011136-1-T 11136-1-T = RL-2208 THE UNIVERSITY OF MICHIGAN COLLEGE OF lENGWRING DEPARTMENT Of ELECTRICAL AND COMPUTER ENGINEERING Rodiation Laboraory ON THE EIGEN-FUNCTION EXPANSION OF DYADIC GREEN'S FUNCTIONS Technical Report By Chen-To Tai Grant GK-36867 April 1973 Prepared for: National 8oienoe Foundation Attn: Frederick H. Abernathy Divilwie Director for Engineering Washington, D. C. 20550 Ann Arbor, Michigan 48105

ABSTRACT This work contains a revision of the treatment of the eigen-function expansion of dyadic Green's functions previously discussed by the author in his book L1]. The singular terms which are missing in the previous treatment have been amended. By starting with the differential equation for the dyadic Green's function of the magnetic type only two sets of solenoidal vector eigenfunctions are needed to determine the complete expressions for both the electric and the magnetic type of dyadic Green's function. ii

TABLE OF CONTENTS Page Number ABSTRACT ii I INTRODUCTION 1 H GENERAL FORMULATION 2 Il RECTANGULAR WAVEGUIDE 7 Cylindrical Waveguide 11 Eigen-function Expansion of Free-space Dyadic Green's Functions Using Cylindrical Vector Wave Functions 13 Eigen-function Expansion of Free-space Dyadic Green's Functions Using Elliptical Vector Wave Functions 15 Perfectly Conducting Wedge 16 Eigen-function Expansion of Free-space Dyadic Green's Functions Using Spherical Vector Wave Functions 16 Cone 18 Rectangular Waveguide with a Moving Isotropic Medium 19 ACKNOWLEDGEMENT 21 REFERENCES 21 iii

I INTRODUCTION Professor Per-Olof Brundell of the University of Lund, Sweden, has kindly called the author's attention to an error in the treatment of the eigenfunction expansion of the dyadic Green's functions described in the author's book [']. In that work only two sets of solenoidal vector wave functions are used in the expansion of the dyadic delta function I 6 (R -R'). Since the latter is nonsolenoidal, the use of solenoidal functions to represent such a quantity is not sufficient or complete. As a result of this error the singular behavior of the dyadic Green's functions is not properly formulated in the author's book. In this work, the correct expressions for various dyadic Green's functions are derived by means of a revised method which removes the shortcomings found in the previous treatment. In the case of a rectangular waveguide the results has been verified by Collin [2] who has independently found the solution for the field in a source region based on the method of potentials. 1

II GENERAL FORMULATION For clarity we introduce two types of dyadic Green's functions designated by G, the electric type, and G, the magnetic type which satisfy the equations VxG x G (1) e m Vx - 6 (-')+k G (2) m e where I denotes the idemfactor defined by ^AA AA AA I= xx+yy+zz and k2 = o o These equations are the dyadic version of Maxwell equations as applied to harmonic fields due to infinitesimal current sources. The relationship between Ge, G, I6(R-R') and E, H, J are m =E(X) x^ (y) +(z) (3) e G iw (H(x) + + z) (4) m o I 6 (R-R') = iw x Y~+J( (5) where E and H represent the electric and the magnetic fields due to an infinitesimal current source with a current density J - )= 6 (R-R')x and similarly for the other triads. By eliminating G or G between (1) and (2) m e we obtain VxVxG -k2 G I 6(R-R') (6) e e VxVxG -k G =Vx I6(R-R'). (7) m m L I 2

Equations (6) and (7) differ from each other in the inhomogeneous term. Furthermore, we have V-G -V 16(R-R' V 6 6(VR-R') e k2 k2 V- G = m thus G is nonsolenoidal while G is solenoidal. e m The dyadic Green's functions are classified according to the boundary conditions which they must satisfy on an assigned surface. The functions of the first kind satisfy the Dirichlet boundary condition A el = (8) A nxG =0 ml and the functions of the second kind satisfy the Neumann condition nxVxGe2= (9) nxVxG O l0 m2 J If the region under consideration is open, it is assumed that the radiation condition prevails at infinity LimR [x - ikxG = R --- oo V el The same applies to Ge2, Gml and Gm2. If the region has a limited open space, e2 ml m2 such as in an infinite waveguide other forms of radiation condition are assumed to be existing at the open ends. Because of (1) and (2), the functions of the first kind of the electric type are related to the functions of the second kind of the magnetic type, thus 3

VxGe Gm (10) m2 VxGm2= T 6(R-R')+k Ge, (11) ni2 el and similarly VxGe2 Gml (12) e m VxGm =f6(R-R')+k2 G (13) ml e2 The dyadic Green's functions are introduced mainly to facilitate the integration of the vector wave equations for E and H: -2 -VxVxE-k E iwlp T, (14) -2 VxVxH-k H VxJ. (15) The integration can be carried out with the aid of the dyadic Green's identity in the form f Pf VxVxQ-(VxVxP)-.Q dv = no. [PxVxQ+(Vx )xQ ds (16) This identity can be derived by superposing three vector Green's identities of the form described by Stratton [3] IflJ VxVxQ-(VxVxf)- Q dv =- n [PxVxQ+(VxP)xQ ds (17) -(x) (Y) (Z) where we let Q be equal to Q, Q( and Q, three distinct vector functions, and then introduce the dyadic function Q defined by - iQ(X) a +Q(y) ^+ 4ZE) With the aid of (16) we can integrate the equation for E, (14), by letting P= E and 4

Q = Ge where G 1 satisfies Eq. (6)and the boundary condition (9). The result is given by E (R)i i/ II (R/R' -T).J(R')dv'In arriving at this expression we have already made use of the symmetry relations el( /) 2(R/R') and v'x G (R*/R) = VxG7 (R/R') el e2 where the sign 'r " denotes the transpose of a dyadic function. The derivation of these symmetrical relations is found in Ref. [1]. If the surface S corresponds to the site of a perfectly conducting surface where nx E = 0 only the volume integral in (18) remains, that is, E(R)= i p Gel (R/R') - (R) dv'. (19) The magnetic field H can be obtained either by using VxEx iw0o H with E given by (18) or by letting P H and 4 x 2 in (16) and the relations (10) to (13). In e2 either case we obtain H(R)= m(R/R' (R') dv' iio- P Vx ml(R/R' nxE (R) ds. (20) S While most of these formulas have been derived in Ref. [1], the presentation here emphasizes the distinction between the two types of dyadic Green's functions which was not stressed before. In fact, neither the subscript notation 'e' and 'm' nor Eqs. (1) and (2) were introduced previously. 5

In the remaining sections we shall present the eigen-function expansions of various dyadic Green's functions. The topics will be arranged in the same order as they appeared in the author's book. Some of the basic formulas, such as orthogonal properties of various vector wave functions, the circulation theorem involving the product of Bessel functions and many other mathematical theorems will not be reviewed here. 6

I RECTANGULAR WAVEGUIDE We start with the equation = 2= F= —J1 VxVxG -k G Vx I6(R-R) (21) m2 m(21) for the magnetic dyadic Green's function of the second kind G 2 which satisfies the boundary condition nxVxG =0 m2 at the walls of a rectangular waveguide corresponding to x z 0 and a; y = 0 and b. The function also satisfies the radiation condition -- A VxG = azxGm2 m2 m2 (22) or G2 x3 zxVxGm2 at z + oo. These conditions correspond to the radiation condition of the TE or TM modes in a rectangular waveguide where a and 13 are two sets of constants. Since V. Vx[6(R-R')j =0, the generalized function V x 6 (R-R') can be expressed in terms of the vector wave functions M (h) and N (h) defined ormn emn by M (h) = Vx; (h) Z omn L omn N" (h) V Vx V (h)z emn K Vemn J where f mr n7 cos xcos- y a b ihz b (h) = e emn sin mr x sin ny 0 a b Y K2= h2+ k2 c }2. 2-) c \a] \b 7

Applying the Ohm-Rayleigh method we let Go Vx ~I6(R- - K M (h) (h) (h ) h)B (h) dh (23) VB6 (RR f omn emn emn (23).=m,n where A, B are two sets of unknown vector coefficients to be determined. By taking the anterior scalar product of (23) with M,,(-h') and integrating through the entire volume of the guide we obtain, as a result of the orthogonal property of the vector wave functions (1+6 ) 7r ab k' V'xM', (-h') 0 c ()A h om'nt 2 om'n' -2 t 2 2 where k'=(-) +(n) c a b l m' or n' = 0 and 6 = m 0, m'andn'O, Hence (- (2-6 )K A (h)= V' xM' N (-h) (24) omn 2 omn 2 omn irabk Xrabk c c where the primed function is defined with respect to the primed variables x', y', z' pertaining to R'o Similarly, by taking the scalar product of (23) with Nem tn -h') and performing the integration, we obtain 2-6 (2-6 )K B (h) = V' xT' (-h)= M — (-h), (25) emn 2 emn 2 emn irabk rT abk c c thus 8

VxT6 ( I ] f C K omn (h) N9mn(-h) + J mn omn omn -a> m,n +N (h) M' (-h) dh (26) emn emn (2-6) 0 where C z mn 2 n rabk c To determine G 2(R/R') we let m2 o00 G (R/R)= I C K a M (h)N' (-h) + m2 J mn omn omn omn *-o m,n +b N (h)M' (-h) dh (27) emn emn emn Substituting (26) and (27) into (21) we find 1 a =b X omn emn 2 2 Having obtained the eigen-function expansion of Gm2 we can determine Ge by means of (10). The term VxG 2 is given by, in view of (27), m2 VxGm2 I c K2 k2 N (h) N' (-h) + m2 mn 2 2 omn omn -aoo m,n +M (h) Mem(-h) dh (28) emn emn J where we have made use of the relations VxM (h) KN (h) omn omn 9

and VxN (h) KM (h) emn emn The expression for VxGm2 as given by (28) has a singular term which can be extracted from the expression. For that reason we split (28) into two parts VxG = m2 dh+ oD + f X +| C mn - m. n K2 k2 k2 K- — 2N ot + K2_k2 h- ot ot +N N' +N N' +N N' dh ot oz oz ot oz oz where Not and Nz denote, respectively, the transversal part and the part of N. It can be shown that 0 (29) longitudinal 6 6(R- R') = C N dh fx mn e e 2 ot ot mn m n as m,n and the second integral in (29) can be closed at infinity in the h-plane which yields a residue series. The final result can be written in the form -m2 -VxGm2 It6(R-R) +k S (R/f') (30) where ab 1 M (+k )Mt ( k )+ / ab k2 emn - g emn g m,n g c m., n g c + N (+k ) N' omn- g omn (+kg)0, z z'. Substituting (30) into (11), we obtain 10

'('- z z 6 (-R) G (/ ') S(/R'I) Zz( ) 2 (31) el k The singular term - z z6 (R-R')/k is missing in the old expression for Gel(R/R?) discussed in Ref. [1]. The residue series S(R/R') is the same as the one defined by Eq. (8), p. 79 of that reference. The singular term vanishes when R i R'. When the point of observation lies inside the source region the singular term must be included in evaluating the electric field for an arbitrary current source; that is, E (R) = i w/11 Gel (R/R') * J(R' ) dv' (32) The method described in this section can be applied to all other dyadic Green's functions. Omitting the details we list below the complete expressions for these functions together with some of the essential formulas. Cylindrical Waveguide 00 D) Vx 6(R-R' = | M (h) N (-h) + -) n o o J n, 0 0en + C K N (h) MN (-hl dh where Me (h) Vx Oe (h) z enX) o Lo N (h) VxVx (h) z enX n) 0 e (h)= J (Xr) 0 e r) n s at r=a; J (0r)= 0 at r a; IKA.2=A+h2 n A 11

M (h) = VxFW (Ih) eih e np e J 0 N (h) 0 1 ih z7 ~VxVx 0(h) e K Ve nl I /2 - 01 (h) =J (jpsr) Cos n0eihz e np n sinn 0 ai n(Pr) 222 ar =0 at r =a; K =/2 + 2 -6 0 cx 2 2 47r /21 \k aI 2 rd rT =JGr) 2 a r a n C = 0 P 4 p A aa 2 2 a 2 1 = i (r) d r= 2 O 0- 2M n2 2 2 n a The origin of these functions or coefficients is found in Ref. [El]. Gm2 RRf= 0 0 n, p K2 -k2 e np e p 0 __ 12

Vx (R/R') I 6(RR') + k S (R/R') A t e (R/R') = (/ff)>-_ 2 f k (33) OD OD (o X- 6 nzO mxl 1 - ID N (+.1 enl — ) L N (+k)2+- 1 M (+k )M' (+k ) en pk2k Ie P e en p z z' -^ 22~ 2 k kk = -p u 1. The residue series S(R/RR) is the same as the one described by Eq. (5), p. 89 of Ref. [1]. Eigen-function Expansion of Free-space Dyadic Green's Functions Using Cylindrical Vector Wave Functions M (h) xVx en) o E e (h) z] Ne nk(h) =K VxVx [pe (h) zx enX KLe 0 0 2 2 bothcontinuous. K2 h + X; h and X both continuous. Vx T6(R-R']) = dh J dX -o00 n M (h) -N' (-h) + N (h) Oen en enX L% 0 0 0 C K x M' (-h) enX 0 - 2-6 0 C - 4~7^ 13

We denote the free-space dyadic Green's function of the magnetic type by G (R/R') and the function of the electric type by Ge (R/R). mo eo G MO(R/R') = h _ n2 00 K n (h) L nX % 0 N' (-h)+ N (h)M' (-h), e nen e n J O Oen (35) VxG (/mo ') mo 0 dh n -OD n. N (h) N' (-h) LeO e n 0 0 + M (h) M' (-h) enX en J O o For cylindrical problems, we remove the X-integration which yields VxG (R/R') k h(R/R') + z 6 (R- R') mo h where Sh(R/R') i(2-6 ) -OD n N' (-h) + M (h) M' (-h) ennr enn enr 0 0 0 r r' (36) N (h). nnr Functions with superscript (1) are defined with respect to the Hankel function of the first kind and r =/k - h 14

- 1t6(R-R') 2G (RIR')=S. (37) eo k2 k The integral of the residue series given by (35) is the same as Eq. (5), p. 96 of Ref. [1]. For flat earth problems, we remove the h-integration in (35) which yields VxG o(R/R) k S(R/R')+t 6 (R-R) hence GO(R/R')?= (R/R- (2R (38) eo = 2R/R)k where OCDr 2-6s n M (h Mr (-h )+N (hjN' (-h ) el nX 1 en 1 en 1e n o0 o o zZ z' (39) M (-h)Me (hh)+N (-hl)N' (hl) enX 1 en 1 e n 1 0 0 0 0 where hl.\. Equation (39) is the same as Eq. (1), p. 103 found in Ref. [1]. Eigen-function Expansion of Free-space Dyadic Green's Functions Using Elliptical Vector Wave Functions VxG (R/R) = k2 (R/R)+ zz 6(R-R') mo I 6(R-R') G (R/R,) = ~(R/R,) - - (40) eo 2 (R/R') is the same as the one given by Eq (3), p. 118 of Ref T(KR/) is the same as the one given by Eq. (3), p. 118 of Ref. [1]. 15

Perfectly Conducting Wedge VxG 2(RT/R') xk S (R/R +22 6 (K7-iT) -i:3;7 I16 (R — R') Gel (R/ff') = g(R/R') 2 (41) S(Rf/R') is the same as the one given by Eq. (9),v p. 123 of Ref. Lii. Eigen-function Expansion of Free-space Dyadic Green's Functions Using Spherical Vector Wave Functions VxG (iR/Rt) f MO0 n1l 0mn mZO K4d K2_2 L~em n 0 (K) MI emr 0 (K) + N (K) N' (K) a emn emnj 0 (42) where (K) em n 0 - Vx qe em(K) R emi 0 1 (K) = -~ VxVx a K Lemn (K)R] Cos m t'(K)= jn (KR) imO Pn(cosO0) Oemn n si n 0 C 0 2 +1 (n -in)! mn 2 7r 2 n(n+l) (n+m),' 6= o O, m~O Equation (42) has a singular part in the integration with respect to K represented by 0n~l mn LemnemnJ 16

OD n 7r (R-R') C m me (43) 2R2 mn en emn nxl mo o0 0 where m pm(cos 0) a pm(cos) ) n sin A n cos m e mn sin 0 cos ae sin o0 It should be pointed out that the function 6 (R - R') is a one-dimensional delta function resulting from K2 (KR) Jn(KR') dK (R-R 2R2 0 Having recognized the singular part of (42), we can evalute the remaining part by contour integration. Thus, we obtain CO ao n VxG (R/R') = K2 CM (K) M' (K dK+ mo mn L mn emn dK SO n m=O o 0 no n - + c K2 L M (K) M' (K) mn 2_ 2 emn e mn n=l mX O O 2 + - -- (K)N' (K) dK K2 -k emn emn (44) S 2 RR 6 — RI) + (45) G (R/R')= S(R/R') -2 R 6(R-R) +T2(R/R (45) eo _ L where aD n s2(R/R')= 2R C (Rxii )(R'x m' 2 2r2 m n emn n m nl miO o o 17

and -(R/ff') is the same as Eq. (18), p. 174 of Ref. [ill. Cone u(7IN9=)T(7/9) el + (R -RI) 27r2 rl1 A A -!it) + 2 R R 6(Rk - m sP 2 -6 A A A(+1 A A 1. il M 0 (46) where R~ff/ffl) 2ik m +pk R~ RI + x 1 p(p+ 1) 'm e.. (1) N!1 (k) enp 0 J P (cos 0 ) = 0,, characteristic equation for p 1.' 0 1=1 (Pn ) sinO dO 0 MP (cos e) si AOPm (cos e) rnsin A Cos A e 1 sine 0 Cos a e sin 06 -0 at 0 = 0, characteristic equation for A. (m2 ( ) sin Od O 18

In Cos M (k) = Vx I x(kR) P (cos 0) s m0R emX I Sf J I rn C Ne () k VxVxj (kR)P (cos) C. m0i. em k p sin Functions with superscript (1) are defined with respect to spherical Hankel functions of the first kind. The residue series S(R/R') is the same as the one given by Eq. (22), p. 191 of Ref. [1]. Rectangular Waveguide with a Moving Isotropic Medium Because of the incomplete sets of functions used the residue series derived in the book was wrong even where it is applied to regions where there are no current sources. The correct expression for Ge (R/R') is found to be el(/ ) /)- 2zz(R-) (47) where /3 2-6 xoYo k k2 m,n g c (+k )b. N' (+k )+ M (+k )b M' (+k omn- g omn g emn - g emn g o z z'. (48) The terms involving the N functions are different from the corresponding terms in the residue series given by Eq. (14), p. 219 of Ref. []. The parameters in (48) are defined as before. They are: k = a2k -ak g \ c 132 1 ^ AA A ^\A a P b - (xx+yy)+ zz n22' c a 19

N (h)=-VxVxkVP (h) -Z omn Ka L- Jm emn h)xx emn Jh r05mix niry k/e (h)M Xxih emn mirx nirv o sin- sin Jx0 0O 2 2 2 2 2 mA(n7r\ K a = h +ak, k + walls of guide: x=O0 and x 0 y = 0 and y0 20

ACKNOWLEDGEMENT The author is most grateful to Professor Brundell for pointing out the error in the author's book and for many valuable discussions which ultimately led to the method described in this report. The technical help which the author received from Professor Olov Einarsson is also very much appreciated. REFERENCES [1] C-T. Tai, Dyadic Green's Functions in Electromagnetic Theory, Intext Educational Publishers, Scranton, Pa., 1971. [2] Robert E. Collin, private communication. [3] J. A. Stratton, Electromagnetic Theory, McGraw Hill Book Company, New York, 1941. 21