Electromagnetic Field Representation in Inhomogeneous Anisotropic Media*

Some of the basic developments in the theory of electromagnetic field representation in terms of Hertz vectors are reviewed. A solution for the field in an inhomogeneous anisotropic medium is given in terms of the two Hertz vectors. Conditions for presentation of the field in terms of uncoupled transverse electric and transverse magnetic modes, in a general orthogonal coordinate system, are derived when the permeability and permittivity tensors have only diagonal components. These conditions are compared with some known special cases. Index Headings: Electromagnetic theory Anisotropic medium Hertzian-vector representation Analysis of electromagnetic field in inhomogeneous anisotropic media has practical applications in antenna, scattering and propagation problems. In the analysis of radiation from an oscillating electric dipole in free space, Hertz [1] introduced a potential of the type which now bears his name. Since that time, considerable use has been made of the Hertz potentials. Field representation, using electric and magnetic Hertzian potentials was investigated by Whittaker [2], Debye [3], Bromwich [4] and Hansen [5]. Extension of Hansen's work for radially inhomogeneous media of constant permeability is due to Bremmer [6] and Tai [7]. A general theory of the Hertz vectors and the associated gauge transformation for inhomogeneous anisotropic media was developed by Nisbet [8], [9] and this was presented in 4-vector form by McCrea [10]. The Whittaker and Debye-Bromwich types of representation, which split the field into so-called transverse electric (TE) and transverse magnetic (TM) parts, have had considerable and fruitful applications for homogeneous media. The search * This work was supported by the National Aeronautics and Space Administration, Langley Research Center, under NASA Grant NGR-23-005-477. for situations giving rise to uncoupled TE and TM fields is justified by the fact that, in such cases, the calculations involved can be considerably simplified. Such situations were investigated in terms of general orthogonal coordinates in homogeneous media by Bromwich [4], by Nisbet [91 for isotropic inhomogeneous media in the presence of sources, and by Friedman [11] for inhomogeneous anisotropic media when the permeability is constant. Some special cases were discussed by Wexler [12] in connection with mode uncoupling in inhomogeneously loaded rectangular and circular waveguides and by Uslenghi [13] for the radially inhomogeneous cylinder. The error in the latter work was pointed out by Parkinson [14]. Mode uncoupling via transformation of coordinates for the radially inhomogeneous cylindrical plasma column was recently considered by Bisbing [15]. Benenson and Zaytsev [16] considered mainly homogeneous anisotropic media and discussed the propagation of TEM, TE and TM waves in cylindrical and spherical coordinate systems. Mention should also be made of contributions by Luneberg [17], Marcuvitz [18], Green and Wolf [19], Samaddar [20], and Felsen and Marcuvitz [2t].