ( rn + Y 0 ) ( r-n + Y 0 ) ( Y Q + Y ^ (l+e - j h a ) (2y 0 ) ( r 0 -y 0 ) ( r i -y 0 ) ( r . 1 -y 0 ) ( y ^ ) ( F ^ ) ( r ^ ) ( 1- e-J^) ( Y o _ Y i ) ( r 0 + Y l ) ( r 1 + Y l ) ( r_ 1 + Y l ) [J ' v V ' W ' W V Table I Amplitude Reflection and Transmission Coefficients, Sub-Optimum Case. (y 0 - r 0)-ln2 sinOha) (2yQ) ( f ^ ) ^ ( y ^ ) <r .-Y()) ( I ^ ) h 2 ( 1 _ e - j h a ) < V r O ) ( r n - r O ) ( r - n - V . ( V r - i > f 1 1 1 2 s l n ( h a > ( 2 V < W V V < r r V - r - r < W < r n - V < r - n - V 3 6 U-e- j h a) ( y ^ r ^ ) (y.+r^) ( i - r ^ ) ( l y r ^ ) \1 <*n + r - i> ( r n - r - i > < r - n " r - i > ( Y Q + T l ) f i n 2 ( i - e - J h a ) ( r Q - Y l ) " ( y ^ ) ( r n ^ ) ( r ^ ) ( l + e ^ h a ) ( r 0 + Y ( ) ) y ' V V ' W ^ + V -e 2 ( y 1)fin2 ( y ^ x r ^ c r ^ x r ^ ) _ (y^ ) ( r -^ ) ( r_ n - Y ] L ) . < V V ? 1 * 2 S ± n ( h a ) ( W - r - r ( W -je ( l + e - J h a ) ( Y ( ) +r 0) \ ( V r O ) ( r n - r O ) ( r - n - V . ( Y r r - i ) f l n 2 sin(ha) (2 Y l) ( Y q + Y i ) (rQ-Y ] L) ( r ^ ) ( Y ^ X r ^ X r - y ) .je •  : | | ( i + e ^ h a ) ( y ^ ) ( y^ r ^ ) ( r y r ^ ) ( i - r ^ ) 2 ' ( V r - i > ( r n " r - i > < r - * - r - i > Table II Amplitude Reflection and Transmission Coefficients, Sub-Optimum Case. CO fo - ( 2 r Q)fin2 (Y Q - r Q ) ( r 1 + r Q ) ( r_ 1 + r Q ) ( Y n - r 0 ) < r n + r 0 ) ( r V r 0 ) " -- < r 0 + r - l ) f l a 2 ( r 0 ) ( Y 0 - r 0 ) ( r i + r 0 > T T <V r0 ) ( rn + r0 ) ( r-n + r0> A Q < V F - 1 > ( V r - 1 > < r r r - l > ' ' < V r - l > ( rn" r-l> ( r-n" r-l> A0 Table III Amplitude Reflection and Transmission Coefficients, Optimum.Case. (2.99) (2.100) f l ( 2 V < V r o > < r i + V ^ r-l + r0> T T ( Y N " R Q ) ( ^ + R Q ) ^ - ^ ^ "(2.10D A Q " < W ( W < r i ^ l > ( r - l V 1 2 ' ( V V ( W ( F - n ^ l > B -r°-= Same as equation (2.81), Table I (2.102) A0 B i = (0 + j0) <2-103> LO A < V V f l n 2 ( * V l ) W W ( W ( r . i ^ T T ( V V ( r n - V ( r - n - y 2,3(Yl+r0)(r1-r0) <VV (VV (r-n-V . V r - i > f l n 2 ( aV 1 )(Y 1 +Y 0)(r 0 -Y 0)(r 1 -Y 0 ) _ ( Y n - ^ 0 ) ( r n - Y 0 ) ( r _ n - Y Q ) 2IR(Y0+r_1)(Y1+r_1)(r1-r_1) 1 1 ( Y / . ^ V r . ^ r - r ^ ) (O+jO) (0+jO) < V ( W (rrYi> T T < W ( r n - Y l ) ( r _ n - Y l ) <W(W(rrrO> V ( V r0 ) ( r n- r0 ) ( r - n- r0 ) ( Y 1 - r _ 1 ) f m 2 ( Y l ) ( Y o + Y l ) ( r ^ ) ^ ( Y n + Y l ) ( r n - Y l ) ( r _ n - Y l ) W W W ! 2 ' W ( r n - r - l ) ( W Table IV Amplitude Reflection and Transmission Coefficients, Optimum Case. However, as h-> — , several factors i n the r e f l e c t i o n and transmission a c o e f f i c i e n t s of tables I and II approach zero. S p e c i f i c a l l y , "*"^ m s i n (ha) = 0 h + - (2.93) a l i m ( i + e - j h a ) = 0 h -> - (2.94) a l i m . ( Y 1 - r 0 ) = o h •> - (2.95) a l i m , „ . „ ( Y i - r _ ! ) = 0 h + - (2.96) a h -»• - (2.97) a Therefore, c o e f f i c i e n t s which contain one or more of these factors i n t h e i r denominator are singular at the optimum angle of incidence and i t i s necessary to evaluate t h e i r l i m i t as h -> — . The c o e f f i c i e n t s obtam-a ed, v a l i d only at the angle 9 i 0 p = s i n ^(X/2a), are displayed i n tables III and IV. 2.7.2 Attenuation of the n=l Mode i n the P a r a l l e l Fin Region For applications-such ras the ILS. problem i t i s necessary to con-s i d e r s i t u a t i o n s i n which the angles of incidence are very close to grazing. However, as equation (2.91) i n d i c a t e s , t h i s requires the use of surfaces with f i n spacings very close to A/2, the cutoff f i n spacing for the n=l mode i n the fi n region. Attenuation of a waveguide mode i s large close to cutoff and i t may not always be possible to consider Yj_ as pure imaginary. Consequently, i t was considered worthwhile to deter-mine the attenuation coefficient, a, of the n=l mode i n the f i n region. Using a well known perturbation technique 16, P.183] i t can be shown that the attenuation of the n=l mode per unit length i n the fi n region i s 2Rm / a *~ nepers/ Z o a ( 1 _ ( | _ ) 2 ) 1 / 2 unit length (2.110) where is the surface r e s i s t i v i t y of the fins. If a r e s i s t i v i t y for copper is assumed, equation (2.110) becomes 2.3965 x 10~ 5 . nepers/ ~ a l X d - ^ ) ] 1 7 2 " ^ " ' - e t e r (2.111) /. a where A and*a'are in meters. Therefore, the true propagation coefficient i s Y i 1 = « + Y X (2.112) where y^ is given by equation (2.17). Equation (2.111) i s known to pre-dict too high an attenuation as the cutoff spacing i s approached and can therefore be regarded as an upper l i m i t under these conditions. 3. Numerical Results 3.1 Introduction An examination of the f i n a l expressions for the overall re-flection coefficients w i l l reveal that they are functions of three para-meters; 1) the f i n spacing, a (or optimum angle of incidence, ®i0p)> 2) the f i n height, d and 3) the angle of incidence, 8^. In most applica-tions, such as the ILS problem mentioned in Chapter 1, the optimum angle of incidence would be known. Therefore, i t was decided that the coefficients would be computed as functions of d and 0^  at fixed a. The major source of error in the numerical computations was the evaluation of the i n f i n i t e products. The Krummer transformation method described by Tseng et a l [8] and the remainder approximation technique described by DeSanto [12,13] were not used. Instead, a fixed ICO products were taken and shown to give two Lo three decimal accuracy at a very reasonable cost. Increasing the number of products to 500 gave, on the average, differences of less than 0.5% i n the magnitudes of the power reflection coefficients and less than 0.05% i n the relative phases of the amplitude reflection coefficients while increasing the cost by nearly four times. The results displayed in this chapter were computed assuming no attenuation of the modes in the f i n region. A discussion of the effects of attenuation of the n=l mode i s given in section 3.5. 3.2 Relative Power as a Function of Fin Height In figures 3.1(a) and (b), the relative power (power reflection coefficients) of the specularly reflected (n=o) and backscattered (n= -1) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 LO FIN HEIGHT d/\ (a) FIN HEIGHT d/\ (b) Figure 3.1 Relative Power of the n=0 and n=-l Modes vs. Fin Height (no attenuation) (a) a=0.578A 6. =59.99° 100% reduction at d=0.559A=0.968a 1op 0 (b) a=0.506A 0 . =81.24 100% reduction at d=0.501A=0.989a modes are p l o t t e d with respect to f i n height, d, for incident angles 60° and 81.2°. Because power i s conserved, a perfect transfer takes place between the two modes and the sum of t h e i r r e f l e c t i o n c o e f f i c i e n t s i s unity for any value of d. Similar r e s u l t s have been obtained by Zaki and Neureuther [10,11] for s i n u s o i d a l surfaces and by DeSanto 112, 13] f o r fin-corrugated surfaces. Note, i n each pl o t there i s a point of 100% power transf e r to the n=-l mode. This i s referred to by the above-mentioned authors as the "Brewster-angle e f f e c t . " I t was found that every f i n spacing has at l e a s t one such point f o r d<A and many more at l a r g e r f i n heights. Figure 3.1(b) has two i n t e r e s t i n g features. F i r s t , there i s a rapid interchange of power with f i n height between the n=o and n=-l modes at about d=0.55A. This phenomenon i s referred to as a "Wood S-anomaly" [8,10-13]. Second, the power i n the n=o mode i s le s s than 0.1 for 0.15A<d<0.5A. This l a t t e r feature could prove extremely u s e f u l , since i t allows the design of near optimum surfaces using f i n heights much smaller than the f i n spacings and without severe tolerances. In fi g u r e 3.2, the n=o and n=-l power r e f l e c t i o n c o e f f i c i e n t s and the r e l a t i v e phases of the corresponding amplitude r e f l e c t i o n coef-f i c i e n t s , are p l o t t e d as functions of f i n height for a case i n which the f i n spacing i s very large. These plo t s were made to compare with those of DeSanto [13,fig.5]. The two are e s s e n t i a l l y identical"'". Figure 3.2 demonstrates the same p e r i o d i c i t i e s with d and the same cor-r e l a t i o n s between the power r e f l e c t i o n c o e f f i c i e n t s and the derivatives of the respective r e l a t i v e phases as described by DeSanto [12,13]. The expressions used to generate the plots of figures 3.1 and 3.2 are not v a l i d near d=o because they do not take into account the 1 DeSanto assumes time dependence e l u t . Figure 3.2 Relative Power and Relative Phase of the n=0 and n=-l Modes vs. Fin Height (no attenuation). a=0.834A e =36.84° 100% reduction at d=1.21X=1.45a 1oP u o contributions from the evanescent modes r e f l e c t e d from the terminating plane. However, f o r TM p o l a r i z a t i o n , i t i s known that when d=o, the power r e f l e c t i o n c o e f f i c i e n t of the n=o mode i s unity and the phase of the respective amplitude r e f l e c t i o n c o e f f i c i e n t i s zero. 3.3 Relative Power as a Function of Incident Angle In figures 3.3(a) and (b), the r e l a t i v e power of the n=o and n=-l modes are p l o t t e d with respect to the angle of incidence, 6^. The f i n spacings used are those of figures 3.1(a) and (b), r e s p e c t i v e l y , and the f i n heights correspond to the 100% transfer points of those p l o t s . Similar r e s u l t s have been described by Tseng 17] and Tseng et a l [8] for a fin-corrugated surface with TE p o l a r i z a t i o n and by Zaki and Neureuther [10,11] f o r a s i n u s o i d a l surface with both,polarizations. The 100% transfer points i n figures 3.3(a) and (b) are located at the optimum angles of incidence given by equation (2.91). Because these angles are known as Bragg-angles i n c r y s t a l s t ructures, the 100% trans-fer points are sometimes referred to as "Bragg-angle anomalies". The widths of these anomalies are d i r e c t l y dependent on the p e r i o d i c i t i e s of t h e i r respective surface with small widths corresponding to small periods. This behaviour i s emphasised i n figures 3.4(a) and (b). The v a r i a t i o n of the r e l a t i v e phase of the n=o and n=-l amplitude coeffiT-cients with 8^ i s also given i n the p l o t s . I t was stated i n section 2.1.2 that the n=-l mode i n the a i r region may be e i t h e r propagating or evanescent, depending on the angle of incidence. The angles where these transformations occur are clear i n figures 3.3 and 3.4. The corresponding points on the n=o curves are referred to as "Rayleigh anomalies" [8,10-13]. Figure 3.3 Relative Power of the n=0 and n=-l Modes vs. Angle of Incidence (no attenuation). (a) a=0.578A 0. =59.99° d=0.559A 100% reduction » •'•op n .- • ~ (b) a=0.506A 8 . =81.24 d=0.501A 100% reduction •"-op (a) 90.0 1 — 1 1 1 1 1 T!T 40.0 A 50.0 50.0 70.0 80.0 80.0 THETA (DEGREES! i n e r a i-9 CCo _|OT. 30.0 40.0 SO.O THETfl (DEGREES) — , , , ! , 10.0 20.0 30.0 40.0 50.0 THETfl (DEGREES) £0.0 •(b) 80.0 90.0 rn*o 70.0 SO.O Figure 3.4 Relative Power and Relative Phase of the n=0 and n=-l Modes vs. Angle of Incidence (no attenuation). (a) a=0.578A Q. =59.99° d-0.559A 100% reduction, (b) a=0.506X 9. =81.24° d=0.501A 100% reduction Lop 1op I—1 42 3.4 Optimum Fin Height as a Function of Fin Spacing In figures 3.5(a) and (b), the f i n height which gives 100% transfer of power to the n=-l mode is plotted as a function of the f i n spacing (or optimum incidence angle). For angles of incidence in the range 50°.'<6j_ <90°', the optimum fin height i s about equal to the f i n spacing. This is an important feature since most ILS reflection problems have incident angles'In this range. It w i l l become apparent in Chapter 4 that these plots may.also be used to predict the optimum performance of surfaces with fins of fin i t e thickness. 3.5 Attenuation of the n=l Mode in the Fin Region The perturbation solution for the attenuation of the n=l mode in the f i n region, as determined in section 2.7.2, was introduced into the analysis and plots similar to those of figures 3.1-3.5 were made. As expected, there were no significant alterations of the results for a>0.53A. However, small changes were noticed at near grazing angles and these deserve some mention. In the plots of relative power with f i n height, i t was found, as expected, that the total power in the two modes was less than unity and decreased with increasing fin-height. However, the positions of 100% power transfer to the n=-l mode remained unaltered. In the plots of relative power with incident angle, i t was found that the relative power of the n=o mode decreased slightly (from unity) with increasing angle of incidence, over the range where the n=-l mode was evanescent. Small changes in the shape of the Bragg-angle anomaly were also detected, but there was no noticeable change in the optimum angle. It should be noted that the a-bove changes were extremely small, in most cases; less than 0.1%. There-ANGLE. OF INCIDENCE 6/ 90.00 ' 65.38' 56.44' 50.28' 45.58* 41.81' 38.68' 36.03' 33.75' 31.78' 30.00' | i i i _ j i I i i i I 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 FIN SPACING a/\ (a) ANGLE OF INCIDENCE 6; 90.00' 65.38' • 56-44° 50.28° 65.58° 41.81° 38.68° 36.03° 33.75° 31.78° 30.00' / g | , , , r , , j , , ! 0.91 1 :—i 1 i i i i i I 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 FIN SPACING a/\ (b) Figure 3.5 Optimum Fin Height vs. Fin Spacing, (a) d/A, (b) d/a fore, i t was concluded that attenuation is not a significant factor the design of a fin-corrugated surface. 4. Experimental Results 4.1 Introduction The numerical results of ^ Chapter 3 are based on the rigorous solution of the problem of plane wave incidence on an i n f i n i t e , f i n -corrugated surface composed of perfectly-conducting, i n f i n i t e l y thin fins. Therefore, as far as this problem is concerned, the results do not require experimental verification. However, as mentioned in Chapter 1, the purpose of the study was to get some idea of how to predict the behaviour of more realizable surfaces. In particular, i t i s desirable to know how the behaviour of the idealized surface i s affected by; 1) fi n i t e size 2) imperfect conductivity 3) f i n i t e l y thin fins 4) non-plane wave illumination 5) slightly oblique illumination Therefore, some experimental results for fi n i t e surfaces were needed to compare with the results of Chapter 3. It was decided that the fi n i t e surfaces would be modelled after a typical ILS problem, since problems of this nature were the main motivation for the study. A model frequency of 35 GHz.(A=8.566 mm.) was chosen so the ILS dimensions would be reduced to a reasonable scale. This i s a scale factor of 1:318 for a 110 MHz.(A=2.726 m.) ILS frequency. The measurements of section 4.3.2 were made at 37 GHz.(A=8.103 mm.) for reasons explained in that section. 4.2 Experimental Arrangement Four experimental surfaces were made in a l l . The f i n spacings a', f i n periods a, f i n heights d and overall dimensions are given with a profile in figure 4.1. The procedure for choosing a, a'' and d w i l l be given in the next section. The f i n thickness t was chosen at the time Plate 3A a = 0.526 cm. d = 0.447 cm. > 26.32 cm. X 11.25 cm. Plate 3B a = 0.526 cm. d = 0.480 cm.j 50 grooves Plate IA a = 0.434 cm. d = 0.432 cm.^ 26.93 cm. X 11.25 cm. Plate IB a = 0.434 cm. d = 0.173 cm. J 62 grooves Figure 4.1 A TM Polarized Plane Wave Incident on a Fin-Corrugated Surface with t=0.028 cm. ±0.002 cm. of the design to be X/30, or t^O.028 cm., about the smallest thickness possible in the milling process used. The overall dimensions corres-pond to a typical hangar wall size of 31A by 13A. One surface was milled on each side of two 2.54 cm. thick brass plates. A photograph of one of the plates is given in figure 4.2. The cover i n (a) is a brass reference plate. In the ILS system on runway 14/32 at Toronto International a Airport, an interfering hangar is located at 0^=81.25° about 3 km. from the transmitter, about half the range necessary for plane wave illumina-tion of the entire hangar surface. This corresponds to a distance of about 10 m. at 35 GHz. Since distances of this size were not available indoors, a range with dimensions shown in figure 4.3(a) was used. This -reduced range is a more severe test of the behaviour of the surfaces under non-piane wave illumination. A photograph of tne actual range used is shown in figure 4.4. Identical pyrimidal horns with 25 dB gain and E-plane 3 dB beamwidths of 9° were used for transmitting and receiv-ing when 0^<7O°. Direct transmission between the horns was blocked by an absorber suspended between the horns (see figure 4.4). For measure-ments near grazing incidence, a paraboloidal reflector transmitting antenna with 2° beamwidth was used. A diagram of the experimental cir c u i t is shown in figure 4.3(b). The receiving antenna crystal current reading with the surface exposed i s returned to the reading obtained with the surface covered by the re-ference plate by adjusting the precision variable attenuator at the transmitter. The reduction i n attenuation i s the difference i n attenuator settings to an accuracy of about +0.1 dB. The klystron output i s monitored continuously to detect output level changes during the measurements. Figure 4.2 Plates IA and IB. (a) T i i t h Reference Plate, (b) Without Reference Plate. 2.55m Plates 1A.1B h = 0.88m Plates 3A,3B h= 0.54m I = 2.27 m at B; = 69.0' 1= 2.51m at 9/=55.5* 2.40m 2.40m (a) \ CRYSTAL CURRENT METER CRYSTAL DETECTOR CRYSTAL DETECTOR PRECISION VARIABLE ATTENUATOR \ CRYSTAL CURRENT METER VARIABLE ATTENUATOR KLYSTRON FIN CORRUGATED SURFACE (b) Figure 4.3 Experimental Arrangement. (a) The: Experimental Range (see Table V, Appendix G, for the transmission distances, SL) (b) The Experimental Circuit 50 To investigate the effects of,oblique illumination, the sur-faces were mounted on a rotatable platform.. A photograph of Plate 3A mounted on the platform with angle, of rotation oLj_=0 is shown in figure 4.5. The perpendicular metal plate attached to. the end of the surface is used to simulate the side of the hangar and prevent transmission underneath the surface. 4.3 Results and Discussion The experimental results which are presented in graph, form in this section are presented in tabular form in Appendix F. 4.3.1 Plates 3A and 3B It was decided that the f i r s t two surfaces O p i a t e s 3A and 3B) should have a f i n spacing a' corresponding to an optimum angle of inci-r dence near 60°, since measurements at this angle w e r e - r e l a t i v e l y easy to -make. Therefore, the f i n spacing was chosen to be a'=0.581A (0^ =59.4°) at 35 GHz. Adding the f i n thickness gives a f i n period of a=0.614A (0iop=54.5°). The relative power vs. f i n height plots of these values are given in figure 4.6. It is obvious that there is a great deal of difference between the two curves, but since the most effective value of d is determined by the waveguide properties, the a' curve was used to determine the f i n heights; Plate 3A (optimum): d=0.561A, Plate 3B (sub-optimum): d=0.522A. Experimental measurements of the relative power of the n=o mode were taken over a range of angles 45°<0i<7O° using the procedure outlined i n section 4.2. The results for the surface of Plate 3A are shown in figures 4.7(a) and (b). A reduction of 23.4 dB or 99.54% is achieved at an angle of 55.5°. There is at least 10 dB reduction over iO' 45' 50' 55" • 60' 65' 70' 75' X' 30' iO" 50' • 60' 70' 60* SO* -ANGLE OF INCIDENCE 8; ANCLE OF INCIDENCE- 6; (a) (b) Figure 4.7 Relative Power of the n=0 Mode vs. Angle of Incidence, Plate 3A (with attenuation). (a) and (b) Experimental: a^O.SSlA a=0.614A d=0.560A at f=35 GHz. 0. =55.5° -23.4 dB or 99.54% reduction xop (b) (solid curve): a'=0.581A 0. =59.35° d=0.560A 99.97% reduction 1op o (broken curve): a=0.614A 0 =54.55 d=0.560A 93.49% reduction L n the range 48°30i<64°. The computed relative power vs. 8^  plots for a' (solid curve) and a (broken curve) with d=0.561A are also shown in figure 4.7(b). It seems that the maximum amount of reduction given by the experimental surface is determined by the proper f i n spacing - f i n height combination' (a',d) while the angle of incidence at which this maximum reduction occurs is determined solely by the f i n period a. This phenonomon suggested the following procedure for predicting the relative power vs. 0.. curve of any fin-corrugated surface composed of f i n i t e l y thick fins. Suppose 100% reduction of the power in the n=o mode is required at 0^  degrees. This angle determines the f i n period a. Substracting the appropriate fin thickness yields the f i n spacing a'. With this value of a', the f i n height d is determined from figure 3.5(a)i Thus, the surface would be constructed with parameters (a, a', d).. Now, to predict the relative power vs. 0^  curve for this surface, the f i n period a i s used with the plot of figure 3.5(a) to determine the "adjusted f i n height", d a. These parameters (a, d a) when used in the theoretical program give the required curve. This procedure was carried out on the parameters of the sur-face of Plate 3A yielding a=0.614A, da=0.595A. The results are shown in figure 4.8. The optimum reduction predicted is 11.9 dB more than observed, but this i s only a difference of 0.43% in relative power. The 1° displacement of the measured curve i s attributed to experimental error. The experimental results for the sub-optimum surface of Plate 3B are shown in figures 4.9(a) and (b). A reduction of 15.4 dB or 97.10% is achieved at an angle of 55.5°. There i s at least 10 dB reduction over the range 49°<0i<64°. Notice that the same situation occurs i n figure 4.9(b) as i t did in 4.7(b), except that now the experimental curve has ANGLE OF INCIDENCE 6; ANGLE OF INCIDENCE 8/ (a) (b) .Figure 4.9 Relative Power of the n=0 Mode.vs. Angle of Incidence, Plate 3B (with attenuation). (a) and (b) Experimental: ^'=0.58^ a=0.614A d=0.522A at f=35 GHz. 6. =55.5° ^ 15.4 dB or 97.10% reduction (b) (solid curve): a'=0.581A 0. =59.35° d=0.522A 88.85% reduction 1op n (broken curve): a=0.614A 8 . =54.55 d=0.522A 75.94% reduction °P \ i i i i i i i i I I I J I t I I I - L -80° -60' -40' -20' -0'. +20' +40' +60' +80' ANGLE OF ROTATION . <Xf Figure 4.11 Relative Power of the n=0 Mode vs. Angle of Rotation, Plate 3A, Sub-Optimum Incidence 8.=61°, f=35 GHz., ( s o l i d curve): Reduction Due to the Fins only, (dashed curve): Reduction Due to Plate Orientation only. Ln VO 33.0 33.5 34.0 34.5 35.0 35.5 36.0 36.5 37.0 FREQUENCY (GHz.)' Figure 4.12 Relative Power of the n=0 Mode vs. Frequency of the Incident Wave, Plate 3A, (with attenuation) Experimental: a'=0.581A a=0.614A d=0.560A at f=35 GHz., &±=&i =55.5° Predicted (broken curve): a=0.614X d =0.595A at f=35 GHz., 6.°B\ =54.55° a 1 1op more reduction than the a' curve. This indicates that the prediction procedure outlined above is -inaccurate except for the case with 100% reduction. The predicted curve, computed using a sub-optimum curve similar to that of figure 3.5(a), i s shown in figure 4.10. Figure 4.11 shows a plot of the relative power of the n=o mode as a function of the angle of rotation for Plate 3A at an angle of incidence 0^=61° (sub-optimum angle). The dashed curve indicates the power reflected from the reference plate as i t i s rotated. The solid curve shows the reduction of power due to the fins alone; Appar-ently, the surface remains effective for oblique, incidence over the range -10°<a^<10°. As ILS glide path angles are about 2 1/2° the sur-face should be essentially as effective in reducing interference along the glide path as in the horizontal plane. The experimental plot of the relative power of the n=o mode vs. frequency of the incident wave for the surface of Plate 3A at a fixed angle of 0i =55.5° is shown in figure 4.12. As the 4 MHz band op over which the ILS systems operate scales to 1.3 GHz., there would be no bandwidth limitation for this surface. Figure 4.12 also contains the predicted curve (broken curve) found using the predicted parameters of figure 4.8. 4.3.2 Plates IA and IB The second two surfaces (Plates IA and IB) were chosen to have a f i n period a, at 35 GHz. corresponding to an optimum angle of incidence near 80°. However, i t was discovered in the preliminary ex-perimental measurements that, because of the f i n i t e thickness of the fins, the f i n spacing a' was well below the cutoff spacing A/2 and that the surfaces would give no reduction. Therefore, the frequency of the FIN HEIGHT d/\ Figure 4.13 Relative Power of the n=0 Mode vs. Fin Height, Plates IA and IB (with attenuation) (solid curve): a'=0.502A 6. =85.42° 100% reduction at d=0.498X=0.992a l oP o (broken curve): a=0.536A. 6. =68.88 100% reduction at d-0.523X=0.975a •'-op Plate IA: d=0.533A Plate IB: d=0.213X ANGLE OF INCIDENCE 8/ ANGLE OF INCIDENCE 8/ (a) (b) Figure 4.14 Relative Power of the n=0 Mode vs. Angle of Incidence, Plate IB (with attenuation). (a) and (b) Experimental: a'=0.502A a=0.536X d=0.213A at f=37 GHz. 6. =69.0° 21.7 dB or 99.33% reduction l oP (b) (solid curve): a'=0.502A 9. =85.42° d=0.213X 98.33% reduction l o P o (broken curve): a=0.536A 9 ^=68.88 d=0.213A 70.69% reduction incident wave was increased to 37 GHz. so that a' = 0.502A (6J =85.42°) and a=0.536A (6+ =68.88°). The relative power vs. f i n 1op xop _ height plots of these values are given in figure 4.13. Because the fin thickness is comparable to the f i n spacing the two curves are very different. The f i n heights were chosen from the a' curve as before, but because of the frequency change to 37 GHz., they both became sub-optimum; Plate IA: d=0.533A, Plate IB: d=0.213A. The experimental results for the surface of Plate IB are shown in figures 4.14(a) and (b). A reduction of 21.7 dB or 99.33% is achieved at an angle of 69°. There is at least 10 dB reduction over the range 62°<0i<79°. Notice the great difference between the a and a' theoreti-cal curves of figure 4.14(b). This indicates that i t would be impossible to have complete reduction at angles 0^  >75°-from any surface unless the fins are made thinner than t or the surface is. modelled at a lower frequency. Measurements taken at the optimum angle of incidence for various values of indicated that the surface of Plate IB remains effective for oblique incidence over a range of angles -10°<ai<10°. The prediction procedure of section 4.3.1 was carried out on the parameters of this surface yielding a=0.536A, da=0.510A. The re-sults are shown in figure 4.15. The optimum reduction predicted is 3.9 dB less than observed, a difference of 0.99% in relative power. The experimental results for the surface of Plate IA are shown in figure 4.16. The reduction was much less than expected with no maxi-mum reduction observed at 69°, the angle of optimum reduction according to the period. Also, there was a variation i n the results at each angle and so only the average reduction could be plotted. It i s believed that these irregularities in behaviour were caused by a Wood S-anomaly lying on 0.35, 35.8 36.0 36.2 36.4 36.6 36.8 37.0 FREQUENCY (GHz.) 37.2 37.4 Figure 4.17 Relative Power of the n=0 Mode vs. Frequency of the Incident Wave, Plate IB (with attenuation) Experimental: a'=0.502A a=0.536A d=0.213A at f=37 GHz., =69.0 , f =36.88 GHz. Predicted (broken curve): a=0.536A d =0.510A at f=37 GHz., 6.=9. =68.88 - a i l o p . or very near the value chosen as the fi n height for this surface (see figure 4.13). Small variations in the klystron frequency change d/A at the anomaly and give large variations in the reflection coefficient. The experimental plot of the,relative power of the n=o mode vs. frequency of the incident wave for the surface of Plate IB at fixed angle 6i0p=69° is shown in figure 4.17. A frequency of 37 GHz. was the highest attainable with the klystron"source used. The frequency fc=36.88 GHz. is the cutoff frequency (corresponding to the cutoff f i n spacing) of the n=l mode in the f i n region. This surface with short fins seems to operate very effectively over a range of frequencies below cutoff. Figure 4.17 also contains the predicted curve (broken curve) found using the predicted parameters of figure 4.15. 5. Conclusions In this study, a rigorous analysis of plane wave scattering from an idealized fin-corrugated periodic surface was presented and numerical results from i t compared with experimental results obtained for f i n i t e surfaces under non-plane wave illumination. Summarized below are the conclusions drawn: 1) An optimum fin^corrugated surface demonstrating complete cancellation of specular reflection can be designed for any fin period in the range \/2<a<\. For angles of i n c i -dence in the range 5 0 ° < 9 ; L < 9 0 O , the usual range for most applications, the optimum f i n height for these surfaces does not need to be any larger than the fi n period. 2) For situations in which the angle of incidence is near grazing ( 9 ^ > 8 0 ° ) , reductions in specular reflection of 9 0 % or greater are possible with f i n heights much shorter than the f i n periods and without severe tolerances on the fin heights. 3) The angular width of the Bragg-angle anomaly demonstrating complete cancellation of specular reflection decreases with the period of the surface. However, i t is s t i l l suf-fi c i e n t l y wide for most applications, even near grazing. 4) The relative power vs. 8^ curve for any optimum, f i n i t e sized, fin-corrugated surface composed of f i n i t e l y thick fins can be predicted using a procedure outlined i n section 4.3.1. However, this procedure is not accurate for sub-optimum surfaces. 5 ) The fin i t e size used for these surfaces, which was a scaled down size of a typical hangar wall i n the ILS problem, had essentially no effect on the performance of the sur-face. Reductions of nearly 100% were achieved with 50 to 60 corrugations. Non-plane wave illumination of these fin-corrugated sur-faces seemed to have very l i t t l e effect on their performances. The experimental range used in this study was about 1/3 the length of an actual ILS range and hence was a very severe test for the surfaces. The experimental fin-corrugated surfaces remained completely effective for angles of rotation in the range -10<a^<10. Since ILS glide path angles are about 2 1/2°, these surfaces should be essentially as effective i n reducing interference glrvno the p'lide path as in the hori.7nr>t^.1 plane. The experimental surfaces used in this study were not very fre quency sensitive. The bandwidths over which they operated effectively were much larger than required when scaled to the ILS frequency range. The frequency sensitivity of the experimental surfaces was predictable using the parameters obtained in the prediction procedure of section A.3.1. The attenuation of the n=l mode in the fin region seems to have l i t t l e effect on the performance of the theoretical surfaces, even when the frequency i s close to cutoff. In addition, i t was found i n the experimental investigation that the fi n i t e surface with short fins would operated very effectively over a range of frequencies below cutoff. With an experimental frequency of 35 GHz. and the f i n thickness used in this study, i t is impossible to construct a surface which w i l l be optimum at an angle of incidence greater than 75°, since i n these cases, the f i n spacing would be too far below cutoff. It is believed that the value of thickness used in this study was the thinnest possible at 35 GHz. This seriously limits any near grazing investigations at this frequency. 72 Appendix A. Determination of the Region of Analytic!ty The following i s a method described by C o l l i n 16 , P.433] i n which the region of a n a l y t i c i t y of the transform function <ji(x,s) i s determined: Let <j>(x,s")_ be represented as the sum of two functions; <(>+(x,s) , which i s analytic i n the right half of the complex s-plane (RHP) and <|>_(x,s) , which i s analytic i n the l e f t half of the complex s-plane (LHP). Then <Kx,s) f P = <|>_(x,s)+ c(>+(x,s) = J e o>(x,z)dz + J e <f>(x,z)dz (A.l) Now consider Problem # 3. In the region z>o, equation (2.33) may be written as <j>(x,z) = A D e Y ° Z + B 0e" Y° Z + B x cos ( ^ e ~ Y l Z (A.2) since the evanescent modes die out within a very short distance of the f i n - a i r interface. Let the t o t a l f i e l d <Kx,z) possess a very small amount of loss so that Vo = T o 1 + J Y o n (A. 3) and Y i = Y l 1 + J Y ] 1 1 ( A- A) where Y 0 1 < <Y 0 1 1'> Y i 1 < < Y i 1 1 and Y Q 1 , Y Q 1 1 * Y^1 and Y i 1 1 are r e a l and po s i t i v e . Substituting (A.2) into the expression for cj>+'(x,s) and i n t e -grating, i t i s obvious the result can only be f i n i t e i f S ^ ^ Y Q - * - , S I> -Y,-,"'" s-^> _ Y j l where s=s-^  + 3 S 2 . Therefore, <}>+(x,s) i s analytic i n the RHP i f and only i f Im s a 1 YQ 1 and Im s z -Yi* (A.5) Similarily, in the region z<o, equation (2.32) may be written as and i t may be assumed that (A.6) and r • _ • 1 11 r - l - r - l + J r _ i (A. 7) (A. 8) 11 where ro1«ro11, r_11«f_111 and VQl, T^1, T^1 and r_1J"t are real and positive. Substituting (A. 6) into the expression for _(x,s) and integrating, i t follows that ^_(x,s) i s analytic in the LHP i f and only i f Im s< T Q 1 and Im s < Y^1 (A.9) Therefore, i f i t is assumed that Ya^~<y±'<T0^'<T t the strip of common analyticity for $ +(x,s) and <j>_(x,s) i s as shown in figure A.l and con-tains the integration contour P. ' j S 2 i i 4> (x,s) ANALYTIC . | ; _4 1 pi pi 0 -1 : r' TTio-iiT-o A . l Th<=> Sf r i ' n nf r.nmmon A n a l v t i c i t V . 74 Appendix B. Determination of g(s) I t was shown i n section 2.4.1 that g(s) i s a meromorphic func-t i o n , that i s , a function which contains only poles. The following i s a method described by C o l l i n [6, P.434] i n which g(s) i s constructed from r a t i o s of en t i r e functions subject to the constraints (2.50) and (2.51): For z>o, the integrand of equation (2.50) i s regular i f g(s)/[cos(ua)-e j n a ] cancels the zeros of [u sin(ua)] corresponding to u = — n= +0,1,2,.i. (B.l) 3. Substituting (B.l) i n t o (2.40) and comparing with (2.17) i t i s seen that s = t y n n=-+0,1,2... (B.2) For n=o, Sj+js., = ty0^:± J Y 0 U ( B > 3 ) or s± = t Y o 1 S 2 = I Y / 1 (B.4) where Y0"'"<<Y0"'"''' as before. S i m i l a r i l y , f o r n=l, where Y i ^ ^ Y ] . 1 1 * and, for n= +2,3,..., - 8! = tyn (B.6) since YN i s pure r e a l f o r n ^ 0 , l . However, because z>o corresponds to the LHP and the n=o mode i s the only mode incident i n the f i n region, the zeros of [u sin(ua)] are n=o n= +1 n= + 2 , 3 , . s = -Y. (B.7) These zeros are pl o t t e d i n figure B . l . For z<o, the integrand of (2.51) i s regular i f g(s)/[cos(ua)-~~ "i hs e ] cancels the zeros of [cos(ha)-cos(ua)] corresponding to u = h + 2mr n= to,1,2, (B.8) Substituting (B.8) in t o (2.40) and comparing with. (2.15) i t can be shown that s = tr n n= to,1,2,.. . (B.9) For n=o, s±+3S2 = ^c; -or s l " ro so = + r 1 1  2 -[o (B.10) ( B . l l ) -where f 0 1 < < r o 1 1 a s b e f o r e » S i m i l a r i l y , f o r n=-l, s l " " r - l s. - -T_± (B.12) where r _ 1 1 « r _ 1 1 1 , and, for n= +1,±2,3,..., (B.13) since r n i s pure r e a l f o r n ^ 0 , - l . However, because z<o corresponds to the RHP, the zeros of [cos(ha)-cos(ua)] are XXX • ' -4 • 1-r l r r 0 - Y i "Y0 n *"~ ~f"2 y 3 j • » • X z>0 JY^ J r o jr" - l Y6 Y i t - j r ^ -JYJ -JYX x r' r 1 ' o - i -X-K*-n n = +1,±2,3, z<0 Figure-B.l Zero P l o t of [u sin(ua)] and [cos(ha)-cos(ua)] with l o s s e s . j s . Wo -1 x x x r n n = +2,3, z>0 f ) - Y l 9"Yo n = +1,±2,3, z<0 Figure B.2 Pole Plot of g(s)/[cos(ua)-e J ] n=o n= -1 n= +1 ,JI2, 3,. . . S = + i T "|I s = +r_ 1 1 + j r ^ 1 1 s = +r_ (B.14) These zeros are also p l o t t e d i n figure B . l . Now assume that the losses Y0^, r 0~^ a n d r - l ~ ^ t e n d t o zero. The zero p l o t of figure B . l then transforms to the zero p l o t of figure B.2. Since the poles of g(s) / [cos(ua)-e ^ a ] must cancel the zeros of [u sin(ua)] and [cos(ha)-cos(ua)], figure B.2 i s therefore the pole p l o t of that function. I t follows that a s u i t a b l e construction f o r g(s) would be g(s) = P(s) (cos(ua)-e - h a} ( s - y 0 ) ( S + Y 0 ) ( s - r Q ) (s+y ) " ( a - r ) ( s - r ) s a v "n nn l — r n -n rnr e I I ———^  —' e (B1, TT 1 ,2mT^  2 a (B.15) where P(s) i s an e n t i r e function yet to be determined. I t w i l l be shown i n Appendix C that the exponential factors and the terms (—) and ( ^ — ) 2 a a i n the denominator of (B.15) are used to ensure the uniform convergence of the i n f i n i t e products. Appendix C. Convergence of the I n f i n i t e Products i n g(s) From equations (2.17) and (2.15), i t i s obvious that, f o r large n, y n + (—) , f +h) and T_n ( ~ I -h) . Therefore, ( n j ) . e ( 1+ e m r -sa nir and (s-r ) ( s - r ) — n -n nir (2n7[) 2 a ( 1 -(s-h)a (s-h)a. 2mr . -. ~i^r) e ( 1 ( C l ) (s+h)a (s+h)a N 2niT -) e 2nir ( C 2 ) However, the r i g h t hand sides of ( C l ) and (C.2) are of the same form as the Euler-Mascheroni expression ( e -z_ n -yz zr(z) Y = constant (C.3) which i s f i n i t e for |z|<°°. Therefore, the i n f i n i t e products of g(s) , equation (B.15) of Appendix B, are uniformly convergent. /y Appendix D. Asymptotic Behaviour of g(s) In section 2.1.1, i t was stated that the t o t a l f i e l d <}>(x,z) 1/2 i s asymptotic to z i n the neighbourhood of a f i n edge. That i s , 1/2 <j>(ma,z) = A(ma) z + higher order terms as z-*o (D.l) where m= ±0,1,2,... . Substituting this expression i n t o equation (2.38) gives oo ;-sz 1/2 e lA(o)z + higher order termsldz (D.2) Watson's Lemma guarantees that i f this r e s u l t i s integrated term by term, the asymptotic expansion of g(s) as |s|-**> w i l l r e s u l t . Consider the f i r s t term only. The integrand has a branch point at z=o and so i f the branch cut i s chosen as shown i n figure D.l and the i n t e g r a l closed i n the upper h a l f plane, by Jordan's Lemma g(s) = j" e ~ S Z A ( o ) z 1 / 2 d z = -2A(o) • j e " S Z z 1 / 2 d z (D.3) Making the s u b s t i t u t i o n t= - j z and using the i n t e g r a l representation for the gamma function, i t i s found that g(s) = -2 A ( o ) r ( | ) s " 3 / 2 as |s| -> - (D.4) Thus, g(s) i s asymptotic to s as |s| -> 80 Figure D.l z-Plane Representation Showing Branch Cut. Appendix E. Determination of P(s) i n g(s) I t was stated i n section 2.4.2 that P(s) must be chosen so.. -3/2 that g(s) has algebraic growth at i n f i n i t y and i s asymptotic to s as |s| -> 0 3. The following approach to t h i s problem Is s i m i l a r to that used by C o l l i n [6, P.436]. Su b s t i t u t i n g the expressions ( C l ) and (C.2) into equation (B.15), i t i s seen that g(s) d i f f e r s by only a bounded function of s . from gi(s) where g^(s) i s given by P(s) jcos(ua)-e~" :' h a} 81 ~~ 0 0 -sa 0 0 (s-h)a « (s-Fh)a s 3 r i (1+ 's> e ™ n c i - ^ Wr>e 2n7r rr <x- e 2n7r 1 1 1 • • ' ' (E.l) E l i m i n a t i n g the i n f i n i t e products i n (E.l) using equation (C.3) with the proper value of z and noting that T (z ) - . , \ p < , - r- (E.2) s m ( T r z ) r ( l - z ) . ' g n ( s ) can be manipulated into the form P(s) {cos(ua)-e~ j h a} (air) .H^L) S l < * S ) = ~ 2 • r(s-h)a, r(s+h)a, ,;f (s-h)a, rT(s+h)a, s sxn{ — } sxn{ —} r { — T { — ^ - } (E.3) Now, the asymptotic expansion of T(z) i s •n r \ / „ 1 Zlnz - 1 / 2 -Z+l 1 1 ^ f n t\ T(z) = / 2Tr x e z e [ z| -* "> (E. 4) I f t h i s expression i s used to replace the T functions i n equation (E.3) and i t i s assumed t h a t h « s as | s j . OT, an a s y m p t o t i c e x p a n s i o n f o r gi (s) o f the form S a l n 2 ( \ = K P(s) {cos(ua)-e j h a } e 7 7 § 1 3/2 . r ( s - h ) a , . r ( s + h ) a i s sxn{- — } s m l 7.—i (E .5) w i l l r e s u l t , where K i s a known constant. Therefore, f o r g-^(s) and -3/2 hence g(s) to be asymptotic to s , i t i s obvious that — S 3. P(s) = C e~T~ln2 (E.6) where C i s a constant. Appendix ,F. Experimental Data The following i s a c o l l e c t i o n of the experimental data taken during the study. Most of t h i s data appears i n graphical form i n Chapter 4. Plates 3A, 3B Plates IA, IB 9 i I ^degrees) (m.) (degrees) (m.) 70.0 2.25 79.0 2.14 67.5 2.29 78.0 2.15 65.0 2.34 77.0 2.16 62.5 2.40 76.0 2.17 60.0 2.47 75.0 2.18 57.5 2.54 74.0 2.19 56.5 2.57 73.0 2.20 55.5 2.61 72.0 2.22 54.5 2.64 71.0 2.23 53.5 2.52 70.0 . 2.25 52.5 2.55 69.0 2.27 50.0 2.66 68.0 2.28 47.5 2.47 67.0 2.30 45.0 2.35 66.0 2.32 65.0 2.34 64.0 2.36 Table V Transmission Distance vs. Angle of Incidence (see figure 4.3(a)) RELATIVE POWER OF THE. n=0 MODE (dB) \^ a. -5.0° -2.5° 0° +2.5° +5.0° 70.0° 67.5° 65.0° 62.5° 60.0° 57.5° 56.5° 55.5° 54.5° 53.5° 52.5° 50.0° 47.5° 45.0° -6.9 -8.0 -9.95 -12.3 -15.5 -20.4 -23.1 -23.4 -22.3 -19.7 -16.6 -11.8 -8.7 -6.05 -6.85 -8.15 -10.05 -12.2 -15.55 -20.5 -23.3 -23.1 -22.3 -19.7 -16.65 -11.9 -8.75 -6.0 -6.75 • -7.9 -9.8 -11.95 -15.15 -20.0 -23.0 -23.4 -23.3 -20.9 -17.55 -12.4 -9.1 -6.35 -6.95 -8.15 -10.1 -12.2 -15.5 -20.5 -23.2 -23.0 -22.3 -19.7 -16.75 -11.9 -8.7 -6.1 -7.1 -8.2 -10.0 -12.2 -15.4 -20.3 -22.8 „-22.4 -22.2 -19.3 -16.65 -11.9 -8.75 -6.05 Table VI Relative Power of the n=0 Mode vs. Angle of Incidence, Plate 3A, f=35 GHz. 85 RELATIVE POWER OF THE n=0 MODE (dB) \ a i -5.0° -2.5° 0° +2.5° +5.0° 70.0° 67.5°. 65.0° 62.5° 60.0° 57.5° 56.5° 55.5° 54.5° 53.5° 52.5° 50.0° 47.5° 45.0° -6.5 -7.55 -9.0 -10.6 -12.55 -14.8 -15.6 -15.5 -15.15 -14.7 -13.75 -10.8 -8.1 -5.65 -6.4 -7.35 -8.75 -10.2 -12.1 -14.2 -15.0 --15.2 -14.8 -14.4 -13.45 -10.65 -8.0 -5.6 -6.3 -7.3 -8.55 -10.2 -12.1 -14.25 -15.1 -15.4 -15.1 -14.65 -13.75 -11.05 -8.3 -5.9 -6.4 -7.4 -8.6 -10.2 -12.2 -14.2 -15.0 -15.25 -14.8 -14.4 -13.45 -10.8 -8.0 -5.65 -6.7 -7.65 -9.0 -10.6 -12.55 -14.7 -15.6 -15.6 -15.15 -14.8 -13.7 -10.9 -8.1 -5.65 Table VII Relative Power of the n=0 Mode vs. Angle of Incidence, Plate 3B, f=35 GHz. 86 FREQUENCY RELATIVE POWER, (GHz) n=0 MODE (dB) 33.00 -9.61 33.20 -10.3 33.40 -10.75 33.60 -11.4 33.84 -12.45 34.00 -13.1 34.20 -14.55 34.40 -16.20 . 34.60 -18.40 34.80 -21.2 35.00 -23.3 35.20 -26.4 35.40 -30.6 35.65 -31.8 35.80 -30.5 36.00 -27.8 36.20 -25.5 36.40 -23.3 36.60 -21.5 36.80 -20.0 37.00 -18.7 Table VIII Relative Power of the n=0 Mode vs. Frequency of the Incident Wave, Plate 3A, a i = 0 ° , 0^=54.5° 0. X (degrees) RELATIVE POWER n=0 MODE (dB) 79.0° -10.15 78.0 -10.89 77.0 -12.25 76.0 -13.06 75.0 -13.80 74.0 -15.5 73.0 -17.4 72.0 -18.7 71.0 -20.75 70.0 -21.7 69.0 -21.7 ( a i= +2.5°) -24.0 (a ±= +5.0°) -25.8 (a ±= -2.5°) -23.9 (a ±= -5.0°) -25.5 68.0 -21.2 67.0 -19.9 . 66.0 -16.78 65.0 -14.5 64.0 -12.41 Table IX. Relative Power of the n=Q Mode vs. Angle Incidence, Plate IB, 0^=0°, f=37 GHz. oo 9. X AVERAGE RELATIVE (degrees) POWER n=0 MODE (dB) 84.0 -0.478 83.0 -0.728 82.5 -0.763 82.0 -0.925 81.5 -0.717 81.0 -0.727 80.0 -0.81, 79.0 -0.777 78.0 -0.64 77.0 -0 i 368 76.0 -0.43 75.0 -0.47 74.0 -0.43 73.0 -0.35 72.0 -0.33 71.0 -0.27 70.0 -0.42 69.0 -0.383 68.0 -0.29 67.0 -0.283 66.0 -0.30 65.0 -0.30 64.0 -0.23 63.0 -0.263 Table X Relative Power of the n=0 Mode vs. Angle of Incidence, Plate IA, a±=0°, f=37 GHz. 89 FREQUENCY RELATIVE POWER (GHz) n=0 MODE -(dB) .35.8 -4.6. 35.9 -6.4 36.0 -8.4 36.2 -12.8 36.4 -16.4 36.6 -22.7 36.8 -25.7 37.0 -21.7 Table XI Relative Power of the n=0 Mode vs. Frequency of the Incident Wave, Plate IB, a± = 0°, e ± = 69.0° 90 References 1. J.F. Carlson and A.E. Heins, "The R e f l e c t i o n of an Electromagnetic Plane Wave by an I n f i n i t e Set of Pla t e s , I", Quart, of App. Math., V o l . VI, No. 1, 1947. 2. A.E. Heins and J.F. 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