Electromagnetic scattering by fractal screens: from diffraction integrals towards a boundary element method

Chadwick, Edmund; Christian, James

Abstract

The scattering of time-harmonic electromagnetic fields by perfectly-conducting plane screens is a fundamental problem in physics. A standard approach to computing the scattered wave is to deploy Rayleigh-Sommerfeld diffraction integrals in conjunction with the ad hoc Kirchhoff approximation to enforce on-screen boundary conditions [1]. These formulations are best suited to high-frequency regimes, where the characteristic aperture size a is typically much larger than the wavelength λ = 2π/k such that ka >> 1. They are also desirable because the well-known Fresnel (near field) and Fraunhofer (far field) integrals follow straightforwardly from Taylor expansions within the Rayleigh-Sommerfeld kernel. However, diffraction integrals must be used with caution when the screen has some kind of fractal (or self-similar) structure— for example, those based on a finite number of stages in the Cantor set or Sierpinski triangle [2]. In such cases, the low-frequency regime ka << 1 may be encountered as a matter of course. A more rigorous approach is to consider scattering as a formal boundary-value problem for the underlying Helmholtz equation. One constructs a boundary integral equation with reference to the free Green’s function, then discretizes the domain of the screen to arrive at a corresponding boundary element formulation. In this way, boundary conditions (typically either Dirichlet or Neumann) are accommodated as a natural part of the numerical scheme. The key advantage over diffraction integrals is that low-frequency regimes can be accessed without difficulty. Our presentation will survey earlier results obtained from a Rayleigh-Sommerfeld prescription, providing background and context. A new boundary element formulation will then be detailed along with some preliminary results.

• Presentation Conference Type: Conference Abstract
• Conference Name: 14th United Kingdom Conference on Boundary Integral Methods
• Start Date: Jul 8, 2025
• End Date: Jul 8, 2025
• Acceptance Date: Jun 3, 2025
• Deposit Date: Jun 20, 2025
• Peer Reviewed: Peer Reviewed