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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 B r i t i s h Columbia, 1972  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  i n 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 an  this  thesis  a d v a n c e d d e g r e e at  the  Library  I further for  shall  agree  the  in p a r t i a l  fulfilment of  University  of  make i t f r e e l y that permission  s c h o l a r l y p u r p o s e s may  by  his  of  this  written  representatives.  be  available  granted  gain  permission.  Depa r t m e n t  Date  for  for extensive by  the  It i s understood  thesis for financial  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  British  Columbia  shall  requirements  Columbia,  Head o f my  be  I agree  r e f e r e n c e and copying of  that  not  the  that  study.  this  thesis  Department  copying or  for  or  publication  allowed without  my  Abstract The use of periodic structures to reduce interference from large r e f l e c t i n g surfaces i s proposed.  Instrument landing system (ILS)  interference from large hangars and terminal buildings i s c i t e d as a t y p i c a l problem.  An a n a l y t i c a l and numerical investigation of an i n f i n i t e  fin-corrugated surface composed of i n f i n i t e l y thin f i n s of spacing \/2<a<\ under TM polarized plane wave i l l u m i n a t i o n i s described.  Specu-  lar r e f l e c t i o n from t h i s surface can be completely converted to backscatter i n a d i r e c t i o n 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 s i z e  fin-corrugated surfaces with f i n s of f i n i t e thickness under non-plane wave illumination and the r e s u l t s indicate that these surfaces behave e s s e n t i a l l y as predicted.  In addition, the experimental surfaces remain  completely e f f e c t i v e for small oblique angles of incidence and have s u f f i c i e n t bandwidth for ILS applications.  ii  Table of Contents Page L i s t of I l l u s t r a t i o n s L i s t of Tables  v  . .  vii  L i s t of Symbols  viii  Acknowledgements  x  1.  Introduction  2.  Theoretical Analysis  5  2.1  5 7 9  4.  1  Formulation of the Overall Problem 2.1.1 Boundary and Edge Conditions 2.1.2 General Solutions  .  2.2  Representation of the Fin-Air Discontinuity by a Scattering Matrix  2.3  Formulation of the Component Problems  2.4  The Transformed Problem 2.4.1 Solution of the Transformed Problem 2.4.2 Determination of the F i n a l Integral Representation  2.5  3.  .  H 15 .  1  7  18 20  Solution of the F i e l d Amplitude C o e f f i c i e n t s 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 2.7.1 Reflection and Transmission Coefficients at the Optimum Angle of Incidence . . 2.7.2 Attenuation of the n=l Mode i n the P a r a l l e l Fin Region  . 28 .28 33  Numerical Results  35  3.1' Introduction  35  3.2  Relative Power as a Function of F i n Height  35  3.3  Relative Power as a Function of Incident Angle  3.4  Optimum F i n Height as a Function of F i n Spacing  3.5  Attenuation of the n=l Mode i n the F i n Region  . . . . . 39 . . . . ^2 ^2  Experimental Results  ^  4.1  Introduction  4.2  Experimental Arrangement  4.3  Results and Discussion . . . . . . . . . . . . . . . . . $2 4.3.1 Plates 3A and 3B 4.3.2 Plates IA and IB •  . A5  iii  5  2  6  1  Page 5.  Conclusions  69 . . . .  72  Appendix A.  Itetermination of the Region of A n a l y t i c i t y  Appendix B.  Determination of g(s)  74  Appendix C.  Convergence of the I n f i n i t e Products i n g(s) . . .  78  Appendix D.  Asymptotic Behaviour of g(s)  79  Appendix E.  Determination of P(s) i n g(s)  81  Appendix F.  Experimental Data  83  References  90  iv  L i s t of I l l u s t r a t i o n s Figure  Page  1.1  Periodic Surface Demonstrating  2.1  F i n Corrugated Structure with Incident TEM Plane Wave . . . .  6  2.2  Single Period with Boundary Conditions  3  2.3  Four Port Scattering Junction Representation  2.A  Component Problems  2.5  Pole Plot of the Integral Representation, Equation (2.54), Problem # 3  21  Pole Plot of the Integral Representation, Equation (2.75), Problem 7/4  26  Pole Plot of the Integral Representation, Equation Problem #1  26  2.6 2.7 3.1 °.2 3.3  Bragg's Law  1  12 . 16  (2.76),  Relative Power of the n=0 and n=-l Modes vs. F i n Height (no attenuation)  36  Rcl-^ivs Pc.rcr and Relative ^has o*~ Modes v s . F i n Height (no attenuation)  38  0  "=Q p.^d •"=—1  Relative Power of the n=0 and n=-l Modes vs. Angle of Incidence (no attenuation)  40  Relative Power and Relative Phase of the n=0 and n=-l Modes vs. Angle of Incidence (no attenuation)  41  3.5  Optimum F i n Height vs. F i n Spacing  43  4.1  A TM Polarized Plane Wave Incident on a Fin-Corrugated  3.4  .  Surface with t=0.028 cm. 1 0.002 cm  46  4.2  Plates IA and IB  48  4.3  Experimental Arrangement  4.4  Experimental Range f o r Plates 3A and 3B. Transmitting  . . . . .  49  Horn i n Foreground  50  4.5  Plate 3A on Mounting Platform, a.j=0  51  4.6  Relative Power of the n=0 Mode vs. F i n Height, Plates 3A and 3B (with attenuation) Relative Power of the n=0 Mode vs. Angle of Incidence, Plate 3A (with attenuation)  53  4.7  v  54  Figure 4.8  4.9  Page  P r e d i c t e d R e l a t i v e Power o f t h e n=0 Mode v s . Angle o f I n c i d e n c e , P l a t e 3A (with a t t e n u a t i o n ) . .  55  R e l a t i v e Power o f the n=0 Mode v s . Angle o f I n c i d e n c e , P l a t e 3B (with a t t e n u a t i o n )  57  4.10 P r e d i c t e d R e l a t i v e Power o f the n=0 Mode v s . Angle of I n c i d e n c e , P l a t e 3B (with a t t e n u a t i o n ) . . .  58  4.11 R e l a t i v e Power o f the n=0 Mode v s . Angle o f R o t a t i o n , P l a t e 3A  59  4.12 R e l a t i v e Power of t h e n=0 Mode v s . Frequency o f the I n c i d e n t Wave, P l a t e 3A, (with a t t e n u a t i o n )  60  4.13 R e l a t i v e Power o f the n=0 Mode v s . F i n H e i g h t , P l a t e s IA and IB (with a t t e n u a t i o n )  62  4.14 R e l a t i v e Power o f the n=0 Mode v s . Angle o f I n c i d e n c e , P l a t e IB (with a t t e n u a t i o n )  63  4.15 P r e d i c t e d R e l a t i v e Power o f the n=0 Mode v s . Angle o f I n c i d e n c e , P l a t e IB (with a t t e n u a t i o n )  64  4.16 R e l a t i v e Power o f the n=0 Mode v s . Angle o f I n c i d e n c e , Plate IA  6  6  4.17 R e l a t i v e Power o f the n=0 Mode v s . Frequency o f the I n c i d e n t Wave, P l a t e IB (with a t t e n u a t i o n )  67  A. l  The S t r i p o f Common A n a l y t i c i t y  73  B. l  Zero P l o t o f [u s i n ( u a ) ] and [ c o s ( h a ) - c o s ( u a ) ] ^6  with losses B.2  Pole P l o t o f g ( s ) / [ c o s ( u a ) - e ~ ^ ]  76  D.l  z-Plane R e p r e s e n t a t i o n Showing Branch Cut  80  h a  vi  List  o f Tables  Table I  II  III  Page Amplitude R e f l e c t i o n and T r a n s m i s s i o n Sub-Optimum Case  Coefficients,  Amplitude R e f l e c t i o n and T r a n s m i s s i o n Sub-Optimum Case  Coefficients,  Amplitude R e f l e c t i o n and T r a n s m i s s i o n  Coefficients,  29  . . . .  Optimum Case IV  30  q-i J  Amplitude R e f l e c t i o n and T r a n s m i s s i o n  X  Coefficients,  Optimum Case  32  V  T r a n s m i s s i o n D i s t a n c e v s . Angle o f Incidence  VI  R e l a t i v e Power o f the n=0 Mode v s . Angle o f I n c i d e n c e , P l a t e 3A, f= 35 GHz R e l a t i v e Power o f the n=0 Mode v s . Angle o f I n c i d e n c e , P l a t e 3B, f=35 GHz  85  R e l a t i v e Power o f the n=0 Mode v s . Frequency o f the I n c i d e n t Wave, P l a t e 3A, 0^=0°, 6 =54.5° . . .  86  VII  VIII  . . . . . . .  1  IX  04=0°, f=37  GHz  . . .  87  R e l a t i v e Power o f the n=0 Mode v s . Angle o f I n c i d e n c e , P l a t e IA,  XI  84  R e l a t i v e Power o f the n=0 Mode v s . Angle o f I n c i d e n c e , P l a t e IB,  X  83  04=0°, f=37  88  GHz  R e l a t i v e Power o f the n=0 Mode v s . Frequency o f the I n c i d e n t Wave, P l a t e IB, a.j=0 , 9i=69.0° o  vii  89  L i s t of Symbols = f i n period = f i n spacing = incident f i e l d amplitude c o e f f i c i e n t i n the o v e r a l l problem = incident f i e l d amplitude c o e f f i c i e n t s i n the component problems = amplitude r e f l e c t i o n c o e f f i c i e n t s , n=0 and n=-l modes = reflected f i e l d amplitude c o e f f i c i e n t i n the o v e r a l l problem l» V l = reflected f i e l d amplitude c o e f f i c i e n t s i n the component problems B  V  B  = transmitted f i e l d amplitude c o e f f i c i e n t i n the o v e r a l l problem = constant  0' - l > V C  c_  1  = transmitted f i e l d amplitude c o e f f i c i e n t s i n 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 r e f l e c t i o n c o e f f i c i e n t s , n=0 and n=-l modes  viii  P(s) R  l» 2 ' R  = s p e c i a l function  V 4,m' V V R  7,m  R  =: residues i n Problem #3 R  = surface r e s i s t i v i t y of the f i n s  m  1  t u  = f i n thickness 2  ,2 2 = k + s o X  x, y, z  = space coordinates  Z  = impedance constant of free space  Q  a  = attenuation c o e f f i c i e n t of the n=l mode i n the f i n region = angle of rotation  Yn  = 0<z<d propagation c o e f f i c i e n t of the n*"* mode, n=*0,l,2,.  Tn  th = propagation c o e f f i c i e n t of the n mode, n=±0,l,2, z<0  T(z)  = gamma function  1  = angle of incidence = optimum angle of incidence  l  op \  = free space wavelength  <|>(x,z)  = magnetic f i e l d component i n the y - d i r e c t i o n , H(x,z)  f(x,s)  = b i l a t e r a l Laplace transform of <j>(x,z)  oi  = angular frequency  ix  Acknowledgements I would l i k e to thank the National Research Council of Canada for the bursary I received i n 1972/73, the Ministry of Transport for their support under Grant MOT  65-8114 i n 1973/74, the B r i t i s h Columbia  Telephone Company for the scholarship I received i n 1973/74 and  the  University of B r i t i s h Columbia for research assistantships received from 1972  to  1974. I am deeply grateful to my research supervisor Dr. E.V.  for his h e l p f u l suggestions l i k e to thank Dr. E.V.  and constant encouragement.  Jull  I would also  Bohn and Dr. M. Kharadly for reading the o r i g i n a l  draft and for their valuable comments.  I would l i k e to give a s p e c i a l  thanks to my office-mate Gary Brooke for his suggestions many f r u i t f u l discussions we  and for the  had.  I would also l i k e to thank Jack Stuber and Derek Daines for the many patient hours they spent m i l l i n g the fin-corrugated surfaces, Brian N u t t a l l for his technical assistance, Herb Black for taking the photos and Miss T i l l y Martens for typing the manuscript. F i n a l l y , I would l i k e 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 l a r g e hangars and terminal b u i l d i n g s . i s a problem confronting many of today's . crowded a i r p o r t i n s t a l l a t i o n s .  Although some problems can be eliminated  by placing the offending structures at n o n - c r i t i c a l angles to the runways, t h i s becomes very d i f f i c u l t i n a multi-runway system.  I t i s sug-  gested that the interference could be eliminated by placing a properly designed p e r i o d i c surface on the r e f l e c t i n g structure so that a l l of the incident energy i s scattered back i n the d i r e c t i o n of the ILS transmitter. In t h i s way, no i n t e r f e r i n g r e f l e c t i o n s would be received along the runway. It i s known from the theory of d i f f r a c t i o n gratings that a maximum amount of energy i s transferred from the specularly r e f l e c t e d mode to one or more backscattered modes when the period of the surface, 'a', s a t i s f i e s 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 i n figure 1.1.  However, the range of periods over which  only one backscattered mode i s generated depends e n t i r e l y on the surface. INCIDENT  2a sin9 = m> i  Figure 1.1  m=0,l,2, ...  /  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 p a r a l l e l conducting plates.  They found that a single  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 s generated for some angles of incidence when the plates have a period i n the range \/2<a<\.  This surface i s not p a r t i c u l a r l y useful i n  i t s e l f , but similar structures, consisting of an i n f i n i t e set of p a r a l l e l plates terminated by a perfectly-conducting plane, are of i n t e r e s t .  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 i n t e g r a l transform .technique similar to that used by C o l l i n [6, P.430] for s e m i - i n f i n i t e plates.  Tseng et a l [8] have shown that, for TE p o l a r i z a t i o n of the  incident wave and the proper choice of corrugation depth, there i s complete concellation of specular r e f l e c t i o n at an angle of incidence 9^=sin~l(X/2a).  Later DeSanto [12,13], using a modified calculus of  residues technique, confirmed  these r e s u l t s and described similar r e s u l t s  for TM p l a r i z a t i o n . 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 i n the range 0<a<X/2 has been analysed rigorously by Hessel and Hochstadt [9] using the above-mentioned scattering matrix and i n t e g r a l transform technique.  While no numerical r e s u l t s were given, this structure may  also be expected, with the proper choice of corrugation depths, to transfer 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  p r i a t e i n t e g r a l equations.  They found, for both TE and TM p o l a r i z a t i o n ,  the same scattering properties as observed by Tseng et a l [8] for f i n corrugated  surfaces.  However, for TM p o l a r i z a t i o n , the reduction of  specular r e f l e c t i o n 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 D e l e u i l [14] and Wirgin  [15,16] for both p o l a r i z a t i o n s .  Numerical r e s u l t s were  obtained by truncating and then solving a set of simultaneous l i n e a r equations for the spectral order amplitudes.  Numerical r e s u l t s at 8^=30°  exhibiting complete cancellation of the specularly r e f l e c t e d mode were confirmed by experiment.  This structure has also been investigated by  Hessel and Schmoys.[17], who  were interested i n i t s a p p l i c a t i o n as a  frequency s e n s i t i v e mirror i n a l a s e r cavity. A s i m i l a r structure to the above, the triangular groove or echelette grating, has been used extensively i n spectroscopy.  However,  t h i s structure i s very d i f f i c u l t to analyse rigorously and only a limited number of accurate numerical r e s u l t s have been obtained  [18].  From p r a c t i c a l considerations, i t i s apparent that the f i n corrugated  periodic surface i s the most promising i n the ILS a p p l i c a t i o n  as i t i s 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 concerned with the i d e a l i z e d 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 t h i n , f i n s under plane wave i l l u m i n a t i o n .  No experimental work has been carried out on  more r e a l i z a b l e surfaces.  The purpose of this thesis i s to provide such  an experimental investigation using numerical r e s u l t s , from the analysis of idealized structures, as a guide i n designing the surfaces. In Chapter 2, a rigorous analysis i s presented for the problem of TM polarized plane wave incidence on an i n f i n i t e fin-corrugated surface composed of perfectly-conducting, i n f i n i t e l y thin f i n s .  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 p o l a r i z a t i o n and Hessel and Hochstadt corrugated surface.  [9] for the modulated f i n -  TM p o l a r i z a t i o n was chosen because this i s the  p o l a r i z a t i o n used i n ILS i n s t a l l a t i o n s .  An investigation of attenuation  i n the p a r a l l e l f i n region concludes the chapter. In Chapter 3, the numerical results f o r some s p e c i f i c cases are presented and compared with those obtained by other workers.  Some  interesting r e l a t i o n s between the f i n height and f i n spacing for optimum cancellation of specular r e f l e c t i o n are included.  The chapter i s con-  cluded with a discussion of the a f f e c t s of attenuation. In Chapter 4, the experimental results f o r four s p e c i f i c 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 f i n s . Conclusions from the investigation are presented i n Chapter 5.  2. 2.1  Theoretical  F o r m u l a t i o n of the O v e r a l l The  fin-corrugated  schematically  i n f i g u r e 2.1.  fectly-conducting ducting plane.  parallel  The  Problem  s t r u c t u r e which s h a l l be a n a l y s e d i s shown I t c o n s i s t s o f an i n f i n i t e s e t of p e r -  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 -  f i n s , which are of h e i g h t "d" and p e r i o d i c s p a c i n g  "a", s h a l l be assumed i n f i n t e l y t h i n . - follows  Analysis  t h a t used by C o l l i n  The n o t a t i o n  and  approach  here  [6, P.430] f o r a s i m p l e r s t r u c t u r e .  C o n s i d e r a y - i n v a r i a n t , TM p o l a r i z e d , TEM plane wave w i t h components H ( x , z ) , E ( x , z ) v  x  and E ( x , z ) i n c i d e n t a t an angle 0^ from z  normal t o the s t r u c t u r e , as shown i n f i g u r e 2.1.  the  From Maxwell's  equations  E ( ,z) = x  x  j|a- 3-<Kx.z) .. o  (2.1)  Z  E ( x , z ) = -i|a-3<K*>z) k dX . o  (2.2)  <j>(x,z) = Hy(x,z)  (2.3)  z  where  and Z  Q  and k  D  of f r e e space.  are the impedance and p r o p a g a t i o n c o n s t a n t s , r e s p e c t i v e l y , The magnetic  f i e l d component i s  •iCx.z) = a e " Q  where  h = k  r  sine^  Q  o = J o k  c  o  s  jhx  " o F  Z  z<o  (2.4)  (2.5)  6  i  (2.6)  V  Figure 2.1  Fin-Corrugated  Structure with Incident TEM Plane Wave.  and  the time dependence e  tinuity  J U , u  has been o m i t t e d .  a t 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  result.  Because o f the d i s c o n and t r a n s m i t t e d wave w i l l  The r e f l e c t e d wave w i l l be composed o f an i n f i n i t e  sum o f  y - i n v a r i a n t , TM p o l a r i z e d TEM modes whereas the t r a n s m i t t e d wave w i l l c o n t a i n one TEM mode p l u s an i n f i n i t e TM modes.  sum o f s i m i l a r , b u t h i g h e r  order,  The t r a n s m i t t e d wave, however, w i l l be r e f l e c t e d by the  t e r m i n a t i n g 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 f u r t h e r r e f l e c t i o n and t r a n s m i s s i o n .  Thus, the t o t a l  reflected  field  above'  the i n t e r f a c e w i l l be governed by m u l t i p l e r e f l e c t i o n s w i t h i n the f i n region. As mentioned i n Chapter 1, a procedure f o r a n a l y s i n g such a problem has been d e s c r i b e d by Tseng [7] and used s u c c e s s f u l l y by Tseng, H e s s e l and O l i n e r [8] and H e s s e l  and H o c h s t a d t  [9].  It consists  essent-  i a l l y o f r e p r e s e n t i n g 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 s c a t t e r i n g matrix,  S, which r e l a t e s , a t the d i s c o n t i n u i t y , the amplitudes  o f the s c a t t e r e d modes t o those  o f the i n c i d e n t modes.  T h i s allows the  o v e r a l l problem t o be broken up i n t o a f i n i t e number o f component p r o b lems, the number depending on the number o f p r o p a g a t i n g f i n r e g i o n , which can be analysed  one a t a time.  modes i n the  The o n l y  assumption  t h a t must be made here i s that the f i n h e i g h t , d, i s l a r g e enough to prevent  r e f l e c t i o n o f evanescent modes.-at-the t e r m i n a t i n g p l a n e . •-  T h i s procedure w i l l 2.1.1  form the b a s i s o f the t h e o r e t i c a l a n a l y s i s t o f o l l o w .  Boundary and Edge  Conditions  Because o f the p e r i o d i c i t y o f the f i n - c o r r u g a t e d s t r u c t u r e , i t i s necessary  to c o n s i d e r o n l y a s i n g l e p e r i o d as d e p i c t e d i n f i g u r e 2.2.  In the r e g i o n 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 F l o q u e t ' s  Theorem  j  m  and equations  -jhma „., 3e(x,z)  e  3d) (x, z) I 3x 'x=ma  cf)(ma,z) = e  [6, P.368J  OX  a  m= t o , 1 , 2 , . . .  x  J  <$>(o, )  z<o  (2.7)  x=o  m=*0,l,2,...  z  (.2.1) to ( 2 . 3 ) ,  z<o.  (2.8)  A l s o , i n the r e g i o n o < z < d, because the t a n g e n t i a l component o f t h e electric  f i e l d i s z e r o a t the s u r f a c e o f a p e r f e c t - c o n d u c t o r ,  9<Hx,z)  m= -0,1,2, ...  o < z < d  (2.9)  x=ma  3<Kx,z) 3z  These e q u a t i o n s  +  = 0  3x  0  z=d  (2.10)  allx  make up the boundary  c o n d i t i o n s o f the o v e r a l l problem.  e  3<> t z<0  |  0<z<d  6  3x  |  J  < > f  - j h a 3ij>  3x  34 3x"  -jha  d±  3x  it=  3x  o  3x  *£- o 3z  '  •  •  z  Figure  2.2  Single Period with  Boundary  Conditions.  Notice that equation (2.9) extended to -°° < z < d.  allows the region of z in.equation  (2.7)  to be  However, because the tangential component of  the magnetic f i e l d i s discontinuous  across  the conducting f i n by  amount equal to the current on the f i n , the region of equation  an (2.8)  cannot likewise be extended. The edge condition, or the behaviour of the f i e l d at the edge of the p a r a l l e l f i n s , i s also of importance. Collin  [6,P. 18]  that  Kma.z) asymptotically,  1/2  ^ z  as z -*- o  (J)(x,z) -*• o  2.1.2  +  m = -0,1,2,...  (2.11)  and hence the magnetic f i e l d component i s f i n i t e i n the  neighbourhood of the f i n edge.  must also be met  I t can be shown as i n  as  z  In addition, the radiation condition,  -°°  (2.12)  i n the s o l u t i o n of the o v e r a l l problem.  General Solutions The expressions for the magnetic f i e l d s above and below the  f i n - a i r interface of figure 2.1 must s a t i s f y the reduced Helmholtz equation (V  2  + k )<f>(x,z) = o  (2.13)  2  0  subject to the conditions of section 2.1.1. sion, (2.1), and the expression  The incident f i e l d expres-  for the t o t a l r e f l e c t e d f i e l d i n the  region z < o, oo <j> (x,z) = ^  b  n  e~  j ( h +  2n7r ~a~  ) 3 d  " n r  z  z<o  (2.14)  satisfy  the boundary c o n d i t i o n s (2.7) and ( 2 . 8 ) .  to h o l d , i t i s n e c e s s a r y  T  = (h+ ^ )  2 n  However, f o r (2.13)  that  -  2  S i m i l a r i l y , the e x p r e s s i o n  k  n= ±0,1,2,...  2 Q  f o r the t o t a l  field  (2.15)  i n the r e g i o n o < z < d,  00  <)> (x,z) = ]>~j n=o  cos(^)e" n Y  t  satisfies  2  = (—) " 2  a.  n Notice that i f Y  2 < n  0  k  n= +0,1,2,...  2  the c o r r e s p o n d i n g  (2.17)  u  f o r some n, Y  n  p o s i t i v e r o o t i s chosen to s a t i s f y  n  (2.16)  ( 2 . 7 ) , (2.8) and (2.13) i f and o n l y i f  Y  Y  o<z<d,  z  i  s  imaginary  ( i f the  the r a d i a t i o n c o n d i t i o n , (2.12)) and  mode i s p r o p a g a t i n g .  However, i f Y ^ 0 2  n  i s r e a l and p o s i t i v e and the c o r r e s p o n d i n g  s i m i l a r s e t o f r u l e s apply t o e q u a t i o n propagating  and p o s i t i v e  f o r some n ,  mode i s e v a n e s c e n t .  (2.15).  A  Thus, the number o f  modes i n the r e g i o n z<o i s determined by the v a l u e s o f a and  h, whereas the number i n the r e g i o n o < z < d depends o n l y on a. As s t a t e d i n Chapter 1, Tseng, H e s s e l and O l i n e r [8] a n a l y s e d the problem o f a TE p o l a r i z e d p l a n e wave i n c i d e n t on a f i n - c o r r u g a t e d structure.  They found, f o r t h i s p o l a r i z a t i o n , the power i n the s p e c u l a r  l y r e f l e c t e d mode can be completely  t r a n s f e r r e d t o a b a c k s c a t t e r e d mode  t r a v e l l i n g i n a d i r e c t i o n o p p o s i t e t o the i n c i d e n t wave i f two p r o p a g a t i n g modes a r e i n the f i n r e g i o n . have confirmed  these  TM p o l a r i z a t i o n ^ corrugated  More r e c e n t works by DeSanto  [12,13]  r e s u l t s and have shown i d e n t i c a l r e s u l t s f o r  A l s o , H e s s e l and Hochstadt  [9] i n v e s t i g a t e d a f i n -  s t r u c t u r e w i t h a modulated c o r r u g a t i o n depth.  ,  They d i s c o v e r e  11  f o r TM p o l a r i z a t i o n and (as i n f i g u r e 2.1),  the g e n e r a l  a backscattered  case of a u n i f o r m  corrugation  mode i s not e x c i t e d i f o n l y  propagating  mode i s i n the  f i n region.  the p r e s e n t  s t u d y , t h a t the n = o and n = 1 modes s h a l l be  propagate I n the f i n r e g i o n . A/2  to the range  I t was  modes i n the a i r r e g i o n to two;  s p e c u l a r l y r e f l e c t e d mode and  the n = -1 b a c k s c a t t e r e d  be n o t e d , however, t h a t the n = -1 mode may  2.2  of  Representation  Hochstadt  A's,  I t should  a Scattering Matrix  approach here f o l l o w s that used by H e s s e l  be  considered  four propagating  as a modal waveguide  and  field  C's  modes under c o n s i d e r a t i o n .  Assume t h a t  at a time as shown i n f i g u r e s 2.3(b) and  o f f i g u r e 2.3(c) are the i n c i d e n t , r e f l e c t e d and  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 , c o r r e s p o n d i n g  these problems.  may  supporting  r e s u l t i n g f o u r problems are termed the component problems and B's  and  as a f o u r p o r t s c a t t e r i n g j u n c t i o n as shown i n f i g u r e  Each p o r t may  of the  mitted  The  matrix  equation  b = S . • a  or  mode.  d i s c o n t i n u i t y o f the f i n - c o r r u g a t e d s t r u c t u r e  each p o r t i s e x c i t e d one The  the n = o  be evanescent f o r some'  of the F i n - A i r D i s c o n t i n u i t y by  fin-air  represented  one  and  [9]. The  2.3(a).  f i n spacing  incidence.  The n o t a t i o n and  be  the  for  assumed to  < a < A, where A i s the f r e e space wavelength,  the number of p r o p a g a t i n g  angles  one'  therefore decided,  This assumption r e s t r i c t s  depth  of the j u n c t i o n i s o f the  2.3(c). the transto  form  (2.18)  z<0 s  z=0 z>0  Y  0  Y  l  (a)  1K  a  1  k •  K I  k  a2  1  • 1  k  }4  3l PROBLEM #1  |  8  PROBLEM #2  PROBLEM #3  k S  h  |  4  PROBLEM #4  (b)  1  ft 1 J  s |  ft s  |  1=1 PROBLEM #1  s  1  W  1  • ft  4  ft PROBLEM #2  PROBLEM #3  1  3  fc PROBLEM #4  (c)  F i g u r e 2.3  Four P o r t S c a t t e r i n g J u n c t i o n R e p r e s e n t a t i o n , (a) O v 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 i s a scattering matrix with elements consisting of the appropriate amplitude r e f l e c t i o n and t r a n s mission c o e f f i c i e n t s . Since the n = o mode i s the only mode incident on the s t r u c t u r e from z<0, (2.20)  In addition, because of the boundary condition (2.10) ,  a  3  - e "  a  4  -  2  ^  (2.21)  (2.22)  These conditions, together with the matrix equation (2.19), constitute a system of seven equations which may be solved uniquely i n terms of the amplitude of the incident f i e l d , a^. The expressions f o r the specular  amplitude  r e f l e c t i o n c o e f f i c i e n t , — • , and the backscatter a  l '  amplitude  bo r e f l e c t i o n c o e f f i c i e n t , — ^ , which result are l a  b  —  i  o  B  =  1  *>2  o " V  c  +  oo  A  e  = B ,+ C_ A  1  A  Q  B  "°  - 2 y d  x  e  H  o  — A  A  Y l  A  0  d  oX  A  (2.25)  6  0  B  o  B  o  ~ V  e  " V 2  4  i  f  i  ~ V  o  2  + — . — e  0  A  B e  B  2  - — A  -l  -2  1  e  1  J  (2.24)  Y l  e  1  (2.23)  V A  Q  Y l  " V  -1  B C -2 d + —.—— e  -2 d e  2  2  e  Q  B  B -  A  A  - 1 -  C  +  ^0  1  A.  +  A q  2  e  A  x  1  " V  V  2  1 A  A  0  -2 d  2  Y l  e  1  0  A  oX  e  (2.26)  The corresponding power r e f l e c t i o n c o e f f i c i e n t s are  b, 2 (2.27)  (2.28) Pi  where  . r  D  '  a i  '  (2.29)  due to the conservation of power i n 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 i n figure 2.4.  Problem #2 has been  omitted because, as equation (2.20) indicates, i t i s not needed i n the solution of the overall problem.  The boundary conditions f o r the re-  maining problems are the same as equations value d i s now replaced by +°°.  (2.7) to (2.9) except that the  The general solutions to these  component  problems, as l i s t e d below, s a t i s f y these boundary conditions and Helmholtz's equation, (2.13), subject to equations (2.15) and (2.17). Problem # 1 z<o  z>o  0(x,z) = A, e ^  <f>(x,z) =  h x  > n=o  - o r  V B n= -°°  z +  cos(  n  e"^  ^ T ^ n  h +  2  (2.30)  ) e 'n (2.31)  a  Problem # 3  z<o  :>o  . e x = ^y Kx,z) *(x,z) = >  „ L n=^-~  <Kx,z) = AQ e ° Y  Z  - j ( h + — )x+r z a n n  e  J V  + ^jT B N cos(^) iPb °  (2.32)  e n Y  z  (2.33)  Problem # 4  z<o  <j>(x,z) =  . " J C h + ^ x + r n Z  C  ^  C —OO  n  e  (2.34)  xu  INCIDENT TEM PLANE WAVE  y  ^Ni*  1  **— a  ^o- X  * (a)  PROBLEM #1 1  i  i  INCIDENT TEM MODE IN EACH REGION WITH PHASE SHIFT e ~  j h a  BETWEEN REGIONS  x  -jha  cj>e  4  4 PROBLEM #3 H  (b)  O  G  H  H y  INCIDENT TL^ MODE IN EACH REGION WITHPHASE SHIFT e ~  jha <f>e  PROBLEM #4  6 H  j h a  BETWEEN REGIONS  -jha  i H  4>e  A  I  6 H  Figure 2.4 -Component Problems, (a) Problem #1, TEM Mode Incident i n Region z<0. (b) Problem #3, TEM Mode Incident i n Region z>0. (c) Problem #4, TM Mode Incident i n Region z>0.  (c)  z>o  cb(x,z) = Ai_ c o s ( - ^ ) e l Y  Z  +  ^> ' B n=o t  n  c o s ( - — ) e 'n 3.  (2.35.)  In the f o l l o w i n g s e c t i o n s , a formal s o l u t i o n f o r the amplitude c o e f f i c i e n t s o f Problem # 3 w i l l be p r e s e n t e d u s i n g an i n t e g r a l t r a n s form c o n s t r u c t i o n technique d e s c r i b e d by C o l l i n Tseng, H e s s e l and O l i n e r come apparent  [8] and H e s s e l and Hochstadt  great d e a l o f unnecessary The Transformed  suppress  t o s u i t the o t h e r s , thus e l i m i n a t i n g a  analysis.  Problem  i n i t i a l s t e p o f the i n t e g r a l t r a n s f o r m technique i s to  the v a r i a b l e z by t a k i n g the b i l a t e r a l L a p l a c e t r a n s f o r m o f  the f i e l d s o l u t i o n satisfies  I t w i l l be-  The i n t e g r a l t r a n s f o r m e x p r e s s i o n o b t a i n e d f o r  Problem # 3 can be transformed  The  [9].  i n s e c t i o n 2.6 that n o t a l l o f the component problems need  to be s o l v e d i n d e t a i l .  2.4  16, P.418] and used by  <}>(x,z).  A s o l u t i o n f o r the transformed problem which  the transformed boundary c o n d i t i o n s and the t r a n s f o r m o f Helm-  h o l t z ' s e q u a t i o n i s then c o n s t r u c t e d .  I n the f i n a l s t e p , the s o l u t i o n  f o r <j>(x,z) i s found by i n v e r t i n g ; the t r a n s f o r m and e v a l u a t i n g the i n v e r s i o n i n t e g r a l i n terms o f i t s r e s i d u e s . ficients field  The d e s i r e d amplitude  coef-  can then be o b t a i n e d by matching t h i s s o l u t i o n w i t h the o r i g i n a l  expansion. L e t the b i l a t e r a l L a p l a c e t r a n s f o r m o f the t o t a l f i e l d <j>(x,z)  be  [6, P.433] oo  (2.36) —CO  The i n v e r s e t r a n s f o r m i s then  -~-r  \ e  S Z  Kx,s)ds  (2.37)  where T i s a s p e c i a l l y chosen contour i n the complex s-plane running p a r a l l e l to the imaginary s-axis through a region i n which cf>(x,s) i s analytic.  I t i s shown i n Appendix A that the region of a n a l y t i c i t y f o r  Problem J 3 i s w e l l defined.  For convenience, a s p e c i a l function g(s)  s h a l l be defined as follows:  oo J e" <),(o,z)dz  g(s)=Ko,s) =  Sz  (2.38)  —CO  2.4.1  Solution of the Transformed  Problem  The Laplace transform of the reduced Helmholtz's equation i s [6, P.434] a Kx,s) 2  3x  u  where  2  +  u  2^  ( x > s ) = 0  ( 2 > 3 9 )  2  = k  2 D  + s  (2.40)  2  The transformed v a r i a b l e , <fi(x,s), of t h i s d i f f e r e n t i a l equation must s a t i s f y the following transformed boundary conditions obtained from equations (2.7), (2.8) and (2.9): 3<Kx,s) I ai  -  3.x  -jhma 3ji(x,s) I  x=ma  Kma,s)= e ~  3$(x,s)j  _  j h m a  °*-  ^(o,s)  = 0 x=ma  m=-0,l,2,... <=°>z>--°  (2.41)  x=o  m=-0,1,2,.. .  m=±0,l,2,...  z>o  z<o  (2.42)  C2.43)  A general solution for this transformed problem i s 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 ^ c o s (ux) J <Kx,s) = -rr[cos(ua)-e ] a  (2.48)  J n a  and the inverse transform, equation (2.37), i s [6, P.434]  . , , 1 <p(x,z) = -z—r  ( e g ( s ) [cos (u(a-x) )-e ^ c o s (ux) ]ds ^TT S Z  s  2 U J  where u  2  = k  2 Q  J  T  3  [cos(ua)-e  J h a  ]  (2.49)  2 + 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 s i n ( u a ) d s Icos(ua)-e  r  j  h  a  =  w i t h m=l,  Z > Q  ]  .  S i m i l a r i l y , s u b s t i t u t i n g equation (2.8)  Q  (2.49)  i n t o the boundary  (2.50)  condition  yields sz e  g(s)[cos(ha)-cos(ua)]ds  T  [cos(ua)-e"  Thus, i t i s apparent regular  g(s)  constructed  of  i s a meromorphic f u n c t i o n  (2.50) and  (2.51) are-  and  would be  [6, P.435]  P(s)  ~~ ,  \/  \  \  {cos(ua)-e~  1—r  0  0  respectively.  to  and  (2.50)  that a s u i t a b l e c o n s t r u c t i o n  )  ( a + y  j h a  -  nTr  0  1  a  can  and for  g(s)  }  -sa  n  (s-y )(s+y )(s-r ) | | - ^ - e f  z<o,  (contains only poles)  from r a t i o s o f e n t i r e f u n c t i o n s s u b j e c t  I t i s shown i n Appendix B  =  (2.51)  s i n g u l a r i t i e s ) f o r z>o  (2.51).  8 ( S )  z < o  ]  the i n t e g r a n d s  ( t h a t i s , c o n t a i n no  Therefore, be  that  j h a  _  i—r  ( s  _  r  _  r  -n  ^ 2 a  1  sa  }  n  | | 1  ;  ) ( s  6  r  }  (2.52)  where P ( s )  i s an e n t i r e f u n c t i o n y e t  to be  I t i s shown i n Appendix C t h a t the  terms (—) a  and  determined.  the e x p o n e n t i a l  D e t e r m i n a t i o n of the F i n a l I n t e g r a l In o r d e r  sary  and  2 *™ (——) i n the denominator o f (2.52) are used to a  s u r e the u n i f o r m convergence o f the i n f i n i t e p r o d u c t s . 2.4.2  factors  en-  [6, P.435]  Representation  to f i n d a s u i t a b l e f u n c t i o n P ( s ) , i t i s f i r s t neces-  to determine the  asymptotic b e h a v i o u r o f g ( s ) .  This  i s done i n 3/ 2 Appendix D where i t i s shown t h a t g(s) i s asymptotic to s as |s|-> « —  If P(s) i s now chosen so that g(s) has algebraic growth at -3/2 i n f i n i t y and i s asymptotic to s  as j s}  ^°» i t i s assured that £(x,z)  has the correct behaviour at the f i n edges, equation (2.11)..  It is  shown i n Appendix E that P(s) i s of the form -sa ln2 P(s) = C e  (2.53)  where C i s a constant. Substituting (2.53) into equation (2.52) and the r e s u l t into equation (2.49) yields the f i n a l i n t e g r a l representation of the t o t a l field,  sz e  >(x,z)  -sa —InZ e {cos(u(a-x))-e  2nj  CO  -Jco  (S+Y) n  (8-Y )(8+Y )(8-r ) 0  0  e  . HI) a  0  nn T  cds(ux)} dS "  r  (s-r ) ( s - r  (  )  — r —e n  -n  a (2.54)  9  o  where u  = k  Q  o  + s . The pole p l o t f o r t h i s problem i s shown i n figure A  3*2  2.5.  P  Y  0 INCIDENT MODE  $ o r  xxx  n = +2,3,.,  z>0  Figure 2.5  *)-^i  n = +1,±2,3,.  z<0  Pole Plot of the Integral Representation, Equation (2.54), Problem #3.  ^  mr  22  2.5  Solution of the F i e l d Amplitude Coefficients by the Method o f Residues The f i n a l step i n the analysis of Problem // 3 i s the evalua-  tion of equation (2.54) using the method of residues.  Once this i s  done, the r e s u l t can be matched with the o r i g i n a l f i e l d expansion f o r <Hx,z) to obtain the desired f i e l d amplitude c o e f f i c i e n t s . For z>o, the integration contour of figure 2.5 i s closed i n the LHP.  Therefore <Kx,z)  (2.55)  C ^> ' Residues  =  If the residues at S=+Y , S=-Y , S=~YI and s=-Y are denoted by Rj_, R > Q  0  N  2  R 3 and R^ , respectively, where m=+2,3,..., then m  •ln2 (l-e^  R, =  h a  )  -Y a Q  % % -  r  o  (  )  W  (1-e  R.  Y  VW  Y  (y  1 +  r ) 0  o  -jha,. )e J  (2.56)  -Y z 0 n  a  (^HI)  a  l  )  (-2HI) 2  e  (HI)  (2.57)  2  a  (In2-1)  H  R.  F  a  In2  2  V  <n-'0>"-n^0 j f  n^T  "(HI) a  e  /-1  >~J  (1+e  ( -Y ) 2  YI  2  0  J  n a  \/\  TTn~V (Y  2  z ^  Tr  71  ) (—) c o s ( — ) e a a  (HI) a  Y-j*  nir  ( r ^ X r . ^ ) (^HI)  a  2  nu  i(2.58)  (In2  )  •  e  -y z  .,  (.cos (rati)-e  j {—)  J  ;  R  4,m o  TTi  (Y - Y )  ,  ( Y + r ) ( Y n ~Y ) m 0 '0 ' m l n  r  1  (  a  —  a  ; <  1  1  }  , .-y a : Fl  OD  „ ( r + y ) ( r  e  nTr. a  1  ) e  a  Y  CO  cos(  + y •)  ~^ z ,2nir. 2 a  {  —  1 1  e  )  (2.59) o  where  9  nnT ^  = (—) - k  Q  . Therefore, from equation (2.55), oo  q.(x,z) = C(R +R +R + ^ R m=2 1  2  3  4 ) m  ) -  (2.60)  However, the o r i g i n a l f i e l d expansion f o r <Kx,z) i n the region z>o, equation (2.33), may be rewritten i n the form (j)(x,z)=A e o Y  + B e" o  z  Y  0  + -& c o s ( ^ ) e ~ l  z  Y  0  z  ±  CO  +  y  B  n cos(—)e  Y  n  ^  n=2 Matching terms i n t h i s expansion with equation (2.60) y i e l d s the following amplitude c o e f f i c i e n t s : A_ = C R i e o o - Y  B  = CR e o Y  Q  l  m  = 4,m ,m7rx, cos ( ) a  Y  B  Z  3  CR  B  (2.63)  z  2  - CR e l . ,TTX^ cos (—) a  n  (2.62)  z  e  (2.64)  m=+2,3,...  (2.65)  For z<o, the i n t e g r a t i o n contour of figure 2.5 i s closed i n  the RHP and (2.66)  <Hx,z) = -C ^> ^ Residues If  the residues at s = r ,  and s = r  s=r_i_  Q  are denoted by R5, Rg and R y ^ ,  m  respectively, where m=+l,-2,3,..., then  •In2  -jhx+r z Q  sin(ha) e  R. = (Y +r )  <o V> r 2  2  n  ^ . ( r _ - r ) ( r „-r ) n " ' 0 - n '0' ne ,2njT^ 2 a  0  " - l r  /vl  n  e  n  a  (2.67)  2TT  (ln2+l) R,  •. -j(h- — ) x + r _ z sin(ha) (—) e 3. -P - a _ i (Y +r .) -=^- °° ( r - r ) (r - r ') — — 'n -1 niT 1 — r n -1 -n -1 n7r e s ?; e ^2mr^ 2 2 i r  2  1  r  < o- -i r  r  ) ( r  f -i r  -no  ) ( r  2 )  "  a  a  a  C2.68)  "^!(ln2+ k j e  L  m  sm(ha) (  R. 7 ,m 2  2  2  S  2  )  -r a —  (r - r ) ( r - r )  r a• — !  2  n  where r =(h+ - - ^-) -k .  3.  (Y +r )  (r -r )(r -r J ( y - r ) 0 m m -m 0 m n  - j (bf ^ f ) x + r z e  n(BE,m  2 nfr T — r n ,2mr^ m -n 1 a ^ I n r\  m  nrr  e  (2.69)  Therefore, from equation (2.66)  2  K x , z ) = "C(R +R + ]T 5  6  7,m)  R  m= -°° m#),-l  (2.70)  1 t  However, the o r i g i n a l f i e l d expansion  for cb(x,z) i n the region z<o,  equation (2.32), may be rewritten i n the form  2TT  <Kx,z) = C e "  o  j h x + r  0  z  + C_  i e  -  j ( h  - ^  )  • /-u_i_  E 0 0  x  +  F  - l  +  z  2njr.  r,- - J (h+ — ) x + r  h  G  e  a  Z  n  (2.71)  n= -°°  n#),-l Matching  terms i n this expansion with equation (2.70) y i e l d s the remain-  der of the amplitude  C  Q  coefficeints.  = -CR e  j h x  5  C-1 = - C R e  " o  j ( h  6  r  -^  • />,+  Cm = -CR e  3Kn  7iXB  2.6  (2.72)  z  ) x  - -l r  (2.73)  Z  2mTT. _  a  )  x  1  m  z m  = +1,±2,3,...  (2.74)  Solution of the Remaining Component Problems As stated i n section 2.3, i t i s not necessary to analyse a l l  of the component problems i n d e t a i l .  The i n t e g r a l transform expression  and pole plot obtained for Problem # 3 can be e a s i l y transformed  to s u i t  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, i n Problem # 4, the incident mode i n the region z>o i s a TMi_ mode.  Therefore, the incident pole would be located at  S=+YI on the complex s-plane, as depicted i n the pole plot of figure 2.6 l  and the i n t e g r a l representation corresponding to this problem would be  26  *>  l  Y  V  INCIDENT MODE  U  _ i  r  XXX n  n. n=+2,3,...  W~ l y  n  =  ' '>  +1  ±2  3  *T 0 Y  z<0  z>0  F i g u r e 2.6  P o l e P l o t o f the I n t e g r a l R e p r e s e n t a t i o n , ^ E q u a t i o n Problem #4.  (2.75),  & o r  r  - l  XXX  >> r n = +2,3,...  z>0  n 0 n = +1,±2,3,...  z<0 INCIDENT MODE  F i g u r e 2.7  P o l e P l o t o f the I n t e g r a l R e p r e s e n t a t i o n , E q u a t i o n Problem #1.  (2.76),  as follows Problem # 4  ,,  .  <!>(x,z) -  „  ,  '-sa. ' . —;ln2 _.. v. ^ {cos(u(a-x))--?. ' cos (tix)} dS , , , -sa , ., \ /* „ v sa (s+Y ) ^ (s-ri ) ( s - i ) ) ( s ) (s-r ) I I e .I I e 1 1 1T> Tl  J  00  (s.  y  0  %  (  (2.75) However, i n Problem #1, the incident mode i n the region z<o i s a TEM mode.  In this case, the incident pole would be located at s = - r  0  on the  complex s-plane, as shown i n the pole plot of figure 2.7, and the i n t e gral representation would be as given i n equation  (2.76).  Problem # 1  (  sz .. — l n 2 • _., c " {c.os(u(a-x))-e cos(ux)} dS ) -D(s-r ) M * 1—r n niT 1 — r n -n n-rr ) (S Y ) (s-r ) T T e U . 2 ^ 2 1 ^ a' 1 ^ a e  (  (s  S  +  T  ( s  6  + r o  +  0  Q  ;  (2.76) The f i e l d amplitude  c o e f f i c i e n t s for Problem # 4 :  A, B , B  l 5  CQ, and C_i  (2.77)  A, B , B  l 5  C , and C_x  (2.78)  x  0  and f o r Problem #1:  Q  D  D  can now be computed using the method of residues and c o e f f i c i e n t matching technique described i n section 2.5. Once this i s complete, the  f i n a l component r e f l e c t i o n and transmission c o e f f i c i e n t s f o r a l l three problems may be determined. 2.7  Determination of the Component Reflection  and Transmission  Coefficients The f i n a l component r e f l e c t i o n and transmission c o e f f i c e i n t s of equation (2.19) are determined by taking the proper ratios of the f i e l d amplitude coefficients obtained i n sections 2.5 and 2.6.  The  results are tabulated i n tables I and I I . I t should be clear from the analysis  carried out i n Appendix C that the i n f i n i t e products contained  i n these c o e f f i c i e n t s are uniformly convergent. Substituting  equations (2.79) to (2.90) of tables I and II  into equations (2.25) and (2.26) and the results into equations (2.23) and (2.24) yields the o v e r a l l amplitude r e f l e c t i o n c o e f f i c i e n t s of the n=o reflected and n=-l backscattered modes.  Substituting  these c o e f f i -  cients into equations (2.27) and (2.28) yields the f i n a l power r e f l e c tion coefficeints of the overall problem. 2.7.1  Reflection and Transmission Coefficients at the Optimum Angle of Incidence The condition,  k^a sinG.= TT ° op  (2.91)  1  relates the angle of incidence giving the maximum reduction i n specular r e f l e c t i o n (the optimum angle of incidence, 6 i p ) to the p e r i o d i c i t y Q  of the corrugated surface (the f i n spacing, a).  Substituting  this value  of sin6^ into the expression f o r h, equation (2.5), gives  h, op  7T  a  (2.92)  -(2r )fm2 0  -e  (  -r )  Y  (Y -r )(r  Q  n  <W  (  V O r  ) ( r  n +  r )(r_ 0  n- O r  2  (  2  F  0  0  r  r  ( P  ) (Y  +r  }  ( -r )fm2 i-e-J )(2r )( -r ) _ h a  0  0  (  Y o  2  M  J £  0  -e  l 0 + r  (  V  r  ' ' n -l (Y  2  (Y -r )(r n  V  T T  )  0  n +  +r  r )(r_ Q  0  ) (T  n +  (  )  n- -l> T  (r  W  I  -n- -l> r  r ) 0  (V^O^W^  (2r ) ( -r ) (r r ) (r_ r ) ( -r ) (r r ) (r_ r ) sinChaJCrQ+^XyQ-y^Cr^^Cr^+y^ ' I ( Y ^ ) ( y ^ ) ^-^J d+e-  j h a  )  Q  2( )fln2 ( r - Y ) Y(J  )  T  0  h sin(ha)  3 6  (VVf  r )  -n- O  r  )  (  Y o  ) ( r  n +  < V 0> < V V ( V V V - l l - l Vl -l>  -^0 -l>^ +T  V  0  YQ  0  1+  (Y +Y ) ( 1 ^ )  Q  n  <W  (  0  VV  ( r  n  + Y  0  ) ( r  Q  1+  Q  Yn  jha  (1  - -J^) e  (Yo  0  0  _ )r Yi  (  0+Yl  0  )(r  n+  Q  n+  Q  (r_ -y ) n  -n  + Y  Q  0  )  (l+e- )(2y )(r -y )(r -y )(r. -y )  ( Y Q + Y ^  Q  i  1+Yl  0  1  )(r_  0  1+Yl  ) [J  (y^)(  F ^ )  ( r ^ )  ' v V ' W ' W V  Table I Amplitude Reflection and Transmission Coefficients, Sub-Optimum Case.  ( y ^ ) <r .- ) ( I ^ )  (y -r )-ln2 sinOha) (2y ) ( f ^ ) ^ 0  Q  0  h  . V -i>f (  2  ( 1  1112  r  U-e-  jha  )fin2 (i- -J ) h a  Y Q + T l  j h a  s l n ( h a  3 6  (  _e-  e  (l e^ +  h a  )  <V O  )  >  r  V <W  ( 2  V  Q  "  Yl  1  n- O r  ) ( r  -n-V  V < r V - r - r <W  0 + Y ( )  \  1  r  <*n -i> +r  ( r  n  r  n  n  - - i > < -n" -i> r  r  r  n  y  )  < -V < - -V  r  ( y ^ )(r ^)( r ^ ) ' V V ' W ^ + V  ( y ^ x r ^ c r ^ x r ^ )  2(y )fin2  ) ( r  ( y ^ r ^ ) (y.+r^) ( i - r ^ ) ( l y r ^ )  (r - )  )(r  Y()  _  (y^)(r-^)(r_ n  Y ] L  )  -e  . <V V ? * 1  -je  2  (l e-J ) h a  +  .  .je  ( Y  r -i f r  )  W -r-r  S ± n ( h a ) (  l n 2  (  Y()+  r ) 0  W  (  \ (V  r  O  ) ( r  n- O r  ) ( r  -n-V  sin(ha) ( 2 ) ( Y q + Y i ) (r Q - Y ] L ) ( r ^ ) • • || ) ( y ^ ) (y^r^) ( r y r ^ ) ( i - r ^ ) 2 '  (Y^Xr^Xr  Yl  (i e^ +  :  h a  (  V -i> r  ( r  -y )  n " - i > < -*- -i> r  r  r  Table I I Amplitude Reflection and Transmission Coefficients, Sub-Optimum Case.  CO  fo  -(2r )fin2 (Y -r )(r  -< 0 r  A  Q  Q  + r  -l f )  Q  l a 2  ( r  0  <V - 1 > F  Q  r ) (r_ r )  1+  Q  ) ( Y  (  1+  0- 0 r  ) ( r  (Y -r )<r n  Q  i  + r  0> T T <V 0 r  V - 1 > < r -l> r  r  0  ) ( r  n +  n  r )(r r ) " 0  + r  V  0  ) ( r  -n  -  0  + r  0>  (2.99)  ' ' < V - l > n " - l > ( -n" -l>  r  r  ( r  r  r  r  (2.100)  fl A  Q  (  2  V <V o > < i V ^ -l 0> T T r  " < W  B -r°-= 0  (  W  r  +  < i^l> r  r  (  r  + r  - l V  1 2  '  (  "  R Q ) (  V V  W  ( Y N  (  ^  +  )  ^ - ^ ^  R  Q  ( F  -n^l>  "(2.10D  Same as equation (2.81), Table I  (2.102)  (0 0)  <-  A  B A  i= 0  2  +j  Table I I I  103  >  Amplitude Reflection and Transmission Coefficients, Optimum.Case.  LO  A  <VVf  l n 2  (  *  V  l  )  W  W  (  W  (  .  r  i  T T(VV(rn-V(r-n-y  ^  2, ( r )(r -r )  <VV VV - -V  3  Yl+ 0  .V -i>f r  1  (  0  ( V )(Y +Y )(r -Y )(r -Y )  (r  n  a  l n 2  1  1  0  0  1  0  _  0  (Y+r_)(Y+r_)(r-r_)  (  2IR  0  1  1  1  1  1  1  Y  n  -  ^  0  )  (  r  n  -  Y  )  0  (  r  _  n  -  Y  Q  )  ( Y / . ^ V r . ^ r - r ^ )  1  (O+jO)  (0+jO)  < V W r i> T T < W (  (r Y  V  <W W r O> (  ( Y  1  - r _  (r  1  r  ) f m 2  (  W  Y  l  )  W  (  (  Y  o  (  r  n  -  Y  )  l  (  r  _  n  -  V 0 n- 0 -n- 0 r  +  Y  r  ) ( r  l  )  W  (  r  ^  r  ) ( r  )  ^  (  ! 2  '  Y  n  +  W  Y  )  l  )  Y  )  l  (  (  r  r  n  -  Y  n -  l  )  r  (  r  _  - l  -  n  )  Y  l  (  )  W  Table IV Amplitude Reflection and Transmission Coefficients, Optimum Case.  However, as h-> — , s e v e r a l f a c t o r s i n the r e f l e c t i o n and a coefficients  o f t a b l e s I and I I approach z e r o .  "*"^  s i n (ha)  m  h+-  l  i  Specifically,  =0 (2.93)  a  (i+ -  m  transmission  j h a  e  )  = 0  h -> a  l i m  (2.94)  .(Y -r ) 1  = o  0  h •> a  (2.95)  lim  , „ . „ (Yi-r_!) = 0 h + a  (2.96)  h -»• a Therefore,  coefficients  (2.97)  which c o n t a i n one  o r more o f these  factors i n  t h e i r denominator are s i n g u l a r at the optimum angle o f i n c i d e n c e and i t i s necessary  to e v a l u a t e  t h e i r l i m i t as h -> — . a  ed, v a l i d only at the angle I I I and 2.7.2  9i p  =  0  s  i  n  ^(X/2a),  The  coefficients  obtam-  are d i s p l a y e d i n t a b l e s  IV. Attenuation  of the n=l Mode i n the P a r a l l e l F i n Region  For applications-such as r  the ILS. problem i t i s n e c e s s a r y  to con-  s i d e r s i t u a t i o n s i n which the angles o f i n c i d e n c e are very c l o s e to grazing.  However, as equation  (2.91) i n d i c a t e s , t h i s r e q u i r e s the  use  of surfaces with f i n spacings very close to A/2, the c u t o f f f i n spacing for the n=l mode i n the f i n region.  Attenuation of a waveguide mode i s  large close to cutoff and i t may not always be p o s s i b l e to consider Yj_ as pure imaginary.  Consequently, i t was considered worthwhile to deter-  mine the attenuation c o e f f i c i e n t , a, of the n=l mode i n the f i n region. Using a w e l l known perturbation technique  16, P.183] i t can  be shown that the attenuation of the n=l mode per u n i t length i n the f i n region i s 2Rm  / nepers/ u n i t length  a *~ Z o a  where  ( _(|_)  2  1  1  )  /  2  i s the surface r e s i s t i v i t y o f the f i n s .  copper i s assumed, equation  I f a r e s i s t i v i t y for  (2.110) becomes  2.3965 x 10~ a l X d - ^ ) ] /. a  ~  (2.110)  1  where A and*a'are i n meters.  7  5  2  . nepers/ "^"'-eter  Therefore, the true propagation  (2.111)  coefficient  is Yi  1  = « + Y  (2.112)  X  where y^ i s given by equation  (2.17).  Equation  (2.111) i s known to pre-  d i c t 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. 3.1  Numerical Results  Introduction An examination of the f i n a l expressions f o r the o v e r a l l re-  f l e c t i o n c o e f f i c i e n t s w i l l reveal that they are functions of three parameters; 1) the f i n spacing, a (or optimum angle of incidence, ®i p)> 0  2) the f i n height, d and 3) the angle of incidence, 8^.  In most a p p l i c a -  tions, such as the ILS problem mentioned i n Chapter 1, the optimum angle of incidence would be known.  Therefore, i t was  decided that the  c o e f f i c i e n t s would be computed as functions of d and 0^ at fixed a. The major source of error i n the numerical computations the evaluation of the i n f i n i t e products.  was  The Krummer transformation  method described by Tseng et a l [8] and the remainder technique described by DeSanto [12,13] were not used.  approximation 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 r e f l e c t i o n c o e f f i c i e n t s and less than 0.05%  i n the r e l a t i v e  phases of the amplitude r e f l e c t i o n c o e f f i c i e n t s while increasing the cost by nearly four times. The results displayed i n t h i s chapter were computed assuming no attenuation of the modes i n the f i n region.  A discussion of the  effects of attenuation of the n=l mode i s given i n section 3.2  3.5.  Relative Power as a Function of Fin Height In figures 3.1(a) and (b), the r e l a t i v e power (power r e f l e c t i o n  c o e f f i c i e n t s ) of the specularly r e f l e c t e d (n=o) and backscattered (n=  -1)  0  0.1  0.2  0.4  0.3  FIN  0.5 HEIGHT  0.6 d/\  0.7  0.8  0.9  (a)  FIN  HEIGHT  d/\  (b) Figure 3.1  Relative Power of the n=0 and n=-l Modes vs. F i n Height (no attenuation) (a) a=0.578A 6. =59.99° 100% reduction at d=0.559A=0.968a op (b) a=0.506A 0 . =81.24 100% reduction at d=0.501A=0.989a 1  0  LO  modes are p l o t t e d w i t h r e s p e c t to f i n h e i g h t , d, f o r i n c i d e n t 60°  and  81.2°.  Because power i s conserved,  p l a c e between the two modes and i s u n i t y f o r any v a l u e of d. Z a k i and Neureuther 13]  the sum  a perfect transfer  of their r e f l e c t i o n  takes  coefficients  S i m i l a r r e s u l t s have been o b t a i n e d by  [10,11] f o r s i n u s o i d a l s u r f a c e s and by DeSanto  for fin-corrugated surfaces.  of 100%  angles  112,  Note, i n each p l o t t h e r e i s a p o i n t  power t r a n s f e r to the n=-l mode.  T h i s i s r e f e r r e d to by  above-mentioned authors as the "Brewster-angle  effect."  I t was  t h a t every f i n s p a c i n g has at l e a s t one such p o i n t f o r d<A  the found  and many  more at l a r g e r f i n h e i g h t s . F i g u r e 3.1(b) has  two  interesting features.  F i r s t , there i s  a r a p i d i n t e r c h a n g e of power w i t h f i n h e i g h t between the n=o modes at about d=0.55A.  and  T h i s phenomenon i s r e f e r r e d to as a "Wood  S-anomaly" [8,10-13].  Second, the power i n the n=o  0.1  T h i s l a t t e r f e a t u r e c o u l d prove extremely  f o r 0.15A<d<0.5A.  n=-l  mode i s l e s s  than useful,  s i n c e i t allows the d e s i g n of near optimum s u r f a c e s u s i n g f i n h e i g h t s much s m a l l e r than the f i n s p a c i n g s and w i t h o u t severe t o l e r a n c e s . In f i g u r e 3.2, and  the n=o  and n=-l power r e f l e c t i o n  the r e l a t i v e phases of the c o r r e s p o n d i n g amplitude  coefficients  reflection  coef-  f i c i e n t s , are p l o t t e d as f u n c t i o n s of f i n h e i g h t f o r a case i n which the f i n s p a c i n g i s v e r y l a r g e . those o f DeSanto [13,fig.5]. F i g u r e 3.2  These p l o t s were made to compare w i t h The two  are e s s e n t i a l l y  identical"'".  demonstrates the same p e r i o d i c i t i e s w i t h d and  the same c o r -  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  o f the r e s p e c t i v e r e l a t i v e phases as d e s c r i b e d by DeSanto [12,13]. The e x p r e s s i o n s used to generate 3.2 1  are not v a l i d near d=o  the p l o t s o f f i g u r e s 3.1  because they do not take i n t o account  DeSanto assumes time dependence e  l u t  .  the  and  Figure 3.2  Relative Power and Relative Phase of the n=0 and n=-l Modes vs. F i n Height (no attenuation). a=0.834A e 1  =36.84° 100% reduction at d=1.21X=1.45a oP  u o  contributions plane.  from the evanescent  modes r e f l e c t e d  However, f o r TM p o l a r i z a t i o n , i t i s known t h a t when d=o,  power r e f l e c t i o n c o e f f i c i e n t o f the n=o the r e s p e c t i v e amplitude 3.3  from the t e r m i n a t i n g the  mode i s u n i t y and the phase o f  r e f l e c t i o n c o e f f i c i e n t i s zero.  R e l a t i v e Power as a F u n c t i o n of I n c i d e n t Angle In f i g u r e s 3.3(a) and  ( b ) , the r e l a t i v e power o f the n=o  and  n=-l modes are p l o t t e d w i t h r e s p e c t t o the angle of i n c i d e n c e , 6^. The  f i n s p a c i n g s used are those of f i g u r e s 3.1(a) and  and  the f i n h e i g h t s correspond to the 100%  (b), respectively,  t r a n s f e r p o i n t s o f those  S i m i l a r r e s u l t s have been d e s c r i b e d by Tseng 17]  plots.  and Tseng e t a l  [8] f o r a f i n - c o r r u g a t e d s u r f a c e w i t h TE p o l a r i z a t i o n and by Neureuther The 100%  [10,11] f o r a s i n u s o i d a l s u r f a c e w i t h b o t h , p o l a r i z a t i o n s .  t r a n s f e r p o i n t s i n f i g u r e s 3.3(a) and  optimum angles o f i n c i d e n c e g i v e n by e q u a t i o n angles are known as Bragg-angles  (b) are l o c a t e d at the (2.91).  o f these anomalies  trans-  anomalies".  The  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 r e s p e c t i v e s u r f a c e w i t h s m a l l widths periods.  Because these  i n c r y s t a l s t r u c t u r e s , the 100%  f e r p o i n t s are sometimes r e f e r r e d to as "Bragg-angle widths  Z a k i and  c o r r e s p o n d i n g to s m a l l  This b e h a v i o u r i s emphasised i n f i g u r e s 3.4(a) and  v a r i a t i o n of the r e l a t i v e phase o f the n=o  and n=-l amplitude  (b).  The  coeffiT-  c i e n t s w i t h 8^ i s a l s o g i v e n i n the p l o t s . I t was r e g i o n may  s t a t e d i n s e c t i o n 2.1.2  that  the n=-l mode i n the a i r  be e i t h e r p r o p a g a t i n g o r evanescent,  depending  on the angle  of i n c i d e n c e .  The angles where these t r a n s f o r m a t i o n s o c c u r are  i n f i g u r e s 3.3  and 3.4.  The  c o r r e s p o n d i n g p o i n t s on the n=o  r e f e r r e d to as " R a y l e i g h anomalies"  [8,10-13].  clear  curves  are  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 .- • ~ (b) a=0.506A 8 . =81.24 d=0.501A 100% reduction •"-op n  90.0  30.0  40.0  SO.O  80.0  THETfl (DEGREES)  90.0  r  in era  1— T!T  40.0  1 A  50.0  THETA (DEGREES!  1  50.0  1  1  70.0  80.0  10.0  1 80.0  —,  ,  20.0  ,  40.0  ,  50.0  THETfl (DEGREES)  £0.0  70.0  SO.O  i-9 CCo  _|OT.  (a) Figure 3.4  !  30.0  n*o  •(b)  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 op op L  1  I—  1  42  3.4  Optimum F i n Height as a Function of F i n Spacing In figures 3.5(a) and (b), the f i n height which gives 100%  transfer of power to the n=-l mode i s plotted as a function of the f i n spacing (or optimum incidence angle). For angles of incidence i n the range 50°.'<6j_ f i n height i s about equal to the f i n spacing.  <90°', the optimum  This i s an important  feature since most ILS r e f l e c t i o n problems have incident angles'In this range.  I t w i l l become apparent i n Chapter 4 that these plots may.also  be used to predict the optimum performance  of surfaces with f i n s of  f i n i t e thickness. 3.5  Attenuation of the n=l Mode i n the F i n Region The perturbation solution f o r the attenuation of the n=l mode  i n the f i n region, as determined i n section 2.7.2, was introduced into the analysis and plots s i m i l a r to those of figures 3.1-3.5 were made. As expected, there were no s i g n i f i c a n t alterations of the results f o r a>0.53A.  However, small changes were noticed at near grazing angles and  these deserve some mention. In the plots of r e l a t i v e power with f i n height, i t was found, as expected, that the t o t a l power i n 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 r e l a t i v e  power with incident angle, i t was found that the r e l a t i v e power of the n=o mode decreased s l i g h t l y (from unity) with increasing angle of incidence, over the range where the n=-l mode was evanescent.  Small changes i n the  shape of the Bragg-angle anomaly were also detected, but there was no noticeable change i n the optimum angle.  I t should be noted that the a-  bove changes were extremely small, i n most cases; less than 0.1%.  There-  ANGLE. OF INCIDENCE 6/ 90.00 '  | 0.50  65.38'  i 0.55  56.44'  i 0.60  50.28'  i 0.65  41.81'  45.58*  _ 0.70  j  i 0.75  38.68'  33.75'  31.78' 30.00'  i  I  i 0.85  i 0.90  0.95  1.00  ANGLE OF INCIDENCE 6; 65.58° 41.81° 38.68° , ,  36.03° j  33.75° ,  31.78° ,  30.00' !  i 0.70  i 0.85  i 0.90  FIN SPACING  I 0.80  36.03'  a/\  (a) / g  90.00' |  0.91 0.50  65.38' • 56-44° , ,  1  0.55  :—i 0.60  50.28° ,  1  0.65  r  i 0.75  FIN SPACING  i 0.80  I 0.95  a/\  (b) Figure 3.5 Optimum F i n Height vs. F i n Spacing, (a) d/A, (b) d/a  1.00  fore, i t was  concluded that attenuation i s not a s i g n i f i c a n t factor  the design of a fin-corrugated  surface.  4. 4.1  Experimental Results  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 fins.  thin  Therefore, as far as this problem i s concerned, the results do  not require experimental v e r i f i c a t i o n . 1, the purpose of the study was  to get some idea of how  behaviour of more realizable surfaces. to know how  However, as mentioned i n Chapter to predict the  In p a r t i c u l a r , i t i s desirable  the behaviour of the i d e a l i z e d surface i s affected by; 1) 2) 3) 4)  f i n i t e size imperfect conductivity f i n i t e l y thin fins non-plane wave illumination  5) s l i g h t l y oblique illumination Therefore, some experimental results for f i n i t e surfaces were needed to compare with the results of Chapter 3. It was decided that the f i n i t e surfaces would be modelled after a t y p i c a l ILS problem, since problems of this nature were the main motivation for the study. was  A model frequency of 35 GHz.(A=8.566  mm.)  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.)  The measurements of section 4.3.2  ILS frequency.  were made at 37 GHz.(A=8.103 mm.)  for  reasons explained i n that section. 4.2  Experimental Arrangement Four experimental surfaces were made i n a l l .  a', f i n periods a, f i n heights d and o v e r a l l dimensions a p r o f i l e i n figure 4.1. given i n the next section.  The f i n spacings are given with  The procedure for choosing a, a'' and d w i l l be The f i n thickness t was chosen at the time  Plate 3A  a = 0.526 cm.  d = 0.447 cm. >  Plate 3B  a = 0.526 cm.  d = 0.480 cm.j  Plate IA  a = 0.434 cm.  d = 0.432 cm.^  Plate IB  a = 0.434 cm.  d = 0.173 cm. J  Figure 4.1  26.32 cm. X 11.25 cm. 50 grooves 26.93 cm. X 11.25 cm. 62 grooves  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 i n the m i l l i n g process used.  The o v e r a l l dimensions corres-  pond to a t y p i c a l hangar wall size of 31A by 13A.  One surface was  m i l l e d on each side of two 2.54 cm. thick brass plates. of one of the plates i s given i n figure 4.2.  A photograph  The cover i n (a) i s a  brass reference p l a t e . In the ILS system on runway 14/32  at Toronto International  a  A i r p o r t , an i n t e r f e r i n g hangar i s located at 0^=81.25° about 3 km. from the transmitter, about h a l f the range necessary f o r plane wave i l l u m i n a tion of the entire hangar surface. about 10 m. at 35 GHz.  This corresponds to a distance of  Since distances of t h i s s i z e were not available  indoors, a range with dimensions shown i n figure 4.3(a) was used.  This -  reduced range i s a more severe test of the behaviour of the surfaces under non-piane wave i l l u m i n a t i o n . used i s shown i n figure 4.4.  A photograph of tne actual range  I d e n t i c a l pyrimidal horns with 25 dB gain  and E-plane 3 dB beamwidths of 9° were used f o r transmitting and receiving 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 r e f l e c t o r transmitting antenna with 2° beamwidth was used. A diagram of the experimental c i r c u i t i s shown i n figure 4.3(b). The receiving antenna c r y s t a l current reading with the surface exposed i s returned to the reading obtained with the surface covered by the r e ference plate by adjusting the p r e c i s i o n 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 k l y s t r o n output i s monitored  continuously to detect output l e v e l changes during the measurements.  Figure 4.2  Plates IA and IB. (a) i i t h Reference Plate, (b) Without Reference Plate. T  Plates 1A.1B  h = 0.88m  I = 2.27 m at B; = 69.0'  Plates 3A,3B  h= 0.54m  1= 2.51m at 9/=55.5*  2.55m  2.40m  2.40m  (a)  \  CRYSTAL CURRENT METER  \  CRYSTAL CURRENT METER  CRYSTAL DETECTOR  CRYSTAL DETECTOR  PRECISION VARIABLE ATTENUATOR  VARIABLE ATTENUATOR  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 C i r c u i t  KLYSTRON  50  To investigate the effects of,oblique i l l u m i n a t i o n , the surfaces were mounted on a rotatable platform.. A photograph of Plate 3A mounted on the platform with angle, of rotation oLj_=0 i s shown i n figure 4.5.  The perpendicular metal plate attached to. the end of the surface  i s 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  i n graph, form i n  this section are presented i n tabular form i n Appendix F. 4.3.1  Plates 3A and 3B It was  Opiates  decided that the f i r s t two surfaces  should have a f i n spacing a' corresponding  3A and  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 make.  Therefore, the f i n spacing was  at 35 GHz.  chosen to be a'=0.581A (0^  easy to =59.4°)  Adding the f i n thickness gives a f i n period of a=0.614A  (0i p=54.5°). o  3B)  The r e l a t i v e power vs. f i n height plots of these values  are given i n figure 4.6.  I t i s obvious that there i s a great deal of  difference between the two curves, but since the most e f f e c t i v e value of d i s determined by the waveguide properties, the a' curve was to determine the f i n heights;  Plate 3A (optimum):  used  d=0.561A, Plate 3B  (sub-optimum): d=0.522A. Experimental measurements of the r e l a t i v e 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 i n figures 4.7(a) and  (b).  achieved at an angle of 55.5°.  A reduction of 23.4  dB or 99.54% i s  There i s at l e a s t 10 dB reduction over  iO'  45'  50' 55" • 60' ANGLE OF INCIDENCE  65' 8;  70'  75'  (a) Figure 4.7  X'  30'  iO" 50' • 60' 70' ANCLE OF INCIDENCE- 6;  60*  SO* -  (b)  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 op (b) ( s o l i d curve): a'=0.581A 0. =59.35° d=0.560A 99.97% reduction op o (broken curve): a=0.614A 0 =54.55 d=0.560A 93.49% reduction x  1  Ln  the range 48°30i<64°.  The computed r e l a t i v e power vs. 8^ plots for a'  ( s o l i d curve) and a (broken curve) with d=0.561A are also shown i n figure 4.7(b). the experimental  I t seems that the maximum amount of reduction given by surface i s 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 i s determined s o l e l y by the f i n period a.  This  phenonomon suggested the following procedure for p r e d i c t i n g the r e l a t i v e power v s . 0.. curve of any fin-corrugated surface composed of f i n i t e l y thick f i n s . Suppose 100% at 0^ degrees.  reduction of the power i n the n=o  mode i s required  This angle determines the f i n period a.  appropriate f i n thickness y i e l d s the f i n spacing a'.  the  With this value of  a', the f i n height d i s determined from figure 3.5(a)i would be constructed with parameters (a, a', d)..  Substracting  Now,  Thus, the surface to predict the  r e l a t i v e power vs. 0^ curve for this surface, the f i n period a i s used with the p l o t of figure 3.5(a) to determine the "adjusted d . a  f i n height",  These parameters (a, d ) when used i n the t h e o r e t i c a l program give a  the required  curve.  This procedure was  c a r r i e d out on the parameters of the sur-  face of Plate 3A y i e l d i n g a=0.614A, d =0.595A. a  i n figure 4.8.  The r e s u l t s are shown  The optimum reduction predicted i s 11.9  observed, but this i s only a difference of 0.43%  dB more than  i n r e l a t i v e power.  1° displacement of the measured curve i s a t t r i b u t e d to experimental The experimental  error.  r e s u l t s for the sub-optimum surface of Plate A reduction of 15.4  3B are shown i n figures 4.9(a) and  (b).  i s achieved  There i s at l e a s t 10 dB  at an angle of 55.5°.  the range 49°<0i<64°.  The  dB or 97.10%  reduction over  Notice that the same s i t u a t i o n occurs i n figure  4.9(b) as i t did i n 4.7(b), except that now  the experimental  curve has  ANGLE OF INCIDENCE  ANGLE OF INCIDENCE 8/  6;  (a) .Figure 4.9  (b)  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) ( s o l i d curve): a'=0.581A 0. =59.35° d=0.522A 88.85% reduction op (broken curve): a=0.614A 8 . =54.55 d=0.522A 75.94% reduction °P 1  n  \  i  -80°  i  i  -60'  i  i  -40'  i  i  -20'  i  I  -0'.  I  ANGLE OF ROTATION F i g u r e 4.11  I  J  +20'  I  +40'  t  I  +60'  I  I  -  L  +80'  . <Xf  R e l a t i v e Power o f the n=0 Mode v s . Angle o f R o t a t i o n , P l a t e 3A, Sub-Optimum I n c i d e n c e 8.=61°, f=35 GHz., ( s o l i d c u r v e ) : Reduction Due t o the F i n s o n l y , (dashed c u r v e ) : Reduction Due t o P l a t e O r i e n t a t i o n o n l y .  Ln VO  33.0 Figure 4.12  33.5  34.0  34.5  35.0  35.5  FREQUENCY  (GHz.)'  36.0  36.5  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., & =& =55.5° ±  i  Predicted (broken curve): a=0.614X d =0.595A at f=35 GHz., 6.°B\ =54.55° op a  1  1  37.0  more reduction than the a' curve.  This indicates that the p r e d i c t i o n  procedure outlined above i s -inaccurate except for the case with 100% reduction.  The predicted curve, computed using a sub-optimum curve  s i m i l a r to that of figure 3.5(a), i s shown i n figure 4.10. Figure 4.11 shows a plot of the r e l a t i v e power of the n=o mode as a function of the angle of rotation of incidence 0^=61° (sub-optimum angle).  for Plate 3A at an angle  The dashed curve indicates  the power r e f l e c t e d from the reference plate as i t i s rotated. The s o l i d curve shows the reduction of power due to the fins alone;  Appar-  ently, the surface remains e f f e c t i v e for oblique, incidence over the range -10°<a^<10°.  As ILS glide path angles are about 2 1/2° the sur-  face should be e s s e n t i a l l y as e f f e c t i v e i n reducing interference along the glide path as i n the h o r i z o n t a l plane. The experimental plot of the r e l a t i v e 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° i s shown i n 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 l i m i t a t i o n 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 incidence near 80°.  to an optimum angle of  However, i t was discovered i n the preliminary ex-  perimental measurements that, because of the f i n i t e thickness of the f i n s , the f i n spacing a' was w e l l below the cutoff spacing A/2 and that the surfaces would give no reduction.  Therefore, the frequency of the  FIN HEIGHT Figure 4.13  d/\  Relative Power of the n=0 Mode vs. F i n Height, Plates IA and IB (with attenuation) ( s o l i d curve): a'=0.502A 6. =85.42° 100% reduction at d=0.498X=0.992a P 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 l o  Figure 4.14  ANGLE OF INCIDENCE 8/  ANGLE OF INCIDENCE 8/  (a)  (b)  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 P (b) ( s o l i d curve): a'=0.502A 9. =85.42° d=0.213X 98.33% reduction l o  l o  (broken curve): a=0.536A 9  P  ^=68.88  o  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°). op op 1  x  The r e l a t i v e power vs. f i n _  height plots of these values are given i n figure 4.13.  Because the  f i n thickness i s 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 optimum;  Plate IA:  d=0.533A, Plate IB:  The experimental  62°<0i<79°.  they both became sub-  d=0.213A.  results for the surface of Plate IB are shown  i n figures 4.14(a) and (b). at an angle of 69°.  change to 37 GHz.,  A reduction of 21.7  dB or 99.33% i s achieved  There i s at l e a s t 10 dB reduction over the range  Notice the great difference between the a and a' t h e o r e t i -  c a l curves of figure 4.14(b).  This indicates that i t would be  to have complete reduction at angles 0^  impossible  >75°-from any surface unless  the f i n s 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  e f f e c t i v e 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 y i e l d i n g a=0.536A, d =0.510A. a  sults are shown i n figure 4.15.  The optimum reduction predicted i s 3.9  dB less than observed, a difference of 0.99% The experimental i n figure 4.16. mum  The r e -  i n r e l a t i v e power.  results for the surface of Plate IA are shown  The reduction was much less than expected with no maxi-  reduction observed at 69°, the angle of optimum reduction according  to the period.  Also, there was  a v a r i a t i o n i n the results at each angle  and so only the average reduction could be p l o t t e d .  I t i s believed that  these i r r e g u l a r i t i e s i n behaviour were caused by a Wood S-anomaly l y i n g on  0.35,  35.8  36.0  36.2  36.4  36.6  36.8  FREQUENCY Figure 4.17  37.0  37.4  37.2  (GHz.)  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. -  a  i  =68.88  l o p  .  or very near the value chosen as the f i n height for this surface (see figure 4.13).  Small variations i n the klystron frequency change d/A  at the anomaly and give large variations i n the r e f l e c t i o n c o e f f i c i e n t . The experimental plot of the,relative power of the n=o mode vs. frequency of the incident wave f o r the surface of Plate IB at fixed angle 6i p=69° i s shown i n figure 4.17. A frequency of 37 GHz. was the 0  highest attainable with the klystron"source used. T h e frequency f =36.88 c  GHz. i s the cutoff frequency (corresponding to the cutoff f i n spacing) of the n=l mode i n the f i n region.  This surface with short fins seems  to operate very e f f e c t i v e l y 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 p e r i o d i c 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 i l l u m i n a t i o n . Summarized below are the conclusions drawn: 1)  An optimum fin^corrugated surface demonstrating complete cancellation of specular r e f l e c t i o n can be designed f o r any f i n period i n the range \/2<a<\. dence i n the range  50°<9 L<90 , O  ;  For angles of i n c i -  the usual range f o r most  applications, the optimum f i n height for these surfaces does not need to be any l a r g e r than the f i n period. 2)  For situations i n which the angle of incidence i s near grazing ( 9 ^ > 8 0 ° ) , reductions i n specular r e f l e c t i o n 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 f i n heights.  3)  The angular width of the Bragg-angle anomaly demonstrating complete cancellation of specular r e f l e c t i o n decreases with the period of the surface.  However, i t i s s t i l l suf-  f i c i e n t l y wide for most applications, even near grazing. 4)  The r e l a t i v e power vs. 8^ curve f o r 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 i s not accurate f o r sub-  optimum surfaces. 5)  The f i n i t e size used f o r these surfaces, which was a scaled  down size of a t y p i c a l hangar w a l l i n the ILS problem, had e s s e n t i a l l y no e f f e c t on the performance face.  of the sur-  Reductions of nearly 100% were achieved with 50 to  60 corrugations. Non-plane wave i l l u m i n a t i o n of these fin-corrugated surfaces seemed to have very l i t t l e e f f e c t on t h e i r  performances.  The experimental range used i n this study was about 1/3 the length of an actual ILS range and hence was a very severe test f o r the surfaces. The experimental fin-corrugated surfaces remained  completely  e f f e c t i v e f o r angles of r o t a t i o n i n the range -10<a^<10. Since ILS glide path angles are about 2 1/2°, these surfaces should be e s s e n t i a l l y as e f f e c t i v e i n reducing interference glrvno the p'lide path as i n the hori.7 r>t^.1 plane. n  The experimental surfaces used i n this study were not very f r e quency s e n s i t i v e .  The bandwidths over which they operated  e f f e c t i v e l y were much l a r g e r than required when scaled to the ILS frequency range.  The frequency s e n s i t i v i t y of the  experimental surfaces was predictable using the parameters obtained i n the p r e d i c t i o n procedure of section A.3.1. The attenuation of the n=l mode i n the f i n region seems to have l i t t l e e f f e c t on the performance  of the t h e o r e t i c a l  surfaces, even when the frequency i s close to c u t o f f .  In  addition, i t was found i n the experimental i n v e s t i g a t i o n that the f i n i t e surface with short f i n s would operated very e f f e c t i v e l y over a range of frequencies below c u t o f f . With an experimental frequency of 35 GHz. and the f i n thickness used i n t h i s study, i t i s 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 c u t o f f .  I t i s believed that the  value of thickness used i n this study was possible at 35 GHz.  the thinnest  This s e r i o u s l y l i m i t s any near  grazing investigations at this  frequency.  72  Appendix A.  Determination of the Region of A n a l y t i c ! t y  The f o l l o w i n g i s a method described by C o l l i n 1 6 , 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 f u n c t i o n s ;  <(> (x,s) , which i s a n a l y t i c i n the r i g h t h a l f of the complex s-plane +  (RHP) and <|>_(x,s) , which i s a n a l y t i c i n the l e f t h a l f of the complex s-plane (LHP). <Kx,s)  Then  P  f  J  = <|>_(x,s)+ c(> (x,s) = +  e  o>(x,z)dz +  Now consider Problem # 3.  J  e  <f>(x,z)dz  (A.l)  In the region z>o, equation (2.33)  may be w r i t t e n as <j>(x,z) = A e ° Y  Z  D  + B e" ° Y  Z  0  + B  cos ( ^ e ~ l Y  x  (A.2)  Z  since the evanescent modes d i e out w i t h i n 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 and  + JY  1  Yi = Y l + J Y ] 1  where Y  1<< 0  Y  positive.  11 0  '> Y i  1 < <  Yi  1 1  (-)  1 1  and Y Q , Y Q 1  (A. 3)  n o  A  1  * Y^  1  1  and Y i  1  1  are r e a l and  S u b s t i t u t i n g (A.2) i n t o the expression f o r cj>'(x,s) and i n t e +  g r a t i n g , i t i s obvious the r e s u l t can only be f i n i t e i f S ^ ^ Y Q - * - , s-^> Y j l where s=s-^ _  and only i f  A  + 3 S 2 .  SI>  -Y,-,"'"  Therefore, <}>(x,s) i s a n a l y t i c i n the RHP i f +  Im s a 1  YQ  and  1  Im s z - Y i *  S i m i l a r i l y , i n the region  (A.5)  z<o, equation (2.32) may be written  as (A.6)  and  i t may be assumed that (A. 7)  and  r r  • _ • 1 - -l +J _i r  r «r , r_ «f_ 1  where  -l  11  o  1  o  and p o s i t i v e .  11  r  (A. 8)  11  1  1  Substituting  i n t e g r a t i n g , i t follows  r_" Jt 11  and V , T^ , l  T^  1  Q  and  1  1 are r e a l  (A. 6) into the expression f o r  _(x,s) and  that ^_(x,s) i s analytic i n the LHP i f and only  if Im s< T Q  1  and  Im s < Y^  (A.9)  1  Therefore, i f i t i s assumed that Ya^~ y±' T ^' T <  <  the s t r i p o f common  <  0  t  a n a l y t i c i t y for $ (x,s) and <j>_(x,s) i s as shown i n figure A . l and con+  tains the integration contour P. '  j  S  2  i 4> ( x , s ) ANALYTIC .  i  |  ; _4 pi 0  :  TTio-iiT-o  A.l  1  pi -1  r'  Th<=> S f r i ' n n f r.nmmon A n a l v t i c i t V .  74  Appendix B. I t was  Determination  shown i n s e c t i o n 2.4.1  of g(s)  t h a t g(s) i s a meromorphic  t i o n , t h a t i s , a f u n c t i o n which c o n t a i n s o n l y p o l e s . a method d e s c r i b e d by C o l l i n  func-  The f o l l o w i n g i s  [6, P.434] i n which g(s) i s c o n s t r u c t e d  from r a t i o s o f e n t i r e f u n c t i o n s s u b j e c t to the c o n s t r a i n t s (2.50) and (2.51): For  z>o, the i n t e g r a n d o f e q u a t i o n  g(s)/[cos(ua)-e  j n a  ]  (2.50) i s r e g u l a r i f  c a n c e l s the zeros o f [u s i n ( u a ) ] c o r r e s p o n d i n g  u = —  n  =  +0,1,2,.i.  to  (B.l)  3.  Substituting  ( B . l ) i n t o (2.40) and comparing w i t h  (2.17) i t i s seen  that  s = ty  n=-+0,1,2...  n  For n=o,  Sj+js., = ty ^:±  or  s  0  ±  = tYo  <<  0  where Y i ^ ^ Y ] . * 1 1  -  since Y the LHP  N  8!  U 0  (  S  1  where Y"'"Y"'"''' as b e f o r e . 0  J Y  (B.2)  2  = IY/  1  B  >  3  )  (B.4)  S i m i l a r i l y , f o r n=l,  and, f o r n= +2,3,...,  = tyn  i s pure r e a l f o r n ^ 0 , l .  (B.6)  However, because z>o corresponds to  and the n=o mode i s the only mode i n c i d e n t i n the f i n r e g i o n ,  the zeros o f [u s i n ( u a ) ] are  n=o  n= +1  (B.7)  n= + 2 , 3 , . s  = -Y.  These zeros are p l o t t e d i n f i g u r e B . l . For z<o, the i n t e g r a n d o f (2.51) i s r e g u l a r i f g ( s ) / [ c o s ( u a ) e  ~~ "i hs ] c a n c e l s the zeros o f [ c o s ( h a ) - c o s ( u a ) ] c o r r e s p o n d i n g to 2mr  u = h +  Substituting  n= to,1,2,  (B.8)  (B.8) i n t o (2.40) and comparing with. (2.15) i t can be shown  that  trn  s =  For  n=o,  s +3S ±  = ^c;  2  l  s  so 2 = -r o  -where f  1  <  0  <  r  o  1  s  r_ «r_ 1  1  " o  s  r  1  l  1  a  s  " 1 1  ,  b  e  f  o  r  e  »  (B.9)  (B.10)  -  +  or  where  n= to,1,2,.. .  [  1  1  (B.ll)  S i m i l a r i l y , f o r n=-l,  " - l r  s. -  -T_  ±  (B.12)  and, f o r 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 o f [ c o s ( h a ) - c o s ( u a ) ] a r e  JY^  x  J o r  j r -" l • '  XXX  r  n *"~ ~f"2 y 3 j  lr  -4 r  •  0  •»•  Y  X  r' r '-X-K*-  1-  - i "0 Y  Y  t-jr^  6  1  Y  o - i  i  n n = +1,±2,3,  -JYJ -JYX  z<0  z>0  Figure-B.l  Zero P l o t o f [u s i n ( u a ) ] and with losses.  [cos(ha)-cos(ua)]  js.  Wo  -1 x x x  r n =  +2,3,  f)- l 9" o  n  n = +1,±2,3,  Y  Y  z>0  F i g u r e B.2  z<0  P o l e P l o t of g ( s ) / [ c o s ( u a ) - e  J  ]  n=o  n=  S  -1  ,JI2, 3,. . .  n= +1  These zeros  The  s =  +r_  + jr^  1 1  r 0  ~^  1  (B.14)  1  a n d  zero p l o t of f i g u r e B . l then transforms Since  -l~^  t  e  n  d  t  o  to the zero p l o t must c a n c e l  a  [ c o s ( h a ) - c o s ( u a ) ] , f i g u r e B.2  of  the  i s therefore  the  I t follows that a s u i t a b l e c o n s t r u c t i o n  be  P(s) = (s-y )(S+Y )(s-r ) 0  r  the p o l e s of g(s) / [cos(ua)-e ^ ]  [u s i n ( u a ) ] and  g(s) would  g(s)  +r_  0  p o l e p l o t of t h a t f u n c t i o n . for  s =  assume that the l o s s e s Y ^,  f i g u r e B.2. zeros o f  "|I  T  are a l s o p l o t t e d i n f i g u r e B . l .  Now zero.  +i  =  0  Q  (cos(ua)-e (s+y ) "n v  B1,  e  - } ha  nn  (  lI —I r  (a-r ) ( s - r  1  a  "  ^ TT ———,2mT^ n  —' 2  -n  )  s a  rnr e  (B.15) where P ( s )  i s an e n t i r e f u n c t i o n y e t  i n Appendix C t h a t the e x p o n e n t i a l in  to be  determined.  f a c t o r s and  I t w i l l be  the terms (—) a  the denominator of (B.15) are used to ensure the u n i f o r m  of the i n f i n i t e  products.  and  shown (^—) a  convergence  2  Appendix C.  Convergence of the I n f i n i t e P r o d u c t s i n g ( s )  From e q u a t i o n s l a r g e n, y  n  (2.17) and (2.15), i t i s obvious t h a t , f o r  + (—) , f  +h) and T_  ( ~ I -h) .  n  Therefore,  -sa  (  nj).  ( 1+  e  e  nir  (Cl)  m r  and (s-r  )(s-r ) — n -n nir (2n7[) 2 a  (s-h)a 2mr  (s-h)a.  ( 1 - ~i^r  )  e  . -. (  1  (s+h)a -) e 2nir N  (s+h)a 2niT (C2)  However, the r i g h t hand s i d e s o f ( C l ) and (C.2) are o f the same form as the E u l e r - M a s c h e r o n i e x p r e s s i o n  (  -z_ e n  -yz  Y = constant  zr(z) (C.3)  which i s f i n i t e f o r |z|<°°. equation  T h e r e f o r e , the i n f i n i t e p r o d u c t s o f g(s) ,  (B.15) o f Appendix B, a r e u n i f o r m l y convergent.  /y  Appendix  D.  Asymptotic Behaviour o f g ( s )  In s e c t i o n 2.1.1, i t was s t a t e d t h a t t h e t o t a l f i e l d <}>(x,z) 1/2 i s asymptotic to z i n t h e neighbourhood  o f a f i n edge.  1/2 <j>(ma,z) = A(ma) z + h i g h e r o r d e r terms where m= ±0,1,2,...  .  That i s ,  as z-*o  (D.l)  S u b s t i t u t i n g t h i s e x p r e s s i o n i n t o e q u a t i o n (2.38)  gives oo  e  -sz  lA(o)z  1/2  + higher order termsldz  (D.2)  ;  Watson's Lemma guarantees that i f t h i s r e s u l t i s i n t e g r a t e d the asymptotic expansion o f g(s) as |s|-**> w i l l Consider the f i r s t  term o n l y .  term by term,  result.  The i n t e g r a n d has a branch  p o i n t a t z=o and so i f the branch cut i s chosen as shown i n f i g u r e D . l and the i n t e g r a l c l o s e d i n the upper h a l f p l a n e , by Jordan's Lemma  g(s) =  j" e ~  S Z  A(o)z  1 / 2  dz  = -2A(o) • j  e"  S Z  Making the s u b s t i t u t i o n t= - j z and u s i n g the i n t e g r a l  z  1 / 2  dz  (D.3)  representation  f o r the gamma f u n c t i o n , i t i s found t h a t  g(s) = - 2 A ( o ) r ( | ) s "  Thus, g(s) i s asymptotic t o s  3 / 2  as |s| -> -  as |s| ->  (D.4)  80  Figure D.l  z-Plane  Representation  Showing Branch  Cut.  Appendix E.  D e t e r m i n a t i o n of P ( s ) i n g ( s )  I t was s t a t e d i n s e c t i o n 2.4.2  t h a t P ( s ) must be chosen so.. -3/2  that as  g ( s ) has a l g e b r a i c growth a t i n f i n i t y and i s a s y m p t o t i c  |s| -> .  The f o l l o w i n g approach  0 3  used by C o l l i n  t o t h i s problem  (B.15), i t i s seen  8  1  Is s i m i l a r  to that  [6, P.436].  S u b s t i t u t i n g the e x p r e s s i o n s  from  to s  ( C l ) and (C.2) i n t o  equation  t h a t g ( s ) d i f f e r s by o n l y a bounded f u n c t i o n o f s .  g i ( s ) where g ^ ( s ) i s g i v e n by  ~~  0 0  s  r i (1+  3  1  P(s) -sa  jcos(ua)-e~" ' } :  ha  s'> ™ n ^Wr> rr <(s-h)a  00  e2n7r  ci-  e  (s-Fh)a  x  1  e 2n7r  1  •  Eliminating  «  •  the i n f i n i t e p r o d u c t s  ' ' (E.l)  i n (E.l) using equation  (C.3) w i t h  the p r o p e r v a l u e o f z and n o t i n g t h a t  T(z)  g (s) n  -  .  \  ,  p  <  , -  r-  (E.2) . '  sm(Trz)r(l-z)  can be m a n i p u l a t e d  into  t h e form  P(s) { c o s ( u a ) - e ~ S l <  *  S )  =  ~ s  2  • r(s-h)a, sxn{ — }  j h a  }  (s+h)a, sxn{ —} r  (air)  .H^L)  ,; ( s - h ) a , (s+h)a, r { — T { — ^ - } f  rT  (E.3)  Now,  the a s y m p t o t i c e x p a n s i o n  •n r \  /„ 1  T(z) = / 2Tr  If  x  t h i s e x p r e s s i o n i s used  o f T(z) i s  Zlnz - 1 / 2 -Z+l e  z  e  1 1 ^ [ z| -* ">  f n t\ ( E . 4)  t o r e p l a c e the T f u n c t i o n s i n e q u a t i o n (E.3)  and  i t i s a s s u m e d 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 t h e f o r m  ln2 ( \ = K P ( s ) {cos(ua)-e } e 1 3/2 . (s-h)a, . (s+h) s sxn{—} sml 7.—i S a  j h a  §  r  77  r  a i  (E.5) will  r e s u l t , where K i s a known c o n s t a n t .  hence g ( s ) to be asymptotic to s —  , i t i s obvious  that  S 3.  P(s) = C e~ ~ T  where C i s a c o n s t a n t .  -3/2  T h e r e f o r e , f o r g-^(s) and  ln2  (E.6)  Appendix ,F. The  Data  f o l l o w i n g i s a c o l l e c t i o n of the e x p e r i m e n t a l d a t a  d u r i n g the study. Chapter  Experimental  Most of t h i s d a t a appears  i n g r a p h i c a l form i n  4.  P l a t e s 3A,  3B  P l a t e s IA,  9i ^degrees)  Table V  (m.)  (degrees)  IB  I  (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  T r a n s m i s s i o n D i s t a n c e v s . Angle Incidence  (see f i g u r e 4.3(a))  of  taken  RELATIVE POWER OF THE. n=0 MODE (dB) \^  a. -5.0°  -2.5°  0°  +2.5°  +5.0°  70.0°  -6.9  -6.85  67.5°  -8.0  -8.15  • -7.9  65.0°  -9.95  -10.05  -9.8  -10.1  -10.0  -6.75  -6.95  -7.1  -8.15  -8.2  62.5°  -12.3  -12.2  -11.95  -12.2  -12.2  60.0°  -15.5  -15.55  -15.15  -15.5  -15.4  57.5°  -20.4  -20.5  -20.0  -20.5  -20.3  56.5°  -23.1  -23.3  -23.0  -23.2  -22.8  55.5°  -23.4  -23.1  -23.4  -23.0  „-22.4  54.5°  -22.3  -22.3  -23.3  -22.3  -22.2  53.5°  -19.7  -19.7  -20.9  -19.7  -19.3  52.5°  -16.6  -16.65  -17.55  -16.75  -16.65  50.0°  -11.8  -11.9  -12.4  -11.9  -11.9  47.5°  -8.7  -8.75  -9.1  -8.7  -8.75  45.0°  -6.05  -6.0  -6.35  -6.1  -6.05  Table VI  R e l a t i v e Power o f the n=0 Mode v s . Angle o f I n c i d e n c e , P l a t e 3A, f=35 GHz.  85  RELATIVE POWER OF THE n=0 MODE (dB) \  a  i -5.0°  -2.5°  70.0°  -6.5  -6.4  67.5°.  -7.55  65.0°  -9.0  62.5°  -10.6  -10.2  60.0°  -12.55  57.5°  0°  +2.5°  +5.0°  -6.3  -6.4  -6.7  -7.35  -7.3  -7.4  -7.65  -8.75  -8.55  -8.6  -9.0  -10.2  -10.2  -10.6  -12.1  -12.1  -12.2  -12.55  -14.8  -14.2  -14.25  -14.2  -14.7  56.5°  -15.6  -15.0 -  -15.1  -15.0  -15.6  55.5°  -15.5  -15.2  -15.4  -15.25  -15.6  54.5°  -15.15  -14.8  -15.1  -14.8  -15.15  53.5°  -14.7  -14.4  -14.65  -14.4  -14.8  52.5°  -13.75  -13.45  -13.75  -13.45  -13.7  50.0°  -10.8  -10.65  -11.05  -10.8  -10.9  47.5°  -8.1  -8.0  -8.3  -8.0  -8.1  45.0°  -5.65  -5.6  -5.9  -5.65  -5.65  Table V I I  R e l a t i v e Power o f the n=0 Mode v s . Angle o f I n c i d e n c e , P l a t e 3B, f=35 GHz.  86  FREQUENCY  RELATIVE POWER, n=0 MODE  (GHz)  (dB)  Table VIII  33.00  -9.6  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  1  R e l a t i v e Power o f the n=0 Mode v s . Frequency o f t h e I n c i d e n t Wave, P l a t e 3A, a i  =0°,  0^=54.5°  RELATIVE POWER  0.  X  (degrees)  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. R e l a t i v e Power o f t h e n=Q Mode v s . Angle Incidence, Plate  IB, 0 ^ = 0 ° , f=37 GHz.  oo  9. X  AVERAGE RELATIVE  (degrees)  POWER n=0  MODE (dB)  Table X  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  R e l a t i v e Power o f the n=0 Mode v s . A n g l e o f I n c i d e n c e , P l a t e IA, a =0°, ±  f=37 GHz.  89  FREQUENCY  n=0  (GHz)  MODE  (dB)  -  Table XI  RELATIVE POWER  .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  R e l a t i v e Power o f the n=0 Mode v s . Frequency o f the I n c i d e n t Wave, P l a t e I B ,  a  ±  = 0°, e = 69.0° ±  90  References 1.  J . F . C a r l s o n and A.E. H e i n s , "The R e f l e c t i o n o f an E l e c t r o m a g n e t i c P l a n e Wave by an I n f i n i t e S e t o f P l a t e s , I " , Quart, o f App. Math., V o l . V I , No. 1, 1947.  2.  A.E. Heins and J.F. C a r l s o n , "The R e f l e c t i o n o f an E l e c t r o m a g n e t i c P l a n e Wave by an I n f i n i t e S e t o f P l a t e s , I I " , Quart, of App. Math., V o l . V, No. 1, 1947.  3.  F. B e r z , " R e f l e c t i o n and R e f r a c t i o n of Microwaves a t a S e t o f P a r a l l e l M e t a l l i c P l a t e s , " J . o f I.E.E. , V o l . 98, No. I l l , P. 47, 1951.  4.  E.A.N. Whitehead, "The Theory o f P a r a l l e l P l a t e Media f o r Microwave L e n s e s , " J . o f I.E.E. , V o l . 98, No. I l l , P. 133, 1951.  5.  H. Gruenberg and R.A. Hurd, " S c a t t e r i n g o f a P l a n e E l e c t r o m a g n e t i c Wave by an I n f i n i t e Stack o f Conducting P l a t e s " , NRC No. 3338, ERA-268 U n c l a s s i f i e d , A p r i l 1954.  6.  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