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Properties of surface waveguides with discontinuities and perturbations in cross-section Brooke, Gary H. 1977

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PROPERTIES OF SURFACE WAVEGUIDES WITH DISCONTINUITIES AND PERTURBATIONS IN CROSS-SECTION  by  Gary H. Brooke 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 DOCTOR OF PHILOSOPHY  in. the Department of E l e c t r i c a l Engineering  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA January, 1977  (c ) Gary H. Brooke, 1977  In p r e s e n t i n g t h i s  thesis  an advanced degree at  further  fulfilment  of  the  requirements  the U n i v e r s i t y of B r i t i s h Columbia, I agree  the L i b r a r y s h a l l make it I  in p a r t i a l  freely  available  for  this  thesis  f o r s c h o l a r l y purposes may be granted by the Head of my Department  of  this  thesis for  It  financial  g a i n s h a l l not  of  The U n i v e r s i t y o f B r i t i s h Columbia  2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date  .ffl-K)  ~2-l ] I  or  i s understood that copying or p u b l i c a t i o n  written permission.  Department  that  reference and study.  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f  by h i s r e p r e s e n t a t i v e s .  for  117  be allowed without my  ABSTRACT  The f i r s t part of t h i s thesis i s concerned with t h e o r e t i c a l and experimental investigations of step d i s c o n t i n u i t i e s on a planar surface waveguide.  An approximate t h e o r e t i c a l solution to the unbounded discon-  t i n u i t y problem i s obtained by bounding the open structure with p e r f e c t conductors, since there i s a d i r e c t r e l a t i o n s h i p between the mode spectra of the two configurations. Mode-matching i s used to solve the bounded case. The method i s "tested" on four d i s c o n t i n u i t y configurations considered by other workers.  Good agreement with the previous r e s u l t s i s obtained i n a l l  cases except one f o r which the o r i g i n a l r e s u l t s are shown to be inaccurate. The experimental i n v e s t i g a t i o n i s carried out on a d i e l e c t r i c coated conductor surface waveguide, supporting the f i r s t TM mode, at a frequency of 30 GHz.  Standing wave measurements are obtained using a  X/2 dipole oriented along the l o n g i t u d i n a l component of e l e c t r i c  field.  The parameters of i n t e r e s t are the magnitude and the phase of the r e f l e c tion coefficient.  The experimental r e s u l t s confirm those obtained theore-  tically. The t h e o r e t i c a l and experimental techniques are l a t e r applied i n an i n v e s t i g a t i o n of a cascaded step d i s c o n t i n u i t y configuration.  The  theo-  r e t i c a l approach involves the use of wave transmission matrices.  Experi-  mental r e s u l t s f o r the magnitude of the r e f l e c t i o n c o e f f i c i e n t are found to be i n reasonable agreement with theory. The second part of t h i s work describes a t h e o r e t i c a l and experimental study of d i e l e c t r i c waveguides, of c i r c u l a r cross-sections, perturbed by a x i a l s l o t s .  In p a r t i c u l a r , the normalized propagation c o e f f i c i e n t  ( i i )  of the dominant modes with each p o l a r i z a t i o n i s determined a n a l y t i c a l l y using a standard perturbation technique and experimentally using an open resonant  cavity.  The perturbation r e s u l t s give quite a good i n d i c a t i o n of  the trends observed experimentally.  I t i s found that there i s an optimum  size of the perturbation which gives the maximum separation between the normalized propagation  c o e f f i c i e n t s of the two p o l a r i z a t i o n s .  (iii)  CONTENTS  FIGURES  .  .  .  .  .  .  TABLES  .-.  .  .  .• .  «• . .  .  .  . . .  PRINCIPAL SYMBOLS . . . . . .  . .  . .  .  .  .  . ... ..  ...  . . .  ACKNOWLEDGEMENT ....... . GENERAL INTRODUCTION AND BACKGROUND . . . PART I SURFACE WAVEGUIDE DISCONTINUITIES  '• . . .  .  .  .  .  , .  . .  . . .  ©  ©  ©  •  Scope o f P a r t I  2.  THEORETICAL INVESTIGATION OF A STEP DISCONTINUITY ON A PLANAR SURFACE WAVEGUIDE  2.1  An a p p r o x i m a t e s o l u t i o n t e c h n i q u e  2.2  Modes o f t h e bounded waveguide  2.3  Mode-matching  2.4  A p p l i c a t i o n to p r e v i o u s l y considered  2.5  Summary  3.  EXPERIMENTAL INVESTIGATION OF STEP DISCONTINUITIES  3.1  E x p e r i m e n t a l waveguide c o n s t r u c t i o n and e x c i t a t i o n  3.2  Measurement a p p a r a t u s and p r o p e r t i e s o f t h e s u r f a c e waveguide . . . . . . . . .  «  e  •  #  «  .  v i .  x  . #  •  x  x i l  v  .  XV 1  •  2 A  .  .  . .  . .  .  . .  .  .  .  .  7 10  . configurations .  .  .  .  .  . . .  .14 19  .  25  3.3  F a c t o r s a f f e c t i n g s t a n d i n g wave measurements  3.4  Experimental determination of the r e f l e c t i o n c o e f f i c i e n t and c o m p a r i s o n w i t h t h e o r y . .  3.5  Summary  4.  APPLICATION TO CASCADEL DISCONTINUITIES  4.1  Wave m a t r i x f o r m u l a t i o n  4.2  Experimental  4.3  Comparison w i t h t h e o r y  4.4  Summary  PART II  .  .  1.2  •  .  .  .  Ssclc^irouiid  *  .  . .  X*X  « •  .  .  INTRODUCTION •  .  . . . .  1.  •  .  27  . . . . .  .  . ..  .  .  .  31  •  • ••  40  .  .  47  .  55  investigation .  .  .  . . .  . . .  .  .  .  . . . .  .  .  .  .  .  .  .  .  .  .  .  .  58  .  .  .  63  .  .  .  65 67  CYLINDRICAL DIELECTRIC WAVEGUIDES WITH PERTURBED CIRCULAR CROSS-SECTION .  5.  INTRODUCTION  5.1  Background  .  (iv)  . . . . . . .  . . . .  .  .  .  .  .  .  .  .68  .  69  5.2  Scope o f P a r t I I  6.  A PERTURBATION METHOD  6.1  Q u a s i - s t a t i c approximation to the perturbed f i e l d s  6.2  A p p l i c a t i o n o f t h e p e r t u r b a t i o n method  76  6.3  Summary  80  7.  AN EXPERIMENTAL INVESTIGATION  7.1  E x p e r i m e n t a l a p p a r a t u s and p r e l i m i n a r y measurements  7.2  Experimental r e s u l t s f o r the s l o t t e d d i e l e c t r i c  .  .  .  70  i  74  .  . 8 2 91  waveguides 7.3  Results  7.4  Summary  8.  SUMMARY AND CONCLUSIONS  101  8.1  S u g g e s t i o n s f o r f u r t h e r work  104  Appendix  of the p e r t u r b a t i o n a l a n a l y s i s  .  .  .  .  .  .  .  .  .  91 99  I-A  FIELD BEHAVIOUR NEAR ANISOTROPIC AND MULTIDIELECTRIC EDGES .  I-A.l  Edge i n v o l v i n g a magnetized  I-A. 2  Edge i n v o l v i n g t h r e e d i e l e c t r i c s .  Appendix I-B  .  ferrite . •  .106  .  .  .  .  .  108  .  -.  . ..  •  112  EFFECT OF DIELECTRIC EDGE CONDITIONS ON MODE-MATCHING SOLUTIONS  120  . . .  Appendix  I-C  INTEGRALS INVOLVED IN EQUATION 2.19  .  Appendix Appendix  I-D I-E  ATTENUATION COEFFICIENTS .• . . . MEASUREMENT OF STANDING WAVES WITH A  .  .  .  .  .  126  . . . . .  12?  X/2 DIPOLE Appendix  I-F  DERIVATION OF WAVE MATRICES w  Appendix  II-A  DERIVATION OF EQUATION 6.1  Appendix  II-B  SELECTED INTEGRALS OF BESSEL FUNCTIONS  REFERENCES .  AND w  ;L  .  & , i = 1, . . . . ±  124  ....8 .  .  .  .  (v)  .  .  .  .  .  .  .  .  .  .  .  .  .  2  .  .  .  .  .  . . .  .  .  .  . .  . .  .  .  .130 .  .  .  131  .  134  .135  FIGURES 2.1  2.2  Step d i s c o n t i n u i t y configurations (a) unbounded (b) bounded  8  Normalized propagation c o e f f i c i e n t versus =2.26, t = 0.079 X B Q r 1 o  15  e  n  2.3  2.4  Normalized ( i ) transmitted ( i i ) radiated ( i i i ) r e f l e c t e d powers. e~/e •= 13.0, e„/e = 12.7, I o Z o e,/e = 2.35, E./e = 2.25, t./X = 0.3, t./X =3.6 3 o ' 4 o l o 4 o a present r e s u l t s f o r the configuration i n i n s e r t A a' o r i g i n a l r e s u l t s from reference (47) b present r e s u l t s f o r the configuration i n i n s e r t B b' o r i g i n a l r e s u l t s from reference (48) Normalized radiated power.  e  i /  e  = 0  1«02, t^/t^  =  20  2.0  present r e s u l t s o r i g i n a l r e s u l t s from reference (49) 2i5(a)  Magnitude of the r e f l e c t i o n c o e f f i c i e n t versus t^/X e,/e  2.5(b)  3.1  = 9.0,  e_/e  = 10.0,  t,/X  22 .  =0.06  l o 2 Q ' l o a present r e s u l t s , N = 50 a' N = 20 a" N = 10 b o r i g i n a l r e s u l t s from reference (52)  23  Percentage power i n the r a d i a t i o n and the transmitted surface-wave modes. Curves "a" and "b" as i n F i g . 2.5(a). .  24  n  A part of the experimental  surface waveguide and of  the measurement apparatus.  . . . . . . -.. . . . .  3.2  E x c i t a t i o n of the surface waveguide  3.3(a)  The dipole used f o r f i e l d probing  3.3(b)  The  3.4  Schematic f o r the complete measurement system  3.5  The decay of the surface wave away from the d i e l e c t r i c surface  3.6  Amplitude and phase c h a r a c t e r i s t i c s (abscissa i s transverse distance from the beam axis) . . . .  3.7  Attenuation measurements ( exponential attenuation)  . . . . . . . . . . . .  28 30 32  coaxial l i n e and waveguide detection unit  (vi)  . . . . .  . . . . .  .  33 35  3.8  Amplitude patterns o nearer to the source x midpoint A f a r from the source . . .'  41  Estimated values of beam width versus distance along the beam axis  42  3.10(a) V a r i a t i o n i n standing wave r a t i o values as a function of the height of the dipole above the d i e l e c t r i c surface . . .  44  3.10(b) A decay curve i n d i c a t i n g heights at which the standing waves were measured  45  3.9  '3.11(a) Example of v a r i a t i o n i n the standing wave r a t i o values as a function of the distance from the step discontinuity ( t h e o r e t i c a l dependence of standing wave ratio)  48  3.11(b) same as (a) but f o r a d i f f e r e n t step height  49  3.12  An example of a measured standing wave pattern  50  3.13  Experimental step discontinuity configuration  51  3.14  Experimental r e s u l t s f o r the magnitude of the reflection coefficient X 29.94 GHz O 30.00 GHz A 30.06 GHz theory (30.00 GHz)  53  4.1  4.2  5.1 6.1  6.2  Cascaded step discontinuity configurations (a) unbounded (b) bounded  -. . .  Magnitude of the r e f l e c t i o n c o e f f i c i e n t versus o experiment theory (30.0 GHz) theory (30.8 GHz)  59  &;A  0  64  Finned d i e l e c t r i c waveguides - possible interconnection and coupling schemes  72  D i e l e c t r i c rod configurations (a) unperturbed (b) perturbed by s l o t s  75  Q u a s i - s t a t i c approximations (a), (b), ( c ) , (d): simple configurations (e) configuration corresponding to a s l o t (f) correction factor K  (vii)  . . . . . . . .  77  7.1  7.2  Experimental configurations of the perturbed rods  dielectric 83  Resonant cavity apparatus ( i n s e r t — t h e coupling probe and the scatterer)  84  Cross-section of the c e n t r a l part of one end plate of the cavity i n d i c a t i n g probe and l o c a t i n g pin arrangement . . . .  85  7.4  Chart recording of the output of the cavity  86  7.5  Schematic of the measurement system  7.6  Resonance curves (horiz. scale = 3.5 MHz/large div. ) (a) frequency measurement using the reference channel trace (b) "double" resonance . . . . . . . . . . . . . .  7.3  7.7  ....  ...  . . . . .  88  90  Normalized propagation c o e f f i c i e n t s of the two p o l a r i z a tions as a function of 6  (fl) Tod A • • • • • • • « e A e-tttte • o e • • »« • • • • 93 (b) rod B (c) rod C  7.8 7.9  11  12  94 95  .  Resonant frequency separation of the two p o l a r i z a t i o n s on a perturbed rod (horiz. scale - 3.5 MHz/large div.) .... Slot configurations (a) approximation of a s l o t i n rectangular coordinates by one i n polar coordinates (b) actual s l o t configuration for rod B . . . . . . . . . Edge configurations (a) dielectric-magnetized f e r r i t e Cb) three i s o t r o p i c d i e l e c t r i c s ...  96  .  98  . . . . . . . . . . .  ^Q7  Behavior of the s i n g u l a r i t y parameter t as a function of $ and <j>„ . . . . . ±  1 1 4  2  13  Region of nonsingularity f o r an edge common to three dielectrics . . . . . , . ... . . . . . . . . . . . . . .  14  Dependence of a region of nonsingularity on the permittivity with 2 i 3* * * * * .... e  >  e  >  e  15  Dependence of A<^  16  Interface between empty and H-plane loaded waveguide e - 8, y = 0.75, 2t/b = 0.7, b/V = 0.3, a/X = 0.<  1  r  • •  on the r a t i o of the highest to lowest  r  Q  (a) waveguide configuration (b) lumped-elemerit equivalent c i r c u i t of the i n t e r f a  (vlii)  H7  2.16  Rate of decay of amplitude c o e f f i c i e n t s of r e f l e c t e d modes i n waveguide A  °  l ml  *  ' ml  modes)  A  A  (  ™  i  m m  Dipole coordinates Volume region  . . . . .  (ix)  o  d  e  s  )  •  TABLES  2.1  Correspondence between the bounded and the unbounded mode spectra «  2.2  Convergence of slow mode scattering c o e f f i c i e n t s as a function of t„/X . e = 2.26, t = 0.079X , B o r 1 o t = 0.555X 2 o Experimental and t h e o r e t i c a l values of the phase angle of the r e f l e c t i o n c o e f f i c i e n t f o r a range of step discontinuity heights 0  3.1  4.1  56  Convergence of slow mode scattering c o e f f i c i e n t s (obtained by wave matrices) as a function of  VX  7.1  18  62  n  B o Values of f  r  and 6/k f o r the unperturbed rods Q  00  92  PRINCIPAL SYMBOLS a^  = complex amplitude coefficient of the k ^ mode i n waveguide A  A  = uniform section of the bounded planar waveguide  fc  An ,Bn,Cn,Dn = amplitude constants for n*"* surface wave mode on a dielectric 1  rod = complex amplitude coefficient of the j * * mode i n waveguide B 1  —1'—1'—2  =  v e c t o r s  °f normalized amplitudes of modes travelling i n the  negative z direction B  = uniform section of the bounded planar waveguide  c^,£^,c_2  vectors of normalized amplitudes of modes travelling i n the  =  positive z direction C  = amplitude constant  d  = distance from a step discontinuity = r. - r 1 o  for a slot  -k Z B  dn  _  o o n A n n  e ,e ,h  = component form of the mode functions for the bounded planar waveguide  ea ,ha  = ex, hy for waveguide A  e^,h^  = e^, h^ for waveguide B  E_,E r Q,Ez  = electric f i e l d components i n polar coordinates (r,0,z)  Ep,E j ,E  = electric f i e l d components i n polar coordinates (p,$,z)  E, H  = total electric and magnetic fields of a perturbed waveguide  E^,  = total electric and magnetic fields of an unperturbed wave-  (  )  z  guide f  r  = resonant frequency  (xi)  J  = Bessel function of the f i r s t  n  k  = 2TT/A  o  0  K  kind  = modified Bessel function of the second kind  n  R ,K r 8  = q u a s i - s t a t i c correction factors f o r the r and 9 f i e l d  Q  components N  = number of modes used i n mode-matching = number of h a l f wavelengths i n the resonant cavity  ^l'^2'^2  =  t  r  a  n  s  v  e  r  s  e  propagation c o e f f i c i e n t s  P ,P a b  = normalization constants i n waveguides A and B  s  = standing wave r a t i o  S  = scattering matrix  t„, t„ £1 n  = e l e c t r i c and magnetic s i n g u l a r i t y parameters  t,,t_,t„,t. = thicknesses of d i e l e c t r i c 1 2 3 4 t„ o  layers  = conductor spacing f  S  t^, t ^  = t r a n s i t i o n values of t ^  tan $ w  = loss tangent • ( 6 , - . e ) ( r - + r )/2 f o r a s l o t 1 z 1 o  W  1  , W  2  1  , W  T  W Y Z  =  w  a  v  e  o  amplit^  6  matrices  = o v e r a l l wave matrix f o r a cascaded section o  = /e /y ' o o  o  - 1/Y o  A,A  M' D a  3,B Bp o>  = attenuation c o e f f i c i e n t s = phase c o e f f i c i e n t s  (xii)  6  r perturbation parameter  tS.. ij  = Kronecker d e l t a = i n t e g r a l of Bessel functions  Ae^  = change i n r e l a t i v e p e r m i t t i v i t y within a region of perturbation  G  Q  e e  = p e r m i t t i v i t y of free space = p e r m i t t i v i t y of a d i e l e c t r i c  r  <= r e l a t i v e p e r m i t t i v i t y  8  = phase angle of the r e f l e c t i o n c o e f f i c i e n t  K  = o f f diagonal term of the permeability  X  Q  tensor  = free space wavelength  X  - waveguide wavelength  u  - diagonal element of the permeability  tensor  = permeability of free space p,p  = magnitude of the r e f l e c t i o n c o e f f i c i e n t  a  = conductivity  (i)  = angular frequency  I  = distance between cascaded d i s c o n t i n u i t i e s  ( x i i i )  ACKNOWLEDGEMENTS The author i s grateful f o r the opportunity of working under the supervision of Dr. M. M. Z. Kharadly, whose i n t e r e s t and expertise have been forwarded i n such a personal manner that they s h a l l not soon be forgotten. Grateful acknowledgement i s made to the National Research Council of Canada f o r a postgraduate bursary i n 1972-1973, f o r a postgraduate scholarship during the period 1973-1976 and f o r support of the project under grant A-3344. The author i s also g r a t e f u l to Dr. W. K. McRitchie f o r h i s c o l l a boration on the work presented i n Appendix I-B, to Mr. J . Stuber and to Mr. D. Daines f o r t h e i r excellent workmanship i n constructing various parts of the experimental apparatus, to Mr. T. K. Chu f o r h i s professional photographic assistance and to Mrs. A. Semmens and Ms. M. E. Flanagan  f o r t h e i r help i n  typing the manuscript. F i n a l l y , I would l i k e to thank my wife, Sharon, f o r typing two drafts and a part of the f i n a l copy of this thesis and, more important, f o r assuming many of my family r e s p o n s i b i l i t i e s throughout the course of this study.  -  (xiv)  GENERAL INTRODUCTION AND BACKGROUND S u r f a c e waveguides a r e used i n a v a r i e t y o f a p p l i c a t i o n s , ranging i n frequency  from u l t r a h i g h t h r o u g h o p t i c a l .  The most common  types o f s u r f a c e waveguides a r e e i t h e r p l a n a r o r c y l i n d r i c a l and may i n v o l v e m e t a l l i c conductors.  However, s i n c e they a r e open ( o r unbounded)  s t r u c t u r e s , s u r f a c e i m p e r f e c t i o n s cause power t o be l o s t i n t o r a d i a t i o n and  they a r e s u b j e c t t o i n t e r f e r e n c e .  R a d i a t i o n and i n t e r f e r e n c e a r e  u s u a l l y c o n s i d e r e d as d e t r i m e n t a l e f f e c t s .  I n some a p p l i c a t i o n s , however,  they can be advantageous. C u r r e n t l y , there i s s u b s t a n t i a l i n t e r e s t i n the use o f s u r f a c e waveguides a t o p t i c a l f r e q u e n c i e s .  I n c y l i n d r i c a l f o r m , t h e s e wave-  guides a r e r e f e r r e d t o as o p t i c a l f i b r e s . optical fibres ( c i v i l been p r o p o s e d .  ( 1 - 3 )  ,  industrial  ( 4  I n d e e d , some e x p e r i m e n t a l  '  Several applications f o r 5 )  , and m i l i t a r y  ( 6 , 7 )  ) have  o p t i c a l communications l i n k s  (8,9)  have a l r e a d y been t e s t e d and s i n c e f i b r e l o s s f i g u r e s o f 2-10db/km are r e a d i l y o b t a i n a b l e , repeater s p a c i n g s ^ * ' ^ ^  o f 5-10km c o u l d be  realized. The b a s i c f i b r e c o n s i s t s o f a c o r e r e g i o n s u r r o u n d e d by a c l a d d i n g medium o f s l i g h t l y l o w e r r e f r a c t i v e i n d e x . serves t o strengthen  The c l a d d i n g  t h e f i b r e and s h i e l d t h e c o r e f r o m t h e e n v i r o n m e n t .  I f t h e c o r e i s chosen such t h a t o n l y one mode p r o p a g a t e s , t h e f i b r e i s c a l l e d s i n g l e mode; whereas, i f t h e c o r e s u p p o r t s many modes, t h e f i b r e i s c a l l e d multimode.  The s i n g l e mode f i b r e o f f e r s t h e l a r g e r b a n d w i d t h  (13) capability  b u t r e q u i r e s a l a s e r l i g h t source  for efficient  excitation.  Such a s o u r c e , does n o t , as y e t , p o s s e s s t h e r e l i a b i l i t y and l o n g - l i f e (14) necessary  f o r economic t r a n s m i s s i o n  (xv)  .  I n a d d i t i o n , s i n g l e mode  f i b r e s require c r i t i c a l alignment a t j o i n t s ( t o loss); this i s further ditions.  prevent excessive  c o m p l i c a t e d when c a r r i e d o u t under f i e l d  A l t e r n a t i v e l y , multimode f i b r e s c a n be e x c i t e d  w i t h i n c o h e r e n t l i g h t s o u r c e s such as l i g h t - e m i t t i n g F u r t h e r m o r e , because o f t h e i r l a r g e r from t h e s t r i n g e n t  con-  efficiently  diodes  c o r e d i m e n s i o n , t h e y do n o t s u f f e r  j o i n i n g l i m i t a t i o n s imposed on s i n g l e mode f i b r e s .  However, because o f t h e many p r o p a g a t i n g modes, each c a r r y i n g  energy  a t a d i f f e r e n t group v e l o c i t y , t h e multimode f i b r e has p o o r e r d i s p e r s i o n c h a r a c t e r i s t i c s t h a n  s i n g l e mode f i b r e s and hence t h e b a n d w i d t h  (19) capability  i s reduced. A t t e m p t s t o overcome t h e d i s p e r s i o n p r o b l e m i n multimode  f i b r e s have r e s u l t e d the  i n the s e l f - f o c u s s i n g  f i b r e ( S E L F O C ) i n which  c o r e has a n e a r p a r a b o l i c r e f r a c t i v e i n d e x p r o f i l e .  a focussing effect at periodic e q u a l i z e t h e group d e l a y s .  The r e s u l t i s  i n t e r v a l s a l o n g t h e f i b r e w h i c h tends t o  I t may be o f i n t e r e s t  imperfections i n the core-cladding i n t e r f a c e  t o n o t e t h a t random  (21 22) ' may a l s o r e d u c e (23)  d i s p e r s i o n i n multimode f i b r e s t h r o u g h mode c o n v e r s i o n e f f e c t s example where i m p e r f e c t i o n s a r e b e n e f i c i a l ) . p r e s e n t , t h e r e may be a need w i t h i n capacity of o p t i c a l f i b r e s .  (an  Although not e s s e n t i a l a t  t h e n e x t decade f o r t h e b a n d w i d t h  Due t o l i m i t a t i o n s on s u i t a b l e  sources  f o r s i n g l e mode f i b r e s , i t appears t h a t multimode f i b r e s c o u l d be u s e f u l for i n i t i a l installations. P l a n a r s u r f a c e waveguides a r e a l s o o f i n t e r e s t , a t o p t i c a l f r e q u e n c i e s , as o p t i c a l d a t a p r o c e s s o r s .  These waveguides f o r m t h e b a s i s  (24) of " i n t e g r a t e d o p t i c s ' optical fibres.  , w h i c h has been d e v e l o p e d c o n c u r r e n t l y w i t h  The p o s s i b i l i t y o f o p t i c a l t r a n s m i s s i o n w o u l d be enhanced  i f d a t a p r o c e s s i n g (e.g.  c o u p l i n g , m o d u l a t i o n , and s p l i t t i n g ) c o u l d be (xvi)  accomplished at o p t i c a l frequencies  (25)  (by eliminating some of the  c i r c u i t r y from repeaters and, to a c e r t a i n extent, from the transmitter and r e c e i v e r ) . Generally speaking,  the devices consist of a thin guiding  layer of d i e l e c t r i c deposited on a substrate. only to support  The substrate may serve  the d i e l e c t r i c waveguide or may be Involved d i r e c t l y i n  s p e c i f i c components (such as mode converters, i s o l a t o r s , etc.) Since the distances involved i n integrated o p t i c a l devices are of the order of centimeters, as compared to kilometers i n o p t i c a l f i b r e s , the loss requirements i n the d i e l e c t r i c materials used need only be of the order of 0.5db/cm.  I t i s i n t e r e s t i n g to note that several passive  devices which incorporate d i s c o n t i n u i t i e s d i r e c t l y i n t o the component design, such as couplers  (27 28) (29) ' , lens and prisms , have been developed.  Again these are examples of applications i n which d i s c o n t i n u i t i e s are used to advantage. At lower frequencies, surface waveguides have been investigated f o r various communications a p p l i c a t i o n s . In the millimeter range two (30 31) proposals  '  , involving waveguides analogous to o p t i c a l f i b r e s but  with loss c h a r a c t e r i s t i c s of 40db/km and bandwidths approaching 1GHz, have been made.  In the microwave range the properties of surface wave(32)  guides used as antennas have long been established  .  Radiation occurs  at abrupt terminations of the waveguide or along continuous the waveguide dimensions. highly d i r e c t i o n a l .  changes i n  Such antennas are of l i g h t weight and are  More r e c e n t l y , i t has been proposed that surface (33)  waveguides be used as antenna feeds  . At u l t r a h i g h frequencies the  increased f i e l d extent of surface waveguides make them natural candidates for high speed v e h i c l e s i g n a l l i n g and control i n connection with r a i l r o a d applications  '  .  The waveguides serve the dual purpose of detecting (xvil)  s t a t i o n a r y and moving o b j e c t s on t h e r a i l s and o f communicating w i t h t h e h i g h speed v e h i c l e s . The  above b r i e f r e v i e w i n d i c a t e s t h a t d i s c o n t i n u i t i e s , whether  i n t e n t i o n a l o r u n i n t e n t i o n a l , o c c u r i n e v i t a b l y i n s u r f a c e waveguide systems and thus need t o be i n v e s t i g a t e d c a r e f u l l y .  Furthermore, there  i s a need f o r l o w - l o s s waveguides f o r t r a n s m i s s i o n systems a t t h e h i g h e r microwave and m i l l i m e t e r wave f r e q u e n c i e s .  C e r t a i n s u r f a c e waveguides  have been shown t o have s u p e r i o r a t t e n u a t i o n and d i s p e r s i o n c h a r a c t e r i s tics be  a t these f r e q u e n c i e s .  P r a c t i c a l s y s t e m s , however, have y e t t o  developed.  Scope o f t h e P r e s e n t Work There a r e two a s p e c t s The  f i r s t was m o t i v a t e d  t o t h e work d e s c r i b e d i n t h i s  thesis.  by t h e need f o r t h e a c c u r a t e d e t e r m i n a t i o n o f  the e f f e c t o f d i s c o n t i n u i t i e s on s u r f a c e waveguides. e f f o r t t o develop a p r a c t i c a l low-loss low-cost  The second i s an  s u r f a c e waveguide t h a t  may have p o t e n t i a l a p p l i c a t i o n s as a t r a n s m i s s i o n medium a t t h e h i g h e r microwave and m i l l i m e t e r wave f r e q u e n c i e s .  Thus, t h e t h e s i s i s d i v i d e d  i n t o two p a r t s . P a r t I d e a l s w i t h a n a l y t i c a l and e x p e r i m e n t a l  determination  o f t h e s c a t t e r i n g parameters o f a b r u p t s u r f a c e waveguide d i s c o n t i n u i t i e s . The  t h e o r e t i c a l approach used i n v o l v e s b o u n d i n g t h e open s t r u c t u r e by  p e r f e c t conductors. techniques.  The bounded p r o b l e m i s t h e n s o l v e d by c o n v e n t i o n a l  T h i s p r o c e d u r e has t h e advantage o f a l l o w i n g cascaded  d i s c o n t i n u i t i e s t o be h a n d l e d by a wave m a t r i x  formulation.  P a r t I I i s concerned w i t h t h e p r o p a g a t i o n  on c y l i n d r i c a l  s u r f a c e waveguides w i t h m o d i f i e d c i r c u l a r c r o s s - s e c t i o n .  (xviii)  In particular,  the phase c o e f f i c i e n t s of d i e l e c t r i c rods, with a x i a l s l o t s , are determined t h e o r e t i c a l l y and experimentally. of several possible configurations that may  This configuration i s one be s u i t a b l e to implement i n  a p r a c t i c a l surface waveguide system, i n which i t i s reasonably easy to connect d i f f e r e n t parts of the system together and where a w e l l defined p o l a r i z a t i o n i s maintained.  (six)  PART I SURFACE WAVEGUIDE DISCONTINUITIES  2.  Chapter 1 INTRODUCTION  1.1  Background Probably the f i r s t i n v e s t i g a t i o n s of surface waveguide  d i s c o n t i n u i t i e s occurred i n the l a t e 1940's, i n connection with (32) dielectric aerials.  Kiely  made a c r i t i c a l review of the approximate  techniques (available at the time) f o r p r e d i c t i n g the r a d i a t i o n patterns of abruptly terminated d i e l e c t r i c waveguides.  He also gave t h e o r e t i c a l  and experimental r e s u l t s f o r the d i e l e c t r i c rod, tube, and horn antennas. (37) In 1954,  Butterfield  measured the r a d i a t i o n patterns of various  tapered d i e l e c t r i c slab waveguides.  The s c a t t e r i n g of surface waves  Incident upon abruptly terminated d i e l e c t r i c waveguides was also con/og\ (39} sidered by Angulo and Chang f o r the c y l i n d r i c a l case and by Angulo for  the planar case.  In references (38) and (39) v a r i a t i o n a l expres-  sions f o r the terminal impedance of the waveguide and f o r the transfer impedance between surface waves and r a d i a t i o n were derived.  The  r e s u l t i n g i n t e g r a l expressions were evaluated assuming that only the incident mode was  excited at the terminal plane.  In 1959, K a y ^ ^ 4  initi-  ated several investigations of reactance surface d i s c o n t i n u i t i e s using the Wiener-Hopf technique. for  Kay obtained u n i v e r s a l r e l a t i v e power curves  the r e f l e c t i o n , transmission and r a d i a t i o n i n the planar case.  Later,  Breithaupt extended the method to the c y l i n d r i c a l structures involving (41) (42) TM modes and hybrid modes . Experimental r e s u l t s , i n d i c a t i n g (43) reasonable agreement, were presented f o r both cases.  Johansen  also  used the Wiener-Hopf technique to obtain approximate solutions to the case of a surface wave incident upon an abrupt step i n a r e a c t i v e plane  and upon an abruptly terminated thick reactive slab applied a Green's function formulation  (44)  .  (45) Felsen  to obtain r e s u l t s for the r a d i a t i o n  from a surface with a gradual taper i n reactance. The f i r s t analysis given to s c a t t e r i n g by abrupt changes i n thickness of d i e l e c t r i c slab waveguides was provided, i n 1966, and I s h i m a r u ^ ^ .  by Cooley  An i n t e g r a l equation technique, involving the s o l u -  tions to two related problems, was used to obtain r e s u l t s f o r the r e f l e c t i o n , transmission,  and r a d i a t i o n .  In addition, an equivalent  circuit  for the d i s c o n t i n u i t y and some measured r a d i a t i o n patterns were presented. (47) C l a r r i c o a t s and Sharpe  used a mode-matching technique to solve f o r  the f i e l d s scattered by a junction between two d i f f e r e n t planar slabs.  dielectric  The same problem was treated by Hockham and S h a r p e u s i n g an  i n t e g r a l formula.  Marcuse investigated the s c a t t e r i n g by small steps  and tapers on both planar  and c y l i n d r i c a l d i e l e c t r i c waveguides.  Some experimental r e s u l t s f o r the radiated power were given i n the l a t t e r case.  Shevechenko^^ introduced  the concept of a relaxed r a d i a t i o n  condition i n obtaining r e s u l t s f o r slowly varying continuous t r a n s i t i o n s on various planar and c y l i n d r i c a l surface waveguide configurations. (52) Mahmoud and Beal  l a t e r applied t h i s concept i n t r e a t i n g a problem  involving an abrupt change i n d i e l e c t r i c waveguide dimensions and composition. F i r s t order perturbational solutions to coupled mode formul a t i o n s of gradual changes i n waveguide cross-section have also been applied by various workers.  S n y d e r u s e d  t h i s technique to  obtain approximate solutions to propagation problems i n connection with tapered r e t i n a l receptors and tapered o p t i c a l f i b r e s . used a s i m i l a r analysis to study mode conversion  Marcuse  on o p t i c a l waveguides  4.  with random imperfections.  He also performed experiments, using  corru-  gated c y l i n d r i c a l d i e l e c t r i c waveguides, which v e r i f i e d the theory A related problem was  also considered  by S t o l l and Y a r i v ^ ^ 1  using (62)  e s s e n t i a l l y the same coupled mode analysis and by Sakuda and Y a r i v using Floquet  expansions.  Tuan  analyzed small Gaussian shaped d i s -  c o n t i n u i t i e s on d i e l e c t r i c waveguides using a Green's function technique. Rawson^ ^ applied an Induced dipole method to the problems i n v o l v i n g 4  small steps, sinusoidal cross-sectional change, and s i n u s o i d a l meandering of d i e l e c t r i c waveguides.  R u l f ^ " ^ presented yet another i n t e g r a l equa-  tion technique v a l i d f o r small d i s c o n t i n u i t i e s . A recent i n v e s t i g a t i o n of a d i e l e c t r i c rod antenna has been made by Yaghjian and K o r n h a u s e r ^ ^ .  Their analysis involves replacing  the unbounded problem by a bounded one.  Several t h e o r e t i c a l r a d i a t i o n  patterns were presented. 1.2  Scope of Part I The review presented i n the previous section Indicates (a)  that:  Theoretical studies of d i s c o n t i n u i t i e s (employing various  problem formulations)  are generally applicable to s p e c i f i c and somewhat  limited configurations.  With the exception  of some of the reactance  surface d i s c o n t i n u i t y studies, approximate solutions are obtained by neglecting parts of the mode spectrum.  In addition, the solutions  ob-  tained contain the continuous spectrum of r a d i a t i o n f i e l d s , making the i n v e s t i g a t i o n of cascaded d i s c o n t i n u i t i e s v i r t u a l l y (b)  impossible.  Experimental investigations of s c a t t e r i n g by  surface  waveguide d i s c o n t i n u i t i e s have generally been r e s t r i c t e d to measurement of r a d i a t i o n patterns. and transmission  Very l i t t l e work has been done on r e f l e c t i o n  studies.  5.  Thus, the work presented i n Part I of this thesis i s an attempt to provide: (1)  an accurate a n a l y t i c a l method, of general a p p l i c a b i l i t y ,  for determining the scattering parameters.  The method i s s u i t a b l e f o r  transmission matrix studies of cascaded d i s c o n t i n u i t i e s .  The approach  used involves bounding the open surface waveguide with perfect conductors, thereby making the entire mode spectrum d i s c r e t e .  The bounded  problem can then be solved by conventional mode-matching techniques. (2)  experimental measurement of the scattering  parameters.  In order to achieve this goal a s p e c i f i c d i s c o n t i n u i t y configuration was  chosen—that provided by an abrupt change i n d i e l e c t r i c  thickness of a d i e l e c t r i c - c o a t e d conductor.  Since TM modes are easier  to excite experimentally on this waveguide, the i n v e s t i g a t i o n i s r e s t r i c t e d to TM modes.  In p r i n c i p l e , the methods are applicable to TE modes, how-  ever. The basis of the t h e o r e t i c a l treatment i s presented i n Chapter 2.  I n i t i a l l y , the s i m i l a r i t y of the f i e l d s i n the open and the  corresponding closed problem i s examined.  Then, the mode-matching  procedure i s described and solutions are presented f o r several cases considered previously i n the l i t e r a t u r e . Chapter 3 i s concerned with the experimental i n v e s t i g a t i o n . The waveguide and i t s construction, the probe  used f o r f i e l d measurement  and the measured properties of the experimental surface waveguide are discussed.  The measurement of the r e f l e c t i o n c o e f f i c i e n t i s described  i n some d e t a i l and the r e s u l t s f o r a range of steps are compared with t h e o r e t i c a l values obtained using the method described i n Chapter  2.  Part I i s concluded with Chapter 4 which describes the exten-  6.  sion of the t h e o r e t i c a l analysis  to include cascaded d i s c o n t i n u i t i e s .  This i s followed by a s p e c i f i c example which i s investigated and experimentally.  theoretically  Chapter 2 THEORETICAL INVESTIGATION OF A STEP DISCONTINUITY ON A PLANAR SURFACE WAVEGUIDE The main o b j e c t i v e o f t h i s i n v e s t i g a t i o n i s t o d e t e r m i n e t h e s c a t t e r i n g , o f a s u r f a c e wave, by t h e s t e p d i s c o n t i n u i t y shown i n F i g . 2.1(a).  In general,  on a u n i f o r m s e c t i o n o f t h e open s t r u c t u r e ,  there  e x i s t s a d i s c r e t e s p e c t r u m o f s u r f a c e wave modes and a c o n t i n u o u s s p e c trum o f b o t h r a d i a t i o n modes and a t t e n u a t e d r a d i a t i o n modes. t i n u o u s mode s p e c t r a  The c o n -  render the exact s o l u t i o n of the d i s c o n t i n u i t y prob-  lem,  shown i n F i g . 2.1(a), d i f f i c u l t , i f n o t  2.1  An A p p r o x i m a t e S o l u t i o n  impossible.  Technique  C o n s i d e r t h e bounded d i s c o n t i n u i t y p r o b l e m shown i n F i g . 2.1(b) In t h i s configuration they c o n s i s t o f :  t h e modes a r e a l l d i s c r e t e .  a f i n i t e number o f s l o w and f a s t modes and an i n f i n i t e  number o f e v a n e s c e n t modes. the mode s p e c t r a  There i s a d i r e c t c o r r e s p o n d e n c e between  o f t h e unbounded and t h e bounded c o n f i g u r a t i o n s  i n F i g s . 2.1(a) and ( b ) , r e s p e c t i v e l y . i n T a b l e 2.1.  shown  T h i s r e l a t i o n s h i p i s summarized  I n t h e l i m i t , as t h e upper c o n d u c t o r ( i n F i g . 2.1(b)) i s  moved t o i n f i n i t y , identical.  Generally speaking,  t h e mode s p e c t r a  o f t h e two c o n f i g u r a t i o n s  must become  F o r p r a c t i c a l p u r p o s e s , i f t h e bound i s moved o n l y a few  w a v e l e n g t h s from t h e d i e l e c t r i c s u r f a c e ,  t h e s l o w modes i n t h e bounded  s t r u c t u r e become v i r t u a l l y t h e same as t h e s u r f a c e wave modes on t h e unbounded s t r u c t u r e .  The r e l a t i o n s h i p between t h e modes  (particularly  t h a t between s l o w and s u r f a c e wave modes) i n t h e unbounded and bounded waveguides s u g g e s t e d t h e p r e s e n t a p p r o x i m a t e s o l u t i o n t e c h n i q u e .  An  a p p r o x i m a t e s o l u t i o n i s o b t a i n e d by s o l v i n g t h e bounded p r o b l e m , w i t h the bound a s u f f i c i e n t d i s t a n c e  from t h e s u r f a c e ,  and t h e n  associating  8.  X  CT= oo (b) fig.  2.1.  Step d i s c o n t i n u i t y (a) unbounded (b) bounded.  configurations  UNBOUNDED PROPAGATION] CONSTANT ft  MODE  SPECTRUM  MODE  SPECTRUM  SURFACE WAVE  DISCRETE  SLOW  DISCRETE  CONTINUOUS  FAST  DISCRETE  o</3<-k  RADIATION  -J </3  ATTENUATED] RADIATION  0  CO  <jo  Table 2.1  BOUNDED.  \  CONTINUOUS EVANESCENT  DISCRETE  Correspondence between the bounded and the unbounded mode spectra.  10.  the s u r f a c e wave, r a d i a t i o n and a t t e n u a t e d r a d i a t i o n modes w i t h  the slow,  f a s t and e v a n e s c e n t modes, r e s p e c t i v e l y . The s c a t t e r i n g i n t h e bounded c o n f i g u r a t i o n i n a straightforward  can be d e t e r m i n e d ,  manner, u s i n g m o d e - m a t c h i n g ^ ^ ' ^ ^ .  Since  t i n u i t i e s i n v o l v i n g m e t a l l i c edges r e q u i r e s p e c i a l c o n s i d e r a t i o n s o l v e d by m o d e - m a t c h i n g ^ ^  disconwhen  i t appeared n e c e s s a r y , f o r t h e p u r p o s e o f  the p r e s e n t work, t o i n v e s t i g a t e two s e p a r a t e p r o b l e m s : (72) 1.  t h e b e h a v i o u r o f e l e c t r o m a g n e t i c f i e l d s n e a r m u l t i d i e l e c t r i c edges ( t h i s i n v e s t i g a t i o n also included  edges i n v o l v i n g an a n i s o t r o p i c  medium). (73) 2.  t h e e f f e c t o f d i e l e c t r i c edge c o n d i t i o n s  These i n v e s t i g a t i o n s a r e i n c l u d e d 2.2  on mode-matching s o l u t i o n s  i n A p p e n d i x I-A and I-B, r e s p e c t i v e l y .  Modes o f t h e Bounded Waveguide C o n s i d e r t h e u n i f o r m s e c t i o n o f waveguide ( z < 0) shown i n F i g .  2.1(b).  The TM f i e l d components f o r t h i s waveguide a r e g i v e n by: ^. ^ j(cot-8z) . propagation f a c t o r e i s assumed) h = cos p.x y 1 c  (the  J  B_ e  x  F ToT_ o"t r  =  c o s  = -.—= k Y e  z  o o r  for 0 <_ x <_ h  p  i  x  s i n p..x 1  9 1  and = C cos p _ ( x - t ) z c  y  e  x  =  e  z  =  CB FT"  C O S  o o  JC p 0 ~  do  P  2  ( x  , x " B t  )  2  S i n  P2 " B (x  t  )  2 , 2  11.  for t  < x < t . 1 — — B  Where e  = e/e  r  d i e l e c t r i c and where p.^ and p  2  o  i s the r e l a t i v e p e r m i t t i v i t y of the  are the transverse propagation c o e f f i c i e n t s  i n the d i e l e c t r i c and free space regions, respectively; given by:  2.3  The f i e l d s  tangential to the d i e l e c t r i c i n t e r f a c e at x = t ^ must be con-  tinuous , hence -  c  C  °  V l  S  P  c o s  ( t 2  •_ J  " B t  1  )  G  l  _  r 2 P  S l  S i n  " p  l *1  P  2 V B (  t  2  4  )  Thus, the eigenvalue equation, obtained by equating the two expressions for C i n equation 2.4, i s : P tanp t 1  1  1  o  c  r  p  2  tan p ( 2  t l  -t ) B  2  There i s an i n f i n i t e number of solutions to equation 2.5 which, i n conjunction with equation 2.3, y i e l d values of the modal propagation c o e f f i cient 8.  The mode type (slow, f a s t or evanescent)  value of B , as shown i n Table 2.1.  i s determined by the  The dominant TM^  (slow) mode corres-  ponds to the dominant TEM mode i n the empty waveguide ( i . e . t ^ = 0) and has no low frequency cutoff.  The number and type of higher order propa-  modes depends on e , t . / X and t - A . This dependence i s best r j. o Jo o seen by f i r s t considering the empty waveguide ( t ^ = 0) with conductor gating  spacing given by t .  For this waveguide the modes consist of the domin-  B ant TEM mode (3 = k "), a f i n i t e number, M, of higher order fast modes o (0 < 3 < k^) and an i n f i n i t e number of evanescent modes ( - j  00  < 3 <  0).  As t ^ increases from a value of zero, the propagation c o e f f i c i e n t s i n crease and, hence, the TEM mode becomes the dominant TM  slow mode.  At  5  12.  certain t r a n s i t i o n values of t ^ , denoted t ^ , higher order f a s t modes pass i n t o the slow mode spectrum while at other t r a n s i t i o n values o f t ^ , denoted t ^ > evanescent modes pass into the f a s t mode spectrum. s  f  t r a n s i t i o n values, t ^ and t ^ , can be found by p l a c i n g 8 = respectively, t  f  1 K  The  and 8 = 0 ,  i n equation 2.5. Hence, k X. =  ° 2v T^l r  k = 1, 2, ...  /  2.6  and ST tan k < t * - O r o L% a  tan k ^ " t . f = o r 18.  .1-1,2,...  2.7  Thus, f o r a p a r t i c u l a r choice of the parameters e , t- and t ^ , the t o t a l r i b number of slow modes i s given by: N  = 1 + K s and the t o t a l number of fast modes i s given by: N  f  2.8  = M - K + L  2.9 s  f  where K and L are the number of the t r a n s i t i o n values t ^ and t ^ , r e s pectively,  i n the range of 0 to t ^ . I t may be worthwhile,  at t h i s point, to examine the r e l a t i o n -  ship between the slow modes of the bounded waveguide and the surface wave modes of the corresponding unbounded waveguide. 2 cay i n the region t- < x < t from equations 2.2  and hence p„ = j y ( Y 0  The slow modes de2 2 =  0  3  ~ k  ) . Thus,  and 2.4, the mode functions i n the region t ^ _< x <_ tg  become (they are the same f o r 0 < x < t ^ ) :  cos p^ t ^ : -, r h = y cosh Y ( - B T  2  T  1  )  cosh Y o ( t ) x -  T >  13.  B cos  ^ e = -—r r-——v x k Y cosh Y->(ti-t„) O O Z 1 D  c  o  s  Y ( i>)  n  x_t  0 2  —JYo Pi t-i e = r - ^ . „ , z k Y cosh Y o ( t , - O O O Z ± D c  o  B  s  sinh Y ( x - t ) '2 B' 0  B  2.10  The eigenvalue equation also changes t o : tan p By comparison  t  x  = -e  1  Y  tanh Y (  2  2  t 1  -  t B  )  2  t  l  1  the mode functions f o r the corresponding surface waves  (denoted by primed quantities) are given b y ^ " ^ h ' = cos p,' x y 1 e  e  B'  i  x  z  t  = -— cos p.* x k.Y e l o o r r  = — s i n p.' x 1 o o r  '  2.12  k Y £  for 0 ^_ x <_ t ^ , and -Y '(x- ) h ' =• cos p.'t e y 1 1 2  B* cos p ^ t e  tl  ^'(x-t^  x  k Y  x  o o  - j Y ' cos  t  2  e  z  kY  2.13  o o  for x j> t ^ . P l  -y^ix-tj  ±  The corresponding eigenvalue equation i s given by '  tan ' t P l  x  = e  2.14  Y '  r  2  where  (Pi') " 2  (Y ') 2  2  -  e  r o k  2  ( 3  '  (3') - k 2  ) 2  2.15  2 Q  I t can be e a s i l y shown that, as  00  , equations 2.1, 2.10 and 2.11  14.  tend to equations 2.12, 2.13 and 2.14, respectively. poses, however, t  need only be a few  For p r a c t i c a l pur-  wavelengths before the slow  modes assume v i r t u a l l y the same form as the corresponding surface wave modes. sidered.  In order to i l l u s t r a t e t h i s , a s p e c i f i c numerical example i s con= 2.26 and t, = 0.079 l...  Let e r  1  The normalized propagation  o  c o e f f i c i e n t 8/k of the TM slow mode i s plotted versus t^/X i n F i g . 2.2. o o " o The value of 3'/k (dashed l i n e ) i s also shown on the same f i g u r e . F i g . o 2.2 indicates that 8/k and 8'/k are the same, to four s i g n i f i c a n t f i g o o ures, i f t_/A > 3.0. This r e s u l t t y p i f i e s the rapid convergence of the B O slow modes to the corresponding surface modes as t ^ i s increased. 2.3  Mode-Matching The  a p p l i c a t i o n of mode-matching to a waveguide d i s c o n t i n u i t y ,  such as that shown i n F i g . 2.1(b), 1.  requires:  expanding the t o t a l f i e l d s on e i t h e r side of the d i s c o n t i n u i t y i n terms of the modes i n the respective  2.  enforcing  uniform waveguides,  the continuity of the tangential f i e l d s i n the plane of  the discontinuity, and 3.  using the orthogonality  relationships between the modes to obtain  a system of l i n e a r equations i n the unknown mode amplitude c o e f f i c i e n t s . For the configuration  of F i g . 2.1(b), the continuity of t o t a l tangential  e l e c t r i c and magnetic f i e l d s i s represented by (for the r*"* mode, i n wave1  guide A, incident on the d i s c o n t i n u i t y ) : CO  (1+a  r  ) e  ~  +  a r  I  i=i  00  a, e .' 1  7 b. e, . j=i • J -*>J  2.16  i^r and a  V  s  «  -  ^ W r  ^  s  *  -  ^ " j f i b j  2  -  1  7  15.  16.  where the c o e f f i c i e n t s { a ^ and {b^} are unknown mode amplitudes and where ( e , h ) and ( e ^ h^) are the transverse mode functions a  a  i n waveguides A and B, respectively.  The orthogonality  (e , h ) x  y  relationships  can be expressed as: /  e .x h , - a d = 6 P. —ai —ak —z x i k ak  a  / * ^bjx h ^ •  u  Q  d = "jA 6.^ 'b£ Pw x "  2  x  ,  1  8  where 6., and 6. are Kronecker deltas and where P , and P, are normallk jl ak hi i z a t i o n constants for the k pectively.  and l  modes i n waveguides A and B, res-  In order to solve equations 2.16 and 2.17,  sions must be truncated.  the f i e l d expan-  In this i n v e s t i g a t i o n an equal number of modes  are considered on either side of the discontinuity ( i . e . gives more rapid convergence for d i s c o n t i n u i t i e s involving d i e l e c t r i c edges —  Appendix I-B)  By applying equations 2.18 to J » (2.16)  x h ^ ' ^ d x  and ^  ^ak  X  *  ( 2  ' Sz  1 7 )  d  X  I t can be shown t h a t ^ ^ 7  N  I =1  rh (e^ x h ^ + e ^ x h y ) 0  b J  • a^x =  2 6  k r  P  a r  2.19  J  and  ^  5  kr  with k = 1, ... N.  _  A  V A ak P  f ^ e 0 ^  X  h ^  • a "*  dx  2.20  The i n t e g r a l s involved i n equations 2.19 and 2.20  are given i n Appendix I-C.  Equation 2.19 represents a system of N l i n -  17. ear equations i n the N unknown mode amplitude c o e f f i c i e n t s b^, j = l , ... N. Solutions of equation 2.19  are then substituted i n equation 2.20  values f o r the mode amplitude c o e f f i c i e n t s i n waveguide A.  to y i e l d  Thus, using  the procedure outlined above, approximations to the scattering c o e f f i c i e n t s of the modes excited at the discontinuity can be obtained.  The  accuracy  of the approximation depends on the number of modes, N, used i n the modematching. I t i s of i n t e r e s t to examine the behaviour of the mode-matching solutions ( f o r the slow modes i n p a r t i c u l a r ) to the d i s c o n t i n u i t y problem, shown i n F i g . 2.1(b), as a function of t_/X .  This can be best achieved  a  = 0.079 A and t„ - 0.555 X r 1 o <£ o mode be incident from waveguide A. For t h i s choice of  by considering an example. and l e t the TM  o = 2.26, t  o  Let e  parameters there i s one slow mode i n waveguide A and two slow modes i n waveguide B.  The r e f l e c t i o n c o e f f i c i e n t a^ and the transmission c o e f f i -  cients b, and b„ are given i n Table 2.2 f o r various values of t^/A . I D O /  Also l i s t e d are the percentage errors i n power conservation and percentage power scattered into the fast modes. Table 2.2 were obtained with N = 50.  total  A l l of the r e s u l t s i n  The s c a t t e r i n g c o e f f i c i e n t s and the  fast mode power converge to two s i g n i f i c a n t figures with t^/X  a  =  3.0.  O  S t r i c t l y speaking, better convergence i n the l e a s t s i g n i f i c a n t d i g i t s could be obtained by increasing N a proportionate amount f o r each i n crease i n t„/A  .  B o value of t /X ). B o B  ( i . e . make the power conservation errors equal at each The increment i n N needed to maintain the same accuracy  was v i r t u a l l y impossible to gauge beforehand, however.  An increase i n N  i s required because there are more propagating modes when t /X  increases.  This may be considered a drawback of the present "bounded" approach. The scattering c o e f f i c i e n t s f o r the corresponding unbounded  SCATTERING COEFFICIENTS a  l  b  l  b  FAST MODE ,. POWER %  2  POWER CONSERVATION ERROR %  1.0  0.1325  /-176.28  0  1.0469  /-0.16°  0.3102  7-178.15°  1.5123  0.0002  2.0  0.1346  /-175.79  0  1.0247  /-0.20°  0.2764  7-176.78°  2.0348  0.0017  3.0  0.1351  7-175.77°  1.0242  7-0.21°  0.2690  7-176.46°  1.9716  0.0098  4.0  0.1349  7-175.77°  1.0243 /-0.22°  0.2674  7-176.35°  2.0277  0.0111  5.0  0.1347  7-175.79°  1.0244  •7-0.23°  0.2665  7-176.30°  2.0744  0.0246  Table 2.2  Convergence of slow mode scattering c o e f f i c i e n t s as a function of t_/X . e - 2.26, t . = 0.079X , t = 0.555X . ° ° 2 o B  r  1  discontinuity problem can be obtained from Table 2.2 by associating the scattering c o e f f i c i e n t s a^, b^ and b^ with the r e f l e c t e d surface wave and the two transmitted surface waves, respectively.  The t o t a l power  radiated can be obtained from the power i n the fast modes. 2.4  Application to Previously Considered Configurations Approximate analyses of several d i e l e c t r i c waveguide step d i s (47 48 49 52)  c o n t i n u i t i e s have been given i n the l i t e r a t u r e  '  '  '  .  In general,  the solutions were obtained by neglecting parts of the continuous mode spectra.  The configurations analyzed i n references (47), (48) and  (49)  were w e l l suited to this approximation since these modes were of secondary importance.  This i s not the case f o r the configuration analyzed i n  reference (52), however.  In this section the application of the present  "bounded" approach i s i l l u s t r a t e d by considering three examples, a l l of ( 76} which have been dealt with i n the l i t e r a t u r e Example 1. Two s l i g h t l y d i f f e r e n t configurations, shown as i n s e r t s A and B i n F i g . 2.3 are considered i n this example.  The r e s u l t s , presented i n  the same figure, are given by the s o l i d and dashed l i n e s , (a) and (b) , (47 48) respectively.  In order to make comparisons,  the o r i g i n a l r e s u l t s  are also given and are denoted by (a') and ( b ) , r e s p e c t i v e l y . T  three d i f f e r e n t sets of curves are shown i n F i g . 2.3.  *  In a l l ,  In each set, the  curves l a b e l l e d (a) were obtained by p l a c i n g the bounds at x = tg =  ±  ^^o  (shown as dashed l i n e s on i n s e r t A) and by choosing an equal number of modes (N = 10) on each side of the discontinuity. l a b e l l e d (b) were obtained with t., = ±6X  S i m i l a r l y , the curves  and N = 20.  The maximum error  o which occurs i n the range ^ / t ^ 1*0,  i s estimated* to be 0.1%  f o r the two sets of curves, (a) and (b), r e s p e c t i v e l y . * by checking convergence as t  and N were increased.  and  1.2%  In both cases  20.  0.01 0.0  F i g . 2.3  i  i  7.0  =  i  :  _i  2.0  i  i _  3.0  N o r m a l i z e d '(i) t r a n s m i t t e d ( i i ) r a d i a t e d ( i i i ) r e f l e c t e d powers, E •/£..•='13.0, E . / E .• = 12.7, E . / E = 2.25, 1 o 2 o 3 o e./e = 2.25* t>/> • = 0'.3, t./X = 3;6". • 4 .o 1 . o 4 o a present r e s u l t s f o r the c o n f i g u r a t i o n i n i n s e r t A a' o r i g i n a l r e s u l t s from r e f e r e n c e (47) b p r e s e n t r e s u l t s f o r the c o n f i g u r a t i o n i n i n s e r t B b' o r i g i n a l r e s u l t s from r e f e r e n c e (48)  21.  power conservation was within 0.005%.  As indicated i n F i g . 2.3 there i s  reasonable agreement between the new r e s u l t s and those i n references (47) and  (48).  Example 2. This example deals with configuration shown as an i n s e r t i n F i g . 2.4.  Although the r e f l e c t e d , transmitted and radiated power have been c a l -  culated, only the radiated power, denoted by the s o l i d l i n e , i s presented (49) i n F i g . 2.4. The o r i g i n a l r e s u l t s  for this configuration, which were  obtained by two d i f f e r e n t methods, are given by the dashed l i n e s .  The  new r e s u l t s were obtained with t„ = ±12A and N = 20 which gave a maximum B o error of 2.0% (occurring i n the range k, t- < 6.0) and power conservation o 1 of 0.01%. I t i s seen, upon comparing the curves presented i n F i g . 2.4, (49) that the approximations made by Marcuse are quite v a l i d over the range &  considered. Example 3. This f i n a l example deals with a d i f f e r e n t type of configuration, shown as an i n s e r t i n Fig.2.5(a).  involves a d i e l e c t r i c waveguide which  has an outer layer of d i e l e c t r i c with greater p e r m i t t i v i t y than that of the inner layer.  The r e s u l t s for t h i s configuration are shown i n F i g s . (52) 2.5(a) and 2.5(b) by the curves l a b e l l e d (a). The o r i g i n a l r e s u l t s are shown as curves l a b e l l e d (b).  Curves (a) were obtained with t =  ±5A , and N = 50, which gave both the estimated maximum error and the power o conservation  to within 1.0%.  I t i s seen that the new r e s u l t s are appreciably d i f f e r e n t from the o r i g i n a l ones; however, i n performing the convergence tests i t was found that the number of modes, N, had considerable e f f e c t on the accuracy of the solutions.  This i s i l l u s t r a t e d i n F i g . 2.5(a) by the dashed curves  22.  23.  F i g 7 2.5(a).  Magnitude of the r e f l e c t i o n c o e f f i c i e n t versus VV  .a  a' a" b  l o ' ' 2 o ' ' V o °' present r e s u l t s , N = 50 N.~= 20 N = 10 o r i g i n a l results from reference (52). e  / e  =  9  0  e  / e  =  1 0  0  X  =  0 6  '  -  100  A  F i g . 2.5(b).  .2  B  -4 C  -6 D  EA  F  LO  1.2  Percentage power i n the r a d i a t i o n and the transmitted surface-wave modes. Curves "a" and "b" as i n F i g . 2.5(a).  25. (a') and (a") f o r which N = 20 and N = 10, respectively.  A  remarkable  s i m i l a r i t y exists between positions of the maxima and the minima i n curve (b) and these i n curves (a') and (a"). Since, i n the bounded configuration, there are 11 r e f l e c t e d and as many as 15 transmitted propagating modes, over the range of t . j A  0  given, these curves suggest that an i n -  s u f f i c i e n t number of terms i n the expansion f o r the continuous spectra was  considered i n reference (52). The o s c i l l a t i o n s i n the curves (a) shown i n F i g . 2.5(b) may  explained as follows.  be  As t ^ A g i s increased, t r a n s i t i o n points ( l a b e l l e d  A, B, C, D, E and F i n F i g . 2.5(b)) are encountered at which a fast mode moves into the slow mode spectrum described i n section 2.2).  ( i . e . i n a s i m i l a r manner to the t ^  Therefore, the fast mode power i s decreased  and slow mode power i s increased.  Thus, corresponding peaks and v a l l e y s  are observed i n the curves (a) for values of t^/X^ near these points. This e f f e c t i s l e s s noticeable at larger values of t^/X  o  since the modes  passing through the t r a n s i t i o n s are of higher order and hence contain less power. 2.5  Summary In t h i s chapter an approximate method f o r obtaining solutions  for the scattering caused by a step discontinuity on a planar surface waveguide has been discussed.  The method involves solving a corresponding  bounded s c a t t e r i n g problem and examining the solutions as the bound i s moved away from the guiding surface.  The configuration analyzed, consisted  of an abrupt change i n thickness of the d i e l e c t r i c on a d i e l e c t r i c coated conductor.  Only the solutions f o r TM modes were considered. The main points covered were:  1.  The basis of the approximate method:  the d i r e c t  correspondence  26.  between the mode spectra i n the unbounded and corresponding bounded waveguides. 2.  The modes of the bounded waveguide:  a numerical example indicated  that the bound need be removed only a few wavelengths before the slow modes become v i r t u a l l y the same as the corresponding surface wave modes. 3.  Mode-matching equations f o r the bounded d i s c o n t i n u i t y problem: a numerical example indicated that the slow mode amplitude c o e f f i c i e n t s converge quite r a p i d l y as the bound i s removed.  4.  Application to previously considered  configurations:  the present  "bounded" approach was t e s t e d ^ ^ on four configurations f o r which 7  approximate solutions were available i n the l i t e r a t u r e . ment was obtained i n a l l cases except that considered (52).  Good agree-  i n reference  However, the present r e s u l t s i n d i c a t e that inaccuracies could (52)  e x i s t i n the o r i g i n a l r e s u l t s  Chapter 3 EXPERIMENTAL INVESTIGATION OF STEP DISCONTINUITIES In  this chapter, an  experimental i n v e s t i g a t i o n of the scat-  tering problem depicted i n F i g . 2.1(a) i s described i n d e t a i l .  The two-  dimensional configuration shown i n F i g . 2.1(a) can be r e a l i z e d i n pract i c e reasonably w e l l .  P r a c t i c a l c r i t e r i a on which the d i e l e c t r i c coated  conductor configuration was chosen were ease of e x c i t a t i o n and ease of support. One of the objectives of the experimental study was to v e r i f y the r e s u l t s of the t h e o r e t i c a l method described i n the previous chapter, particularly  since the application of t h i s method to the configuration  considered i n reference (52) yielded r e s u l t s which were s i g n i f i c a n t l y (52) d i f f e r e n t from the o r i g i n a l ones  (this was discussed i n section 2.4).  In order that a wide range of step discontinuity heights ( t ^ -  i n Fig.  2.1(a)) could be measured, the i n v e s t i g a t i o n was carried out at an experimental frequency of 30 GHz.  Commercially  available low-density polyethy-  lene sheet was found to be a suitable d i e l e c t r i c . 3.1  Experimental Waveguide Construction and E x c i t a t i o n A part of the waveguide assembly and of the associated measure-  ment apparatus are shown p i c t o r i a l l y i n F i g . 3.1. The surface waveguide consisted of a smooth f l a t m e t a l l i c surface (3 f t . x 8 f t . ) covered with a sheet of polyethylene 1/32 i n . thick (at a frequency of 30 GHz t h i s thickness allowed only one mode to propagate).  The m e t a l l i c surface was  constructed by coating a sheet of plate glass (3/8 i n . thick) with aluminum f o i l .  Some d i f f i c u l t y was encountered i n making the polyethylene  l i e f l a t on the m e t a l l i c surface.  However, by forcing the a i r out from  beneath the polyethylene and clamping along the sides of the waveguide  an adequate surface was obtained.  Despite these precautions, pockets of  a i r formed occasionally beneath the d i e l e c t r i c and had to be forced out before measurements were made. The surface waveguide was shielded from the surrounding  environ-  ment, p a r t i c u l a r l y the measurement apparatus, by a plywood box covered, on the inside, with absorbing material, as shown i n F i g . 3.1. This box was constructed i n sections and could e a s i l y be removed and be rearranged. Wooden wedges, having t y p i c a l l y 20 db attenuation and l e s s than 5% r e f l e c t i o n , served as absorbers  (placed on top of the polyethylene) around  the edges of the waveguide. A matched load was constructed from s t r i p s of graphite-loaded absorbing material which, when s u i t a b l y arranged  on top of the guiding  surface, yielded a standing wave r a t i o of 1.03 or b e t t e r . ment of these absorbers  The arrange-  can be seen i n F i g . 3.1.  The e x c i t a t i o n of the surface waveguide was achieved i n two stages. 2.75  F i r s t , a small H-plane s e c t o r a l horn (aperture dimensions of  i n . x 0.14 in.) was used to e x c i t e a section of p a r a l l e l p l a t e  waveguide incorporated with one end of the 3 f t . x 8 f t . m e t a l l i c surface.  Second, the p a r a l l e l plate waveguide was used to excite the sur-  face waveguide by extending the polyethylene i n t o t h i s s e c t i o n .  A flared  section was bolted onto the p a r a l l e l plate waveguide and provided a smooth t r a n s i t i o n to the surface waveguide.  The separation between conductors  i n the p a r a l l e l plate section was made equal to 0.14 i n . (to coincide with the narrow dimension of the H-plane horn) using aluminum spacers. The main dimensions of the p a r a l l e l plate region are sketched i n F i g . 3.2. In an attempt to obtain a beam with a wide, plane wavefront two other types of e x c i t a t i o n f o r the p a r a l l e l plate region were also considered:  H-PLANE HORN  PARALLEL PLATE  FLARED  EXTENSION  7  PLATE GLASS F i g . 3.2  POLYETHYLENE  __  7  ALUMINUM FOIL  E x c i t a t i o n of the surface waveguide  SHEET  31. a)  a resonant waveguide s l o t array with 56 s l o t s covering 12 i n . i n the  broad w a l l of a rectangular waveguide normally used i n the 26-40 GHz band of frequencies, and b)  a large H-plane s e c t o r a l horn with aperture dimensions 24.0 i n x 0.14 i n .  These types of e x c i t a t i o n were found unsuitable however, due to higher o r der mode e x c i t a t i o n i n the p a r a l l e l plate region. 3.2  Measurement Apparatus and Properties of the Surface Waveguide In an attempt to provide maximum discrimination against r a d i a t i o n  fields,the longitudinal  component of the e l e c t r i c f i e l d (of the "TM " o  mode) was chosen f o r measurement.  For t h i s purpose a A/2 dipole was used.  I t w i l l be shown i n the next section that the correct values o f the measurement parameters can be obtained using t h i s r e l a t i v e l y large ( i n terms of wavelength) probe.  The dipole was connected to an 18.0 i n .  length of coaxial l i n e which coupled the dipole to a section of waveguide. The d e t a i l s of the dipole are shown i n F i g . 3.3(a).  The coaxial l i n e and  the waveguide detection unit are sketched i n F i g . 3.3(b).  The coaxial  section (and hence the dipole) was held r i g i d by an expanded cone which was attached The polystyrene 1.03,  polystyrene  to an x-y-z micropositioner as shown i n F i g . 3.1.  support, which has a r e l a t i v e p e r m i t t i v i t y of approximately  did not cause any s i g n i f i c a n t disturbance of the f i e l d . The micropositioner was r i g i d l y supported independently of the  surface waveguide.  However, provisions were made to allow the p o s i t i o n e r  to be adjusted r e l a t i v e to the plane of the surface waveguide. was  Thus, i t  possible to have the dipole movement aligned p a r a l l e l to t h i s surface.  The micropositioner was oriented such that i t provided 8 cm o f continuous t r a v e l (0.01 mm accuracy) i n the v e r t i c a l and transverse axis) d i r e c t i o n s .  (to the waveguide  A t o t a l of 22.5 cm of t r a v e l was provided along the  32.  \~0.062-\  0.195 0.011 -  i  \ -  0.005  0.394  © © TEFLON INSULATION © TEFLON SUPPORT ® QUARTER WAVE SLOTS © SL/DJNG BALUN © COPPER WIRE © BRASS  \—0-031 F i g . 3.3(a)  The dipole used f o r f i e l d probing.  4  COUPLINGPROBE\©  CRYSTAL  5 BNC ADAPTER  HI  JE  ur  ur  JTL  0.005 dm. CENTER CONDUCTOR OF THE COAXIAL LINE  3=Q  ffl TUNABLE O.U" SHORT CIRCUIT  FLEXIBLE WAVEGUIDE 0.062  ®  0.265  (diet)'*  =z CO AX AIL LINE DIPOLE  F i g . 3.3(b)  The coaxial l i n e and waveguide detection u n i t . co  34.  waveguide axis but with the r e s t r i c t i o n that, at 2.5  cm i n t e r v a l s (contin-  uous t r a v e l ) , a part of the p o s i t i o n e r had to be unfastened and moved manually. The f i n a l stages of detection were accomplished by feeding the output of the mixer, shown i n F i g . 3.3(b), i n t o a Model 1752 Atlanta  Scientific  Phase/Amplitude receiver. The receiver was phase locked to a  reference s i g n a l which also provided a reference f o r the phase measurements. With the above arrangement, detection could be achieved over a 60 db dynamic range. Atlanta  Monitoring of the phase and amplitude was provided by S c i e n t i f i c Amplitude (Model 1832)  and Phase (Model 1822)  which could read to an accuracy of 0.01  d i g i t a l displays  db and 0.1°, respectively.  A block  diagram of the complete measurement system i s shown i n F i g . 3.4. A preliminary check on the measurement system and on the surface waveguide was obtained by measuring the decay of the surface wave away from the polyethylene surface.  Thus, the amplitude of the wave was measured  as a function of the height above the surface. shown i n F i g . 3.5.  A t y p i c a l decay curve i s  For comparison a t h e o r e t i c a l decay rate was  assuming a frequency of 30 GHz,  a 1/32  calculated  i n . thickness of the polyethylene  and a value of the r e l a t i v e p e r m i t t i v i t y of the polyethylene equal to The r e s u l t i s represented by the dashed l i n e i n F i g . 3.5.  2.26.  Apart from the  s l i g h t o s c i l l a t o r y nature of the experimental p o i n t s , there i s e x c e l l e n t agreement between the measured and the calculated rates of decay.  The os-  c i l l a t i o n s were attributed to the i n t e r a c t i o n of the dipole with the surface waveguide.  By adjusting the balun on the d i p o l e , the magnitude of  these o s c i l l a t i o n s was  minimized.  Before any scattering measurements could be made, i t was sary to determine two other properties  neces-  of the experimental surface wave-  POWER SUPPLY  ISOLATOR I KLYSTRON  E-H TUNER  CALIBRATED ATTENUA TOR  /  —1....  FREQUENCY ME TER  —  SURFACE  J  SLOTTED LINE  MIXER REFERENCE PHASE DISPLAY  AMPLITUDE DISPLAY AMPLITUDE/ RECEIVER F i g . 3.4.  PHASE  Schematic f o r the complete measurement  system.  WAVEGUIDE  36.  o.o] \ -4.0\  -6.01 CO  UJ Q  \  -12.01  o\ o.  a.  o  V  -16.0]  \  o  -20.0]  o.o  4.0  8.0 HEIGHT  F i g . 3.5.  12.0  (mm)  The decay of the surface wave away from the d i e l e c t r i c surface.  37. guide:  the d i r e c t i o n of propagation and the attenuation.  The d i r e c t i o n  of propagation of the wave was determined by measuring i t s amplitude and phase c h a r a c t e r i s t i c s .  The probe movement was adjusted u n t i l the measured  c h a r a c t e r i s t i c s were symmetric.  An example of the amplitude and the phase  c h a r a c t e r i s t i c s i s shown i n F i g . 3.6.  There was, t y p i c a l l y , l e s s than  0.5 db v a r i a t i o n i n the wavefront over a width of three wavelengths (3.0 cm at a frequency o f 30 GHz).  This v a r i a t i o n was comparable or b e t t e r  than that obtained by other w o r k e r s ^ ' ^ who found that beams with s i m i l a r v a r i a t i o n adequately represented the two dimensional case.  For this ex-  perimental waveguide i t was found that the best patterns were measured withi n 60 cm  of the source aperture (edge of the f l a r e d extension shown i n  F i g . 3.2) and, thus, a l l measurements described hereafter were made i n t h i s range. The attenuation was determined by measuring the amplitude of the wave (at a constant height above the polyethylene) at 1 cm i n t e r v a l s over a 22 cm distance along the beam axis.  The results are shown i n F i g . 3.7.  A straight l i n e (dashed) drawn through these points corresponds to an exponential attenuation factor of 0.059 db/cm.  This factor was much higher  than expected by considering the loss tangent of the polyethylene; the approximate t h e o r e t i c a l attenuation factor was estimated to be 0.0021 db/cm. This value was derived f o r the corresponding bounded waveguide configuration (neglecting conductor losses) and assuming a loss tangent (tan 6) f o r polyethylene equal to 0.0005. The d e t a i l s of the t h e o r e t i c a l c a l c u l a t i o n are given i n Appendix I-D. The "apparent" attenuation can, however, be attributed  to the  beam divergence., The amplitude patterns were measured at 2 cm i n t e r v a l s over the same 22 cm distance as the attenuation measurements.  The patterns  38.  DISTANCE F i g . 3.6  (cm)  Amplitude and phase c h a r a c t e r i s t i c s (abscissa i s . transverse distance from the beam a x i s ) .  o  •8  O  -8.01  .o  o  Uj  x  S  o  o  o o o  •701  °yo  0  2  4  8  10  DISTANCE F i g . 3.7  Attenuation measurements (  12  14  16  18  (cm) - exponential attenuation)  20  y  40.  representing the midpoint and the two extremes of this 22 cm range are shown i n F i g . 3.8.  By drawing the "best" curves (dashed p r o f i l e s )  through  the experimental points, an estimate of the divergence was obtained.  The  estimated values of beam width, at the 0.5 db points, are plotted i n F i g . 3.9.  The r e s u l t s indicate that the beam was divergent at an angle of  approximately 5.3° and, thus, the apparent attenuation was estimated to be 0.052 db/cm.  Since there were some o s c i l l a t i o n s i n the attenuation  measurements ( F i g . 3.7) and the beam patterns ( F i g . 3.8) the measurements were repeated at two other frequencies close to the i n v e s t i g a t i o n frequency, namely:  29.94 GHz and 30.06 GHz. The measured values of attenuation at these two frequencies  were 0.054 db/cm and 0.072 db/cm, respectively.  The corresponding c a l -  culated attenuations due to beam divergence (pattern measurements were made at each frequency) were, respectively, 0.057 db/cm and 0.067 db/cm. The measurements were done at three c l o s e l y spaced frequencies i n an attempt to guard against possible spurious frequency dependence i n the r e s u l t s .  The r e s u l t s obtained f o r the three frequencies were  averaged,  thus, the average value of attenuation, which was used i n the l a t e r stages of the experimental i n v e s t i g a t i o n , was 0.062 db/cm.  The average value of  the apparent attenuation due to beam divergence was 0.059 db/cm.  I f the  difference between these two values, i s attributed to attenuation i n the d i e l e c t r i c , i t y i e l d s a value of tan 6 = 0.00077, i n reasonable agreement (79)  with the published 3.3  'range of values o f l o s s tangent f o r polyethylene.  Factors A f f e c t i n g Standing Wave Measurements The main parameter of i n t e r e s t i n the experimental i n v e s t i g a t i o n  was the r e f l e c t i o n c o e f f i c i e n t .  This parameter can be determined by measur-  ing the standing wave pattern and, thus, various factors which affected  °\  v  r  DIRECTION  OF  PROPAGATION  v  1  I  / i o  2.0\  *  \> \-— o  x V  *  V  \  *  BEAM WIDTH (0.5 cto POINTS)/  I  /, —  / .  <0 •CJ  \  \  *\  A  A  V  I  LO oO °  L  X  /  /  /  /  f  A  /  \  v  y  x  /  \  f  /  A  A  ao -4.0  -2.0  0 DISTANCE  F i g . 3.8  2.0 (cm)  Amplitude patterns o - nearer to the source X - midpoint A •-. f a r from the source  4.0  9.0  x  00  40  8.0  12.0 DISTANCE  Fig. 3.9  (cm)  16.0  ^  20.0 ,  Estimated values of beam width versus distance along the beam axis.  ^  to  43.  the standing wave measurements had to be considered.  As discussed pre-  viously, a dipole was used to probe the l o n g i t u d i n a l component of e l e c tric field.  Also, the decay measurements ( F i g . 3.5) indicated that there  was some i n t e r a c t i o n between the dipole and the surface waveguide.  Fur-  thermore, there was a p o s s i b i l i t y of interference between the surface wave and the radiation caused by the discontinuity.  The above consider-  ations posed the following questions regarding the measured value of standing wave r a t i o : a)  what i s the e f f e c t of having the A/2 dipole oriented i n the d i r e c t i o n  of propagation? b)  and,  what e f f e c t s do the i n t e r a c t i o n , between the dipole and the surface,  and the interference between the surface wave and the r a d i a t i o n from the discontinuity have? The f i r s t question has been investigated and i t was found  theor-  e t i c a l l y , not accounting for the f i n i t e s i z e of the coaxial feed, that the A/2 dipole should measure the correct value of standing wave. of  The d e t a i l s  the analysis are included i n Appendix I-E. The second question was investigated experimentally.  Standing  wave patterns, over a 2.5 cm length o f l i n e , were measured (20 cm from the discontinuity) at 1.0 mm i n t e r v a l s above the polyethylene surface. of standing wave r a t i o  Values  between successive maxima and minima were calcu-  lated from each pattern.  These values of standing wave r a t i o are p l o t t e d ,  at regular i n t e r v a l s corresponding to quarter wavelengths, i n F i g . 3.10(a). There are f i v e curves ( l a b e l l e d A, B, C, D, and E) shown i n t h i s f i g u r e . One curve i s shown for each height of the dipole. A part of the corresponding decay curve measured with the step discontinuity i n place i s shown i n F i g . 3.10(b).  The height at which the  44.  C  L.  "A.  D  • • "A  o —  221  .A.  A T  /v V  \  5 CO  5 QUARTER F i g . 3.10(a).  WAVELENGTH  ;o INTERVALS  V a r i a t i o n i n standing wave r a t i o values as a function of the height of the dipole above the d i e l e c t r i c surface.  45.  0.0  A  10  Uj  o B  -4.0  § >0  -8.0  , o  0  4.0 HEIGHT  F i g . 3.10(b).  8.0 (mm)  A decay curve i n d i c a t i n g heights at which the standing waves were measured.  46. various standing waves were measured can be found from the l a b e l l e d points on the decay curve.  The curves i n F i g . 3.10(a) are t y p i c a l of the v a r i -  ation which was observed  ( f o r a l l the step d i s c o n t i n u i t i e s considered) as  the distance between the dipole and the surface waveguide was increased. This behaviour was attributed to two factors:  near the polyethlene sur-  face, the i n t e r a c t i o n of the dipole with the surface appeared to s i g n i f i cantly d i s t o r t the standing wave measurements (curve A)  and f a r from the  surface, there appeared to be s i g n i f i c a n t interference of the surface wave with the radiation from the step (curve E ) . Taking a l l the measurements, on several steps, into consideration i t appeared that a height of 3 to 4 mm above the surface (curve D) offered the best compromise between these two effects.  Thus, the "best" height of the dipole above the surface of the  waveguide was taken to l i e between 3 and 4 mm from the surface i n subsequent standing wave measurements. I t i s i n t e r e s t i n g to note that, whereas the standing wave incurred large o s c i l l a t i o n s at the l a r g e r distances from the surface, the decay curve did not (and should not).  The change of phase  i n the radiation f i e l d , over a v e r t i c a l distance of a few m i l l i m e t e r s , at a distance of 20 cm from the step i s very small and, thus, the interference of the surface wave f i e l d and the radiation f i e l d did not e x h i b i t much o s c i l l a t o r y behaviour i n the v e r t i c a l d i r e c t i o n .  On the other hand, f o r  the standing wave measurements t h i s was obviously not the case. Further investigations of the interference with the r a d i a t i o n from a step d i s c o n t i n u i t y were made by examining the standing wave as a function of the distance from the step.  The dipole was placed at the  "best" height and the values of the maxima and minima of the standing wave were measured over a 22.5 cm distance from the step. of  Two examples  the standing wave r a t i o values obtained i n these measurements are  47.  plotted (at regular i n t e r v a l s corresponding to quarter wavelengths) i n Figs. 3.11(a) and (b).  These plots c l e a r l y indicate the decaying nature  of the standing wave due to the attenuation (the ordinate axis represents the p o s i t i o n of the step).  In addition, the interference of the surface  wave with the radiation from the step appears to d i s t o r t the standing wave measurements within, approximately,  a 10 cm range from the step (the t o t a l  distance covered by the points i s roughly 22.5 cm).  For this reason, a l l  r e f l e c t i o n c o e f f i c i e n t s were calculated from standing wave measurements i n the range 20 cm. to 22.5 cm from the step.  An example of the q u a l i t y  of the standing wave patterns measured i n this range i s shown i n F i g . 3.12. 3.4  Experimental Determination of the R e f l e c t i o n C o e f f i c i e n t and Comparison With Theory Using the general procedures outlined i n the previous section,  standing wave patterns were measured f o r a range of step heights (1/32 i n . , 1/16 i n . , 1/8 i n . , 3/16 i n . , 1/4 i n . , 3/8 i n . , and 1/2 i n . ) .  The steps  were constructed from separate sheets of polyethylene and were placed f l a t on top of the surface waveguide, as shown i n F i g . 3.13.  The f a r end of  these sheets were tapered down to "zero" thickness i n an e f f o r t to eliminate multiple r e f l e c t i o n s within each sheet. c o e f f i c i e n t s f o r each step were determined separate frequencies:  Values of the r e f l e c t i o n  from measurements at the three  29.94 GHz, 30.00 GHz and 30.06 GHz f o r the same  reason mentioned i n section 3.2.  Since the standing waves were measured  i n the range of 20.0 cm to 22.5 cm from the steps a correction was made to account f o r the e f f e c t s of attenuation.  The attenuation causes the value  of the standing wave r a t i o , s, to decrease with increasing distance from the step.  In f a c t , a l i n e a r relationship i s obtained by p l o t t i n g s ( i n db)  against the distance, d, from the step.  The slope depends on:  the value  °°o°  3.0  00 k o  10  00^ o  2.  o o" 0  o  0  o ° o o  s  o o o  0  s. -  I  0  0  . ° o  Q  0  > o  o  o  °o  o  ft0o  o oo°°  w<J  o o  00  o o  o o  °o  o  2.0  O  c> *««  CD  .0  20  40 QUARTER  F i g . 3.11(a)  WAVELENGTH  60  80  INTERVALS  Example of v a r i a t i o n i n the standing wave r a t i o values as a function of the distance from the step discontinuity ( t h e o r e t i c a l dependence of standing wave r a t i o ) .  00  o o  2.5  10  O §  ho  UJ  1 o  CD  o o o  ;.5  o  ^ o °o  o  s  o  o —  w  o « o O  to  ^. o° — o ° o  o  o  o o  © o  o  0 ° 0 o  o  °o  °-  o  r>^P o  0  o ^ ° ° ^ c ° o  o  0  20  °  40 QUARTER  F i g . 3.11(b)  WAVELENGTH  60  o  80  INTERVALS  Same as (a) but for a d i f f e r e n t step height!  o  o  o  n  o  oo  Oo  aoi  J  .4.0  2.0  0  I  6.0 DISTANCE  JL_:  1_  I  L.  10.0  8.0  12.0  (mm)  01  oo  o ° °o  Uj  5: 1~0\  o  00\ 12.0  1  1-  14.0  o  ' 16.0 DISTANCE  F i g . 3.12  o  o  o  ° o 0  J  I  18.0  I  20.0  J  L  22.0  (continued)  An example of a measured standing wave pattern.  I  24.0  ±  13 in STEP  DISCONTINUITY  32m TAPERED  SECTION  POLYETHYLENE LAYER  /3 PLATE  F i g . 3.13.  GLASS  ~  ALUMINUM FOIL  Experimental step d i s c o n t i n u i t y configuration.  GRAPHITE LOADED ABSORBER  52. of the magnitude of the r e f l e c t i o n c o e f f i c i e n t at the step, p, and on the attenuation c o e f f i c i e n t , a.  The r e l a t i o n between s, p, a and d i s w e l l  known and given by^ ~^ : 7  i  =  8  _i_  -2ad  1 P * , -2ad 1 - p e  3.1  +  Equation 3.1 was  used to calculate the value of p from the standing wave  measurements as follows: a)  an average value of the magnitude of the r e f l e c t i o n c o e f f i c i e n t s , p,  was  determined  to 22.5 b)  from the standing wave measurements i n the range of 20.0  cm  cm from the step.  p and a = 0.00713 np/cm (0.062 db/cm) were used i n equation 3.1 to  calculate the slope of the l i n e a r dependence of s ( i n db) on d. c)  the measured values of the standing wave r a t i o were plotted, as i n  F i g . 3.11,  and the t h e o r e t i c a l l i n e (dashed i n F i g . 3.11)  such that i t passed through the points i n the range 20.0  was drawn cm to 22.5  cm  from the step, and d)  the intercept of this l i n e with the ordinate (location of the step)  determined  the value of the standing wave r a t i o at the step and, hence,  P. The values of p determined by the above procedure are plotted i n F i g .  3.14.  An experimental point i s shown f o r each, step and f o r each of the three frequencies.  The t h e o r e t i c a l dependence of p ( s o l i d curve) i s also shown  for comparison.  The t h e o r e t i c a l results were obtained, by the approximate  method described i n Chapter 2, using a frequency of 30 GHz of the parameters e , t , and N equal to 2.26, r a  5X , and 50,  and using values respectively.  o  The t h e o r e t i c a l curves f o r frequencies of 29.94 GHz and 30.06 GHz  are  n e g l i g i b l y d i f f e r e n t than that f o r 30 GHz and, thus, are not shown.  The  (t -tj)A 2  F i g . 3.14.  0  Experimental results f o r the magnitude of the reflection coefficient" X - 29.94 GHz O - 30.00 GHz A - 30.06 GHz - theory (30.00 GHz).  differences between the experimental values at the three frequencies are attributed to small r e f l e c t i o n s from the tapered sections.  Some d i f f i c u l -  ty was, experienced i n getting the tapers to l i e f l a t on the surface waveguide (due to stresses, within the polyethylene, possibly caused by machining  the tapers) and also, there must be some r e f l e c t i o n , however small,  due to these tapers. The phase angle,  6, of the r e f l e c t i o n c o e f f i c i e n t can be  de-  termined from the r e l a t i o n ^ " ^ : 7  3.2  e = ^ g where d i s the distance between the step and the location of the f i r s t minimum i n the standing wave pattern and where X length.  i s the waveguide wave-  Thus, the following measurements, at a frequency of 30.0  GHz,  were made f o r each step: a)  the positions of the minima i n the standing wave i n the range  to 22.5 b)  20.0  cm from the step, and  the distance, d, between each of the above minima and the step.  The measurements i n (a) above were used to calculate an average value of the waveguide wavelength f o r the standing waves caused by each step. These values varied between a maximum of 9.651 mm.  mm  and a minimum of 9.548  Since the frequency was held constant, this v a r i a t i o n was  unreasonable  and was  considered  attributed to s l i g h t asymmetries i n the pattern.  Therefore, the average values of X  calculated f o r the i n d i v i d u a l steps  were i n turn averaged, y i e l d i n g a value of 9.608 mm. better agreement with the t h e o r e t i c a l value of 9.610  This value was i n mm.  The average  value of X^ (9.608 cm) was used i n conjunction with the measurements i n (b) above to obtain the p o s i t i o n of the f i r s t minimum i n the standing  55. waves corresponding to each minimum i n the measurements (a).  Equation  was  The r e s u l t s  then used to obtain an average value of 6 f o r each step.  are shown i n Table 3.1.  For comparison,  3.2  t h e o r e t i c a l r e s u l t s which were  calculated using the same parameters as those used to obtain the s o l i d curve i n F i g . 3.14  are also shown.  The difference between the t h e o r e t i c a l  and the experimental values shown i n Table 3.1 i s not viewed as being that great considering that the coaxial feed to the dipole had an outside diameter of 0.021  i n . This diameter corresponds to 0.11  A /2 (40° i n phase)  at a frequency of operation equal to 30.00 GHz. The r e s u l t s shown i n F i g . 3.14  and Table 3.1 indicate that, i n  general, there i s good agreement between the experimental and the theore t i c a l values of the r e f l e c t i o n c o e f f i c i e n t .  These r e s u l t s i n s p i r e con-  siderable confidence i n the "bounded" approach to solving the d i s c o n t i n u i t y problem and i n the experimental techniques used to measure the scattering from such d i s c o n t i n u i t i e s . 3.5  Summary In t h i s chapter an experimental i n v e s t i g a t i o n of step d i s c o n t i n -  uties on a d i e l e c t r i c coated conductor surface waveguide was discussed. Three closely spaced frequencies, 29.94 GHz,  30.0 GHz,  and 30.06 GHz were  used to guard against possible spurious dependence of the r e s u l t s on f r e quency.  • The main points were . :  1.  The construction and e x c i t a t i o n of the experimental waveguide,  2.  The use of a A/2 dipole (oriented i n the d i r e c t i o n of propagation) as a probe,  3.  Properties of the surface wave on the experimental waveguide: decay of the surface wave away from the polyethylene was  the  found to  STEP HEIGHTS (In.)  1/32  1/16  1/8  3/16  1/4  3/8  1/2  EXPERIMENT  177.6  206.2  181.7  172.0  189.1  179.6  191.4  THEORY  189.7  188.5  186.2  184.2  182.8  182.2  182.4  Table 3.1  Experimental and t h e o r e t i c a l values of the phase angle of the r e f l e c t i o n c o e f f i c i e n t for a range of step discontinuity heights.  57.  be i n e x c e l l e n t agreement with theory; a high value of attenuation was measured and explained i n terms o f beam divergence, 4.  Factors a f f e c t i n g standing wave measurements:  two factors were con-  s i d e r e d — i n t e r a c t i o n of the dipole with the surface waveguide and interference of the surface wave with radiation from the step; these factors determined  the best location f o r the standing wave measure-  ments. 5.  Determination of the r e f l e c t i o n c o e f f i c i e n t : waveguide was accounted  the attenuation of the  f o r i n the measurements of the magnitude of  the r e f l e c t i o n c o e f f i c i e n t , 6.  Comparison with t h e o r e t i c a l r e s u l t s f o r step d i s c o n t i n u i t i e s :  good  agreement between theory and experiment was obtained f o r the magnitude and the phase of the r e f l e c t i o n c o e f f i c i e n t f o r a wide range of step heights.  58.  Chapter 4 APPLICATION TO CASCADED DISCONTINUITIES In t h i s chapter the t h e o r e t i c a l and experimental techniques presented i n Chapters 2 and 3, respectively, are applied to the d i s c o n t i n u i t y problem shown i n F i g . 4.1(a). bounded configuration, rices.  I n i t i a l l y , the solution to the corresponding  shown i n F i g . 4.1(b), i s obtained using wave mat-  This s o l u t i o n i s then examined as the bound i s moved away from the  o r i g i n a l guiding  surface.  For comparison,  experimental r e s u l t s f o r the  magnitude of the r e f l e c t i o n c o e f f i c i e n t of the dominant mode incident on one such discontinuity are presented. 4.1  Wave Matrix Formulation Wave amplitude matrices  r e l a t e the wave (mode) amplitudes on  one side of a discontinuity to those on the other side. these matrices are useful f o r analysing  For t h i s reason  cascaded d i s c o n t i n u i t i e s ;  the wave  matrices f o r the i n d i v i d u a l d i s c o n t i n u i t i e s can be m u l t i p l i e d together ( i n the same order as the d i s c o n t i n u i t i e s ) to y i e l d an o v e r a l l wave matrix f o r the cascaded sections.  Thus, the same number of modes on either side of a  discontinuity must be considered.  Normally, only propagating modes are used.  I f , however, evanescent mode i n t e r a c t i o n between successive d i s c o n t i n u i t i e s i s important, then the wave matrices can be extended to include modes as w e l l , ^ ^  evanescent  The configuration shown i n F i g . 4.1(b) represents an  i n t e r e s t i n g s i t u a t i o n regarding the a p p l i c a t i o n of these matrices;  dif-  ferent numbers of propagating modes may e x i s t i n waveguides A and B. In such cases, the wave matrix must include a s u f f i c i e n t number of evanescent modes (even though the Interaction of these modes may not be important) to render the number of modes considered on either side of the d i s c o n t i n u i t i e s  z=z,  -  z - z  2  (b) F i g . 4.1  Cascaded step d i s c o n t i n u i t y configurations (a) unbounded (b) bounded •  60.  equal. The application of wave matrices to the problem shown i n F i g , 4.1(b) i s as follows.  Wave matrices, w^ and w,,, are defined  f o r each of the two  step d i s c o n t i n u i t i e s at planes z = z^ and z = z , respectively. 2  A wave  matrix w^, i s also defined  f o r the length of uniform waveguide, B, between  the two d i s c o n t i n u i t i e s .  Let c^ and b^ be vectors representing  l i z e d amplitudes  the norma-  of modes (each element corresponding to a mode) propa-  gating i n the p o s i t i v e and negative z directions, respectively, i n waveguide A to the l e f t of the discontinuity.  Let c  2  and b  waveguide A to the r i g h t of the d i s c o n t i n u i t y .  V  =  W  be s i m i l a r l y defined i n  2  Then  V 4.1  b„ -2  b, -1  where W i s the o v e r a l l wave matrix for the cascaded d i s c o n t i n u i t i e s and i s given by: W  =  W  lT 2 W  W  In general, i f N modes are considered on either side of the d i s c o n t i n u i t y then  4.3 normalized such that the power contained i n the i -ii  1  t  h  mode i s given by  61.  where 8^, i = 1  N are the propagation c o e f f i c i e n t s of the modes con-  sidered i n waveguide B and where I i s the distance between the two steps, as shown i n F i g . 4.1(b). The mode-matching solutions, discussed i n Chapter 2, define e l e ments i n a scattering matrix.  The scattering matrix f o r a d i s c o n t i n u i t y  can be used to obtain the elements of the corresponding wave matrix. the configuration considered here, w^ and scattering matrix.  For  can be derived from the same  This derivation i s given i n Appendix I-F.  To i l l u s t r a t e the application of the above procedure and to i l l u s t r a t e the convergence of the solutions f o r the bounded configuration, a s p e c i f i c numerical example i s considered. ^A = q  12.6 and  = 2.26.  Let  t]_A  = 0  0.079,  ^ A Q  0.555,  =  For t h i s choice of parameters there i s one slow  mode propagating i n waveguide A and two slow modes propagating i n waveguide B.  The large value of &A  Q  i s chosen to s i m p l i f y the analysis since i t  eliminates the p o s s i b i l i t y of s i g n i f i c a n t evanescent mode i n t e r a c t i o n . scattering of the T M  q  The  mode incident upon the d i s c o n t i n u i t y from waveguide A,  as shown In F i g . 4.1(b), i s examined as a function of mode c o e f f i c i e n t s , b ^ and c^^, of the T M  q  tgA . Q  Values f o r the  modes i n waveguides A to the l e f t  and r i g h t of the d i s c o n t i n u i t y , respectively, are presented i n Table 4.1. They are obtained by setting b  = 0 (wave incident only from z < 0 ) and  ?  1 0 Si L  o  J  4.4 i n equation 4.1.  SCATTERING COEFFICIENTS b  11  C  CONSERVATION ERROR %  21  0.87064  /-98.82°  0.0004  0.94901  /-91.62°  0.0033  /176.23°  0.92544  /-90.46°  0.0162  0.27551  /179.01  0.93598  /-91,12°  0.0218  5.0  0.27617  /177.77°  0.93719  /-90.93  0  0.0480  6.0  0.26996  /179.17°  0.94149  /-90.81  0  0.1058  7.0  0.27051  /179,37°  0.94123  /-91.02°  0.1593  8.0  0.27091  7177.95°  0.94031  /-90.82°  0.1171  1.0  0.26327  /176.33  2.0  0.26850  /179.56  3.0  0.27070  4.0  T a b l e 4.1  0  0  0  Convergence o f s l o w mode s c a t t e r i n g c o e f f i c i e n t s ( o b t a i n e d b y wave m a t r i c e s ) as a f u n c t i o n o f t A  63.  Also given i n Table 4.1 are values of the percentage error i n power conservation at each value of t„/X . The r e s u l t s i n d i c a t e that the mode D O  c o e f f i c i e n t s converge as t /X i s increased. 15 O  Again convergence should be  examined on the basis of equality i n power conservation errors, as discussed i n Chapter 2.  Since t h i s i s not the case f o r the r e s u l t s shown, convergence  of the c o e f f i c i e n t s i s obtained only to two s i g n i f i c a n t f i g u r e s . 4.2  Experimental Investigation To v e r i f y the r e s u l t s obtained by the method presented i n the  previous section, the magnitude of the r e f l e c t i o n c o e f f i c i e n t of the dominant mode incident on a step d i s c o n t i n u i t y , such as the one shown i n F i g . 4.1(a), was measured using the procedures outlined i n Chapter 3.  The step  d i s c o n t i n u i t y consisted of a slab of polyethylene 0.476 cm thick and of length & = 12.62 cm placed on top of the basic polyethylene guiding layer of the experimental waveguide described i n Chapter 3 (the slab extended the f u l l width of the surface waveguide).  At a frequency of 30.0 GHz t h i s  configuration corresponds to that of the numerical example considered i n section 4.1. The i n t e r a c t i o n of the two step d i s c o n t i n u i t i e s causes the r e f l e c t i o n c o e f f i c i e n t to exhibit an o s c i l l a t o r y dependence on SL.  This  v a r i a t i o n could be observed by changing the p h y s i c a l length, I, of the slab or by changing the frequency. l e r and more accurate.  Changing the frequency was r e l a t i v e l y simp-  Thus, the frequency was v a r i e d from 30.0 GHz to  30.8 GHz i n steps of 0.1 GHz. Approximately  one cycle of the v a r i a t i o n i n  the r e f l e c t i o n c o e f f i c i e n t was observed over the above frequency range. The experimental values of the magnitude of the r e f l e c t i o n c o e f f i c i e n t are plotted versus Z/\ i n F i g . 4.2.  65.  4.3  Comparison with Theory Before any comparison with theory could be made, the e f f e c t s of  attenuation  ( i n waveguide B) should be included i n the wave matrix formula i This i s done by including attenuation f a c t o r s , e , i n the wave  lation.  where o i . i s an estimated value of the attenuation c o e f f i c i e n t of x  matrix w ; T m  the i  t  b  mode.  Two  separate contributions to each  were considered:  1.  that due to bulk material loss (denoted by  2.  that due to beam divergence (denoted by  a^)  a^)  In an attempt to model the unbounded configuration, these two attenuation effects were only considered f o r the slow modes i n the corresponding bounded configuration ( i . e . the f a s t modes to the r a d i a t i o n modes  i n the bounded structure  correspond  i n the unbounded structure which are not substan-  t i a l l y affected by these attenuations). Using the analysis given i n Appendix I-D and assuming a value of tan 6 = 0.00077, the values of  f o r the  two slow modes (of waveguide B shown i n F i g . 4.1(b)) were calculated to be 0.00368np/cm and 0.00210np/cm, respectively. calculations was  equal to 5.0X^. o  The value of t,, used i n these  Since the material attenuation i s con-  sidered separately here, the value of  was assumed to be 0.00675np/cm (as  discussed i n Chapter 3). The t h e o r e t i c a l v a r i a t i o n i n the magnitude of the r e f l e c t i o n coe f f i c i e n t (with the above attenuation e f f e c t s included) i s given by the s o l i d curve i n F i g . 4.2.  For reference purposes, the t h e o r e t i c a l curve  (dashed) corresponding to a frequency of 30.8 GHz i s also shown.  These  curves were obtained (at each frequency) by varying the parameter £.  The  following observations can be made of the curves shown i n F i g . 4.2: a)  the "period", AJ2,/X , of the experimental points, A Z / \ q  Q  0.32,  66.  i s shorter than that of either of the t h e o r e t i c a l curves, A £ / X o b)  the experimental points are s h i f t e d  by approximately, 0.025 i n terms of £ A , Q  c)  0.36.  (to the l e f t i n F i g . 4.2)  and  the amplitude of experimental o s c i l l a t i o n i s lower than  predicted by the theory. The observation (a) i s due to the d i f f e r e n t frequencies used to determine each experimental point ( i . e . the experimental point at &A  Q  = 12.62  should l i e on the s o l i d curve corresponding to a frequency of  30 GHz, whereas, the experimental point at £ / X = 12.955 should l i e on the o dashed curve corresponding to a frequency of 30.8 GHz). The observation (b) could be accounted f o r by an increase i n the r e l a t i v e p e r m i t t i v i t y , e , of less than 0.5%. r  Such a small increase changes  the values of the propagation c o e f f i c i e n t s i n waveguide B only s l i g h t l y ; however, since &A  Q  i s large, these small changes have considerable e f f e c t .  The observation (c) i s attributed mainly to the value used for otp since i t was an estimate based on the measured properties of the basic experimental surface waveguide.  S t r i c t l y speaking, i n order that t h i s value  of dp be a p p l i c a b l e to the thicker section of waveguide (waveguide B) r e quires that the slab be perfect ( i . e . constant length £ over the f u l l width of the experimental waveguide, interfaces perpendicular to the d i r e c t i o n of propagation and interfaces perpendicular to the guiding surface). not seem unreasonable  I t does  to suggest that, i n p r a c t i c e , there must be some v a r i a -  t i o n i n the above f a c t o r s .  Since there were two surface modes propagating  on the slab section, attenuation measurements on t h i s section would have been d i f f i c u l t .  67.  4.4  Summary In t h i s chapter an i n v e s t i g a t i o n of the scattering from two cas-  caded step d i s c o n t i n u i t i e s was discussed.  The "bounded" approach, outlined  i n Chapter 2, was used i n conjunction with the wave matrix formulation to obtain values of the scattering c o e f f i c i e n t s .  The magnitude of the r e f l e c -  t i o n c o e f f i c i e n t f o r one s p e c i f i c discontinuity configuration was measured, using the techniques discussed i n Chapter 3. The main points were: 1.  the wave matrix formulation of the cascaded  discontinuity:  a numerical example indicated that the s c a t t e r i n g c o e f f i c i e n t s obtained using wave matrices converge as the bound i s removed, 2.  experimental i n v e s t i g a t i o n :  an e f f e c t i v e check on the be-  haviour of the r e f l e c t i o n c o e f f i c i e n t was obtained by changing the e l e c t r i c a l length between the two steps rather than the p h y s i c a l length, 3.  comparison between theory and experiment:  Reasonable agree-  ment was obtained between the experimental and the t h e o r e t i c a l r e s u l t s when the l a t t e r included the e f f e c t s of estimated values of attenuation.  PART I I CYLINDRICAL DIELECTRIC WAVEGUIDES WITH PERTURBED CIRCULAR CROSS-SECTION  Chapter 5 INTRODUCTION 5.1  Background One  of the f i r s t investigations of c y l i n d r i c a l d i e l e c t r i c  surface waveguides of non-circular cross-section was  presented by Weiss  / o i \  and Gyorgy  v  ' i n 1954.  They discussed the r e l a t i v e merits of the d i e -  l e c t r i c rod and d i e l e c t r i c tube and concluded that the tube possessed better dispersion c h a r a c t e r i s t i c s , whereas, the rod confined the f i e l d s better.  Both of these guides were considered  unsuitable.for p r a c t i c a l  transmission purposes, however, because any imperfection or bend i n the waveguide resulted i n power transfer between the two possible p o l a r i z a tions.  They also presented the r e s u l t s of an experimental i n v e s t i g a t i o n  concerning  attenuation, launching, s h i e l d i n g and guide supports,  d i e l e c t r i c waveguides of rectangular cross-section.  for  Rectangular d i e l e c t r i c  waveguides have also been the subject of several more recent i n v e s t i g a (82) tions.  In 1966  Schlosser and Unger  derived dispersion curves f o r the  f i r s t few low order modes and discussed the mode designations.  Their  method involved placing the d i e l e c t r i c waveguide i n a s u f f i c i e n t l y large rectangular m e t a l l i c waveguide; the r e s u l t i n g cross-section was subdivided subregion.  and point-matching was  applied along the boundaries of each  They also gave experimental r e s u l t s for attenuation ( 83}  dispersion curves.  then  Later, M a r c a t i l i  considered  and  the problem of a  rectangular d i e l e c t r i c waveguide surrounded by several d i f f e r e n t d i e l e c t r i c s of lower r e f r a c t i v e index.  He applied a f i e l d matching technique  over the rectangular core boundary which resulted i n approximate eigenvalue equations f o r the propagation constant.  Dispersion curves and  f i e l d patterns were derived using these equations.  M a r c a t i l i then  extended the method to the case of two rectangular d i e l e c t r i c waveguides i n close proximity, with a view to d i r e c t i o n a l coupler design. also used a point-matching propagation  technique  Goell^^  to c a l c u l a t e approximate values of  constants f o r the rectangular d i e l e c t r i c waveguide. His ( 85)  results compared favourably with the previous i n v e s t i g a t i o n s .  Pregla  used a v a r i a t i o n a l technique s i m i l a r to the Rayleigh-Ritz method and calculated dispersion curves f o r the f i r s t few low order modes on a rectangular d i e l e c t r i c waveguide. The f i r s t comprehensive i n v e s t i g a t i o n of d i e l e c t r i c waveguides (Of.)  of e l l i p t i c cross-section was presented i n 1961, by Yeh  . He derived  dispersion curves and f i e l d patterns for the dominant modes with each of the two possible p o l a r i z a t i o n s . These dispersion curves were v e r i f i e d by experiment.  Later Yeh  Investigated the attenuation properties of  e l l i p t i c a l d i e l e c t r i c waveguides. 5.2  Scope of Part I I The b r i e f review presented above indicates that to date, there  have been few investigations of d i e l e c t r i c waveguides with non-circular cross-section.  Such waveguides are deemed necessary because of two prac-  t i c a l considerations which occur i n connection with d i e l e c t r i c rod and tube waveguides of c i r c u l a r cross-section.  The two problems are:  (1)  lack of a w e l l defined p o l a r i z a t i o n , and  (2)  d i f f i c u l t y i n connecting components together.  The f i r s t consideration i s important because any s l i g h t  ellip-  t i c i t y i n the d i e l e c t r i c rod or tube cross-section causes the crosspolarized modes to propagate with s l i g h t l y d i f f e r e n t phase and group velocities.  Thus, e i t h e r surface imperfections or d i s t o r t i o n s (e.g.  71.  bends) of the waveguide r e s u l t i n mode conversion between these modes and, hence, poorer dispersion c h a r a c t e r i s t i c s are obtained and  difficulties  i n detection are encountered. The second consideration i s one which i s common to most waveguide systems (metallic or d i e l e c t r i c ) ; an e f f i c i e n t and r e l a t i v e l y easy way  of coupling between components i s desired.  however, the problem i s more complicated radiation.  With open structures,  because of the p o s s i b i l i t y of  Also, i n connection with the f i r s t consideration, i f s l i g h t  e l l i p t i c i t i e s occur i n d i e l e c t r i c rods or tubes then alignment between components may  have to be  considered.  Some of the methods envisaged for achieving p r a c t i c a l surface waveguide systems involve the use of f i n s on the waveguide, be i t of c i r c u l a r , e l l i p t i c a l , or rectangular cross-sections, as shown i n F i g . These f i n s would make the p o l a r i z a t i o n more d e f i n i t e and may  5.1.  be u s e f u l  i n interconnection and coupling schemes. For the purposes of the present work, a d i e l e c t r i c waveguide with c i r c u l a r cross-section has been chosen.  The reasons are:  the r e l a -  t i v e s i m p l i c i t y of the analysis and :the a v a i l a b i l i t y of such waveguides commercially.  The waveguide i s then perturbed by a x i a l s l o t s .  configuration was  A slot  chosen f o r this i n v e s t i g a t i o n , rather than a f i n con-  f i g u r a t i o n , again because i t was  easier to construct.  The analysis and  procedures for the i n v e s t i g a t i o n of a finned waveguide would be s i m i l a r to those used f o r the s l o t t e d waveguides, however. In Chapter 6, a perturbation method f o r determining separation i n the propagation caused by two symmetrically presented.  the  c o e f f i c i e n t s of the cross-polarized modes  placed s l o t s (with polar geometry), i s  FINNED DIELECTRIC WAVEGUIDE CROSS - SECTIONS  INTERCONNECTION WITH KEYS F i g . 5.1.  Finned d i e l e c t r i c waveguides - possible interconnection and coupling schemes.  COUPLING  In Chapter 7 an experimental i n v e s t i g a t i o n of the e f f e c t of rectangular s l o t s i s presented.  Although the t h e o r e t i c a l and experi-  mental configurations are only approximately comparable, Chapter 7 i s concluded with a discussion of the trends indicated by the t h e o r e t i c a l and experimental r e s u l t s .  74.  Chapter 6 A PERTURBATION METHOD This chapter i s concerned with the a p p l i c a t i o n of a perturbation method f o r finding the change i n the propagation c o e f f i c i e n t of the dominant H E ^ mode on the d i e l e c t r i c rod (Fig. 6.1(a)), when the rod i s modified by a x i a l s l o t s ( F i g . 6.1(b)).  The approach used here i s the same as that given  (88) by Harrington.  Let E  q  and H  be the dominant mode f i e l d s of the unper-  q  turbed rod ( F i g . 6.1(a)) and l e t E and H be the dominant mode f i e l d s of the (88) perturbed rod (Fig. 6.1(b)). Then, i t can be shown // E-E dS B - B Ae S ~ ~°  - V — o  £  - ? -  *  £  o  *  6  // (E x H + E x H )-a dS -o -o z  g  -  1  w  where B and B are the propagation c o e f f i c i e n t s of the modes on the P o perturbed and unperturbed waveguides, r e s p e c t i v e l y .  Ae  r  i s the change i n  r e l a t i v e p e r m i t t i v i t y within the region of the perturbation, cross-section of the perturbation and section of the waveguide.  i s the  i s the t o t a l ( i n f i n i t e ) cross-  Equation 6.1 i s exact;  i t s derivation i s  given i n Appendix II-A. In order to evaluate the i n t e g r a l s i n equation 6.1, i t i s necessary to approximate the unknown f i e l d s E and H.  In the denominator i t i s (88)  usual to approximate E and H by the unperturbed f i e l d s E  q  and H . q  In  the numerator, however, a more accurate approximation i s normally required. 6.1  Quasi-Static Approximation  to the Perturbed F i e l d s  A useful approximation to the perturbed f i e l d s E can be obtained assuming E to have the same f u n c t i o n a l form as E  q  and by assuming that the  magnitude of each component of E i s m u l t i p l i e d by a correction f a c t o r  75.  (a)  F i g . 6.1.  D i e l e c t r i c rod configurations (a) unperturbed (b) perturbed by s l o t s .  76.  which i s related to the behaviour of s t a t i c f i e l d s i n the region of the (88")  perturbation.  This i s known as a q u a s i - s t a t i c approximation.  q u a s i - s t a t i c correction factors are shown f o r several u s e f u l  The configurations  i n F i g s . 6.2(a), (b), ( c ) , and (d). The configuration of i n t e r e s t i n t h i s investigation ( F i g . 6.2(e)) i s not the same as any of the above.  However,  i n t h i s work an approximate q u a s i - s t a t i c correction i s obtained as follows. Consider, f o r the moment, the r-component of e l e c t r i c f i e l d . Let d = r.. - r and w = (r, + r ) ( 0 , - 9 )/2 where r , r , 6 , , and 9 are de1 o 1 o 1 o 1' o' 1' o n  fined i n F i g . 6.2(e).  The configurations  (and the behaviour of the s t a t i c  f i e l d s ) i n Figs. 6.2(a), (b), and (c) correspond approximately to the conf i g u r a t i o n shown i n F i g . 6.2(e) i f d/w tively.  0, d/w -> <*>, and d/w  1, respec-  This correspondence suggests an exponential dependence of the quasi-  s t a t i c correction f a c t o r , as shown i n F i g . 6.2(f).  Let  be the c o r r e c t i o n  factor f o r the r-component of e l e c t r i c f i e l d , then from F i g . 6.2(f)  K  r  = 1 + (e / 2  - l)e"  a  e i  d  /  6.2  w  where a = l n ( l + e /e' ) 2  6.3  1  The correction factor f o r the 6 component of e l e c t r i c f i e l d , K , may be 9  obtained from equation 6.2 by substituting w/d f o r d/w.  F i g . 6.2(d) i n d i -  cates that no correction i s necessary f o r the z-component of e l e c t r i c  field,  i n t h i s case. 6.2  Application of the Perturbation Method The perturbational analysis and the q u a s i - s t a t i c approximations  discussed  i n section 6.1 are now applied to the configuration shown i n  F i g . 6.1(b).  The surface wave mode f i e l d s of the unperturbed d i e l e c t r i c  int  Eext \A  CD ^ext  ^ext  •int  C-  -/nf -  e  7  Fig  6.2  Quasi-static approximations (a),(b),(c) and (d): simple configurations (e) configuration corresponding to a s l o t (f) correction factor K..  78.  waveguide (Fig. 6.1(a)) are given  E  z  = A  by^^  J (p,r) cos n9 n l  n  r  r " - ^ l n V l > " ^ TP," rnV*l » l  E  (  A  (p  r  B  P  H  z  ( p rr  3^ ^ + —2-9. J ' ( r ) ) s i n n9 l p, n n 1 i  )  B  P l  r  p P  r  6  j n n  A  = B J (p.r) s i n n9 n n l  H  B  C O S n 9  n  = (J^S0 2 p r  E  r  n n  1  z  2 o l  n n  r  1  r  V , < V " FFT V n ' «»!'» «- "  " ^ Pn r  6  6  -  *o l r  for 0 < r < r ^ , and  E  z  E  =  C  j n  = r  Y  2  Y  H  H  z  r  K  n  ( Y O I ) cos -  n '2 C K  n  n  1  n9  (y_r) + y 2  9  0  0  2  n  n  2  2  r  2  2  = D K ( r ) s i n n9 n n 2 Y o  - (IB. K ' ( r ) + Y n n 2 D  Y o  2  H  jk Z n % D K ( y r ) ) cos n9 r  "  Y  2  r  Vn<V>  z  +  ° , 2 o 2  C K ( ~ r ) ) s i n n9 n n 2 Y  r  TT n o'2 C  0  K  n' < V »  C O S  "  9  6  '  5  4  79.  for r < r < <=°, where e = e/e and p.. and y 1 — — r o J. z  a  r  e t  n  transverse propa-  e  9  gation c o e f f i c i e n t s given by: p. *1  2  2  Y  2  i = e k ro  2  - 8R  1,  -e- k C  2  2  2  6.6  o  Matching the f i e l d s given by equation 6.4 and 6.5 at r = ^  y i e l d s the  following eigenvalue equation:  ' W l V P  l l n l l r  J  ( p  nB,  ko ( P  r  K  VlVVl*  }  +  )  6.7  —L  1  v  i r i  , n ' <Vl>  (Y r ) _  2  2  2  1  J  and the following r e l a t i o n between the c o e f f i c i e n t s A , B n  B  A  Q  , C^, and D ; n  n _ V 2 l> n C ~ D n l l n n Y  J  B  r  ( p  r  }  -8n n d " k z n " k Z n o o .o 6  A  +—i.  1  n  2 ' (p-^) ^ 2  l  r  )  2  PlVnW  Y  2 l n r  K  ( Y  2  6.8 For the dominant HE^^ mode n = 1 i n equations 6.4 - 7.8.  Thus, f o r the  configuration shown i n F i g . 6.1(b), equation 6.1 becomes :  . , *  p  6 +A0 o  - 3  As ' r / e ° r  ,(K lE l r  r  where K  r  2  +  e  9  < A* - 9 r* E  0 0  K l E l 4 - | J ) rd9dr  2  r  E  e  H  a  2  E  6.9  ) r d 6 d r  r  and K , defined by equation 6.2 (with e^E-j^ = E ) , are functions Q  of r , r , 9 , and A9. o 1 o  R  Using equations 6.4 and 6.5, and upon evaluation  80.  of the i n t e g r a l s , equation 6.1 becomes P  _  t  P  o  ...  o  l  E  1  f.  Pi !  +  fri l 2' 2  J  r  )E  Y  where N  l  =  F  iPl V 2  B 2  E  E  l =  2  =  <  ( 1  e  +  r  +  K  F r  2 B  +  d  l  (  d  l  2  A  2  A  3  ^ o  +  +  " lV 2 d  +  K  9 2 F  ( A  2  +  d  l 3 A  "  2 d  lV  2  l •  ^ i " o 2 2 B d  )  A  s  5 -  2  d  l  T Tk " 7 " l f ~ 2k )A  2 d  (  o  (  2  2 +  1 ) A  o  E  r  )  A  6  8  6  -  U  and F, = A6 + % ( s i n 2(0 + AG) - s i n 29 ) 1 o o 6.12  F- = A9 - % ( s i n 2(9 + AG) - s i n 29 ) I o o The terms A^, i = 1,  8 are i n t e g r a l s of Bessel and modified  Bessel  functions, and are given e x p l i c i t l y i n Appendix II-B. I t i s of i n t e r e s t to determine the separation i n the  propagation  c o e f f i c i e n t s of the cross-polarized modes which propagate on the s l o t t e d d i e l e c t r i c rod shown i n F i g . 6.1(b). 6.10 6  o  This can be determined using  by taking the d i f f e r e n c e between the r e s u l t obtained  + AG/2  = 0° and that obtained with 9  o  + AG/2  = 90°.  with  Numerical r e s u l t s  which correspond to configurations investigated experimentally are at the end of the next 6.3  equation  presented  chapter.  Summary (88) In t h i s chapter a w e l l known perturbational a n a l y s i s  used to obtain an approximation to the propagation  has been  c o e f f i c i e n t of the domi-  81.  nant mode on an a x l a l l y - s l o t t e d d i e l e c t r i c rod.  Slots with polar geometry  were considered. The main points were: 1.  Quasi-static approximation  perturbed waveguide:  by approximating  to the e l e c t r i c f i e l d on the  some l i m i t i n g forms of a region with  polar geometry with simpler configurations f o r which the s t a t i c f i e l d behaviour i s known, q u a s i - s t a t i c correction factors f o r the e l e c t r i c f i e l d components i n polar coordinates were obtained. 2.  A p p l i c a t i o n to an a x i a l l y - s l o t t e d d i e l e c t r i c rod:  an ex-  p l i c i t equation f o r the propagation c o e f f i c i e n t of such a waveguide was derived.  82.  Chapter 7 AN EXPERIMENTAL  INVESTIGATION  In t h i s chapter an experimental study of the d i e l e c t r i c waveguides configurations shown i n F i g . 7.1 i s described.  The parameter of  primary i n v e s t i g a t i o n was the normalized phase c o e f f i c i e n t , 3/k , of each Q  (89) of the cross-polarized modes.  A suitable resonant cavity apparatus  was r e a d i l y a v a i l a b l e f o r the measurement of t h i s  parameter.  Commercially a v a i l a b l e polyethylene rods (0.5 i n . nominal  dia-  meter) and frequencies i n the X-band range (for single mode propagation) were compatible with the e x i s t i n g apparatus and, hence, were used  through-  out . 7.1  Experimental Apparatus and Preliminary Measurements The resonant cavity apparatus i s shown p i c t o r i a l l y i n F i g . 7.2.  The end plates of the cavity were f l a t aluminum discs 0.5 i n . thick and 24.0 i n . i n diameter.  These plates were i n d i v i d u a l l y supported by angle  i r o n constructions, as shown i n F i g . 7.2.  Central 6.0 i n . diameter sec-  tions of each end p l a t e were removable, allowing d i f f e r e n t coupling schemes to be used with the same cavity.  The length of the cavity was  fixed at 15.0 i n . by three separate aluminum spacers.  These spacers also  kept the end plates aligned p a r a l l e l to each other and perpendicular to the axis of the d i e l e c t r i c waveguide.  Small removable locating pins,  passing through holes i n the center of each end p l a t e were used to support and a l i g n the d i e l e c t r i c rod.  These pins also allowed the d i e l e c t r i c rod  to be rotated on i t s axis without requiring any realignment.. Energy was coupled into and out of the l o n g i t u d i n a l component of e l e c t r i c f i e l d within the cavity by two i d e n t i c a l c o a x i a l probes (one  FIG. 7.2  THE RESONANT CAVITY APPARATUS (INSERT--THE  COUPLING PROBE AND THE SCATTERER)  CO  85.  on each end p l a t e ) .  A cross-sectional sketch of the c e n t r a l part of  end plate, i n d i c a t i n g the probe and in Fig.  locating p i n configurations,  i s shown  7.3. At resonance, 3 was  determined by counting the number of h a l f  wavelengths within the cavity ( i . e . the number of nodes i n the component of the e l e c t r i c f i e l d d i s t r i b u t i o n ) . cording,  one  This was  longitudinal  achieved by r e -  on a chart recorder, the output of the cavity as a thin m e t a l l i c  s t r i p , positioned  s l i g h t l y above the d i e l e c t r i c rod, was  length of the cavity.  The m e t a l l i c s t r i p was  moved over the  attached, by nylon thread, to  a motor-driven pulley system on the sides of the cavity (shown i n F i g . and,  thus, continuous movement was  provided.  7.2)  The magnitude of the s c a t t e r -  ing caused by the m e t a l l i c s t r i p depends on the f i e l d i n t e n s i t y at each point i n the standing wave pattern on the waveguide.  Thus, the output of  the cavity exhibited an o s c i l l a t o r y behaviour as the m e t a l l i c s t r i p moved.  A sample chart recording  i s shown i n F i g . 7.4.  was  The propagation co-  e f f i c i e n t can be determined by N  7.1  where L i s the length of the cavity and where N i s the number of o s c i l l a tions (half wavelengths i n the cavity) i n the recorded output o f the cavity. In order to completely specify 3/k ,  the resonant frequency, f ^ ,  Q  of the cavity was  determined ( i . e . k  Q  =  2irf  v r  'lJ^)  •  This was  by the standard procedures given by Sucher and Fox. diagram of the measurement system shown i n F i g . 7.5. of the cavity and  accomplished  Consider the block The resonance curve  the output of the reference channel were displayed  taneously on the oscilloscope, as shown i n F i g . 7.6(a).  simul-  The p o s i t i o n of  the small dip ( i n the trace f o r the reference channel) i s determined by  the  86.  ®  ®  '"5 UD  32  In-)  ©  ALUMINUM END PLATE POLYETHYLENE ROD LOCATING PIN (BRASS) @ TAPERED COAXIAL CENTER CONDUCTOR (BRASS. © TEFLON INSULATION © PROBE (0.005'dia) ® COAXIAL ADAPTER BRASS Fig. 7 . 3 .  Cross-section of the c e n t r a l part of one end plate of the cavity i n d i c a t i n g probe and l o c a t i n g p i n arrangement .  87.  90  — l —  F i g . 7.4  Chart recording of the output of the cavity.  POWER SUPPLY CALIBRATED ATTENUATOR X-13 [KLYSTRON  CALIBRATED FREQUENCY METER  ISOLATOR  CRYSTAL DETECTOR  WAVEGUIDE TO COAXIAL ADAPTER  DIELECTRIC WAVEGUIDE RESONATOR CRYSTAL DETECTOR  (REFERENCE JCHANNEL AMPLIFIER  SAWTOOTH MODULATION OSCILLASCOPE Fig. 7 . 5 .  Schematic of the measurement  CHART RECORDER system.  89.  calibrated frequency meter i n the reference channel.  The resonant frequency  of the cavity was read d i r e c t l y o f f the frequency meter when the small dip was aligned with the peak of the resonance trace (as they are, approximately, i n F i g . 7.6(a)). ±0.5 MHz  It i s estimated that the accuracy of this procedure  i n the range of frequencies 8.8 GHz to 9.5  was  GHz.  In order to obtain a reference f o r the measurements on d i e l e c t r i c waveguides with a x i a l s l o t s , the normalized phase c o e f f i c i e n t s of three unperturbed polyethylene rods (denoted A, B, and C, corresponding to the notation i n F i g . 7.1) were measured.  These rods were a l l cut from one  ori-  g i n a l piece i n an attempt to provide a basis f o r meaningful comparisons tween them.  be-  Micrometer measurements on the diameter of the rods indicated  that they had s l i g h t l y e l l i p t i c a l cross-section with major and minor axes given by 0.499 i n . and 0.495 i n . , r e s p e c t i v e l y .  Thus as expected, the  cross-polarized modes were observed on the same resonance curve but with s l i g h t l y d i f f e r e n t resonant frequencies, as shown i n F i g . 7.6(b).  This  "double" resonance occurred at four separate angles of r o t a t i o n (roughly separated by 90°) of the polyethylene rods.  A s i n g l e resonance curve was  obtained, however, when the rods were rotated such that the coupling probe lay on the l i n e of either the major or the minor axes.  Using the analysis  ( 86} presented by Yeh,  i t was determined that the higher resonant frequency  of the "double" resonance corresponded to having the probe on the l i n e of the minor axis.  Thus, i t was possible to approximately determine the p o s i -  tions of the major and minor axes experimentally. In an attempt to guard against any spurious frequency dependence of  the r e s u l t s , the values of 8/k  Q  for the cross-polarized modes were  measured (for each of the rods A, B, and C) at three separate values of 8  F i g . 7.6  Resonance curves ( h o r i z . s c a l e = 3.5 MHz/large d i v . ) (a) frequency measurement using the  reference  channel t r a c e (b) "double" resonance  91.  corresponding to N = 23, 24, and 25 i n equation 7.1. i n Table 7.2  The r e s u l t s are given  7.1.  Experimental Results f o r the Slotted D i e l e c t r i c Waveguides Using the above technique,  was measured (for each of the  cross-polarized modes) as the rods A, B, and C, were modified i n the manner shown i n F i g . 7.1.  The perturbations were aligned with the minor axis of  the e l l i p t i c a l cross-section of each rod i n an attempt to accentuate the i n i t i a l separation i n normalized propagation c o e f f i c i e n t s f o r the two polarizations.  Values of 3/k  Q  f o r each of the two p o l a r i z a t i o n s are plotted  i n F i g s . 7.7(a), (b) and ( c ) . The p o l a r i z a t i o n s corresponding to each curve are defined by the i n s e r t s i n each f i g u r e .  Although the measurements were  done at three d i f f e r e n t values of B (as discussed i n the previous section) only the r e s u l t s f o r N = 24 are shown i n F i g s . 7.7(a), (b), and behaviour of the curves with N = 23 and N = 25 i s s i m i l a r . rod A show, as expected, that the separation  i n B/k  Q  polarized modes increases with increasing values of 6.  (c).  The  The r e s u l t s f o r  between the crossA typical  resonance  curve i n d i c a t i n g the separation of cross-polarized modes (for small 5) i s shown i n Fig. 7.8. v  For small values of 6, the r e s u l t s f o r rods B and C  i n d i c a t e the same trends as rod A;  however, the separation between the  cross-polarized modes attains a maximum and then decreases f o r increasing values of 6.  In f a c t , there i s a value of 6 f o r rod C f o r which the  separation i s zero ( i . e . B/k  Q  i s the same f o r the two p o l a r i z a t i o n s ) .  appears that rod B would also exhibit t h i s phenomenon i f 6 was  It  further  increased. 7. 3 Results of the Perturbational Analysis The perturbational analysis presented i n Chapter 6 i s not d i r e c t l y  ELLIPTICAL AXES  RESONANT FREQUENCY (GHz)  MINOR  MAJOR  MINOR  MAJOR  MINOR  MAJOR  N  8.7900  8.7880  8.7905  8.7885  8.7905  8.7890  23  9.1075  9.1050  9.1080  9.1065  9.1090  9.1070  24  9.4165  9.4135  9.4170  9.4150  9.4175  9.4150  25  1.0302  1.0304  1.0301  1.0303  1.0301  1.0303  23  1.0375  1.0378  1.0374  1.0376  1.0373  1.0375  24  1.0452  1.0456  1.0452  1.0454  1.0451  1.0454  25  and 8/k  f o r the unperturbed rods  e/k  o (rad/cm)  ROD C  ROD B  ROD A  Table 7.1  Values of f  93.  ROD  A COUPLING  /  PROBE  1.042  0.032  0.024 &  F i g . 7.7(a)  0.096  0.128  (inches)  Normalized propagation c o e f f i c i e n t s of the two p o l a r i z a t i o n s as a function 6. .. a, b  experimental r e s u l t s f o r the configurations shown i n i n s e r t s a and b  ROD  B COUPLING  /  i  F i g . 7.7(b)  PROBE  \  (inches)  Normalized propagation c o e f f i c i e n t s of the two p o l a r i zations as a function of 6. a, b  experimental r e s u l t s f o r the configurations shown i n i n s e r t s a and b  . a',b'  perturbation r e s u l t s corresponding to a, b  a",b"  perturbation r e s u l t s f o r a 10° symmetry error i n the s l o t s  \ \ \  ROD ^  COUPLING  &  F i g . 7.7(c)  C PROBE  (inches)  Normalized propagation c o e f f i c i e n t s of the two polarizations,as a function of 6. a, b a',b'  (^experimental r e s u l t s f o r the shown i n i n s e r t s a and b perturbation  configurations  r e s u l t s corresponding to a, b  Fig.  7.8  The r e s o n a n t  frequency separation of  polarizations  on a p e r t u r b e d  rod.  (horiz.  = 3.5MHz/large  div.)  scale  the  two  97.  applicable to the experimental configurations because of the difference i n the geometry of the s l o t s and because of the i n i t i a l e l l i p t i c i t y of the experimental rods.  Perturbation r e s u l t s were obtained, however, under the  following conditions: a)  the diameters of the unperturbed rods were taken as 0.5 i n . ,  b)  the frequency used i n the c a l c u l a t i o n s was chosen to be  s l i g h t l y lower than the resonant frequencies* of the cross-polarized modes  (Qf.\ on the unperturbed experimental rods (the e l l i p t i c i t y  tends to increase  hoth resonant frequencies from that of a perfect rod), c)  the r e l a t i v e p e r m i t t i v i t y , e^, used i n the c a l c u l a t i o n s was  varied between a value of 2.26 and a value of d)  2.30,  the experimental s l o t s were approximated by t h e o r e t i c a l s l o t s  according to the construction shown i n F i g . 7.9(a), and e)  the parameters d and w ( i n equation 6.2) f o r the t h e o r e t i c a l  s l o t s were approximated by the depth and width, r e s p e c t i v e l y , of the experimental s l o t s . Perturbation r e s u l t s f o r rods B and C are shown as the dashed curves i n F i g s . 7.7(b) and ( c ) , r e s p e c t i v e l y . using a frequency of 9.105  These curves were derived  GHz and a value of  = 2.29.  No attempt  was  made to obtain perturbation r e s u l t s for rod A because of the obvious d i f f i c u l t y i n representing the rod cross-section i n t h i s case by s l o t s with polar geometry.  The t h e o r e t i c a l curves l a b e l l e d a' and b' correspond to  the experimental curves l a b e l l e d a andb, r e s p e c t i v e l y .  Due to some  machining problems, the experimental s l o t s on rod B had the configuration shown i n F i g . 7.9(b).  For t h i s reason the perturbation theory was modified  such that nonsymmetric s l o t s could be analyzed.  These perturbation r e s u l t s  98.  (b)  F i g . 7.9.  Slot configurations (a) approximation of a s l o t i n rectangular coordinates by one i n polar coordinates (b) actual s l o t configuration f o r rod B.  99.  for a 10° symmetry error, are shown as curve b" i n F i g . 7.7(b).  For the  scale used i n F i g . 7.7(b), the curve a" i s the same as curve a'. Based on the curves presented i n F i g s . 7.7(b) and (c) the following  observations can be made: 1.  f o r small values of 6 the experimental r e s u l t s are lower than  the perturbation r e s u l t s because the perturbational analysis does not account for  the i n i t i a l e l l i p t i c i t y of the experimental rods. 2.  since curve b' i n F i g . 7.7(c) was derived with 0 < d/w < 1  (in equation 6.2), the good agreement between curves b' and b suggests that the q u a s i - s t a t i c approximation may be quite good i n t h i s range. 3.  the propagation c o e f f i c i e n t i s not very s e n s i t i v e to symme-  try errors i n the s l o t s . Generally speaking, the perturbation r e s u l t s give a reasonably good i n d i c a t i o n of the trends i n the experimental r e s u l t s .  In p a r t i c u l a r ,  they confirm the experimental observation (rod C) that a value of 5 exists for which the two p o l a r i z a t i o n s have the same normalized propagation coefficient. 7.A  Summary In t h i s chapter an experimental i n v e s t i g a t i o n of propagation on  a x i a l l y s l o t t e d d i e l e c t r i c rods was described.  The measurements were  carried out at X-band using polyethylene rods with nominal diameters of 0.5 i n . The s l o t configurations considered were of rectangular geometry. The main points were: 1.  Experimental apparatus:  the measurement of 3/k using a  resonant cavity apparatus was described.  Q  Preliminary r e s u l t s obtained f o r  unperturbed d i e l e c t r i c rods indicated that they had, i n f a c t , s l i g h t l y  100  e l l i p t i c a l cross-sections. 2.  Measurement of perturbed rods:  tions were considered. the separation i n B/k  Q  three d i f f e r e n t configura-  Generally, the e f f e c t of the s l o t s was between the cross-polarized modes.  was not i n d e f i n i t e i n the case of rods B and C, however. C, there was  to increase  This increase In f a c t , f o r rod  a non-zero value of s l o t depth for which the separation  was  zero. 3.  Perturbation r e s u l t s :  the perturbational a n a l y s i s f o r s l o t s  with polar geometry (given i n Chapter 6) was fashion to the experimental  applied i n an approximate  s l o t configurations.  The perturbation r e s u l t s  gave a reasonably good i n d i c a t i o n of the trends i n the experimental  results.  101. Chapter 8 SUMMARY AND CONCLUSIONS The material i n this thesis has been presented  i n two parts.  Part I deals with the s c a t t e r i n g by step d i s c o n t i n u i t i e s on a planar surface waveguide.  The main objectives were (a) to provide an  v  accurate a n a l y t i c a l method, of general a p p l i c a b i l i t y and s u i t a b l e f o r transmission matrix studies of cascaded d i s c o n t i n u i t i e s , for  determining  the s c a t t e r i n g parameters and (b) to provide accurate experimental  measur-  ment of the s c a t t e r i n g parameters, p a r t i c u l a r l y since the t h e o r e t i c a l r e (52) s u i t s i n this work did not agree with some recently published r e s u l t s Of the many surface waveguide configurations possible, a d i e l e c t r i c coated conductor supporting TM modes was chosen for the i n v e s t i gation because of the advantages this configuration offered experimentally. The d i s c o n t i n u i t i e s consisted of abrupt changes i n the d i e l e c t r i c thickness. The a n a l y t i c a l method involved bounding the open structure with a perfect conductor.  Using the d i r e c t r e l a t i o n s h i p between the mode  spectra of the bounded and the unbounded waveguides, the s o l u t i o n to the unbounded configuration was obtained d i r e c t l y from the s o l u t i o n of the bounded configuration. matching.  The closed boundary problem was solved using mode-  P r i o r to the application of the above method, two separate i n -  vestigations were deemed necessary, namely ( i ) the behaviour of e l e c t r o magnetic f i e l d s near a m u l t i - d i e l e c t r i c edge (this i n v e s t i g a t i o n also i n cluded edges which involved an anisotropic medium) and ( i i ) the e f f e c t of d i e l e c t r i c edge conditions on mode-matching. The experimental of 30.00 GHz.  i n v e s t i g a t i o n was c a r r i e d out at a frequency  A surface waveguide was constructed and i t s properties  were determined experimentally.  The surface wave which was excited on  102.  this waveguide had less than 0.5  db v a r i a t i o n over a width of three wave-  lengths and a decay away from the d i e l e c t r i c surface which was ent agreement with theory. i n reasonable  in excell-  Measured values of attenuation were also found  agreement with theory when conductor losses, material losses  and beam divergence were taken into consideration.  A l l f i e l d probing  was  done using a A/2 dipole oriented along the l o n g i t u d i n a l component of the electric field.  This type of "large" probe has proved useful i n such  measurements on planar surface waveguides when the probe i s placed away from the d i e l e c t r i c surface and thus does not s i g n i f i c a n t l y disturb the fields. Both the magnitude and the phase of the r e f l e c t i o n c o e f f i c i e n t of the "TM " o  mode were measured for a range of step heights (these measurements  were also done at 29.94 GHz spurious frequency  and 30.06 GHz  i n an attempt to guard against  dependence i n the r e s u l t s ) .  any  Only the magnitude of the r e -  f l e c t i o n c o e f f i c i e n t was measured for a cascaded d i s c o n t i n u i t y . However, a useful check on the results was  obtained i n this case by varying the e l e c -  t r i c a l distance between the cascaded steps. The o v e r a l l contributions of the work contained i n Part I may  be  summarized as follows: 1.  The s o l u t i o n of planar surface waveguide d i s c o n t i n u i t y problems using the "bounded" approach.  The method.has been shown to be  simple,  accurate and applicable to a v a r i e t y of configurations and to cascaded discontinuities.  Also, the correct r e s u l t s f o r the configuration an-  alysed i n reference (52) are given. 2.  The experimental v e r i f i c a t i o n of the r e s u l t s obtained  3.  Certain aspects of the experimental of the r e f l e c t i o n c o e f f i c i e n t :  theoretically.  technique used f o r the measurement  (a)  the use of a A/2 dipole to measure the l o n g i t u d i n a l component of the e l e c t r i c f i e l d i n the standing wave and  (b)  the i n t e r p r e t a t i o n of the experimental r e s u l t s i n the context of the properties of the experimental waveguide and of the various factors which affected the standing wave measurements.  4.  The application of the wave amplitude matrix approach i n s i t u a t i o n s where the numbers of propagating modes on e i t h e r side of a discont i n u i t y are unequal.  5.  Two side issues which were relevant to t h i s i n v e s t i g a t i o n : (a)  the behaviour of electromagnetic f i e l d s near an edge common -  to three d i e l e c t r i c s was considered and i t was found that the f i e l d s are not necessarily singular at such an edge and (b)  the e f f e c t of d i e l e c t r i c edge conditions on mode-matching was also considered and i t was shown that these edge conditions do not require any s p e c i a l consideration when using mode-matching.  Part I I deals with surface waveguides with perturbed c i r c u l a r cross-sections.  The main objective was to determine what r e l a t i v e e f f e c t  the perturbations have on the cross-polarized modes, with a view to making the p o l a r i z a t i o n more d e f i n i t e .  D i e l e c t r i c rods, of c i r c u l a r cross-section,  perturbed by a x i a l s l o t s were considered t h e o r e t i c a l l y and experimentally. The parameter of i n t e r e s t was the normalized propagation c o e f f i c i e n t . The a n a l y t i c a l treatment involved a standard perturbational ana l y s i s applied to s l o t s with cross-sections i n polar geometry. The experimental study was carried out at X-band frequencies using an open resonant cavity.  Polyethylene rods with perturbations i n  rectangular geometry were considered.  104. The r e s u l t s indicate that the p o l a r i z a t i o n i s made more d e f i n i t e i f the perturbations are small ( i . e . separation of the normalized gation c o e f f i c i e n t s ) . there was  I f the perturbation consisted of a s l o t , however,  an optimum value of s l o t depth which gave the largest separation  of the two p o l a r i z a t i o n s . decreased separation. 8.1  propa-  S l o t depths beyond the optimum resulted i n  This fact was  shown experimentally  and  theoretically.  Suggestions for Further Work The work presented  i n this thesis suggests several possible  extensions: (a)  t h e o r e t i c a l and experimental  analysis of mode conversion on multimode  surface waveguides (with applications to o p t i c a l f i b r e s ) with imperfections.  The  surface  theory would involve the use of wave transmission  matrices. (b)  a more generally applicable analysis of tapered surface waveguides by approximating the tapers with a s e r i e s of small steps (the works of Marcuse  (49 50) (53 54) ' and Snyder ' are r e s t r i c t e d to gradual tapers of  d i e l e c t r i c waveguides used i n o p t i c a l f i b r e applications where the r e f r a c t i v e index differences between the core and cladding are small), (c)  the determination  of the r a d i a t i o n pattern of a step d i s c o n t i n u i t y  by considering the f i e l d s i n the plane of the d i s c o n t i n u i t y found from the "bounded" approach, (d)  t h e o r e t i c a l and experimental  investigations of finned d i e l e c t r i c  waveguides, (e)  the a p p l i c a t i o n of the f i n i t e element method to f i n d i n g more accur a t e l y the propagation cross-sections. f i e l d patterns.  properties of d i e l e c t r i c waveguides with various  This approach would also y i e l d information about the  105.  (f)  the i n v e s t i g a t i o n of f e a s i b l e interconnection and coupling schemes for surface waveguide components.  106. Appendix I-A FIELD BEHAVIOUR NEAR ANISOTROPIC  AND  MULTIDIELECTRIC EDGES  In the v i c i n i t y of an edge, the electromagnetic f i e l d s which are transverse to the edge can become singular  ' ^'^"""^ whereas the f i e l d  components i n the d i r e c t i o n of the edge are always nonsingular.  The  field  s i n g u l a r i t y i s determined by the edge condition which requires that the energy contained i n a small volume about the edge must be f i n i t e . e x p l i c i t l y , singular f i e l d components must be of order p  f c  0 < t < 1, where p i s the r a d i a l distance from the edge.  More  ^ as p -> 0 with The  singularity  parameter, t, depends on the geometry and the properties of the media surrounding the edge.  In a l l cases, however, two types of s i n g u l a r i t y must  be considered: (1)  that of the e l e c t r i c f i e l d  (due to d i f f e r e n t p e r m i t t i v i t i e s ) ,  characterized by a s i n g u l a r i t y parameter t„, and (2)  that of the magnetic f i e l d  (due to d i f f e r e n t p e r m e a b i l i t i e s ) ,  characterized by a s i n g u l a r i t y parameter t„. n There may be more than one value of t_ or t i n the range of 0 to 1. su  If  ti  both e l e c t r i c and magnetic type s i n g u l a r i t i e s e x i s t at an edge, the dominant s i n g u l a r i t y i s due to the smallest value of e i t h e r t  or t„.  c>  ri  The purpose of t h i s i n v e s t i g a t i o n i s to extend the e x i s t i n g t h e o r e t i c a l treatment of electromagnetic f i e l d behaviour near d i e l e c t r i c edges to include two cases of p r a c t i c a l i n t e r e s t : (a) the case where one of the two media surrounding an edge i s a magnetized f e r r i t e , shown i n F i g . 11(a), and (b) the case where three i s o t r o p i c d i e l e c t r i c s have a common edge, shown i n F i g . 11(b).  107.  e  2  M  £; JJ 2  MEDIUM 1  MEDIUM 2  0,2TC  (a) h  2^2 MEDIUM 2 6  MEDIUM 1 0, 2n '3,^3 MEDIUM 3  0;+  t  ?  (b)  F i g . I I . Edge configurations (a) dielectric-magnetized f e r r i t e (b) three i s o t r o p i c d i e l e c t r i c s .  108.  The basics of the underlying theory f o r the derivation of the s i n g u l a r i t y parameters can be found e l s e w h e r e F o r  case (a), however,  the theory must be modified s l i g h t l y to account f o r the permeability sor i n medium 1 and, thus, i t i s outlined for this case.  ten-  In case (b) the  emphasis i s on the dependence of the f i e l d s i n g u l a r i t y on geometric v a r i a tions of the edge configuration.  This dependence i s somewhat complex and,  thus, i s i l l u s t r a t e d by a. numerical example.  I-A.l  Edge Involving a Magnetized F e r r i t e Consider the edge configuration shown i n F i g . 11(a) where medium  1 i s magnetized f e r r i t e and medium 2 i s i s o t r o p i c d i e l e c t r i c . steady magnetic f i e l d , B » z  With a  applied i n the d i r e c t i o n of the edge ( z -  d i r e c t i o n ) , the permeability of the f e r r i t e i s given  by:  0 -5K  L  0  V  0  0  y  °J  In a homogeneous region, the electromagnetic of Maxwell's  f i e l d s are solutions  equations:  V x IS = -jcoyH 7 x H =  jtoeE  12  For the configuration shown i n F i g . 11(a) y i s given by equation I I , a tensor, i n medium 1 and by y , a s c a l a r , i n medium 2. f i e l d s can be expanded i n a power series as follows:  Near the edge, the  109. u p E  =  i  E  a  0  t-1 , l  p  +  " 0 b  p t _ 1  a  +  p  l  b  t , 2  +  p  t  a  +  p  b  2  +  t+1 , ~  p t + 1  ~  +  . t , t+1 + p + c p +  t _ 1  E  z  = c p  n H  p  t—1 t , t+1 , = ap + o^p + a p +  H  J  Q  C;L  2  Q  f  u  t-1 , „ 1  V  6  z  Substituting  0  P  t+1 ,  n  l  Y  t , 2  P  t-1 ' Y  —  2  „  =  —  3  t , 2 p Y  "  p  t+1 , P  3  equation 13 into the component form of equation 12 and then  equating c o e f f i c i e n t s of l i k e powers of p y i e l d s f o r medium 1: (t-1) = 0  C()  Y (t-D = 0 0  0 3z  3 P  j o e1* 0 =  J t o e  i o b  ---34  - 3 T - **!  =  -J»( y a + j B ) = K  Q  3c  0  o —  -  3b  3 a  ^ ( - j ^  t  b  3 a  o  3 a  o  0 - - l ?  + y6 ) Q  =  -  t  C  l  0  14  Rearranging equation 14 r e s u l t s i n the following equations f o r the c o e f f i cients ( a , b , c.p a , 3 Q , y^) i n medium 1: Q  Q  Q  110.  OQ  =  s i n tty + A^ cos t<j>  =  s i n tty +  B  cos tty  2  b^ = A^ cos t<j> - A  s i n tty  2  3Q = B . ^ cos t<j> - B 9 B  tYj^ =  1  3A t  c  i  =  . . . s i n tty +  2  9 B  s i n t<j>  2  1  3A  1  Ti"  s i n  ^  +  H  cos ti}) - ja)e (A  1  cos t<J> - A^ s i n tty)  2  c o s  + jco((ucos tty - J K s i n tcJ))B - (ysin t<j> + j< cos t<fi)B ) 1  2  15 A s i m i l a r procedure y i e l d s f o r medium 2: (prime denotes medium 2) a^ = Aj^ s i n t$ + A^ cos t<j> = B^ s i n tty + B^ cos t<j> b^ = A j cos t<j> - A^ s i n tty 6^ = B| cos t<{) - Bj", s i n 3B|  3BJ  ty* = - 7 — s i n t<() H — — '  3A^  t((i  cos t<j> - ja)e_(A' cos tty - A\ s i n t<}>)  3A^  t c ^ = —g^- s i n t<f> + — — cos t<j> - jo)y (B^ cos tty - B 2  2  s i n t(j>) 16  In order to s a t i s f y the boundary conditions, the components of the IS and H f i e l d s tangential to the d i e l e c t r i c interfaces must be continuous. These conditions, when applied to the c o e f f i c i e n t s i n equations 15 and 16 at ty = ty^ and ty = 2ir, i n F i g . 11(a), r e s u l t i n a homogeneous system of eight equations i n the eight unknown c o e f f i c i e n t s (A^, A B  2 >  2 >  A^, A , B^, 2  B^, B ) . This system of equations i s separable into the following 2  111. two 4 x 4 homogeneous systems: a) with  =  =  =  = 0  -S,  -C,  -C  '27T  2ir = 0  e  l 2  "h 2  - 1 2  C  £  S  C  - 2 e  C 2 l  e  r  e  A'  2 2 S  2  S 2 l  A  l  A'  r  A  2  17 b) with Aj^ = A  2  = Aj^ = A J = 0  -S -S  B,  2 : -C 2v  2-rr  = 0 yc2-jKs2  -yS -JKC 2  -y2C2  2  - 2 y  C 2 l  y2s  r  y  2  S  2  2 ,  B  2 18  Where S, = s i n t<j>..  C_ = cos t(f>  1  S- = s i n t2ir 2ir  C  2lr  = cos t2ir  19  The above homogeneous systems of equations are consistent i f the determinant of each matrix i s zero.  Setting the determinant of the matrix i n  equation 17 equal to zero y i e l d s : s i n t„ir Ei  e i n t ((j> - TT) E  1  e  l  £  1  = +  - 2 e  +  £  2  110  112.  Repeating  t h i s f o r equation 18 y i e l d s : 2 sin  t (<J  sin  H  Equation 110 was was  tjjTT )]L  -  IT)  " W  /(y  -  y )  (u  +  y )  2  2  2  2  +  K  +  k  111  2  o r i g i n a l l y derived i n reference (71).  Equation  derived i n connection with some recent work by McRitchie  111  and  (91) Kharadly  i n v o l v i n g a homogeneous to inhomogeneous nonreciprocal wave-  guide i n t e r f a c e .  In that work, the s i n g u l a r i t y parameter, t^, was (73)  to check mode-matching solutions non-singular.  used  since the e l e c t r i c f i e l d s were  In general, however, both t  and t„ would have to be  considered. I-A.2 Edge Involving Three D i e l e c t r i c s In this section, the behaviour of the f i e l d s near an edge common  to three i s o t r o p i c d i e l e c t r i c media, shown In F i g . 11(b), i s examined.  Let 2 E  >  E  i  >  £  3 k  generality.  this assumption does not r e s u l t i n any loss i n  u t  Here, only the e l e c t r i c f i e l d s (hence t„) need be  considered,  since the magnetic f i e l d s i n g u l a r i t y behaves i n a s i m i l a r fashion.  Using  the procedure outlined i n the previous section, i t can be shown that t„ i s a solution of: E e  £  where  l< 2  +  3 1  +  G  ( E  E  e  3 1 21 )C  S  2 l 21 ) S  S  S 27  C 2 l  :2  +  r2  +  e  2  2 e  ( e  3  +  e  l l 21 2rr2  l^2 3 e  ) S  ( 1  C  S  +  " l 21 2ir2 C  C  C  )  1 1 2  113.  S  = sin t $  1  S  E  21 °  S  i  =S i n  S 2 l T  2  n  C  1  V2  V  2ir  C  --'•2 "  *1  }  C  21  =  =C O S  2u2  » cos t $  1  E  C  O  1  S  V *" 2  *2 " * 1  }  113 With  ^  - 0 or 2ir - <|>^, equation 112 reduces to the expression f o r an  edge common to two d i e l e c t r i c media as given by equation 110. A -Numerical Example To i l l u s t r a t e the behaviour of t„ i n equation 112, consider the CJ  case where E , = 2E , e = 10e, and e- = e . A plot of t„ versus <(>.. + <j> , X O / O J O il. J . Z 0  for various values of <J>^, i s shown i n F i g . 12. For comparison purposes, three s p e c i a l cases involving only two d i e l e c t r i c s are also plotted i n the same f i g u r e .  These are shown as the dashed curves, representing  solutions of equation 110 f o r the following choices of parameters: a) upper curve:  e, = e 1 .0 b) middle curve: e, = 2e 1 o c) lower curve: e, = E r r  1  o  , e~ = 2e 2 o , E = 10E 2 o , e _ = 10E 0  2  o  One might expect, on f i r s t examination, that the dashed curves would provide bounds on the s o l i d curves. behaviour of t  Instead, i t i s observed that the  i s more complex than can be accounted f o r by such bounds.  E  In f a c t , F i g . 12 indicates that f o r <j>^ = 45° there i s a range of angles <|> f o r which t i s greater than unity, i . e . the f i e l d s are nonsingular. 2 E This s i t u a t i o n does not occur i n the two-dielectric case^ however, i n a 0  recent i n v e s t i g a t i o n  (92)  ~ involving a p e r f e c t l y conducting wedge, loaded  with at l e a s t two d i e l e c t r i c s , a s i m i l a r phenomenon has been observed.  115.  Region of Nonsingularity The range of nonsingularity i s now boundaries  examined i n more d e t a i l .  The  can be obtained by setting t_ = 1 i n equation 112 and solving E  for <j>2 In terms of <J>^.  Results indicate that there i s a region of non-  s i n g u l a r i t y i n the <J>^, <t>^ plane f o r any t h r e e - d i e l e c t r i c configuration. For the example considered, this region i s shown i n F i g . 13.  The boun-  dary locus passes through the point ( l a b e l l e d A i n F i g . 13)  corresponding  to an i n t e r f a c e configuration for which there i s obviously no s i n g u l a r i t y . This configuration i s i l l u s t r a t e d by i n s e r t (a) i n F i g . 13 whereas an a r b i t r a r y point inside the region i s i l l u s t r a t e d by i n s e r t (b) i n the same f i g u r e . The extent and shape of the region depend on the r e l a t i v e values of e^, e^, and e^. function of c  1  For example, i n F i g . 14, the region i s shown as a  i n the range of e  two l i m i t i n g values of shown i n F i g . 14.  1  = e  3  = e  o  to e  = e  = 10e  2  o  .  At these  the region collapses to s t r a i g h t l i n e s , as  These l i n e s correspond to the s p e c i a l t w o - d i e l e c t r i c  case i l l u s t r a t e d by i n s e r t (a) i n F i g . 13.  A study of the r e s u l t s of  F i g . 14 reveals that there i s some value of for which the area of the region i s maximum. the maximum occurs at a value of the i n t e r v a l of angles vary with e^.  1  i n the range of  to  For the parameters considered, equal to 4 E  approximately  o <  However,  spanned by the region, denoted by A<j>^, does not  Further i n v e s t i g a t i o n shows that A<j>^ depends s o l e l y on the  r a t i o of the highest to lowest p e r m i t t i v i t i e s , 2^ 3' £  i s given i n F i g . 15.  I t may  e  ^ ^-  r  l  s  dependence  also be seen from F i g . 15 that there i s  some minimum value of ty^ associated with the region given a set of permittivities.  Numerical results indicate that t h i s minimum value,  ^ » depends on  i n a manner c l o s e l y related to the dependence  116.  0  10  20 ^  Fig. 14  30  40  50  ANGLE IN DEGREES  Dependence of a region of nonsingularity on the p e r m i t t i v i t y  • with  >  > e^*  • M  80  0 .  F i g . 15  20  40  .  60  . 80  Dependence of A<J>.. on the r a t i o of the highest to lowest p e r m i t t i v i t i e s ,  100  e^/  of Ad>, on e„/e„. l 2 3 Y  substituting  In f a c t , (<j>„) . can be obtained from F i g . 15 simply by 2 mm  £^ 3 e  f  o  r  2^ 3  e  e  a  n  d  ~ ^2^min  f  °  rA  ^l"  T  h  i  s  m  e  a  n  s  that  the wedge angle, ty^, of the medium with the highest p e r m i t t i v i t y must always be greater than 90° i f f i e l d s i n g u l a r i t i e s are to be eliminated, Based on the numerical example considered above, the following conclusions may be drawn: (a)  loading a d i e l e c t r i c edge with a t h i r d d i e l e c t r i c can, i n  some cases, eliminate the f i e l d s i n g u l a r i t y at the edge, (b)  there i s a region i n the <{>^, ^  plane f o r which the f i e l d s  are nonsingular, (c) e  f o r a given choice of  >  >  E  3» there i s some value of  ^ f o r which the area of the region i s maximum, (d)  the i n t e r v a l of <f)^ spanned by the region i s determined  s o l e l y by the r a t i o of the highest (z^) and the lowest (e^) p e r m i t t i v i t i e s , (e)  the wedge angle ty^ of the medium with the highest permit-  t i v i t y must be greater than 90° i n order that f i e l d s i n g u l a r i t y can be eliminated.  In f a c t , the minimum wedge angle i s determined  s o l e l y by the  l ratio — . 3 e  £  The points l i s t e d above are consistent with those found i n r e f erence (92) f o r the s p e c i a l case when medium 2 i s a perfect conductor. This configuration can be achieved by l e t t i n g  00  - .  I t would not be  reasonable to suggest that the region of nonsingularity exists f o r edges common to more than three d i e l e c t r i c s . much more complex.  However, the analysis could be  Appendix I-B EFFECT OF DIELECTRIC EDGE CONDITIONS ON MODE-MATCHING SOLUTIONS The configuration used i n t h i s analysis i s shown i n F i g . 16(a). The discontinuity i s an interface between an empty rectangular waveguide and a waveguide of the same dimensions inhomogeneously loaded with a d i e l e c t r i c f o r which e i s considered  and u ^ are d i f f e r e n t from unity.  r  here.  Only H-plane loading  Since two types of modes are excited i n both wave-  guides A and B (see F i g . 16(a)) the e f f e c t of the d i e l e c t r i c edge condition may be observed on the r a t i o of these modes as w e l l as the t o t a l number of modes on e i t h e r side of the d i s c o n t i n u i t y . A T E ^ Q mode incident on the i n t e r f a c e from waveguide A excites ((La)  the following sets of modes TE  1 Q  LSE  l m  + TE  : + TM  l m  + LSM _ lm  1  i n waveguide A, and  i n waveguide B,  where m = 2,4,6.... The equivalent c i r c u i t of such an i n t e r f a c e i s shown  (f.Q\ i n F i g . 16(b)  .  In order to i l l u s t r a t e the e f f e c t of the edge condi-  tions a numerical example (e^ =8.0, u choice of the parameters, e  r  r  = 0.75) i s discussed.  With t h i s  and u ^ , there are both e l e c t r i c and magnetic  type s i n g u l a r i t i e s at the edges, as discussed  i n Appendix I-A.  Using  equations 19 and 110, i t can be shown that the e l e c t r i c f i e l d s i n g u l a r i t y i s dominant.  The edge condition determines the rate of decay of the ampli-  tude c o e f f i c i e n t s obtained by m o d e - m a t c h i n g T h u s , using the analysis given i n reference modes, |A | and E  large m.  (71), the amplitude c o e f f i c i e n t s of the T E  l m  and T M  lm  respectively, should decay at the rate m * ^ f r 2  D  0  The correct r a t i o of modes to be used i n mode-matching can be  122.  detemined by examining the rate of decay of the c o e f f i c i e n t s  (93)  .  For the  example considered here, t y p i c a l values of | A | and | A ^ | are shown as E  functions of m i n F i g . 17. A straight l i n e of slope -2.746 i s also shown for comparison.  The results shown i n F i g . 17 indicate that there i s  reasonable agreement between the predicted and calculated decay i n the coefficients.  The scatter i n the calculated points make i d e n t i f i c a t i o n  of the decay rate d i f f i c u l t , however,  For t h i s reason the e f f e c t of the (73)  choice of modes on the equivalent  c i r c u i t parameters has been considered  These parameters are derived from the mode-matching solutions and, hence, provide an a d d i t i o n a l check on the e f f e c t of the edge conditions. B r i e f l y , the r e s u l t s indicate that: (a) the equivalent (b)  i r r e s p e c t i v e of the choice of modes used i n mode-matching, c i r c u i t parameters eventually  converge to the same value.  f a s t e r convergence i s obtained with fewer TE modes than  TM modes ( s i m i l a r l y , fewer LSE modes than LSM modes) but with the same t o t a l number of modes on either side of the d i s c o n t i n u i t y .  Appendix I-C INTEGRALS INVOLVED IN EQUATION 2.19 Let superscripts a and b denote waveguides A and B, respectil  t i v e l y , and l e t subscripts k and j denote the k  til  and j  modes,  respectively. Normalization constant P , ak For the k  P  ak  til  2iTT o o  =  mode i n waveguide A  ( A  ~ ^ < 2k (  2  A  lk  +  s  i  +  i  n  A  s  n  A  2k  lk  ) / 2  ) / 2 p  lk r e  P2k  where _ a a ^ i k h  A A  A  lk  =  2k - ^  1  - V  The corresponding expression f o r the j  mode i n waveguide B, P^.can  found from equations 114 and 115 by substituting 3  f  P 3 r  e  t  p  b  a a , P , p ^ , and C*, respectively.  f c  l k  Integrals  L  B  ' \ Q  lkj  A  A  A  2kj  k  =  =  A A  3kj  A  A  =  x h^. • a^dx and  Plk P l j +  a b Plk " l j p  a P k 2  +  p  , b l j  b 4kj ~ P k "Plj 3  2  ^  t . h ^ •^ d x  , p . , and C b  b  125.  A A  5kj  =  P  =  P  A A  6kj  a , b 2k 2 j + P  a b 2k " 2 j  116  P  Then 0 —ak  0 % j  X  l  /  -Hn .  —z  ^ k '^ —  J  d  2k Y oo  x =  1  2kY l oo  117  T  where T  The terms T  =  T  2 T  2 >  3 >  E  r  +  C  k  3 -  T  C  k  C  j  T  A  1  .t /A  . + sin A ^ t J / A ^  2  = sin A  T  3  = ( s i n ( A . t ^ - p ^ t ) - s i n ( A ^ . t * - p ^ ) ) /A^.  4  8  and T^ are given by:  T  T  1  a  l k  l k  3 k  B  + (sin ( A  4 k j  = sin A  (t> - t )/A  5 k j  t^ - p  B  a k  t ) - sin ( A B  5 k j  + sin A  4 k j  f i k j  t * - p * ^ ) ) /A^ ^4kj (t> - t ^ / A ^ 119  The expressions f o r the slow modes can be obtained from equations 114-19 with p  2 k  = jy  2 k  and p!^ = j y ^ j •  126.  Appendix I-D ATTENUATION COEFFICIENTS An approximation to the attenuation c o e f f i c i e n t of the k  t b  mode i n waveguide A may be found from a °k K  • &k 2P~~ P  =  ak  120  where P ^ i s power/unit width absorbed by the d i e l e c t r i c . by  P ^ i s given  a  tan 6 , . 2 a , ,a. 2 ., a ..2, . „ a a,„ a .. Ak Y e <r Ao tli ' [(8.) ^ k ' - *(lpk ' ) 1 ° sxn 2p_. *'lkt./2p..) l" 'lk o or r//  e  v  +  l  v  J  fc  w  t  n i  121 th The corresponding expressions f o r the j  mode i n waveguide B may be  obtained from equations 120 and 121 by s u b s t i t u t i n g t , 8 , P^j» a. a a b  for t ^ , 8^, P ^ j and ^ » respectively. p  a  P ^ may be found i n Appendix I-C.  b  a n  ^  The expressions f o r P ^ and  127.  Appendix I-E MEASUREMENT OF STANDING WAVES WITH A A/2 DIPOLE Consider the general case of a wave propagating i n the direction.  z-  Assume that a standing wave e x i s t s i n t h i s d i r e c t i o n and  that the e l e c t r i c f i e l d has  z-component given  E (x,y,z) = E ( x , y ) ( e " z o  j e z  + re  j g Z  by:  )  122  where T i s the r e f l e c t i o n c o e f f i c i e n t and where E (x,y) remains unspecified Q  by e s s e n t i a l l y depends on the source of the wave and tions.  the boundary condi-  Let the dipole be oriented as shown i n F i g . 18.  coordinate terminals  z  1  = z - a i s introduced  i s given b y ^ ^  I f the dipole  then the current, I , at the antenna  :  L I (x,y,a) = c'.jE^x.y^YCO.z^dz' .  123  L  where C i s a constant which depends on the impedances of the load and dipole. and  Y(0,z') i s the transfer admittance of the dipole between z  z' ( i . e .  the current at z' due  to a unit voltage at z' = 0 ) .  the = 0  1  Thus,  Y(0,z') i s j u s t the current d i s t r i b u t i o n of the dipole of length 2L when used as a transmitter.  Hence  Y(0,z ) = s i n k (L-z') 0  z'>  0  = s i n k (L+z')  7. <  0  f  0  Substituting equations 122 and 124  y  into equation 123 y i e l d s :  I (x,y,a) = CE (x,y){ j ^ - j e C z ' - h x ) L  o  124  +  ^jSCz'-ta))  s ± n  k  ( L  _ . z  ) d z  .  125  128.  129,  Upon evaluation of the i n t e g r a l s i n equation 125 one obtains: cos BL - cos k L I (x,y,a) = CE (x,y)2k ^ 2 ~ k - B L  o  o  ^  e  +  T  e  *  o  Thus, the current delivered to the load depends on a i n true standing wave fashion.  Therefore, one may conclude that a dipole of length 2L  (or 2L = ^Q/ which i s of i n t e r e s t here) measures the correct value of 2  the standing wave pattern.  126  130.  Appendix I-F DERIVATION OF WAVE MATRICES  AND  w  2  Consider the discontinuity at z = z^, as shown i n F i g . 4.1(b) Let c_| and b_^ be defined i n waveguide B i n a s i m i l a r manner to c.^ and b_^ i n waveguide A. defined b y  ( 8 0 )  The scattering matrix, S, f o r the d i s c o n t i n u i t y i s  :  *1 =  S 127  where i n d i v i d u a l elements of S are found from the mode-matching  solutions  to the discontinuity problem with i n d i v i d u a l modes incident from either side.  S can be partitioned  s: S  =  128 Then w^ and w^ are defined by: ,-1  -S S 3 4 _1  b  W  l  b  =  1 3  S  S  S  2~ l 3 S  S  l s  4  129  and ,-1 " 2 S  S  4 2 S  1  S  1 S  1  3~ 4 2 1 S  S  1 S  130  131.  APPENDIX I I - A DERIVATION OF EQUATION 6.1 The f i e l d s E , H , E and H are solutions of Maxwell's equations: —o ~o V x E —o  =  V x H —o  =  V x E  =  V x H  =  -ja>u  H o —o  j u e E, —o  HI  and  Consider  -JWV H 0  ja)(e + Ae) E  V • (H x E *) — —o  and  112  V • (H * x E); from equations I I I and 112 —o —  V . (H x E *) + V • (H * x E) = jco Ae E • E * — —o —o — — —o  113  Define a volume T as shown i n F i g . I l l . Applying the divergence theorem to the volume i n t e g r a l of equation 113 / / (H x E * + H * x E) • dS S ~ -o -o -  results i n =  jcoAe / E • E * dx -o  114  where / / dS S  =  / / S  w  +  S  s  +  S  dS  115  W  Since the surface wave f i e l d s decay to zero as R / / s Also  (H x E * + H * x E) • dS  =  0  +  " ( i n F i g I I I ) then  1  1  6  132.  Fig. I l l  Volume region.  133.  And since  Ac  =  / dx =  0  except i n the perturbed regions then  dz / /  S  T  dS  118  P  Thus, using equations 116 , 117 , and 118  |9z  equation 114  / / ( H x E * + H * x E ) « a dS ' * — —o —o — —z w  =  J  becomes  jojAe / / E • E * dS ~° p  119  c  where S i s the cross-section of the perturbations. P The z-dependence i n the integrand of the left-hand side of equation I I 9 -j(B - B ) z i s given by e . Thus, equation II9 reduces to 0  -0 ) / / (H x E * + H * x E) • a  s  Q  q  z  dS  =  juAe / / E • E * dS  s  w  p  and, hence (equation 6.1) B -B  / / E • E * dS S °  Ae  0  k  Z  o  / / (H x E * + H * x E) • a J  J  —  SW  where  — n  —  — n  — Z  r Ae  r  = — e  , Z o  o  =  / e  , k o ' o  =  (o/y e oo  dS  q  m o  134.  Appendix II-B SELECTED INTEGRALS OF BESSEL FUNCTIONS L  i = 1,. . . . .8  ±t  Define  n  p  =  i  p  r  r  p  io  =  i V  p  A  N  P  D  I  2  =  Y  2 i r  t  h  e  n  ll 2 J,(x) x dx 10 P  A- = 2J 1  P  1  = Pn^(Pii))  2  +  =  J  (Pir ^!^ 1  - Pio< i io J  ( p  "  ) ) 2  fcio-^frio*  J (x) 2  A  2  =  2 J  p'  ~7~~  1 0  d x  x<PlO  )  " ^ l l  +  " "'frll*  5  1  1  1  2  ll 2 A- = 2J " ( J ( x ) ) x dx 10 P  J  L  Z  1  3  p  - A  A  4  1  - A  x  2  +  U  2 l (x)) + - i — J  l i  n  p  1  + (p  2  6  = 2;Q J (x)J^(x)dx  A  ?  = 2r  11  1  " 1 J  J 1 0  ( P  i<Pio> i PlO J  10  (  )  1  )  1  1  1  1  3  4  dx  ir 1 ) j 2 ( p ii ) + 2 p ii J i ( p ii ) J i ( p ii )  1115  = 4(P )  H16  n  IH7  2  1  2 1  x(K'(x)) P ^ 1  K (x) 2  +  Z  2  .- - < P 2 i ) i 2 1 + 1  1  )  K (x)Kj_(x)dx = - K ( P ) 9  = 2*  x  )  " P  X  A  a  (  Z  1  - P u ^ i ^ n ^  ft  (  1  p  A  4 Pll  = 2J^J (x)Jj_(x)dx =  ll Ar = 2 X x ( J 10 5  PuVPu)J1<P >  K  ( p  )  dx  x  " P 1 2  ( K  1  ( P  21  ) ) 2  "  2 p  21 l K  ( p  21  ) K  l  ( p  21  )  1  1  1  8  135.  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