Fundamentals of a Null Field Method-Surface Equivalence Principle Approach for Scattering by Dielectric Cylinders

TL;DR

The paper investigates combining the null-field method with the surface equivalence principle (NFM-SEP) to solve dielectric-cylinder scattering with internal/external line sources. It proves that discretized NFM-SEP currents converge to the corresponding continuous current densities as $N\to\infty$, avoiding the divergence and oscillations seen in the method of auxiliary sources (MAS). For circular geometry, analytic and numerical results show that discrete currents reproduce exact fields and extend to noncircular (elliptical) shapes, with auxiliary-surface choices having no detrimental effect on NFM-SEP. A detailed comparison with MAS highlights fundamental differences in stability, while numerical experiments demonstrate robust convergence and competitive speed. The approach has potential for stable computation in dielectric waveguides and related eigenproblem studies.

Abstract

The null-field method (NFM) and the method of auxiliary sources (MAS) have been both used extensively for the numerical solution of boundary-value problems arising in diverse applications involving propagation and scattering of waves. It has been shown that, under certain conditions, the applicability of MAS may be restricted by issues concerning the divergence of the auxiliary currents, manifested by the appearance of exponentially large oscillations. In this work, we combine the NFM with the surface equivalence principle (SEP) and investigate analytically the convergence properties of the combined NFM-SEP with reference to the problem of (internal or external) line-source excitation of a dielectric cylinder. Our main purpose is to prove that (contrary to the MAS) the discrete NFM-SEP currents, when properly normalized, always converge to the corresponding continuous current densities, and thus no divergence and oscillations phenomena appear. The theoretical analysis of the NFM-SEP is accompanied by detailed comparisons with the MAS as well as with representative numerical results illustrating the conclusions.

Figures (10)

• Figure 1: Scattering of a cylindrical wave by an infinitely-long dielectric cylinder. (a) Original problem; (b) first equivalent problem for the external field in $R_1$; and (c) second equivalent problem for the internal field in $R_2$.
• Figure 2: Discretization of (a) the external and (b) the internal NFM equivalent problem for a dielectric cylinder of arbitrary cross section. Black dots represent the discrete electric and magnetic sources, and black squares represent the collocation points.
• Figure 3: Scattering of a cylindrical wave by an infinitely long circular dielectric cylinder. (a) Original problem; (b) first equivalent problem (for external field in $R_1$); and (c) second equivalent problem (for internal field in $R_2$).
• Figure 4: Discretization of (a) the external and (b) the internal NFM equivalent problem for the circular dielectric cylinder.
• Figure 5: Real (red solid lines) and imaginary parts (blue dashed lines) of (a) continuous electric current density $J_z^s(\phi_{\mathrm{cyl}})$ and (b) continuous magnetic current density $M_\phi^s(\phi_{\mathrm{cyl}})$ together with real and imaginary parts (circles) of normalized NFM currents $NI_l/2\pi\rho_{\mathrm{cyl}}$ and $NK_l/2\pi\rho_{\mathrm{cyl}}$ for external $\rm{TM_z}$ excitation. (c) and (d) are as (a) and (b) but for internal $\rm{TM_z}$ excitation. For (a)-(d): $k_1\rho^{(1)}_{\mathrm {aux}}=1.5$, $k_1\rho^{(2)}_{\mathrm {aux}}=2.5$ (or $k_1\rho^{(1)}_{\mathrm {aux}}=0.5$, $k_1\rho^{(2)}_{\mathrm {aux}}=10$) and $N=40$. (e) and (f) show the oscillations of the MAS currents of $C^{(1)}_{\mathrm{aux}}$ for internal and $C^{(2)}_{\mathrm{aux}}$ for external excitation with $k_1\rho^{(1)}_{\mathrm {aux}}=0.5$, $k_1\rho^{(2)}_{\mathrm {aux}}=10$ and $N=40$.