THE UNIVERSITY OF MICHIGAN 4386-1 -T WAVE PROPAGATION IN AN ANISOTROPIC COLUMN WITH RING SOURCE EXCITATION by Surendra Nath Samaddar December 1961 This work was submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at The University of Michigan The work described in this report was partially supported by the ADVANCED RESEARCH PROJECTS AGENCY, ARPA Order Nr. 187-61, Project Code Nr. 7200 and ARPA Order Nr. 147-60, Project Code Nr. 7600 ARPA Order Nr. 187-61, Project Code Nr. 7200 Contract AF 19(.6Q4)-8032 and ARPA Order Nr. 147-60, Project Code Nr. 7600 Contract AF 19(604)J-,7428 prepared ltt Electronics Researtch Directorate Air Force Cambridge Research Laboratories Office of Aei ospace Research Laurence G. Hanscotn Field Bedford, Massachusetts

THE UNIVERSITY OF MICHIGAN 4386-1-T Requests for additional copies by Agencies of the Department of Defense, their contractors, and other Government agencies should be directed to: ARMED SERVICES TECHNICAL INFORMATION AGENCY ARLINGTON HALL STATION ARLINGTON 12, VIRGINIA Department of Defense contractors must be established for ASTIA services or have their "need-to-know" certified by the cognizant military agency of their project or contract. All other persons and organizations should apply to: U, S. DEPARTMENT OF COMMERCE OFFICE OF TECHNICAL SERVICES WASHINGTON 25, D. C. I —----------— ii —------------

THE UNIVERSITY OF MICHIGAN 4386-1-T ACKNOWLEDGEMENT The author wishes to thank the members of his doctoral committee for their helpful suggestions and criticisms. He is especially grateful to Professor K. M. Siegel, Chairman of the committee, for arranging the support for this investigation and for his continued interest and his invaluable assistance and comments during the course of this work. The author also wishes to express his appreciation to Professor C-M Chu for many enlightening discussions and criticisms of the subject matter. In addition, the author wishes to express his appreciation to Professor L. B. Felsen of Polytechnic Institute of Brooklyn, New York, for suggesting the problem and for his interest and guidance. To Drs. R. F. Goodrich and R. E. Kleinman of the Radiation Laboratory, Electrical Engineering Department, The University of Michigan, the author extends deep gratitude for their many stimulating discussions and invaluable criticisms. Finally, he would like to acknowledge the help of his colleagues, Mr. O. G. Ruehr for formulating numerical procedures, and Mr. H. E. Hunter who performed the tedious task of numerical calculations........V_ —-^ —-------- iii -

THE UNIVERSITY OF MICHIGAN 4386-1 -T TABLE OF CONTENTS ABSTRACT v INTRODUCTION 1 I GENERAL PROBLEM 8 II WAVE PROPAGATION IN AN ANISOTROPIC PLASMA: SLOW SURFACE WAVES 50 III PROPAGATION OF SLOW WAVES IN AN ANISOTROPIC FERRITE 82 IV CONCLUSIONS 86 APPENDIX A: Maxwell's Equations for Anisotropic Medium 89 APPENDIX B: Construction of Dyadic Green's Functions 106 APPENDIX C: Dispersion Relations for Various Special Cases 118 APPENDIX D: A Numerical Example 133 BIBLIOGRAPHY 148 iv

THE UNIVERSITY OF MICHIGAN 4386-1-T ABSTRACT Propagation of electromagnetic waves through a homogeneous anisotropic column of a medium of infinite length is considered. The anisotropy of the medium is characterized by the dyadic form of the permittivity 6 and permeability P. This anisotropic column is surrounded by a coaxial homogeneous isotropic medium characterized by scalars ~ 2 and A, this complete structure being enclosed by a perfectly conducting metallic circular cylindrical waveguide. A magnetic current ring source is inserted symmetrically in the isotropic medium. Throughout the analysis the strength of the source is considered to be an arbitrary function of the polar angle 0. For this general problem the complete expressions for fields (due to the source), power flow, and the dispersion relation have been studied. To solve a source problem, dyadic Green's functions for both point electric current source and point magnetic current source have been constructed in a formal way from the source free solutions of the appropriate Maxwell's equations. These dyadic Green's functions can be used for any arbitrary source. From the general expressions for the transverse fields in terms of the longitudinal fields, in any arbitrary cylindrical region (unbounded) the propagatlon wave number of a TEM mode travelling in the longitudinal direction z has _ v Y....._...........................

THE UNIVERSITY OF MICHIGAN 4386-1-T been obtained. The wave numbers for a TEM wave propagating in a direction perpendicular to z, can be obtained from the general expressions for the transverse wave numbers, using a: _j= o az The results of the above general problem have been used to study the wave propagation in an anisotropic plasma column and an anisotropic ferrite column separately. The various possible passbands for the propagation of electromagnetic waves in an anisotropic plasma column have been obtained and a special case Is considered for numerical computation of the longitudinal electric field in a plasma. The analysis for the plasma problem emphasizes the slow wave propagation.......- - - -- - - - viS

THE UNIVERSITY OF MICHIGAN 4386-1-T I INTRODUCTION Electromagnetic wave propagation through an anisotropic medium has been studied by many authors in different situations - from ionosphere to conventional waveguides. Ionospheric anisotropy is due to the presence of the earth's static magnetic field. Ionized, but macroscopically neutral gases of any kind are known as plasmas. An ionized and neutral stationary plasma in a weak electromagnetic field can be represented as an equivalent dielectric medium. Moreover, when this plasma is situated in a uniform static magnetic field, the strength of which is not necessarily small, its equivalent dielectric "constant" behaves as a dyadic (tensor). It is well known that radio wave propagation through the ionosphere depends on the frequency of the electromagnetic wave, electron density, the collision frequency, the ion gyrofrequency and the electron gyrofrequency. The same is true for a wave propagating through a plasma waveguide. This knowledge of the wave propagation is essential for satisfactory long-distance radio communication through the ionosphere. The study of a plasma is also of vital importance in the field of thermonuclear reactions. In a thermonuclear reactor a static magnetic field is used to confine and to heat a plasma. Many times it is desirable to obtain information on the temperature, density, etc. of the plasma. Such investigations which ___________________________________ ____ 1 _______________________________________

THE UNIVERSITY OF MICHIGAN 4386-1-T determine the characteristics and behavior of the plasma are known as "plasma diagnostics." Besides the above, the propagation of electromagnetic waves in plasmafilled or partially plasma-filled waveguides has aroused considerable interest in recent years [6] 7 ] 8], primarily because of possible applications to the generation or amplification of microwaves. A medium whose dielectric "constant" is a tensor is called gyroelectric. On the other hand, there is another class of materials known as ferrites which exhibit ferro-magnetic properties. The chemical composition of the ferrites may be expressed [3] [18 by the formula MO Fe2 03, where M represents a metal, such as Mn, Fe, Ni, Cu, Mg, Al, Co, etc. Although ordinary iron (Fe) and nickel (Ni) possess ferromagnetic properties, they are of little use as microwave components due to their high losses. But the ferrite materials mentioned above, whose specific resistances are above 106 times higher than those of the metals, with relative permeabilities ranging up to several thousands and relative dielectric constants varying from 5 to 25, have extensive use in microwave devices. In the presence of a static magnetic field the permeability t of a ferrite becomes anisotropic, i. e., P becomes a dyadic, which is the characteristic of a gyromagnetic medium. The medium whose dielectric constant and permeability both are tensors, is known as gyrotropic. 2 2

THE UNIVERSITY OF MICHIGAN 4386-1-T In connection with the Faraday rotation of guided electromagnetic waves in a gyromagnetic medium with a uniform static magnetic field in a circular cylindrical waveguide (the axis of the waveguide coincides with the gyro-axis which is also the direction of the static magnetic field and has been taken as the z axis), Suhl and Walker [2i have shown that only circularly polarized modes exist, if Ez / 0, and Hz $ 0, and pure TE and TM modes do not exist. However, pure TE and TM modes can exist if - = 0. It should be noted that in this case as well as throughout the present work, the anisotropic medium under consideration is homogeneous. For an anisotropic plasma medium TE and TM modes can exist [ 7 independently in another special case when the axial-static magnetic field is infinite. Besides the above mentioned work, a number of investigators including Van Trier [1, Gamo 19, Fainberg and Gorbatenko 4], Agdur [6], Epstein [3], Trivelpiece [8] etc. have carried out research in connection with wave propagation in gyroelectric, gyromagnetic, or gyrotropic media with various configurations. All of the research work cited above except that in [7] considered only the source free resonance behavior of electromagnetic waves. In the present problem, however, a source of electromagnetic waves which interact with the anisotropic medium is included. In Appendix A, a general formulation of the source-free problem is presented for any cylindrical geometry with arbitrary cross section. This formulation is __________________________________ 3

THE UNIVERSITY OF MICHIGAN 4386-1-T suitable even for an unbounded anisotropic medium, provided cylindrical symmetry is assumed. In Appendix B, dyadic Green's functions for a point source (electric current or magnetic current or both) are constructed from the general source free solutions of the Maxwell's equations for both dissipative and nondissipative media. For such a construction of Dyadic Green's functions references [9], 11, and 12j have been found very useful. An alternative method using a transmission line formulation can be devised for the construction of Dyadic Green's functions. In chapter I the problem considered is to find the dispersion relations and the complete fields due to an excitation by a magnetic current ring source situated in a cylindrical isotropic homogeneous medium characterized by a relative dielectric constant 2 and a relative permeability /2, which encloses a central cylindrical column of a homogeneous anisotropic medium characterized by dyadics 6 and p, this whole structure being enclosed by a perfectly conducting cylindrical waveguide. This general analysis has avoided specifying any particular medium, say a plasma or a ferrite, and also it does not necessarily consider a ring source of constant strength. A ring source (magnetic current) represents an idealization of a possible excitation, for example, a circumferential slot in the waveguide wall, or an annular slot on a thin metallic disc-* fitting In this example one must also consider the boundary condition for the conducting metallic disc. 4

THE UNIVERSITY OF MICHIGAN 4386-1-T tightly across the waveguide. Although a ring source is taken for analysis, any other type of source can also be handled adequately since the formal expressions for dyadic Green's functions for an electric current source and a point magnetic current source are given in Appendix B. A magnetic current ring source is more appropriate for a plasma problem, whereas for a ferrite problem an electric dipole at the center of a cross section of a circular waveguide is more appropriate. In Appendix C, the general dispersion relation of Chapter I has been evaluated in a number of interesting special situations with appropriate limiting processes. Although in these special cases the procedures are also applicable to obtain expressions for the total fields due to the source from the general expressions given in Chapter I, no attempt has been made to obtain these expressions owing to the laborious task they involve. Chapter II deals with a problem in which the anisotropic column is taken to be a plasma in an axial static magnetic field, using the results of Chapter I. In this case necessary conditions for slow wave propagation (which give maximum passbands) have been obtained. The sufficient condition and hence the actual passbands can be obtained from the solution of the dispersion relation. Since it is not possible to study a dispersion relation in general, a few special cases have been discussed. In addition to these a more general dispersion relai-i-5-i-i-i-iii-i-i —5

THE UNIVERSITY OF MICHIGAN 4386-1-T tion for slow wave propagation has been considered for numerical computation In Appendix D. In this case the lowest eigenvalues are found and the corresponding longitudinal electric field is calculated. In Chapter III, the results of Chapter I have been applied to study wave propagation through a ferrite column with a uniform axial static magnetic field. It may be noted here that results for a ferrite problem can be obtained by using duality on the corresponding results for a plasma problem when the boundary conditions on H in the ferrite are the same as those on E in the plasma, for example an unbounded plasma and a ferrite. From the formal expressions of the transverse electric and magnetic fields as functions of the longitudinal fields Ez and Hz, and using the expressions for the transverse propagation wave numbers obtained in Appendix A, conditions for TEM wave propagation in the direction parallel to or perpendicular to the d. c. magnetic field have been obtained. These conditions provide expressions for the propagation wave numbers in the respective cases. The conditions which give the possibility of a TEM wave propagation in an unbounded medium cannot be valid in a bounded medium or in waveguides (except those bounded by two non-connecting metallic boundaries). Therefore these discussions suggest that the study of a dispersion relation for a bounded anisotropic medium, under the condition of TEM wave propagation, is meaningless and ___________________- -- _ 6 ------------------------

THE UNIVERSITY OF MICHIGAN 4386-1-T Inconsistent. It can be shown that the condition of TEM wave propagation in the direction of the static magnetic field, is equivalent to zero-value of the product of the two transverse wave numbers. However, results for such a situation have been presented in the literature mistakenly.* A reason for such inconsistent results may be that the authors overlooked the direct equivalent relations between the conditions of TEM waves travelling in the direction of the static magnetic field and the vanishing conditions of the product of the transverse wave numbers. Agdur, 6 pages 183 to 185, also 4, page 497. ________________________ 7 __ —---------

THE UNIVERSITY OF MICHIGAN 4386-1-T I GENERAL PROBLEM Statement of the Problem An infinitely long column of an anisotropic medium characterized by tensors 6 and p, of radius a, is situated coaxially inside a perfectly conducting circular cylindrical waveguide of radius b. The annular space between the column of the anisotropic medium and the cylindrical waveguide is filled with an isotropic medium characterized by scalars 62 and m2. The electromagnetic fields are introduced into this system by a magnetic current ring source of radius c, the center of which lies on the axis of the waveguide, such that a < c ~ b. The total fields and their behaviors are studied. Figure 1 shows the geometry of the problem. General Formulation of the Problem For convenience the plane of the ring source will be chosen as z = 0, where the axis of the cylinder lies along the z-axis. Due to the cylindrical symmetry of the structure, cylindrical coordinates r, 0, and z will be used here. If the ring source is very thin both in the radial and in the axial direction, it can be represented in the following way I 0 I eJWt = 0 m(0) S(r -c) (z) eJWt (1) -m -o m -o __________________________________ 8 __________________________________

THE UNIVERSITY OF MICHIGAN 4386-1-T where w = exciting angular frequency 0 = unit vector in the 0-direction -O m(O) = strength of the source in volts S (r - c) and S(z) are well known Dirac-delta functions. Magnetic Ring Current Region 2 / An isotropic medium Metallic waveguide Region 1 - An anisotropic medium FIGURE 1 Although ultimately m(0) will be chosen as a constant for numerical computational facility, the present formulation of the problem is valid for m(0), any arbitrary function of its argument 0. The Maxwell's equations for this problem can be expressed (the time dependence is assumed to be ejWt) as _________________________ 9 ______________I

THE UNIVERSITY OF MICHIGAN 4386-1-T x E(r) = - jcqud(r)' H(r) - 0o I (2) x H (r) = joW -(r) E (r) where the relative dielectric constant (r) and the relative permeability p(r) are defined in the following way. E(r) |= -~Or &60, for 0$raa (3a) 0 0 ~ zz with constant elements = 62 (constant), for a5 r < b (3b) rr 00 (3c) /rr r0 /(r) = -Jr 66 0 | for 0 r a (4a) 0 0 ~zz with constant elements _________________________ 10 _____________

THE UNIVERSITY OF MICHIGAN 4386-1-T = J2 (constant), for at4 r b (4b) Prr = P0O (4c) Pre = Or r in eqs. (2) represents a three dimensional position vector, and r is the radial coordinate. The results developed in Appendices A and B will be frequently used in the following. A method of solving any source problem in terms of Green's function will be presented here. To construct a Green's function for a problem with some given boundary conditions, it is sufficient to find corresponding eigenfunctions which form a complete orthogonal set. These eigenfunctions are solutions of the source-free problem subject to the same boundary conditions. In the present problem where the waveguide is uniform (Independent of z) and the medium is also homogeneous In the axial direction z (i. e., components of 6 and i are not functions of the coordinate z, with e and pu having forms shown in (3) and (4) respectively), one can assume that there will be waves propagating in the z-direction, having z-dependence as e-Z, where X is a propagation wave number for a particular mode. This assumption leads Maxwell's equations with appropriate boundary conditions, to an eigenvalue problem, withX as an eigenvalue (see Appendix B and L91 to [13] ). I_________________________ 11 ___I —-------

THE UNIVERSITY OF MICHIGAN 4386-1-T Thus the source-free solutions _ (r) and (r) satisfying the following Maxwell's equations (5), form a complete orthogonal set of eigenfunctions. V x 6(r) - jo&I,(r) X (r) (5)+ Vx X)(r) = jwc6(r) ~ (r) The orthogonality relation can be obtained by choosing another set -,~ g(r) and X'(r), which satisfy the same boundary conditions and the following Maxwell's equations, v x,e,(r) = jWuo (r) X (r),, +*,, (6)+ Vx ),(r) = -jwCEo (r) *' (r) where: denotes complex conjugate, + (or,u+) = adjoint of E (or iu) = complex conjugate of the transpose of 6 (or p). Now It can be shown (see the above mentioned references) that the required orthogonality relation is t(r - t (r)x o ds = N, =i (r) ~ 5o xt (r) ds (7) s s +In general the single Index ~ (or ) is actually a double index in (or i'n') corresponding to radial and angular variations. ~~~____~~__________. 12 ______________

THE UNIVERSITY OF MICHIGAN 4386-1-T where I and 1' correspond to X -th and -X th modes (eigenvalues) respectively. z = unit vector in z-direction -0 Nn = normalization constant i = 1, forX = 0, for)~ Xg* I' Ec and are transverse components of! and 6 respectively. s is Lea t.9e the cross section of the waveguide. Since in the present problem the only source is a magnetic current, the total fields can be expressed in the following way, using the appropriate dyadic Green's functions (eqs. (16b) and (17a) in Appendix B): E (r) S= - y \ \ \ ^l). go Im (rL) r dz1 dr dO (8) i J J J ZN 0 O -o and Zir b oo H(r) = - \e (I()r1) 0 (r' ) r' dz' drT dO (9) 0 0 -co r = observation position vector r_ = source position vector (i.e., primed coordinates refer to source) and the time dependence eWt is suppressed everywhere. ____________________________________ 13 __________ —-----------

THE UNIVERSITY OF MICHIGAN 4386-1-T Using eq. (1), the above two expressions can be reduced to the following forms 27r E(r) = - c E (r) \ (, (c, 0, 0)m(0') dO' (10) 2N J 0 and 2'r H(r) = - c (T) \ t 4(c, 0, 0)m(0') dO' (11) ZN 0 where 0 (c,', 0) is the 0-component of) (c 0', 0). If _ (r), (r) c 0' 0 0) Isthe and X'? (r) can be expressed as (the single index I is replaced by the double index in): in (r) A1 in f in (r) ^f^ —)/iL[nin (r) n (r) = A n (r 12) = " -in = iin and moreover if'n (r) = gW n (r z) e n0 i = 1, 2,3, (13) n 0O, -, t,. then the above expressions (10) and (11) can be reduced further to the following 14 _

THE UNIVERSITY OF MICHIGAN 4386-1-T E (r) - E(r) (c) m (14).... f in - 0in n i, n in H (r) - N g An (r) g" (c) ^n (15) c Kin goin i, n \ in where 27r mn = \ e m(0') dO' (16a) nn 0 Ain and A'' are constants and f (r), and in (r) are known functions. uin in-L' — If m(O') = m, a constant, then n = 0 i.e., - _ 0] and consequently m =27rm (16b) n For n = 0, the above summation is no more than a single summation over i. Now using (12) and the orthogonality relation — S tin(r) ~ tin (r) x n z ds = Ni (17) s it can be shown easily that A A" lin An 1 1 in S f tin(r)' g'tin(r)x Z ds 2= b s 5 |prinf(r)gin(.) fin()m- rin(r rdrdO 0 ff (18) _____________________________ 15 ____________

THE UNIVERSITY OF MICHIGAN 4386-1-T Now using the above relation (18) in the expressions (14) and (15), it can be shown that the complete fields have the following forms cE(r) I -_ ii: Zr mn fin (r) gjin (c) iLn Cr [fC; ( rgin - in) fin(r)grin(r)] rdrdO (19) H(r) = - (r) g () — r) 2 ZTr b i,n Of\ [ (r g n(r) - ()gi (r) rdrd (20) Lrn- - 1n rdrn Although the above two expressions (19) and (20) are valid for both dissipative and non-dissipative media, the relation between gin and grn becomes simple in the case of non-dissipative medium. Thus in a non-dissipative medium (see Appendix B and [9] ) it can be shown that Rin = i con = complex conjugate of gin (21) ft = f = complex conjugate of fn -rin - n -in Therefore, for non-dissipative medium, the total electromagnetic fields are given by the following expressions (using (21) ): _______________________________ 16 _______ —,, —-,,

THE UNIVERSITY OF MICHIGAN 4386-1-T E(r) c m-_ n nfin (r) gjin (c) Z n'nIn 9rb i,n,T\I 0 in(r)g'in(r) f n(r)g n(r) rdrd (23) H(r) = - z E Z7 b m n &In (r) g'n (c) 27r b n f fr n(r)gin(r) - f0in(r)gn() rdrd (23) It may be noted here that for non-dissipative medium the propagation wave number Xin (eigenvalue) In the z-directlon is a real number. An alternative set of expressions for total electromagnetic fields which are particularly suitable for dissipative media (although valid for non-dissipative media also), can be obtained from (19) and (20) using the following transformations in= - in f-" i = - r- -in - (r) (24) gin (r' r r ) ~ln = (r, -' -er' -r) ~- (r) Thus the total fields, which are particularly suitable for a dissipative medium, can be expressed in the following way (using (24) ): 17 _

THE UNIVERSITY OF MICHIGAN 4386-1-T E(r) = 2c 2b m fn (r) gin (c) E(r) -c n in in.....) (r 2 r. b i —(r)-5- ( )(25) n 0 0 frin(r)goin(r)- f n(r)grn(r) rdrdO n in - go[n H(r) C Z(26), rn()gn(r-) - f(r)r n(r)rdrd0 jj\ rin- gn 1n- grin - j It should be pointed out here that for a dissipative medium, the propagation wave number x)n (eigenvalue) in the z-direction is complex. Solutions of the Homogeneous (source-free) Maxwell's Equations It has been demonstrated in Appendix A that for source-free and homogeneous (or for piecewise constant P and ~ ) medium the longitudinal (z-component) components of electric and magnetic fields obey the following two equations 726 + Cz je U O ^ x z t z ai z a3 (z (27)'' V2 )2 + r -e -J z (28) t z tr/Ur 1 z Er/tr a3 z To simplify notation, indices are omitted from both field quantities and. _______________________ 18 ______ —----

THE UNIVERSITY OF MICHIGAN 4386-1-T where a' = k2 r (r -/2) - /AX a, = k2 (z2 _-'2) ~ Z r r r arr = 0 = C(r E (r0 = I r r r HrO Or (0 = = Ml zz z k =L 0, E (see Appendix A) in terms of 6 and XZ in the following manner zz ~t ~P1 [+qa4a7tz- )oQ~vt) - p^xzk'a30tz+jy uoalVt<] (29) and t 1 a4tz + Woa2tz- X [k2 a37 - jw a V (30) 49t + a PI 3 t z o 1 tz _ _ _ _ _ _ _ _ _ _ _ _ _ 1 9 a 3 ___/z -_ __ z_

THE UNIVERSITY OF MICHIGAN 4386-1-T, where a' = k2, 2_z -z2)+ xZ a (Pr - /' 2 r a = k2'(2 _ e,2) +~2~, a = k2 rr +'c ) -2 P1 = k4a -a4 It is observed here that if a3 = 0, the two equations (27) and (28) become uncoupled.* Although this Is a necessary condition that the conventional E-type and H-type modes separate, it is not sufficient. The sufficient condition depends on the boundary conditions. For example, if the medium completely fills a perfectly conducting waveguide, and if a3 = 0, E-type and H-type modes can exist separately. But on the other hand, if there are two coaxial media, (the outer one may or may not be bounded by a perfect conductor) E-type and H-type modes can exist separately if and only if,- = 0 (with a3 = 0). But if d / 0, and even if a3=, in the above two coaxial media-system E-type and H-type modes cannot exist separately. A similar discussion for Isotropic media where a3 = 0, can be found In i16], sec. 11.6. For the solution of (27) and (28) z and z can be eliminated yielding a single 4th degree equation In each of Ez and )z which satisfy both of the second degree equations (27) and (28). If a choice is made such that 0=&z+ja4, *TE and TM modes also decouple, i.e., they can exist separately if there is a constant line source in the z-direction. In this case O = 0. L zo I~~~~~~~2

THE UNIVERSITY OF MICHIGAN 4386-1-T then the equations (27) and (28) can be reduced to the following equation (for detail see Appendix A): t2 0 + 12 0 = O (31) where ~z = aa 1 (32) Esr [a; - woea3a] = 7' = erg [a1 t a (3Z) Solving equation (32) for a, one obtains 1, 2 W C C a3 o z 3 Therefore, the roots of 7'2 can also be expressed in the following way 2= V + - U (34) 1)2 where V a=..ai. ='1+?2 (35a) r r and U Ez/Lz a -k4a ] = -k a3 ~r/r Ez,AZ,2 2 ~= E r P~i ~= T 1 2 1 (35b) ________ 21 ___ 21

THE UNIVERSITY OF MICHIGAN 4386-1-T Equation (31) has the following form in polar coordinates r r r Z +1 (r) 0= O (36) rr Br 02 a 2 where,2,2 r7 (r) = 12 for 0O r a (37a) = 2 = k 2 for a r b (37b) The general solution of (36) in the region containing the origin can be written as 01,2 = Al Jn ( 2 r) en n = 1,. (3 8) for 0O r /a and the solutions of (36) in the region a~4 r b, which correspond to longitudinal fields, are given by' Z= {Bi' J(n (r)+ C1' Nn (r)] ejn0, a. r b (39), Xz = B"2 Jn ( 7r) + C' Nn (?r) e jn a, r 4 b (40)* A1, A2, B"1, B"2, C1' and C"2 are arbitrary constants which depend on n and Y]2 and -7. If not clearly indicated, these constants and radial propagation wave numbers are understood to have the double index in. ___________________________ 22 _

THE UNIVERSITY OF MICHIGAN 4386-1-T where J and N are Bessel's functions of the 1 st kind and 2 nd kind of order n n n respectively. Since 0 = +j1^ = AJ(r7/ r) ejnO, and 03 + j 2) = A2J (72 r) ejn0, 2 z n it is easy to verify that - a,[ jn C^ = - -- a: l A2Jn(12 r) -a2Al J(l r) ]forO r a, (41) and. nO dz al - aa LA2 \ n ('2 r) - A1 Jn ( r) for a, Since the boundary condition requires that(42) Since the boundary condition requires that (r)| =b = rr r=b r=b the equations (39) and (40) can be rewritten in the following way &,z = B n(r) e jnO, for a,< rb (43) Z = B2 Gn(r) ejn0 for a rb (44) where _______________________________ 23 ________________________________

THE UNIVERSITY OF MICHIGAN 4386-1-T W = J (Ib) N n (Ir) - ~ (r) N ( b) (45a) G (r) = J1 (7vr) NA (lb) - Jn (vb) N1 (77r) (45b) B11 B, = -- -(45c) N (ilb) B211'2 = (45d) N' (Vb) n 11, (71 r=b,, ~dJn (71 r) J' (nb) = dn (r) (45f) n d (y r) (1r) r=b Now using equations (41) to (44), in the relations (29) and (30), the transverse components of the source —free solutions of Maxwell's equation (5) can be expressed as -je211O j r{77R() r)n er=~ - AI<^ ~ ^'1j R ^ ^ ^ W ^ —Er P (Cel _a2)jn z 1 1 0 6' Jr 12 r 6. L~o r (46a) r 2rnw Jn (?27 r) + A2 I T J r) + nwp i zATJ r)nr --- j for 0< r< a ___________________________ _^_ 24 _4

THE UNIVERSITY OF MICHIGAN 4386-1-T n 2 12 e jnO (r) J B ejn C (r), for as r b (46b) ejnO' I- Jn (r Ir)i r p(^2 lltl'12 J11l rJ+nzIR a \ +- A2 ~, L 722 Ti'Jn r1) r r0ZZ <r for 0O r a (47a) j ]~PtIJ~P2 eJne n 4 jn( Jn(r) - iB02 n () j B2 e S (, for a r b (47b) o 2, 2 r Jne r2 me2 1 )r =jn )[1{ n1 ('2M rr Jn( -A T''Pr r'1 + (Jn (r+2T J r)+ r for0 r a (48a) -ns ~:z j ~Ue z 3 r I -,A 0( 2B1 ej nCn (r)+ B G(r), for a, r b (48b) r =.2r r2 2 2 vt -j A' f ejni nJr)+,nr r)' Pi (el - ^2 A3 a az- 3 ^/^ 2^n r L z~za3 r J - (49a) _ -^JCEoE2B, ejne (r) + n B2 ejn Gn(r), for a, r<b j2r (49b) _________________________ 25 _________________________

THE UNIVERSITY OF MICHIGAN 4386-1-T where z a S = - - (50a) 1/r Er rr E a,a 2 M = - 2 (50b) ^rr ",4 aia4 ~r/ rM - k- fCz a3 a. R =Xa 4a4r2 - o Eza3 (50c) k2 z a3 a2 - a4/ r S T = wOu a2 - A a4 c-l = (50d) o 4 a3 - v' = ao a - a4 = M a2rEr, a3a4E (50e) T1= Xa - S 2LaqE - a-a= (50f) 3 z S(r) = J' (7b)N1 (1r)-Jn(lr)Nn(I b) = - (r) (50g) C(r) J b) N r ( r)-J(r)Nn(7b) = d /(r) (50h) Djspers on Relation Since In the present problem the eigenvalueso are discrete, the boundary conditions satisfied by the total fields (i. e., the fields due to the presence of the source) are the same as those satisfied by any individual fields (i. e., sourcefree field). Therefore, the following boundary conditions can be imposed upon 26

THE UNIVERSITY OF MICHIGAN 4386-1-T the source-free fields, 0z, 4z, ~e and e: z(al) = ~(a1) atr= a (51a) 0(a) = 60(a+) at r = a (51b) 6 (b) = 0 at r = b (51c) &Q(b) = 0 atr = b (51d) gz (a-) = z (a+) at r = a (51e) e (a-) = o (a+) at r = a (51f) The constructions of;z and 6Q in a, r.b are made in such a way that the boundary conditions (51c) and (51d) are now automatically satisfied, since:,jn(b)=0=Sn(b). If the remaining boundary conditions in (51) are imposed upon the source-free fields expressed in the equations (41) to (44) and (47) and (49), the following relations among the arbitrary coefficients A1, A2, B1 and B? are obtained: 2 — LalA2Jn( a)-a2AlJn(7ia)] = BA1n(a), (52a) p )e[ A1 Y {/ -rr a t2 J(7' a )+n R J7n( a) } (5Zb)'P1 (A-l Pr"r A - ~ 12 5Jn(T a)+nezR a (52b),r2, o~~n (12 a) +A2 UPo~rJr7rl2 J )O(12 a)+ncz T a -JI~~3~2 J -jP(a) B nor Y,(a) 7 B2 S(a)+ a.B1 72 a _______________________________ 7 ______________________________

THE UNIVERSITY OF MICHIGAN 4386-1-T ^ — A in a)-Al a) E BG (a), (52c) al1 -n a) - A Jn(7a B2n(a)' 12 2 2 12 _ *j'E rR' A 1 2 ^ nR2 S rrl 2 Jt I~~~ a)+ a)+r~ t 1a) ~Pl(al-a2)[1I-za3 J nR' n a j z a j P1(CYYrr r 2, aT j n(a)}a'2 ~ ~ ~a l s s fr t c a 1 C, (a) + B exi, (52d) Non-trivial solutions for the constants A1, A2, B1, and B2 exist, if and only if the determinant of the coefficients of these constants appearing in equations (52) vanishes. The vanishing condition of the determinant gives the characteristic equation (or dispersion relation). Instead of calculating the determinant of the coefficients and then equating it to zero, one can also eliminate B1 and B2 from (52) and from the remaining equations it is easy to obtain two independent values of the ratio A2/A1. Now equating these two values of A2/A1, one obtains the desired dispersion relation expressed in the following: 28

THE UNIVERSITY OF MICHIGAN 4386-1-T Cc N' ^ ^3 3' m cd cd c eVN N 4) c c ce ed -e 3 Co f Coo cdcf ^- -...Ic^ ^c^~~ "c P- cd cd cd<- c-v ( —Z - ~"3 bI ~ ) c'M Sfi -- o 0 NC't CC~ O Uv_ t kc o % Nf C\ ^ + + 0 Io c d -'- -C3;.! c. O'l cq c t' - J^ ad N N Co Co'- N a) N rNT c,"' 3~~c C^' C'; c4b 4)N 4)N w w} k~~~~~~~~~~~~~~~~~~~ — -C' - ~ c -c- r 1 z- c ~ "- cuc, F` O rl C~~~~~~~~~~~~~~~~~~~c F~~~II

THE UNIVERSITY OF MICHIGAN 4386-1-T It should be noted here that to obtain the expressions (53) and (54), some useful relations (tabulated in Appendix A) have been used. These relations will be used frequently in subsequent derivations. When the ring-source m(O) is constant, n=O (i.e., 0 -0). In this case it can be shown that the dispersion relation simplifies to S{Pz G(a)Jl('1 a)-rl2So(a)Jo(lla)] z/(a)J1(ra)+21l Co(a)Jo(lla) 7 a (55) Mf l /z Go(a)Jl( a) - 2So(a)Jo(712a)} 1zo(a))Jl(ja)+E2i C o(a)Jo(ia) Alternatively, equation (55) can also be written in the following manner: (1 2 12),122 (22 - G (a) (a)Jl(7'a) J(1aa) - 22I- ) Co(a)So(a)Jo(Ila)Jo(12la) J- @a)J1(-a- [2 Az M Co(a)Go(a)+P2Czs o(a)So(a) (56) J (a)J1(J=a) + L 2/zs Co(a)Go(a)+P/2EzM o (a)So (a) A number of dispersion relations for various special cases has been developed in Appendix C. The solution of the dispersion relation (54) together with the relations (34) and (37b) gives an infinite number of discrete values of 1 I, 12, q (and hence ( also). The radial wave numbers Y] and q27 can also be called eigenvalues.in the, domain O f r~ a, and similarly r is the eigenvalue in the domain a r _b, of the ___________________ 30

THE UNIVERSITY OF MICHIGAN 4386-1-T 0 tL0 LO 0 o0 -2*c -cd ^0 N?4 4-4 1-1-1 S1 r^ r3 T2J,,^r' ^.-| o cii co a) a) k a FN WN?^ 33) 1 N 0N N 0 N ~ ^ ^^'c?'^ ^S,~L CV vc9 C Cl) 0 v 0 a ed dlC4 a) II || SAP a) CV N >I xo N| | 0 c I —---------------------— 31 -CF ------- 4-)'-4 Fl 0o a)\ \ 4 -) 0.P -q - 4, -. a )4

THE UNIVERSITY OF MICHIGAN 4386-1-T Now using (59) and the relations (52a), (52c) of this chapter and (38 (4)), (38 (10)), and (38(11)) of Appendix A, one can express B1 and B2 in the following manner: B1 = A 2 (60) B2= A13 (61) where where S S Jn' a) -MJn(' )a) (2 -,) J'(a),i ) 1 Jn (j2 a) - Jn(l1a) (63) 3 (2 _- l2)rrGn(a) Expressions for Source-Free Fields In Terms of Only One Unknown Constant Al Since all the unknown coefficients A2, B1 and B2 are now expressible in terms of the only one unknown Al, the source-free fields can be written in the following way (e -jz-wt)ls assumed to multiply all the expressions for the fields): Al eJnO z, = - S -l n (1 2 r) - MJn(1r) = Al fz, for 0 r a (64a) z H(,1-2 1 2) 2- J z - Al 2 ejn0/n (r) = Al f, for an r b, (64b) ________________________ 32

THE UNIVERSITY OF MICHIGAN 4386-1-T ~J6^^eZw6/ae^i GV) RJ r) - fW/oL P,J2(r r'2 22 2, 2 i 22 {1RJnr) -onwuor 2. rALr7rl 2 (2 -1 ) (65a) +' 2 T~- (72r)+n o rl r } = Alfr,... for 0< r a Er= Alej 2 2- g(GnI() ~ O 2 Cn = Afr,...for a< r4b (65b) r _ ___r r C zifzwwe~ a3ej" A, Jn2 Ij1-) 2 2 12 2 2 12 r1r Gr 1r tr) + n1R ( ) r (66a) I + ljoPe4 712 J+n(72r) + nEzT n }j] = Alf, for 0_< r a,; L r Aejn0 [jWI noj (26; -Ae -e S3n(r) + 2 (r) Alf0, for a<rLb, (66b) 0 7? -- jwe ga3E Al e,r0,F X - 2-,2,2 L Jn (72 r) - Jn(71ir) = Alg, for 0 R r a, (67a) k= A1 S'e j0Gn(nr) = Alg, for a rb, (67b) _________________________ 33 n

THE UNIVERSITY OF MICHIGAN 4386-1-T W ez Al ejn r T A,2 [ii M, Jn r) f i,3,z 2,2 I 1 Rtz/,uza3J (91lr) + n12 MEr/r 7 ir r?1 12 (12 -1i L ~r I (69a) {',,,2 S2 2 Jn(2r) 12 r 1 q n( T' zza3 ] = A1gr, for On r 4 a. rjnO r Ae o2 rI |o = A^l -er~ Cn2(r)+ 1 r n(r Algr, for ac r b, (68b) valid for both disspative and non-diss pative meda, they are particularly L 7y r 7 jne jwsole Alfe dispate m. (r 2__ _ _ _ 234 2 2 12 t2 12,2 6 Pr72'MJ ( r) + nRt~ Zza3 (69a) El rPr7^ S J" r)nT'~ef 3- =za Al g, for 0 r 4 a Ale - - C (r)+ Gr) Algo. for a r Sb, (69b) Determination of the Constant A1 Determination of the unknown constant Al depends on the orthogonality condition satisfied by the source-free fields. Since orthogonality conditions are different-r: for a dissipative and a non-dissipative medium, the constant Al will also Although the forms of the orthogonality relation given in (Z2) of Appendix B are suitable for dissipative media......... 34............

THE UNIVERSITY OF MICHIGAN 4386-1-T be different for the above-mentioned two media. Here A1 will be calculated for non-dissipative medium only. For dissipative media the corresponding A1 can be calculated using equation (18) together with the transformations given In (24). The primary aim here Is to calculate the total fields with amplitudes due to a given source. Therefore, to calculate total fields it Is only necessary to find the ratio I A1 n2 for any mode (eigenvalue) in. This ratio can be determined from the Nin knowledge of source-free fields, as suggested in equation (18) together with (21). A J2 In other words, the ratio - Is given by Nin 32 Ni 27r b The above integral in (70) can be expressed!Pn the following compact form 27r b - n io ~~io r r'Ln ein~r) g:~in~I r dr 327 Nl 2 d rn_ d b' Ff:n(r) -f0 (r) g:rin(r r dr = 27r FinIn S dOrin )o 0 g. =1 (71) where F~', 2:-2 R.. - (72-1) l. T'I. nWPa inl = LLncr/r ~lin |z ~14ar72LnMLn - c/o 2in Rin in (72-1) FnZ = Lin6#rlin72Ln 9linez Lori2n TmLniiuza3 - RintrlnLnj (72-2) 35 __

THE UNIVERSITY OF MICHIGAN 4386-1-T F =L.C p Z. n~2[nl.ET.CrLr~[nM. + 4,o0[ Rtfiza (72-3) in3 = Lin r r fz1 fin7 l L rMiiner+rj72in Mn n - F =-LV 1ai2 ^l Po nTin za +r Tn71n S.in] (72-4) [n4 in r r 1 l4 n [n in z 3 r Fin5 6 i [ l| in 2 N (linb) | +l 13in INn (linb) I2] (7Z-5) F^6 -— T? [S~~in 2 6 2o n ( bnPV2 (72-6) Fn7 C111L2I 2inl Jn(1b)N(nb)+ 3[nI N(1 b)J1(711b)] (72-7) - W 2 2 2 n8 In [ E J(2inl 1J ib) +PL 2 1 n IJ nb) (72-8) nL P a R~3[. R.M. F nLI. PL r 3 in2 i2 2' (72-10) inlO = in l in iz or 12 1r in /i - R1 r 1 din ai (72- I ) Finl in innL 2 in 2i nin z 3 in i nJ (72-11) F12 = n L L [R. 12 M. -ci q 2in r -in Rid.Xi za3 (72-12) FinIi2 in r r zin, in in 3 3n F nL*1 f 33 i2 2*2 * 2 in13 nZLn inLPo% E r1in'2 in zR inneinpza3J (72-13) *' ~2 * n~in/a3 31, 2~,

THE UNIVERSITY OF MICHIGAN 4386-1-T 2 * t2 * T *t2 * in14 = n L r S1 iinO/po72inn T. iza3 -Rin.1in in, A] (72-14) m in r r z i n in in i r r FT nL *n3 3' T2'T2 T Finl5 = nLin7 lin[ 1ioEsr'l1 in2 in Min + TinRin inz/tza3] (72-15) Fin16 =_nLinL7q1'in1 in2 2 T i a 3 3'2 2 (72-1 in17 in1 in in z TinRin intza3 +owo r/rin R 2in inJ (72-17) 2 f~ * t2' ~2 4' 1 inl8 in inin z r riin 2 in + o in ] j 2F in19 -nLin2 IJ ICr z TinTin in/nta3 + wE 71ir S 1 n ] (72-10) 2,~3 3 t' 2 "., 2 2 in2O i'2 t2 *' (72-21) F =n2 i n. N (7 b)N (1. b)-k22 N(. b) (Y. b)] Fin22 Tnin L in 2in 3n n inin n in 3mi 2in n mi n m (72-22) 2in3' Fjn L. 2; T? 2 + T i S (72-2) F. 2N = -of,..zTi nib)n. b)1 in23 12.i Ln2in2 3m n1in b)Jnn(inb)-k nlTin 2 inin)jr n(in (72-23) _____________________________________ 37 __

THE UNIVERSITY OF MICHIGAN 4386-1 -T Fin24 *Nn( - )n2 b)nib) in24 - [3k2 2in nJn( in)Nn()in22iJ3inJn in nb)1 inb( n7minn (72-24) FFi~.n-.n 2 t in25 in ^2in 3i Jn(qlinb)Jn( inb)-k P2t2 3i2 2inJn(in b)Jn(inb)'in in (72-25) jn, 2, Fin26,2 Lk2Cn2 23inbn in bn( inb)-in 2 in 3inJn(( inb)Nn( inb) in2 7 -In[i 3 in 2-in n n n( inb) iJ 2 i3inJn( i bN in 4n~~~~~in ~~~(72-26) F.27 2 k2 3in 2in ln (i) 2in 3in Jnb)N n(n b)J nin (72-27) jn 2 2 * * F = * r ~ N (7 b)N (r1 b)-k 2Cg2. N (5q b)N (7' b) in28 in in28 2 3in n 2in n in 3 n'in in in (72-28) n 2 2 t F 2= in Fr-z(2 2 J (1 b)2l +-k P2 in. inJn. b)|2l (72-29) in29 3y2 [2 L o 1 2in | n ino n 2 I inl 2 n w*4. F in C2 - in 112 N ( b)2+ I i2312 I N( b)i (7230) in3F0 n2 2 2o| 2in n iin o 3in n nin F m! I in in3l K 2in n "inniO b)N 2 2 -- F = —, 12 j 2 J (7n b)N (7. b)+l 81?. I J (n. b)N (7l. b) in31 n 12 o | in n in n in 2~~~I l~~~inl~ ~(72-31) 2Wd~n wiJ. F =-_ __ I C [X IE |J (4 b)N ( 7 b)+,u r2lO. | J (. b)N (9. b) in3 2 2 2 2 in n in n in on n nn in (72-32) ______2ininb o 38in I _ ___

THE UNIVERSITY OF MICHIGAN 4386-1-T o z Az jina3 L. (73) in 4 4 t2,2,2 r2 (73) rPr I'lin 12in(12in-'in)I Iinl =) jn(linr) Jn(7linr)] rdr (74-1) in ~n inj(74-1) I1 iJ r) J () r) rdr (74-2) in2 nL lin n 2fin I. n ( r) Jn ( r) rdr (74-3) in3 n 2in n n J 0 Iin5 = Jn( -in r) J ( i nr) rdr (74-4) a b S nin 5r) J-(Iinr( rdr T74-5) in6 nI 1 rdr (74-6) a b t:c t Ii = J (7in r) N (in r) rdr (74-6) a b Iin7 s j n1 r) (in( r) rdr (74-7) a Iin8 Nin r) rdr (74-8) an iNn inr J a ________________________ 39 ____________

THE UNIVERSITY OF MICHIGAN 4386-1-T a'ing 5 3 ( ir j n(qrin ] dr (74-9) Iin = [(inr) n(linr dr (74-1) 0 a in1 ninr) J(2inr) dr (74-10) 0 l7 dr <74-r1),.c'rvKI inl 1 Iin12 i L —— r) - in dr (74-12) 0 a Iin13 = j ($[ inI r) J (2inr)] dr (74-13) a r4jn(~2inr) Jirn ( in I14 = r-) —— J — dr (74-14) in14 J~ rr 0 a In15 (j 2inr) Jn( linr) dr (74-15) 0 a'in16 = \ (2ir) J (2 inr dr (74-16) a I1nl 7 = L 2inr) Jnlhir) dr (74-17)

THE UNIVERSITY OF MICHIGAN 4386-1-T n 1n n lin _ SL - Jn( 2inr) Jn(inr)] dr (74-18) in18 = r = LJn(2in r) Jn( inr dr (74-19) 0 Iinl 9 = Jn( 2in r) n( 2in ) = ( r) (.r) dr (74-20) Iin20 = Jn( inr) n(inr a b Iin23 S [LN1(in r) N.(lir)j dr (74-23) Iin24 = n( in r) N(in r) dr (74-24) a b Ii 25 = E [N (, r)7( r) Ndr (74-25) Iin22 I Jn(linr) n(n inr) dr a b Iin2= [ n r) N n(inr) dr (74-26) a b I. ~ = p (n. r)NJ (Tinr) dr (74-24) in264 n in. n a a_____n 4

THE UNIVERSITY OF MICHIGAN 4386-1-T b 1n2 7 = S LJ(in r) N 7r) dr (74-27) in27 \ n nm n'in a b in28 8= Jn(inr) Jn( inr) dr (74-28) a = \! n(''in ) n (I ) 1r'029 J dr (74-29) Iin2 9 r a b n. j in r) N n ( ir) d in31O r series, or they ma be calculated numerically also. a It may be noted here that for n = 0 (i.e., when 3-=0), Fi =0, for cay in~ 9 and all the integrals I. i, for I 4 8 can be evaluated in closed form. But when n ^ 0, the above integrals can be expressed partly in closed form, partly in series, or they may be calculated numerically also. __________________________ 42 _________________________

THE UNIVERSITY OF MICHIGAN 4386-1-T Expressions for Total Fields (for loss-less media) Now substituting the expressions (64) to (69) and (71), in the equations (22) and (23), the complete fields due to the magnetic current ring source can be expressed in the following way (suppressing the time dependence factor e ): J inz + jnO me in c n. E = I --------, in1 Sin Jn(2in r) - M. J n( r) l z 4 Zi (i n 2 F l n'nnj 1n in n frin i X 2in in Finintin (75a) ".'f- ^, Ai^+ for 0 rfa in in 3in n fr \7 i e e 3 i _JQinz + jnO E 47 in n z 4 i, n ~ FinLIine, (75b) JWC2%in i+C nde 3in cin) for L. InJ^ ~~~12for a4 r ~b in c lin _________________________ 43 ____________

THE UNIVERSITY OF MICHIGAN 4386-1 -T jwE Cpu a'7. i e e+ jin n _ _ - o z z 3 n in r + t- 222?2,2 2 r r i, n n iinl 2ill(in F2int Iin) F ~ 0 inC' in 2-1 7 J"E 2 o. (c) n G. (c).o r 22in in in 3 in in mC n inn- i nr)C 2 Jn(-inr) i in R Jn(l2n in for r) a n r r) r r..9 J ( inr) +^o.;2 fin * T. ~2J (q^ r) + nt,~ ~ /~ri1 - for 0 r a. ~2im i' in D.n r11 o r r" r'in r (76a) +J<. z + jn+ I Z <^ n Fin'ine dJje d. (c) n,(. i?G (c) in in i n 2 G (r) -E c (rC)(, for nr. G ( E =in +n in 3in in r 4z F I L-'in 4riJn (76b) ~~~~4

THE UNIVERSITY OF MICHIGAN 4386-1-T jdi. z+jnO jci2C C2 C. (c) na. G. (c) +ih z^ n 0 o 2 inm in 3 in in [{ rIin r noLrr + i in 32inn i —n ) l (7/a) 6' = a i'i nO in z in r o In n in n - -n I ~2 2,r,2,2, i r -' + r2 i t f or Jnn r) ], for a2 (77a) C ^- m e j C "i2n C. (c) n n. G (c) E L \ n ___ on 2in ini n in in 0 - + 47Tr r i- -2 in 2in In o 3in in 2 fo' r b in in (77b) r o i inJ inr45

THE UNIVERSITY OF MICHIGAN 4386-1-T Tjaf z —ja0 + kinz t jO jcwC C za3 e n H + oz3 z 47ru ~C T2 i2 i 4rr r i,n (12in-lin) Fin IinL.nc E C.mn (C) n'. G. (C). 0 22in in, j 5 in. n for 0~ rS a 4w -, m e mjoCc2,2 C. (c) n?e. G (C) i n 2 ini in 7in C in for a0 rb (78b) Tj_ z+jnO cwc me (n + mn cC)a, G. (r) _ o z _ nin in in 3i in in in I 2 2 in1 3 i i R. aei. e M a J (i r)+n2. --- -, L n in in z 3 n in in C in iil r CZ~~~~~~~~~, in~~~

THE UNIVERSITY OF MICHIGAN 4386-1-T I, 2 2 21J r Or i-n* j2. d. C A a J ( 2. r)+nli S. C for 0 r <a in 1 in in in z z 3 n n lin r r J o ra (79a) +ji z+j n0 m e j-^ f e 2lC (JoC) n~ G. (c) H = c m n[. oC 2in in in 3in i n( 3 r 3 ~ - -,- + —2-t-r(79b) Fin z+jIin jin n in ~:1 ~-wcC j s. (r) 0r nin n inS in n inJ m rin (79b) n'j 0(z+jn0 jcu~ e ^ —7 mII e juC 0CI o 22. (C) _ o z__________ _ oI 2 in H0 o - 2 2t ~2 t t2 t2 t 47Te iu 4-2^ n1(12,l n 2, -1l.i FFIR a r r i, n n n n in pI A in 4 in nP. (C) 2 2 t t2 t r)' p2n-1 r'-.tl., ztinM. r)+nT.. — a for;i n )in in - in r r1 in2i i n in i.n n(in n fr 4 r a (80a) ____~._______ ____ ~47

THE UNIVERSITY OF MICHIGAN 4386-1-1 Tjc. z+jnO in... m e e 22 C (c) n G (c) i, n. C inin eI in 7in -jwe C (r) nG. (r) X r C \ 2in in (r i n I inin( 4- 2- for 0~ rx b. in r o 2 in, inr The average power flow is defined as av Re E x H z dS av 2 - -- S (81) 1= Re E 1 xz dS 2 Re S where Re means real part of E = total electric field due to the source at any point H_ = total magnetic field due to the source at any point Now using equations (18), (21), (22) and (23) in (81), the expression for P can be written in the following manner _________________________ 48 ______________

-. THE UNIVERSITY OF MICHIGAN 4386-1 —T 2 2 n2 I I i P = Re (82) av 16w2 3 (C) )inn in In particurh n 0 i, 3in the r.' ins of c — 2 ( 22 in7 0 CM.. (83) In particular when n O0, i.e., when the ring soutrce is of constant apnlyTitude, the expression for the power flow redtuces) to tohe following formn: cm k —- 7 ^ ^ 1'o Pa = -m 2 Re 2 - (84) av 4 87 It may be noted that for non-dissipative media the quantity inside the square bracket is real. Any individual term in the series in (82) or (83) represents average power flow corresponding to that particular mlode in (or i when n = 0). ______________________ __49 ____________

THE UNIVERSITY OF MICHIGAN 4386-1-T II WAVE PROPAGATION IN AN ANISOTROPIC PLASMlIA: SLOW SURFACE WAVES Introduction In this chapter the general results of the previous chapter will be applied to the study of propagation of electromagnetic waves in an infinitely long anisotropic plasma column enclosed by a dielectric cylinder, which is also enclosed by a perfectly conducting metallic cylindrical waveguide; io e. the geometry and the source of excitation are the same as those of the general probelm except that in the present situation the anisotropic medium is represented by a plasma column with a uniform static magnetic field in the axial direction z, The relative permeability I7 2 of the dielectric medium which encloses the plasma column is assumed to be unity. The plasma is considered to be fully ionized (i. e. macroscopically neutral) and there is no drift velocity (d. c. ) of electrons or of ions, i. e. the plasma is also stationary. If one also assumes that the illuminating electromagnetic waves are weak, then it is possible to describe a plasma as a dielectric medium. In this analysis it will be assumed that the plasma is homogeneous, i. e. its density (and hence dielectric constant) is not a function of space. 50 _____________

THE UNIVERSITY OF MICHIGAN 4386-1-T In the presence of a static magnetic field, the dielectric constant of the plasma becomes a tensor, which means it is an anisotropic medium. if the static magnetic field is applied in the axial direction, it can be shown L2]31[55 that the plasma has the following dielectric tensor rr ~rO 0 E - -j Er E 0 (1) Or 00 0 0 E zz where 2 iv F C 12a) C C.'rr=er=r2,Ev )'0 2a2 2 -c ~p rO Or w - 2 (1(t W W W9 c U 2 CO and E - p1- - (2c) zz z 2 j (1 ) v -- collision frequency (radian) (3a) qeBo WC = cyclotron frequency (radian) (3b) e qe = charge of an electron (3c) me = mass of an electron, (3d)

..... - THE UNIVERSITY OF MICHIGAN 4386-1 -T Bo = d. c. magnetic induction (3e) 2q N 1/2 (%= Ne = electron-plasma frequency (radian) (3f) p \COme/ N = electron density (3g) c = free-space dielectric constant (3h) In the above analysis the motion of an ion due to a disturbance is neglected in comparison to that of an electron. The relative permeability of the plasma is assumed to be unity. It can be shown from the relations in (2) that the components of ~ satisfy the following relation 2 = (1 - ~r) (~( - ~c) (4) An interesting conclusion can be made from the relation (4), namely' =- 0, for either e -- c or e = 1 The physical interpretation of these results can be given in the following way. For isotropic plasma (i. e when B - 0), e = c, and:' - 0. On the other hand when — + o (i.e. Uo -) co), c -+ 1 and e' -> 0 The above statements can also be verified directly from (2). It may be noted here that the collision term v in the expression for e in (2), represents loss in the plasma. Although the various dispersion relations developed in Chapter I anrd in Appendix C are valid for a lossy plasmla, the complete field expressions obtained in the previous chapter are not. On the other hand if it is desirable to find the expressions for a dissipative medium (i. e. plasma), one must use the appropriate _______________________________ I52

THE UNIVERSITY OF MICHIGAN 4386-1 -T orthogonality condition and the resulting Green's functions which have been discussed in Chapter I as well as in Appendix B. But here only the loss-free (v = 0) plasma will be considered. Although the results obtained in the previous chapter are valid for all possible modes of propagation in the structure, in the present chapter attention will be directed to the analysis connected with slow surface waves. The slow surface waves are those waves which decay radially in the dielectric region and propagate along the interface of the plasma column and the dielectric, in other words for such slow waves one finds --- > 1, where E2 is the relative dik 4 I electric constant of the medium surrounding the plasma column. This medium imay represent a glass tube, Various passbands for slow-wave propagation will be obtained in the following investigation. Some of the passbands depend on the range of the values of the ratio -I > 1, and some passbands do not depend on k C2 the particular values of this ratio, provided -- > 1, the condition for the k 12 existence of slow surface waves. Finally numerical computation will be made for a special case. Conditions for Slow-Wave Propagation and Determination of Various Passbands In the following analysis only the expressions for the radial wave rnumbers, rl' and ri', will be considered. These quantities do not depend explicitly on any 1 2 particular boundary, except that the geometry is cylindrical and uniform in the

THE UNIVERSITY OF MICHIGAN 4386-1-Ti z-direction, thus the results will be true for any such structure, closed or open, which can support slow waves. For a particular structure, one must consider the solutions of the respective dispersion relation together with the following analysis. Since the solution of any dispersion relation, determines only a particular set of values of r and r'2, the limitation imposed on the values of r and r72, which are obtained from the expressions for r' 2 and r22 (valid for unbounded medium also) is furthermore narrowed. Therefore the requirements which are obtained from a study of the expressions for r712 and 7'22 alone, are nothing but l necessary conditions of wave propagation. The sufficient conditions for propagation of waves arce provided (only )y the silmultanerous solutions of the respective di;lpersion relation and the expressions for r2 and ri2 The expressioi.s for radial propagation wave numbers rl' and r2f can be written here in the following way nt2 =v+ v -u (5a) 1,2 1J2 x y 2-1 k2(1-y)(02-1)(1-x) +y} k2 xy(S2-_) x+y - 1 2(x+ y - 1) (5sb) 2 (21)x +4 2x(1-y) +ky - 2(x+ y - i) where 54 _____________

--- THE UNIVERSITY OF MICHIGAN 4386 —1-T| 2 v= ---- _ (/32-l){xy-2(x+y- 1 )}+2y(1 - y) (6a) 2(x+y- 1) L )1 k4(1-y) 2 22 29 u=- ( -1) (1-x)+2y( _2-1)+y (6b) x+y - 1 2 WC x- c2 (7a) 2 y 2 c~2 (7b) ez = 1 - y (7c) x+y-l e = (7d) r x-1 e'- Y'f- (7e) x -1 - =, (7f) k In the above definitions of ez, er' and e', the collision frequency v is neglected. It can also be shown that k2 2 v2 - u = [ ] (2 - 1) x2 + 432 x(l -y) (8) L2(x+ y - 1) J In the dielectric region 55

THE UNIVERSITY OF MICHIGAN 4386-1-T 972 2 2 2 k2 e2 - (9) since P2 = 1. 2 Therefore for slow surface waves, rn < 0, and 2 2 - - - 1>0 2 2 k E2 k 2 i.e. ~2 k E2 12 or - = (I > E2 (10) k where - = -jS, S > 0. It will also be assumed that E2 > 1. Since it is assumed that both plasma and dielectric are non-dissipative, the propagation wave number De is always real. Although the expressions for r7'2 and 1 r,2 show that these radial wave numbers may be complex, on physical grounds 2 only the real values of rt2 and r'2 will be allowed. For example* if complex values of ri2 and rf?2 (which are complex conjugates of one another) are allowed, this means that there exist growing waves showing instability of the plasma in the *The primary reason for allowing only the real values of 2 and U'2 in a nondissipative medium, is that the power flow, the characteristic impedance, etc. must be real for such a medium. 56 _

THE UNIVERSITY OF MICHIGAN 4386-1-T radial direction. Therefore this apparent inconsistant situation will be avoided by allowing only the real values of r'2 and r7f2 in the following analysis. The following three cases specify the conditions for which r-'2 and r22 are 1 2 realo Case I u < 0 in this case r1l2 > 0 and (lla) r'2 < 0. 2 Case II u > 0 in this case v >0, rj2 > 0, and (llb) and v2-u> 0 r)'2 > 0o 2 Case III u > 0, in this case v <0, r 2 < 0, and and v2-u > 0'2 < 0. (lic) 2 The regions of u and v in the above three cases are shown in Figure 2. 2 rf'2 and r1'2 become equal on the parabola v = u and both of them assume complex 1 2 values inside parabola, which contains the positive axis of u. Therefore this is the forbidden zone for u and v. When y < 1 (i. e. when cp < c), it is easy to show from (5) that rj2 and ryT2 are both real. So even if one considers instability in plasma, this does not occur for wp < w, provided the medium is loss-free, 57

THE UNIVERSITY OF MICHIGAN 4386-1 -T A A A='0=^ Xv --------------------— 58 ____ C_______

THE UNIVERSITY OF MICHIGAN 4386-1-T In the following detailed analysis the possibility of the above three individual cases, i. e. the conditions under which the parameters x, y, /3, etc. have to be chosen, will be shown. These conditions on the parameters, which are necessary, will give the maximum passbands of slow wave propagation. Before starting the actual analysis, it will be convenient to introduce 2 = 32 - 1 in the equations (6a), (6b) and (8) which can be rewritten in the following manner 2 (for slow wave 3 > 2 >_ 1): k2 (1 -y) u =y) L -(1 -x) + 2y + y2] (12a) x+y - 1 v = L xy - 2(x + y - 1) + 2y(l - y) (12b) 2(x +y - 1) 2 _ _ _ __k y 2c v - u = | -- *.. x + 4(o + 1) x(l - y) (12c) _ 2(x + y -1) L 2 + 1 = > E2 (13) Case I u < 0 r2 > o0,,2 < 0 1 2 This situation can be satisfied in the following three ways. 59

THE UNIVERSITY OF MICHIGAN 4386-1 -T 1 -x> 0 1 - y > 0 (14a) 1> x+y-1> 0 1 -x<O 1- y < 0 (14b) ~> Y (1+ 4T) x-1 1 - x< 1 - y > 0 (14c) < C Y (1 + J-) x-l1 Although the condition in (14a) does not explicitly depend on I, for slow wave rb must satisfy the condition (13). On the other hand the conditions in (14b) and (14c) show that besides the restriction imposed on x and y, (/ must satisfy two simultaneous conditions. More precisely, the inequalitites in (14b) suggest that $ must be greater than a certain value, i. e. the wave propagation which satisfies (14b) is possible for a value of / above a certain value. In other words this passband of wave propagation depends on the degree of slowness of the waves. On the other hand, the condition in (14c) can be met only for a definite range of I, namely 2 -1 < ~ < -Y (1+ J) x-01....____________________ 60 ________________________

THE UNIVERSITY OF MICHIGAN 4386-1-T where 1 - x< 0, and 1 - y> 0. Before investigating Case II and Case III, it is desirable to find conditions 2 2 under which u> 0, v - u > 0, v >, and v < 0, respectively. For u > 0, one finds the following possibilities 1 -y< (15a) l-x> 0 1 -y<0 1 -x< 0 (15b) < Y (1+ -x x-1 +o 1 -y> 0 1 -x< 0 { (15c) >,Y (1+ Y) x-1 1-x> 0 1 -y> 0 (15d) x+y - 1 < 0 2 For v - u > 0, the following conditions must be satisfied 1 -y> 0 (16a) 61

THE UNIVERSITY OF MICHIGAN 4386-1-T (here one may choose either x + y- 1 > 0, or x+y - 1 0) 1 -y 0 (16b) > 2(y- 1) + 2Y - )(x + - X X The following three situations satisfy V > 0: x+y- 1 > 0 l-y > 0 xy 4 2(x + y - 1) <.2y(l - y) IJ (2(x + y - ) -xy (17a) x+y- 1> 0 1-y> O xy> 2(x+y- 1) (17b) l-y(o xy > 2(x + y - 1) > 2y(y- 1) It > xy-2(x+y- 1) (17c) 62

THE UNIVERSITY OF MICHIGAN 4386-1-T Finally one obtains the following four possibilities for V 4 0: 1-y O 1 1f-yK I (18a) 2(x + y - 1) > xy 1-y <0 xy> 2(x + y -1) (18b) < -— 2y(l - y) xy - 2(x+y-1) x+y- 1 > 0 1 - y > O (18c) 2(x+ y - ) > xy $1> ~ 2y(l - y) 2(x + y - 1) - xy x+y- 1 0 (18d) l-y> To satisfy the requirements for Case II and Case III, it is necessary to satisfy inequalities (llb) and (llc) respectively. It can be shown by a little analysis that the following are the conditions by which Case II and Case III can be realized. 63

THE UNIVERSITY OF MICHIGAN 4386-1-T Case II This situation can be obtained if the following conditions are met: 1 -y 0 1 -x< (19) 2(x+y- 1)+x xy- 2(x+y- 1) > xy + 2(x+y- 1) > xy + 2(x - 1) ~ (y- 1) (x+y- 1) The inequalities in (19) are obtained from (15b), (16b) and (17c). There is no other possibility which can satisfy Case II. Case III In this case one can show that there are only four possible conditions as follows: 1-y >0 1 -x < 0 (20a) > x-1 (1+ fx) Note that this condition is the same as (15c), which also satisfies conditions (16a) and (18c) automatically. 64 _____

THE UNIVERSITY OF MICHIGAN 4386-1-T 1-x >O 1 - y > 0 (20b) x+y-1 < 0 The condition (20b) is the same as (15d) which also satisfies conditions (16a) and (18d). l-y < - x > 0 (20c) >, > 2+ ( (y- 1)(x+y- 1) X X This condition (20c) is obtained by combining the conditions (15a), (16b) and (18a). It may be noted that condition (15a) automatically satisfies (18a) also. The fourth possibility of realizing Case III is obtained by combining the conditions (15b), (16b) and (18b). Since it is not obvious that these three conditions can be satisfied simultaneously, it is necessary that they must meet the following requirements (a detailed analysis is omitted for the sake of brevity). 1-y< 0 1 - x < 0 1-x(O xy> 2(x + y- 1) (20d) 2(x + y - 1) + xy x > xy + 2(x - 1) \(y- l)(x+y- 1) > 2(x +y - 1) + x xy - 2(x +y - 1 ___________________________________ 65

THE UNIVERSITY OF MICHIGAN 4386-1-T Thus it is found that when the condition (10) and any of the following conditions (14), (19) and (29) are satisfied simultaneously, one obtains maximum passband of slow wave propagation. These conditions are necessary for slow waves. Study of Dispersion Relation for Slow Wave Propagation Under Various Special Situations Although the various results together with dispersion relations obtained in the preceding chapter are valid for any arbitrary angular variation of the magnetic current ring source, in this section only those dispersion relations which are independent of angular variation (i. e., n = 0, for constant amplitude of the ring source) will be considered. Since the solution of the dispersion relation, appropriate for any particular case, together with the expressions in (5) for r)'l2 and r'22 gives exact information of the propagation of waves, and as this solution cannot be obtained analytically in general, the information obtained here without actual solutions will give only necessary conditions for slow wave propagation. In general, the actual solution can be obtained only by numerical computation. Static limit: This static limit is a good approximation in the following situations: 1) circumference of the plasma column is much shorter than the wavelength of the operating frequency Trivel piece in his work [8] discusses this problem in detail and his method of solving this problem is different. Here his results are obtained as a limiting case. _____ 66 i

THE UNIVERSITY OF MICHIGAN 4386-1-T 2) for extremely slow wave, i.e., k2 >> 1. E2 E 1 This dispersion relation in this case for n = 0, can be obtained from the relation (3) of Appendix C, and it reduces to the following form: E rl'l Jjl('l0a) Ii( a) Ko(eb)+ I b)+Ib)Kl(a) (21) e2 d J (r'1a) I YCb)K (ea) - I (Wa)K ()b) 0 0 0 0 0 I where m 2 - (c/c) 2 I~~~~~~! } (22) i 2 _ _ x2 The above relations show that the dispersion relation and hence fields do not depend on 7'2 in this limit. Moreover, in this limit the magnetic current ring source excites only E-type mode. But in a general anisotropic medium characterized by e of the form shown in (1), pure E-type and H-type modes do not exist, i.e., they are coupled to each other. From the properties of modified Bessel's functions it can be shown that the right-hand side of (21) is positive and greater than unity (it approaches unity as |->G oo). If e /c < 0, r1' is real, a solution of (21) is possible. z r Since e / = (1 -(x- 1) < 0 z r x+y-1 either x > (23a) y>l ________________________ 67 ______l_____

THE UNIVERSITY OF MICHIGAN 4386-1-T or x(l y 1 (23b) 1 <x+yy 2 If e /e > 0, rl'2 < 0, and rl' is purely imaginary, the dispersion relation (21) z r becomes /eZ/eC Ii (9 a 1/e) 1W a)K (db)+I b)K (a) -E r > 1 (24) r eI (9a /Cr ) I (0b)K(da)-I ((a)K (db) A solution of (24) is possible if e < 0, e < 0 (since 2 > 0). As it is also known that Il(z) < I (z), for any z > 0, it is necessary that O ~r zr -r ez /~ -r > e2 or x+ y -1.(1 - y)(x -1) > E2 (25) -x V x+y-1 In particular if e2 = 1 (i. e., when the plasma column is surrounded by air), the inequality (25) reduces to the following (x+y- l)(y- 1) > 1 (26) 1 -x Finally the inequality (26) can be shown to be equivalent to the following two passbands for slow waves when the dielectric surrounding the plasma column is air: 68

THE UNIVERSITY OF MICHIGAN 4386-1-T y- 1 >0 1 - x >0 0 (27a) x+y > 2 y- 1 0 1 - x 0 (27b) x+y 2 It may be noted here also that when the plasma column is surrounded by a dielectric e2 > 1, the passbands are reduced further. When b = a, i. e., when the plasma completely fills the waveguide, it can be shown that the corresponding dispersion relation (in the static limit) reduces to the following J (tjia) = 0 (28) It is easy to show that only real values of rYl' can satisfy the above equation (28). Therefore, in this case also it is necessary that /e < 0. A study of the relations (21) and (22) reveals that for x = 1, y = 1 or x + y = 1, the dispersion relation (21) does not possess any solution, which is equivalent to saying that these points represent cut-off for the slow wave propagation. It may be remarked here that the passbands for slow wave propagation when the plasma completely fills the waveguide, give maximum range for the case when ________________________ 69 ____________

THE UNIVERSITY OF MICHIGAN 4386-1-T the plasma column partially fills the waveguide (provided rY'1 is real) in the static limit. These maximum passbands are depicted in the following Figures 3 and 4, subject to the conditions (23a) and (23b). (2 +k2 - u2)V2'1 + = E + J-,r/ = + 2 2)( c 2) (29)+ - z _ (W -~/.:. - W p c Since group velocity is defined as dw/dR, Figures 3 and 4 show that this value can also assume negative values - which proves the existence of backward wave in such a structure. With a few more remarks, the discussion of the static limit case will be concluded. The static limit results are reasonably valid for extremely slow wave propagation, as pointed out in the beginning of this section. In this limit r7'2 does not appear in the dispersion relation, showing that the field components for extremely slow wave propagation do not depend on r'2, when the source of excitation is a magnetic ring current. In other words in this limit an H-type mode is not excited. This does not mean, however, that in this limit' = 0. In fact, 21 =- j X, a large imaginary number, which shows that a wave dependent on 2 decays away very rapidly. So it may be conceived that in this limiting condition the H-type mode is very weakly coupled with the E-type mode and the components representing H-type mode are also very highly attenuated. It may be noted here + In 18 1 Trivelpiece has also obtained similar results. 70 ___

THE UNIVERSITY OF MICHIGAN 4386-1 -T + 3, 71,_1

THE UNIVERSITY OF MICHIGAN... 4386-1-T also that the results for /2/E2 >> 1, essentially correspond to those for static limit. General Dispersion Relation (when n = 0) for Slow Wave Propagation It can be shown from equation (561) of Chapter I, that for slow waves (when r = -j, S > 0), the dispersion relation becomes Ec (22 - r W2),f (a)G (a)Jl(riTa)Jl(r'2a) E2(rT'2 - nl)C (a)S (a)J (rTla)J (rT2a) z 0 o o 0 o0o?1 r+'2 k2 (32-_ 2) -J (r1'2a)Jl(r1'2a) ~ e MC (a)G(a) - eoS (a)G (a)- S(a ky'2 y^ -— x-~c2 J (r1'2a)J1 (ri'la) k7+ 2~' ^[2 S C (a)G (a) - e M (a)S (a)] (30) where 2 = 6 2 = k22- d2 (31a) R/k = / (31b) M = k2(c/ )(er- 32)_- 12 (31c) k2E (31d) S Z (e (c - P2) 2,' E r r J(a) = I (S b)K ( a) - I (S a)K ( b) (31e) 0 0 0 0 0

THE UNIVERSITY OF MICHIGAN 4386-1-T S (a) = Ii( b)K(S a) - I( a)Kl( b) (31g) o 1 C (a) = Ii( a)Ko( b)+ I ( b)Kl( a) (31h) In connection with the solution of the dispersion relation (30), appropriate for slow waves, no general discussion can be made. Only a numerical solution subject to the expressions for r1'12 and r122 given in (5), can give the actual nature of slow wave propagation. The dispersion relation (30) will be solved subsequently for a special case stated in (14a), for which u ( 0, r1t is real and rl'2 is imaginary. For Zero Magnetic Field (with n = 0) It has been shown in Appendix C that in this special case the dispersion relation has the following particular form (for surface waves) - S J(rt'la) I1(S a)K (S b) + I (S b)Kl(S a) = (32)+ rlT e2Jo(r1'la) I (S b)K ( a) - I ( a)K (< b) 0 ~~0 0 0 0 where r7'2 = k2 -2 = p2 (33a) z n k2= k2e2-2 = 2 (33b) +This result agrees with that obtained in [7. 73 [~~~~~~~~~~~~~~~i 7bm

THE UNIVERSITY OF MICHIGAN 4386-1-T In this case of isotropic plasma only an E-type mode is excited, due to the type of the source of excitation chosen. Since e is always less than 1 and e2 > 1, it can be shown from (33a) and (33b) that when rl is imaginary, Yr)l is also imaginary. In this circumstance a surface wave is possible if e < 0, moreover, since 1 (pa) the left hand side of (32) is greater than unity, p/d' ( 1, and () < 1, it is w Ig~~I(pa) also necessary that IE /E2 > 1, i.e., w <(/1. For Infinite D. C. Magnetic Field in the z-Direction In this case the dispersion relation for slow waves has the same form as (32), with 12 = E r72 = -e 6 2.It can be shown, [73 that slow wave is possible if E < 0, which makes 7'1 real. For zero or infinite d.c. magnetic field in the z axial direction, an elaborate investigation has been made in [7] Propagation of Slow Waves in an Infinitely Long Column of Plasma Embedded in an Unbounded Medium, with an Axial Uniform Static Magnetic Field To investigate all related results and the nature of slow wave propagation, in this case, it is only necessary to let b - oo in the corresponding results of the waveguide problem, with r72 = -< 2 = k2e2-_2. In this case the dispersion relation takes the following form which is equivalent to equation (22b) of Appendix C (with r = A= 1' = 0, P2 =1) __________________________ 74 _

THE UNIVERSITY OF MICHIGAN 4386-1-T e J (r'1a)J (r'2a) aE K 2(' a) z 1 rTl n'2 J ('a)J (('a)J (r2 K2 ( a) 0 0 0 E2 K1(S a) r Jl(r'a) J1(r2a) + T r'2 S(71'22-_T'12) K (S a) l' J (?'la) -('1 a) j n1n'2 S (r7'2-r17'2) K (8 a) ) J (rT'2a) J (r't2a) 1 ~0 0 0 (34) The following identities are found useful E2(1 - E )(Snt'22 + Mr7'12) = e'2 (k2 +02)(k2E + d2) (35a) and 2(1- e )(Sr'22- MrT'12) = e (S - M) (k2 -X2)(1+e ) - k2E2 (35b) r z xz z r -^~~~ 1 - To derive relation (34) it also has been assumed that - = 0, which may be interpreted as following from taking the excitation to be a constant magnetic current ring source. When the d. c. magnetic field is either zero or infinity, only the E-type mode is excited due to the type of source chosen here. In this particular case the dispersion relation takes the following form This result agrees with the corresponding result in [4, when the identities (35a) and (35b) are used (with c2 = 1). It should be noted, however, that to obtain radiated fields in this configuration the present limiting process is not valid. _____________________ ~75 ___________l__

THE UNIVERSITY OF MICHIGAN 4386-1-T 6E Jl(r1la) K1( a) riW ~~~i^ = -'~~~ Kj(36) nl~c2Jo(Nila) K (5 a) If the axial static magnetic field is zero, e = e, and r'l2 = ke - d2, the r z z condition of slow wave is exactly the same as stated in connection with the similar situation in the metallic wave guide, namely, e < 0, and Ic | /2 > 1. z z On the other hand, if the magnetic field is infinity, a slow wave is possible if e < 0 and r'l = e S2. z z Static limit: It can be shown that in the static limit with b - oo, the dispersion relation (21) reduces to the following expression E ~t JY (r'1a) K (ha) ~~~~~r ___ - _/(37) 2 J (t'1a) Ko(a) i where 1 2 _Z 2 r (38) If E /e < 0, then r7' is real and a solution to (37) is possible for a slow wave. If e /e > 0, then rt'l is imaginary, i.e., rt'l = -jp, where p > O. In this z r case (37) transforms to the following form l __________________________________~ 76 ____

THE UNIVERSITY OF MICHIGAN 4386-1-T c2pIl(ap) K (4a) I Catl(ap) KJta) (39) K (a) I (ap) Since K a) > 1 and I (p) 1, it is necessary in order for (39) to K a) 0 Eap) have any solution that E < 0 and also > 1. It has already been stated r c2 above that e /e > 0, therefore, e < O and e ( 0. z r z r If one writes x+y-1 If one writes = -, e = 1-y, as defined in (7), then the conditions r x- 1 z e 40, e < 0 and L >rl 1, become equivalent to z r C2 (y- )(x+y- 1) 1 - x 2> ( 1-x l-x>0O y - 1 > o 1 - x > 0 (40) Note: The discussion on page 497 of [4 of the situation where u = 0, i.e., r'2 = 0 seems to be inconsistent. The first reason is that when r7'2 = 0, it can be shown from the general expressions for E and H appearing in Chapter I, as well as in Appendix A, that electromagnetic waves which can exist under such a situation are TEM only. In this statement it is also assumed that the diagonal components of ~ are finite and non-zero. But a TEM wave cannot exist in a structure considered by the authors of [4J. A similar inconsistency appears on pp 183 - 185 of [6], discussed by Agdur. _________________________ 77 ____________

THE UNIVERSITY OF MICHIGAN 4386-1-T Secondly, the authors of [4] consider a case when y-+ o (and r'2 = 0), and simplify the dispersion relation to an expression containing a logarithmic term, although the modified Bessel's functions Kl(z) and K (z) do not behave as logarithmic functions for large arguments. Any other interesting situation can be studied by considering the corresponding dispersion relation given in Appendix C. Expressions for E Which is Independent of Angular Variation (i.e., n = 0) Since E plays an important role in a plasma, its expression will be given z here for n = 0. |E I will also be calculated numerically as a function of r for a special case of slow wave. +ji.z * Em e.2iC' (c) E =-jWEe2 2cI E 27 -- (c)I [Si.gJ (rji'.r) - M.J(r'1r) E 0 ~ 2 2 I ii o 21 10 for 0 r a (41) S2 J (lil'.a) - M.J (rnla), 1 li 0 2i — I_ (42a).2i (rt2 - t2 (a) S. = - (k2E -d<.)-r'2 (42b) 1i r 1 i i r _M = <-_(k2 -o)-,2E (42c) i e r i -2i r ____ 78. _ _ _ _ _ _

THE UNIVERSITY OF MICHIGAN 4386-1-T Fil = L.ce I',2 12 MR.- R r'2' p (42d) 1 r z r ci r 2i i I o 2i 1 ic Fi i r z |r172i |li| L"'ot'^i Ti to' + e'-ll] (g' ) F = 12 I 2 l il27 N ( 2l7 l | lE R.i E |1 i j (42e) i2 I rzl L r 02i 1i o ii iJ 1 |=l| [- O*f I 2 | 2i| 2J~(rib)NO(r'ib) + po lg3i|2Nl(r'ib)* b>J (42j) | Fi8 = L.c e r'2 r 2i| |JO ibw +O|3i|2 lW''' l2] (42f) 12 ir l - i li. r12a) J (i1r lia)i (421) 13 1 r z lir12i ii rl (42m) i i r z 2i li 1 i r 1 i e M Eke -^2N) - k e 2 b2 (43a) w4. 1.6 2 E 01 2 L 2 121i 1 0 1 —^ ---- z- -oib) + 0 3(423) 1 2 _F i_ E i -Iiib)N0 79b) + pIN3ii N(i b)J 1(ib)j (42j) 17 H2 L2 1211001 oJ32 1o 2 18 = 1 Lc2 j%12 j (nib)j2+p 112 Ii (riob)12J (42k) =i5 i1i J r1'2 a)JWr1 (421) (r 12 -r'2 )E G (a) 110 21 0 1 J o2i ir o ~L.= ~ ~ ~ I (42m) 1i 4,12'o12 (r'2 r+2i)'r I o 2i 2 2 ii c M.(k2E rz - k c R.= = (43a) _ _ _ _ _ _ _ _ _ _ _ _ _ __ j79 _

THE UNIVERSITY OF MICHIGAN 4386-1-T E MX. -.E (k2E -A2) r 1 1i z r 1 ( ) R (43b) C z (r) = J (r7ib)N(r7ir) - Jo(r7ir)N o(rib) (43c) Si (a) = J (r7ib)Nli(ria) - J1(r7ia)N1 (rib) (43d) Gio(a) = Ji(rib)No(r7ia) - J (ria)Nl(rib) (43e) Cio() = Jl(ric)No(rib) - J (r7ib)Nl(ri c) (43f) 0 al g Jl~nlir)J(?7tlir~rdr (44a) i2 = Ii J1(r1ir)Ji (r7n2r) rdr (44b) 1i3 =\ Ji(n' r)Ji 67.r)rdr (44c).., Ii4 = J(rT r)Ji (r7l2ir)rdr (44d) i15 = Jl(rlir)J1(rir) rdr (44e) ~0~ ~80 1i5 = i i(n i( ii ir)rdr (44e) a I16 = J1(rn.r)N1(ri.r)rdr (44f) _______________________________ 80 _______________________________

, THE UNIVERSITY OF MICHIGAN 4386-1-T Ii7 = J i(r.r)N1(r' r)rdr (44g) a Ii8 = Ni(r)ir)N1(nir)rdr (44h)

THE UNIVERSITY OF MICHIGAN 4386-1-T III PROPAGATION OF SLOW WAVES IN AN ANISOTROPIC FERRITE In this chapter the general problem solved in Chapter I will be applied to the study of wave propagation in an infinitely long anisotropic ferrite column enclosed in a dielectric medium, which is again enclosed by a perfectly conducting metallic cylindrical waveguide. All other conditions are similar to those described in the previous chapters. In a ferrite medium with an axial static magnetic field, anisotropy is exhibited by the following dyadic form [23 of /rr rO ~ O e-1 r 0 0 (1) 0 0 1Uzz P where Mr = A or 1 - g2 (2a) rO - fOr t - --- _ (2b) 1-o0 Pu = 1 (2c) M P | i ~ (3a) H i)= 07 |-~ (3b),____________________________,,82

THE UNIVERSITY OF MICHIGAN 4386-1-T M = d.c. magnetization H = d.c. magnetic intensity 7 = gyromagnetic ratio for electron It can be shown from (2) that l and p' are related by the following relation - 1 = lT' (4) r It will be assumed that the relative dielectric constant of the ferrite is e 1, a scalar quantity and the medium surrounding the ferrite has relative dielectric constant C2 and relative permeability 1. With the above assumptions, the magnetic fields, dispersion relations etc. for this problem under any limiting conditions can be easily derived from the corresponding results given in Chapter I and Appendix C. Therefore, no detailed discussion will be given in this chapter. It may be noted here that when a ferrite column or a plasma column is situated in an unbounded dielectric medium, the boundary conditions for E and H in both cases are identical, consequently any general expression for one situation can be derived from the other, using the duality, provided p is not replaced by unity in any general expression. ____________________________ 83

THE UNIVERSITY OF MICHIGAN 4386-1-T Expressions for Transverse Wave Numbers If the values for p and C given above are substituted in equations (34) and (35) of Chapter I, one obtains the following expressions t2 -- 1,2 = V+JV2- U (5a)+ k^~2 -E I 2) - r_ = - 2 2C -'_ ~ [ ke1(Ur-A') 2t +4k2^2e /2 (5b) k2C(u-2 +,2 + ) - 2 + 1) (6a) r 1r 22 7 = -j, > o,2 -- — ( - x') - k 2 (5b) + For ferrite problem the relation follow= ing2 cannot be satisfied, since V2- U> 0. L~~____~~____2 84 ____2 (7) wr J~~~8

THE UNIVERSITY OF MICHIGAN 4386-1-T T2,2 Since p = 1, equations (5) show that V2- U > 0 and 717 and 1 2 are real for real /u, 1 and C 2 Therefore, no instability phenomena appears in the case T! of a ferrite column. It may be noted that 1 and 12 may assume purely imaginary values as follows: T2 Case I 71 > 0 if U<0, V>0, or V<0 (8a) t2 12 < J t2 Case II 11 > 0 if U> 0, and V >0 (8b),2 2 > 0 f2 Case III oi < 0 2 if U>0, and V<0 (8c) 2 < 0 The above information and the general results given in Chapter I and Appendix C are sufficient to obtain any particular result for a ferrite column. It may be noted that an electric current dipole source is more appropriate for a ferrite problem than a magnetic current ring source. The field expressions for an electric dipole source can be obtained easily by using the appropriate dyadic Green's functions developed in Appendix B. 85 _

THE UNIVERSITY OF MICHIGAN 4386-1-T IV CONCLUSIONS In conclusion it may be mentioned that the work described here gives a systematic rigorous approach of solving a source problem (not necessarily a ring source) involving a homogeneous anisotropic cylindrical structure bounded by conductors. Since the source free solutions are capable of representing all possible modes for the structure of the problem, total fields due to any arbitrary source of any kind (namely electric or magnetic current source) can be calculated by using the appropriate dyadic Green's function. If there is more than one source, the total fields can be obtained by using the superposition theorem, provided there are no interactions among the individual sources. As the present analysis considers a magnetic current ring source of arbitrary angular variation, the results can be used for any given angular variation of the source. From the general dispersion relation which is an eigenvalue equation and independent of source, various interesting special cases, some of which are already known, have been studied. The limiting procedures used in obtaining the dispersion relations for these special cases can also be used to obtain the expressions for the fields in the corresponding situations. The analysis for the plasma problem which is a special case of a general anisotropic medium characterized by dyadics C and,i, emphasizes the slow wave propagation. Here the necessary conditions for slow wave propagation, including 86 ___

THE UNIVERSITY OF MICHIGAN 4386-1-T a number of special cases, have been obtained from the expressions for the transverse wave numbers. These necessary conditions also give maximum passbands. The sufficient conditions and hence the actual passbands can be obtained from the solution of the dispersion relation. For slow wave propagation, some passbands depend on the degree of slowness of the waves. The degree of slowness of the waves depends on the relative dielectric constant, C 2, of the medium surrounding the plasma column, when all other parameters are kept constant. It has been shown that the higher the value of 6 2, the slower the phase velocity of the wave. In other words, for a given C 2 there is a minimum phase velocity for which a corresponding slow-surface wave can propagate. Most of the energy of the slow waves considered here is confined into the anisotropic plasma. In general, the lower the phase velocity of the surface wave, the lower the amplitude. Not all the various passbands for slow wave propagation in an anisotropic plasma column, mentioned above, are known in the literature, at least to the best knowledge of the author. Although these passbands could be obtained without any consideration of the presence of a source. In the case of an unbounded homogeneous anisotropic medium where a TEM wave can propagate, conditions of TEM wave propagation in the direction parallel to or perpendicular to the static magnetic field are obtained from the general expressions of the transverse wave numbers. It is also shown that the condition of 87 _

THE UNIVERSITY OF MICHIGAN 4386-1-T TEM wave propagation in the direction of the static magnetic field is equivalent to the vanishing condition of the product of the transverse wave numbers. This also establishes the fact that the consideration of the situation under which the product of the transverse wave numbers vanishes is not justified in connection with the wave propagation in a bounded medium which cannot support TEM waves. In other words, if a bounded isotropic medium cannot support TEM waves, so also is the case for a bounded anisotropic medium. For not being able to recognize the fact that the condition of TEM wave propagation in the direction of the static magnetic field, is equivalent to the zero-value of the product of the two transverse wave numbers, some authors discussed the possibility of wave propagations in a bounded anisotropic plasma column, under the situation for which the product of the two transverse wave numbers vanishes. + Agdur, pages 183-185 of Ref. [6] and the authors of Ref. [4, page 497. ____ 88 ______

THE UNIVERSITY OF MICHIGAN 4386-1-T APPENDIX A MAXWELL'S EQUATIONS FOR ANISOTROPIC MEDIUM The medium to be described here is characterized by a relative dielectric (permittivity) tensor C and a relative permeability tensor pu having the following particular form 11l1 jE12 ~ -.jC^12 22 ~ (1) 0~ ~ C33, 11 22 P11 Pl2 ~ M I= -12 p22 0 (2) 0 0 P33 11. 11 P=22 The above representations show, in both cases, that the transverse comp|nents and longitudinal components of the tensors are uncoupled, where C and L33 correspond to the longitudinal, a preferred direction (say z-direction) components of the tensors C and p respectively. Therefore, e and p can be written ___________________________ 89 ______l_____

THE UNIVERSITY OF MICHIGAN 4386-1-T in the following manner also = C +z z (3) ~tt - oC 33( AP = tt + ZoZo 33 (4) where 6tt and /tt are tensors (transverse to z-direction) and z is a unit vector in the z-direction. The Maxwell's equations for anisotropic media with sources have the following form (the time dependence being ejt). V x E(r) = - j0 tp(r) H(r) -I (r') (5) Vx H(r) = jw6 6(r) E(r)+I (r") (6) V,u(r) H(r) = 0, for r A r' (7) V' 6(r) E (r) = 0, for r A r" (8) where, r = observation position vector (3-dimensional) I (r') = magnetic current source at r = r' I (r") = electric current source at r = r" -e -

THE UNIVERSITY OF MICHIGAN 4386-1-T In the following discussion the transverse vector and transverse operators (which are designated by the subscript t) correspond to any plane transverse to the zdirection, and the transverse plane may have any arbitrary cross section. That is, the following derivations are suitable for any cylindrical geometry having the z-direction as its axis. First of all, it will be shown (see [10o, [ll], [12], and [14] ) that the longitudinal fields, E and H, can be expressed in terms of the transverse fields, E and H t. Secondly, it will be demonstrated that the transverse fields can also be expressed from the knowledge of the longitudinal fields. In the particular problem discussed in the text, the latter method has been adopted for the solutions of Maxwell's equations. If equations (5) and (6) are multiplied by z in a scalar product fashion, E and H can be expressed in the following way z z E = V H x z - (9) Z JW 6 33 t~ - 33 1 _mz H = 1 V, z x E - -mz (10) j 33 t0 33 where I and I are the z-components of I and I respectively. ez mz -e -m Now taking the vector product of (5) and (6) with z, one obtains ________________________ 91 _____,______l

THE UNIVERSITY OF MICHIGAN 4386-1-T z x (V x E) = - jwp/ z x *' H + I x z (11) -o oo - - -m -o z x (V x H) = jwe z x e E + z xI (12) -o oo - -o -e Introducing v='o a * t -o az E = E +z E - -t -o z -0 z and H= H + zH - — t -o z one obtains z x (Vx E) = VtE - E -o - z zand z x (Vx H) = VH - -o t z az- t Now equations (11) and (12) can be rewritten as V E - - E = -jwt z x L H+ I x z (13) t z az-t o-o t I -ot -mt -o and 7VH- HH jwe z x E+z x I t z az -t o -o t -o -et (14) where I t and It are transverse components of I and I respectively. -mt -et -m -e Again operating (14) by jwpu z x and (13) by a from the left one obtains the following expressions 92

THE UNIVERSITY OF MICHIGAN 4386-1-T a a2 a a aZ VtE - -- E - jwjs a (Z X ut Ht) a-tI x z (15) az t z OT a~K -0 o _z -o %t - z -mt -o and jw zO x (tA V H jwPo aL (z x p H) = x t _ x t ot + o x t -o -et (16) where k2 = 6 Adding (15) and (16), one can show that tE + jw.u zO x (P VH) + -z x I + (p 1- Jl2 xlt 1et = aEt -jka3z x E (17) where 1, = transverse unit dyadic (18a) a3 A11612 + 126 1 (18b) a4 = k2(E11p11+ (12-12)+ a2 (18c) \1- 1 x )zx z xIi) (18d) ll-t - 12-o x t) I et Zo x (t' z x et) (18d) Taking the vector product of (17) with z, one obtains the following independent equation: ________________________ 93 _____________

THE UNIVERSITY OF MICHIGAN 4386-1-T x z-jw x tE -j / 7H I +(jt 1 + xlI az -o tz t z - Iz mt + 12t 11o x t) -et =a4z x t+jk2a3E (19) Eliminating z x E from (17) and (19) by multiplying (17) by a4 from the left and (19) by j k2a and then adding the results, Et can be expressed in terms of E H, I and I in the following way: z z -mt -et PE:a -a E - wp a? z x a E + j aIV t =L a4 az t z 0 2 VtHzJ-o Lj 3 az t z +j ol t -a^o -t a - a z x 1 -j - a4 z z t 3 az ~t -mt (20) where the following relations have been used. 42 2 p = k a3 - a (21a) 2 2 2 2 a2 (21b) a1 = 411 a3 12 11( 11 12) (21b) 3z 2 2 2 2 a a = k2a3 l - a412 = k 2( 12 1 (21c) 2 3 11 4 12 12 1 2 2 z and the identities _________________________ 94 ________________________

THE UNIVERSITY OF MICHIGAN 4386-1-T t'' z = /ll tHz- jm12- o x VtH (22a) z x Ht = / Z xVH +j1t jH/1 -o x t tH x Vt j2 tz (22b) By a similar method (or by duality of (20) ), one can obtain the expressions for Ht in terms of H, E, It and Iet in the following form: z z -mt -et P H = [-a4 a vtH + ea2 tE z] - o 3z tL4z Z t2t z - a1t + j a z 1t Imt 1-t * 02-o ^J Mt + 4 az _t-o Xt 3 az ztz -et (23) where 2 2 2 a2 al k 1l (e + 6 (24a) 1 11 11 12 11 2 az 2 2 2 a2 a2 = k2 12(6l- )612- (24b) 8az Although the medium is anisotropic, if it is homogeneous (i. e., components of ~ and pu are not functions of position, although they may be piecewise constants) and source free (i.e., I = 0 = I ), one can show that ~ and ^ -m -e z z _______________________ 95 II9______________________

THE UNIVERSITY OF MICHIGAN 4386-1-T satisfy the following equations [using (20) and (23) in (9) and (10)], V2 6 + -33 aj =a t-33 a (25) t z <1ll a 1 z 611 M z (25) V2 + /33 a, =- J~33 a6 (26)3 t z G11P11 az 11P11 3 z where C and i are solutions of homogeneous (source-free) Maxwell's equations. To obtain the above two expressions, it has been assumed that a = -jP(, az where S- is the propagation wave number in the z-direction. This assumption is permissible in the situations where both the 6, p and the geometry of the problem are independent of z, subject to another restriction, that tne transverse anisotropy of e and,u are not coupled to the longitudinal anisotropy of E and u | respectively. To obtain solutions for u and L, it is possible to have 4th degree z z equations in C and X from (25) and (26) by elimination. Since it is a tedious 6z z task to solve such equations, one can alternatively find a function 0 which is a linear combination of C and, satisfying a two dimensional wave equation. Let such a choice be 0 = & -IC a(l,

THE UNIVERSITY OF MICHIGAN 4386-1-T Now multiplying (26) by jot and then adding the result to (25), one obtains the following relations 2 I/ e jaj~)c 33 (6+ j ) - [al - wdoa3-f ] a t + 33 [a - C3] ja = 0 (28) t~l z 1/ J l 11 11 The above equation can be represented as a two dimensional wave equation in 0 of the following form \7 0+120 = 0 (29) where 633 r - w6a 1 2 332 a 03 (30) C- l7l^ 1 ~ 3 _ 1 P' Ll' c (3a ) Solving equation (30) for a, one obtains r 2 22 1/2 a atp + I(at6 -ap) +4k2eau I33a - zzal + 1 al 33-al133) + 4k a3 333 3 al 12 2we=a (31) 0 oz 3 2 Therefore, the roots r'can also be expressed in the following form: 2 331 33a 1 a33 1 B 33 a ) +4k2e2a23333 ]1/2 2 =11p11 261111 2a 1111 (32) The equation (32) can also be rewritten in the following form 1l,2 = V~/V-U (33) 97 _

THE UNIVERSITY OF MICHIGAN 4386-1-T where at + 2 2 = 133 a1p33 _ 2+; +T v 2 -- 2E/1 3 1 3 2 (34a) =633 a-k42] = -333 = X,27,2 ( a33 33 r2 42 3333 2 2 /, U = 611/ 11 P it (34b) 1 1P 1 P 1 1/7 1 1 2 Relations Between Various Parameters Introduced in the Above Analysis First of all, the following new parameters are introduced and defined: E a' L E 6a a /1111 11 11ll __= _ 17k = weA6 (35a) M =33;1 _ 1,2 33 3 (35b) /11411- 2 = /1161 R = a qde2- - ao2 (36a) T = wpa- dat Xa (36b) R' = woa2a2 - a 4 (37a) T'= we6a2 - Xa4 (37b) 98

THE UNIVERSITY OF MICHIGAN 4386-1-T It will be found convenient to carry out algebraic operations in the sequel using the relations listed in the following table. In tabulating the following results, no attempt has been made to write them in such a way that any result is a consequence of those preceding it. ael S.1. S (38-1) a,2 M a' w6 6 dea a,2 33 _ o 33 3 1(38-2) F2 33a W Fo33 ea3 a,2 331 - 0 33 3 2 (38-3) 2 ml61 1 11611l 1 11 12 - =..) (38-4)| a 1-a2 = —---- - ~(38-4) 0 3 33 0 P/33 1a2 = - (38-5) o 33 E a; a t e 9a a s= 331 - 2 = o33 3 1 (38-6) ~11Q 11 1 ^111 11 11S A11 iiS 11 A a1 = w ad (38-7) ~oa3 ~3399

THE UNIVERSITY OF MICHIGAN 4386-1-T M 33 1 2 o 33 3 38-8) 11 11 pll111 a2 =~- 11P11M (38-9) 0 3 33 a E S t = _ 2(38-10) 1 - a2 12 72 1 - e112 a2 M (38-11) 1 -a2 12 _ 2 2 2 a t E33 + al/P33,2 +'2 = 133 +a133 (38-12) E -M'2 2 S- = 2 - (38-13) ~.2p2 2 3' 33 33 MS = - (38-14) 2 2 11 /11 42 2 1111 2 2 P1 = ka3 a4 11- 1a2 1/? (38-15) 333/33 1 2 SR + MT = -wpa (12 - 1 2) (38-16) a4M 2 - a3k a (3R =aa -w at 4 3 2 33 (38-17) 0 33 A00

THE UNIVERSITY OF MICHIGAN 4386-1-T 2 k 33a3a - aa4Sl/11 T = wp a- - ga = - 33 3 2 4 n n l(38-18) o2 41 W oa3633 Ma2 11/11- 3a4 33 RI= we a - 4 = 33 (38-19) Sa2611/p11 - d2 a3a4633 (38-20) a 3 33 T26 - a (38-20) a I _l - e All 1 2 = l llM (38-21) e33al - 611112 11p11 (38-22) 33a? - 2l /1 )2 = 1s - 1122 (38-22)!33 1 - ll l = S = - ll (38-24) k a3e33 33 d33 a - 11p11 a- Pal (38-23) 33 2 11 112 2 al = 3a l - k 2 M12 (38-26) 1111111 2 222 a3 = a2Pll - a1l12 (38-27) a1 = a411 k a12 (38-28) _______________________ 101 ___

THE UNIVERSITY OF MICHIGAN 4386-1-T a= k 2aAl' a 1 (38-29)= 2 a311- a412 (38-29) a2 -kaG -a (38-30) 2 aa2 a 4 aa pll (38-32) a4a3 - a2al = P1 11 12 k a3a2 a 11 a (38-33) 3 2 4l 11 Pi 2,O2 2 k a3- aa; = 1p11 (38-34) a k 2A 2 2 2 (38-35) 2 2 2 2 a1 =k11( 11 12) - 11 (38-35) a' = 2k261lUll ~- "'12> - 21 (38-36) a2 = k 12(6- 62 + 2 (38-37) a? = k2612(11 - 12)+ X 2 (38-38) a3 = 611/12+ 11612 (38-39) a4 = k2 If ll + -12612-) (38-40) 102 ______...

THE UNIVERSITY OF MICHIGAN 4386-1-T It may be noted that all of the analyses and results obtained in this Appendix are based on the assumption that the problem has a cylindrical geometry with axis in the z-direction and having an arbitrary cross section which is independent of the coordinate z. Propagation of TEM Waves in an Unbounded Homogeneous Anisotropic Medium Since the foregoing analysis does not include any particular boundary, it is valid for an unbounded medium also. Hence, it is possible to obtain conditions for TEM wave propagation in a direction parallel or perpendicular to the z-axis. It may be mentioned here that a plasma and a ferrite with a static uniform magnetic field in the z-direction will have tensor permittivity and tensor permeability respectively. The forms of these tensors are given in equations (1) and (2). As mentioned earlier, here also the medium considered will have both E and pt as tensors with constant elements. TEM Wave Parallel to the Magnetic Field For a TEM wave in the z-direction both E = 0 and H = 0. If the source z z terms in equations (20) and (23) are equated to zero, non-vanishing values of E and Ht are possible if and only if pl= 0, when E = 0 = H, provided the elements - 11 E and 33 are finite and non-zero. The condition =, gives two 6 11' wa33e pr and at33 are finite and non-zero. The condition pa= 0, gives two TEM waves propagating in the z-direction. These two waves are characterized by 103 _

THE UNIVERSITY OF MICHIGAN 4386-1-T the following expression of the propagation wave number d( in the z-direction x - (6E E+6 ) (/t + A(39) k2 ( E11 12) (11 12' (39) Since p is proportional to 2 1 2 the condition 1 172 2 = 0 is equivalent to TEM wave propagation in the z-direction, provided the diagonal elements of E and p, are finite and non-zero. TEM Wave in the Direction Perpendicular to the Static Magnetic Field The conditions for a TEM wave propagating in the direction perpendicular (i. e., perpendicular to the z-axis) to the static magnetic field can be obtained upon substitution of ~ = 0 (i.e., 0 = 0) in the expression (32). This substitution gives at two propagation wave numbers ~ 1)2 which represent two TEM waves in the transverse plane of the z-axis,2 _=1 33 2 2) (40a) k2 S ( ( - P )'1 (40a) k2 p1 1 1l 12 and 12 pA33 2 2 (6 - 6 ) (40b) 2^~-' =^ 6 11 12(40b) k2 11 12 The results obtained in (39) and (40) agree with those obtained by Van Trier __.__________________________ 104 ____________

THE UNIVERSITY OF MICHIGAN 4386-1-T l1] by an entirely different approach. When pl= 0, a study of the expressions (33) and (34) shows that 2= 0, provided the diagonal components of G and p are finite and non-zero. It can be shown also that such TEM modes in an unbounded homogeneous anisotropic medium do not vary in the transverse plane, but in a coaxial waveguide TEM waves behave as 1/r in the transverse plane. The above statement follows from the fact that for a TE M wave the transversely varying part of the transverse fields can be derived from - Vt0(o),7 where 0 () is a scalar potential dependent on the transverse coordinate p. The two waves given by (39) are known as ordinary and extraordinary waves in the literatures of the ionospheric wave propagation. The permeability of the ionosphere is a scalar quantity and equal to that of free space. These two equations also explain the phenomena known as Faraday Rotation. The two waves represented by (40a) and (40b) can be said to explain [1] the phenomena known as magnetic and electric Cotton-Mouten effect..______________________________ 105 _

THE UNIVERSITY OF MICHIGAN 4386-1-T APPENDIX B CONSTRUCTION OF DYADIC GREEN'S FUNCTIONS Although a construction of Dyadic Green's functions from the source-free solutions of Maxwell's equations for inhomogeneous anisotropic non-dissipative media in a uniform waveguide of arbitrary cross section bounded by a perfect conductor, has been discussed in [i3], they will be also briefly presented here for the sake of completeness of this work. In this appendix the corresponding results for anisotropic dissipative medium will also be obtained. The most important technique involved in the construction of dyadic Green's functions is the determination of an appropriate orthogonality condition among the source-free solutions (i. e., eigenfunctions) of Maxwell's equations. Methods of finding such orthogonality conditions have been discussed elaborately by the authors in [9, under different situations. Here an indirect method will be presented for the construction of dyadic Green's functions 12]. In this method it will be assumed that the sources are due to some discontinuities, which causes discontinuities in the fields also. Dyadic Green's functions Z(r r'), T (r, r') Y(r, r')and T (r, r') - - em- me - are defined by the following expressions: E(r) = - Z(r,' I (r'))'T (r') I (r) dV' (1) -e - — m -- V V l___________________________________ 106 _

THE UNIVERSITY OF MICHIGAN 4386-1-T H(r) = -S C Y(r,r') I (r')dV' - Tme(r, r' I (r') (2) in. jJ1T —me - -eV V Where the E(r), H(r), I, I have the same significance as given by the Maxwell's equations (5) to (8) of Appendix A. Instead of volume currents, if I -e and I represent surface currents, the volume integrals in (1) and (2) should be -m replaced by surface integrals (over the regions of surface currents). In the following are given the physical meanings of dyadic Green's functions: - Z(r, r') * u = electric field at r due to a point electric current source at r', directed along the unit vector u. - T (r, r')' v electric field at r due to a point magnetic current source at r', directed along the unit vector v. - Y(r, r') * v = magnetic field at r due to a point magnetic current source 1IN-' ~ at r' directed along the unit vector v. - T (r, r')' u = magnetic field at r due to a point electric current source at r' directed along the unit vector u. In the above statements the point source means a source which has spatial variation as a Dirac delta function S (r - r'). Let 8 (r) and _(r) be the solutions of the homogeneous (source-free) Maxwell's equations (3) V x (r) = -jwpp(p) 1(r) (3a) 107

THE UNIVERSITY OF MICHIGAN 4386-1-T x _(_r) = jwe e(p) _ (r) (3b) where pu(p) and e (p) are functions of transverse coordinate p only. Since under appropriate boundary conditions, _ (r) and (r) form a complete orthogonal set, the total fields E(r) and H(r) due to any arbitrary source can be expressed as a superposition of _ (r) and ~(r) in the following way: E(r) = A c (r) (4a) a and H(r) = A (r) (4b) a where A is the coefficient of expansion corresponding to a-th-mode (eigenvalue). Reciprocity Relations for Homogeneous Maxwell's Equations To establish Lorentz's reciprocity relation and hence an orthogonality condition it is desirable to consider another set of Maxwell's equations. This new set of equations is sormetirn s called the Adjoint-Maxwell's equations [9]. After taking complex-conjugates of these so-called adjoint equations, the resulting Maxwell's equations have the following forms: V x: W jp * I (5a) 108 _

THE UNIVERSITY OF MICHIGAN 4386-1-T V x - je6 a (5b) -- O, —.3 + + where p, 6 = adjoint of p., and FE respectively = complex conjugate of the transpose of /u and 6 respectively. + p = p = transpose of p ov / ^_ P11 -J12 0 j~12'22 ~ (6) 0 0 p0 0 ~ ^33.1i +, and 6 = ~ = transpose of e 611 -j 12 0 j12 6 22 0 (7) 0 0 6 0,~ ~ 633 Although the authors in [9] haveused the symbols +5& for ~ and I for _, ingeneral and _ have no simple relations with the solutions of equations (3a) and (3b). Thus, to avoid confusion, the symbols 109

THE UNIVERSITY OF MICHIGAN 4386-1-T and _ have been used here. Here 3 and only mean the solutions of (5a) and (5b), subject to some appropriate boundary conditions. Now multiplying (3a) by and (3b) by A and (5a) by __ and (5b) by in a scalar product fashion from the left and then subtracting, one can show that V [fi%" - (x 0]= O (8) Since + * " A"; 6,% * 6d (9a) and Xa P ^/ =,. (9b) Let 7 = V t z where z is the unit vector in the z-direction. v t -o z -o Now using a 6 = J 6, a -- Ja l 3z -a a -a az - a a- a and az -/ =' / az /t - / - equation (8) can be rewritten in the following way v - ~ x - _i x ~] - j (t -) 9 x z + z x 1 t -a X -3 a -o -a -o p (10) _________________________ 110 _________

THE UNIVERSITY OF MICHIGAN 4386-1-T Now integrating over the cross-section of the waveguide one obtains (using the two dimensional divergence theorem) [ vx' + v x 6ds = j( - 5A\ xz + z x dS...a a 0 - 3 — 0 -= S S where v is a unit outward normal vector on the boundary curve s of the waveguide cross section S. The left-hand side of the above expression vanishes on the boundary of a perfect conductor the orthogonality relation becomes a Acz * x z,+ * z0 x dS =2N (11) S where N is a normalization constant and a 8 - =1, for X = d af3 a 3 = 0, for X When the waveguide has a reflection symmetry, i. e., when the properties of the waveguide are independent of the coordinate z, the orthogonality relation can be rewritten as ________________________ 111 ______1_________________

THE UNIVERSITY OF MICHIGAN 4386-1-T x z dS = N S = z x dS (12) 3 5X - 3 -o c c J -o - dS S S or -z adS =NS' z x d dS (13) ta to t3 -o a af3o J ta -o to S S where the suffix t represents the transverse components of the fields. Construction of Dyadic Green's Functions (see Li] and [12] ) Knowing the total fields E(r') and H(r') which one may consider are due to discontinuity at some cross section S of the waveguide, one can write, using z equation (4) _ (r') x z E(r') = - (r') x z 3 A (r' -- -- o.. -o a -a a and _z x &(r') H(r') = z x (r') * A (r') -o -- a0 aIntegrating the sum of the above two equations over the cross section at S and using (11), one obtains the value of the coefficient A as \ E[ir') ((r') x z + H(r') z x (r' dS' J- -" - -o -- -o — J A = -Z —-- (14) 2N......._________________________ 112

THE UNIVERSITY OF MICHIGAN 4386-1-T Therefore, the total electric field at any point r due to E(r') and H(r') is given by dS' E(r') (r') xz + H(r') z x"(r' (r) - -o — -o - a- - aE>(r) = A j (r) = 2N a a a which can also be written E(r) = -5 [ z x H(r') dS' L- J a 2N -o - L a2N -o 5 a, " 2N ] ^E(r') x z dS' (15) z Since z x H(r') and E(r') x z represent sources due to discontinuities -o - -- o in fields at r', one can represent z x H(r') = I (r') and E(r') x z = I (r') -o -et - -o -mtAlternatively one can consider the discontinuities in z x H(r') and E(r') x z -o -- -- - at r' are due to actual sources I (r') and I (r') respectively. Now with the et- -mt above assumption if one compares equation (15) with equation (1), one finds a (r) a (r') Z (r, r') = 2Na (16a) 11 a3 _______________________________ 113 ________________________________

THE UNIVERSITY OF MICHIGAN 4386-1-T (rr) = (r'),Tem(rr') =2N (16b) Zem^n ^- - 2N a a Similarly if one expresses the total magnetic field as H(r) = A % (r) and uses equation (14), following the above procedure, it can be shown that ^ (r) (r 2N Y(r') - E'tt^"" (17a) 7 (r) (r') T (r,r') 27 (17b) mne - =(b2N a a Special Cases It has been pointed out before that in general there are no simple relations between 6 and n and X and, but in some special cases these -a -- a a" relations simplify. For example, in non-dissipative anisotropic media = e and p + = P, i.e., 6 and p are hermitean (self-adjoint) dyadics. In this case 8 = 6, " =', and a is real. a a a a Cam Therefore, for non-dissipative medium, the dyadic Green's functions can be expressed in the following way,.< 6 (r) _ (r') Z(r r') = - (18a) 1 2N _____.....______________ 114 ________________________

THE UNIVERSITY OF MICHIGAN 4386-1-T 6 (r) I T (r, r) 7- > a -(18b) -em - 2N a a (r)^ (r' a a Y(r ra) (18c) -J, - 2N W (r) (r ) eT r. a Z " N - (18d) me - 2N a a Another kind of orthogonality relation and hence dyadic Green's function can be constructed in the following way (see 9 ). These results are particularly suitable for dissipative medium. In this method the following replacement is made p - p = transpose of p (19a) C- = transpose of 6 (19b) + = ^ -) - (19c) a a a 6^ ~~~~~~~+ ^ ~(19d) +t-3 ~(19e) Due to the transformations given in (19) and the Maxwell's equations (3) and (5), the following simple relations can be established _____________________________ __ 115 _______________________________

THE UNIVERSITY OF MICHIGAN 4386-1-T 6 (d, 12' 12 r) =+ (-a a 12 -12' r) (20a) 6 1 r2' ) 2' - - c - -p r) and (dee E U r) =- -A< r) (20b) fa z12' p12 ) a a 1 2 r12' 2' r) Therefore, for the dissipative - anisotropic medium the orthogonality relation and dyadic Green's functions can be expressed in the following way x z -' z x 3]dS 2N = a (21) -o -a -~ -o c a ao3 If there is reflection symmetry in the waveguide then _cS' z dSpx dS (22) Ji'^; x z dS = N zo x dS (22) -f -3 -o a a 13 - Z (r r'a )2 Z(r, r') = - ~~ --- 12' / 12 -" a'12 -12) (23a) 2N a a ^(r, ), -d - 2 ) T (r, r') =' -1 -- 1 2 (-r - 12- (23b) I-em —') = 2N a a (rdE,' 6 p ), (r', f,-E,) Y(, r,') -= a' (r 12' 12' -a — 12' 12) Y(r, r') c= E2N (23c) a a ________________________ 116

THE UNIVERSITY OF MICHIGAN 4386-1-T __ (r ) _(r I?,X -/, -6 P T (r, r') -12' 12 ) (- a' 612' 12(2d)'me (-}r) 2N a a For dissipative medium X and N are complex. a a 117

--- -THE UNIVERSITY OF MICHIGAN. 4386-1-T 14~~~~~~~~~~~~~~~~~~~~~.C 0t C\] 0~~~~~~~~~~~~~~~C 0 ~,1.. N __^ ^?r -. -I, 1 (M c^ ccj c-^ ^v.c c 0 ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~,. 4-~~~~~~~~t,1.~ O0 4-4 >' o~ i 0 cq Qa -r- 1%.^ cd Qd, cr 0 ~~~~~~~~~~~~~~~~~~~~C- 0 ~ q *~i (..S^^ )N o O'-a ^ |^ T^~ F -T-I &. |r^ ^^ c - i 2 co~^ ^.N ~^ 01 M ^ —. fi<M'~ ar~ *.~~~~~-~ r^ ^ ^^-I~1 CM~~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~.3..-<-I~~~~~~~0 0 Q' ^~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~..,-. AJI LO z~~~ od ^ L^J *i-l ^r-'^ —4 cd c 3~~~~~~~~~~~~~- -T- si > ^ ^ r~ bo g i)" ^io 0 o _- ^ * d"^a, o o j ~~~~~~~~~~~~~~L, I'~ IQ^^ ~ 4 —,.z 0 ~-~- cd 0 >~~~~~~~ |'~ tq cq Q'~0 4 D~ 0:_ t~~~~ ku o* r,,.t/ 2 ^? -1^! ^...... 118............

THE UNIVERSITY OF MICHIGAN 4386-1-T can exist independently. This approximation is also valid when (ka) < 1, where 2a is the diameter of the anisotropic column. For static approximations it is easy to show that E a'1 Ed2l2 2 z z,2 Z Z 2 ~ E P rr r S 0 M ^ - ('22 - r72) (2) Now using these relations and the assumption k/, ( 1, it can be shown from the dispersion relation (1) that for the H-type mode the right-hand side of the expression (1) vanishes and for the E-type mode the denominator of the left-hand side of (1) vanishes. For a magnetic current ring source an E-type mode will be excited in this static-limit situation. Whereas for an electric dipole source an H-type mode can be excited in the static limit. It is of practical interest to consider the E-type modes in a plasma and the H-type modes in a ferrite. The following dispersion relations for these two limiting cases can be expressed in the following way: __________________________________ 119 _________________________

THE UNIVERSITY OF MICHIGAN 4386-1-T for E-type modes and n n n n n Jr (r'2a) ( [I (db)K0 (a) - I (Va)K' (4b) n [n n n n n for H-type modes When the anisotropic medium completely fills the waveguide, i.e., when a = b, the above two relations reduce to the following: J (11la) = 0, for n 0l and for n = 0 E-type mod) (5) It should be noted also that in this case a change in sign of n does not effect any result. J' (l' a) ("nair J'a = n' (rl,2a)/,r n for H-type modes (6) and for n = 0, Jl(rl'2a) = 0 When the radius of the waveguide b -+ oo, the above equations (3) and (4) reduce to the following simple forms: +These results agree with those obtained by Trivelpiece [8] except for a change in sign in the term ni' of equation (4). 120 ____

THE UNIVERSITY OF MICHIGAN 4386-1-T k~ cq d 4-4 oI o -- ~'"~ -_ a) a) +_ I w H(D I'1 3 r ~'~ ~' L~- c'.-~'"- 0 co Cl) 0''Z OQ~~3+ U) 0. o YJ Cd Cd 0 ^-^ ^> 0 co _j?-( ^^ —'^j ^ _,'-.~d cI c^ I a) 0 c-.j o a )0 0 I' I,-I.- (i% i E ) i cd.11- a) U) 0 c- 0, C 4 <c~d 1 c* ~4 dc i~Jaa __'-)'cd ccd ccd cj S S1 _z Cz I'd 4J a ={~ -'' <' 0' — 4 ILI _v: i4'^ c ^,~ 0 *_ +3 121 c- II 9i 0 ^ |'3 ^ | a) 8c0 c^'-%~ c 49 PI CI 0 +3 V u-4 + 0 0 d a) CY' 0, t r i cCd L 12 1 ^ ^ + I —-------------- 12 —-----------

THE UNIVERSITY OF MICHIGAN 4386-1-T For n = 0, the above relation becomes k272X2a2e22[2J (ra)J(la)-ar al{J2 (rlla)+J2(r,?la)} G (a) (a) 3 Z ZL 0 1 - 0 0 0 2[rcrJ(a{J1 (r a)J(rlIa))l[r?2-S]+anr2rl SJ I a)+J(r a 2 }+2e2r 3C(a)J2(qlIa) oG (a)2J (r a)J1 (r la) 1 2+s] -aTl S(J(ra)+J2(a9) -22p2' S (a)J2(rlta) (10) For n = 0, the dispersion relations (551) and (561) can also be derived from the relation (1). It should be mentioned here that all the following dispersion relations for various special cases can be derived from any of the general relations (55I), (56I), and (1) of this Appendix. The purpose of writing these various forms of the general. dispersion relation is that it is found more convenient to use one particular form rather than another for some special cases. When the anisotropic medium completely fills the waveguide (i. e., when a = b), the corresponding dispersion relation (without using any approximation) becomes (since G (b) = C (b) = = S (b) at ab): n n ( r) b)' n n a? (r/l''a) -1 Lj a) a~E Ml 2 L> ) - a rr r 2 = a = a nE a'2(r2- 2), for n 0 xa r 1 7122 r702.....rr 1 2 (( ii)22 (11) I 122

THE UNIVERSITY OF MICHIGAN 4386-1-T riy2S Jl(7'2a)Jo(rl'a) and'M = for n=0 (12) rl'1M J (rl'2a)Jl(r7'ia) 0 If one analyzes equations (341) and (351) or equation (32a) of Appendix A, it is easy to show that in the limit e' = t' = a 3 0 E aT1 a1 =2 z z s=0 rl a='2 rr r 12 =T r Pr S - 0 M - (n' 2 - n'12), ea4 -'1Er e S S a z 4 r r Pa3 C )'0 ea4- er'l'C - arn'1 z_ a 2 z a3 E (13) Now if one divides the numerators of both sides of (541) by a3 and uses relations (13), the dispersion relation can be shown to have the following form, after rearranging terms:,,_____ 123 ____

THE UNIVERSITY OF MICHIGAN 4386-1-T 2 E rWiJf (ri'a) E2a Cn(a)1 rrrl'2Jn(r'2a) 2a4S(a)1 n(a- 2 Fr n 4n j r n 4n a4 LJ(rl'ia) Jn(a) J J (n'2a) G (a) L ka1 n n n n - (14) For an isotropic medium, the dispersion relation can be obtained from (14) letting'2 = r'22 = a4 = ke -_2 and e = e, = p. For axially symmetric fields (i.e., when the ring source is of constant strength, - 0), n = 0, and one obtains from equation (14) the following dispersion relations c rr7'lJl(r7'la) E2a4C (a) J J(r7'a) 1(a) for E-type mode (15a) and W r rl2 J1 (rT'2a) p2a4So(a) a) for H-type mode (15b) i (7312a) G (a) O O The relations (15a) and (15b) can also be obtained from (551) with appropriate limiting procedures. It should be noted here that for axially symmetric fields and E' = 0 = /t', E-type modes and H-type modes can exist separately. But in the present problem where the source is a magnetic current ring source, only E-type modes will be excited for c' = 0 = p'. Further it may be stated that even in an isotropic medium -_ 124

THE UNIVERSITY OF MICHIGAN 4386-1-T (i.e., e == p and E = e and = ) if 0, E-type and H-type modes cannot exist independently. If n 0, but a = b and E' = 0 = p', it can be shown either from (11) or from (14) that the dispersion relations become J (r7'la) = 0, for E-type modes (16a) and J' (r'2a) = 0, for H-type modes (16b) n It is now trivial to see that for n = 0, a = b and E' = 0 =,', one obtains the following dispersion relations J (r'la) = 0, for E-type modes (17a) Ji(r7'2a) = 0, for H-type modes (17b) Relations (16) and (17) are valid for isotropic media as well as diagonally anisotropic media. However, in our present problem, we consider only (16a) and (17a), restricting our consideration to E-type modes (due to choice of the source). When an infinite column of an anisotropic medium is placed in another unbounded isotropic medium, the corresponding dispersion relation for surface It should be noted, however, that to obtain radiated fields in the present situation, this limiting process (namely b -+ oo) is not valid. 125

THE UNIVERSITY OF MICHIGAN 4386-1-T wave s can be obtained either from (531) or from (541) with b -- oo. To study surface wave propagation (which is also slow wave), for which de/k > 1, one can show that rl is purely imaginary. So putting rl = -j, < > 0, one obtains 4 (a) = -(2/7r).[I (6b)K (6a) - I (S a)K (6b)] (18a) n n n n n S (a) = -(2/7r).[I ( a)K' (S b) - I'($b)K'( a) (18b) C (a) = (j2/r). I'(6a)K (6b) - I (5b)K'(6a)] (18c) n n n n n G (a) =(j2/7) [In (b)K (6a)- I (Sa)K'(b)] (18d) n L- n n n n n for convenience let j (r) = I ($b)K (Sr) - I (r)K (6b) (19a) n n n n n S (r) = I' (r)K' (Sb) - I' (b)K' ( r) (19b) n n n n n C (r) = I' (r)K (bb)-I (bb)K' (Gr) (19c) n n n n n G (r) = I(Sb)K (er) - I(r)K' (^b) (19d) n n n n n Also it is easy to show that (if n < S b, 6 may be finite or very large), K (^ a)e b K a~-, as b - oo (20a) n 1/2r 5 b, ______ 126

THE UNIVERSITY OF MICHIGAN 4386-1-T ca 0O CD cd —'- c r-. cd' 4- -oQ?1 cd ~ o.-t Cr 0ac co N oa. rI p- j'o + CC N cc m ( c<DN ef C_^cu + m tZs. N CQ cd -.. C) r a) 4 c Ky< C? *V " cc K- CI cc cNt o'~ ^i m 0' ur' I-1 C 0 N Ujj C S. Ij 3 CV ^ "I-. vr 0 } 0 ^ i.,-\ 4-~ CI Nk ^. a) a) a) cd 0 0 4 —------------— 2 ------- -r0 m mS ~r4 "-4 ci 0

THE UNIVERSITY OF MICHIGAN 4386-1-T When n = 0, the above relation becomes SLz K (6 a)J (r1ja)+r)'l,2K j ( a)j (r',)a)+ Se K (6 a)J('a)+2r'iK(ea)J (r1a) _ _Z 0 ____ _ _______ z o 0 Mfu K ( I a)Jl (r'2a)+'K21'2K ( Sa)J (ri2a)} SE K (6 a)Jl (r'2a)+E2r'2K1 (6a)J(r1 2a) L Z 0 0 ~' Z 0 0 (22a) The equation (22a) may also be written in the following form: e z('22-r'1i2) E2/62('22-r'12) --- - -2 -- K2 (5a)Jl(r7'la)Jl(rT2a) + ^ 2 Kl2( a)J (rT'la)J (r7'2a) ~r? i'2 o o E2P M-A2E S] [e2, S-~2E M]l - rl' K (6a)Kq(6a)J (rija)Jl(rt 2a)+ l K (oa)Kl(6a)J (rj2a)Ji (rl a)=o (22b) When the anisotropic medium completely fills the waveguide in such a way that a = b > 1, the corresponding dispersion relation can be obtained from (11) letting a > > 1. It should be noted here that if a > > 1 in equations (21) where the limit b -- oo has been taken the result will be different from that obtained from (11) with a > 1. The reason is that if a >> 1 in (11), it also means a = b ~) 1, but if a )> 1 in (21), it means that b >> 1, a ~) 1, yet a f b. The difference between the above two results can be shown rather easily for n = 0 as follows: either from (11) or (12) one obtains the following dispersion relation +When a = b - oo, there is no dispersion due to the boundary. 128

THE UNIVERSITY OF MICHIGAN 4386-1-T r7'2S tan(r7'2a - r/4). = M tan(7r7ia - 7/4) as a = b > 1 (23) Y],M tan67t'a - 7r/4) and from (22a) one obtains |S @uzSin(rt a-7r /4)9+t2 cos(ra-7r/4) SE Sin(rl( la- 7r/4)+e2r'lcos(rlta-7r/4) 6E cSin(rt a-7r/4+r E2rcos(rlka-?r/4) ML $zSin(r7 a-7 /4)+t'22cos((r'2a-7r/4)} e Sin(r2a-7r/4)+e2r'2cos( 2a- r/4) 2z ^ z 2 for a >> 1, b > 1, but a / b. (24) For E' = 0 =' = a3, the dispersion relation (21) becomes V| ertJp'(ta) E2aKn 2Jn(raa) + p2a4K(S2a)6a n(a4+S2) J (n'la) + K ( a) J L J(r'2a) K a) + L ka ( - ~n n J~n n J~/uaK(alO (2 5) i Equation (25) can also be obtained from (14) directly using rO = -j6, and letting b- >o. For n = 0, the E-type and H-type modes separate and one obtains the following two relations from (25) e y lWJl(7'la) e2a4K ( a) J-(rt- ) K( -a) for E-type modes (26a) J (r'lia) K (S a) O O and u S r'2Jl (7'2a) p2a4K1 ( aa) Ka a -for H-type modes (26b) J (r'2a) K (6 a) It should be noted here that for isotropic media, rT'i and l'2 in (25) and (26) are 129

THE UNIVERSITY OF MICHIGAN 4386-1-T given by 7'2 = 12 = 1'2 = a = k2eru - 2, where e = e * / = u. 4 rr'r z r z For e' = 0 = U', n = 0 and also a = b >> 1, it can be shown either from (16) or from (23) that the dispersion relation reduces to cos(r;'1a - 7r/4) = 0, for E-type modes, a >> 1 (27a) and sin(rW'2a - 7/4) = 0, for H-type modes, a>> 1 (27b) Now it is also natural to discuss the dispersion relation for a (( 1 and b finite. To do this the following approximations will be used (see [17 ): J(x) 2)n for 0x < 1 (28a) n n N (x) (2/7r)log (a ), for 0 x C< 1 (28b) where a = 1.781072 N (x) - ( - )! ( ), n 0, 0<x ( 1 (28c) n T \x / K (x) - lo ax 0x 1 (28d) K (x) (n - " n f 0, 0 4 x 4< (28e) n 2 x n- J'(x) n l, (28f) n 2n(n_-)! 130

- THE UNIVERSITY OF MICHIGAN 4386-1-T n! 2n N'(x) --, for n ) 1, 0 x z 1 (28g) n n+l 7TX n- 1 n!2 K' (x) -!, for n> 1, 0 < x < 1 (28h) n n+1 x It will be assumed that ria 4 1, r'1a < 1, r2 a < 1, in addition to a < 1. Then, one obtains the following expressions: G (a) (na) N' (rb) + (n- 1) J (rb) n 2nn n 2 n a n J (a) - Jn (b) - N &1b) n 7r (nan n 2n n, n S(a) n-! J' (b)- - a) N' (nb) n, r n+1 n 2n(, 1) n -(Jna) 2 (n -)! C (a) n +l!2 J(r1b) - (na)- N (nb) for n 1 n n+1 n n n 0 2T 2 o w nb 2 2nb obr _ ( C (a) ir nb Jn(nb) y N0(rb) for n = O (30) t_________G_()_-N__b____j__blog 131 ____

THE UNIVERSITY OF MICHIGAN 4386-1-T Thus, for a < 1, ra <.< 1, r1 a << 1 and r7 a << 1, various forms of the 1 2 dispersion relation can be obtained merely by substituting these expressions in (541), (551) etc...... —- 132

THE UNIVERSITY OF MICHIGAN 4386-1-T APPENDIX D A NUMERICAL EXAMPLE In this appendix, results of numerical calculations will be presented for a special case of the physical situation described in (14a) of Chapter II. A normalized value of the electric field, E, will be computed as a function of r for the smallest eigenvalue, r'l, where we are treating the case a r'2 > 0 r'22 < 0 (la) a ao 1 - x > 0 l- y > 0 (lb) 2 > x+y > 1 with the specific parameter choices ka > 102 E2 = 6 (which represents a glass tube) (2) u2 = 1,/k = /3 = 2.5(> /6 / ) 133 _____

- THE UNIVERSITY OF MICHIGAN 4386-1-T 2 Ea —2 n l 2 a = ka 3 - 2p2 = ka -3 t (2) a = 103 c = b = 2 x 103 First the lowest eigenvalue must be determined. To do this, first some necessary constants must be computed as follows: o = c1 = I (6b) K(6 a)-I (6a)K (6b) > 0 O O O O O G = c2 = I1(6b)K (Sa)+I (6a) K(Sb);- 0 (3) S = c3 =I (6b) K(6a)- I (6a) K1(6b) 0 C = c4 = I l(a) K (Sb) + I (6b) K1 ( a) > 0 0 14 O 0 rT'l (y) will then be computed from the following expression 7'12 1 -- = --— 2( 1 21(1-y) - x)+y} - /xy+yf(y), (4a) k2 2(x + y - 1) where - = 2 1, (4b) and f2(y) = 2 x2 +42 x(- y), (4c) For a range of values of y (where 1 - x < y < 1). Starting with y corresponding to the smallest value of rt'l obtained from (4a), the function G(r'l, y) given below will be computed for neighboring y until 134

THE UNIVERSITY OF MICHIGAN 4386-1-T a change in sign is obtained. [i.e., we want to find lowest value of rli/k which satisfies G(r', y) = 0] r(l-y)c c Jl(ria)I,(pa) E2c3C4J (r1T7a)I (pa) G(r'1, y) = (p2 +r2). 2... + 02 J (rT 1a) I(pa) [ 4 + 6 - E-2 Mc4c2- (1 -y)Sclc3 r- n I(pa)J (r7'ia) + (l - y)Mc -c3 - C E2Sc4c2] I6 0r (5) where 2 P (y) 1'22 Yf(Y) r12(y) 2 2 2 k k x+y-1 ka) 2 k (1 y)(1 -x). [x+ y- 1 + 2 +2 M = ~ (6b) x+y-1 1 -x S = M- (p2 + n'2) It should be noted that neither r7'1 = 0, nor p = 0, can be a solution to (5). When the lowest eigenvalue rn'l is determined, | E I will be calculated as a function of r from the following expression where all relevant parameters are evaluated at the previously determined lowest r'1 c I Ez WEoe C | 2 [SiI (0pr) - MJ (1'ir)] m T 2 8 2 (p2+r1T'12 ) F2 fo r 0<ra (7a) 135

THE UNIVERSITY OF MICHIGAN 4386-1-T 2wec3c 2 2 C(r) 2 - -8 A — ~ - |. for a c r < b (7b) =1 where p [6c2Ji(r1n'a) + r)'lc3J(r7'ia1 r1 =' 6c2I(pa) +pc3Io(pa) (8a) =T FS Io(pa) - MJo (ria)( O2 = (8b) 2c (p +ntl12) 2 r 2 2 FL = Le e r'2 p MR + p R' k 3' (8c) 1 r z L rM o -/ F2 = -j L e e rY'1 P il [w o p2T' k 3 e'+ ERr'i' S] (8d) F3 = - j Le e PrT' 1 u Wu p7'12R'k 3' - eMT, (8e) 3 rz 1 LO F5 = wk [e`2 22{I2(6b) + 42 Ko(8b)} + {lJ(6b)+2 K1(&b)} (8g) F =- wk [E 22I (b){jI (sb)+ 2 K(6b)} -j 3 Il,(&b){j Kl(6b) +I,(Sb)}] (8h) F = F (8i) ~7 6 c I 2 (8b)+ 4 K2'$b)) + /'136

THE UNIVERSITY OF MICHIGAN 4386-1-T F8 = ok C 2 I (6b) + o 2 12 (6)], (8j) 3 > / i 2 \ O 0 a) 222 w e e e'k 3 0 Z L = (9a) 4 2 [ 2 j2(2+ 2 Er[p'r (P +r 7 k2e M(e 32) - k4e e2! R = r r z,(9b) WE e e' 0 z _k LeM - k2 ( - 32)] R' z (9c) C z k4 e e'2- k2e S(e - 32) T z r r (9d) WE C C' 0 Z k3 [e S- k2 (e - 2) T' =:, (9e) z S (r) = I (6b) K (6r) - I ( r) K (b), (9f) w = (9g) -9 l0 = (in M. K. S. unit) (9h) 0 367r p = 47 x 10-7 (in M.K.S. unit) (9i) ________________________ 137 ___________

THE UNIVERSITY OF MICHIGAN 4386-1-T e = -y, (9j) Z c = X+- 1 (9k) r x-1'= y (91) x-1 I1 = 2 [J1'a)- -- a J (nl^a)Ji^(ra) + J (rla), (lOa) I2 = I3 = -; - ) pJp(i7'ia)I2(pa) + riT I1(pa)J2(r?'ja) (1 Ob) 2 3 (2 2 a2 F2 2I (pa) Il(pa) 2 = 2 o 2 (b) I(6b35b) 2 I2 3 1 (p2+,2 + 6b) (lOd) I I jI + -1(6b)K (6b) + Io(6b)Ka b) + 12 FI(6 a) ( 1 IO(6a)K (( a) + 15 2 LiI 6b) -Ii(6b) =T Sb 0 (lOe) _____ 138 ___ ___ __ ___ __ ___ __ _,,__ _ 138 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

THE UNIVERSITY OF MICHIGAN 4386-1-T 2b2 r2 2K (6b)K1(6b) 2 I I + 2 K (6 b) - K2(6^b) 8 5 i5 262b 0 2a2 r 2 2K ( a)K(6 a) 2^-K^Sa)- K —-- --- - K a) T- _ ~ 6 a o (1Of) Discussion of the Computation Procedure and Results Starting with y values near 1.0, it is found that r1'1 increases very rapidly as 1 - y increases. An analysis of the expression for G shows that the first zero of G will occur shortly after J (rg1'a) changes sign, i.e., when r'1 a is slightly greater than 2.4, i.e., r'1 > 2.4 x 10 (with a = 10 ). With the given parameters it is found y is very near 1.0. After finding r7'1, and the corresponding x and y, for which G = 0, various expressions in equations (8), (9) and (10) have been computed. Since the parameters 6 a, 6 b are large in this calculation, the asymptotic formulas for evaluating I (z), I,(z), K (z) and K1(z) have been used (where z stands for either 6 a or 6 b). With the parameter used, K (6 b) and K (6b) are very small and nearly all terms involving them were insignificant. However, the form used included all operations and can be used, without alteration, for any set of parameters. It should be noted that, although some of the F and Ie in (8) and (10) are real, some pure imaginary, and some complex, all imaginary terms in the summation cancel out, and hence F is real. This fact is in agree139 ____

THE UNIVERSITY OF MICHIGAN 4386-1-T ment with the theory as discussed in connection with the equations (82) - (84) of Chapter I, where it is stated that the power-flow in a non-dissipative medium is real. Computation has been completed for the following cases: 2 2 Case I) x = 0. 7 andx = 0.5, with ka = 2 x10 and 3 = 2.5(or 3 =6.25) Case II) x = 0.7 andx = 0.5, with ka = 10, 12 = 6.000025 Case III) x = 0.7 andx = 0.5, withka = 7, j3 = 6.005102041 The following pages show tables of values of many of the variables involved, ci El c El and the values of, for 0 < a < b. Also graphs of in the range m rm 0 < r/a < 1.0 corresponding to the above cases have been shown. For the range ClEzI r > a, the values of are too small to show on the graph. The behavior of all the graphs plotted here is more or less the same. It is the nature of the slow waves. The higher the value of P = -, the slower the k wave. Moreover, the higher the value of e2, there is a minimum value of 3, for which a corresponding slow-surface wave can propagate. Since in the above comuptation E2 is chosen to be 6, the minimum value of 3 is greater than 2.45. On the other hand a larger value of ~2 will permit a lesser slow wave to propagate. Although the above statements show that the degree of slowness of the surface waves is markedly influenced by the value of ~2 and hence 3, the strength or amplitude, however, of these waves depends on various other parameters. For ________________________ 140 ____________

THE UNIVERSITY OF MICHIGAN 4386-1-T example, in the Cases I and III the graphs show that higher the value of x = lc |E |2 the lower the amplitudes of. On the other hand, in the Case II the c Ez graphs show that the higher the value of x, the larger the amplitudes of m although the values of 3 in all of the above cases are of the same order. Moreover, c[E i 299 the graphs of the Case II show that the amplitudes of is about 102 m times higher than that of the Case I and is about 10 times higher than that of the Case III. Therefore, the above discussions of the numerical results suggest that! for any practical purposes the results of Case II will be of greater significance. All the graphs plotted here change monotomically, because of the higher value of 3. On the other hand, if 3 is small (and hence the smaller value of the parameters (pa), 6 a etc.), it is expected that there will be a few oscillations of c |EZ Z in the range 0 < r ~ a. In support of this statement reference may be made elsewhere [7. Finally, it should be noted here that is is the value of 6 a and S b which played the significant role in producing tremendous difference of amplitudes of c E| in Case I and either Case II or Case III. In the former case, the value m of 6 is much higher than that of either of the later cases.....__________________ 141 _

THE UNIVERSITY OF MICHIGAN 4386-1-T CASE I 2 cE Summary of Values for ak = 2x102, =2.5 Values of m x |.7.5 r/a For x=.7 Forx=.5 y.9999731125.9999849842 0 8. 7559211 x 10-299 1. 1286428 x 10-298 f 3.675064017 2.625035752.1 8.6297345 x 10-299 1. 1123792 x 10-298 rl'1 2. 405305063 x 103 2.405160330 x 103.2 8.2566279 x 10-299 1.0642912 x 10-298 P 4. 582578887 x 105 4. 582578191 x 105.3 7. 6526916 x 10-299 9. 8645246 x 10-299 M 2,100042490 x 1011 2.100045835 x 1011.4 6.8438881x 10~299 8.8220867 x 10-297 S -1. 8291 x 106 -1. 4301 x 106.5 5.8648043 x 10-299 7.5601706 x 10-299 ^1 9. 062918575 x 10-197 9. 064671742 x 10-197. 6 4. 7570032 x 10-299 6.1323402 x 10-299 1. 574477758 x 10-45 8. 863836389 x 10-46. 7 3. 5670449 x 10-299 4. 5985988 x 10-299 C2 -2.333243708 -.9999699684.8 2. 3442840 x 10-299 3.0225528 x 10-299' -2. 788791769 -1. 414192327.9 1. 1385489 x 10299 1. 4684199 x 10-299 R -4. 228947963 x 1024 -5. 405747531 x 1024 1.0 8. 3359942 x 10-303 6. 0491791 x 10-303 T -4. 524434966 x 1019 -4. 107746983 x 1019 1.1 3. 6088193 x 10307 2. 6188111 x 10-307 R' -9. 111703604 x 1021 -6. 992432040 x 1021 1.2 1. 5687972 x 10-311 1. 1384287 x 10-311 T' 2. 510277019 x 1017 1. 926178097 x 1017 1.3 6.843 557 x 10-316 4. 9660890 x 10-316 -1. 187508895 x 10-47 -7. 847883607 x 10-48 1.4 2.9941111 x 10-320 2. 1727359 x 10-320 L -1. 470483272 x 10-58 -6. 895189885 x 10-58 1.5 1. 3133097 x 10-324 9.5302916 x 10-325 F1 3.396169086 x 10-8 1. 896654271 x 10-8 1.6 5. 7733840 x 10-329 4.1895703 x 10-329 F2 5.107586708j x 10-207 2. 230758055j x 10-207 1.7 2. 5429686 x 10-333 1.8453555 x 10-333 F3 -8. 171358617j x 10-208 -3. 568967890j x 10-208 1.8 1. 1220024 x 10-337 8. 1421868 x 10-338 F4 2. 429623822 x 10-405 1. 061285373 x 10-405 1. 9 4. 9582959 x 10-342 3. 5980855 x 10-342 I1 1. 347570276 x 10-7 1. 347570418 x 10-7 2.0 0 0 12 2. 205105178 x 10188 2. 205083423 x 10188 13 -2. 205105178j x 10188 -2. 2050834233 x 10188 14 4. 126696590 x 10385 4.126122615 x 10385 F5 3. 842092989 x 1080 1. 481363914 x 1080 Re F6 1. 474033506 x 10-93 5. 68546845 x 10-94 Im F6 -5. 592484518 x 1079 -4. 409151006 x 1079 F8 3.842092989 x 1080 1. 481363914 x 1080 5 4. 139482976 x 10162 4.139482976 x 10162 Re I6 3. 183039180 x 10-9 3. 183039180 x 10-9 18 4. 139482976 x 10162 4. 139482976 x 10162 F1 I1 4. 576576513 x 10-15 2. 555875189 x 10-15 F2 I2 -1. 126276590 x 10 18 -4. 919007608 x 1019 F3 13 -1.801870520 x 10-19 -7.869871931 x 10-20 F4 14 1. 002632034 x 10-19 4. 378993579 x 10-20 F 5 5 1.590427852 x 10243 6.132080703 x 10242 Re F6 I6 2. 314999446 x 10242 1. 825160553 x 10242 F 8 8 1.590427852 x 10243 6. 132080703 x 10242 EFe 18 3.643855593 x 10243 1. 591448251 x 10243 Coeff 1 4.169401884 x 10-310 5. 374372065 x 10-310 Coeff 2 8. 776518558 x 10-344 6. 368854307 x 10-344 Eo e2 2 Coeff1 = 7r (p2+r2)! 2w Eo 2 c Coeff2 2 2 8 where = i F1I 142 --

THE UNIVERSITY OF MICHIGAN 11 - 4386-1-T CASE I 10x=.5 9 - 299 Ez 10 x - vs - 8 - m a 7f x=7 7 - 6 - 5 143 r/a

THE UNIVERSITY OF MICHIGAN 4386-1-T CASE II Summary of Values for ak = 102, 32 = 6. 000025 6=5x102 c ~Ez 6; = 5 x 102 Values of --- m x.7.5 r/a For x=.7 For x.5 y.9998905959.9999384817 0 1.8096432 x 102 5.2136187 x 10 f 3.500280060 2.500160139.1 1.7835734 x 102 5. 1385115 x 10 2. 404840387 x 103 2.404835008 x 103.2 1.7064903 x 102 4. 9164345 x 10 p 2.236080556 x 105 2.236079069 x 105.3 1. 5817168 x 102 4. 5569609 x 10 M 5. 000447026 x 1010 5. 000480279 x 1010.4 1. 4146147 x 102 4. 0755384 x 10 S -1.87552 x 106 -1.47648 x 106.5 1.2123270 x 102 3.4927458 x 10 1 4. 638354547 x 10-97 4.639144141 x 10-97.6 9.8343796 x 10 2. 8333140 x 10 2 3.646241238 x 10-6 3. 532368146 x 10-7.7 7. 3756387 x 10 2. 1249478 x 10 er -2.332968653 -.9998769634.8 4. 8489892 x 10 1. 3970166 x 10 E' -2.788561642 -1. 414126562.9 2.3573572 x 10 6.7917354 R -1.201246835 x 1023 -1. 516656482 x 1023 1.0 3. 0356136 x 10-4 8 4724674 x 10-6 T -5. 556824512 x 1018 -5 011324661 x 1018 1.1 2.5978231 x 10-4 7. 2505840 x 10-6 R' -2. 611720668 x 1020 -1. 990641619 x 1020 1. 2 2. 2038224 x 10-4 6. 1509190 x 10-6 T' 3. 020818349 x 1016 2. 302445054 x 1016 1.3 1. 8470813 x 10-4 5.1552463 x 10-6 3 -3. 676783183 x 10-8 -2. 446310694 x 10-8 1.4 1. 5209538 x 10-4 4. 2450169 x 10-6 L -9. 287120418 x 10-56 -4.413530397 x 10-55 1.5 1. 2211892 x 10-4 3.4083672 x 10-6 F1 1. 421535942 x 10-7 7. 993544921 x 10-8 1.6 9.4365436 x 10-5 2. 6337610 x 10-6 F2 2. 299760241j x 10-106 1. 018226137j x 10-106 1.7 6. 8530496 x 10-5 1. 9127019 x 10-6 F3 -3.832039965j x 10-107 -1. 696764786j x 10-107 1.8 4. 4333925 x 10-5 1. 2373700 x 10-6 F4 1. 147162520 x 10-204 5. 079880743 x 10-205 1.9 2. 1556305 x 10-5 6. 0164144 x 10-7 I1 1. 347570470 x 10-7 1. 347570466 x 10-7 2.0 0 0 2 8. 015023185j x 1086 8. 013856901j x 1086 13 -8. 015023185i x 1086 -8. 013856901i x 1086 14 2. 653635970 x 10182 2. 652850350 x 10182 F5 5. 800314707 x 10-8 1. 063203035 x 10-8.91 2. 1139941 x 10 6.0906023 ReFg 1. 784938139 x 10-8 4. 853114861 x 10-9.92 1. 8720750 x 10 5.3936289 ImF6 1. 728284623 x 10-8 -6. 748558216 x 10-9.93 1. 6316944 x 10 4.7010883 F8 4. 918169934 x 10-8 7. 372337806 x 10-9.94 1.3929462 x 10 4.0132505 |1I5 2. 787895257 x 10-7 2. 787895257 x 10-7.95 1. 1559233 x 10 3.3303830 ReI6 3 605254500 x 10-7 3.605254500 x 10-7.96 9.2071729 2.6527501 18 8.762842367 x10-7 8.762842367 x 10-7.97 6.8741914 1.9806136 Fl1 1.915619857 x 10-14 1.077186505 x 10-14.98 4.5611931 1.3142336 F2I2 -1. 843263165 x 10-19 -8.159918555 x 10-20.99 2.2691461.65388399 F313 -3.071388917 x 10-20 -1.359763019 x 100.99' 1.1314051.32608387 F4I4 3. 044151727 x 10-22 1. 347616341 x 10-22.99 4. 5160701 x 10-1.13021180 F5I5 1.617066986 x 10-14 2.964098699 x 10-15.99 2. 2552059 x 10-1 6. 5065232 x 10-2 ReF6I6 1. 616879755 x 10-15 3. 631098763 x 10-15 F8I8 4. 309714787 x 10-14 6. 460263407 x 10-15 FeIe 8. 165756107 x 10-14 2. 745832962 x 10-14 Coeff1 3. 618962810 x 10-9 1. 042623596 x 10-9 Coeff2 4. 200826659 x 10-4 1. 172460375 x 10-5 eoe2c |2 Coeff= 2(p2+2 2 2w Ec 2c o 2 2 Coeff2=: 2 | 144

THE UNIVERSITY OF MICHIGAN 4386-1-T CASE II 180 clEzI r VS - m a 160x=.7 \/ 14 120100 80 60 - 40 — 20 — 0.1.2.3.4.5.6.7.8.9 1.0 r/a 145

THE UNIVERSITY OF MICHIGAN 4386-1-T CASE III ka = 7 x 103, j = 6. 005102041 7, 9 2 c IE zi = 5 x 102 Values of - x |_.7 |.5 r/a Forx.7 | For x=.5 y.9779975594.9876519579 0 7.125499252 8.421984918 f 3.555976389 2.532007921.1 7.022644030 8.300472142 t1 2. 407251260 x 103 2. 406684664 x 103.2 6.718529845 7.941190526 P 1. 566993507 x 104 1. 566800030 x 104.3 6.226294257 7 359653509 M 2. 494896933 x 108 2.498167021 x 108.4 5.567130934 6. 580887224 S -1.8520304 x 106 -1. 4616625 x 106.5 4.769275220 5.638235455 1l 2.343388642 x 10-7 2. 353220927 x 10-7.6 3.866659658 4.571767509 ~2 2.793267998 x 10-4 1. 570659319 x 10-4.7 2.897321346 3.426394461 Er -2.259991865 -.9753039158.8 1.901712998 2.249848394 E' -2. 727504880 -1.396750794.9 9. 213164880 x 10-1 1.090918066 R -2. 045747178 x 1017 -2. 600475240 x 1017 1.0 9. 224264276 x 10-4 6.120993715 x 10-4 T -1. 873920396 x 1015 -1. 703190155 x 10-5 1.1 7.89395 x 1 0 10-4 5. 238235476 x 10-4 R' -4 326419710 x 1014 -3.326038291 x1014 1.2 6.696715401 x 10-4 4.443774772 x 10-4 T' 1.021026696 x 1013 7. 847629400 x 1012 1.3 5.612692559 x 10-4 3.724444016 x 10-4 C3 -6. 973370110 x 10-6 -4, 646927041 x 10-6 1.4 4. 621694767 x 10-4 3.066842382 x 10-4 L -2. 340389651 x 10-45 -1. 090450853 x 10-44 1.5 3. 710805671 x 10-4 2. 462398898 x 10-4 F1 2.794648081 x 105 1. 575790576 x 10-5 1.6 2. 867465463 x 10-4 1.902779187 x 10-4 F2 3. 235412563j x 10-13 1. 444443912j x 10-13 1. 7 2. 082423802 x 104 1. 381844950 x 10-4 F3 -5. 153970917j x 10-14 -2. 323110925j x 10-14 1.8 1. 347166981 x 10-4 8. 939467016 x 10-5 F4 1. 153494515 x 10-20 5. 155845188 x 10-21 1.9 6. 550275578 x 10-5 4. 346600927 x 10-5 I1 1.347567668 x 10-7 1. 347568769 x 10-7 2.0 0 0 2 2. 099601048j x 10-2 2. 096263246j x 10-2 13 -2.099601048j x 10-2 -2. 096263246j x 10-2 14 1. 256431006 x 104 1. 251876324 x 104 F5 5.104127906 x 10-6 2. 138824275 x 106 ReF 2. 109232969 x 10-6 9.107530442 x 10-7 ImF6 -1 856345443 x 10-6 -9. 465968673 x 10-7 F8 3.768417259 x 10-6 1.551161837 x 10-6 15 2. 787895257 x 10-7 2. 787895257 x 10-7 ReI6 3.60525450 x 10-7 3. 60525450 x 10-7 18 8.762842367 x 10-7 8. 762842367 x 10-7 F1I1 3. 765977397 x 10-12 2.123486167 x102 F2I2 -6. 793075608 x 10-15 -3.027934684 x 10-15 F3I3 -1. 082128274 x 10-15 -4.869852048 x 10-16 F44 1. 449286274 x 10-16 6.454480521 x 10-17 F5I5 1. 422977398 x 10-12 5. 962818052 x 10-13 ReF6I6 1. 277961831 x 10-12 5. 922509428 x 10-13 F8I8 3.302204641 x 10-12 1.359258666 x 10-12 FeIe 1.103935282 x 10-11 5 260078149 x 10-12 Coeff1 2. 856029500 x 10-8 3.371265747 x 10-8 Coeff2 1.276497612 x 10-3 8.470522556 x10-4 W o2c 2 Coeffl = - -— 7-2' —2a~eoec |2 ~ 2 Coeff2 =: | 2 --------- 146 - 146

THE UNIVERSITY OF MICHIGAN 4386-1-T CASE III 9..I IVS - 8 - m a 7 /- x-.5 i i x-.7 0.1.2.3.4.5.6.7.8.9 1.0 1.1 1.2 r/a 147

THE UNIVERSITY OF MICHIGAN 4386-1-T BIBLIOGRAPHY 1. Van Trier, A, A. Th. M. "Guided Electromagnetic Waves in Anisotropic Media", Appl. Sci. Research B., 3, 4-5, 305 (1953). 2. Suhl, H. and Walker, L. R. "Topics in Guided Wave Propagation through Gyromagnetic Media", B. S. T. J. (1954), 33, Part I, 579-659, Part II, 939-986; Part III, 1133-1194. 3. Epstein, Paul S., "Theory of Wave Propagation in a Gyromagnetic Medium" Rev. Mod. Phys., Vol. 29, No. 1, Jan. 1956. 4. Falnberg, Y. B. and Gorbatenko, M. F., "Electromagnetic Waves in a Plasma Situated in a Magnetic Field", Soviet Physics Technical Physics, ~I Vol. 4, No. 5, pp 487-500, Nov. 1959. A translation of the Journal of Technical Physics of the USSR; Russian original Vol. 29, No. 5, May 1959. 5. Marcuvitz, N., "General Electronic Waveguides", Microwave Research Institute, Polytechnic Institute of Brooklyn, Research Report R-692-58, P. I.B. 620. 6. Proceedings of the Symposium of Electronic Waveguides, New York, April 1958. Polytechnic Press of the Polytechnic Institute of Brooklyn. 7. Samaddar, S. N. "Study of Surface Waves Along the Boundary of Two Coaxial Cylindrical Dielectric Columns In a Circular Cylindrical Metallic Waveguide with Excitation by a Magnetic Ring Current", Microwave Research Institute of Brooklyn, Research Report PIBMRI-857-60, 7 October 1960. 8. Trivelplece, A. W,, "Slow Wave Propagation in Plasma Waveguides", Technical Report No. 7, May 1958, California Institute of Technology. 9. Bresler, A. D., Joshi, G. H. and Marcuvitz, N. "Orthogonality Properties for Modes in Passive and Active Uniform Waveguides", Research Report R-625, 57, PIB-553, Microwave Research Institute, Polytechnic Institute of Brooklyn. 10. Bresler, A. D., "Vector Formulations for the Electromagnetic Field Equations in Uniform Waveguides Containing Anisotropic Media", Research Report R-676-58, PIB-604, Microwave Research Institute, PIB. 148

THE UNIVERSITY OF MICHIGAN 4386-1-T 11. Bresler, A. D. and Marcuvitz N. "Operator Methods in Electromagnetic Field Theory", Research Report R-495-56, PIB-425, Microwave Research Institute, Polytechnic Institute of Brooklyn. 12. Marcuvitz, N. and Schwinger, "On the Representation of the Electric and Magnetic Fields Produced by Currents and Discontinuities in Waveguides", Journal of Applied Physics, Vol. 22, No. 6, 806-819, June 1951. 13. Samaddar S. N. "Construction of Dyadic Green's Functions from the Sourcefree Solutions of Maxwell's Equations for Inhomogeneous Anisotropic Nondissipative Media in a Uniform Waveguide of Arbitrary Cross Section Bounded by a Perfect Conductor", The University of Michigan Radiation Laboratory Memo 2987-518-M, 13 June 1960. 14. Felsen, L. B., and Marcuvitz, N. "Modal Analysis and Synthesis of Electromagnetic Fields", Research Report R-446-55(a) and (b), and R-726-59, PIB-654, Microwave Research Institute, Polytechnic Institute of Brooklyn. 15. McLachlan, N.W., "Bessel Functions for Engineers", Clarendon Press, Oxford, 1934. 16. Collin, Robert E., "Field Theory of Guided Waves", McGraw-Hill Book Company, Inc. New York, 1960. 17. E. Jahnke and F. Emde, "Tables of Functions with Formulae and Curves", Dover Publications, New York. 18. Bozorth, Richard M. "Ferromagnetism", D. Van Nostrand Co., Inc., New York, 1953. 19. Gamo, H. J. Physics Soc. Japan 8, (1953) 176. ------------------------ 149

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