Cấu trúc sóng chức năng trong điện lý thuyết P4

We shall consider the scattering of a plane, linearly polarized monochromatic wave by a perfectly conducting prolate spheroid immersed in a homogeneous isotropic medium. Solution of the EM scattering by the oblate spheroid can be obtained by the transformations 5 --+ it and c --+ -ic. It is assumed that the surrounding medium is nonconducting and nonmagnetic. The geometry of the configuration is shown in Fig. 4.1, and the surface of the spheroid is given by
Nội dung trích xuất từ tài liệu:
Cấu trúc sóng chức năng trong điện lý thuyết P4 Spheroidal Wave Functions in Electromagnetic Theory Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong Copyright  2002 John Wiley & Sons, Inc. ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic) E.M Scattering by a 4 Conducting Spheroid4.1 GEOMETRY OF THE PROBLEMWe shall consider the scattering of a plane, linearly polarized monochromaticwave by a perfectly conducting prolate spheroid immersed in a homogeneousisotropic medium. Solution of the EM scattering by the oblate spheroid canbe obtained by the transformations 5 --+ it and c --+ -ic. It is assumed thatthe surrounding medium is nonconducting and nonmagnetic. The geometryof the configuration is shown in Fig. 4.1, and the surface of the spheroid isgiven by (4 . 114.2 INCIDENT AND SCATTERED FIELDSWithout loss of generality, the direction of propagation of the linearly polar-ized monochromatic incident wave is assumed to be in the z, z-plane, makingan angle 80 with the z-axis, as shown in Fig. 4.1. At an oblique incidence (00 # 0), the polarized incident wave is resolvedinto two components: the TE mode, for which the electric vector of the in-cident wave vibrates perpendicularly to the X, z-plane, and the TM mode, inwhich the electric vector lies in the x, z-plane. Thus the plane-wave expres-sions for both modes are given by -jk.r (4.2a) ETE = ETEO@e 7 8990 EM SCATTERING BY A CONDUCTING SPHEROID Incident Wave (#o=4 Scattered Wave Fig. 4.1 Geometry of EM scattering by a conducting prolate spheroid. INCIDENT AND SCATTERED FIELDS 91 ETM = ETM~ (-ii? cos 80 + 2 sin 0e)e -jk*r 3 (4.2b)where ETEO and ETMO are the amplitudes of the TE and TM fields respec-tively, and -k l r = Ic(zsin& + ZCOS~~), (4 . 3)with k being the wave number of the monochromatic radiation. Flammer [l] has obtained the plane-wave expansion in terms of prolatespheroidal wave functions aswhere N mn(~) is the normalization constant given in Eq. (3.9) and em is theNeumann number, em = 1 for m = 0 and em = 2 for m > 0. For simplicityin what follows, the argument c will be suppressed in the description of thefunctions. From the equations above the excitation can be described in terms of vectorwave functions [l] as follows: ETE = ETEo fJ 2 Amn(~O)M~~~(C; ?j, 0. In other words, by Eq. (4.5c),A mn is non-zero only for m = 0 and the expansions are defined only form = 0 for an axial incidence: Ei = (4.6)92 EM SCATTERING BY A CONDUCTING SPHEROIDwhere m-1 AOn = k Son(l)* To describe the scattered fields, we can use radial functions of the fourthkind, because the radiation condition at infinity must be satisfied. Radialfunctions of the fourth kind are suitable because of their asymptotic behavior.This ensures that at large distances from the spheroid the scattered wave be-haves as a spherical diverging wave, emanating from the center of the spheroid.The components of the scattered field must also have the same +-dependenceas the corresponding components of the incident field (&matching). To satisfy these requirements, we can write ESTE = arnn M+(4) + YrnnM~$f& (4.7a) e,m+l,n n=m m=O &TM = 2 2 (~~n”i$?t-l,n + Pm+l,n+l Mz!z+l,n+l) n=m m=O ...