Vector Green Functions versus Dyadic Green Functions C.T. Tai Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor MI 48109-2122 November, 1994 Technical Report RL 910 RL-910 = RL-910 1

Dyadic Green functions, denoted by G, are introduced to integrate the vector wave equa[-.A tions for E and H to provide an integration solution of these differential equations [1]. A dyadic Green function is made of three vector Green functions, denoted by G with i = 1, 2, 3 such that (i) where xi with i =- 1, 2, 3 correspond to x, y, z, the unit vectors in cartesian coordinates. In [1], the eigenfunction expansion of various G's are found by dealing directly with the differential equation of G. In this note we are going to find the expansions for the vector Green function by means of (1). We consider the eigenfunction expansion for the vector Green function in a rectangular waveguide as an example. Maxwell equations for the field quantities E and H in a region assumed to be occupied by air (vacuum) are: VE ioRoH (2) H = J- icoE (3) In this note we are using the new notations [1] for the divergence and the curl in writing the equations. The dot and the cross signs are used exclusively for the scalar and the vector products. For an infinitesimal current source with current moment c^ix located at R' we write J = CiXi^(R- ') (4) We introduce the vector Green functions Ge and Gm such that - -(i) (5) i0oH1 = G- (6) iwg0oJ = icogoCiXi (R - R) (7) 2

and let the current moment be so normalized that icoIoci =, then the equations for Ge -(i) and G, are: VGe G, (8) - R' VGm)= Xi8 (R-R) +k Ge' (9) k2_=_W2 -(i) -(i) where k2 = O)2E0. he functions Ge and Gm with i = 1, 2, 3 are called, respectively, the electric and magnetic vector Green functions. There are three functions for each set corresponding to three different orientations of the infinitesimal current source. By eliminating G or G between (8) and (9) we obtain -(i) the equations for Ge and Gm. They are G(i) -k2Gi) = x (R- R) (10) (i) 2- (i) VVG - k2G = [xVi5(R-R)] (11) To find Ge To find Ge it is more convenient to find Gm first because G 's are solenoidal while -(i) i) G 's are not. The eigenfunction expansion of Gm requires only the solenoidal vector - (i) - wave functions. Once Gm 's have been found we can use (9) to find G() '. The vector wave functions appropriate for the rectangular waveguide are Me (h) = Vx eihz L Sx Sy (12) and Ne (h) = VMe (h) (13) om n K 'ni where 3

mn mr/ C= cos -x S, sin -x a X a nit nit C!, = cos y Sy = sin-y 2 1/2 K= (h +kc) 2 ml 2 n 2 kc = ( —) + (-) a, b = width and height of guide To solve (11) by the Ohm-Rayleigh method we let [x (R-R')] = fdh [Ne (h)A ) (h) + M(h) B (i)] (14) -00 m, n where Ne (h) and MO (h) represent, respectively, Nemn (h) and Mom,, (h),and A () and B (i) are two scalar coefficients to be determined. The reason that we place these coefficients after the two vector functions will become clear later. The orthogonal properties of the vector wave functions are discussed in detail in [1, pp. 102-103]. By taking the scalar products between (14) and Nemn' (-h') and Mem'n' (-h'), respectively, and integrating throughout the entire volume of the waveguide we can determine the coefficients A (i) and B(i) they are (2 - 80)K _ A(t) (h) = (M', (-h) (i) (15) nabkc W (2 - 50)KU B(i) (h) (2 - o)KN(-h) x(i) (16) 7nabk2 where 60 = 1 when m or n equal to one and zero for other integers. The primed functions in (15) and () 16) are defined with respect to primed variables x' associated with R', the position vector of the infinitesimal source. The eigenfunction expansion of V [xi6S (R - R') ] is therefore given by 4

- - (2-60)K - __ V[i(R - R')] Idh - 2 [Ne,(h)M'e(-h) ~ xi I mz 7cabk+ Mo (h) N'o (-h) x (17) - (i) We let G( have a similar expansion with unknown coefficients ae and be attached to the eigenfunctions. By means of (11) we find ae = be = K2e e k2 thus (i) = d(2 - 60)K Gm idhCe, nbk2(-k2) (18) [Ne (h) M'e (-h) x + M (h) N', (-h) x The integration with respect to h can be evaluated by means of a contour integration that yields ((i) - G = G - = ~ C k [Ne (~k+) Mo k) x +M (+kgO'() N O (:kg) x ] z' m, n (19) The top line applies to GM for z > z' and the bottom line to GM for z < z', and kg (k2 2 k12 -kc) To find Ge,we make use of (9). The function G however, is discontinuous at z = z. If we write -(i) - (i) + Gm = Gm U(z-z') +Gm U(z'-z) (20). where U (z - z') and U (z' - z) are two step functions defined at z = z', then 5

VG = [VG ]U(z-z) +z6(z-z) xG,) - - (i)+- )+ [VG) ] U (z' - z) - ^ ( z - z') x Gi) [Gm ]U (z- z) + [G () ] U(z -z) -+x [(i)+ - (i)- z) +zx [G, GM ]z-z) (21) Equation (21) can be simplified by applying the boundary condition for the G, ) 's. We start with -+ - (i) 2x [H -H ] = J (22) - (i) where Js denotes the surface current density at z = z'. For an infinitesimal source (22) can be converted to the form 2x [G'-Gm ] = x 6(R-R') (23) for i = 1,2 and - (3)+ -(3)-x [GM -GM ] =0 (24) for i = 3. With the aid of (9),(21),(23), and (24) we find - () (i) Ge = S e e = Cmn [Me (~kg)M' (Tkg) No (~kg) N+ (,g) ] Xi, m, n z' (25) for i = 1, 2 and -G(3) = 1 - - (3) Ge = — 6 (R- R') + S k (26) for i = 3 where i(2 -50) mn abk abk2kg c 9 6

To find the dyadic Green function Ge we use (1) that yields Ge = — ZZ6 (R - R') + Se (27) kC where Se = Crn [Me (+kg) M' (kg) + N (+kg) No (Tkg) ], z in, In By comparing the formulation used in [1] and the present formulation it is seen that the dyadic Green function approach is more direct and perhaps simpler than the vector Green function method even though dyadic analysis is being used therein. References [1] C.T. Tai, Dyadic Green Functions in Electromagnetic Theory, Second Edition, IEEE Press, Piscatawnay, NY. 1994. [2] C.T. Tai, Generalized Vector and Dyadic analysis, IEEE Press, Piscataway, NJ. 1992. 7