Recurrent relationships for reflection coefficients of electromagnetic wave potential representation modes in manylayered spherical environment excitation problem
Authors: Colichev S.A. , Camalova N.S. 
Published: November 2021

In theoretical assessments of the electrodynamic characteristics of multilayer screens of electromagnetic waves, the concept of the input impedance of a wave on the surface of the screen is used [1]. In [2], an approach was proposed that is not based on this characteristic, and direct recurrence relations were obtained both for the reflection coefficient of a plane wave from a multilayer screen and for the coefficient of transmission of such a wave through this screen. In [3], this approach was applied to solving problems of excitation of a layered medium by localized sources. In this paper, we generalize the approach to the case of a multilayer medium with another simple layer geometry (with interface in the form of concentric spherical shells). The solution of excitation problems is based on the representation of the Hertz potentials of waves of electric (magnetic) types in the form of series in spherical harmonics. Each harmonic defines the angular distribution of radially converging and diverging modes of the representation of potentials in layers free from external sources. The recurrence relations for the reflection coefficients of convergent (diverging) modes of representation when sequentially adding interfaces are a consequence of the boundary conditions. One of the equivalent forms of these relations is derived from the condition for the existence of a nontrivial solution of a system of linear equations for the amplitudes of converging (diverging) modes in adjacent layers. The other is based on a simple physical model of the process of interaction of a converging (diverging) mode in a layer with a layered structure, which it covers (which covers it). For a field source inside one of the layers, one can determine the representation of potentials that this source would excite in an infinite medium with permeabilities similar to a layer. The amplitudes of the modes of representation of this «primary» field and the reflection coefficients of the modes of representation from structures at the internal and external boundaries of the layer, found from recurrence relations, determine the amplitudes of the modes of the total field of the layer incident on these boundaries. To determine the amplitudes of the modes of representation of the potentials of the fields in the remaining layers, recurrence relations can also be used, in a certain sense inverse to the recurrence relations for reflection coefficients. If a layered structure is excited by a plane wave (scattering problem), the total field potentials (of the electric and magnetic types) are the sum of radially converging modes in the potentials describing the incident plane wave and radially diverging modes formed by the reflection of these converging mod from structure. With the proposed approach, the complexity of solving the excitation problems of the considered layered structure remains almost unchanged with an increase in the number of layers. In essence, only the number of repetition cycles of the obtained recurrence relations increases. The proposed approach is conceptually simple, the final recurrence relations are physically transparent. The work does not represent the full facilities of the approach. Within the framework of this approach, recurrence relations for the permeability coefficients of multilayer spherical shells of the structure under consideration, recurrence relations during its layer-by-layer formation or formation from several multilayer shells can be found.