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Waves in inhomogeneous isotropic media 1961

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WAVES IN INHOMOGENEOUS ISOTROPIC MEDIA by CHRISTOPHER ROBERT JAMES B.A.Sc., University of British Columbia, 1960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Electrical Engineering We accept this thesis as conforming to the standards required from candidates for the degree of Master of Applied Science Members of the Department of Electrical Engineering THE UNIVERSITY OF BRITISH COLUMBIA August 1961 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. ABSTRACT 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 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 in 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 is developed which, i f 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 f i e l d 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 i s carried out into the behaviour of plane waves in a medium with the permeability constant and the permittivity varying in the direction of propagation. i i i TABLE OF CONTENTS page L i X S * t O f I X X U S "fcr&'t/X 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 "V 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 V I X o I l l t rO d-UC " 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 X 2o GGjlXGrflfX Th.6 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 7 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 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 for Case of Constant Permeability and of z-Dependent Permittivity and the Inter- Cll.£tXl££@CL 0SLS6 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 X8 3 o X 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 X 8 3.2 Differential Field Equations for 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 X 9 3.3 Differential Field Equations for Waves with Longitudinal Components....................... 22 3.4 Summary of z-Dependent Equations.............. 25 3.5 Oscillation Theorem Due to Sturm.............. 25 4. Field Problem in 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 30 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 4.2 Functional Behaviour of the Permittivity...... 30 4.3 Field Restrictions Due to E-Mode Condition.... 33 4.4 Unified Differential Equations................ 33 iv 4 o 5 E-Mode Solution of Unified Differential Equations for Matched Case, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 o P l a n e ¥ a v e s i n a M e d i u m w i t h P e r m i t t i v i t y a C o n t i n u o u s F u n c t i o n o f Zo o o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o o o 5 o1 I n t r o d u c t i o n 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 2 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 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 6 0 Manufacturing of Inhomogeneous Dielectric Media. .<> 7 o 0 OnC Xu S1 OnS o oo o ooo O O O O , O O O O O O O O O - O O O oo o o ooo o. o o o o o o o o Aj)J)©ndlX l 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 ^jP^)6.ndX3C 2o 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 B l f o l X O ̂ r a j ) l i y 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 F i g u r e Page 1. C r o s s S e c t i o n o f D i e l e c t r i c Loaded P e r i o d i c S"fcrUC t\ir© o o o e e o o o-o o o o e o o o o © © O O O O O O O O O O O O O O O O O O O 2 2. S k e t c h o f F u n c t i o n a l V a r i a t i o n s o f P e r m i t t i v i t y 31 3o C r o s s S e c t i o n o f S o l i d D i s c P e r i o d i c St rUC tUT 6o0aoeo»oooooooooooooooo'ooooooooooeoa'oo 37 2 2 2 2 4« Diagram o f u u e— M and k e V e r s u s z» » o .-••.••.» 38 5. A Diagram o f / w ? | i Q e - M 2 dz V e r s u s z« • « » • < > . • o «?..•<. 40 6. S k e t c h o f P e r m i t t i v i t y V e r s u s z . . . >»„ * < , » . o . . o. 45 7. An Inhomogeneous S l a b Between Two Homogeneous MecLX £to © 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 51 v i 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 this studyo To Mr. EoL. Lewis, Mr. D»B. McDiarmid, and other colleagues, the author would like to express his appreciation for their many useful suggestions on this work. Acknowledgement is gratefully given to the National Research Council for the assistance received from a Bursary awarded in I960 and a Studentship awarded in 1961. WAVES IN INHOMOGENEOUS ISOTROPIC MEDIA lo INTRODUCTION The purpose of this thesis is to investigate theoretically the behaviour of electromagnetic waves in lossless inhomogeneous isotropic media containing no free charges. Throughout this thesis inhomogeneous isotropic media will be called inhomogeneous media. Inhomogeneous media are of interest because they may be used to make slow wave structures which in turn can be used in linear accelerators 9 traveling wave tubes, backward-wave oscillators, and microwave f i l t e r s . Besides this, inhomogeneous media may be used in 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 cross- sectional density and with an axial density variation of the form /D(l+a sin ^z) for a slow wave structure.^" The axial variation in plasma density is to be achieved by propagating sound waves down the waveguide. In this case 1. G.S. Kino, A Proposed Millimeter-wave Generator,, Microwave Laboratory, W.W. Hanson Laboratories of Physics, Stanford University, Stanford, California. the plasma forms an inhomogeneous medium. The topic of this thesis arose during an investi- gation into the solution for the wave functions in a dielectric loaded periodic structure,. Such a structure is shown in Figure 1. z Ir / / / / / Cylindrical Waveguide T ~ T T / / / / / / / / I I I I Z V - " I ' V ' Dielec /Disc • ('i i • 111 ii> l>l | ' l I Hi; " ' >ric Center ^Hole Air Region y / / / / /TT I I I M M i / zrz a Center . ;ine I I I I I L ' M i ' i 11 I 1 " J ' ;!','!; i ' ' " . 1 111 * • 1' 11 I / I I Fig. = Permittivity of dielectric material > Cylindrical Coordinates tQ = Permittivity of air z = Longitudinal Coordinate r = Radial coordinate /> =Angular coordinate Notes Discs are solid when a equals 0 . 1. Cross Section of Dielectric Loaded Periodic Structure For a periodic structure similar to the one shown in Figure 1, the exact wave functions can be readily deter- mined provided the dielectric discs are solid. These wave functions are derived in Appendix 1 for the Eg^-mode. If the periodic structure is used in the usual fashion for 3 accelerator or traveling wave tube applications, electrons must pass along the axis of the waveguide0 To make this possible, a hole must exist through the center of each disc. With the hole present the problem of finding the wave functions becomes exceedingly complex. In principle, this problem can be solved by solving in each homogeneous region the differential wave equations developed from Maxwell's equations and by matching the solutions for the different 2 regions at the boundaries. Also, i t should be noted that Floquet's Theorem must be used in the same manner as i t i s used in Appendix 1. Due to the excessive labour involved in 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 solid disc theory and/or the anisotropic theory approximations in the design of periodic structures having 2 center holes in the dielectric discs. Since previous techniques used in attempting to solve the problem in which the dielectric discs have center holes are not entirely satisfactory, i t was thought that a different approach might be useful. Instead of placing the emphasis on the medium inside the waveguide being made up of homogeneous sections, i t was decided that an investiga- tion 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-Dielectrie- Loaded Linear Electron Accelerator", Proc. of I.E.E., vol. 104, Part B, 1957. "~*"~ 4 fact that the medium as a whole is inhomogeneous. In other words, the permittivity is a function of the spatial parameterso Three reasons can be advanced for following this approacho One reason is that a different approach at times reveals new information about a problem, and a second reason is that only one vectorial differential wave equation has to be solved, as will be shown in section 2.2, to obtain a f i e l d solution which holds throughout the waveguideo Also, since the permittivity in the neighbourhood of the boundaries between the air and dielectric regions as well as elsewhere is assumed continuous and sufficiently differentiable, the theory developed to attack the problem in which the dielectric discs have center holes can be expanded to include inhomogeneous media in general. The continuity and differentiability assumptions w i l l be i discussed in section 4.2. The one vectorial differential wave equation, which is valid throughout the waveguide, yields three scalar partial differential equationso These scalar partial differential equations will hereafter be referred to as the unified differential equations. As i t turned out, when the dielectric discs have center holes, no technique was devised to find the general solution for any of the unified differential equations. Consequently, the original objective was not achieved. However, this problem initiated the following work in this thesis. 5 For the case where both the permittivity and permeability are continuous and sufficiently differentiable functions of the spatial parameters, the electric and magnetic vectorial differential wave equations are derived 0 Through the use of these equations, necessary conditions for E- and H-modes to exist in a waveguide are f o u n d o An example of a use of the E-mode condition is shown,. For the case where either the permittivity or ^ permeability is a function only of the axial parameter z and the remaining characteristic of the medium is constant, the pertinent unified differential equations are separated into ordinary differential equations» A test i s developed which, i f met, assures that the solutions are oscillatory for the ordinary differential 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 in the dielectric loaded periodic structure are investigated using the theory for inhomogeneous media* Also, when the dielectric discs are solid, provided the dielectric regions are matched into the air regions, for the E-mode case a solution for the pertinent unified differential equation is given 0 To provide a better physical understanding of the behaviour of electromagnetic waves in an inhomogeneous medium, an investigation i s carried out into the behaviour of plane waves in a medium with the permeability constant 6 and with the permittivity varying in the direction of propagation,. In this thesis the behaviour of E-modes i s investigated far more thoroughly than the behaviour of H-modeso The reason for this i s that the dielectric loaded periodic structure discussed in this thesis is primarily used for linear accelerator and traveling wave tube applications and in these applications E-raodes and not H-modes are excited* 7 2. GENERAL THEORY 2.1 I n t r o d u c t i o n Through t h e use o f M a x w e l l ' s e q u a t i o n s , t h e f o l l o w i n g wave t h e o r y w i l l be d e v e l o p e d f o r the case o f a l o s s l e s s medium c o n t a i n i n g no f r e e c h a r g e s . To b e g i n w i t h , the s i t u a t i o n where the 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 o f the medium a r e g e n e r a l f u n c t i o n s of the s p a t i a l c o o r d i n a t e s w i l l be c o n s i d e r e d . F o l l o w i n g t h i s , the case w i t h the p e r m e a b i l i t y c o n s t a n t and p e r m i t t i v i t y a f u n c t i o n o f z w i l l be t r e a t e d a l o n g w i t h the i n t e r c h a n g e o f t h i s c a s e . M a x w e l l ' s e q u a t i o n s i n a medium c o n t a i n i n g no f r e e c h a r g e s and w i t h z e r o c o n d u c t i v i t y are Vx E V x s V.B (1) 5¥ (2) o (3) (4) where E = E l e c t r i c f i e l d i n t e n s i t y v e c t o r , 13 = E l e c t r i c f l u x d e n s i t y v e c t o r , 5 = M a g n e t i c f i e l d i n t e n s i t y v e c t o r , S = M a g n e t i c f l u x d e n s i t y v e c t o r . A l s o , 15 and E are r e l a t e d by the e q u a t i o n (5) * I n t e r c h a n g e d case i s t h a t o f c o n s t a n t p e r m i t t i v i t y and of z-dependerit p e r m e a b i l i t y . 8 and B* and S are related by the equation S = uB (6) vhere e is the permittivity and \i is the permeability. 2.2 Permeability and Permittivity, Functions of the Space Coordinates To obtain an expression for the electromagnetic f i e l d in a homogeneous medium, the differential wave equation which has to be solved is the standard equation where u-e i s constant. When the medium is not homogeneous, the permeability and permittivity being continuous and sufficieatly differentiable functions of the space coordinates, the differential wave equations from which the f i e l d expressions can be obtained are somewhat more complex. These more complex differential equations can be arrived at in the following manner. For the electric 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 2 E , 9 then V ( V . E ) « V 2 E = (Vx u.H) or V( V.E) - V 2 E = - | r ( ^ ( e E ) + Vlt X S)0 Since i t has been assumed that the permeability and permittivity are not functions of time, V(V .E) - V ^ ^ - i x e ^ f - VnxSf bt^ o x b 2 E i n T bB = " M , E b t 1 ~ ^ ^ = - jie & + iVn X (Vx E ) b 2 E bt^ + ^ [ V R O V E - (VtioVJE where in rectangular coordinates with I, j and k" being the unit vectors in the x,y, and z directions respectively. Prom equations (3) and (5) VoD" = V « e E = 0. Consequently, V e . E + e V»E = 0 and thus V .E = 4 V e . E c Therefore, the vectorial differential wave equation for the electric f i e l d is V 2 E + V l J V e . E l = u e ̂ | - i f o . V E - (Vn-V)*!- (7) bt Similarly, the vectorial differential wave equation for the magnetic f i e l d is V 2H +V(f Vix.s) = v£% - ifVc.Va - (Ve.V)sl. * & t (8) At this point sufficient theory has been developed to establish a necessary condition for the existence of an E-mode in a waveguide and a dual condition for the H-mode case. 2.21 E-Mode Condition i If the permeability and permittivity are continuous and sufficiently differentiable for a l l interior points in a waveguide, then for an E-mode to exist where z is the coordinate in the direction of propagation. Proof: From equation (8) i t can be seen that the scalar equation obtained when the coefficients of the component vector in the z direction are equated i s where H z is the z component of H0 Since for an E-mode )z H - 0, z - 9 then dz J V M ] + J V . . g - o . 11 The dual for this condition is the following one. 2.22 H-Mode Condition If the permeability and permittivity are continuous and sufficiently differentiable for a l l interior points in a waveguides, then for an H-mode to exist (10) Proof: The proof for this condition is the same as for the previous condition except that equation (7) is used instead of equation (8), and also. instead of H - 0 z - where E z is the z component of E» 2.23 Example of Use of a Mode Condition The mode conditions can be used to determine certain restrictions which must be imposed upon the fields before E- and H-modes can exist in a waveguide f i l l e d with an inhomogeneous medium. An example which demonstrates this use is the case where an E-mode exists i n a waveguide which is f i l l e d with a medium having a permeability tha-J; is constant and a permittivity that i s sufficiently well defined and satisfies the equation e = f(r,z) (11) where r is the radial parameter0 For a waveguide f i l l e d with a medium which behaves in this manner, from the E-mode condition U oH A Oz = or in cylindrical coordinates ^ e ^ r + 1 $ e ^ + de ^ _ 0 (12) dr dz + r 57 T i + dz T i ~ ° U 2 J where is the radial component and i s the angular r component of S« Since the permittivity does not have angular dependence, Also, for an E-mode de - n 37 - 0 < H - Oo (13) z — Hence, from equation (12) de dr b z > dH - „ r - 0. Since d r ~ then for an E-mode to exist in the waveguide ^ H r = Oo (14) dz ~ As a result of identity (14), further restrictions on the fields can be found through the use of equations (l) and (2)o Prom equations (1) and (2) 1 dE z dEl_ b ^ r ~S7 ' bz ~ " dt ' . bH bH/ bD 1. z p _ r r I ? " bz ~ bt ' and (15) bE„ bE„ OB/ i * I 0 E v bB^ i § ? < * . V - i T f — * f • ( 1 7 ) (18) bH bH, bD / i > , bH bD i I? <r V - J -si. - -rf ( 2 0 ) where E r is the radial component and E^ is the angular component of The substitution of identities (13) and (14) into equation (19) gives t If the f i e l d varies in time as e J W \ with u> being the frequency in radians per second, then E / - Oo (21) Now, i t can be seen from equations (15), (16), (17), (18), and (20) that i T7 = - ^ E T > < 2 2 ) bE bE bi-Ti- ' ( 2 3 ) b E r I 0 , (24) bl ~ bK, - r f = > e E r ' ( 2 5 ) and (rHv) - ± M T = jo>eE . (26) r dr p r J z If equation (22) i s differentiated with respect to z and identity (14) is substituted into the resulting equation, then * 6 E z = 0. (27) "5z ~£T? Through the differentiation of equation (25) with respect to p1 and the use of identity (24), i t is found that b ^ : o . (28) "bp7 b z ~ If identity (14) is integrated, H r = f^r . t f ) , (29) and i f identity (24) is integrated, E r = g 1(r,z). (30) 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 = f2 ( r'*^ + g 2 ( r ? z ) o ^ 3 1 ) Similarly, identity (28) can be integrated to give = f 3(r,j^) + g3(r,z)» (32) Prom the substitution of expressions (29) and (31) into equation (26) r fe (rV = r + ^ t ( x 9 z ) f 2 ( v f f i ) + j«e(r,z)g2(r,z) = s 1(r,^) + s 2(r,p\z) + s 3(r,z) (33) where S i ( r ^ ) = r — : ' s 9(r,jf,z) = otoe(r,z)f 2(r,^) t (34) s,j(r,z) = jwe(r,z)g 2(r,z). and However, from equation (32) where and = t^lj) + t 2 ( r , z ) (35) t 2(r,z) = i | j ( r g3 ( r' z )) * Since Ĥ  must behave in the manner described in equation (32) the functional form given for r dr p' in equation (33) must be in agreement with equation (35). Such an agreement is f u l f i l l e d only i f or i f s 2(r,^,z) = s 2(r,z) (36) s 2(r,0\z) = s 2 ( r , ^ ) 0 (37) Equation (37) cannot be satisfied because e = f(r,z)o Hence, equation (36) must be satisfied. Therefore,.from equation (34) i t can be seen that i t is necessary that fgU.jrf) = f 2 ( r ) . With this being the case, from equation (31) E becomes z E z = f 2 ( r ) + g 2 ( r , z ) o ( 3 8 ) b E z ~ 0, (39) Therefore, "b7 and thus from equation (22) H Z Oo (40) r - Furthermore, i f equation (23) is differentiated with respect to £ 9 from identities (24) and (39) u_^£ - Oo (41) It can be concluded from identities (24), (39), and (41) that the fields have no angular dependence0 A point to note is that the restrictions 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 restrictions result from the E-mode condition because in equation (12) H "- 0 z — which forces be ^ H z - 0. bz b z and thus a l l terms in 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 permittivity- is a function of z and the permeability is constant w i l l be considered along with the interchanged case. For these problems the unified differential equations, which result from the vectorial differential wave equations, are sufficiently separable. If the permittivity i s a function of z only and the permeability i s constant and Vl* = O O Hence, equation (7) becomes V 2E +V(i ft N = ,egf , (42) v-/2n d 2 H A l ' d e oB 1 d e V H . (43) V H = ne — + e a i ^ - e a i * and equation (8) becomes I dt Similarly, i f the permeability is a function of z only and the permittivity is constant, Ve = 0 and Vu- = djij £ o dz Hence, equation (7) becomes and equation (8) becomes 3.2 Differential Field Equations, for Transverse Waves For transverse waves E = H - 0. z z - Therefore, equation (42) becomes V 2 E = (46) and equation (45) becomes V 2H = ne ̂ | . (47) For the case where the permittivity is dependent on z, the f i e l d equations can be determined by f i r s t solving for the two f i e l d components in equation (46) and then by using equations (l) and (2) in the usual manner. Hence, i f the rectangular components of the f i e l d are to be determined, the partial differential equations b 2E V \ = lie — f (48) Ot and d 2E V \ = ^ "^2 (49) have to be solved where E^ and Ê . are the components of E in the x and y directions respectively. Similarly, for the case where the permeability is dependent on z, the f i e l d equations can be determined by f i r s t solving for the two f i e l d components in 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 partial differential equations x ^ ~ ^ 7 2 and V^H = |xe 1 (50) x dV v 2 \ = ^ % ( » ) have to be solved where H and H are the components of H x y in the x and y directions respectively. Equations (48), (49), (50), a n d (51) have the general form V 2 G = u.e (52) d t 2 where is a function of z. Consequently, equation (52) wi l l be considered for the remainder of this section. Equation (52) can. also, be written as V7 2 R . d 2G d 2G where \^.2 denotes the part of which operates in the transverse plane of a rectangular coordinate system. If the fields vary in time as e1* , then G can be expressed as G = G 0 ( x , y f * ) e J t t t t and thus \ Gn + 1 = > G . (53) t o ^ Z 2 ** o The variables can be separated by letting G q = P(x,y) T(z). (54) Once equation (54) is substituted into equation (53), 2„ „ 2 dz where M i s i>he separation constant. Hence, \^.2F + M2P = 0. (55) Equation (55) is the ordinary differential equation confronted when the transverse dependence of a transverse wave in a homogeneous medium is investigated. As well as equation (55), the differential equation ^ | + (a>2|ie - M2)T = 0 (56) dz^ has to be solved to obtain the solution for the transverse waves under consideration^, For plane waves equation (56) is slightly simpler due to the fact that for plane waves V t 2 E x . V t \ - V t 2H x . \ \ E o • which forces M = Oo Consequently, equation (56) becomes 2 ^ + «2|icT = Go (57) dz 3 o 3 Differential Field Equations for Waves with Longitudinal Components When the fields have longitudinal components, the wave solutions can be found by f i r s t solving for the longitudinal f i e l d components and then by using equations (l) and (2)o This section is concerned with the differential equations arising in the solution of the longitudinal f i e l d componentso When the permittivity is dependent on z, from equation (45) the longitudinal component of the magnetic f i e l d must satisfy the differential equation V 2H = |« jffs (58) a t 2 23 and from equation (44) the longitudinal component of the electric f i e l d must satisfy the differential equation v z oz 1 e dz zl r ^ 2 Equation (58) is the same type of differential equation as equation (52), and thus can be treated in a similar manner. However, the restriction that \^ must i- operate in a rectangular coordinate system no longer applies. Equation (59) can be simplified by replacing 1 > E by —D . In terms of D equation (59) becomes e z le zl bzle2 dz z> r ,̂2 b2D bV or 0 0 V+2D + - i f£ 5-£ = u-e . (60) t z £ 22 e dz bz >*. b t2 The variables can be separated by letting D = P ( x , y ) T C » ) e > t . ( 6 1 ) If this expression for 0 is substituted into equation (60), z the result i s I v t 2 p = 4 d"T 1 de dT ,2 e dz dz CLZ tu2jxe = -M2 where once again M i s the separation constant. The equation \^.2F + M2F = 0 (62) i s identical to the equation which contains the transverse dependent part of the longitudinal component of the f i e l d in a homogeneous medium and can be solved for a number of boundary value problems using well known techniqueso The equation for the z-dependent part of can be written as g - i f e g + - m 2 » t = »• < « > When the permeability is dependent on z, from equation (46) the longitudinal component of the electric f i e l d must satisfy the differential equation d2E S}\ = lie — f . (64) ot and from equation (47) the longitudinal component of the magnetic f i e l d must satisfy the differential equation V V ^ & B . h " - ^ . .<«> Equation (64) i s the frequently occurring type of differential equation given in equation ,(52), and thus can be treated accordingly. The only change i s that \ ] 2 i s no longer restricted to operate in a rectangular coordinate l system. In the same manner as equation (59) was simplified by using D , equation (65) can be simplified by using B . Prom equation (65) i t can be seen that B can be expressed z i n the form B7 = F ( x 9 y ) T ( Z ) e J W t w i t h F ( x , y ) s a t i s f y i n g e q u a t i o n (62) and T ( z ) s a t i s f y i n g dl _ 1 M dT + ( w2 _ M 2 ) T = Q o ,2 u dz dz r dz r 3o4 Summary o f z-Dependent E q u a t i o n s I f by d e f i n i t i o n q ( z ) = fie, t h e n the d i f f e r e n t i a l e q u a t i o n s c o n t a i n i n g the z - d e p e n d e n t p a r t o f the s o l u t i o n f o r the wave f u n c t i o n s c o n s i d e r e d i n s e c t i o n s 3.2 and 3.3 can be summarized by the f o l l o w i n g 'i t h r e e d i f f e r e n t i a l e q u a t i o n s . 2 + o>2qT = 0, (66) ^ | + ( a > 2 q - M 2 ) T = 0, (67) dz and d z2 q dz dz ; 1 u ' 3.5 O s c i l l a t i o n Theorem Due t o Sturm E q u a t i o n s (66) and (67) can be e x p r e s s e d i n the form 2 + h ( z ) v = 0, (69) dz and equation (68) can be transformed into the form of equation (69) by the following transformation. I f T i s expressed as T = etfq dt d z W = e * l n 4 V = ^ ¥ , then equation (68) becomes 2 2 - 2-i 2 + dz^ 2 w2 , 1 1 O . 3 1 fdjgf ^ ~ 2 q T~2 ~ 4 ~2 Id3 -̂ dz q ¥ = 0 (70) which i s of the form of equation (69). Owing to a theorem by Sturm, i t i s possible to show that for a physically r e a l i z a b l e situation the solutions for equation (66) are o s c i l l a t o r y , and the solutions for equation (67) are o s c i l l a t o r y provided w2|xe - M 2>0. Besides t h i s , the theorem offers a possible test for showing whether or not the solutions for equation (68) or (70) are o s c i l l a t o r y . Theorem: The functions u(z) and v(z) are the respective solutions of the d i f f e r e n t i a l equations ^ § + g(z)u = 0 (71) d z Z ^ | + h(z)v = 0 (72) dz 3. L.R. Ford, D i f f e r e n t i a l Equations. McGraw-Hill, Inc., New l o r k , 1955, p. 169. in an interval in which the coefficients of the equations are continuouso If 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: First of a l l , 2 2 d | . du dvl __ d u d v d z l v dz " u dzJ - v - u dz' dz' = (h - g)uv which after integration becomes z=b b̂ du dz u dv dz (h - g)uv dz. (73) z=a a Now, the supposition that v(z) has no root between a and b is made. Without loss of generality u(z) and v(z) may both be considered positive in the interval a^z<b; either one can be replaced by i t s negative, i f necessary. Consequently, and du(a) dz du(b) dz > 0 < 0 , Hence, du _ dv dz dz z=b z=a = v(b) - v( a) ^ a i a l ^ o , dz dz However, the right hand side of equation (73) i s positive. Therefore, a contradiction exists, and thus the theorem is proved. Prom this theorem i t follows that i f the solution to equation (71) is oscillatory over some interval, 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, oscillatory over the same interval. For a physically realizable medium |ie^k = Constant >0 where normally ro o' and for the non-trivial cases to be considered p,e •£ k. Since the solution for ^ | + u>2kT = 0 is oscillatory, the solution for equation (66) i s , also, oscillatory. 2 9 In a similar manner, vhen w2|Ae - M 2^k' = Constant>0, (74) then since the solution for d 2T + k'T = 0 dz^ is oscillatory, the solution for equation (67) i s oscillatory. 4. FIELD PROBLEM IN A PERIODIC STRUCTURE LOADED WITH DIELECTRIC DISCS 4.1 General As mentioned in the introduction, the topic of this thesis arose from the problem of finding the vave functions for a periodic structure of the type shown in Figure 1 Section 4 w i l l expand upon this problem through the use of the fact that inside the waveguide the medium as a whole is inhomogeneous. The permittivity is a function of the spatial parameters and the permeability is constant. 4.2 Functional Behaviour of the Permittivity Before the unified differential wave equations can be dealt with, the functional behaviour of the permittivity must be specified. Since the permittivity in the interior of the air regions equals e Q and in the interior of the dielectric regions equals e ^ , the functional form of the permittivity minus e Q approaches the product of a rectangular wave variation in the z direction times a step variation in the radial direction. Consequently, i f cylindrical coordinates are used, the permittivity can be expressed as e - e Q = h(r)g(z) where h(r) and g(z) are sketched in Figure 2. The reasons that the curves in Figure 2 are shown as continuous and smooth are given in the following paragraphs. Before the theory so far developed can be applied to the f i e l d problem in a periodic structure loaded with dielectric 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 interior points in the waveguide. The following argument is given to justify this assumption. g U ) e l ' e d cr h(r) a Fig. 2. Sketch of Functional Variations of Permittivity At a l l points except those in the regions of the boundaries between the air and the dielectric medium, there is no doubt as to the existence of the permittivity and a l l its derivatives. If in the neighbourhood of the boundaries 32 the point of view of mathematical physics, which is that matter is continuous, is 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 satisfies the assumptions made about the behaviour of the permittivity. It is worth noting that the step function used to describe the boundaries in 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 11 provided m is small but f i n i t e . For equation (75) s = Coordinate in the normal direction to the boundary 4. A.G. Webster, Partial Differential Equations of Mathema- ti 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 0 , and for s>a, e - * - e ^ . In the limit as m-»-0f with an increasing s, the permittivity changes discontinously through the boundary from e Q to e ^ . 4.3 Field Restrictions Due to E-Mode Condition Now that i t has been assumed that the permittivity behaves sufficiently well to ensure that the theory so far developed i s applicable, the material in section 2 can be used to find the necessary conditions for the existence of E-modes in the periodic structure shown in Figure 1. When the dielectric discs have center holes, e sr h(r)g(z) + e Q = f( r , z ) . Therefore, the restrictions on the f i e l d components found in section 2.23 must hold. If these restrictions are not met and a hybrid mode results, i t is 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 permittivity has no radial dependence, and thus, the E-mode condition is satisfied without imposing any restrictions on the f i e l d components. 4.4 Unified Differential Equations When the discs have center holes, the unified differential 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 H - 0 , r - and the f i e l d components have no angular variation, the unified differential equations are d_ dr and d dr mS44 r drt dr I ^2 dz d2E r d dz 1/dl r dr ( r E r r dr p' , 5 — E + §r E e i dr r oz • z< i f e - E + & E ) e I Or r dz zJ d 2E z d t ^ d2H I d2H 2 = [ie - ~ d z 2 dt 2 e d e dr d r r/ d 2E 3 d"? de ̂ f L dz d z . As previously mentioned, the techniques applied so far to these differential equations, as well as the ones arising in the H-mode case, have not yielded general solutions. However, numerical methods could be devised to calculate specific solutions for these differential equations. S i n c e general solutions have not been found, these differential equations w i l l not be considered further in this thesis. For the case where the discs are solid, the 35 permittivity is a function of z only. Consequently, the unified differential equations for the longitudinal f i e l d components can be separated as was seen in section 3o Also, the complete wave solutions for this case can be readily solved, as done in Appendix 1 for the EQ^-mode, by determining the fields in the dielectric regions and in the air 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 satisfy the unified differential equations in the limit as the permittivity approaches a rectangular waveshape, the viewpoint taken in this thesis would be verified as applicable for finding the wave equations to the solid disc case. However*, this was not shown in 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 is to find an infinite series expression for the permittivity which converges absolutely and yet has an nth order term which is manageable. This d i f f i c u l t y is discussed by L. Brillouin in "Wave Propa- 5 gation in Periodic Structures". Secondly, even i f such a series is obtained, since the known wave solutions are expressed in terms of infinite series in z, single and 5. L . Brillouin, Wave Propagation in Periodic Structures, Dover Publications, Inc., New York, 1953, p. 186. 36 double infinite series appear in the z-dependent part of the unified differential equations. This can readily be seen from the equations derived in section 3. Consequent- ly, the problem of establishing that the coefficients of the z-dependent series in the known solutions satisfy the unified differential equations is very awkward. However, when the dielectric regions are matched into the air regions, i t is possible to find the wave solution for an E-mode from the unified differential equations. As w i l l be shown, this solution agrees with the solution obtained in Appendix 1. 4.5 E-Mode Solution of Unified Differential Equations for Matched Case For the case where the dielectric discs are solid and matched into the air 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 differential 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 the permeability is constant, the z-dependent portion of D can z be found from equation (63), d ? r _ i de a r + ( w 2 _ M 2 ) T = O o j 2 e dz dz dz The cross section of a periodic structure with solid dielectric discs is shown in Figure 3. For such a structure 37 the permeability equals u Cylindrical Waveguide . r 1 z _ z : Y Z Z Z Z Z Z T Didlectri Disc Center Line Region (1) i i i ' I i TT ^-Origin g i i O r i g i n 1 i I i f I i i ' V , ' . y / / / 7-7 z (5) " 7 V -7-7- .V, 1 1 2 Z Z = Permittivity of dielectric medium c Q = Permittivity of air Note: For simplicity Origin 1 is used in Appendix 1 and Origin 2 is used in the body of the thesis. Pig. 3. Cross Section of Solid Disc Periodic Structure Since the permittivity very closely approximates a rectangular waveshape in the z direction, i t follows that 2 2 2 2 eo n Qe - M and k e , k being a constant, also, very closely approximate rectangular waveshapes as shown in Figure 4. For the matched case 38 2 M2 ' O w2u e,-M2 r o- 1 — <o2u e -M2 " O 0 ->- z 2 2 2 2 F i g o 4. Diagram of (D u^e-M and k e Versus z Consequently, when a match exists, the identity u)2u e-M2 •- k 2 e 2  ro - (77) holds except in the transition region at a boundary between an air region and a dielectric region., The transition 39 region i s defined by (5 in Figure 4. As (5 tends to zero, identity (77) tends to hold for a l l values of z. Consequently, the differential equation d 2T 1 de dT 2 2 T n , _ R v ^ 2 " F d 7 d i + k e T = 0 ( 7 8 ) approximates equation (63) to any required degree of accuracy for the matched case. The solution for equation (78) is = A i e - j / k e d z+ A 2 e j / k e d ! where A^ and A 2 are arbitrary constants. Through the use of identity (77), T becomes -j/^ 2u, oe-M 2 d 2 + a ejJ\4>2,xoe-M2 dz. (79) T = A^e ^ . - 2 As (5approaches zero, the solution for T given by equation (79) approaches the solution for the entire waveguide. -i/~2 2 In the limit V(o |xQe-M approaches a rectangular waveform, and thus for the limiting case the integral /V w 2|i e-M2 dz can be evaluated graphically by integrating the rectangular waveform as illustrated in Figure 5. From Figure 5 i t can be seen that, in the limiting case j V c A e-M2 dz = S z + X(z) 40 where S Qz = Ramp function and X(z) = Periodic function oscillating about the ramp function, >- z 41 The slope S Q of the ramp can be found in the following manner. Prom Figure 5 at z equals / V a ^ e-M 2 dz = £=3A/u,2lioeo-M2 and thus at z equals \ H\0*-*2 dz = ^ 2 , Q e o - M 2 + [* - ^ f f L \ ^ S p o^ Hence, S = \4 2n e -M2 + ^l\L2ti e,-M2 - Vto2u e -M2 |. o ro o pi , 0 1 ' O O / ¥hen only the incident wave is present, T - V - f l S . » + X U > ' • and thus D r J L P d ^ l * * - ^ " ^ (80) Z J. where r is the radial and the ^ the angular variable 0 In the limit as the permittivity approaches a rectangular waveform, i t can be seen from Figure 5 that in an air region z and in a dielectric region - i / V w 2 L i e,-M2 dz - j ( V t o 2 M e,-M2] z = D a e ^ *o 1 = e ° I ro 1 ' These results coincide with the results obtained by- solving the wave equation in each of the homogeneous regions separately. To check that the solution given by equation (80) has the same phase shift per section as found in Appendix 1, the expression for T(z+p) should be considered, namely. T( Z +p) = A 1 e - j ( S o ( z + P ) +*< Z +P)) 0 Since X(z+p) =*(z), then T(z+p) = A i e - J S o 4 " 3 ' l S o Z + X i z ) \ = e"J So p T(z). Hence, the phase shift per section 0 is given by 0 = S op = (p-q) V w 2 { i oe o-M 2 + q V ^ u ^ - M 2 . This i s the same value for 0 as is found in Appendix 1. Therefore, since in the limit the f i e l d in each section behaves in the same fashion as i t was found to behave in Appendix 1 and the phase shift per section is identical to the value found in Appendix 1, the two approaches are in agreement. When the limit 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 in the limit 9 s — s o o X ' U ) — ~ X { z ) The phase velocity for D can be approximately determined by z differentiating 9 cot - S z - XT(z) = C o n s t a n t . o ' Hence, the phase velocity v^ is dz ~ to P " At " s»+-JJC o dz Consequently, the phase velocity is modulated by the periodic term d ) C ' .' dz 5. PLANE WAVES IN A MEDIUM WITH PERMITTIVITY A CONTINUOUS FUNCTION OF z 5.1 Introduction To obtain a better understanding of the behaviour of electromagnetic waves i n an inhomogeneous medium, i t was decided that an investigation should be made into the behaviour of plane waves i n a medium with a perm 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 d i f f e r e n t i a b l e function of z. This problem tends to be simpler than ones dealing with longitudinal f i e l d components. At the same time, the techniques used i n solving the d i f f e r e n t i a l equation a r i s i n g from plane wave considerations have only to be s l i g h t l y modified for 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) with equation (66). 5.2 Problem In p r i n c i p l e a complete solution for a plane wave can readily be obtained for any s u f f i c i e n t l y well behaved z-dependent functional form of the p e r m i t t i v i t y . Various pa r t i c u l a r forms were considered, and i t was found that the form e = l ^ z 2 8 " " 2 +^2 (81) 1 2 z where = Constant k„ = Constant and s = VI - 4co 2 u Q k 2 , 0< s < c l , (82) y i e l d s s o l u t i o n s w h i c h are e a s i l y i n t e r p r e t e d i n p h y s i c a l termso C o n s e q u e n t l y , a p e r m i t t i v i t y s a t i s f y i n g e q u a t i o n (81) w i l l be u s e d . A s k e t c h o f the p e r m i t t i v i t y e x p r e s s e d i n e q u a t i o n (81) i s g i v e n i n F i g u r e 6. A l s o , f o r the medium 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 e q u a l s n Q. e A F i g . 6. S k e t c h o f P e r m i t t i v i t y V e r s u s z F o r a p l a n e wave w i t h from equation (57), since E = T, xo ' d 2E dz k 22s-2 + _| 1 z\ E = 0 xo (83) where E x q i s the component of the e l e c t r i c f i e l d i n the x d i r e c t i o n w i t h time dependence suppressed. Equation (83) can be solved i n the f o l l o w i n g manner. F i r s t of a l l , the transformation u = — z =pz , p = a>.yjTk (84) i s made. From t h i s transformation dE xo dz -P sz , dE s-1 xo du and d 2E o o o o d 2E - dE — ^ a ' = p 2 s 2 z 2 s " 2 —Jsa + p s ( s - i ) z s - 2 ^ dz du u (85) Consequently, by the s u b s t i t u t i o n of expression (84) i n t o equation (85) „ . d 2E dE « 2 2 xo , / ', \ xo . , 2 du k^u P 2" + k. E X Q = 0. (86) If E x q is transformed into E = u 2 s I, xo 1 (87) then and dE xo 1_ 2s dY l-2s du = u " « + J - u  2 s Y du 2s (88) d 2E xo u2s d£l + 1 u 2s dY + 1 d u2 s du 2s I 2s l-2s du' l-4s II u 2 s Y, (89) If expressions (87), (88), and (89) are substituted into equation (86), the result is 2 d 2 I x dY . , 2 l w n u ^2 + u dH + ( u ° 4 ) Y = 0 which is Bessel's equation for n = 2„ Hence, Y = C ^ U ) + C 2J 1(u) 2 "2 with and being arbitrary constants, and thus Exo = ^ N 1 J 1 ( P Z 8 ) + N 2 J 1 ( p z s y 2 ~2 where and N 2 are arbitrary constants, Since and J± (u) J x ( u ) T I U sin u nu cos u, 48 E = \£ xo u N. 1\ „n s s i n P z S + N 2 \ f ^ l c o s P or 1-s E z 2 xo VTI/O A , e ^ P z S + A 0e^P z S l 1 2 (90) where A^ and A 2 are arbitrary constants<> Hence, r A i e 3 ( - t ^ z s ) + A 2 j ( ^ z s ? E = E e ^ * = U , 2 x xo HP (91) Consequentlyo the incident electric wave i s 1-s TT _ /iT 2 . ej(tot-pz S) E x i =\f^p z A l e H ' and the reflected electric wave is xr ~V*P Since ^ E - - |5 , bE b B "ST = Tt = j^o Hy< Therefore . e J w t fT [ i-s ^e'P** + A 2 e ^ z S + z 1-s 2 gpsz 8" 1 A i e " j P z S + j p s z 8 " 1 A 2 e J P z ! 49 or S S i 1-s 2 . n 2 z -jpsz s S 1 1-s 2..~ 2 — z +opsz (92) Hence, the incident magnetic wave is H = - i - . f l i y i to>jxo Vrtp y/z _s_ s. 1-s 2 .~ 2 -~2~ 2 " O p s z V J < " t ^ " ) and the reflected magnetic wave is H _ _ i _ . /_2 I . yr a)ji o V*p ft s l^s 12 2 z " + j p s z Consequently, the wave impedance seen in the medium by the incident wave is Z . = E . xi fa>jXoZ oi H . ~ s .. I 1-sl ' y i p s z — and by the reflected wave is E Zor ~ ~H xr = z . r, S • f 1 -S \ Oi ° yr p s z ' It can be noted that provided z>0 the imaginary part of ZQ^ is negative, implying that the reactance is capactive, while the imaginary part of Z Q r i s positive, implying that the reactance is inductive» The phase velocity of the incident f i e l d can be calculated by letting a>t - p z = Constant (93) and by differentiating equation (93)o If this is done. P s—1 dz /•» sz a£ = Oo Hence, the phase velocity v is dz «o 1-s /0/J\ v p = d t = / > z ' ( 9 4 ) Since 0<s the phase velocity increases as z increases. If an E-mode f i e l d can be set up such as to have a phase velocity increasing with z, the f i e l d may possibly be very useful in pre-accelerator applications. Equation (94) tends to point towards the possibility of obtaining such an E-mode f i e l d through the use of an inhomogeneous medium of this 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 this section is to demonstrate this use for the case where the inhomogeneous slab has a permittivity that is functionally described in equation (81). In Figure 7 the f i e l d i s assumed to originate in medium 1, pass through the inhomogeneous mediufho and enter medium 2. Also, i t i s assumed that the f i e l d in medium 29 i s totally absorbed, none of the energy being reflected back toward the source» In medium 1 the f i e l d equations are E x l = N^^-P.!** + N 2 e J ( w t + P l z ) and y l V u- where P i = w V v on and N^ and Ng are constantso The equations representing Surface 1 Inhomogeneous — Medium z Fields are and H Fig. 7» An Inhomogeneous Slab Between Two Homogeneous Media 6. So Bamo and J 0R 0 Whinnery, Fields and Waves in Modern Radio» John Wiley & Sons, Inc 0, New York, 1953, p»28lo the fields E and H in the inhomogeneous medium are given x y by equations (91) and (92). Since only the transmitted wave is present in medium 2, the f i e l d equations are E x 2 = C leJ ( w t"P2 z ) and _./!2 n i(«t-e.,*> H _ =J— C^e y2 V | i 0 i where P 2 = W ^ 2 and is a constant< At z equals a in Figure 7S and At z equals b, and E , = E (95) xl x V - V ( 9 6 ) E x - Ex2 ( 9 7 ) H y = H y 2 . (98) From equations (95), (96), (97), and (98) the values for and N 2 can be determined, as done in Appendix 2, from the parameters of the media, the frequency, and the constant Ĉ <, Once and N 2 are known, the reflection coefficient 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 reflection at surface 1 in Figure 79 a l l the energy can be transferred from medium 1 to medium 20 For this situation to occur, R = 0„ As shown in Appendix 2, in order that R = 0, two equations must be satisfied. These equations are V^ jV^ a 8 " 1 - V^i b ^ c o s ( b s „ a S ) 1-s a V j i 0 b V | i Q / ( b s - a 8 ) (101) and l v / ^ l 1-s / b 8 " 1 s-1 cos- k ^ a b ) s-1 2 u < o u , a b 1 - s 2 ( b s - a s ) s i n ( b s - a s ) (102) T h i s means t h a t two o f the p a r a m e t e r s must be d e t e r m i n e d by- e q u a t i o n s (101) and (102)o Due to the p e r i o d i c i t y o f e q u a t i o n s (101) and (102), t h e s e two p a r a m e t e r s , e x c l u d i n g and e 2 , have an i n f i n i t e number of d i s c r e t e v a l u e s . S i n c e e q u a t i o n s (101) and (102) are not p e r i o d i c w i t h r e s p e c t t o and e^, and e 2 have one s o l u t i o n each i f t h e y are d e t e r m i n e d by t h e s e e q u a t i o n s . I f f o r a p a r t i c u l a r f r e q u e n c y two o f the r e m a i n i n g parame te rs are e v a l u a t e d by e q u a t i o n s ( l O l ) and (102) and the o t h e r s are a s s i g n e d c o n v e n i e n t v a l u e s , the matched c o n d i t i o n a t s u r f a c e 1 i s e s t a b l i s h e d . I f t h e p e r m i t t i v i t y o f t h e inhomogeneous medium i s s p e c i f i e d , the f i e l d s o l u t i o n s o b t a i n e d f o r the inhomogeneous 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 . The r e a s o n i s t h a t once the p e r m i t t i v i t y i s s p e c i f i e d , k ^ , k 2 , and s have 55 fixed values, and thus from the equation = V l~4w 2 u k o"2 the frequency i s determined. Therefore, when the permittivity is specified, the frequency cannot be evaluated from equations (lOl) and (102). However, f i e l d solutions for an inhomogeneous region which hold for any frequency after the permittivity is specified can be obtained quite readily. For example, the medium with a permittivity behaving as e = k^z where and k^ = Constant s = ™ (n—X y 3 p 5 o o o o o o Sn^X} has such f i e l d solutions. 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 is a continuous function of z, three possible methods have been considered,, The f i r s t method is to construct the medium by using thin sheets of homogeneous dielectric material<> If thin sheets having different values of permittivity are cemented together in some desired order, the resulting laminated dielectric medium varies functionally with z<> Provided the thickness of the sheets is small compared to the wavelength of the f i e l d being propagated in the medium and provided the change in dielectric constant between adjacent sheets i s small, i t is believed that the medium as seen by the f i e l d effectively varies in the desired continuous manner with z. Another possible method for manufacturing a medium with a permittivity which is a continuous function of z is by varying the density of the medium in the z direction,. For example, the porosity of the medium could be varied in the z direction as in the case of some types of foam rubber. It might be possible to use a centrifugal process while the dielectric material is solidifying to establish a variable density. Through the use of plasmas, a medium can be obtained with a permittivity which varies continuously in the z direction. 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 in 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 sufficiently differentiable spatially' dependent permeability and permittivity, two vectorial differential wave equations, V 2 E + V £ Ve.E) and were derived from Maxwell's equations<> The magnetic vectorial differential wave equation was used to find the necessary condition, / iv^H) + i V e 0 ^ = o, dz \ \i r I . e oz for an E-mode to exist in a waveguide, and the electric vectorial differential wave equation was used to find the necessary condition, for an H-mode to exist in a waveguide0 Through the use of the E-mode condition, an investigation was carried out to determine certain restrictions which must be imposed upon the fields before E-modes can exist in a waveguide f i l l e d ^ T § ~ JIViiVE - (Vn-.V)E bt lie ^ - J ( V e . V H - (Ve.V)H) b t ' with a medium whose permeability is constant and permittivity i s e = f(r,z)o The restrictions were found to be that *V z <>> and the fields have no angular variation. Also, i t was noted that the E-mode condition did not impose any restriction 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 restrictions imposed upon the fields when the permittivity has other functional variations is suggested for future study along with a complimentary investigation using the H-mode condition. It is worth pointing out that these restrictions are the duals to the restrictions on the fields for the interchanged cases, namely, the cases having the permeability spatiallycdependent and the permittivity constant. An investigation into the f i e l d restrictions 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 is constant and the permittivity varies with z and for the case where the interchanged situation is true. In particular, the f i e l d equations for transverse waves and waves with longitudinal components were considered. After the variables of the pertinent unified differential f i e l d equations were separated, the differential 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 for homogeneous media. For the different cases considered, the z-dependent part of the f i e l d components were found to satisfy one of the following differential equations; i) ^ § + o>2qT = 0, dz i i ) & + ( t t2 q _ M 2 ) T = Q > dz z »*> 7§ " \ ft S + <*2*-m2>t - 0 dz where M is 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 for the f i r s t of these equations in T i s oscillatory, and the solutions for the second one is oscillatory provided W 2fA6 - M2 >0o Besides this, the theorem offered a possible test for showing whether or not the solutions for the third equation are oscillatory,, The fields in a dielectric loaded periodic structure were considered from the viewpoint that the medium as a whole is inhomogeneous inside the waveguide0 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 interior points in the waveguide was discussed from the point of view taken in mathematical physics, which is that a l l matter i s continuous„ After this assumption was made, an investigation into the restrictions on the fields when E- modes are present was carried outo For the case where the dielectric discs have center holes, since there is a radial variation in the permittivity as well as a longitudinal variation, the restrictions were recognized to be the same as those discovered for the example where e = f ( r , z ) o When the discs are solid, i t was noted that the E-mode condition is satisfied without imposing any restrictions on the fieldso For a periodic structure with solid dielectric discs the theory developed for inhomogeneous dielectric media was used to find the E-mode f i e l d expressions when the dielectric regions are matched into the air regions 0 In the limit as the permittivity approaches a rectangular waveshape, these f i e l d expressions were shown to be in agreement with the expressions derived by solving for the fields in each of the homogeneous regions and by matching the fields at the boundaries» It i s f e l t that further effort should be made to use the theory for inhomogeneous media to find the f i e l d equations for the solid dielectric 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 limit is known, there should be some method for showing that this known solution satisfies in the limit the differential wave equations resulting from the theory for inhomogeneous media. If this problem could be solved, i t may shed some light on how to solve the f i e l d problem using the theory for inhomogeneous media when the dielectric 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 interest. Also, i t is f e l t that a further attempt should be made to find accurate and manageable f i e l d solutions for the case where the dielectric discs have center holes. It has not yet been possible to attempt a thorough investigation of this problem. An investigation was carried out into the behaviour of plane waves in a medium whose permittivity is e = k , z 2 s ~ 2 + ^2 (103) 1 2 z where and kg are c o n s t a n t s and s s Vx - - 4a> 2 u ok 2. 0 < s < X The e X e c t r i c and magnetic f i e X d s were caXcuXated t o be X - s and H - i - X i _S S i 1-s ~2 .~ 2 — z -opsz _s X^s _2 I 2 z" + apsz A 2 e J ( w t H P z S > where ; s The wave impedance seen i n t h e medium by the i n c i d e n t wave was shown t o be z , = toty 01 and the wave impedance seen i n the medium by the r e f l e c t e d wave was shown t o be Z o r ^ o z = z * s . | l - s \ ' o i The phase v e l o c i t y was caXcuXated t o be Also, a slab of dielectric 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 brief discussion i s made on the possible methods of manufacturing inhomogeneous media with a constant permeability and a permittivity varying in the direction of propagation., APPENDIX 1* For a circular waveguide loaded with solid dielectric discs as shown in Figure 3 , page 37, the f i e l d patterns for an Eg^-mode can be determined by matching at the boundaries the fields found in each homogeneous region 0 Through the use of Maxwell's equations, the f i e l d components for an Eg^-mode are found to be E = z -JP-.Z dP-. z A l e + A 2 6 J 0(Mr)eJ w t, (1) E = ^ r Mv, -OP-jZ jp, z Aĵ e 1 - A2e l J^Mr)^?*, (2) and 3^1 Hjrf = ~M~ -OPiZ OP, z A l e + A 2 e J ^ M r J e ^ (3) in the dielectric region (2), and E = z - j p z dp z • C^e 0 + C2e 0 J J 0 ( M r ) e > t (4) and E - 22-r ~ Mv L 1 o - D P 0 Z dp Z C,e 0 - C2e 0 J, (Mr)^"*, H 3<»>e, p1 ~ M -dP0z dP0z C i e 0 + C2e 0 (5) (6) * The treatment in this Appendix is based on the paper presented by G.B= Walker and C.G. Englefield, "Some Properties of Dielectric Loaded Slow Wave Structures", PGMTT Symposium. San Diego, May, I960. in the air region (3), where 2 2 2 P i = w ^ o e l " M ' (7) Po = » Vo"^ » ( 8 ) v.. = ^— = Phase velocity in dielectric region, 1 Pi v = ^— = Phase velocity in air region, 0 Po s l ^ = K » s 1 = First root of JQ(Mb) =0 , 7 and A^,Ag»C^fand Cg are related constants 0 The f i e l d in region (4) can be determined by using Floquet's Theorem which states that in a given mode of oscillation 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 is observed a distance g of one period down the structure. Consequently, the f i e l d components in region (4) are given by the multiplication of • A the f i e l d components in equations (l),(2), and (3) by e""*. 7. Go Ramo and J.R. Vhinnery, Fields and Waves in Modern Radio John Wiley & Sons, Inc., New York, 1953, p. 375» 8o J.C. Slater, Microwave Electronics. D0 van Nostrand Co., Inc., Princeton, New Jersey,' Toronto, London, New^York, 1950, p. 170. where 0 i s the phase change per section,, If the f i e l d s are matched at z equals 0::and^p-q, the equations obtained are V l - V o A 2 ~ V1 C1 + V1 C2 = °> ( 9 ) e 1A 1 + e xA 2 - e oC 1 - e QC 2 = 0, (10) j(2O 1-0) -j(2©,+0) -J29 j2© A l " v o e A 2 ~ v l e C l + V l e C2 = °» (11) and ••3(20,-0) -0(2©,-^) - j 20 j2© e l e h + t l * A2 " e o e C l " e o e C 2 = ° (12) where the phase change in the air region is 2© Q ^ 80(p-q) and the phase change in the dielectric region i s 261 =8^ q o Equations (4), (10), (.11), and (12) have unique solutions for three of the constants A^,A2,C^0 and C 2 in terms of the remaining constant only i f 68 - V , -e. v e o j(2© 1 - 0 ) -v e o .3(20^+0) •320, 3 20f " v l e v l e e l e jUOj-0) -3'(2©1+0) -J2© 320, -eoe = Oo (13) Prom the expansion of equation (13) an expression for 0 can be determined. This expression is 4 cos 0= + 0 1 cos (2© 1+2© Q) - Z 1 ITT z~ "vzT, o 1 ' cos(2©,-2© ) 1 o or cos 0 = cos 2© cos 29, o l Z Z, - ° 4 ~ 1 .Z,*Z^ ' 1 o' sin 2© sin 2©, o I where Zrt is the wave impedance for an E^-mode in the air region, Z = v e o o and Ẑ  is the wave impedance for an E^-mode in the dielectric region, Z, = 1 v, e 1"! Through the use of any three of the equations (9),(lO),(ll), and (12), the constants A^.Ag* and C^ can be evaluated in terms of Cgo If this is done, the 69 r e l a t i o n s h i p s o b t a i n e d are A, e c o s 2 6 o + 3'z7 S i n 2®o " e JL _ f o J0 " " "~° VZ1 G 2 " e, e +j(0-2© ) • z7 e - cos 2©̂ ^ - j^rr s i n 2© 1 Z 0 j (20 ,-0)- Ag ^ C O S 2 Q o - J f r - s i n  2 V 6 c 2 " e i 6 " T ^ o T ~"~ cos 29, - j — s i n 2©, 1 Z~ 1 a n d j(0+2© o ) Z, c e + j - ^ s i n 2© 1 - cos 2© x CT = 3*(0-2©ft) " "zl ° - c o * 2 6 l - ^ s i n 2 9 i Nov t h a t the f i e l d components have been d e t e r m i n e d over one p e r i o d p , e x p r e s s i o n s f o r t h e f i e l d components can be found which h o l d t h r o u g h o u t the waveguide. F o r example, f o r E such an e x p r e s s i o n can be f o u n d by z t a k i n g the f o l l o w i n g s t e p s . The f i r s t s t e p i s t o d e f i n e t h e f u n c t i o n •i0z E ( r , z , t ) e p F ( z ) = -r-i . (14) J ( M r j e ^ o From e q u a t i o n (14) E ( r , z+p, t ) e P e ^ F(z+p) = z J ( M r ) e J t t t and from Floquefs Theorem E (r, zip, t) = E(r,z,t)e" JJ0 Hence, F(z+p) = F(z) , and thus F(z) is a periodic function in z with a period p< Consequently, F(z) can be expressed as the Fourier sum o o -I -.i2nnz P where F(z) = / a ne n— - o o p-q .j 2rcnz a n = i-/F(z)e p dz Therefore, o o ., 0+ 2 7m _ .1 0+27tn \ , E = J (Mr)eJ t t t ^ =- 1 * J z o n=—oo where P-1 //7<, 1 fVr,»,t) • a = i / e * dz . f 1 1 P J J n(Mr)e3 w t -q u ¥hen the dielectric regions are matched into the air regions, 1 A 2 = C 2 = 0. Consequently, equations ( l ) , (2), (3), (4), (5), and (6) 71 become respectively j(o)t-p,z) E = A1Jr.(Mr)e 1 z 1 0 E r =M? 1 V l < 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 . = Z, , o 1 the phase shift per section i s given by cos 0 = cos 26 cos 2©, - sin 2© sin 2©,• 0 1 O 1 Therefore, 0 = 2© Q + 2© x , = (p-t)P0 + qB x = (p-q) Va,2noco-M2 + q V ? ^ I ? APPENDIX 2 The fields for the different regions shown in Figure 7, page 51, are: i) in medium 1 E and x l y i v |i -JB,z 30,z Nje x + N2e L -J0 - .Z 0*01 z N l e ~ N 2 e Jut i i ) in the inhomogeneous medium 1-s E z 2 x V^P V - * > » + A 2 e ^ and H _J_ G T i y"*«n0vipyz | r S S" »>"• + _ s sr 1-s 2,.n 2 -jjpa +opsz A.e3Pz' 1 i i i ) in medium 2 x2 1 and y2 Vp I where s = Vl - 4 u ) 2 j i ok 2 , 0<s<l , *2 = W#o*2 » and A^BAg.N^.Ng, and are related constants. At z equals a the boundary conditions are and E , = E xl x H , = H y i y Therefore, - J P - i a O K I N l e + N 2 e and •o 1-s = V * p a A,e"JPAS + A^P*' N l e ~ N 2 e J3 £[1 1-s 2 . n e Q2 S S i l-s_2 pa*' A, At z equals b and E = E 0 x x2 H = H _, y y2 Consequently, 1-s / 2 ' v 2 A le-JP B S .+ A^P*' = C l e (3) and _ s s_- _s s_- ^ b 2-jpsb 2 ^ b 2+ 3Y>b 2 !2 R - J P 2 B — O, e 1 (4) At surface 1 the reflection coefficient R is defined as N 2 R = | j - (5) One approach to finding R i s to express both and Ng in terms of C^. To do this, A^ and Ag must be found in terms of C, o Prom equations (3) and ( 4 ) , j(pb s - f i 2b)| , ft h = and A = £e ^ >b g^b s-rB 2b), /e * V | * o / (6) Cj (7) where * 2 p s b 2 (8) 75 and s-2 2 \ = / > * > 2 + * ( ¥ ) » > (9) From equations (l) and (2) both N^ and are solved for in terms of A^ and Ago The results are -3 (pf-Pi*) A 2 (10) and -j(pa s+p i a) ^ jOQa^S.a) / * /eTIl A 2 (11) where and i / ^ V sVuDe, a o ^ 1 2ai^ s-2 a r o ^ ^ L P s a 2 + 3 | ¥ ) a 2 J (12) (13) Now, the values for A^ and A 2 found respectively i n equations (6) and (7) are substituted into equations (10) and ( l l ) to gxve K 1 = j(B i a-8 2b) -dp(b s-a 3) e (14) and N 2 = C4°i6 =j(p1a+p2b) S _ S ) —a a w **o / I (15) If expressions (14) and (15) are substituted into equation (5). i t i s found that (16) If there i s no reflection at surface 1, R = 0< Therefore, from equation (16) b s-a s) or * , S S \ b -a ; • jp(b S-a s) (17) Now, i f the following definitions are made: i) T =p(b s-a s), i i ) 7 ^ = v a + j v a , the substitution of these newly defined quantities into equation (17) yields ( r i + i%u*r = <r 2 + w 2 ) * ~ * r o ( i s ) Once the real parts of equation (18) are equated, the resulting equation i s (1^ - f ^ c o s T = (% +%)sinT . (19) Similarly, from the imaginary parts of equation (18) (\ -%)COBT = - ( [ ^ + Q ) s i n T o ( 2 0 ) 78 In terms of the original parameters, equations (19) and (20) are respectively W k ^ ^ a 8 " 1 b ^ j c o s - 9 ^ 1 ( b s - a s ) 1 / i S _ S ' 1 r o *o/ and 1 k l 1-s I b 8 " 1 a 8 ' 1 HQ 2 \ a b ^ v / ^ ^ i cos (b s«a s) 79 BIBLIOGRAPHY 1. Adler, R.B., "Waves on Inhomogeneous Cylindrical Structures*", Proc. I.R.E.. vol. 40, March 1952, No. 3, p. 339. 2. Brillouin, L., Wave Propagation in Periodic Structures. Dover Pub., Inc., 1953. 3. Chambers, L.G., "Propagation in Waveguides Pi l l e d Longitudinally with Two or More Dielectrics", B.J.A.P.. vol. 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.. vol, 3, January 1952, p„ 19, 5. Chambers, L.G., "An Approximate Method for the Calculation of Propagation Constants for Inhomogeneous F i l l e d Waveguides", Quart. J. Mech. Appl. Math., vol. 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., Differential Equations, McGraw-Hill, Inc., New York, Toronto, London, 1955. 8. Hildebrand, F.B., Advanced Calculus for Engineers, Prentice- Hall, Inc., Englewood C l i f f s , New York, 1949. 9. Ince, E.L., Ordinary Differential 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, California, 1960. 11. Liebowitz, B., "Development of Electromagnetic Theory for Non-Homogeneous Spaces", Phys. Rev.. vol. 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 in Inhomogeneous Media", J. Opt. Soc. Amer., vol. 48. August 1958, p. 513. 14. Pincherle, L. "Electromagnetic Waves in Metal Tubes Pilled Longitudinally with Two Dielectrics", Phys. Rev., vol. 66, No. 5 and 6, September 1944, p. 118. 15. Ramo, S. and Whinnery, J.R., Fields and Waves in Modern Radio. John Wiley & Sons, Inc., New York, Chapman & Hall, Ltd., London, 1953. 16. Ro-Shersby-Harvie, R.B., et a l , "A Theoretical and Experimental Investigation of Anisotropic- Dielectric-Loaded Linear Electron Accelerator" Proc. of I.E.E.. vol. 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 Englefield, C.G., "Some Properties of Dielectric Loaded Slow Wave Structures", PGMTT Symposium. Sah Diego, May I960. 19. Webster, A.G., Partial Differential 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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