PROPERTIES OF SURFACE WAVEGUIDES WITH DISCONTINUITIES AND PERTURBATIONS IN CROSS-SECTION by Gary H. Brooke B.A.Sc. University of British Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1977 (c ) Gary H. Brooke, 1977 in. the Department of E l e c t r i c a l Engineering In present ing th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f r ee ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date .ffl-K) ~2-l ] I 1 1 7 ABSTRACT The f i r s t part of this thesis i s concerned with theoretical and experimental investigations of step discontinuities on a planar surface waveguide. An approximate theoretical solution to the unbounded discon-tinuity problem i s obtained by bounding the open structure with perfect conductors, since there i s a direct relationship 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 discontinuity configurations considered by other workers. Good agreement with the previous results i s obtained i n a l l cases except one for which the original results are shown to be inaccurate. The experimental investigation i s carried out on a di 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 longitudinal component of electric f i e l d . The parameters of interest are the magnitude and the phase of the reflec-tion coefficient. The experimental results confirm those obtained theore-t i c a l l y . The theoretical and experimental techniques are later applied i n an investigation of a cascaded step discontinuity configuration. The theo-r e t i c a l approach involves the use of wave transmission matrices. Experi-mental results for the magnitude of the reflection coefficient are found to be i n reasonable agreement with theory. The second part of this work describes a theoretical and experi-mental study of dielectric waveguides, of circular cross-sections, per-turbed by axial slots. In particular, the normalized propagation coefficient ( i i ) of the dominant modes with each polarization i s determined analytically using a standard perturbation technique and experimentally using an open resonant cavity. The perturbation results give quite a good indication of the trends observed experimentally. It i s found that there i s an optimum size of the perturbation which gives the maximum separation between the normalized propagation coefficients of the two polarizations. ( i i i ) CONTENTS FIGURES . . . . . . «• . . . . . . . . . . . . . . v i TABLES . - . . . . • . . . . . . . . ... .. . . . . . . . x PRINCIPAL SYMBOLS . . . . . . . . . . . . . . . . . . . x i ACKNOWLEDGEMENT ....... . '• . . . , . # . # x l v GENERAL INTRODUCTION AND BACKGROUND . . . . . . . . . . . XV PART I SURFACE WAVEGUIDE DISCONTINUITIES 1 1. INTRODUCTION X*X Ssclc^irouiid • • « • * • « e • © © © • « • • 2 1.2 Scope of P a r t I A 2. THEORETICAL INVESTIGATION OF A STEP DISCONTINUITY -ON A PLANAR SURFACE WAVEGUIDE 2.1 An approximate s o l u t i o n technique . . . . . . . . . . . 7 2.2 Modes of the bounded waveguide 10 2.3 Mode-matching . . . . . . . 1 4 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 c o n f i g u r a t i o n s . . . . . 19 2.5 Summary . 25 3. EXPERIMENTAL INVESTIGATION OF STEP DISCONTINUITIES 3.1 Experimental waveguide c o n s t r u c t i o n and e x c i t a t i o n 27 3.2 Measurement apparatus and p r o p e r t i e s of the su r f a c e waveguide . . . . . . . . . . . . . . . . 31 3.3 F a c t o r s a f f e c t i n g standing wave measurements • • •• 40 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 w i t h theory . . . . .. . . . . 47 3.5 Summary 55 4. APPLICATION TO CASCADEL DISCONTINUITIES 4.1 Wave ma t r i x f o r m u l a t i o n . . . . . . . . 58 4.2 Experimental i n v e s t i g a t i o n . . . . . . . . . . . . . 63 4.3 Comparison w i t h theory . . . . . . . . . . . . . 65 4.4 Summary 67 PART II CYLINDRICAL DIELECTRIC WAVEGUIDES WITH PERTURBED CIRCULAR CROSS-SECTION . . . . . . . . . . . 6 8 5. INTRODUCTION 5.1 Background . . . . . . . . . . . 69 (iv) 5.2 Scope of P a r t I I . 7 0 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 i 74 6.2 A p p l i c a t i o n of the p e r t u r b a t i o n method 7 6 6.3 Summary 80 7. AN EXPERIMENTAL INVESTIGATION 7.1 Experimental apparatus and p r e l i m i n a r y measurements . . . . 8 2 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 waveguides 91 7.3 R e s u l t s 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 7.4 Summary 99 8. SUMMARY AND CONCLUSIONS 101 8.1 Suggestions f o r f u r t h e r work 104 Appendix I-A FIELD BEHAVIOUR NEAR ANISOTROPIC AND MULTIDIELECTRIC EDGES . . . 1 0 6 I - A . l Edge i n v o l v i n g a magnetized f e r r i t e . . . . . . 108 I-A. 2 Edge i n v o l v i n g three d i e l e c t r i c s . • . -. . . . • 112 Appendix I-B EFFECT OF DIELECTRIC EDGE CONDITIONS ON MODE-MATCHING SOLUTIONS . . . 120 Appendix I-C INTEGRALS INVOLVED IN EQUATION 2.19 . . . . . . 124 Appendix I-D ATTENUATION COEFFICIENTS .• . . . . . . . . 126 Appendix I-E MEASUREMENT OF STANDING WAVES WITH A X/2 DIPOLE 12? Appendix I-F DERIVATION OF WAVE MATRICES w;L AND w2 . . . . . 1 3 0 Appendix I I - A DERIVATION OF EQUATION 6.1 . . . . . . . . . 131 Appendix I I - B SELECTED INTEGRALS OF BESSEL FUNCTIONS &±, i = 1, . . . .8 . . . . . . . . . . . . 134 REFERENCES . . . . . . . . . . . . . . . . . . . . 1 3 5 (v) FIGURES 2.1 Step discontinuity configurations (a) unbounded (b) bounded 8 2.2 Normalized propagation coefficient versus e =2.26, t n = 0.079 X 15 B Q r 1 o 2.3 Normalized (i) transmitted ( i i ) radiated ( i i i ) reflected 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 results for the configuration i n insert A a' original results from reference (47) b present results for the configuration i n insert B b' original results from reference (48) 20 2.4 Normalized radiated power. e i / e 0 = 1«02, t^/t^ = 2.0 present results original results from reference (49) 22 2i5(a) Magnitude of the reflection coefficient versus t^/X . e,/en = 9 .0 , e_/e = 10 .0 , t,/X = 0 . 0 6 l o 2 Q ' l o a present results, N = 50 a' N = 20 a" N = 10 b original results from reference (52) 23 2.5(b) Percentage power in the radiation and the transmitted surface-wave modes. Curves "a" and "b" as i n Fig. 2.5(a). . 24 3.1 A part of the experimental surface waveguide and of the measurement apparatus. . . . . . . -.. . . . . . . . . . 28 3.2 Excitation of the surface waveguide . . . . . . . . . . . . 30 3.3(a) The dipole used for f i e l d probing 32 3.3(b) The coaxial line and waveguide detection unit . 33 3.4 Schematic for the complete measurement system 35 3.5 The decay of the surface wave away from the dielectric surface 3.6 Amplitude and phase characteristics (abscissa i s transverse distance from the beam axis) . . . . 3.7 Attenuation measurements ( exponential attenuation) . . . . . (vi) 3.8 Amplitude patterns o nearer to the source x midpoint A far from the source . . .' 41 3.9 Estimated values of beam width versus distance along the beam axis 42 3.10(a) Variation i n standing wave ratio values as a function of the height of the dipole above the dielectric surface . . . 44 3.10(b) A decay curve indicating heights at which the standing waves were measured 45 '3.11(a) Example of variation in the standing wave ratio values as a function of the distance from the step discontinuity ( theoretical dependence of standing wave ratio) 48 3.11(b) same as (a) but for a different step height 49 3.12 An example of a measured standing wave pattern 50 3.13 Experimental step discontinuity configuration 51 3.14 Experimental results for 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 Cascaded step discontinuity configurations (a) unbounded (b) bounded -. . . 59 4.2 Magnitude of the reflection coefficient versus &;A0 o experiment theory (30.0 GHz) theory (30.8 GHz) 64 5.1 Finned dielectric waveguides - possible interconnection and coupling schemes 72 6.1 Dielectric rod configurations (a) unperturbed (b) perturbed by slots 75 6.2 Quasi-static approximations (a), (b), (c), (d): simple configurations (e) configuration corresponding to a slot (f) correction factor K . . . . . . . . 77 (vii) 7.1 Experimental configurations of the perturbed dielectric rods 83 7.2 Resonant cavity apparatus (insert—the coupling probe and the scatterer) 84 7.3 Cross-section of the central part of one end plate of the cavity indicating probe and locating pin arrangement . . . . 85 7.4 Chart recording of the output of the cavity 86 7.5 Schematic of the measurement system . . . . . . . . . . . . 88 7.6 Resonance curves (horiz. scale = 3.5 MHz/large div. ) (a) frequency measurement using the reference channel trace (b) "double" resonance . . . . . . . . . . . . . . 90 7.7 Normalized propagation coefficients of the two polariza-tions as a function of 6 (fl) Tod A • • • • • • • « e A e-tttte • o e • • »« • • • • 93 (b) rod B 94 (c) rod C . 95 7.8 Resonant frequency separation of the two polarizations on a perturbed rod (horiz. scale - 3.5 MHz/large div.) . . . . 96 7.9 Slot configurations (a) approximation of a slot in rectangular coordinates by one in polar coordinates (b) actual slot configuration for rod B . . . . . . . . . . 98 11 Edge configurations (a) dielectric-magnetized f e r r i t e Cb) three isotropic dielectrics . . . . . . . . . . . . . . Q^7 12 Behavior of the singularity parameter t as a function of $ ± and <j>2„ . . . . . 1 1 4 13 Region of nonsingularity for an edge common to three dielectrics . . . . . , . . . . . . . . . . . . . . . . . . 2.16 14 Dependence of a region of nonsingularity on the permittivity with e2 > e i > e3* * * * * . . . . • • H7 15 Dependence of A<^ 1 on the ratio of the highest to lowest 16 Interface between empty and H-plane loaded waveguide e r - 8, y r = 0.75, 2t/b = 0.7, b/V = 0.3, a/XQ = 0.< (a) waveguide configuration (b) lumped-elemerit equivalent c i r c u i t of the interfa ( v l i i ) Rate of decay of amplitude coefficients of reflected modes in waveguide A ° lAml modes) * 'Aml ( ™ i m m o d e s ) • Dipole coordinates Volume region . . . . . (ix) TABLES 2.1 Correspondence between the bounded and the unbounded mode spectra « 2.2 Convergence of slow mode scattering coefficients as a function of t„/X . e = 2.26, t = 0.079X , B o r 1 o t 0 = 0.555X 18 2 o 3.1 Experimental and theoretical values of the phase angle of the reflection coefficient for a range of step discontinuity heights 56 4.1 Convergence of slow mode scattering coefficients (obtained by wave matrices) as a function of V X n 62 B o 7.1 Values of f r and 6/kQ for the unperturbed rods 92 00 PRINCIPAL SYMBOLS a^ = complex amplitude coefficient of the kfc^ mode in waveguide A A = uniform section of the bounded planar waveguide A ,B ,C ,D = amplitude constants for n*"*1 surface wave mode on a dielectric n n n n rod = complex amplitude coefficient of the j * * 1 mode in waveguide B —1'—1'—2 = v e c t o r s °f normalized amplitudes of modes travelling in the negative z direction B = uniform section of the bounded planar waveguide c^ ,£^ ,c_2 = vectors of normalized amplitudes of modes travelling in the positive z direction C = amplitude constant d = distance from a step discontinuity = r. - r for a slot 1 o d -k Z B _ o o n n A n n e ,e ,h = component form of the mode functions for the bounded planar waveguide e ,h = e , h for waveguide A a a x y e^,h^ = e^, h^ for waveguide B E_,E ,E = electric field components in polar coordinates (r,0,z) r Q z Ep,E(j),Ez = electric field components in 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-guide f r = resonant frequency (xi) J = Bessel function of the f i r s t kind n k = 2TT/A 0 o K = modified Bessel function of the second kind n R , KQ = quasi-static correction factors for the r and 9 f i e l d r 8 components N = number of modes used in mode-matching = number of half wavelengths i n the resonant cavity ^l'^2'^2 = t r a n s v e r s e propagation coefficients P , P = normalization constants in waveguides A and B a b s = standing wave ratio S = scattering matrix t„, t„ = electric and magnetic singularity parameters £1 n t,,t_,t„,t. = thicknesses of dielectric layers 1 2 3 4 t„ = conductor spacing o S f t^, t^ = transition values of t^ tan $ = loss tangent w • ( 6 1 , - . e o ) ( r - + r )/2 for a slot 1 z 1 o W 1 , W 2 , W T = w a v e a m p l i t ^ 6 matrices W = overall wave matrix for a cascaded section Y = /e /y ' o o o Z - 1/Y o o A , AM' aD = attenuation coefficients 3,B o >Bp = phase coefficients (xii) 6 r perturbation parameter tS.. = Kronecker delta i j = integral of Bessel functions Ae^ = change in relative permittivity within a region of pertur-bation G Q = permittivity of free space e = permittivity of a dielectric e r <= relative permittivity 8 = phase angle of the reflection coefficient K = off diagonal term of the permeability tensor X Q = free space wavelength X - waveguide wavelength u - diagonal element of the permeability tensor = permeability of free space p,p = magnitude of the reflection coefficient a = conductivity (i) = angular frequency I = distance between cascaded discontinuities ( x i i i ) ACKNOWLEDGEMENTS The author is grateful for the opportunity of working under the supervision of Dr. M. M. Z. Kharadly, whose interest and expertise have been forwarded in such a personal manner that they shall not soon be forgotten. Grateful acknowledgement is made to the National Research Council of Canada for a postgraduate bursary in 1972-1973, for a postgraduate scholar-ship during the period 1973-1976 and for support of the project under grant A-3344. The author i s also grateful to Dr. W. K. McRitchie for his c o l l a -boration on the work presented in Appendix I-B, to Mr. J. Stuber and to Mr. D. Daines for their excellent workmanship in constructing various parts of the experimental apparatus, to Mr. T. K. Chu for his professional photographic assistance and to Mrs. A. Semmens and Ms. M. E. Flanagan for their help in typing the manuscript. Finally, I would l i k e to thank my wife, Sharon, for typing two drafts and a part of the f i n a l copy of this thesis and, more important, for assuming many of my family responsibilities throughout the course of this study. -(xiv) GENERAL INTRODUCTION AND BACKGROUND Surface waveguides are used i n a v a r i e t y of 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 through o p t i c a l . The most common types of sur f a c e waveguides are e i t h e r p l a n a r or 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 are open (or unbounded) s t r u c t u r e s , surface i m p e r f e c t i o n s cause power to be l o s t i n t o r a d i a t i o n and they are su b j e c t to 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 are u s u a l l y considered as d e t r i m e n t a l e f f e c t s . In 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 of sur f a c e waveguides at o p t i c a l f r e q u e n c i e s . In c y l i n d r i c a l form, these wave-guides are r e f e r r e d to as o p t i c a l f i b r e s . S e v e r a l a p p l i c a t i o n s f o r (8,9) o p t i c a l f i b r e s ( c i v i l ( 1 - 3 ) , i n d u s t r i a l ( 4 ' 5 ) , and m i l i t a r y ( 6 , 7 ) ) have been proposed. Indeed, some experimental o p t i c a l communications l i n k s have already 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 ^ * ' ^ ^ of 5-10km could be r e a l i z e d . The b a s i c f i b r e c o n s i s t s of a core r e g i o n surrounded by a cl a d d i n g medium of s l i g h t l y lower r e f r a c t i v e index. The c l a d d i n g serves to strengthen the f i b r e and s h i e l d the core from the environment. I f the core i s chosen such that only one mode propagates, the f i b r e i s c a l l e d s i n g l e mode; whereas, i f the core supports many modes, the 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 the l a r g e r bandwidth (13) c a p a b i l i t y but r e q u i r e s a l a s e r l i g h t source f o r e f f i c i e n t e x c i t a t i o n . Such a source, does not, as y e t , possess the 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 . In a d d i t i o n , s i n g l e mode (xv) f i b r e s r e q u i r e c r i t i c a l alignment a t j o i n t s ( t o prevent e x c e s s i v e l o s s ) ; t h i s i s f u r t h e r complicated when c a r r i e d out under f i e l d con-d i t i o n s . A l t e r n a t i v e l y , multimode f i b r e s can be e x c i t e d e f f i c i e n t l y w i t h incoherent l i g h t sources such as l i g h t - e m i t t i n g diodes Furthermore, because of t h e i r l a r g e r core dimension, they do not s u f f e r from the s t r i n g e n t 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 of the many propagating modes, each c a r r y i n g energy at a d i f f e r e n t group v e l o c i t y , the multimode f i b r e has poorer 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 the bandwidth (19) c a p a b i l i t y i s reduced. Attempts to overcome the d i s p e r s i o n problem i n multimode f i b r e s have r e s u l t e d 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 the core has a near p a r a b o l i c r e f r a c t i v e index p r o f i l e . The r e s u l t i s a f o c u s s i n g e f f e c t at p e r i o d i c i n t e r v a l s along the f i b r e which tends to e q u a l i z e the group delays. I t may be of i n t e r e s t to note that random (21 22) im p e r f e c t i o n s i n the c o r e - c l a d d i n g i n t e r f a c e ' may a l s o reduce (23) d i s p e r s i o n i n multimode f i b r e s through mode conversion e f f e c t s (an example where im p e r f e c t i o n s are b e n e f i c i a l ) . Although not e s s e n t i a l a t present, there may be a need w i t h i n the next decade f o r the bandwidth c a p a c i t y of o p t i c a l f i b r e s . Due to 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 that multimode f i b r e s could be u s e f u l f o r i n i t i a l i n s t a l l a t i o n s . P l a n a r surface waveguides are 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 data p r o c e s s o r s . These waveguides form the b a s i s (24) of " i n t e g r a t e d o p t i c s ' , which has been developed c o n c u r r e n t l y w i t h o p t i c a l f i b r e s . The p o s s i b i l i t y of o p t i c a l t r a n s m i s s i o n would be enhanced i f data p r o c e s s i n g (e.g. c o u p l i n g , modulation, and s p l i t t i n g ) could be (xvi) (25) accomplished at optical frequencies (by eliminating some of the circuitry from repeaters and, to a certain extent, from the transmitter and receiver). Generally speaking, the devices consist of a thin guiding layer of dielectric deposited on a substrate. The substrate may serve only to support the dielectric waveguide or may be Involved directly in specific components (such as mode converters, isolators, etc.) Since the distances involved in integrated optical devices are of the order of centimeters, as compared to kilometers i n optical fibres, the loss requirements in the dielectric materials used need only be of the order of 0.5db/cm. It i s interesting to note that several passive devices which incorporate discontinuities directly into the component (27 28) (29) design, such as couplers ' , lens and prisms , have been developed. Again these are examples of applications i n which discontinuities are used to advantage. At lower frequencies, surface waveguides have been investigated for various communications applications. In the millimeter range two (30 31) proposals ' , involving waveguides analogous to optical fibres but with loss characteristics 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 changes i n the waveguide dimensions. Such antennas are of light weight and are highly directional. More recently, i t has been proposed that surface (33) waveguides be used as antenna feeds . At ultrahigh frequencies the increased f i e l d extent of surface waveguides make them natural candidates for high speed vehicle signalling and control i n connection with railroad applications ' . The waveguides serve the dual purpose of detecting (xvil) s t a t i o n a r y and moving ob j e c t s on the r a i l s and of communicating w i t h the high speed v e h i c l e s . The above b r i e f review 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 or u n i n t e n t i o n a l , occur i n e v i t a b l y i n su r f a c e waveguide systems and thus need to 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 low-loss waveguides f o r t r a n s m i s s i o n systems a t the higher microwave and m i l l i m e t e r wave fr 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 to 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 -t i c s at these f r e q u e n c i e s . P r a c t i c a l systems, however, have y e t to be developed. Scope of the Present Work There are two aspects to the work des c r i b e d i n t h i s t h e s i s . The f i r s t was motivated by the need f o r the accurate determination of the e f f e c t of d i s c o n t i n u i t i e s on su r f a c e waveguides. The second i s an e f f o r t to develop a p r a c t i c a l l o w -loss low-cost 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 at the hi g h e r microwave and m i l l i m e t e r wave fr e q u e n c i e s . Thus, the 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 deals w i t h a n a l y t i c a l and experimental determination of the s c a t t e r i n g parameters of abrupt 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 bounding the open s t r u c t u r e by p e r f e c t conductors. The bounded problem i s then solved by co n v e n t i o n a l techniques. This procedure has the advantage of a l l o w i n g cascaded d i s c o n t i n u i t i e s to be handled by a wave ma t r i x f o r m u l a t i o n . P a r t I I i s concerned w i t h the propagation on c y l i n d r i c a l s u r face 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 . In p a r t i c u l a r , ( x v i i i ) the phase coefficients of d i e l e c t r i c rods, with axial slots, are determined theoretically and experimentally. This configuration is one of several possible configurations that may be suitable to implement in a practical surface waveguide system, in which i t i s reasonably easy to connect different parts of the system together and where a well defined polarization i s maintained. (six) PART I SURFACE WAVEGUIDE DISCONTINUITIES 2. Chapter 1 INTRODUCTION 1.1 Background Probably the f i r s t investigations of surface waveguide discontinuities occurred in the late 1940's, in 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) for predicting the radiation patterns of abruptly terminated dielectric waveguides. He also gave theoretical and experimental results for the dielectric rod, tube, and horn antennas. (37) In 1954, Butterfield measured the radiation patterns of various tapered dielectric slab waveguides. The scattering of surface waves Incident upon abruptly terminated dielectric waveguides was also con-/og\ (39} sidered by Angulo and Chang for the cylindrical case and by Angulo for the planar case. In references (38) and (39) variational expres-sions for the terminal impedance of the waveguide and for the transfer impedance between surface waves and radiation were derived. The resulting integral expressions were evaluated assuming that only the incident mode was excited at the terminal plane. In 1959, Kay^ 4^ i n i t i -ated several investigations of reactance surface discontinuities using the Wiener-Hopf technique. Kay obtained universal relative power curves for the reflection, transmission and radiation i n the planar case. Later, Breithaupt extended the method to the cylindrical structures involving (41) (42) TM modes and hybrid modes . Experimental results, indicating (43) reasonable agreement, were presented for 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 reactive plane (44) (45) and upon an abruptly terminated thick reactive slab . Felsen applied a Green's function formulation to obtain results for the radiation from a surface with a gradual taper i n reactance. The f i r s t analysis given to scattering by abrupt changes i n thickness of dielectric slab waveguides was provided, i n 1966, by Cooley and Ishimaru^^ . An integral equation technique, involving the solu-tions to two related problems, was used to obtain results for the reflec-tion, transmission, and radiation. In addition, an equivalent c i r c u i t for the discontinuity and some measured radiation patterns were presented. (47) Clarricoats and Sharpe used a mode-matching technique to solve for the fields scattered by a junction between two different planar dielectric slabs. The same problem was treated by Hockham and S h a r p e u s i n g an integral formula. Marcuse investigated the scattering 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 results for the radiated power were given in the latter case. Shevechenko^^ introduced the concept of a relaxed radiation condition in obtaining results for slowly varying continuous transitions on various planar and cylindrical surface waveguide configurations. (52) Mahmoud and Beal later applied this concept i n treating a problem involving an abrupt change in die l e c t r i c waveguide dimensions and composition. F i r s t order perturbational solutions to coupled mode formu-lations of gradual changes in waveguide cross-section have also been applied by various workers. S n y d e r u s e d this technique to obtain approximate solutions to propagation problems i n connection with tapered retinal receptors and tapered optical fibres. Marcuse used a similar analysis to study mode conversion on optical waveguides 4. with random imperfections. He also performed experiments, using corru-gated cylindrical dielectric waveguides, which verified the theory A related problem was also considered by S t o l l and Y a r i v ^ 1 ^ using (62) essentially the same coupled mode analysis and by Sakuda and Yariv using Floquet expansions. Tuan analyzed small Gaussian shaped dis-continuities on dielectric waveguides using a Green's function technique. Rawson^4^ applied an Induced dipole method to the problems involving small steps, sinusoidal cross-sectional change, and sinusoidal meandering of dielectric waveguides. Rulf^"^ presented yet another integral equa-tion technique valid for small discontinuities. A recent investigation of a d i e l e c t r i c rod antenna has been made by Yaghjian and Kornhauser^^ . Their analysis involves replacing the unbounded problem by a bounded one. Several theoretical radiation patterns were presented. 1.2 Scope of Part I The review presented in the previous section Indicates that: (a) Theoretical studies of discontinuities (employing various problem formulations) are generally applicable to specific and somewhat limited configurations. With the exception of some of the reactance surface discontinuity studies, approximate solutions are obtained by neglecting parts of the mode spectrum. In addition, the solutions ob-tained contain the continuous spectrum of radiation f i e l d s , making the investigation of cascaded discontinuities v i r t u a l l y impossible. (b) Experimental investigations of scattering by surface waveguide discontinuities have generally been restricted to measurement of radiation patterns. Very l i t t l e work has been done on reflection and transmission studies. 5. Thus, the work presented in Part I of this thesis i s an attempt to provide: (1) an accurate analytical method, of general applicability, for determining the scattering parameters. The method i s suitable for transmission matrix studies of cascaded discontinuities. The approach used involves bounding the open surface waveguide with perfect con-ductors, thereby making the entire mode spectrum discrete. 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 specific discontinuity config-uration was chosen—that provided by an abrupt change i n d i e l e c t r i c thickness of a dielectric-coated conductor. Since TM modes are easier to excite experimentally on this waveguide, the investigation i s restricted to TM modes. In principle, the methods are applicable to TE modes, how-ever. The basis of the theoretical treatment i s presented i n Chapter 2. I n i t i a l l y , the similarity of the fields in the open and the corresponding closed problem i s examined. Then, the mode-matching procedure i s described and solutions are presented for several cases considered previously i n the literature. Chapter 3 is concerned with the experimental investigation. The waveguide and i t s construction, the probe used for f i e l d measurement and the measured properties of the experimental surface waveguide are discussed. The measurement of the reflection coefficient i s described i n some detail and the results for a range of steps are compared with theoretical 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 theoretical analysis to include cascaded discontinuities. This i s followed by a specific example which i s investigated theoretically and experimentally. Chapter 2 THEORETICAL INVESTIGATION OF A STEP DISCONTINUITY ON A PLANAR SURFACE WAVEGUIDE The main o b j e c t i v e of t h i s i n v e s t i g a t i o n i s to determine the s c a t t e r i n g , of a su r f a c e wave, by the step d i s c o n t i n u i t y shown i n F i g . 2.1(a). In g e n e r a l , on a uniform s e c t i o n of the 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 spectrum of surface wave modes and a continuous spec-trum of both r a d i a t i o n modes and attenuated r a d i a t i o n modes. The con-tinuous mode s p e c t r a 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 not i m p o s s i b l e . 2.1 An Approximate S o l u t i o n Technique Consider the bounded d i s c o n t i n u i t y problem shown i n F i g . 2.1(b) In t h i s c o n f i g u r a t i o n the modes are a l l d i s c r e t e . G e n e r a l l y speaking, they c o n s i s t of: a f i n i t e number of slow and f a s t modes and an i n f i n i t e number of evanescent modes. There i s a d i r e c t correspondence between the mode s p e c t r a of the unbounded and the bounded c o n f i g u r a t i o n s shown i n F i g s . 2.1(a) and ( b ) , r e s p e c t i v e l y . This r e l a t i o n s h i p i s summarized i n Table 2.1. In the l i m i t , as the upper conductor ( i n F i g . 2.1(b)) i s moved to i n f i n i t y , the mode sp e c t r a of the two c o n f i g u r a t i o n s must become i d e n t i c a l . For p r a c t i c a l purposes, i f the bound i s moved only a few wavelengths from the d i e l e c t r i c s u r f a c e , the slow modes i n the bounded s t r u c t u r e become v i r t u a l l y the same as the s u r f a c e wave modes on the unbounded s t r u c t u r e . The r e l a t i o n s h i p between the modes ( p a r t i c u l a r l y that between slow and su r f a c e wave modes) i n the unbounded and bounded waveguides suggested the present approximate s o l u t i o n technique. An approximate s o l u t i o n i s obtained by s o l v i n g the bounded problem, 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 the s u r f a c e , and then a s s o c i a t i n g 8. X CT = oo (b) fig. 2.1. Step discontinuity configurations (a) unbounded (b) bounded. UNBOUNDED BOUNDED. PROPAGATION] CONSTANT ft MODE SPECTRUM MODE SPECTRUM SURFACE WAVE DISCRETE SLOW DISCRETE o</3<-k0 RADIATION CONTINUOUS FAST DISCRETE -JCO</3 <jo ATTENUATED] RADIATION \ CONTINUOUS EVANESCENT DISCRETE Table 2.1 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 attenuated r a d i a t i o n modes w i t h the slow, f a s t and evanescent 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 the bounded c o n f i g u r a t i o n can be determined, i n a s t r a i g h t f o r w a r d manner, u s i n g mode-matching^^'^^. Since d i s c o n -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 when solved by mode-matching^^ i t appeared necessary, f o r the purpose of the present work, to i n v e s t i g a t e two separate problems: (72) 1. the behaviour of electromagnetic f i e l d s near 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 a l s o i n c l u d e d 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. the e f f e c t of d i e l e c t r i c edge c o n d i t i o n s on mode-matching s o l u t i o n s These i n v e s t i g a t i o n s are i n c l u d e d i n Appendix I-A and I-B, r e s p e c t i v e l y . 2.2 Modes of the Bounded Waveguide Consider the uniform s e c t i o n of 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 are given by: (the ^. c ^ j(cot-8z) . propagation f a c t o r e J i s assumed) h = cos p.x y 1 ex = F T T " c o s p i x B_ _ t o o r = -.—= s i n p..x 9 1 z k Y e 1 o o r for 0 <_ x <_ and h = C cos p _ ( x - t ) y z c CB , x ex = FT" C O S P 2 ( x " t B ) o o JC p 2 ez = 0 ~ S i n P2 ( x" tB ) 2 , 2 do 11. for t < x < t . Where e = e/e i s the relative permittivity of the 1 — — B r o dielectric and where p.^ and p 2 are the transverse propagation coefficients in the dielectric and free space regions, respectively; given by: 2.3 The fields tangential to the dielectric interface at x = t^ must be con-tinuous , hence c - C ° S V l •_ J l _ S l " P l *1 2 4 c o s P 2 ( t 1 " t B ) G r P2 S i n p 2 ( V t B ) Thus, the eigenvalue equation, obtained by equating the two expressions for C i n equation 2.4, i s : P 1 t a n p 1 t 1 o c r p 2 tan p 2 ( t l - t B ) 2 5 There is an i n f i n i t e number of solutions to equation 2.5 which, i n con-junction with equation 2.3, yield values of the modal propagation c o e f f i -cient 8. The mode type (slow, fast or evanescent) i s determined by the value of B , as shown in Table 2.1. The dominant TM^ (slow) mode corres-ponds to the dominant TEM mode in the empty waveguide (i.e. t^ = 0) and has no low frequency cutoff. The number and type of higher order propa-gating 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 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 0 0 < 3 < 0). As t^ increases from a value of zero, the propagation coefficients i n -crease and, hence, the TEM mode becomes the dominant TM slow mode. At 12. certain transition values of t^, denoted t^ , higher order fast modes pass into the slow mode spectrum while at other transition values of t^, denoted t^> evanescent modes pass into the fast mode spectrum. The s f transition values, t^ and t^ , can be found by placing 8 = and 8 = 0 , respectively, i n equation 2.5. Hence, k X. t f = ° k = 1, 2, ... 2.6 1 K 2v /T^l r and tan k ^ " t.f = ST tan k <t* - O . 1 - 1 , 2 , . . . 2.7 o r 18. r o L% a Thus, for a particular choice of the parameters e , t- and t^, the total r i b number of slow modes i s given by: N = 1 + K 2.8 s and the total number of fast modes i s given by: N f = M - K + L 2.9 s f where K and L are the number of the transition values t^ and t^ , res-pectively, i n the range of 0 to t^. It may be worthwhile, at this point, to examine the relation-ship between the slow modes of the bounded waveguide and the surface wave modes of the corresponding unbounded waveguide. The slow modes de-2 2 2 cay in the region t- < x < t and hence p„ = j y 0 (Y 0 = 3 ~ k ). Thus, from equations 2.2 and 2.4, the mode functions i n the region t^ _< x <_ tg become (they are the same for 0 < x < t ^ ) : cos p^ t^ h = : -, r cosh Y o ( x - t T > ) y cosh Y 2 ( T 1 - T B ) 13. B cos ^ e = -—- r r -— — v c o s n Y 0 ( x _ ti>) x k Y cosh Y->(ti-t„) 2 B O O Z 1 D —JYo c o s Pi t-i e = r - ^ . „ , sinh Y 0(x-t B) 2.10 z k Y cosh Yo(t,-O '2 B' O O Z ± D The eigenvalue equation also changes to: tan p x t 1 = -e Y 2 tanh Y 2 ( t 1 - t B ) 2 t l 1 By comparison the mode functions for the corresponding surface waves (denoted by primed quantities) are given by^"^ h ' = cos p,' x y 1 i B' t e = -— cos p.* x x k.Y e r l o o r e ' z k Y o o r for 0 ^ _ x <_ t^, and =— sin p.' x 2.12 £ 1 -Y 2'(x- t l) h ' =• cos p.'t e y 1 1 B* cos p ^ t x ^ ' ( x - t ^ e x k Y o o - j Y 2 ' cos t± -y^ix-tj e z k Y o o 2.13 2.14 for x j> t^. The corresponding eigenvalue equation i s given by P l ' tan P l ' t x = e r Y 2 ' where (Pi') 2 " e r k o 2 - ( 3 ' ) 2 ( Y 2 ' ) 2 - (3') 2 - k Q 2 2.15 It can be easily shown that, as 0 0, equations 2.1, 2.10 and 2.11 14. tend to equations 2.12, 2.13 and 2.14, respectively. For practical pur-poses, however, t need only be a few wavelengths before the slow modes assume vi r t u a l l y the same form as the corresponding surface wave modes. In order to i l l u s t r a t e this, a specific numerical example i s con-sidered. Let e = 2.26 and t, = 0.079 l... The normalized propagation r 1 o coefficient 8/k of the TM slow mode is plotted versus t^/X in Fig. 2.2. o o " o The value of 3'/k (dashed line) i s also shown on the same figure. Fig. o -2.2 indicates that 8/k and 8'/k are the same, to four significant f i g -o o ures, i f t_/A > 3.0. This result typifies the rapid convergence of the B O slow modes to the corresponding surface modes as t^ is increased. 2.3 Mode-Matching The application of mode-matching to a waveguide discontinuity, such as that shown in Fig. 2.1(b), r e q u i r e s : 1. expanding the total fields on either side of the discontinuity in terms of the modes in the respective uniform waveguides, 2. enforcing the continuity of the tangential fields i n the plane of the discontinuity, and 3. using the orthogonality relationships between the modes to obtain a system of linear equations in the unknown mode amplitude coefficients. For the configuration of Fig. 2.1(b), the continuity of total tangential ele c t r i c and magnetic fields i s represented by (for the r*"*1 mode, i n wave-guide A, incident on the discontinuity): CO 00 (1+a ) e + I a, e .' - 7 b. e, . 2.16 r ~ a r i=i 1 j=i • J -*>J i ^ r and a V s « - ^ ^ s * - ^ " j f i b j 2 - 1 7 W r 15. 16. where the coefficients { a ^ and {b^} are unknown mode amplitudes and where (e a, h a) and ( e ^ h^) are the transverse mode functions (e x, h y) in waveguides A and B, respectively. The orthogonality relationships can be expressed as: / a e . x h , - a d = 6 P . — a i —ak —z x ik ak / * x h ^ • d x = 6.^ P w Q ^bj u x " "jA 'b£ 2 , 1 8 where 6., and 6. are Kronecker deltas and where P , and P, are normal-lk jl ak hi ization constants for the k and l modes in waveguides A and B, res-pectively. In order to solve equations 2.16 and 2.17, the f i e l d expan-sions must be truncated. In this investigation an equal number of modes are considered on either side of the discontinuity (i.e. gives more rapid convergence for discontinuities involving dielectric edges — Appendix I-B) By applying equations 2.18 to J » (2.16) x h ^ ' ^ d x and ^ ^ak X ( 2 * 1 7 ) ' Sz d X It can be shown t h a t ^ 7 ^ N rh I b (e^ x h ^ + e ^ x h y ) • a^x = 2 6 k r P a r =1 J 0 J 2.19 and 5 _ V A f ^ e X h • a dx 2.20 ^ k r A Pak 0 ^ ^ "* with k = 1, ... N. The integrals involved i n equations 2.19 and 2.20 are given in Appendix I-C. Equation 2.19 represents a system of N l i n -17. ear equations in the N unknown mode amplitude coefficients b^, j=l, ... N. Solutions of equation 2.19 are then substituted i n equation 2.20 to yield values for the mode amplitude coefficients in waveguide A. Thus, using the procedure outlined above, approximations to the scattering coefficients 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 mode-matching. It i s of interest to examine the behaviour of the mode-matching solutions (for the slow modes in particular) to the discontinuity problem, shown in Fig. 2.1(b), as a function of t_/X . This can be best achieved a o by considering an example. Let e = 2.26, t = 0.079 A and t„ - 0.555 X r 1 o <£ o and l e t the TM mode be incident from waveguide A. For this choice of o parameters there i s one slow mode in waveguide A and two slow modes in waveguide B. The reflection coefficient a^ and the transmission c o e f f i -cients b, and b„ are given in Table 2.2 for various values of t^/A . I / D O Also l i s t e d are the percentage errors in power conservation and total percentage power scattered into the fast modes. A l l of the results in Table 2.2 were obtained with N = 50. The scattering coefficients and the fast mode power converge to two significant figures with t^/X = 3.0. a O S t r i c t l y speaking, better convergence in the least significant digits could be obtained by increasing N a proportionate amount for each i n -crease in t„/A . (i.e. make the power conservation errors equal at each B o value of t B/X ). The increment i n N needed to maintain the same accuracy B o was v i r t u a l l y impossible to gauge beforehand, however. An increase in 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 coefficients for the corresponding unbounded SCATTERING COEFFICIENTS a l b l b2 FAST MODE ,. POWER % POWER CONSERVATION ERROR % 1.0 0.1325 /-176.280 1.0469 /-0.16° 0.3102 7-178.15° 1.5123 0.0002 2.0 0.1346 /-175.790 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 coefficients as a function of t_/X . e - 2.26, t. = 0.079X , t = 0.555X . B ° r 1 ° 2 o discontinuity problem can be obtained from Table 2.2 by associating the scattering coefficients a^, b^ and b^ with the reflected surface wave and the two transmitted surface waves, respectively. The total power radiated can be obtained from the power in 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 dis-(47 48 49 52) continuities have been given i n the literature ' ' ' . In general, the solutions were obtained by neglecting parts of the continuous mode spectra. The configurations analyzed in references (47), (48) and (49) were well suited to this approximation since these modes were of second-ary importance. This i s not the case for the configuration analyzed in reference (52), however. In this section the application of the present "bounded" approach i s illustrated by considering three examples, a l l of ( 76} which have been dealt with in the literature Example 1. Two slightly different configurations, shown as inserts A and B in Fig. 2.3 are considered in this example. The results, presented in the same figure, are given by the solid and dashed lines, (a) and (b) , (47 48) respectively. In order to make comparisons, the original results * are also given and are denoted by (a') and ( b T ) , respectively. In a l l , three different sets of curves are shown i n Fig. 2.3. In each set, the curves labelled (a) were obtained by placing the bounds at x = tg = ±^^o (shown as dashed lines on insert A) and by choosing an equal number of modes (N = 10) on each side of the discontinuity. Similarly, the curves labelled (b) were obtained with t., = ±6X and N = 20. The maximum error o which occurs in the range ^ / t ^ 1*0, i s estimated* to be 0.1% and 1.2% for the two sets of curves, (a) and (b), respectively. In both cases * by checking convergence as t B and N were increased. 20. 0.01 i i = i : _ i i i _ 0.0 7.0 2.0 3.0 F i g . 2.3 Normalized '(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 reference (47) b 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 B b' o r i g i n a l r e s u l t s from reference (48) 21. power conservation was within 0.005%. As indicated in Fig. 2.3 there i s reasonable agreement between the new results and those in references (47) and (48). Example 2. This example deals with configuration shown as an insert in Fig. 2.4. Although the reflected, transmitted and radiated power have been cal-culated, only the radiated power, denoted by the solid line, i s presented (49) in Fig. 2.4. The original results for this configuration, which were obtained by two different methods, are given by the dashed lines. The new results were obtained with t„ = ±12A and N = 20 which gave a maximum B o & error of 2.0% (occurring in the range k, t- < 6.0) and power conservation o 1 of 0.01%. It i s seen, upon comparing the curves presented in Fig. 2.4, (49) that the approximations made by Marcuse are quite valid over the range considered. Example 3. This f i n a l example deals with a different type of configuration, shown as an insert i n Fig.2.5(a). involves a dielectric waveguide which has an outer layer of dielectric with greater permittivity than that of the inner layer. The results for this configuration are shown i n Figs. (52) 2.5(a) and 2.5(b) by the curves labelled (a). The original results are shown as curves labelled (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%. It i s seen that the new results are appreciably different from the original ones; however, i n performing the convergence tests i t was found that the number of modes, N, had considerable effect on the accuracy of the solutions. This i s illustrated in Fig. 2.5(a) by the dashed curves 22. 23. Fig 7 2.5(a). Magnitude of the reflection coefficient versus V V e l / e o = 9 ' 0 ' e 2 / e o = 1 0 ' 0 ' V Xo = ° ' 0 6 . a present results, N = 50 a' N.~= 20 a" N = 10 b original results from reference (52). ' - 100 A .2 B -4 C -6 D E A F LO 1.2 Fig. 2.5(b). Percentage power i n the radiation and the transmitted surface-wave modes. Curves "a" and "b" as i n Fig. 2.5(a). 25. (a') and (a") for which N = 20 and N = 10, respectively. A remarkable similarity exists between positions of the maxima and the minima in curve (b) and these in curves (a') and (a"). Since, in the bounded configura-tion, there are 11 reflected and as many as 15 transmitted propagating modes, over the range of t . j A 0 given, these curves suggest that an i n -sufficient number of terms in the expansion for the continuous spectra was considered i n reference (52). The oscillations i n the curves (a) shown in Fig. 2.5(b) may be explained as follows. As t ^ A g i s increased, transition points (labelled A, B, C, D, E and F in Fig. 2.5(b)) are encountered at which a fast mode moves into the slow mode spectrum (i.e. in a similar manner to the t ^ described i n section 2.2). Therefore, the fast mode power is decreased and slow mode power i s increased. Thus, corresponding peaks and valleys are observed i n the curves (a) for values of t^/X^ near these points. This effect i s less noticeable at larger values of t^/X o since the modes passing through the transitions are of higher order and hence contain less power. 2.5 Summary In this chapter an approximate method for 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 scattering problem and examining the solutions as the bound i s moved away from the guiding surface. The configuration analyzed, consisted of an abrupt change in thickness of the di e l e c t r i c on a di e l e c t r i c coated conductor. Only the solutions for TM modes were considered. The main points covered were: 1. The basis of the approximate method: the direct correspondence 26. between the mode spectra in 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 vir t u a l l y the same as the corresponding surface wave modes. 3. Mode-matching equations for the bounded discontinuity problem: a numerical example indicated that the slow mode amplitude coefficients converge quite rapidly as the bound i s removed. 4. Application to previously considered configurations: the present "bounded" approach was t e s t e d ^ 7 ^ on four configurations for which approximate solutions were available i n the literature. Good agree-ment was obtained i n a l l cases except that considered in reference (52). However, the present results indicate that inaccuracies could (52) exist in the original results Chapter 3 EXPERIMENTAL INVESTIGATION OF STEP DISCONTINUITIES In this chapter, an experimental investigation of the scat-tering problem depicted i n Fig. 2.1(a) i s described i n det a i l . The two-dimensional configuration shown in Fig. 2.1(a) can be realized in prac-tice reasonably well. Practical c r i t e r i a on which the die l e c t r i c coated conductor configuration was chosen were ease of excitation and ease of support. One of the objectives of the experimental study was to verify the results of the theoretical method described in the previous chapter, particularly since the application of this method to the configuration considered in reference (52) yielded results which were significantly (52) different from the original 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 investigation was carried out at an experi-mental frequency of 30 GHz. Commercially available low-density polyethy-lene sheet was found to be a suitable di e l e c t r i c . 3.1 Experimental Waveguide Construction and Excitation 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 in Fig. 3.1. The surface waveguide consisted of a smooth f l a t metallic surface (3 f t . x 8 ft.) covered with a sheet of polyethylene 1/32 i n . thick (at a frequency of 30 GHz this thickness allowed only one mode to propagate). The metallic surface was constructed by coating a sheet of plate glass (3/8 i n . thick) with alum-inum f o i l . Some d i f f i c u l t y was encountered in making the polyethylene l i e f l a t on the metallic surface. However, by forcing the air out from beneath the polyethylene and clamping along the sides of the waveguide an adequate surface was obtained. Despite these precautions, pockets of air formed occasionally beneath the dielectric and had to be forced out before measurements were made. The surface waveguide was shielded from the surrounding environ-ment, particularly the measurement apparatus, by a plywood box covered, on the inside, with absorbing material, as shown in Fig. 3.1. This box was constructed in sections and could easily be removed and be rearranged. Wooden wedges, having typically 20 db attenuation and less than 5% re-flection, served as absorbers (placed on top of the polyethylene) around the edges of the waveguide. A matched load was constructed from strips of graphite-loaded absorbing material which, when suitably arranged on top of the guiding surface, yielded a standing wave ratio of 1.03 or better. The arrange-ment of these absorbers can be seen i n Fig. 3.1. The excitation of the surface waveguide was achieved in two stages. F i r s t , a small H-plane sectoral horn (aperture dimensions of 2.75 i n . x 0.14 in.) was used to excite a section of paral l e l plate waveguide incorporated with one end of the 3 f t . x 8 f t . metallic sur-face. Second, the parallel plate waveguide was used to excite the sur-face waveguide by extending the polyethylene into this section. A flared section was bolted onto the parallel plate waveguide and provided a smooth transition to the surface waveguide. The separation between conductors i n the parallel 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 parallel plate region are sketched i n Fig. 3.2. In an attempt to obtain a beam with a wide, plane wavefront two other types of excitation for the parallel plate region were also considered: H-PLANE HORN PARALLEL PLATE FLARED EXTENSION POLYETHYLENE SHEET 7 PLATE GLASS __ 7 ALUMINUM FOIL Fig. 3.2 Excitation of the surface waveguide 31. a) a resonant waveguide slot array with 56 slots covering 12 i n . i n the broad wall of a rectangular waveguide normally used in the 26-40 GHz band of frequencies, and b) a large H-plane sectoral horn with aperture dimensions 24.0 in x 0.14 i n . These types of excitation were found unsuitable however, due to higher or-der mode excitation in the parallel plate region. 3.2 Measurement Apparatus and Properties of the Surface Waveguide In an attempt to provide maximum discrimination against radiation fields,the longitudinal component of the ele c t r i c f i e l d (of the "TM " o mode) was chosen for measurement. For this purpose a A/2 dipole was used. It w i l l be shown in the next section that the correct values of the measurement parameters can be obtained using this relatively large (in terms of wavelength) probe. The dipole was connected to an 18.0 i n . length of coaxial line which coupled the dipole to a section of waveguide. The details of the dipole are shown in Fig. 3.3(a). The coaxial line and the waveguide detection unit are sketched in Fig. 3.3(b). The coaxial section (and hence the dipole) was held r i g i d by an expanded polystyrene cone which was attached to an x-y-z micropositioner as shown i n Fig. 3.1. The polystyrene support, which has a relative permittivity of approximately 1.03, did not cause any significant disturbance of the f i e l d . The micropositioner was ri g i d l y supported independently of the surface waveguide. However, provisions were made to allow the positioner to be adjusted relative to the plane of the surface waveguide. Thus, i t was possible to have the dipole movement aligned par a l l e l to this surface. The micropositioner was oriented such that i t provided 8 cm of continuous travel (0.01 mm accuracy) i n the vert i c a l and transverse (to the waveguide axis) directions. A total of 22.5 cm of travel was provided along the 32. \~0.062-\ 0.394 0.195 0.011 - i \ -0.005 © © TEFLON INSULATION © TEFLON SUPPORT ® QUARTER WAVE SLOTS © SL/DJNG BALUN © COPPER WIRE © BRASS \—0-031 Fig. 3.3(a) The dipole used for f i e l d probing. 4 CRYSTAL BNC ADAPTER 5 HI ur COUPLING-PROBE\© JE 0.265 ur FLEXIBLE WAVEGUIDE 0.062 (diet)'* ® 0.005 dm. CENTER =z CONDUCTOR OF THE CO AX AIL COAXIAL LINE LINE DIPOLE JTL 3=Q ffl TUNABLE O.U" SHORT CIRCUIT Fig. 3.3(b) The coaxial line and waveguide detection unit. co 34. waveguide axis but with the restriction that, at 2.5 cm intervals (contin-uous travel), a part of the positioner 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 in Fig. 3.3(b), into a Model 1752 Sc i e n t i f i c Atlanta Phase/Amplitude receiver. The receiver was phase locked to a reference signal which also provided a reference for the phase measurements. With the above arrangement, detection could be achieved over a 60 db dynamic range. Monitoring of the phase and amplitude was provided by S c i e n t i f i c Atlanta Amplitude (Model 1832) and Phase (Model 1822) d i g i t a l displays which could read to an accuracy of 0.01 db and 0.1°, respectively. A block diagram of the complete measurement system i s shown in Fig. 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. A typical decay curve i s shown i n Fig. 3.5. For comparison a theoretical decay rate was calculated assuming a frequency of 30 GHz, a 1/32 i n . thickness of the polyethylene and a value of the relative permittivity of the polyethylene equal to 2.26. The result i s represented by the dashed l i n e i n Fig. 3.5. Apart from the slight oscillatory nature of the experimental points, there i s excellent agreement between the measured and the calculated rates of decay. The os-cilla t i o n s were attributed to the interaction of the dipole with the sur-face waveguide. By adjusting the balun on the dipole, the magnitude of these oscillations was minimized. Before any scattering measurements could be made, i t was neces-sary to determine two other properties of the experimental surface wave-POWER SUPPLY CALIBRATED ISOLATOR ATTENUA TOR E-H TUNER SURFACE WAVEGUIDE KLYSTRON FREQUENCY ME TER MIXER / — J I —1.... SLOTTED LINE REFERENCE AMPLITUDE DISPLAY PHASE DISPLAY AMPLITUDE/ PHASE RECEIVER Fig. 3.4. Schematic for the complete measurement system. 36. o.o] -4.0\ -6.01 \ CO U J Q a . -12.01 -16.0] -20.0] \ o \ o. o V \ o o.o 4.0 8.0 HEIGHT (mm) 12.0 Fig. 3.5. The decay of the surface wave away from the dielectric surface. 37. guide: the direction of propagation and the attenuation. The direction of propagation of the wave was determined by measuring i t s amplitude and phase characteristics. The probe movement was adjusted u n t i l the measured characteristics were symmetric. An example of the amplitude and the phase characteristics i s shown i n Fig. 3.6. There was, typically, less than 0.5 db variation in the wavefront over a width of three wavelengths (3.0 cm at a frequency o f 30 GHz). This variation was comparable or better than that obtained by other w o r k e r s ^ ' ^ who found that beams with similar variation adequately represented the two dimensional case. For this ex-perimental waveguide i t was found that the best patterns were measured with-in 60 cm of the source aperture (edge of the flared extension shown in Fig. 3.2) and, thus, a l l measurements described hereafter were made in this range. The attenuation was determined by measuring the amplitude of the wave (at a constant height above the polyethylene) at 1 cm intervals over a 22 cm distance along the beam axis. The results are shown i n Fig. 3.7. A straight line (dashed) drawn through these points corresponds to an ex-ponential attenuation factor of 0.059 db/cm. This factor was much higher than expected by considering the loss tangent of the polyethylene; the approximate theoretical attenuation factor was estimated to be 0.0021 db/cm. This value was derived for the corresponding bounded waveguide configuration (neglecting conductor losses) and assuming a loss tangent (tan 6) for poly-ethylene equal to 0.0005. The details of the theoretical calculation are given in Appendix I-D. The "apparent" attenuation can, however, be attributed to the beam divergence., The amplitude patterns were measured at 2 cm intervals over the same 22 cm distance as the attenuation measurements. The patterns 38. DISTANCE (cm) Fig. 3.6 Amplitude and phase characteristics (abscissa is. transverse distance from the beam axis). o . o y •8 -8.01 O o Uj x o o S o o •701 o °yo 0 2 4 8 10 12 14 16 18 20 DISTANCE (cm) Fig. 3.7 Attenuation measurements ( - exponential attenuation) 40. representing the midpoint and the two extremes of this 22 cm range are shown in Fig. 3.8. By drawing the "best" curves (dashed profiles) 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 in Fig. 3.9. The results 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 oscillations i n the attenuation measurements (Fig. 3.7) and the beam patterns (Fig. 3.8) the measurements were repeated at two other frequencies close to the investigation 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 ca 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 closely spaced frequencies i n an attempt to guard against possible spurious frequency dependence in the results. The results obtained for the three frequencies were averaged, thus, the average value of attenuation, which was used in the later stages of the experimental investigation, was 0.062 db/cm. The average value of the apparent attenuation due to beam divergence was 0.059 db/cm. If the difference between these two values, i s attributed to attenuation i n the d i -electric, i t yields a value of tan 6 = 0.00077, in reasonable agreement (79) with the published 'range of values of loss tangent for polyethylene. 3.3 Factors Affecting Standing Wave Measurements The main parameter of interest i n the experimental investigation was the reflection coefficient. This parameter can be determined by measur-ing the standing wave pattern and, thus, various factors which affected 2.0\ v °\ DIRECTION OF PROPAGATION 1 r v I / <0 •CJ ao i o * x \> BEAM I /, V* V \-— WIDTH — / \ * o (0.5 cto POINTS)/ . * \ A L O O / / A \ I X \ A L o ° / / V f / \ v y x / f \ A A / -4.0 -2.0 0 2.0 4.0 DISTANCE (cm) Fig. 3.8 Amplitude patterns o - nearer to the source X - midpoint A •-. far from the source 9.0 x ^ 00 40 8.0 12.0 16.0 20.0 DISTANCE (cm) , ^ to Fig. 3.9 Estimated values of beam width versus distance along the beam axis. 43. the standing wave measurements had to be considered. As discussed pre-viously, a dipole was used to probe the longitudinal component of elec-t r i c f i e l d . Also, the decay measurements (Fig. 3.5) indicated that there was some interaction between the dipole and the surface waveguide. Fur-thermore, there was a possibility 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 ratio: a) what i s the effect of having the A/2 dipole oriented i n the direction of propagation? and, b) what effects do the interaction, between the dipole and the surface, and the interference between the surface wave and the radiation 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 size of the coaxial feed, that the A/2 dipole should measure the correct value of standing wave. The details of the analysis are included in Appendix I-E. The second question was investigated experimentally. Standing wave patterns, over a 2.5 cm length of l i n e , were measured (20 cm from the discontinuity) at 1.0 mm intervals above the polyethylene surface. Values of standing wave ratio between successive maxima and minima were calcu-lated from each pattern. These values of standing wave ratio are plotted, at regular intervals corresponding to quarter wavelengths, in Fig. 3.10(a). There are five curves (labelled A, B, C, D, and E) shown i n this figure. One curve i s shown for each height of the dipole. A part of the corresponding decay curve measured with the step discontinuity in place is shown i n Fig. 3.10(b). The height at which the 44. 221 C D L . • • "A " A . o — . A . A T /v V \ 5 CO 5 ;o QUARTER WAVELENGTH INTERVALS Fig. 3.10(a). Variation in standing wave ratio values as a function of the height of the dipole above the dielectric surface. 4 5 . 10 U j § 0.0 -4.0 -8.0 A o B 0 > 0 , o 4.0 8.0 HEIGHT (mm) Fig. 3.10(b). A decay curve indicating heights at which the standing waves were measured. 46. various standing waves were measured can be found from the labelled points on the decay curve. The curves in Fig. 3.10(a) are typical of the v a r i -ation which was observed (for a l l the step discontinuities 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 interaction of the dipole with the surface appeared to s i g n i f i -cantly distort the standing wave measurements (curve A) and far from the surface, there appeared to be significant 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 in subse-quent standing wave measurements. It i s interesting to note that, whereas the standing wave incurred large oscillations at the larger distances from the surface, the decay curve did not (and should not). The change of phase in the radiation f i e l d , over a ver t i c a l distance of a few millimeters, 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 exhibit much oscillatory behaviour in the vertical direction. On the other hand, for the standing wave measurements this was obviously not the case. Further investigations of the interference with the radiation from a step discontinuity 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. Two examples of the standing wave ratio values obtained i n these measurements are 47. plotted (at regular intervals corresponding to quarter wavelengths) i n Figs. 3.11(a) and (b). These plots clearly indicate the decaying nature of the standing wave due to the attenuation (the ordinate axis represents the position of the step). In addition, the interference of the surface wave with the radiation from the step appears to distort the standing wave measurements within, approximately, a 10 cm range from the step (the total distance covered by the points i s roughly 22.5 cm). For this reason, a l l reflection coefficients were calculated from standing wave measurements in the range 20 cm. to 22.5 cm from the step. An example of the quality of the standing wave patterns measured in this range i s shown i n Fig. 3.12. 3.4 Experimental Determination of the Reflection Coefficient and Comparison With Theory Using the general procedures outlined in the previous section, standing wave patterns were measured for a range of step heights (1/32 i n . , 1/16 i n . , 1/8 i n . , 3/16 i n . , 1/4 in . , 3/8 i n . , and 1/2 in . ) . 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 Fig. 3.13. The far end of these sheets were tapered down to "zero" thickness in an effort to elim-inate multiple reflections within each sheet. Values of the reflection coefficients for each step were determined from measurements at the three separate frequencies: 29.94 GHz, 30.00 GHz and 30.06 GHz for the same reason mentioned in section 3.2. Since the standing waves were measured in the range of 20.0 cm to 22.5 cm from the steps a correction was made to account for the effects of attenuation. The attenuation causes the value of the standing wave ratio, s, to decrease with increasing distance from the step. In fact, a linear relationship is obtained by plotting s (in db) against the distance, d, from the step. The slope depends on: the value 3.0 k o 10 o s. I 2. CD 0 °°o° 00 .0 0 0 ^ 2. o 0 o ° o s o 0o" o o o o w < J o o 0 o 0 0 0 o > ft0o . o o Q ° ° o oo°° 0 o - o o 0 o o o °o o O c> *«« 20 40 60 QUARTER WAVELENGTH INTERVALS 80 Fig. 3.11(a) Example of variation in the standing wave ratio values as a function of the distance from the step discontinuity ( theoretical dependence of standing wave ratio). 0 0 o o 2 . 5 10 O § UJ 1 CD s t o ho ; . 5 o o o o ^ o °o o o o « o O © ^. w o o o° — — °o o o o o o o o ° 0 ° 0 - r>^P o o o o o ° - o 0 o ^ ° ° ^ c ° o ° o o o 0 20 40 60 QUARTER WAVELENGTH INTERVALS 80 Fig. 3.11(b) Same as (a) but for a different step height! o n o O o o o aoi J I 1_ JL_: I L. 0 2.0 .4.0 6.0 8.0 DISTANCE (mm) 10.0 12.0 01 Uj 5: 1~0\ o o o ° ° o 00\ 1 1 -12.0 14.0 o o o o o ' ° 0 o J I I L J I 16.0 18.0 20.0 DISTANCE (continued) 22.0 24.0 Fig. 3.12 An example of a measured standing wave pattern. 13 in STEP DISCONTINUITY TAPERED SECTION / 3 ± 32m POLYETHYLENE LAYER PLATE GLASS ~ ALUMINUM FOIL GRAPHITE LOADED ABSORBER Fig. 3.13. Experimental step discontinuity configuration. 52. of the magnitude of the reflection coefficient at the step, p, and on the attenuation coefficient, a. The relation between s, p, a and d i s well known and given by^7~^ : i _i_ -2ad 8 = 1 + P * 3.1 , -2ad 1 - p e 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 reflection coefficients, p, was determined from the standing wave measurements in the range of 20.0 cm to 22.5 cm from the step. b) p and a = 0.00713 np/cm (0.062 db/cm) were used i n equation 3.1 to calculate the slope of the linear dependence of s (in db) on d. c) the measured values of the standing wave ratio were plotted, as i n Fig. 3.11, and the theoretical line (dashed i n Fig. 3.11) was drawn such that i t passed through the points i n the range 20.0 cm to 22.5 cm from the step, and d) the intercept of this line with the ordinate (location of the step) determined the value of the standing wave ratio at the step and, hence, P. The values of p determined by the above procedure are plotted i n Fig. 3.14. An experimental point is shown for each, step and for each of the three frequencies. The theoretical dependence of p (solid curve) is also shown for comparison. The theoretical results were obtained, by the approximate method described in Chapter 2, using a frequency of 30 GHz and using values of the parameters e , t , and N equal to 2.26, 5X , and 50, respectively. r a o The theoretical curves for frequencies of 29.94 GHz and 30.06 GHz are negligibly different than that for 30 GHz and, thus, are not shown. The ( t 2 - t j ) A 0 Fig. 3.14. Experimental results for 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 reflections 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 wave-guide (due to stresses, within the polyethylene, possibly caused by machin-ing the tapers) and also, there must be some reflection, however small, due to these tapers. The phase angle, 6, of the reflection coefficient can be de-termined from the relation^ 7"^ : e = ^ 3.2 g where d i s the distance between the step and the location of the f i r s t minimum in the standing wave pattern and where X is the waveguide wave-length. Thus, the following measurements, at a frequency of 30.0 GHz, were made for each step: a) the positions of the minima in the standing wave in the range 20.0 to 22.5 cm from the step, and b) the distance, d, between each of the above minima and the step. The measurements in (a) above were used to calculate an average value of the waveguide wavelength for the standing waves caused by each step. These values varied between a maximum of 9.651 mm and a minimum of 9.548 mm. Since the frequency was held constant, this variation was considered unreasonable and was attributed to slight asymmetries in the pattern. Therefore, the average values of X calculated for the individual steps were in turn averaged, yielding a value of 9.608 mm. This value was i n better agreement with the theoretical value of 9.610 mm. The average value of X^ (9.608 cm) was used i n conjunction with the measurements in (b) above to obtain the position of the f i r s t minimum in the standing 55. waves corresponding to each minimum in the measurements (a). Equation 3.2 was then used to obtain an average value of 6 for each step. The results are shown in Table 3.1. For comparison, theoretical results which were calculated using the same parameters as those used to obtain the solid curve in Fig. 3.14 are also shown. The difference between the theoretical and the experimental values shown in 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 results shown in Fig. 3.14 and Table 3.1 indicate that, i n general, there is good agreement between the experimental and the theor-et i c a l values of the reflection coefficient. These results inspire con-siderable confidence in the "bounded" approach to solving the discontinuity problem and i n the experimental techniques used to measure the scattering from such discontinuities. 3.5 Summary In this chapter an experimental investigation of step discontin-uties on a di 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 results on fre-quency. • The main points were: . 1. The construction and excitation of the experimental waveguide, 2. The use of a A/2 dipole (oriented i n the direction of propagation) as a probe, 3. Properties of the surface wave on the experimental waveguide: the decay of the surface wave away from the polyethylene was found to STEP HEIGHTS (In.) 1/32 1/16 1/8 3/16 1/4 3/8 1/2 EXPERIMENT THEORY 177.6 189.7 206.2 188.5 181.7 186.2 172.0 184.2 189.1 182.8 179.6 182.2 191.4 182.4 Table 3.1 Experimental and theoretical values of the phase angle of the reflection coefficient for a range of step discontinuity heights. 57. be in excellent agreement with theory; a high value of attenuation was measured and explained in terms of beam divergence, 4. Factors affecting standing wave measurements: two factors were con-sid e r e d — interaction of the dipole with the surface waveguide and interference of the surface wave with radiation from the step; these factors determined the best location for the standing wave measure-ments. 5. Determination of the reflection coefficient: the attenuation of the waveguide was accounted for in the measurements of the magnitude of the reflection coefficient, 6. Comparison with theoretical results for step discontinuities: good agreement between theory and experiment was obtained for the magni-tude and the phase of the reflection coefficient for a wide range of step heights. 58. Chapter 4 APPLICATION TO CASCADED DISCONTINUITIES In this chapter the theoretical and experimental techniques pre-sented in Chapters 2 and 3, respectively, are applied to the discontinuity problem shown i n Fig. 4.1(a). I n i t i a l l y , the solution to the corresponding bounded configuration, shown i n Fig. 4.1(b), i s obtained using wave mat-rices. This solution i s then examined as the bound i s moved away from the original guiding surface. For comparison, experimental results for the magnitude of the reflection coefficient of the dominant mode incident on one such discontinuity are presented. 4.1 Wave Matrix Formulation Wave amplitude matrices relate the wave (mode) amplitudes on one side of a discontinuity to those on the other side. For this reason these matrices are useful for analysing cascaded discontinuities; the wave matrices for the individual discontinuities can be multiplied together (in the same order as the discontinuities) to yield an overall wave matrix for the cascaded sections. Thus, the same number of modes on either side of a discontinuity must be considered. Normally, only propagating modes are used. If, however, evanescent mode interaction between successive discontinuities i s important, then the wave matrices can be extended to include evanescent modes as w e l l , ^ ^ The configuration shown i n Fig. 4.1(b) represents an interesting situation regarding the application of these matrices; d i f -ferent numbers of propagating modes may exist i n waveguides A and B. In such cases, the wave matrix must include a sufficient 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 discontinuities z = z , - z - z 2 (b) Fig. 4.1 Cascaded step discontinuity configurations (a) unbounded (b) bounded • 60. equal. The application of wave matrices to the problem shown in Fig, 4.1(b) i s as follows. Wave matrices, w^ and w,,, are defined for each of the two step discontinuities at planes z = z^ and z = z 2, respectively. A wave matrix w^, i s also defined for the length of uniform waveguide, B, between the two discontinuities. Let c^ and b^ be vectors representing the norma-lized amplitudes of modes (each element corresponding to a mode) propa-gating in the positive and negative z directions, respectively, i n waveguide A to the l e f t of the discontinuity. Let c 2 and b 2 be similarly defined i n waveguide A to the right of the discontinuity. Then V = W V b, b„ -1 -2 4.1 where W i s the overall wave matrix for the cascaded discontinuities and i s given by: W = W l WT W2 In general, i f N modes are considered on either side of the discontinuity then 4.3 normalized such that the power contained i n the i t h mode i s given by - i i 1 61. where 8^, i = 1 N are the propagation coefficients of the modes con-sidered in waveguide B and where I i s the distance between the two steps, as shown in Fig. 4.1(b). The mode-matching solutions, discussed in Chapter 2, define ele-ments in a scattering matrix. The scattering matrix for a discontinuity can be used to obtain the elements of the corresponding wave matrix. For the configuration considered here, w^ and can be derived from the same scattering matrix. 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 il l u s t r a t e the convergence of the solutions for the bounded configuration, a specific numerical example i s considered. Let t]_A 0 = 0 . 0 7 9 , ^ A Q = 0.555, ^ A q = 12.6 and = 2.26. For this choice of parameters there i s one slow mode propagating in waveguide A and two slow modes propagating i n waveguide B . The large value of &AQ i s chosen to simplify the analysis since i t eliminates the possi b i l i t y of significant evanescent mode interaction. The scattering of the T M q mode incident upon the discontinuity from waveguide A, as shown In Fig. 4.1(b), i s examined as a function of tgA Q. Values for the mode coefficients, b ^ and c^^, of the T M q modes i n waveguides A to the l e f t and right of the discontinuity, respectively, are presented i n Table 4.1. They are obtained by setting b ? = 0 (wave incident only from z < 0 ) and S i 1 0 L o J i n equation 4.1. 4.4 SCATTERING COEFFICIENTS b 11 C 2 1 CONSERVATION ERROR % 1.0 0.26327 /176.33 0 0.87064 /-98.82° 0.0004 2.0 0.26850 /179.56 0 0.94901 /-91.62° 0.0033 3.0 0.27070 /176.23° 0.92544 /-90.46° 0.0162 4.0 0.27551 /179.01 0 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 Table 4.1 Convergence of slow mode s c a t t e r i n g c o e f f i c i e n t s (obtained by wave matrices) as a f u n c t i o n o f t A 63. Also given in Table 4.1 are values of the percentage error in power conservation at each value of t„/X . The results indicate that the mode D O coefficients converge as t /X i s increased. Again convergence should be 15 O examined on the basis of equality in power conservation errors, as discussed in Chapter 2. Since this i s not the case for the results shown, convergence of the coefficients i s obtained only to two significant figures. 4.2 Experimental Investigation To verify the results obtained by the method presented i n the previous section, the magnitude of the reflection coefficient of the domi-nant mode incident on a step discontinuity, such as the one shown i n Fig. 4.1(a), was measured using the procedures outlined i n Chapter 3. The step discontinuity 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 this configuration corresponds to that of the numerical example considered in section 4.1. The interaction of the two step discontinuities causes the reflection coefficient to exhibit an oscillatory dependence on SL. This variation could be observed by changing the physical length, I, of the slab or by changing the frequency. Changing the frequency was relatively simp-ler and more accurate. Thus, the frequency was varied from 30.0 GHz to 30.8 GHz in steps of 0.1 GHz. Approximately one cycle of the variation i n the reflection coefficient was observed over the above frequency range. The experimental values of the magnitude of the reflection coefficient are plotted versus Z/\ in Fig. 4.2. 65. 4.3 Comparison with Theory Before any comparison with theory could be made, the effects of attenuation (in waveguide B) should be included in the wave matrix formu-l a i lation. This i s done by including attenuation factors, e , i n the wave matrix wm; where o i . i s an estimated value of the attenuation coefficient of T x the i t b mode. Two separate contributions to each were considered: 1. that due to bulk material loss (denoted by a^) 2. that due to beam divergence (denoted by a^) In an attempt to model the unbounded configuration, these two attenuation effects were only considered for the slow modes in the corresponding boun-ded configuration (i.e. the fast modes in the bounded structure correspond to the radiation modes in the unbounded structure which are not substan-t i a l l y affected by these attenuations). Using the analysis given in Appen-dix I-D and assuming a value of tan 6 = 0.00077, the values of for the two slow modes (of waveguide B shown in Fig. 4.1(b)) were calculated to be 0.00368np/cm and 0.00210np/cm, respectively. The value of t,, used i n these calculations was equal to 5.0X^. Since the material attenuation i s con-o sidered separately here, the value of was assumed to be 0.00675np/cm (as discussed in Chapter 3). The theoretical variation i n the magnitude of the reflection co-efficient (with the above attenuation effects included) i s given by the solid curve i n Fig. 4.2. For reference purposes, the theoretical 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 in Fig. 4.2: a) the "period", AJ2,/Xq, of the experimental points, A Z / \ Q - 0.32, 66. i s shorter than that of either of the theoretical curves, A £ / X - 0.36. o b) the experimental points are shifted (to the l e f t in Fig. 4.2) by approximately, 0.025 in terms of £A Q, and c) the amplitude of experimental os 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 different frequencies used to determine each experimental point (i.e. the experimental point at & A Q = 12.62 should l i e on the solid 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 for by an increase i n the relative permittivity, e r, of less than 0.5%. Such a small increase changes the values of the propagation coefficients in waveguide B only slig h t l y ; however, since &AQ is large, these small changes have considerable effect. 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, in order that this value of dp be applicable to the thicker section of waveguide (waveguide B) re-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 direction of propagation and interfaces perpendicular to the guiding surface). It does not seem unreasonable to suggest that, i n practice, there must be some varia-tion i n the above factors. Since there were two surface modes propagating on the slab section, attenuation measurements on this section would have been d i f f i c u l t . 67. 4.4 Summary In this chapter an investigation of the scattering from two cas-caded step discontinuities was discussed. The "bounded" approach, outlined in Chapter 2, was used in conjunction with the wave matrix formulation to obtain values of the scattering coefficients. The magnitude of the reflec-tion coefficient for one specific 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 scattering coefficients obtained using wave matrices converge as the bound i s removed, 2. experimental investigation: an effective check on the be-haviour of the reflection coefficient was obtained by changing the e l e c t r i -cal length between the two steps rather than the physical length, 3. comparison between theory and experiment: Reasonable agree-ment was obtained between the experimental and the theoretical results when the latter included the effects of estimated values of attenuation. PART II CYLINDRICAL DIELECTRIC WAVEGUIDES WITH PERTURBED CIRCULAR CROSS-SECTION Chapter 5 INTRODUCTION 5.1 Background One of the f i r s t investigations of cylindrical d i e l e c t r i c surface waveguides of non-circular cross-section was presented by Weiss / o i \ and Gyorgyv ' i n 1954. They discussed the relative merits of the die-l e c t r i c rod and dielectric tube and concluded that the tube possessed better dispersion characteristics, whereas, the rod confined the fields better. Both of these guides were considered unsuitable.for practical transmission purposes, however, because any imperfection or bend in the waveguide resulted i n power transfer between the two possible polariza-tions. They also presented the results of an experimental investigation concerning attenuation, launching, shielding and guide supports, for dielectric waveguides of rectangular cross-section. Rectangular dielectric waveguides have also been the subject of several more recent investiga-(82) tions. In 1966 Schlosser and Unger derived dispersion curves for the f i r s t few low order modes and discussed the mode designations. Their method involved placing the dielectric waveguide in a sufficiently large rectangular metallic waveguide; the resulting cross-section was then subdivided and point-matching was applied along the boundaries of each subregion. They also gave experimental results for attenuation and ( 83} dispersion curves. Later, Marcatili considered the problem of a rectangular dielectric waveguide surrounded by several different dielec-trics of lower refractive index. He applied a f i e l d matching technique over the rectangular core boundary which resulted i n approximate eigen-value equations for the propagation constant. Dispersion curves and f i e l d patterns were derived using these equations. Marcatili then extended the method to the case of two rectangular di e l e c t r i c waveguides in close proximity, with a view to directional coupler design. G o e l l ^ ^ also used a point-matching technique to calculate approximate values of propagation constants for the rectangular dielectric waveguide. His ( 85) results compared favourably with the previous investigations. Pregla used a variational technique similar to the Rayleigh-Ritz method and calculated dispersion curves for the f i r s t few low order modes on a rectangular dielectric waveguide. The f i r s t comprehensive investigation of dielectric 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 polarizations. These dispersion curves were verified by experiment. Later Yeh Investigated the attenuation properties of e l l i p t i c a l dielectric waveguides. 5.2 Scope of Part II The brief review presented above indicates that to date, there have been few investigations of dielectric waveguides with non-circular cross-section. Such waveguides are deemed necessary because of two prac-t i c a l considerations which occur in connection with di e l e c t r i c rod and tube waveguides of circular cross-section. The two problems are: (1) lack of a well defined polarization, 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 slight e l l i p -t i c i t y i n the dielectric rod or tube cross-section causes the cross-polarized modes to propagate with slightly different phase and group velocities. Thus, either surface imperfections or distortions (e.g. 71. bends) of the waveguide result i n mode conversion between these modes and, hence, poorer dispersion characteristics are obtained and d i f f i c u l t i e s in detection are encountered. The second consideration i s one which i s common to most wave-guide systems (metallic or d i e l e c t r i c ) ; an e f f i c i e n t and relatively easy way of coupling between components i s desired. With open structures, however, the problem is more complicated because of the p o s s i b i l i t y of radiation. Also, i n connection with the f i r s t consideration, i f slight 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 practical surface waveguide systems involve the use of fins on the waveguide, be i t of circular, e l l i p t i c a l , or rectangular cross-sections, as shown in Fig. 5.1. These fins would make the polarization more definite and may be useful in interconnection and coupling schemes. For the purposes of the present work, a di e l e c t r i c waveguide with circular cross-section has been chosen. The reasons are: the rela-tive simplicity 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 axial slots. A slot configuration was chosen for this investigation, rather than a f i n con-figuration, again because i t was easier to construct. The analysis and procedures for the investigation of a finned waveguide would be similar to those used for the slotted waveguides, however. In Chapter 6, a perturbation method for determining the separation in the propagation coefficients of the cross-polarized modes caused by two symmetrically placed slots (with polar geometry), i s presented. FINNED DIELECTRIC WAVEGUIDE CROSS - SECTIONS INTERCONNECTION WITH KEYS COUPLING Fig. 5.1. Finned dielectric waveguides - possible interconnection and coupling schemes. In Chapter 7 an experimental investigation of the effect of rectangular slots i s presented. Although the theoretical and experi-mental configurations are only approximately comparable, Chapter 7 i s concluded with a discussion of the trends indicated by the theoretical and experimental results. 74. Chapter 6 A PERTURBATION METHOD This chapter i s concerned with the application of a perturbation method for finding the change in the propagation coefficient of the dominant HE^ mode on the dielectric rod (Fig. 6.1(a)), when the rod i s modified by axial slots (Fig. 6.1(b)). The approach used here i s the same as that given (88) by Harrington. Let E q and H q be the dominant mode fie l d s of the unper-turbed rod (Fig. 6.1(a)) and let E and H be the dominant mode fields of the (88) perturbed rod (Fig. 6.1(b)). Then, i t can be shown // E-E dS B - B Ae S ~ ~° - V — £ - ? - £ * * 6 - 1 o o // (E x H + E x H )-a dS g - -o -o - z w where B and B are the propagation coefficients of the modes on the P o perturbed and unperturbed waveguides, respectively. Ae r i s the change in relative permittivity within the region of the perturbation, i s the cross-section of the perturbation and i s the total (infinite) cross-section of the waveguide. Equation 6.1 i s exact; i t s derivation i s given in Appendix II-A. In order to evaluate the integrals in equation 6.1, i t i s neces-sary to approximate the unknown fields E and H. In the denominator i t i s (88) usual to approximate E and H by the unperturbed fi 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 Fields A useful approximation to the perturbed fields E can be obtained assuming E to have the same functional form as E q and by assuming that the magnitude of each component of E i s multiplied by a correction factor 75. (a) Fig. 6.1. Dielectric rod configurations (a) unperturbed (b) perturbed by slots. 76. which i s related to the behaviour of static f i e l d s i n the region of the (88") perturbation. This i s known as a quasi-static approximation. The quasi-static correction factors are shown for several useful configurations in Figs. 6.2(a), (b), (c), and (d). The configuration of interest i n this investigation (Fig. 6.2(e)) i s not the same as any of the above. However, in this work an approximate quasi-static correction i s obtained as follows. Consider, for the moment, the r-component of electric f i e l d . Let d = r.. - r and w = (r, + r ) (0 , - 9 )/2 where r n , r , 6 , , and 9 are de-1 o 1 o 1 o 1' o' 1' o fined i n Fig. 6.2(e). The configurations (and the behaviour of the static fields) in Figs. 6.2(a), (b), and (c) correspond approximately to the con-figuration shown in Fig. 6.2(e) i f d/w 0, d/w -> <*>, and d/w 1, respec-tively. This correspondence suggests an exponential dependence of the quasi-static correction factor, as shown i n Fig. 6.2(f). Let be the correction factor for the r-component of electric f i e l d , then from Fig. 6.2(f) K r = 1 + ( e 2 / e i - l ) e " a d / w 6.2 where a = l n ( l + e2/e'1) 6.3 The correction factor for the 6 component of electric f i e l d , K , may be 9 obtained from equation 6.2 by substituting w/d for d/w. Fig. 6.2(d) i n d i -cates that no correction i s necessary for the z-component of electric f i e l d , in this case. 6.2 Application of the Perturbation Method The perturbational analysis and the quasi-static approximations discussed in section 6.1 are now applied to the configuration shown i n Fig. 6.1(b). The surface wave mode fields of the unperturbed dielectric Eext \A C-int CD ^ext ^ext •int -/nf - e7 Fig 6.2 Quasi-static approximations (a),(b),(c) and (d): simple configurations (e) configuration corresponding to a slot (f) correction factor K.. 78. waveguide (Fig. 6.1(a)) are given by^^ E = A J (p,r) cos n9 z n n r l E r " (- ^ A n V (plr> " ^ T " BnV*l r» C O S n 9 n l P l P, r 3 ^ ^ E = (J^S- A j ( p r ) + —2-9. B J ' ( P l r ) ) s i n n9 0 2 n n r l p, n n 1 p r i H = B J (p.r) sin n9 z n n r l H r P p n n 1 z 2 r n n 1 o r l B6 " ^ V , < V " FFT V n ' «»!'» «- " 6 6 - 4 Pn r * o r l for 0 < r < r^, and E = C K (YOI -) cos n9 z n n '2 j n jk Z n E = C K 1 (y_r) + % 0 D K (y 0r)) cos n9 r Y2 n n 2 y 2 2 r n n 2 Y 2 r 2 H = D K ( Y o r ) sin n9 z n n 2 H - (IB. D K ' ( Y o r ) + ° , C K ( Y~r)) sin n9 r Y 2 n n 2 z 2 r n n 2 o 2 H9 " V n < V > + TT0 Cn Kn' < V » C O S " 9 6 ' 5 Y 2 r o'2 79. for r < r < <=°, where e = e/e and p.. and y 9 a r e t n e transverse propa-1 — — r o J. z gation coefficients given by: 2 i 2 R 2 p. = e k - 8 *1 r o 2 C 2 1, 2 Y 2 - e - ko 6.6 Matching the fields given by equation 6.4 and 6.5 at r = ^ yields the following eigenvalue equation: P l r l J n ( p l r l } V l V V l * ' W l V , Kn' <Vl> nB, k v 1 + —L o ( P i r i ) 2 ( Y 2 r 1 ) 2 _ J 6.7 and the following relation between the coefficients A n, B Q , C^, and Dn; A n B n _ V Y 2 r l > C n ~ D n J n ( p l r l } B n -8n d " 6 n A n - k z o o n " k Z . o 1 + — i . 2 ' 2 (p-^) ^ 2 r l ) P l V n W Y 2 r l K n ( Y 2 6.8 For the dominant HE^^ mode n = 1 in equations 6.4 - 7.8. Thus, for the configuration shown in Fig. 6.1(b), equation 6.1 becomes : . , 6o+A0 * p - 3 As r ' r / e ° ,(K rlE rl 2 + K e l E 9 l 2 4 - | E J 2 ) rd9dr <EA* - E9 Hr* ) r d 6 d r 0 0 r e a r 6.9 where K r and KQ, defined by equation 6.2 (with e^E-j^ = E R ) , are functions of r , r , 9 , and A9. Using equations 6.4 and 6.5, and upon evaluation o 1 o 80. of the integrals, equation 6.1 becomes P _ P t . . . 1 f. o o E l + P i J ! frirl)E2'Y2 where N l = F i P l 2 V B 2 + K r F l ( d l 2 A 2 + A3 " 2 d l V + K 9 F 2 ( A 2 + d l A 3 " 2 d l V 2 2 B d l • s 2 E l = < e r + ^ i " ) A 5 - 2 d l ( ^ + E r ) A 6 o o 2 2 B d 2 E2 = ( 1 + TT" ) A7 " 2 dl ( f~2 + 1 ) A 8 6 - U k k o o and F, = A6 + %(sin 2(0 + AG) - sin 29 ) 1 o o F- = A9 - % ( s i n 2(9 + AG) - sin 29 ) 6.12 I o o The terms A^, i = 1, 8 are integrals of Bessel and modified Bessel functions, and are given e x p l i c i t l y in Appendix II-B. It i s of interest to determine the separation i n the propagation coefficients of the cross-polarized modes which propagate on the slotted dielectric rod shown in Fig. 6.1(b). This can be determined using equation 6.10 by taking the difference between the result obtained with 6 + AG/2 = 0° and that obtained with 9 + AG/2 = 90°. Numerical results o o which correspond to configurations investigated experimentally are presented at the end of the next chapter. 6.3 Summary (88) In this chapter a well known perturbational analysis has been used to obtain an approximation to the propagation coefficient of the domi-81. nant mode on an axlally-slotted dielectric rod. Slots with polar geometry were considered. The main points were: 1. Quasi-static approximation to the electric f i e l d on the perturbed waveguide: by approximating some limiting forms of a region with polar geometry with simpler configurations for which the static f i e l d be-haviour i s known, quasi-static correction factors for the electric f i e l d components in polar coordinates were obtained. 2 . Application to an axially-slotted d i e l e c t r i c rod: an ex-p l i c i t equation for the propagation coefficient of such a waveguide was derived. 82. Chapter 7 AN EXPERIMENTAL INVESTIGATION In this chapter an experimental study of the die l e c t r i c wave-guides configurations shown in Fig. 7.1 i s described. The parameter of primary investigation was the normalized phase coefficient, 3/kQ, of each (89) of the cross-polarized modes. A suitable resonant cavity apparatus was readily available for the measurement of this parameter. Commercially available polyethylene rods (0.5 i n . nominal dia-meter) and frequencies in the X-band range (for single mode propagation) were compatible with the existing 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 in Fig. 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 . in diameter. These plates were individually supported by angle iron constructions, as shown in Fig. 7.2. Central 6.0 in. diameter sec-tions of each end plate were removable, allowing different 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 dielectric waveguide. Small removable locating pins, passing through holes i n the center of each end plate were used to support and align the die l e c t r i c rod. These pins also allowed the dielectric rod to be rotated on i t s axis without requiring any realignment.. Energy was coupled into and out of the longitudinal component of electric f i e l d within the cavity by two identical coaxial probes (one FIG. 7.2 THE RESONANT CAVITY APPARATUS (INSERT--THE COUPLING PROBE AND THE SCATTERER) CO 85. on each end plate). A cross-sectional sketch of the central part of one end plate, indicating the probe and locating pin configurations, i s shown in Fig. 7.3. At resonance, 3 was determined by counting the number of half wavelengths within the cavity (i.e. the number of nodes in the longitudinal component of the electric f i e l d distribution). This was achieved by re-cording, on a chart recorder, the output of the cavity as a thin metallic strip, positioned slightly above the dielectric rod, was moved over the length of the cavity. The metallic strip was attached, by nylon thread, to a motor-driven pulley system on the sides of the cavity (shown in Fig. 7.2) and, thus, continuous movement was provided. The magnitude of the scatter-ing caused by the metallic strip depends on the f i e l d intensity at each point in the standing wave pattern on the waveguide. Thus, the output of the cavity exhibited an oscillatory behaviour as the metallic strip was moved. A sample chart recording i s shown in Fig. 7.4. The propagation co-efficient 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 in the cavity) in the recorded output o f the cavity. In order to completely specify 3/kQ, the resonant frequency, f^, of the cavity was determined (i.e. k Q = 2 i r f r v ' l J ^ ) • This was accomplished by the standard procedures given by Sucher and Fox. Consider the block diagram of the measurement system shown in Fig. 7.5. The resonance curve of the cavity and the output of the reference channel were displayed simul-taneously on the oscilloscope, as shown in Fig. 7.6(a). The position of the small dip (in the trace for the reference channel) i s determined by the 8 6 . ® ® '"5 U D 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 central part of one end plate of the cavity indicating probe and locating pin arrange-ment . 87. 90 — l — Fig. 7.4 Chart recording of the output of the cavity. POWER SUPPLY CALIBRATED ATTENUATOR WAVEGUIDE TO COAXIAL ADAPTER DIELECTRIC WAVEGUIDE RESONATOR X-13 [KLYSTRON CALIBRATED FREQUENCY METER ISOLATOR CRYSTAL DETECTOR (REFERENCE JCHANNEL SAWTOOTH MODULATION CRYSTAL DETECTOR AMPLIFIER OSCILLASCOPE CHART RECORDER Fig. 7 . 5 . Schematic of the measurement system. 89. calibrated frequency meter in the reference channel. The resonant frequency of the cavity was read directly off the frequency meter when the small dip was aligned with the peak of the resonance trace (as they are, approximately, in Fig. 7.6(a)). It i s estimated that the accuracy of this procedure was ±0.5 MHz in the range of frequencies 8.8 GHz to 9.5 GHz. In order to obtain a reference for the measurements on di e l e c t r i c waveguides with axial slots, the normalized phase coefficients of three un-perturbed polyethylene rods (denoted A, B, and C, corresponding to the notation in Fig. 7.1) were measured. These rods were a l l cut from one o r i -ginal piece i n an attempt to provide a basis for meaningful comparisons be-tween them. Micrometer measurements on the diameter of the rods indicated that they had slightly e l l i p t i c a l cross-section with major and minor axes given by 0.499 in . and 0.495 i n . , respectively. Thus as expected, the cross-polarized modes were observed on the same resonance curve but with slightly different resonant frequencies, as shown in Fig. 7.6(b). This "double" resonance occurred at four separate angles of rotation (roughly separated by 90°) of the polyethylene rods. A single resonance curve was obtained, however, when the rods were rotated such that the coupling probe lay on the line 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 line of the minor axis. Thus, i t was possible to approximately determine the posi-tions of the major and minor axes experimentally. In an attempt to guard against any spurious frequency dependence of the results, the values of 8/kQ 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 . sca le = 3.5 MHz/large d i v . ) (a) frequency measurement using the reference channel t race (b) "double" resonance 91. corresponding to N = 23, 24, and 25 in equation 7.1. The results are given in Table 7.1. 7.2 Experimental Results for the Slotted Dielectric 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 in Fig. 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 in an attempt to accentuate the i n i t i a l separation i n normalized propagation coefficients for the two polarizations. Values of 3/kQ for each of the two polarizations are plotted in Figs. 7.7(a), (b) and (c). The polarizations corresponding to each curve are defined by the inserts in each figure. Although the measurements were done at three different values of B (as discussed in the previous section) only the results for N = 24 are shown in Figs. 7.7(a), (b), and (c). The behaviour of the curves with N = 23 and N = 25 i s similar. The results for rod A show, as expected, that the separation in B/k Q between the cross-polarized modes increases with increasing values of 6. A typical resonance curve indicating the separation of cross-polarized modes (for small 5) i s shown in Fig. v 7.8. For small values of 6, the results for rods B and C indicate the same trends as rod A; however, the separation between the cross-polarized modes attains a maximum and then decreases for increasing values of 6. In fact, there i s a value of 6 for rod C for which the separation i s zero (i.e. B/k Q i s the same for the two polarizations). It appears that rod B would also exhibit this phenomenon i f 6 was further increased. 7. 3 Results of the Perturbational Analysis The perturbational analysis presented i n Chapter 6 i s not directly ROD A ROD B ROD C ELLIPTICAL AXES MINOR MAJOR MINOR MAJOR MINOR MAJOR N 8.7900 8.7880 8.7905 8.7885 8.7905 8.7890 23 RESONANT 9.1075 9.1050 9.1080 9.1065 9.1090 9.1070 24 FREQUENCY (GHz) 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 e/k o 1.0375 1.0378 1.0374 1.0376 1.0373 1.0375 24 (rad/cm) 1.0452 1.0456 1.0452 1.0454 1.0451 1.0454 25 Table 7.1 Values of f and 8/k for the unperturbed rods 9 3 . ROD A 1.042 COUPLING PROBE / 0.032 0.024 0.096 & (inches) 0.128 Fig. 7.7(a) Normalized propagation coefficients of the two polarizations as a function 6. .. a, b experimental results for the configurations shown in inserts a and b ROD B COUPLING PROBE / \ i (inches) Fig. 7.7(b) Normalized propagation coefficients of the two polari zations as a function of 6. a, b experimental results for the configurations shown in inserts a and b . a',b' perturbation results corresponding to a, b a",b" perturbation results for a 10° symmetry error in the slots \ \ ROD C \ ^ COUPLING PROBE & (inches) Fig. 7.7(c) Normalized propagation coefficients of the two polarizations,as a function of 6. a, b (^experimental results for the configurations shown in inserts a and b a',b' perturbation results corresponding to a, b F i g . 7 . 8 The resonant f requency s e p a r a t i o n o f the two p o l a r i z a t i o n s on a per turbed r o d . ( h o r i z . s c a l e = 3 .5MHz/ large d i v . ) 97. applicable to the experimental configurations because of the difference i n the geometry of the slots 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 results 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 calculations was chosen to be slightly 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 relative permittivity, e^, used in the calculations was varied between a value of 2.26 and a value of 2.30, d) the experimental slots were approximated by theoretical slots according to the construction shown in Fig. 7.9(a), and e) the parameters d and w (in equation 6.2) for the theoretical slots were approximated by the depth and width, respectively, of the experi-mental slots. Perturbation results for rods B and C are shown as the dashed curves i n Figs. 7.7(b) and (c), respectively. These curves were derived using a frequency of 9.105 GHz and a value of = 2.29. No attempt was made to obtain perturbation results for rod A because of the obvious d i f -f i c u l t y in representing the rod cross-section i n this case by slots with polar geometry. The theoretical curves labelled a' and b' correspond to the experimental curves labelled a andb, respectively. Due to some machining problems, the experimental slots on rod B had the configuration shown in Fig. 7.9(b). For this reason the perturbation theory was modified such that nonsymmetric slots could be analyzed. These perturbation results 98. (b) Fig. 7.9. Slot configurations (a) approximation of a slot i n rectangular coordinates by one in polar coordinates (b) actual slot configuration for rod B. 99. for a 10° symmetry error, are shown as curve b" i n Fig. 7.7(b). For the scale used in Fig. 7.7(b), the curve a" i s the same as curve a'. Based on the curves presented i n Figs. 7.7(b) and (c) the follow-ing observations can be made: 1. for small values of 6 the experimental results are lower than the perturbation results 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 Fig. 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 quasi-static approximation may be quite good in this range. 3. the propagation coefficient i s not very sensitive to symme-try errors in the slots. Generally speaking, the perturbation results give a reasonably good indication of the trends in the experimental results. In particular, they confirm the experimental observation (rod C) that a value of 5 exists for which the two polarizations have the same normalized propagation co-efficient. 7.A Summary In this chapter an experimental investigation of propagation on axially slotted dielectric rods was described. The measurements were carried out at X-band using polyethylene rods with nominal diameters of 0.5 in . The slot configurations considered were of rectangular geometry. The main points were: 1. Experimental apparatus: the measurement of 3/kQ using a resonant cavity apparatus was described. Preliminary results obtained for unperturbed dielectric rods indicated that they had, i n fact, 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: three different configura-tions were considered. Generally, the effect of the slots was to increase the separation i n B/k Q between the cross-polarized modes. This increase was not indefinite in the case of rods B and C, however. In fact, for rod C, there was a non-zero value of slot depth for which the separation was zero. 3. Perturbation results: the perturbational analysis for slots with polar geometry (given i n Chapter 6) was applied in an approximate fashion to the experimental slot configurations. The perturbation results gave a reasonably good indication of the trends in 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 scattering by step discontinuities on a planar surface waveguide. The main objectives were (a) to provide an v accurate analytical method, of general applicability and suitable for transmission matrix studies of cascaded discontinuities, for determining the scattering parameters and (b) to provide accurate experimental measur-ment of the scattering parameters, particularly since the theoretical re-(52) suits in this work did not agree with some recently published results Of the many surface waveguide configurations possible, a d i -electric coated conductor supporting TM modes was chosen for the investi-gation because of the advantages this configuration offered experimentally. The discontinuities consisted of abrupt changes in the dielectric thickness. The analytical method involved bounding the open structure with a perfect conductor. Using the direct relationship between the mode spectra of the bounded and the unbounded waveguides, the solution to the unbounded configuration was obtained directly from the solution of the bounded configuration. The closed boundary problem was solved using mode-matching. Prior to the application of the above method, two separate i n -vestigations were deemed necessary, namely (i) the behaviour of electro-magnetic fields near a multi-dielectric edge (this investigation also i n -cluded edges which involved an anisotropic medium) and ( i i ) the effect of dielectric edge conditions on mode-matching. The experimental investigation was carried out at a frequency of 30.00 GHz. 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 variation over a width of three wave-lengths and a decay away from the dielectric surface which was in excell-ent agreement with theory. Measured values of attenuation were also found in reasonable 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 longitudinal component of the el e c t r i c f i e l d . This type of "large" probe has proved useful in such measurements on planar surface waveguides when the probe i s placed away from the dielectric surface and thus does not significantly disturb the fi e l d s . Both the magnitude and the phase of the reflection coefficient of the "TMo" mode were measured for a range of step heights (these measurements were also done at 29.94 GHz and 30.06 GHz i n an attempt to guard against any spurious frequency dependence in the results). Only the magnitude of the re-flection coefficient was measured for a cascaded discontinuity. However, a useful check on the results was obtained in this case by varying the elec-t r i c a l distance between the cascaded steps. The overall contributions of the work contained in Part I may be summarized as follows: 1. The solution of planar surface waveguide discontinuity problems using the "bounded" approach. The method.has been shown to be simple, accurate and applicable to a variety of configurations and to cascaded discontinuities. Also, the correct results for the configuration an-alysed i n reference (52) are given. 2. The experimental verification of the results obtained theoretically. 3 . Certain aspects of the experimental technique used for the measurement of the reflection coefficient: (a) the use of a A/2 dipole to measure the longitudinal component of the electric f i e l d in the standing wave and (b) the interpretation of the experimental results in 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 in situations where the numbers of propagating modes on either side of a discon-tinuity are unequal. 5. Two side issues which were relevant to this investigation: (a) the behaviour of electromagnetic fields near an-edge common to three dielectrics was considered and i t was found that the fields are not necessarily singular at such an edge and (b) the effect of dielectric edge conditions on mode-matching was also considered and i t was shown that these edge conditions do not require any special consideration when using mode-matching. Part II deals with surface waveguides with perturbed circular cross-sections. The main objective was to determine what relative effect the perturbations have on the cross-polarized modes, with a view to making the polarization more definite. Dielectric rods, of circular cross-section, perturbed by axial slots were considered theoretically and experimentally. The parameter of interest was the normalized propagation coefficient. The analytical treatment involved a standard perturbational an-alysis applied to slots with cross-sections in polar geometry. The experimental study was carried out at X-band frequencies using an open resonant cavity. Polyethylene rods with perturbations in rectangular geometry were considered. 104. The results indicate that the polarization is made more definite i f the perturbations are small (i.e. separation of the normalized propa-gation coefficients). If the perturbation consisted of a slot, however, there was an optimum value of slot depth which gave the largest separation of the two polarizations. Slot depths beyond the optimum resulted i n decreased separation. This fact was shown experimentally and theoretically. 8.1 Suggestions for Further Work The work presented in this thesis suggests several possible extensions: (a) theoretical and experimental analysis of mode conversion on multimode surface waveguides (with applications to optical fibres) with surface imperfections. The 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 series of small steps (the works of (49 50) (53 54) Marcuse ' and Snyder ' are restricted to gradual tapers of dielectric waveguides used in optical fibre applications where the refractive index differences between the core and cladding are small), (c) the determination of the radiation pattern of a step discontinuity by considering the fields in the plane of the discontinuity found from the "bounded" approach, (d) theoretical and experimental investigations of finned d i e l e c t r i c waveguides, (e) the application of the f i n i t e element method to finding more accu-rately the propagation properties of d i e l e c t r i c waveguides with various cross-sections. This approach would also yield information about the f i e l d patterns. 105. (f) the investigation of feasible 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 fields which are transverse to the edge can become singular ' ^ '^ """^ whereas the f i e l d components in the direction of the edge are always nonsingular. The f i e l d singularity i s determined by the edge condition which requires that the energy contained in a small volume about the edge must be f i n i t e . More exp l i c i t l y , singular f i e l d components must be of order p f c ^ as p -> 0 with 0 < t < 1, where p is the radial distance from the edge. The singularity parameter, t, depends on the geometry and the properties of the media sur-rounding the edge. In a l l cases, however, two types of singularity must be considered: (1) that of the electric f i e l d (due to different permittivities), characterized by a singularity parameter t„, and (2) that of the magnetic f i e l d (due to different permeabilities), characterized by a singularity parameter t„. n There may be more than one value of t_ or t in the range of 0 to 1. If su ti both electric and magnetic type singularities exist at an edge, the domin-ant singularity i s due to the smallest value of either t or t„. c> r i The purpose of this investigation i s to extend the existing theoretical treatment of electromagnetic f i e l d behaviour near di e l e c t r i c edges to include two cases of practical interest: (a) the case where one of the two media surrounding an edge is a magnetized f e r r i t e , shown in Fig. 11(a), and (b) the case where three isotropic dielectrics have a common edge, shown in Fig. 11(b). 107. e 2 M 2 MEDIUM 2 (a) £; JJ MEDIUM 1 0,2TC 62^2 MEDIUM 2 0;+ t? '3,^3 MEDIUM 3 (b) h MEDIUM 1 0, 2n Fig. II. Edge configurations (a) dielectric-magnetized f e r r i t e (b) three isotropic dielectrics. 108. The basics of the underlying theory for the derivation of the singularity parameters can be found e l s e w h e r e F o r case (a), however, the theory must be modified slightly to account for the permeability ten-sor in medium 1 and, thus, i t i s outlined for this case. In case (b) the emphasis i s on the dependence of the f i e l d singularity on geometric varia-tions of the edge configuration. This dependence is somewhat complex and, thus, is illustrated by a. numerical example. I-A.l Edge Involving a Magnetized Ferrite Consider the edge configuration shown in Fig. 11(a) where medium 1 i s magnetized fe r r i t e and medium 2 is isotropic d i e l e c t r i c . With a steady magnetic f i e l d , B z» applied i n the direction of the edge (z-direction), the permeability of the ferrite i s given by: 0 -5K V 0 0 0 y L °J In a homogeneous region, the electromagnetic fields are solutions of Maxwell's equations: V x IS = -jcoyH 7 x H = jtoeE 12 For the configuration shown in Fig. 11(a) y i s given by equation II, a tensor, i n medium 1 and by y , a scalar, in medium 2. Near the edge, the fields can be expanded in a power series as follows: 109. u t-1 , t , t+1 , Ep = a 0 p + a l p + a 2 p + ~ E i " b 0 p t _ 1 + b l p t + b 2 p t + 1 + ~ t _ 1 . t , t+1 J E z = c Qp + C ; Lp + c 2p + — n - t—1 t , t+1 , H p = a Qp + o^p + a 2p + — „ t-1 , „ t , n t+1 , H f = V 6 1 P 3 2 p " u - t-1 ' t , t+1 , z Y 0 P Y l p Y 2 P 3 Substituting equation 13 into the component form of equation 12 and then equating coefficients of like powers of p yields for medium 1: C ( )(t-1) = 0 Y 0 ( t -D = 0 j o e * = 3 P0 1 0 ---34 3z J t o e i b o = -3T- **! 3c 3b -J»( ya Q + j K B 0 ) = -3 a o ^ ( - j ^ + y 6 Q ) - — - t C l 3 a o 3 a o t b 0 - - l ? = 0 14 Rearranging equation 14 results i n the following equations for the coeffi-cients ( a Q , b Q, c.p a Q, 3Q , y^) i n medium 1: 110. = sin tty + A^ cos t<j> O Q = sin tty + B 2 cos tty b^ = A^ cos t<j> - A 2 sin tty 3Q = B . ^ cos t<j> - B 2 sin t<j> 9 B 1 . . . 9 B2 tYj^ = sin tty + cos ti}) - ja)e 1(A 1 cos t<J> - A^ sin tty) 3A 1 3A 2 t c i = T i " s i n ^ + H c o s + jco((ucos tty - J K sin tcJ))B1 - (ysin t<j> + j< cos t<fi)B2) 15 A similar procedure yields for medium 2: (prime denotes medium 2) a^ = Aj^ sin t$ + A^ cos t<j> = B^ sin tty + B^ cos t<j> b^ = Aj cos t<j> - A^ sin tty 6^ = B| cos t<{) - Bj", sin t((i 3B| 3BJ ty* = - 7 — sin t<() H — — cos t<j> - ja)e_(A' cos tty - A\ sin t<}>) ' 3A^ 3A^ t c ^ = —g^- sin t<f> + — — cos t<j> - jo)y 2(B^ cos tty - B 2 sin t(j>) 16 In order to satisfy the boundary conditions, the components of the IS and H fields tangential to the dielectric interfaces must be continuous. These conditions, when applied to the coefficients i n equations 15 and 16 at ty = ty^ and ty = 2ir, in Fig. 11(a), result in a homogeneous system of eight equations i n the eight unknown coefficients (A^, A 2 > A^, A 2, B^, B 2 > B^, B 2). This system of equations i s separable into the following 111. two 4 x 4 homogeneous systems: a) with = = = = 0 e l C 2 - £1 S2 -S, '27T " h C 2 - e 2 C 2 l r -C, -C 2ir e2 S2 e 2 S 2 l r A ' A l A' A2 = 0 17 b) with Aj^ = A 2 = Aj^ = A J = 0 y c 2 - j K s 2 -yS 2-JKC 2 Where -S 2 : -S 2-rr - y 2 C 2 - y 2 C 2 l r -C 2v y 2 s 2 y 2 S 2 , B, B2 = 0 18 S, = sin t<j>.. C_ = cos t(f>1 S- = sin t2ir 2ir C 2 l r = cos t2ir 19 The above homogeneous systems of equations are consistent i f the deter-minant of each matrix i s zero. Setting the determinant of the matrix i n equation 17 equal to zero yields: sin t„ir Ei ein tE((j>1 - TT ) = + e l - e2 £1 + £2 110 112. Repeating t h i s for equation 18 yields: 2 2 sin tjjTT /(y - y 2) + K sin tH(<J ) ] L - IT) " W (u + y 2 ) 2 + k 2 111 Equation 110 was originally derived in reference (71). Equation 111 was derived in connection with some recent work by McRitchie and (91) Kharadly involving a homogeneous to inhomogeneous nonreciprocal wave-guide interface. In that work, the singularity parameter, t^, was used (73) to check mode-matching solutions since the electric fields were non-singular. In general, however, both t and t„ would have to be considered. I-A.2 Edge Involving Three Dielectrics In this section, the behaviour of the fields near an edge com-mon to three isotropic dielectric media, shown In Fig. 11(b), is examined. Let E2 > E i > £3 k u t this assumption does not result i n any loss i n generality. Here, only the electric fields (hence t„) need be considered, since the magnetic f i e l d singularity behaves i n a similar fashion. Using the procedure outlined i n the previous section, i t can be shown that t„ i s a solution of: E el< G2 + E3 ) C1 S21 S 2 7:2 + e 2 ( e 3 + e l ) S l C 2 1 S 2 r r 2 + £ 3 ( E 1 + e 2 ) S l S 2 1 C 2 l r 2 + 2 e l ^ 2 e 3 ( 1 " C l C 2 1 C 2 i r 2 ) 1 1 2 where 113. S1 = sin tE$1 C1 » cos tE$1 S21 ° S i n V2 C21 = C O S S2 l T2 = S i n V 2 i r --'•2 " *1 } C2u2 = C O S V 2* " *2 " *1 } 113 With ^ - 0 or 2ir - <|>^, equation 112 reduces to the expression for an edge common to two dielectric 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 0 = 10e, and e- = e . A plot of t„ versus <(>.. + <j> , X O / O J O il. J . Z for various values of <J>^, i s shown in Fig. 12. For comparison purposes, three special cases involving only two dielectrics are also plotted i n the same figure. These are shown as the dashed curves, representing solutions of equation 110 for the following choices of parameters: a) upper curve: e, = e , e~ = 2e r r 1 .0 2 o b) middle curve: e, = 2e , E 0 = 10E 1 o 2 o c) lower curve: e, = E , e_= 10E 1 o 2 o One might expect, on f i r s t examination, that the dashed curves would pro-vide bounds on the solid curves. Instead, i t i s observed that the behaviour of t i s more complex than can be accounted for by such bounds. E In fact, Fig. 12 indicates that for <j>^ = 45° there i s a range of angles <|>0 for which t i s greater than unity, i.e. the fields are nonsingular. 2 E This situation does not occur in the two-dielectric case^ however, i n a (92) ~ recent investigation involving a perfectly conducting wedge, loaded with at least two dielectrics, a similar phenomenon has been observed. 115. Region of Nonsingularity The range of nonsingularity is now examined in more detail. The boundaries can be obtained by setting t_ = 1 in equation 112 and solving E for <j>2 In terms of <J>^. Results indicate that there is a region of non-singularity in the <J>^, <t>^ plane for any three-dielectric configuration. For the example considered, this region is shown in Fig. 13. The boun-dary locus passes through the point (labelled A in Fig. 13) corresponding to an interface configuration for which there is obviously no singularity. This configuration i s illustrated by insert (a) i n Fig. 13 whereas an arbitrary point inside the region is illustrated by insert (b) in the same figure. The extent and shape of the region depend on the relative values of e^, e^, and e^. For example, i n Fig. 14, the region i s shown as a function of c in the range of e = e = e to e = e = 10e . At these 1 1 3 o 1 2 o two limiting values of the region collapses to straight lines, as shown in Fig. 14. These lines correspond to the special two-dielectric case illustrated by insert (a) in Fig. 13. A study of the results of Fig. 14 reveals that there is some value of i n the range of to for which the area of the region i s maximum. For the parameters considered, the maximum occurs at a value of approximately equal to 4 E o < However, the interval of angles spanned by the region, denoted by A<j>^, does not vary with e^. Further investigation shows that A<j>^ depends solely on the ratio of the highest to lowest permittivities, £2^e3' r^l^-s dependence i s given in Fig. 15. It may also be seen from Fig. 15 that there i s some minimum value of ty^ associated with the region given a set of permittivities. Numerical results indicate that this minimum value, ^ » depends on i n a manner closely related to the dependence 116. 0 10 20 30 40 50 ^ ANGLE IN DEGREES • • Fig. 14 Dependence of a region of nonsingularity on the permittivity with > > e^* M 80 0 . 20 40 . 60 . 80 100 Fig. 15 Dependence of A<J>.. on the ratio of the highest to lowest permittivities, e^/ of Ad>, on e„/e„. In fact, (<j>„) . can be obtained from Fig. 15 simply by Y l 2 3 2 mm substituting £^e3 f o r e2^e3 a n d ~ ^2^min f ° r A ^ l " T h i s m e a n s that the wedge angle, ty^, of the medium with the highest permittivity must always be greater than 90° i f f i e l d singularities are to be eliminated, Based on the numerical example considered above, the following conclusions may be drawn: (a) loading a dielectric edge with a third d i e l e c t r i c can, i n some cases, eliminate the f i e l d singularity at the edge, (b) there i s a region in the <{>^, ^ plane for which the fields are nonsingular, (c) for a given choice of > > E3» there i s some value of e ^ for which the area of the region i s maximum, (d) the interval of <f)^ spanned by the region i s determined solely by the ratio of the highest (z^) and the lowest (e^) permittivities, (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 singularity can be eliminated. In fact, the minimum wedge angle i s determined solely by the e l ratio — . £3 The points l i s t e d above are consistent with those found in ref-erence (92) for the special case when medium 2 is a perfect conductor. This configuration can be achieved by letting 0 0 - . It would not be reasonable to suggest that the region of nonsingularity exists for edges common to more than three dielectrics. However, the analysis could be much more complex. Appendix I-B EFFECT OF DIELECTRIC EDGE CONDITIONS ON MODE-MATCHING SOLUTIONS The configuration used in this analysis i s shown in Fig. 16(a). The discontinuity i s an interface between an empty rectangular waveguide and a waveguide of the same dimensions inhomogeneously loaded with a die-l e c t r i c for which e r and u^ are different from unity. Only H-plane loading is considered here. Since two types of modes are excited in both wave-guides A and B (see Fig. 16(a)) the effect of the die l e c t r i c edge condition may be observed on the ratio of these modes as well as the total number of modes on either side of the discontinuity. A T E ^ Q mode incident on the interface from waveguide A excites ((La) the following sets of modes : T E 1 Q + T E l m + TM in waveguide A, and LSE l m + LSM l m_ 1 i n waveguide B, where m = 2,4,6.... The equivalent c i r c u i t of such an interface i s shown (f.Q\ i n Fig. 16(b) . In order to i l l u s t r a t e the effect of the edge condi-tions a numerical example (e^ =8.0, u r = 0.75) i s discussed. With this choice of the parameters, e r and u^, there are both ele c t r i c and magnetic type singularities at the edges, as discussed in Appendix I-A. Using equations 19 and 110, i t can be shown that the electric f i e l d singularity i s dominant. The edge condition determines the rate of decay of the ampli-tude coefficients 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 (71), the amplitude coefficients of the T E l m and TM l m modes, |AE| and respectively, should decay at the rate m 2 * ^ D f 0 r large m. The correct ratio of modes to be used i n mode-matching can be 122. (93) detemined by examining the rate of decay of the coefficients . For the example considered here, typical values of | A E | and | A ^ | are shown as functions of m in Fig. 17. A straight line of slope -2.746 i s also shown for comparison. The results shown in Fig. 17 indicate that there i s reasonable agreement between the predicted and calculated decay i n the coefficients. The scatter in the calculated points make identification of the decay rate d i f f i c u l t , however, For this reason the effect 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 additional check on the effect of the edge conditions. Briefly, the results indicate that: (a) irrespective of the choice of modes used i n mode-matching, the equivalent c i r c u i t parameters eventually converge to the same value. (b) faster convergence i s obtained with fewer TE modes than TM modes (similarly, fewer LSE modes than LSM modes) but with the same total number of modes on either side of the discontinuity. Appendix I-C INTEGRALS INVOLVED IN EQUATION 2.19 Let superscripts a and b denote waveguides A and B, respec-t i l t i l tively, and l e t subscripts k and j denote the k and j modes, respectively. Normalization constant P , ak t i l For the k mode i n waveguide A Pak = 2iTT ( A l k + s i n A l k ) / 2 p l k e r o o where ~ ( ^ 2 < A 2 k + s i n A2k ) / 2P2k A _ a a A l k = ^ i k h A2k - ^ 1 - V The corresponding expression for the j mode in waveguide B, P^.can found from equations 114 and 115 by substituting 3 B p b , p b., and C b a a f P r 3 f c , P l k, p ^ , and C*, respectively. Integrals ' Q \ k x h^. • a^dx and ^ t . h ^ • ^ d x L e t A l k j = Plk + Plj A a b A2kj = Plk " p l j A a , b A3kj = P 2k + p l j A - 3 b A4kj ~ P 2k " Plj 125. Then A a , b A5kj = P2k + P 2 j A a b A6kj = P2k " P 2 j 116 0 —ak -Hn —z 2k Y 1 . o o 0 % j X ^ k ' ^ d x = 2kY T l 117 J — o o where T l = T 2 / E r + C k T 3 - C k C j T A 1 1 8 The terms T 2 > T 3 > and T^ are given by: T 2 = sin A l k . t a / A l k . + sin A ^ t J / A ^ T 3 = (sin ( A 3 k . t ^ - p ^ t B ) - sin ( A ^ . t * - p ^ ) ) /A^. ^4kj + (sin ( A 4 k j t ^ - pak t B ) - sin ( A 4 k j t * - p * ^ ) ) /A^ T 4 = sin A 5 k j ( t > - t B ) / A 5 k j + sin A f i k j ( t> - t ^ / A ^ The expressions for the slow modes can be obtained from equations 114-19 with p 2 k = j y 2 k and p!^ = j y ^ j • 119 126. Appendix I-D ATTENUATION COEFFICIENTS An approximation to the attenuation coefficient of the k t b mode in waveguide A may be found from a • P&k °k = 2P~~ K ak 120 where P ^ i s power/unit width absorbed by the diele c t r i c . P ^ i s given by a tan 6 , . 2 a , r / /,a. 2 ., a ..2, . „ a a,„ a .. < e A t i + [(8.) - (p n i) 1 sxn 2p_. t./2p..) Ak Y e v r o l ' l ^ k ' v * l k ' J ° fc*'lkwl"t'lk o o r 121 th The corresponding expressions for the j mode in waveguide B may be obtained from equations 120 and 121 by substituting t b , 8 b, P^j» a n ^ a. a a for t^, 8^, P ^ j and p a^» respectively. The expressions for P ^ and P^ may be found in Appendix I-C. 127. Appendix I-E MEASUREMENT OF STANDING WAVES WITH A A/2 DIPOLE Consider the general case of a wave propagating in the z-direction. Assume that a standing wave exists i n this direction and that the electric f i e l d has z-component given by: E (x,y,z) = E ( x , y ) ( e " j e z + r e j g Z ) 122 z o where T is the reflection coefficient and where E Q(x,y) remains unspecified by essentially depends on the source of the wave and the boundary condi-tions. Let the dipole be oriented as shown in Fig. 18. If the dipole coordinate z 1 = z - a i s introduced then the current, I , at the antenna terminals i s given b y ^ ^ : L I L(x,y,a) = c'.jE^x.y^YCO.z^dz' . 123 where C is a constant which depends on the impedances of the load and the dipole. Y(0,z') i s the transfer admittance of the dipole between z 1 = 0 and z' (i.e. the current at z' due to a unit voltage at z' =0). Thus, Y(0,z') i s just the current distribution of the dipole of length 2L when used as a transmitter. Hence Y(0,z f) = sin k 0(L-z') z'> 0 = sin k0(L+z') 7.y< 0 124 Substituting equations 122 and 124 into equation 123 yields: I L(x,y,a) = CE o(x,y){ j^ - j e C z'-hx) + ^ jSCz'-ta)) s ± n k ( L _ z . ) d z . 125 128. 129, Upon evaluation of the integrals in equation 125 one obtains: cos BL - cos k L I L(x,y,a) = CE o(x,y)2k o ^ 2 ~ ^ e + T e * k - B 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/ 2 which i s of interest here) measures the correct value of the standing wave pattern. 126 130. Appendix I-F DERIVATION OF WAVE MATRICES AND w2 Consider the discontinuity at z = z^, as shown in Fig. 4.1(b) Let c_| and b_^ be defined in waveguide B in a similar manner to c.^ and b_^ in waveguide A. The scattering matrix, S, for the discontinuity i s defined b y ( 8 0 ) : *1 = S 127 where individual elements of S are found from the mode-matching solutions to the discontinuity problem with individual modes incident from either side. S can be partitioned S = s: 128 Then w^ and w^ are defined by: W l = ,-1 S1 S3 -S _ 1S b3 b4 S 2 ~ S l S 3 l s 4 129 and ,-1 S 4 S 2 1 " S 2 1 S 1 S3~ S4 S2 1 S1 130 131. APPENDIX II - A DERIVATION OF EQUATION 6.1 The fields E , H , E and H are solutions of Maxwell's equations: —o ~o V x E = -ja>u H —o o — o V x H = ju e E, H I —o —o and V x E = -JWV0 H V x H = ja)(e + Ae) E 112 Consider V • (H x E *) and V • (H * x E); from equations III and 112 — —o —o — V . (H x E *) + V • (H * x E) = jco Ae E • E * 113 — —o —o — — —o Define a volume T as shown i n Fig. I l l . Applying the divergence theorem to the volume integral of equation 113 results i n / / (H x E * + H * x E) • dS = jcoAe / E • E * dx 114 S ~ -o -o - - -o where dS 115 / / dS = / / S Sw + S s + SW Since the surface wave fields decay to zero as R + " (in Fig III ) then / / (H x E * + H * x E) • dS = 0 1 1 6 s Also 132. Fig. I l l Volume region. 133. And since Ac = 0 except in the perturbed regions then / dx = dz / / dS 118 T S P Thus, using equations 116 , 117 , and 118 equation 114 becomes |- / / ( H x E * + H * x E ) « a dS = jojAe / / E • E * dS 119 9z ' * — —o —o — —z J c ~° w p where S is the cross-section of the perturbations. P The z-dependence in the integrand of the left-hand side of equation II 9 -j ( B - B 0)z is given by e . Thus, equation II9 reduces to -0 ) / / (H x E Q* + H q* x E) • a z dS = juAe / / E • E q * dS s s w p m o and, hence (equation 6.1) / / E • E * dS B -B 0 Ae S ° k Z o / / (H x E * + H * x E) • a dS J J — — n — n — — Z S W where r Ae = — , Z = / e , k = (o/y e r e o o ' o o o o 134. Define Appendix II-B SELECTED INTEGRALS OF BESSEL FUNCTIONS L±t i = 1,. . . . .8 p n = p i r r p i o = p i V A N D P 2 I = Y 2 r i t h e n P l l 2 A- = 2J J,(x) x dx 1 P10 1 = P n ^ ( P i i ) ) 2 + ( P i r 1 ^ ! ^ - P i o < J i ( p i o ) ) 2 " fcio-^frio* J 2(x) A2 = 2 Jp' 0 1 ~ 7 ~ ~ d x = Jx<PlO ) + " ^ l l 5 " " ' f r l l * 1 1 1 2 P l l 2 A- = 2J J " L ( J 1 ( x ) ) Z x dx 3 p10 1 - A x - A 2 + PuVPu)J1<PU> " P 1 0 Ji<Pio> Ji (PlO ) 1 1 1 3 A 4 = 2J^J 1(x)Jj_(x)dx = 4 (Pll ) " J 1 ( P 1 0 ) 1 1 1 4 p l l 2 J l ( x ) Ar = 2 X n l i x ( J 1 (x)) Z + - i — dx 5 p10 1 X - P u ^ i ^ n ^ 2 + ( p ir 1 ) j 2 ( p ii ) + 2 pii Ji ( pii ) Ji ( pii ) 1115 A 6 = 2;Q 1 1J 1(x)J^(x)dx = 4(P n) H16 A ? = 2r K1(x)Kj_(x)dx = -K 2(P 2 1) IH7 9 K 2(x) Aft = 2* x(K'(x)) Z + dx a P 2^ 1 x .- - < P 2 i + 1 ) K i ( p 2 1 ) " P 2 1 ( K 1 ( P 2 1 ) ) 2 " 2 p 2 1 K l ( p 2 1 ) K l ( p 2 1 ) 1 1 1 8 135. REFERENCES 1. Sandbank, CP.: "The prospects of fibre-optic communication systems", IEE Conf. 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Trans, on Antennas and Propagation, 1957, AP-5, pp. 100-109. 40. Kay, A.F.: "Scattering of a surface wave by a discontinuity in reactance", I.R.E. Trans, on Antennas and Propagation, 1959, AP-7, pp. 22-31. 41. Breithaupt, R.W.: "Diffraction of a cylindrical surface wave by a discontinuity in surface reactance", Proc. IEEE, 1963, 51, (11), pp. 1455-1463. 42. Breithaupt, R.W.: "Diffraction of a general surface wave mode by a surface .reactance discontinuity", IEEE Trans, on Antennas and Propa-gation, 1966, AP-14, (3), pp. 290-297. 43. Johansen, E.L.: "Surface wave scattering by a step", IEEE Trans, on Antennas and Propagation, 1967, AP-15, (3), pp. 442-448. 44. Johansen, E.L.: "Surface wave radiation from a thick semi-infinite plane with a reactive surface", IEEE Trans, on Antennas and Propa-gation, 1968, AP-16, (4), pp. 391-398. 138. 45. Felsen, L.B.: "Radiation from a tapered surface wave antenna", I.R.E. Trans, on Antennas and Propagation, 1960, AP-8, pp. 577-585. 46. Cooley, W.W., and Ishimaru, A.: "Scattering of a surface wave from an abrupt change in thickness on a dielectric slab", Contract No. AF19(628)-2763, Technical Report No. 105, March 1966. 47. Clarricoats, P.J.B., and Sharpe, A.B.: "Modal matching applied to a discontinuity in a planar surface waveguide", Electron. Lett., 1972, 8, (2), pp. 28-29. 48. Hockham, G.A., and Sharpe, A.B.: "Dielectric waveguide discontinui-ties", Electron. Lett., 1972, 8, (9), pp. 230-231. 49. Marcuse, D.: "Radiation losses of tapered dielectric slab waveguides", Bell Syst. Tech. J., 1970, 49, (2), pp. 273-290. 50. Marcuse, D.; "Radiation losses of the dominant mode in round dielec-t r i c waveguides", Bell Syst. Tech. J., 1970, 49, (10), pp. 1665-1693. 51. Shevechenko, V.V.: "Continuous transitions in open waveguides", Golem Press, 1971. 52. Mahmoud, S.F., and Beal, J.C: "Scattering of surface waves at a dielectric discontinuity on a planar waveguide", IEEE Trans, on Microwave Theory and Techniques, 1975, MTT-23, (2), pp. 193-198. 53. Snyder, A.W.: "Surface mode coupling along a tapered dielectric rod", IEEE Trans, on Antennas and Propagation, 1965, AP-13, (9), pp. 821-822. 54. Snyder, A.W.: "Coupling of modes on a tapered dielectric cylinder", IEEE Trans, on Microwave Theory and Techniques, 1970, MTT-18, (7), pp. 383-392. 55. Snyder, A.W.: "Radiation losses due to variations of radius on dielectric or optical fibres", IEEE Trans, on Microwave Theory and Techniques, 1970, MTT-18, (9), pp. 608-615. 56. Snyder, A.W.: "Mode propagation in.a nonuniform cylindrical medium", IEEE Trans, on Microwave Theory and Techniques, 1971, MTT-19, (4), • pp. 402-403. 57. Marcuse, D,: "Mode conversion caused by surface imperfections of a dielectric slab waveguide", Bell Syst. Tech. J., 1969, 48, (10), pp. 3187-3215. 58. Marcuse, D., and Derosier, R.M.: "Mode conversion caused by diameter changes of a round dielectric waveguide", Bell Syst. Tech. J., 1969, 48, (10), pp. 3217-3243. 139. 59. Marcuse, D.: "Coupling coefficients for imperfect asymmetrical slab waveguides", Bell Syst. Techn. J., 1973, 52, (1), pp. 63-82. 60. Marcuse, D.: "Coupled mode theory of round optical fibers", B e l l Syst. Tech. J., 1973, 52, (6), pp. 817-842. 61. S t o l l , H. , and Yariv, A.: "Coupled mode analysis of"periodic dielec-t r i c waveguide", Opt. Commun., 1973, 8, (1), pp. 5-8. 62. Sakuda, K., and Yariv, A.: "Analysis of optical propagation in a corrugated dielectric waveguide", Opt. Commun., 1973, 8, (1), pp. 1-4. 63. Tuan, H.: "The radiation and reflection" of surface waves at a discontinuity", IEEE Trans, on Antennas and Propagation, 1973, AP-21, (3), pp. 351-356. 64. Rawson, E.G.: "Analysis of scattering from fiber waveguides with irregular core surfaces", Appl. Opt., 1974, 13, (10), pp. 2370-2377. 65. Rulf, B.: "Discontinuity radiation i n surface waveguides", J. Opt. Soc. Amer., 1975, 65, (11), pp. 1248-1252. 66. Yaghjian, A.D., and Kornhauser, E.T.: "A modal analysis of the dielectric rod antenna excited by the HE^ mode", IEEE Trans, on Antennas and Propagation, 1972, AP-20, (2), pp. 122-128. 67. Clarricoats, P.J.B., and Slinn, K.R.: "Numerical solution of wave-guide-discontinuity problems", Proc. IEE, 1967, 114, (7), pp. 878-886. 68. McRitchie, W.K., and Kharadly, M.M.Z.: "Properties of interface between homogeneous and inhomogeneous waveguides", Proc. IEE, 1974, 121, (11), pp. 1367-1374. 69. Masterman, P.H., and Clarricoats, P.J.B.: "Computer field-matching solution of waveguide transverse discontinuities", Proc. IEE, 1971, 118, (1), pp. 51-63. 70. Lee, S.W., Jones, W.R., and Campbell, J.J.: "Convergence of numerical solutions of iris-type discontinuity problems", IEEE Trans., 1971, MTT-19, (6), pp. 528-536. 71. Mittra, R., and Lee, S.W.: "Analytical techniques i n the theory of guided waves", MacMillan Co., New York, 1971. 72. Brooke, G.H., and Kharadly, M.M.Z.: "Field behavior near anisotropic and multi-dielectric edges", IEEE Trans, on Antennas and Propagation, (to be published in July issue, 1977). 140. 73. Brooke, G.H., and McRitchle, W.K.: "Effect of dielectric edge conditions on mode-matching solutions", Electron. Lett., 1975, 11, (17), pp. 422-423. 74. Meixner, J.: "The behavior of electromagnetic fields at edges", IEEE Trans, on Antennas and Propagation, 1972, AP-20, (4), pp. 442-446. 75.. Collin, R.E.: "Foundations of microwave engineering", McGraw-Hill, New York, 1966. 76. Brooke, G.H., and Kharadly, M.M.Z.: "Step discontinuities on dielec-t r i c waveguides", Electron. Lett., 1976, 12, (18), pp. 473-475. 77. El-Kharadly, M.M.Z.: "Some experiments on a r t i f i c i a l dielectrics at centimetre wavelengths", Proc. IEE, 1955, 102B, (1), pp. 17-25. 78. 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J., 1969, 48, (9), pp. 2133-2161. 85. Pregla, R.: "Analysis of rectangular dielectric waveguides required in integrated optics by use of a variational method", Proceedings of European Microwave Conference, 1973, Paper B.5.2. 86. Yeh, C.: " E l l i p t i c a l dielectric waveguides", J. Appl. Phys., 1962, 33, (11), pp. 3235-3243. 87. Yeh, C : "Attenuation i n a dielectric e l l i p t i c a l cylinder", IEEE Trans, on Antennas and Propagation, 1963, AP-11, pp. 177-184. 141. 88. H a r r i n g t o n , R.F.: "Time-harmonic electromagnetic f i e l d s " , McGraw-H i l l , New York, 1961. 89. Makino, I . : "Measurement of the propagation c h a r a c t e r i s t i c s of s h i e l d e d and unshielded d i e l e c t r i c - t u b e waveguide", MaSc. T h e s i s , December 1970. 90. Sucher, M., and Fox, J . : "Handbook of microwave measurements - v o l . I I " , P o l y t e c h n i c P r e s s , New York, 1963. 91. 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Properties of surface waveguides with discontinuities and perturbations in cross-section Brooke, Gary H. 1977
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Title | Properties of surface waveguides with discontinuities and perturbations in cross-section |
Creator |
Brooke, Gary H. |
Publisher | University of British Columbia |
Date Issued | 1977 |
Description | The first part of this thesis is concerned with theoretical and experimental investigations of step discontinuities on a planar surface waveguide. An approximate theoretical solution to the unbounded discontinuity problem is obtained by bounding the open structure with perfect conductors, since there is a direct relationship between the mode spectra of the two configurations. Mode-matching is used to solve the bounded case. The method is "tested" on four discontinuity configurations considered by other workers. Good agreement with the previous results is obtained in all cases except one for which the original results are shown to be inaccurate. The experimental investigation is carried out on a dielectric coated conductor surface waveguide, supporting the first TM mode, at a frequency of 30 GHz. Standing wave measurements are obtained using a X/2 dipole oriented along the longitudinal component of electric field. The parameters of interest are the magnitude and the phase of the reflection coefficient. The experimental results confirm those obtained theoretically. The theoretical and experimental techniques are later applied in an investigation of a cascaded step discontinuity configuration. The theoretical approach involves the use of wave transmission matrices. Experimental results for the magnitude of the reflection coefficient are found to be in reasonable agreement with theory. The second part of this work describes a theoretical and experimental study of dielectric waveguides, of circular cross-sections, perturbed by axial slots. In particular, the normalized propagation coefficient of the dominant modes with each polarization is determined analytically using a standard perturbation technique and experimentally using an open resonant cavity. The perturbation results give quite a good indication of the trends observed experimentally. It is found that there is an optimum size of the perturbation which gives the maximum separation between the normalized propagation coefficients of the two polarizations. |
Subject |
Surface waves |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-19 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0065621 |
URI | http://hdl.handle.net/2429/20551 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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