WAVES IN INHOMOGENEOUS ISOTROPIC MEDIA by CHRISTOPHER ROBERT JAMES B.A.Sc., University of B r i t i s h Columbia, 1960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept this thesis as conforming to the standards required from candidates f o r the degree of Master of Applied Science Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA August 1961 In p r e s e n t i n g the t h i s thesis i n p a r t i a l f u l f i l m e n t of requirements f o r an advanced degree a t t h e British Columbia, I agree t h a t the a v a i l a b l e f o r reference and study. University of L i b r a r y s h a l l make i t f r e e l y I f u r t h e r agree t h a t permission f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may g r a n t e d by the Head o f my It i s understood t h a t f i n a n c i a l gain Department Department o r by h i s be representatives. copying or p u b l i c a t i o n o f t h i s t h e s i s f o r s h a l l not be a l l o w e d w i t h o u t my of The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada. Columbia, written permission. ABSTRACT For the case of a lossless medium containing no free charges and possessing a continuous and s u f f i c i e n t l y differentiable s p a t i a l l y dependent permeability and permittivity, two v e c t o r i a l d i f f e r e n t i a l wave equations, one for the e l e c t r i c and one f o r the magnetic f i e l d , are derived through the use of Maxwell's equations. From these two equations necessary conditions,for E-.ahd H-modes ., to exist i n a waveguide are established,. The f i e l d equations for the case of constant permeability and z-dependent permittivity as well as the interchanged case are investigated,, A test i s developed which, i f met, assures that the solutions are o s c i l l a t o r y for the ordinary d i f f e r e n t i a l equations containing the z-dependent part of the wave function. For the d i e l e c t r i c loaded periodic structure the theory for inhomogeneous isotropic media i s used to determine the r e s t r i c t i o n s on the f i e l d components which are necessary before E—modes can exist and to f i n d the E- mode wave solutions for the s o l i d disc case when the d i e l e c t r i c regions are matched into the a i r regions. An investigation i s carried out into the behaviour of plane waves i n a medium with the permeability constant and the permittivity varying i n the d i r e c t i o n of propagation. iii TABLE OF CONTENTS LiXS*t O f I X X U S "fcr&'t/X page O U S o c o o o o o c o o o o o o o o o o o o o o o o o o o o o o o o e A c i d l O V X © C L ^ G I U G JD."t O « O * 0 O 0 O O O O O O O 0 O O 0 0 0 0 0 0 0 O 0 0 0 0 0 0 0 O 0 0 0 0 0 0 Xo I l l t r O d-UC 2o GGjlXGrflfX Th.6 " f c X O H . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o o o o o o o o o 03?yo o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 2oX IH"trod\lC "tl O H o © o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o "V V I X 7 T 2.2 Permeability and Permittivity, Functions of tbe Space Coordinatesoooooooooo.©o.ooooooooo.. 3 2 o 21 E~°Mode COndjL tXOn ooo. .oooeo...o. .......... 10 2. 22 H—Mode Conditiono ooooooo..e.oooo.oo.oo..o 11 2o23 Example of Use of a Mode Condition....oo. 11 3. Wave Equations f o r Case of Constant Permeability and of z-Dependent Permittivity and the InterCll.£tXl££@CL 0SLS6 3 oX O 0 O O O O 4 O O O 0 0 S 0 O 0 O O 0 0 0 O O 0 O 0 0 0 0 0 0 0 O 0 0 0 0 0 G r 6 H G I* £tX o o o o o o o o o o o o ^ o o o o o a o o o o o o o o o o o o o o o o o o o o 3.2 D i f f e r e n t i a l F i e l d Equations f o r Transverse Wfl/V O S O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O X8 X 8 X9 3.3 D i f f e r e n t i a l F i e l d Equations f o r Waves with Longitudinal Components....................... 22 3.4 Summary of z-Dependent Equations.............. 25 3.5 O s c i l l a t i o n Theorem Due to Sturm.............. 25 4. F i e l d Problem i n a Periodic Structure Loaded with D X ©X © , C "bl* X C D X S C S o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o f r 4 o 1 G"6 n©r£tX o o o o o o 6 ' O e o o o o c o o o o o o o o o o o o o o o o o o o o o o o o o 30 30 4.2 Functional Behaviour of the Permittivity...... 30 4.3 F i e l d Restrictions Due to E-Mode Condition.... 33 4.4 Unified D i f f e r e n t i a l Equations................ 33 iv 4 o 5 E-Mode Solution of Unified D i f f e r e n t i a l Equations f o r Matched Case,0 0 0 0 0 0 0 0 0 0 0 0 0 0 5o Plane ¥aves Continuous 5 o1 i n a Medium Function of with Zo Permittivity 0 0 0 0 0 a oooo0000000000000000oooo Introductionooooooooooooooooooooooo'oooooooooo 5 o2 P r o b X em © o o o o o o o o o o o o o o o o o o o O O O O O o' O O O O O O. O O O O O o 5 o 3 An Inhomogeneous Slab Between Two Homogeneous M e d i a o oooo 6 0 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O Manufacturing of Inhomogeneous D i e l e c t r i c Media. .<> 7 o 0 OnC X u S1 O n S o oo o o o o Aj)J)©ndlX l o o O O O O O O O O O . ^jP^)6.ndX3C 2o Blfol O O XO^raj)liy O O O O O O O O O O , O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O - O O O O O O O O oo o o o o o o. o o o o o o o o O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O V L I S T OP ILLUSTRATIONS Figure Page 1. C r o s s S e c t i o n S"fcrUC t\ir© 2. Sketch 3o C r o s s of D i e l e c t r i c o o o e e o o o-o o o o e o o o o © © Loaded P e r i o d i c O O O O O O O O O O O O O O O O O O O of Functional Variations Section of S o l i d Disc of P e r m i t t i v i t y 4« D i a g r a m o f u u e—M 2 2 2 and k e V e r s u s 5. A Diagram o f / w ? | i e - M 6. Sketch Q dz V e r s u s of P e r m i t t i v i t y Versus 7. An Inhomogeneous MecLX £ t o 2 © O O O O O S l a b Between O O O O O O O O O O O O O O O O 31 Periodic S t rUC t U T 6o0aoeo»oooooooooooooooo'ooooooooooeoa'oo 2 2 z» » o .-••.••.» z« • «»•<>.• o «?..•<. z . . . >»„ * < , » . o . . o. 37 38 40 45 Two Homogeneous o o o o o o o o o o o o o o o o o o o 51 vi ACKNOWLEDGEMENT The author wishes to express his thanks to Dr 0 G.B. Walker, the supervisor of this project, for his invaluable guidance and encouragement throughout t h i s studyo To Mr. EoL. Lewis, Mr. D»B. McDiarmid, and other colleagues, the author would l i k e to express his appreciation f o r their many useful suggestions on this work. Acknowledgement i s gratefully given to the National Research Council for the assistance received from a Bursary awarded i n I960 and a Studentship awarded i n 1961. WAVES IN INHOMOGENEOUS ISOTROPIC MEDIA lo INTRODUCTION The purpose of t h i s thesis i s to investigate t h e o r e t i c a l l y the behaviour of electromagnetic waves i n lossless inhomogeneous isotropic media containing no free charges. Throughout this thesis inhomogeneous isotropic media w i l l be c a l l e d inhomogeneous media. Inhomogeneous media are of interest because they may be used to make slow wave structures which i n turn can be used i n l i n e a r accelerators 9 traveling wave tubes, backward-wave o s c i l l a t o r s , and microwave f i l t e r s . Besides t h i s , inhomogeneous media may be used i n pre-accelerator designs. At Stanford University, G.S. Kino has considered using a waveguide f i l l e d with a plasma of uniform crosssectional density and with an a x i a l density v a r i a t i o n of the form /D(l+a s i n ^z) for a slow wave structure.^" The a x i a l v a r i a t i o n i n plasma density i s to be achieved by propagating sound waves down the waveguide. In t h i s case 1. G.S. Kino, A Proposed Millimeter-wave Generator,, Microwave Laboratory, W.W. Hanson Laboratories of Physics, Stanford University, Stanford, C a l i f o r n i a . the plasma forms an inhomogeneous medium. The topic of this thesis arose during an investigation into the solution for the wave functions i n a d i e l e c t r i c loaded periodic structure,. Such a structure i s shown i n Figure 1. Cylindrical Waveguide I r z / / VI -' V" ' / / / T ~ T T Dielec >ric /Disc l>l |'l I y / / / / / / / I II I 'Mi 'i 1 I 1 " J' ;!','!; • ('i i • 111 ii> Z / 1 a Center ^Hole Hi; " ' / III M A i r Region / / / M i /TT Center . ;ine / i ' ' " . 111 * 1 1 • ' 11 zrz I I I I I LI / I I t = Permittivity of d i e l e c t r i c material = Permittivity of a i r z = Longitudinal r = Radial coordinate Q /> =Angular Coordinate coordinate > Cylindrical Coordinates Notes Discs are s o l i d when a equals 0 . Fig. 1. Cross Section of D i e l e c t r i c Loaded Periodic Structure For a periodic structure similar to the one shown in Figure 1, the exact wave functions can be r e a d i l y determined provided the d i e l e c t r i c discs are s o l i d . These wave functions are derived i n Appendix 1 f o r the Eg^-mode. I f the periodic structure i s used i n the usual fashion f o r 3 accelerator or traveling wave tube applications, electrons must pass along the axis of the waveguide 0 To make t h i s possible, a hole must exist through the center of each d i s c . With the hole present the problem of finding the wave functions becomes exceedingly complex. In p r i n c i p l e , t h i s problem can be solved by solving i n each homogeneous region the d i f f e r e n t i a l wave equations developed from Maxwell's equations and by matching the solutions for the d i f f e r e n t 2 regions at the boundaries. Also, i t should be noted that Floquet's Theorem must be used i n the same manner as i t i s used i n Appendix 1. Due to the excessive labour involved i n any numerical work carried out to establish a match at a l l the boundaries, the results have only formal significance. Consequently, i t has been necessary to use the s o l i d disc theory and/or the anisotropic theory approximations i n the design of periodic structures having 2 center holes i n the d i e l e c t r i c d i s c s . Since previous techniques used i n attempting to solve the problem i n which the d i e l e c t r i c discs have center holes are not e n t i r e l y satisfactory, i t was thought that a different approach might be u s e f u l . Instead of placing the emphasis on the medium inside the waveguide being made up of homogeneous sections, i t was decided that an investigat i o n should be carried out with the emphasis shifted to the 2. B.B.Ro-Shersby-Harvie et a l , "A Theoretical and Experimental Investigation of Anisotropic-DielectrieLoaded Linear Electron Accelerator", Proc. of I.E.E., v o l . 104, Part B, 1957. "~*"~ 4 fact that the medium as a whole i s inhomogeneous. In other words, the permittivity i s a function of the spatial parameterso Three reasons can be advanced for following this approacho One reason i s that a d i f f e r e n t approach at times reveals new information about a problem, and a second reason i s that only one v e c t o r i a l d i f f e r e n t i a l wave equation has to be solved, as w i l l be shown i n section 2.2, to obtain a f i e l d solution which holds throughout the waveguideo Also, since the permittivity i n the neighbourhood of the boundaries between the a i r and d i e l e c t r i c regions as well as elsewhere i s assumed continuous and s u f f i c i e n t l y d i f f e r e n t i a b l e , the theory developed to attack the problem in which the d i e l e c t r i c discs have center holes can be expanded to include inhomogeneous media i n general. The continuity and d i f f e r e n t i a b i l i t y assumptions w i l l be i discussed i n section 4.2. The one v e c t o r i a l d i f f e r e n t i a l wave equation, which i s v a l i d throughout the waveguide, y i e l d s three scalar p a r t i a l d i f f e r e n t i a l equationso These scalar p a r t i a l d i f f e r e n t i a l equations w i l l hereafter be referred to as the u n i f i e d d i f f e r e n t i a l equations. As i t turned out, when the d i e l e c t r i c discs have center holes, no technique was devised to find the general solution for any of the unified d i f f e r e n t i a l equations. was not achieved. Consequently, the o r i g i n a l objective However, t h i s problem i n i t i a t e d the following work i n t h i s thesis. 5 For the case where both the permittivity and permeability are continuous and s u f f i c i e n t l y differentiable functions of the spatial parameters, the e l e c t r i c and magnetic v e c t o r i a l d i f f e r e n t i a l wave equations are derived 0 Through the use of these equations, necessary conditions for E- and H-modes to exist i n a waveguide are foundo An example of a use of the E-mode condition i s shown,. For the case where either the permittivity or ^ permeability i s a function only of the axial parameter z and the remaining characteristic of the medium i s constant, the pertinent unified d i f f e r e n t i a l equations are separated into ordinary d i f f e r e n t i a l equations» A test i s developed which, i f met, assures that the solutions are o s c i l l a t o r y for the ordinary d i f f e r e n t i a l equations containing the axial dependent portion of the wave function„ For the E-mode case certain limitations which must be imposed upon the f i e l d components i n the d i e l e c t r i c loaded periodic structure are investigated using the theory for inhomogeneous media* Also, when the d i e l e c t r i c discs are s o l i d , provided the d i e l e c t r i c regions are matched into the a i r regions, for the E-mode case a solution f o r the pertinent u n i f i e d d i f f e r e n t i a l equation i s given 0 To provide a better physical understanding of the behaviour of electromagnetic waves i n an inhomogeneous medium, an investigation i s carried out into the behaviour of plane waves i n a medium with the permeability constant 6 and with the permittivity varying i n the d i r e c t i o n of propagation,. In t h i s thesis the behaviour of E-modes i s investigated f a r more thoroughly than the behaviour of H-modeso The reason for t h i s i s that the d i e l e c t r i c loaded periodic structure discussed i n this thesis i s primarily used f o r linear accelerator and traveling wave tube applications and i n these applications E-raodes and not H-modes are excited* 7 2. GENERAL THEORY 2.1 Introduction Through the use f o l l o w i n g wave t h e o r y w i l l lossless the of Maxwell's equations, be d e v e l o p e d medium c o n t a i n i n g no f r e e f o r the charges. the case of To b e g i n with, s i t u a t i o n where t h e p e r m e a b i l i t y and p e r m i t t i v i t y medium a r e g e n e r a l will functions be c o n s i d e r e d . permeability be t r e a t e d charges the spatial Following t h i s , constant the interchange equations of case w i t h the of t h i s z will case. i n a medium c o n t a i n i n g no and w i t h z e r o c o n d u c t i v i t y the coordinates and p e r m i t t i v i t y a f u n c t i o n o f a l o n g w i t h the Maxwell's of a free are Vx E (1) Vxs (2) 5¥ (3) V.B o (4) where E = Electric field 13 = E l e c t r i c f l u x Also, density 5 = Magnetic field S = Magnetic flux 15 and E a r e r e l a t e d b y t h e intensity vector, intensity density vector, vector, vector. equation (5) * Interchanged case i s t h a t of z-dependerit p e r m e a b i l i t y . constant permittivity and o f 8 and B* and S are related by the equation S = uB (6) vhere e i s the permittivity and \i i s the permeability. 2.2 Permeability and Permittivity, Functions of the Space Coordinates To obtain an expression f o r the electromagnetic f i e l d i n a homogeneous medium, the d i f f e r e n t i a l wave equation which has to be solved i s the standard equation where u-e i s constant. When the medium i s not homogeneous, the permeability and permittivity being continuous and s u f f i c i e a t l y differentiable functions of the space coordinates, the d i f f e r e n t i a l wave equations from which the f i e l d expressions can be obtained are somewhat more complex. These more complex d i f f e r e n t i a l equations can be arrived at i n the following manner. For the e l e c t r i c f i e l d the vectorial differential wave equation can be derived by f i r s t taking the curl of equation ( l ) . Vx E) = -|rVx B 0 Since and Vx (Vx E ) = V ( V . E ) - V E , 2 9 then V(V.E) «V E = (Vx u.H) 2 or V( V.E) - V E = - | r ( ^ ( e E ) + Vlt X S) 2 0 Since i t has been assumed that the permeability and permittivity are not functions of time, VnxSf V(V.E) - V ^ ^ - i x e ^ f bt^ o x i n b E 2 = " M , E bt 1 b E 2 bt^ bB ~ ^ & = - jie T ^ + iVn X (Vx + ^[VROVE - E) (VtioVJE where i n rectangular coordinates with I , j and k" being the unit vectors i n the x,y, and z directions respectively. VoD" Prom equations = V«eE = (3) and (5) 0. Consequently, Ve.E + e V»E = 0 and thus V.E = 4 V e . E c Therefore, the v e c t o r i a l d i f f e r e n t i a l wave equation f o r the electric f i e l d i s V E + V l J V e . E l = u e ^ | - i f o . V E - (Vn-V)*!- (7) 2 bt Similarly, the v e c t o r i a l d i f f e r e n t i a l wave equation for the magnetic f i e l d i s V H V(f Vix.s) = v£% - ifVc.Va - (Ve.V)sl. 2 + * & (8) t At t h i s point s u f f i c i e n t theory has been developed to establish a necessary condition for the existence of an E-mode i n a waveguide and a dual condition f o r the H-mode case. 2.21 E-Mode Condition i If the permeability and permittivity are continuous and s u f f i c i e n t l y d i f f e r e n t i a b l e f o r a l l i n t e r i o r points i n a waveguide, then f o r an E-mode to exist where z i s the coordinate i n the d i r e c t i o n of propagation. Proof: From equation (8) i t can be seen that the scalar equation obtained when the c o e f f i c i e n t s of the component vector i n the z d i r e c t i o n are equated i s )z where H z i s the z component of H 0 Since f o r an E-mode H - 0, z 9 then dz JVM] JV..g-o. + 11 The 2.22 dual for t h i s condition i s the following one. H-Mode Condition If the permeability and permittivity are continuous and s u f f i c i e n t l y differentiable for a l l i n t e r i o r points i n a waveguides, then for an H-mode to exist (10) Proof: The proof for t h i s condition i s the same as for the previous condition except that equation (7) i s used instead of equation (8), and also. instead of H where E 2.23 z - 0 z - i s the z component of E» Example of Use of a Mode Condition The mode conditions can be used to determine certain r e s t r i c t i o n s which must be imposed upon the f i e l d s before E- and H-modes can exist i n a waveguide f i l l e d with an inhomogeneous medium. An example which demonstrates t h i s use i s the case where an E-mode exists i n a waveguide which i s f i l l e d with a medium having a permeability tha-J; i s constant and a permittivity that i s s u f f i c i e n t l y well defined and s a t i s f i e s the equation e = f(r,z) (11) where r i s the r a d i a l parameter For a waveguide f i l l e d 0 with a medium which behaves i n t h i s manner, from the E-mode condition U oH Oz A = or i n c y l i n d r i c a l coordinates ^e^r 1 $ e ^ dr dz r 57 T i ++ + + de ^ _ dz T i ~ ° 0 (12) U 2 J where i s the radial component and i s the angular r component of S« Since the permittivity does not have angular dependence, de 37 - n 0 < Also, f o r an E-mode Hz — - Oo (13) Hence, from equation (12) > dH - „ de de r - 0. dr b z Since dr ~ then f o r an E-mode to exist i n the waveguide ^ r = Oo dz ~ H (14) As a result of identity (14), further r e s t r i c t i o n s on the f i e l d s can be found through the use of equations ( l ) and (2)o Prom equations (1) and (2) 1 dE r dE _ z ~S7 b^ l ' (15) bz ~ " d t ' bE„ bE„ i * OB/ I 0 E v bB^ i§?<*.V-iTf—*f • (17) . bH/ bDr 1. bHz p _ r I ? " bz ~ b t ' bH and bH, i > bD / , bH bD i I? < V - J -si. - -rf r where E r (18) (20) i s the r a d i a l component and E^ i s the angular component of The substitution of i d e n t i t i e s (13) and (14) into equation (19) gives t If the f i e l d varies i n time as e J W \ with u> being the frequency i n radians per second, then E/ - Oo (21) i t can be seen from equations (15), Now, (16), (17), (18), and (20) that i T7 = bE > -^ T E < 22) bE bi-Ti- ' ( 2 3 ) r I 0 , bl ~ (24) b E bK, - r f = > e E r' ( 2 5 ) and r dr If equation (22) identity (14) (rHv) - ± p r M T = jo>eE . z (26) J i s differentiated with respect to z and i s substituted into the resulting equation, then * (27) z = 0. 6 E ~£T? "5z Through the d i f f e r e n t i a t i o n of equation (25) to p and the use of identity (24), 1 b with respect i t i s found that ^ : o . (28) "bp b z ~ 7 If identity (14) i s integrated, H = f^r.tf), r (29) and i f i d e n t i t y (24) i s integrated, E = g (r,z). r (30) 1 Identity (27) can be integrated f i r s t with respect to z to give and then with respect to f£ to give E z = f 2 '*^ (r + g ^ ( r ? z ) o 2 3 1 ) Similarly, i d e n t i t y (28) can be integrated to give = f ( r , j ^ ) + g (r,z)» 3 (32) 3 Prom the substitution of expressions (29) and (31) into equation (26) r fe V (r = r ^t(x z)f (v fi) + 9 2 f + j«e(r,z)g (r,z) 2 = s ( r , ^ ) + s (r,p\z) + s ( r , z ) 1 2 (33) 3 where S i ( r ^ ) = r — : ' s (r,jf,z) = otoe(r,z)f (r,^) 9 and 2 t s,j(r,z) = jwe(r,z)g (r,z). 2 (34) However, from equation (32) = t^lj) + t (r,z) (35) 2 where and t (r,z) = i | j 2 ( r g 3 ( r ' z ) ) * Since H^ must behave i n the manner described i n equation (32) the functional form given for r dr p' i n equation (33) must be i n agreement with equation (35). Such an agreement i s f u l f i l l e d only i f s (r,^,z) = s (r,z) (36) s (r,0\z) = s ( r , ^ ) (37) 2 2 or i f 2 2 0 Equation (37) cannot be s a t i s f i e d because e = f(r,z)o Hence, equation (36) must be s a t i s f i e d . Therefore,.from equation (34) i t can be seen that i t i s necessary that fgU.jrf) = f ( r ) . 2 With this being the case, from equation (31) E z becomes E z = 2 f ( r ) g2 + ( r , z ) o ( 3 8 ) Therefore, z ~ 0, (39) H (40) b E "b7 and thus from equation (22) Furthermore, respect to £ Z Oo r - i f equation (23) i s differentiated 9 from i d e n t i t i e s (24) and with (39) (41) _^£ - Oo u It can be concluded from i d e n t i t i e s (24), (39), and that the f i e l d s have no angular dependence (41) 0 A point to note i s that the r e s t r i c t i o n s imposed upon the f i e l d components are i n i t i a l l y caused by the radial dependence of the permittivityo If the permittivity i s only a function of z, no r e s t r i c t i o n s result from the E-mode condition because i n equation (12) H z"— 0 which forces be ^ z bz b z H - 0. and thus a l l terms i n equation (12) vanish,, 3. WAVE EQUATIONS FOR CASE OF CONSTANT PERMEABILITY AND OF z-DEPENDENT PERMITTIVITY AND THE INTERCHANGED CASE 3ol General At this point the problem where the permittivityi s a function of z and the permeability i s constant w i l l be considered along with the interchanged case. For these problems the u n i f i e d d i f f e r e n t i a l equations, which result from the v e c t o r i a l d i f f e r e n t i a l wave equations, are s u f f i c i e n t l y separable. If the permittivity i s a function of z only and the permeability i s constant and Vl* OO = Hence, equation (7) becomes V E + V(i ft N 2 = ,egf , (42) and equation (8) becomes v-/2n dI H l ' d e oB 1 deVH V H = ne — + e a i ^ - e a i 2 A dt . * (43) Similarly, i f the permeability i s a function of z only and the permittivity i s constant, Ve = 0 and Vu- = djij £ dz o Hence, equation (7) becomes and equation (8) becomes 3.2 D i f f e r e n t i a l F i e l d Equations, f o r Transverse Waves For transverse waves E z = H - 0. z - Therefore, equation (42) becomes V 2 E = (46) and equation (45) becomes V H = ne ^ | . (47) 2 For the case where the permittivity i s dependent on z, the f i e l d equations can be determined by f i r s t solving f o r the two f i e l d components i n equation (46) and then by using equations (l) and ( 2 ) i n the usual manner. Hence, i f the rectangular components of the f i e l d are to be determined, the p a r t i a l d i f f e r e n t i a l equations bE 2 V \ = lie — f Ot (48) and dE 2 V \ = ^ (49) "^2 have to be solved where E^ and E^. are the components of E i n the x and y directions respectively. Similarly, f o r the case where the permeability i s dependent on z, the f i e l d equations can be determined by f i r s t solving f o r the two f i e l d components i n equation (47) and then by using equations ( l ) and (2). Hence, i f the rectangular components of the f i e l d are to be determined, the p a r t i a l d i f f e r e n t i a l equations V^H x x 1 dV = ^|xe ~^72 = ^ % (50) and v 2 \ (») have to be solved where H and H are the components of H x y i n the x and y directions respectively. Equations (48), ( 4 9 ) , (50), and (51) have the general form V G = u.e (52) 2 where i s a function of z. dt 2 Consequently, equation (52) w i l l be considered f o r the remainder of this section. Equation (52) can. also, be written as V7 2 . R d G d G 2 2 where \^. denotes the part of which operates i n the 2 transverse plane of a rectangular coordinate system. the f i e l d s vary i n time as e* If , then G can be expressed 1 as G = G (x,y *)eJ 0 f t t t t and thus \ n G t 1= + o . > G (53) ** o ^ 2 Z The variables can be separated by l e t t i n g G q = P(x,y) T(z). (54) Once equation (54) i s substituted into equation ( 5 3 ) , „ 2„ dz where M i s i>he separation constant. \^. F + M P 2 2 2 Hence, = 0. Equation (55) i s the ordinary d i f f e r e n t i a l (55) equation confronted when the transverse dependence of a transverse wave i n a homogeneous medium i s investigated. equation ( 5 5 ) , the d i f f e r e n t i a l As well as equation ^ | + (a>|ie - M )T dz^ 2 2 = 0 (56) has to be solved to obtain the solution f o r the transverse waves under consideration^, For plane waves equation (56) i s s l i g h t l y simpler due to the fact that f o r plane waves V 2 t E x . V \ t - V H . \ 2 t x \ Eo • which forces M = Oo Consequently, equation (56) becomes 2 ^ dz + « |icT = Go (57) 2 3 o 3 D i f f e r e n t i a l F i e l d Equations f o r Waves with Longitudinal Components When the f i e l d s have longitudinal components, the wave solutions can be found by f i r s t solving f o r the longitudinal f i e l d components and then by using equations (l) and (2)o This section i s concerned with the d i f f e r e n t i a l equations a r i s i n g i n the solution of the longitudinal f i e l d componentso When the permittivity i s dependent on z, from equation (45) the longitudinal component of the magnetic f i e l d must s a t i s f y the d i f f e r e n t i a l equation VH 2 = |« j f f s at 2 (58) 23 and from equation (44) the longitudinal component of the e l e c t r i c f i e l d must s a t i s f y the d i f f e r e n t i a l equation oz z v 1 e dz zl ^2 r Equation (58) i s the same type of d i f f e r e n t i a l equation as equation (52), and thus can be treated i n a similar manner. However, the r e s t r i c t i o n that \^ must i- operate i n a rectangular coordinate system no longer applies. E 1 Equation (59) can be simplified by replacing by —D . e z > In terms of D equation (59) becomes bD 2 le zl bzle2 dz or z> r ^,2 bV 0 0 Vt D z + £ 2 2 + 2 -i f£ 5-£ = u -e e dz bz >*. 2 . (60) b t The variables can be separated by l e t t i n g = P(x,y)TC»)e> . D (61) t If t h i s expression f o r 0 i s substituted into equation (60), z the r e s u l t i s I v t 2 p = d"T 4 ,2 1 de dT e dz dz tu jxe = -M 2 2 CLZ where once again M i s the separation constant. The equation \^. F + M F = 0 2 2 (62) i s identical to the equation which contains the transverse dependent part of the longitudinal component of the f i e l d i n a homogeneous medium and can be solved f o r a number of boundary value problems using well known techniqueso equation for the z-dependent part of g - i f e g can be written as - + The m 2 » t = »• <«> When the permeability i s dependent on z, from equation (46) the longitudinal component of the e l e c t r i c f i e l d must s a t i s f y the d i f f e r e n t i a l equation S}\ dE = lie — f . ot 2 (64) and from equation (47) the longitudinal component of the magnetic f i e l d must s a t i s f y the d i f f e r e n t i a l equation V V ^ & B . h " - ^ . .<«> Equation (64) i s the frequently occurring type of d i f f e r e n t i a l equation given i n equation ,(52), and thus can be treated accordingly. The only change i s that \ ] 2 is no longer r e s t r i c t e d to operate i n a rectangular coordinate l system. In the same manner as equation (59) was by using D , equation (65) can be simplified Prom equation (65) i t can be seen that B simplified by using B . can be expressed z in the form B with F(x,y) = 7 satisfying F(x y)T( )eJ 9 W t Z equation and T ( z ) (62) dl _ 1 M dT ,2 u dz d z dz + (w 2 _ 2 M ) T satisfying = Q o r r 3o4 Summary o f If z-Dependent by Equations definition q(z) then the part differential o f the sections equations solution for and 3.3 3.2 = fie, the c a n be c o n t a i n i n g the wave f u n c t i o n s z-dependent considered summarized by the in following 'i three differential equations. 2 + o> qT = 0, (66) 2 ^| (a> q-M )T dz 2 + = 0, 2 (67) and d z 3.5 2 1 O s c i l l a t i o n Theorem Due t o Equations form q d z dz ; (66) ' u Sturm and (67) c a n be e x p r e s s e d in the 2 + h(z)v dz = 0, (69) and equation (68) can be transformed i n t o the form of equation (69) by the f o l l o w i n g t r a n s f o r m a t i o n . expressed If T i s as T = etfq dt d W = e* z l n 4 V =^¥ , then equation (68) becomes 2 2 dz^ 2 + ^ ~ 2 w2 - 2-i ,11 O. 3 1 fdjgf ¥ = 0 2 q T~2 ~ 4 ~2 Id3 ^- dz q which i s of the form of equation (70) (69). Owing to a theorem by Sturm, i t i s p o s s i b l e to show that f o r a p h y s i c a l l y realizable s i t u a t i o n the solutions f o r equation (66) are o s c i l l a t o r y , and the s o l u t i o n s f o r equation (67) are o s c i l l a t o r y p r o v i d e d w |xe - M > 0 . 2 2 Besides t h i s , the theorem o f f e r s a possible test f o r showing whether or not the s o l u t i o n s f o r equation (68) or (70) are o s c i l l a t o r y . Theorem: The f u n c t i o n s u(z) and v ( z ) are the r e s p e c t i v e s o l u t i o n s of the d i f f e r e n t i a l equations ^§ dz + g(z)u = 0 (71) + h(z)v = 0 (72) Z ^ | dz 3. L.R. Ford, D i f f e r e n t i a l E q u a t i o n s . New l o r k , 1955, p. 169. McGraw-Hill, Inc., i n an i n t e r v a l i n which the c o e f f i c i e n t s of the equations are continuouso I f a and b are consecutive roots of u(z) with a < b and i f h(z) >g(z) h(z) I g(z) in the closed interval ja,b , then there exists a root of v(z) between a and b.» Proof: F i r s t of a l l , d | . du d z l dz " v u 2 2 dvl __ d u d v = (h - g)uv - u dzJ dz' dz' -v which after integration becomes z=b dv du u dz dz z=a Now, the supposition between a and b i s made. b^ (h - g)uv dz. (73) a that v(z) has no root Without loss of generality u(z) and v(z) may both be considered positive i n the interval a ^ z < b ; either one can be replaced by i t s negative, i f necessary. Consequently, du(a) > 0 dz and du(b) <0, dz Hence, z=b du _ dz dv dz - v( ) ^ a i a l ^ o , = v(b) z=a a dz dz However, the right hand side of equation (73) i s positive. Therefore, a contradiction exists, and thus the theorem i s proved. Prom this theorem i t follows that i f the solution to equation (71) i s o s c i l l a t o r y over some i n t e r v a l , then, provided that over this interval g(z) and h(z) are continuous and h(z)^g(z) h(z) £ g(z), the solution to equation (72) i s , also, o s c i l l a t o r y over the same i n t e r v a l . For a physically realizable medium |ie^k = Constant >0 where normally r o o' and f o r the n o n - t r i v i a l cases to be considered p,e •£ k. Since the solution f o r ^ | + u>kT = 0 2 i s o s c i l l a t o r y , the solution f o r equation (66) i s , also, oscillatory. 29 In a similar manner, vhen w |Ae 2 - M ^k' 2 = Constant>0, then since the solution for dT 2 dz^ + k'T = 0 i s o s c i l l a t o r y , the solution f o r equation (67) i s oscillatory. (74) 4. FIELD PROBLEM IN A PERIODIC STRUCTURE LOADED WITH DIELECTRIC DISCS 4.1 General As mentioned i n the introduction, the topic of t h i s thesis arose from the problem of finding the vave functions for a periodic structure of the type shown i n Figure 1 Section 4 w i l l expand upon t h i s problem through the use of the fact that inside the waveguide the medium as a whole i s inhomogeneous. The permittivity i s a function of the spatial parameters and the permeability i s constant. 4.2 Functional Behaviour of the Permittivity Before the unified d i f f e r e n t i a l wave equations can be dealt with, the functional behaviour of the permittivity must be specified. Since the i n the i n t e r i o r of the a i r regions equals e permittivity Q and i n the i n t e r i o r of the d i e l e c t r i c regions equals e ^ , the form of the permittivity minus e Q functional approaches the product of a rectangular wave variation i n the z direction times a step variation i n the radial d i r e c t i o n . Consequently, i f c y l i n d r i c a l coordinates are used, the permittivity can be expressed as e - e Q = h(r)g(z) where h(r) and g(z) are sketched i n Figure 2. The reasons that the curves i n Figure 2 are shown as continuous and smooth are given i n the following paragraphs. Before the theory so f a r developed can be applied to the f i e l d problem i n a periodic structure loaded with d i e l e c t r i c discs, i t must be assumed that the permittivity and a l l i t s f i r s t and second order derivatives are defined for a l l i n t e r i o r points i n the waveguide. argument i s given to j u s t i f y this The following assumption. gU) e l ' e d cr h(r) a F i g . 2. Sketch of Functional Variations of Permittivity At a l l points except those i n the regions of the boundaries between the a i r and the d i e l e c t r i c medium, there i s no doubt as to the existence of the permittivity and a l l i t s derivatives. If in the neighbourhood of the boundaries 32 the point of view of mathematical physics, which i s that matter i s continuous, i s taken, then the properties of 4 matter can also be regarded as continuous<> Therefore, the permittivity can be considered continuous but changing very rapidly through the boundaries. Consequently, a continuous function can be used to describe the permittivity through the boundaries which both approximates the situation as closely as desirable and s a t i s f i e s the assumptions made about the behaviour of the permittivity. I t i s worth noting that the step function used to describe the boundaries i n the standard approach for finding the wave solutions i s , also, an approximation, although a very good one, of the actual situation. The step function i s an approximation because at the boundaries there exists not one big discontinuity, but rather a large number of discontinuities which arise from the discontinuities between the atoms and, also, between the separate parts of the atoms. In the region of a boundary the permittivity could be represented by the function i m provided m i s small but f i n i t e . 11 For equation (75) s = Coordinate i n the normal direction to the boundary 4. A.G. Webster, P a r t i a l D i f f e r e n t i a l Equations of Mathemat i c a l Physics, Dover Publications, New York, 1955, p.2. and a = s at the boundary<> As m decreases, for s«=ca, e - > - e , and for s > a , e - * - e ^ . 0 In the l i m i t as m-»-0 with an increasing s, the p e r m i t t i v i t y f changes discontinously through the boundary from e Q to e ^ . 4.3 F i e l d Restrictions Due to E-Mode Condition Now that i t has been assumed that the p e r m i t t i v i t y behaves s u f f i c i e n t l y well to ensure that the theory so far developed i s applicable, the material i n section 2 can be used to f i n d the necessary conditions for the existence of E-modes i n the periodic structure shown i n Figure 1. When the d i e l e c t r i c discs have center holes, e sr h(r)g(z) + e Q = f(r,z). Therefore, the r e s t r i c t i o n s on the f i e l d components found in section 2.23 must hold. met I f these r e s t r i c t i o n s are not and a hybrid mode r e s u l t s , i t i s worth noting that the hybrid mode may very closely approximate an E-mode provided the longitudinal component of the magnetic f i e l d has only a secondary effect on the remaining f i e l d components. When no center hole exists, the p e r m i t t i v i t y has no r a d i a l dependence, and thus, the E-mode condition i s s a t i s f i e d without imposing any r e s t r i c t i o n s on the f i e l d components. 4.4 Unified D i f f e r e n t i a l Equations When the discs have center holes, the u n i f i e d d i f f e r e n t i a l equations for an E-mode can be found from equations (7) and (8). Since for an E-mode H - 0 z - and since from the E-mode condition = ^ 0 Hr -0, and the f i e l d components have no angular v a r i a t i o n , the unified d i f f e r e n t i a l equations are mS44 r drt dr I ^2 d E 2 1/dl — E + §r E e ,i 5dr r oz • z< z dt^ dz d E 2 d_ dr r dr ( r E r ife- E + & ) e I Or r dz zJ E dE 2 r and d dr dz dH 2 r dr p' 2 dz 2 I = 3 d"? d dH 2 [ie - ~ dt 2 de e dr d r r/ de ^ f L dz d z . As previously mentioned, the techniques applied so f a r to these d i f f e r e n t i a l equations, as well as the ones arising i n the H-mode case, have not yielded general solutions. However, numerical methods could be devised to calculate specific solutions for these d i f f e r e n t i a l equations. Since general solutions have not been found, these d i f f e r e n t i a l equations w i l l not be considered further i n t h i s thesis. For the case where the discs are s o l i d , the 35 permittivity i s a function of z only. Consequently, the u n i f i e d d i f f e r e n t i a l equations for the longitudinal f i e l d components can be separated as was seen i n section 3o Also, the complete wave solutions for this case can be readily solved, as done i n Appendix 1 for the EQ^-mode, by determining the f i e l d s i n the d i e l e c t r i c regions and i n the a i r regions separately and by matching these f i e l d solutions at the boundaries with the help of Floquet's Theorem. Consequently, i f i t could be shown that these known solutions s a t i s f y the unified d i f f e r e n t i a l equations i n the l i m i t as the permittivity approaches a rectangular waveshape, the viewpoint taken i n this thesis would be v e r i f i e d as applicable for finding the wave equations to the s o l i d disc case. However*, t h i s was not shown i n general because no method was devised to overcome two d i f f i c u l t i e s simultaneously. One d i f f i c u l t y i s to f i n d an i n f i n i t e series expression for the permittivity which converges absolutely and yet has an nth order term which i s manageable. This d i f f i c u l t y i s discussed by L. B r i l l o u i n i n "Wave Propa5 gation i n Periodic Structures". Secondly, even i f such a series i s obtained, since the known wave solutions are expressed i n terms of i n f i n i t e series i n z, single and 5. L . B r i l l o u i n , Wave Propagation i n Periodic Structures, Dover Publications, Inc., New York, 1953, p. 186. 36 double i n f i n i t e series appear i n the z-dependent part of the unified d i f f e r e n t i a l equations. This can readily be seen from the equations derived i n section 3. Consequent- l y , the problem of establishing that the c o e f f i c i e n t s of the z-dependent series i n the known solutions s a t i s f y the unified d i f f e r e n t i a l equations i s very awkward. However, when the d i e l e c t r i c regions are matched into the a i r regions, i t i s possible to find the wave solution for an E-mode from the u n i f i e d d i f f e r e n t i a l equations. As w i l l be shown, t h i s solution agrees with the solution obtained i n Appendix 1. 4.5 E-Mode Solution of Unified D i f f e r e n t i a l Equations for Matched Case For the case where the d i e l e c t r i c discs are s o l i d and matched into the a i r regions, a complete wave solution for an E-mode can be determined by f i r s t solving for the f i e l d component D through the use of the pertinent z unified d i f f e r e n t i a l equation and then by finding the other f i e l d components from equations (l) and (2). Since the permittivity i s a function of z and permeability the i s constant, the z-dependent portion of D can z be found from equation (63), d?r j 2 dz The de a r e dz dz _ i + ( w 2 _ 2 M ) T = O o cross section of a periodic structure with s o l i d d i e l e c t r i c discs i s shown i n Figure 3. For such a structure 37 the permeability equals u Cylindrical Waveguide . r Didlectri Disc Center Line i i i YZZZZZZT 1 z_z: i I i i .V, fI i ' TT g i i V,'. ^-Origin Region ' I i (1) Origin y / / / z (5) 1 7-7 "7 V 11 -7-7- 2ZZ = Permittivity of d i e l e c t r i c medium c Q = Permittivity of a i r Note: For s i m p l i c i t y Origin 1 i s used i n Appendix 1 and Origin 2 i s used i n the body of the thesis. Pig. 3. Cross Section of Solid Disc Periodic Structure Since the permittivity very closely approximates a rectangular waveshape i n the z d i r e c t i o n , i t follows that eo2n e - M 2 and k 2e 2 , k being a constant, also, very Q closely approximate rectangular waveshapes as shown i n Figure 4. For the matched case 38 2 M 2 ' O w u e,-M r o- 1 — 2 2 <ou e -M 2 2 " O 0 ->- z F i g o 4. Diagram of 2 2 22 u^e-M and k e Versus z (D Consequently, when a match exists, the identity u) u e-M •- k e o 2 r 2 2 (77) 2 holds except i n the t r a n s i t i o n region at a boundary between an a i r region and a d i e l e c t r i c region., The t r a n s i t i o n 39 region i s defined by (5 i n Figure 4. As (5 tends to zero, identity (77) tends to hold for a l l values of z. Consequently, the d i f f e r e n t i a l equation dT 1 de dT ^ 2 " F d 7 d i 2 + k e 2 2 T = , _ n T 0 ( 7 8 ) approximates equation (63) to any required degree of accuracy for the matched case. The solution for equation (78) i s = A where A^ and A 2 i e -j/ k e d z + A ej/ k e d ! 2 are arbitrary constants. Through the use of identity (77), T becomes -j/^ u, e-M T = A^e ^ 2 o 2 d 2 + a jJ\4> ,x e-M dz. 2 e .- 2 o (79) 2 As (5approaches zero, the solution for T given by equation (79) approaches the solution for the entire waveguide. -i/~2 2 In the l i m i t V(o |xe-M Q approaches a rectangular waveform, and thus for the l i m i t i n g case the integral /V |i e-M dz 2 2 w can be evaluated graphically by integrating the rectangular waveform as i l l u s t r a t e d i n Figure 5. From Figure 5 i t can be seen that, i n the l i m i t i n g case j V c A e-M dz = S z + X(z) 2 R v 40 where Sz Q = Ramp function and X(z) = Periodic function o s c i l l a t i n g about the ramp function, >- z 41 The slope S of the ramp can be found i n the following manner. Q Prom Figure 5 at z equals /Va^e-M 2 dz = £=3A/u, i e -M 2 l o 2 o and thus at z equals \ H\ *-* dz = ^ , e - M 2 2 Q 0 + [* - 2 o S ^ f f L \ ^ p o^ Hence, S = \4 n e -M o o 2 o + ^l\L ti pi , 0 2 e,-M 1 2 r - Vto u e -M |. 2 2 2 'O O / ¥hen only the incident wave i s present, T -V-fl .» S + X U > ' • and thus D r J L P d ^ l * * - ^ " ^ Z (80) J. where r i s the r a d i a l and the ^ the angular v a r i a b l e 0 In the l i m i t as the permittivity approaches a rectangular waveform, i t can be seen from Figure 5 that i n an a i r region z and i n a d i e l e c t r i c region D a -i/V e ^ 2 w e,-M *o 1 Li 2 dz -j(V = e °I 2 t o Mr ,-M ] z o 1 ' 2 e = These results coincide with the results obtained bysolving the wave equation i n each of the homogeneous regions separately. To check that the solution given by equation (80) has the same phase s h i f t per section as found i n Appendix 1, the expression for T(z+p) should be considered, namely. T( p) = A e - j ( o S Z+ ( z + 1 P ) + *< P)) Z + 0 Since X(z+p) =*(z), then T(z+p) = A - J o 4 " ' l o S 3 S i e Z + X i z ) \ = e"J o T(z). S p Hence, the phase s h i f t per section 0 i s given by 0 = S p = (p-q)V o 2 w {io e -M o 2 + qV^u^-M 2 . This i s the same value for 0 as i s found i n Appendix 1. Therefore, since i n the l i m i t the f i e l d i n each section behaves i n the same fashion as i t was found to behave i n Appendix 1 and the phase s h i f t per section i s identical to the value found i n Appendix 1, the two approaches are i n agreement. When the l i m i t has not been taken, then D has z the form to within any required degree of accuracy V [ r , f l e i K ^ W ] where i n the l i m i t s 9 —s o o X ' U ) — ~ X { z ) The phase v e l o c i t y for D can be approximately determined by z differentiating 9 cot - S z - X (z) T = Constant. o ' Hence, the phase v e l o c i t y v^ i s dz ~ to P " At " s » + - J J C o dz Consequently, the phase v e l o c i t y i s modulated by the periodic term d ) C ' .' dz 5. PLANE WAVES IN A MEDIUM WITH PERMITTIVITY A CONTINUOUS FUNCTION OF z 5.1 I n t r o d u c t i o n To o b t a i n a b e t t e r understanding of the of electromagnetic was behaviour waves i n an inhomogeneous medium, i t decided t h a t an i n v e s t i g a t i o n should be made i n t o the behaviour of plane waves i n a medium with a p e r m i t t i v i t y which i s a continuous and s u f f i c i e n t l y f u n c t i o n of z. differentiable T h i s problem tends to be simpler than ones d e a l i n g with l o n g i t u d i n a l f i e l d components. At the same time, the techniques used i n s o l v i n g the d i f f e r e n t i a l equation a r i s i n g from plane wave c o n s i d e r a t i o n s have o n l y to be s l i g h t l y m o d i f i e d f o r E- and H-mode problems i n which equation (67) a r i s e s . This can e a s i l y be seen by comparing equation (67) w i t h equation (66). 5.2 Problem In p r i n c i p l e a complete s o l u t i o n f o r a plane wave can r e a d i l y be obtained f o r any s u f f i c i e n t l y w e l l behaved z-dependent f u n c t i o n a l form of the p e r m i t t i v i t y . p a r t i c u l a r forms were considered, and i t was Various found t h a t the form e = l^z 2 8 "" 2 +^2 1 z 2 where = Constant k„ = Constant (81) and s = VI - 4 c o u k 2 yields s o l u t i o n s which are e a s i l y termso Consequently, will used. be A sketch equation (81) is , 0<s<cl, 2 satisfying permittivity i n Figure t o be c o n s i d e r e d t h e p e r m e a b i l i t y (82) interpreted in a permittivity o f the given Q 6. physical equation expressed Also, equals for in the medium n . Q e A Fig. 6. Sketch of P e r m i t t i v i t y Versus F o r a p l a n e wave w i t h z (81) from e q u a t i o n (57), since E d E xo = T, ' 2 k dz where E x q 2 2s-2 + _| z\ 1 Exo = (83) 0 i s t h e component o f the e l e c t r i c f i e l d i n t h e x d i r e c t i o n w i t h time dependence suppressed. Equation manner. (83) can be s o l v e d i n t h e f o l l o w i n g F i r s t o f a l l , the t r a n s f o r m a t i o n — u = i s made. z =pz ,p= a>.yjTk (84) From t h i s t r a n s f o r m a t i o n dE dz xo , dE s-1 xo sz du -P and d E o o o o d E - dE — ^ a ' = p s z " —Jsa + p ( s - i ) z ^ dz du 2 2 2 2 2 s 2 s s 2 u (85) Consequently, by t h e s u b s t i t u t i o n o f e x p r e s s i o n (84) i n t o e q u a t i o n (85) „ . d E 2 2 xo , 2 du / ', \ dE « xo . , 2 k^u P" 2 + k. E X Q = 0. (86) If E x q i s transformed into E xo = u I, 2 s (87) 1 then l-2s 1_ dE dY xo = u2s " du « + J Y 2s- u du (88) 2 s and d E l-2s 1 2s dY 1 s du 2s I 2s 2 xo du' u 2 s d£l 2 + u d u l-4s II u + 2 s If expressions (87), (88), and (89) are substituted equation (86), the result i s 2d I ^2 dY . , 2 l dH °4 = 2 u x + u + ( u ) Y which i s Bessel's equation for n = „ with and and N + C J (u) 2 1 2 "2 being arbitrary constants, and thus E where n Hence, 2 Y = C^U) w 0 2 ^N J (PZ )+N J (pz y 2 ~2 xo = 1 1 8 2 1 are arbitrary constants, Since J (u) ± TIU sin u and J x (u) nu cos u, s Y, into (89) 48 E = \ £ N. xo u 1\ „n s s i P n z S + N 2 \ f ^ l c o s P or 1-s E E x = E xo 2 are arbitrary e^* =U HP , Hence, 3(-t^z ) s A i e A + 2 j ( ^ 2 and the reflected = z S e A e H e l e c t r i c wave i s xr ~V*P Since ^ bE "ST E - - |5 , b B = Tt = ^o y< j H Therefore . J fT [ i - s w t e 1-s 2 + z gpsz " 8 1 A i e z s ? (91) Consequentlyo the incident e l e c t r i c wave i s 1-s TT _ /iT . j(tot-pz ) x i \f^p l ' E (90) z S l 0 constants<> r 2 + A e^P 2 z S 1 VTI/O xo where A^ and A A,e^P 2 z "jP ^e'P** z S + jpsz " 8 1 + A e^ 2 A eJP 2 z ! z S 49 or S 1-s S i 2 . 2 z -jpsz n S 1 s 1-s 2..~ 2 — z +opsz (92) Hence, the incident magnetic wave i s H = - i - . f l i y i to>jx Vrtp y/z 1-s -~2~ 2 _s_ s. 2 .~ 2 V "Opsz J < " t ^ " ) o and the reflected magnetic wave i s H . yr _ _ i _ . /_2 I l ^ s a)ji V*p ft 2 s 12 z " + j p s z o Consequently, the wave impedance seen i n the medium by the incident wave i s E . xi Zo i. = H . yi fa>jX Z o ~ s .. p s z I 1-sl ' — and by the reflected wave i s E Z xr o r ~ ~H yr r, = S • f 1-S p s z \ z. Oi ° ' It can be noted that provided z>0 the imaginary part of Z ^ Q i s negative, implying that the reactance i s capactive, while the imaginary part of Z Q r i s positive, implying that the reactance i s inductive» The phase v e l o c i t y of the incident f i e l d can be calculated by l e t t i n g a>t - p z = Constant (93) and by d i f f e r e n t i a t i n g equation (93)o I f this i s done. Psz s—1 a£ = Oo dz /•» Hence, the phase v e l o c i t y v i s v p = dz «o dt=/> z 1-s ' / \ 0/J ( 9 4 ) Since 0<s the phase v e l o c i t y increases as z increases. If an E-mode f i e l d can be set up such as to have a phase v e l o c i t y increasing with z, the f i e l d may possibly be very useful i n pre-accelerator applications. (94) Equation tends to point towards the p o s s i b i l i t y of obtaining such an E-mode f i e l d through the use of an inhomogeneous medium of t h i s type. 5.3 An Inhomogeneous Slab Between Two Homogeneous Media An inhomogeneous slab can be used to effect a match between two different homogeneous semi-infinite media The purpose of t h i s section i s to demonstrate t h i s use f o r the case where the inhomogeneous slab has a permittivity that i s functionally described i n equation (81). In Figure 7 the f i e l d i s assumed to originate i n medium 1, pass through the inhomogeneous mediufho and enter medium 2. Also, i t i s assumed that the f i e l d i n medium 2 i s t o t a l l y absorbed, none of the energy being reflected 9 back toward the source» In medium 1 the f i e l d equations are E x l = N^^-P.!** + N e J 2 ( w t + Pl z ) and yl V u- where P i and N^ and Ng are constantso = w V on v The equations representing Surface 1 — Inhomogeneous Medium z Fields are and H F i g . 7» An Inhomogeneous Slab Between Two Homogeneous Media 6. So Bamo and J R Whinnery, Fields and Waves i n Modern Radio» John Wiley & Sons, I n c , New York, 1953, p»28lo 0 0 0 the f i e l d s E and H i n the inhomogeneous medium are given x y by equations (91) and (92). Since only the transmitted wave i s present i n medium 2, the f i e l d equations are E x 2 = eJ (wt Cl "P2 z) and H _ _./!2 =J— n C^e i(«t-e.,*> y2 V | i i 0 where P 2 = and ^ 2 W i s a constant< At z equals a i n Figure 7 S E x l, = Ex (95) V -V (96) and At z equals b, E x - x2 E ( 9 7 ) and H y = H y 2 . (98) From equations (95), (96), (97), and (98) the values for and N 2 can be determined, as done i n Appendix 2, from the parameters of the media, the frequency, and the constant C^<, Once and N 2 are known, the r e f l e c t i o n c o e f f i c i e n t E at surface 1 can be readily found since E = • (99.) In Appendix 2 i t i s shown that If there i s no r e f l e c t i o n at surface 1 i n Figure 7 9 a l l the energy can be transferred from medium 1 to medium 2 0 For t h i s situation to occur, R = 0„ As shown i n Appendix 2, i n order that R = 0, two equations must be s a t i s f i e d . These equations are V^jV^ 1-s a " 8 a V ji - V^i 1 b ^ c o s ( a S ) (101) (b -a ) s b V |i / 0 s„ b Q 8 and /b " 1-s lv / ^ l k^ab) 8 s-1 1 (b -a ) cos- s-1 s 1-s sin 2 2 u <o u , a b s (b -a ) s s (102) T h i s means t h a t two o f the equations (101) and (102)o equations (101) and (102), t h e s e two p a r a m e t e r s , excluding values. and e , 2 Since with respect if they have equations to p a r a m e t e r s must be d e t e r m i n e d byDue t o an i n f i n i t e number o f and (102) (101) and e^, and e a r e d e t e r m i n e d by t h e s e If are evaluated the are assigned c o n d i t i o n at If specified, 2 surface field one s o l u t i o n two of the (lOl) convenient values, once t h e solutions permittivity remaining and (102) the and matched of the inhomogeneous medium i s o b t a i n e d f o r the r e g i o n o n l y h o l d f o r one p a r t i c u l a r f r e q u e n c y . that each established. the p e r m i t t i v i t y the periodic equations. by e q u a t i o n s 1 is of discrete are not have f o r a p a r t i c u l a r frequency parameters others the p e r i o d i c i t y is specified, k^, inhomogeneous The r e a s o n k , 2 and s have is 55 fixed values, and thus from the equation = Vl~4w u k o"2 2 the frequency i s determined. Therefore, when the permittivity i s specified, the frequency cannot be evaluated from equations (lOl) and (102). However, f i e l d solutions f o r an inhomogeneous region which hold f o r any frequency after the permittivity i s specified can be obtained quite readily. For example, the medium with a permittivity behaving as e = k^z where k^ = Constant and s = ™ has such f i e l d solutions. (n—X y 3 p 5 o o o o oo Sn^X} 6. MANUFACTURING OF INHOMOGENEOUS DIELECTRIC MEDIA Although not top much thought has been given to the possible ways of manufacturing a medium which has a permittivity that i s a continuous function of z, three possible methods have been considered,, The f i r s t method i s to construct the medium by using thin sheets of homogeneous d i e l e c t r i c material<> If thin sheets having different values of permittivity are cemented together i n some desired order, the resulting laminated d i e l e c t r i c medium varies functionally with z<> Provided the thickness of the sheets i s small compared to the wavelength of the f i e l d being propagated i n the medium and provided the change i n d i e l e c t r i c constant between adjacent sheets i s small, i t i s believed that the medium as seen by the f i e l d e f f e c t i v e l y varies i n the desired continuous manner with z. Another possible method f o r manufacturing a medium with a permittivity which i s a continuous function of z i s by varying the density of the medium i n the z direction,. For example, the porosity of the medium could be varied i n the z d i r e c t i o n as i n the case of some types of foam rubber. It might be possible to use a centrifugal process while the d i e l e c t r i c material i s s o l i d i f y i n g to establish a variable density. Through the use of plasmas, a medium can be obtained with a permittivity which varies continuously i n the z d i r e c t i o n . For exampl already mentioned, at Stanford i t has been proposed that sound waves be propagated down a waveguide f i l l e d with plasma to vary i n a periodic fashion the density of the medium and thus the effective permittivity. 7c CONCLUSION For the case of a lossless medium containing no free charges and possessing a continuous and s u f f i c i e n t l y differentiable spatially' dependent permeability and permittivity, two v e c t o r i a l d i f f e r e n t i a l wave equations, V E 2 + V £ ^ T § ~ JIViiVE - (Vn-.V)E Ve.E) bt and lie ^ - J(Ve.VH - (Ve.V)H) ' bt were derived from Maxwell's equations<> The magnetic v e c t o r i a l d i f f e r e n t i a l wave equation was used to f i n d the necessary condition, /iv^H) i V e ^ + dz \ \i r I . e 0 = o, oz for an E-mode to exist i n a waveguide, and the e l e c t r i c v e c t o r i a l d i f f e r e n t i a l wave equation was used to f i n d the necessary condition, for an H-mode to exist i n a waveguide 0 Through the use of the E-mode condition, an investigation was carried out to determine certain r e s t r i c t i o n s which must be imposed upon the f i e l d s before E-modes can exist i n a waveguide f i l l e d with a medium whose permeability i s constant and permittivity i s e = f(r,z)o The r e s t r i c t i o n s were found to be that *V z <>> and the f i e l d s have no angular v a r i a t i o n . Also, i t was noted that the E-mode condition did not impose any r e s t r i c t i o n upon the f i e l d components when the permittivity i s a function of z only. Through the use of the E-mode condition, an investigation into the r e s t r i c t i o n s imposed upon the f i e l d s when the permittivity has other functional variations i s suggested for future study along with a complimentary investigation using the H-mode condition. It i s worth pointing out that these r e s t r i c t i o n s are the duals to the r e s t r i c t i o n s on the f i e l d s f o r the interchanged cases, namely, the cases having the permeability spatiallycdependent and the permittivity constant. An investigation into the f i e l d r e s t r i c t i o n s using the mode conditions i s recommended for cases where both the permeability and permittivity have various functional forms. The f i e l d equations were investigated for the case where the permeability i s constant and the permittivity varies with z and for the case where the interchanged situation i s true. In p a r t i c u l a r , the f i e l d equations f o r transverse waves and waves with longitudinal components were considered. After the variables of the pertinent unified d i f f e r e n t i a l f i e l d equations were separated, the d i f f e r e n t i a l equations containing the transverse dependent part of the f i e l d components were found to be the same as the corresponding equations found f o r homogeneous media. For the d i f f e r e n t cases considered, the z-dependent part of the f i e l d components were found to s a t i s f y one of the following d i f f e r e n t i a l equations; i ) ^ § + o>qT = 0, dz 2 ii) & dz + ( 2 _ 2 tt q M ) T = Q > z »*> 7 § " \ ft S + <**- > 2 m2 t 0 dz where M i s a separation constant and q = |xe o Owing to a theorem by Sturm, i t was possible to show that for a physically realizable situation the solutions f o r the f i r s t of these equations i n T i s o s c i l l a t o r y , and the solutions f o r the second one i s o s c i l l a t o r y provided W fA6 2 - M 2 >0o Besides t h i s , the theorem offered a possible test for showing whether or not the solutions for the t h i r d equation are oscillatory,, The f i e l d s i n a d i e l e c t r i c loaded periodic structure were considered from the viewpoint that the medium as a whole i s inhomogeneous inside the waveguide 0 The assumption that the permittivity and a l l i t s f i r s t and second order derivatives are defined for a l l i n t e r i o r points i n the waveguide was discussed from the point of view taken i n mathematical physics, which i s that a l l matter i s continuous„ After t h i s assumption was made, an investigation into the r e s t r i c t i o n s on the f i e l d s when Emodes are present was carried outo For the case where the d i e l e c t r i c discs have center holes, since there i s a r a d i a l v a r i a t i o n i n the permittivity as well as a longitudinal variation, the r e s t r i c t i o n s were recognized to be the same as those discovered for the example where e = f(r,z)o When the discs are s o l i d , i t was noted that the E-mode condition i s s a t i s f i e d without imposing any r e s t r i c t i o n s on the fieldso For a periodic structure with s o l i d d i e l e c t r i c discs the theory developed for inhomogeneous d i e l e c t r i c media was used to f i n d the E-mode f i e l d expressions when the d i e l e c t r i c regions are matched into the a i r regions 0 In the l i m i t as the permittivity approaches a rectangular waveshape, these f i e l d expressions were shown to be i n agreement with the expressions derived by solving for the f i e l d s i n each of the homogeneous regions and by matching the f i e l d s at the boundaries» It i s f e l t that further e f f o r t should be made to use the theory for inhomogeneous media to find the f i e l d equations for the s o l i d d i e l e c t r i c disc case when a match does not exists In that the behaviour of the f i e l d as the permittivity approaches i t s l i m i t i s known, there should be some method for showing that this known solution s a t i s f i e s i n the l i m i t the d i f f e r e n t i a l wave equations resulting from the theory for inhomogeneous media. be solved, i t may I f this problem could shed some l i g h t on how to solve the f i e l d problem using the theory for inhomogeneous media when the d i e l e c t r i c discs have center holes. Furthermore, a second approach for finding the f i e l d solutions would be established, which i s at least of academic i n t e r e s t . Also, i t i s f e l t that a further attempt should be made to f i n d accurate and manageable f i e l d solutions for the case where the d i e l e c t r i c discs have center holes. I t has not yet been possible to attempt a thorough investigation of t h i s problem. An investigation was carried out into the behaviour of plane waves i n a medium whose permittivity i s e = k,z 1 2 s ~ 2 + ^2 2 z (103) where and kg a r e c o n s t a n t s s s Vx - - The e X e c t r i c and m a g n e t i c and 4a> u k . 2 o 2 f i e X d s were 0<s<X caXcuXated to be X-s and H - i - X i _S Si 1-s ~2 .~ 2 — z -opsz _s X^s _2 z" + apsz I 2 A eJ ( w t H 2 P z S > where ; The wave impedance was shown t o s e e n i n t h e medium by t h e i n c i d e n t wave be z , 01 and t h e wave impedance wave was s shown t o to seen i n the ty medium by t h e be Z The phase v e l o c i t y = or ^ o s z = . |l-s\' was c a X c u X a t e d t o be z * oi reflected Also, a slab of d i e l e c t r i c material having a permittivity satisfying equation (103) was placed between two different homogeneous semi-infinite regions and used to effect a match between these two regions 0 A b r i e f discussion i s made on the possible methods of manufacturing inhomogeneous media with a constant permeability and a permittivity varying i n the direction of propagation., APPENDIX 1* For a c i r c u l a r waveguide loaded with s o l i d d i e l e c t r i c discs as shown i n Figure 3 , page 37, the f i e l d patterns for an Eg^-mode can be determined by matching at the boundaries the f i e l d s found i n each homogeneous region Through the use of Maxwell's equations, the f i e l d components for an Eg^-mode are found to be E E z = = r -JP-.Z A l e dP-. z + ^ Mv, Aj^e A 2 -OP-jZ 1 J (Mr)eJ , wt 6 0 - Ae jp, z J^Mr)^?*, l 2 (1) (2) and 3^1 jrf = ~M~ l A H -OPiZ e OP, z + A 2 J^MrJe^ e (3) i n the d i e l e c t r i c region (2), and E E -jpz = • C^e z dp z J (Mr)e> + Ce 0 - 22r ~ Mv LC,e 1 o 0 J 2 -DP Z dp Z 0 - Ce 0 t 0 0 2 J, (Mr)^"*, (4) (5) and H * 3<»>e, C p ~ M 1 -dP z 0 0 i e dP z 0 + Ce 2 0 (6) The treatment i n this Appendix i s based on the paper presented by G.B= Walker and C.G. Englefield, "Some Properties of D i e l e c t r i c Loaded Slow Wave Structures", PGMTT Symposium. San Diego, May, I960. 0 i n the a i r region (3), where 2 Pi 2 = w Po = ^o l" e 2 (7) ' M » Vo"^ » (8) v.. = ^— Pi = Phase v e l o c i t y i n d i e l e c t r i c region, v = Phase v e l o c i t y i n a i r region, 1 = ^— 0 Po s ^ s 1 = l K » = F i r s t root of J (Mb) = 0 , Q 7 and A^,Ag»C^ and Cg are related constants f 0 The f i e l d i n region (4) can be determined by using Floquet's Theorem which states that i n a given mode of o s c i l l a t i o n of a periodic structure, at a specific frequency, the wave function i s multiplied by a given complex phase constant when the f i e l d i s observed a distance g of one period down the structure. Consequently, the f i e l d components i n region (4) are given by the multiplication of • A the f i e l d components i n equations ( l ) , ( 2 ) , and (3) by e""*. 7. Go Ramo and J.R. Vhinnery, Fields and Waves i n Modern Radio John Wiley & Sons, Inc., New York, 1953, p. 375» 8o J.C. Slater, Microwave E l e c t r o n i c s . D van Nostrand Co., Inc., Princeton, New Jersey,' Toronto, London, New^York, 1950, p. 170. 0 where 0 i s t h e phase change p e r section,, If t h e f i e l d s a r e matched a t z e q u a l s 0::and^p-q, the e q u a t i o n s o b t a i n e d a r e V l e A 1 - o 2 ~ 1 1 V V + e A 1 x j(2O -0) C + - e C 2 o V 1 2 = °> C - e C 1 Q -j(2©,+0) 1 A A l " v o e ( 9 ) = 0, 2 (10) -J29 2 ~ A l v e j2© C l + V l e C 2 = °» (11) and ••3(20,-0) e l e h - j 20 -0(2©,-^) + t l * A 2 " o e e C j2© l" o e e C 2 = ° (12) where the phase change i n the a i r region i s 2© Q ^ 8 (p-q) 0 and the phase change i n the d i e l e c t r i c region i s 26 1 Equations =8^ q o (4), (10), (.11), and (12) have unique solutions for three of the constants A^,A ,C^ 2 the remaining constant only i f 0 and C 2 i n terms of 68 -V, -e. j(2© -0) 1 v e o e l -v e o jUOj-0) .3(20^+0) 3 20 •320, " l v -3'(2© +0) 1 f e v -J2© l = Oo e 320, -e e e o (13) Prom the expansion of equation (13) an expression f o r 0 can be determined. 4 cos 0= This expression i s + 0 cos (2© +2© ) 1 Q 1 Z ITT z~ "vzT, cos(2©,-2© 1 o) o 1' 1 or cos 0 = cos 2©o cos l 29, where Z rt Z Z, sin 2©o s i n 2©, I .Z,*Z^ ' 1 o' -°4~1 i s the wave impedance for an E^-mode i n the a i r region, Z = v e o o and Z^ i s the wave impedance f o r an E^-mode i n the d i e l e c t r i c region, Z, = 1 v, e 1"! Through the use of any three of the equations ( 9 ) , ( l O ) , ( l l ) , and (12), the constants A^.Ag* and C^ can be evaluated i n terms of Cgo I f t h i s i s done, the 69 relationships A, JL G 2 e _ f o J0 " e, e obtained are c o " " "~° o s 2 6 3'z17 V + Z +j(0-2© ) - Ag c a n 2 ^ " i e S o e c CT = o - 2 ®o " 0 2 Jfr- s i n ft cos " co* Nov t h a t 2 6 - l the f i e l d - 1 Z, + j - ^ s i n 2© - z7 j^rr s i n 2© j — s i n 2©, Z~ 1 c o s 2© "zl ^ x ° s i n 2 9 i components have d e t e r m i n e d o v e r one p e r i o d p , e x p r e s s i o n s components for E s u c h an e x p r e s s i o n been for the f i e l d c a n be f o u n d w h i c h h o l d t h r o u g h o u t F o r example, 1 j (20,-0)- V 1 3*(0-2© ) e 6 ~"~ 29, "T^oT 6 j(0+2© ) d 2 Q n c o s 2©^^ - Z O i • e C S the waveguide. c a n be f o u n d b y z taking the f o l l o w i n g the steps. The f i r s t step i s to define function F(z) From e q u a t i o n = E J (r,z,t)e •i0z p . -r-i o (Mrje^ (14) (14) E F(z+p) = J z ( r , z+p, t ) e (Mr)eJ t t t P e^ and from F l o q u e f s Theorem E ( r , z i p , t ) = E ( r , z , t ) e " J0 J Hence, F(z+p) = F(z) , and thus F(z) i s a periodic function i n z with a period p< Consequently, F(z) can be expressed as the Fourier sum oo -.i2nnz P F(z) = / a e -I n n—-oo p-q where a .j 2rcnz = i-/F(z)e dz p n ., 0 + 7 m _ .1 0+27tn \ , oo Therefore, E z = J (Mr)eJ o 2 ^ =- ttt 1 * J n=—oo where P-1 f a 11 = 1 fVr,»,t) • / P J J (Mr)e3 -q i wt e <, //7 * dz . n u ¥hen the d i e l e c t r i c regions are matched into the a i r regions, 1 A 2 = C 2 = 0. Consequently, equations ( l ) , (2), (3), (4), (5), and (6) 71 become respectively E E j(o)t-p,z) = A J .(Mr)e z 10 1 r =M? 1 r 1 Vl< >* f c jwe, j(tot-8,z) 3(«t-p z) E z = C ^ M r J e , 0 i(wt—8 z) -o and , v j(»t-p z) Since for the matched case Z. o = Z, , 1 the phase s h i f t per section i s given by cos 0 = cos 26 cos 2©, - s i n 2© 0 1 s i n 2©,• 1 O Therefore, 0 = 2© Q + 2© , x = (p-t)P + qB 0 x = (p-q) Va, n c -M 2 o o 2 + qV?^I? APPENDIX 2 The f i e l d s f o r the different regions shown i n Figure 7, page 51, are: i ) i n medium 1 E Nje xl -JB,z x 30,z + Ne L 2 and 0*01 z -J0-.Z v |i yi N l ~ 2 e N e Jut i i ) i n the inhomogeneous medium 1-s z E 2 x and H _J_ G T i y"*«n0vipyz | V -*>» + A r S e ^ _s sr 1-s 2,. »>"• + -jjpa +opsz2 A.e3P z ' S" n i i i ) i n medium 2 x2 1 y2 Vp and where 2 V^P I 1 s = Vl - 4u) ji k , 2 o *2 and A^BAg.N^.Ng, and 2 0<s<l , = #o*2 » W are related constants. At z equals a the boundary conditions are E , =E xl x and H , = H y i y Therefore, -JP-i N l e 1-s •o OKI a + N 2 e A,e"JP + A^P*' AS = V * p a and N J3 1-s 2 l ~ 2 e N S £[1 . n e Q e l-s_2 2 At z equals b E x = E x2 0 and H y = H _, y2 S i pa*' A, Consequently, 1-s / 2 'v 2 Al and e-JP _s ^ .+ A^P*' BS = C l _s s_- b -jpsb 2 (3) e ^ 2 !2 -JP R — 2 s_- b + Y>b 2 3 2 B (4) O, e 1 At surface 1 the r e f l e c t i o n c o e f f i c i e n t R i s defined as 2 N (5) R = |jOne approach to finding R i s to express both terms of C^. and Ng i n To do t h i s , A^ and Ag must be found i n terms of C, o Prom equations (3) and ( 4 ) , j(pb -fi b)| , s 2 h (6) = and g^b -rB b), /e s A ft ^ = £e >b 2 * V|*o/ Cj (7) where (8) * 2psb 2 75 s-2 2 and \ />*> = 2 (9) *(¥)»> + From equations ( l ) and (2) both N^ and for i n terms of A^ and Ago The results are -3 (pf-Pi*) and are solved A -j(pa +p ) ^ jOQa^S.a) / * /eTIl s ia (10) 2 A 2 (11) where i a (12) s-2 and ^ Now, / ^ V sVuDe, o ^ 1 2ai^ ^ a Lr o P the values for A^ and A (6) and (7) are substituted s 2 a 2 + 3 | ¥ ) a J 2 (13) found respectively i n equations into equations (10) and ( l l ) to gxve K j(B -8 b) ia 2 1 = -dp(b -a ) e s 3 (14) and N 2= C4°i 6 =j(p a+p b) 1 a w S 2 —a_ S ) I **o / (15) If expressions (14) and (15) are substituted into equation (5). i t i s found that (16) If there i s no r e f l e c t i o n at surface 1, R = 0< Therefore, from equation (16) b -a ) s or * s S\ , S b -a ; •jp(b -a ) S s (17) Now, i f the following d e f i n i t i o n s are made: i) T =p(b -a ), s ii)7^ = v a s + jv , a the substitution of these newly defined quantities into equation (17) y i e l d s ( r i + i%u* r = <r + w 2 2 )*~* r o (is) Once the r e a l parts of equation (18) are equated, the r e s u l t i n g equation i s (1^ - f ^ c o s T = (% +%)sinT . (19) Similarly, from the imaginary parts of equation (18) (\ -%)COBT = -([^ +Q)sinT o (20) 78 In terms of the o r i g i n a l parameters, equations (19) and (20) are respectively Wk^^ 1 a " 8 r b ^ j c o s -9^1 1 o 1 / i S _ Ss ' (b -a ) s *o/ and 1 l H k Q 1-s I b " 2 \ a 8 1 a ' ^v/^^i cos b 8 1 (b «a ) s s 79 BIBLIOGRAPHY 1. Adler, R.B., "Waves on Inhomogeneous C y l i n d r i c a l Structures*", Proc. I.R.E.. v o l . 40, March 1952, No. 3, p. 339. 2. B r i l l o u i n , L., Wave Propagation i n Periodic Structures. Dover Pub., Inc., 1953. 3. Chambers, L.G., "Propagation i n Waveguides P i l l e d Longitudinally with Two or More D i e l e c t r i c s " , B.J.A.P.. v o l . 4, February 1953,' p. 39. 4. Chambers, L.G., "Compilation of the Propagation Constants of an Inhomogeneous F i l l e d Waveguide", B.J.A.P.. v o l , 3, January 1952, p„ 19, 5. Chambers, L.G., "An Approximate Method for the Calculation of Propagation Constants f o r Inhomogeneous F i l l e d Waveguides", Quart. J . Mech. Appl. Math., v o l . 7, Pt. 3, September 1954, p. 299. 6. Cunningham, W.J., Introduction to Nonlinear Analysis, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1958. 7. Ford, L.R., D i f f e r e n t i a l Equations, McGraw-Hill, New York, Toronto, London, 1955. Inc., 8. Hildebrand, F.B., Advanced Calculus for Engineers, PrenticeH a l l , Inc., Englewood C l i f f s , New York, 1949. 9. Ince, E.L., Ordinary D i f f e r e n t i a l Equations. Dover Publications, Inc., New York, 1956. 10. Kino, G.S., A Proposed Millimeter-wave Generator, Microwave Laboratory, W.W. Hanson Laboratories of Physics, Stanford University, Stanford, C a l i f o r n i a , 1960. 11. Liebowitz, B., "Development of Electromagnetic Theory for Non-Homogeneous Spaces", Phys. Rev.. v o l . 64, No. 9 and 10, July 1943, p. 294. 12. McLachlan, N.W., Theory and Application of Mathieu Functions. Clarendon Press, Oxford, 1947. 13. Osterberg, H„, "Propagation of Plane Electromagnetic Waves i n Inhomogeneous Media", J . Opt. Soc. Amer., v o l . 48. August 1958, p. 513. 14. Pincherle, L. "Electromagnetic Waves i n Metal Tubes P i l l e d Longitudinally with Two D i e l e c t r i c s " , Phys. Rev., v o l . 66, No. 5 and 6, September 1944, p. 118. 15. Ramo, S. and Whinnery, J.R., Fields and Waves i n Modern Radio. John Wiley & Sons, Inc., New York, Chapman & H a l l , Ltd., London, 1953. 16. Ro-Shersby-Harvie, R.B., et a l , "A Theoretical and Experimental Investigation of AnisotropicDielectric-Loaded Linear Electron Accelerator" Proc. of I.E.E.. v o l . 104, Part B, 1957. 17. Slater, J.C., Microwave Electronics, D. van Nostrand Co Inc., Princeton, New Jersey, Toronto, New York, 1950. 18. Walker, G.B. and E n g l e f i e l d , C.G., "Some Properties of D i e l e c t r i c Loaded Slow Wave Structures", PGMTT Symposium. Sah Diego, May I960. 19. Webster, A.G., P a r t i a l D i f f e r e n t i a l Equations of Mathematical Physics, Dove Publications Inc., New York, 1955. 20, Whittaker E.T. and Watson, G.N., A Course of Modern Analysis, University Press, Cambridge, 1915.
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Waves in inhomogeneous isotropic media James, Christopher Robert 1961
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Title | Waves in inhomogeneous isotropic media |
Creator |
James, Christopher Robert |
Publisher | University of British Columbia |
Date | 1961 |
Date Issued | 2011-11-29 |
Description | For the case of a lossless medium containing no free charges and possessing a continuous and sufficiently differentiable spatially dependent permeability and permittivity, two vectorial differential wave equations, one for the electric and one for the magnetic field, are derived through the use of Maxwell's equations. From these two equations necessary conditions for E- and H-modes to exist in a waveguide are established,. The field equations for the case of constant permeability and z-dependent permittivity as well as the interchanged case are investigated. A test is developed which, if met, assures that the solutions are oscillatory for the ordinary differential equations containing the z-dependent part of the wave function. For the dielectric loaded periodic structure the theory for inhomogeneous isotropic media is used to determine the restrictions on the field components which are necessary before E-modes can exist and to find the E-mode wave solutions for the solid disc case when the dielectric regions are matched into the air regions. An investigation is carried out into the behaviour of plane waves in a medium with the permeability constant and the permittivity varying in the direction of propagation. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2011-11-29 |
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 | 10.14288/1.0105059 |
Degree |
Master of Applied Science - MASc |
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 |
URI | http://hdl.handle.net/2429/39365 |
Aggregated Source Repository | DSpace |
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