A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media

Abstract

We present a fast method for numerically solving the inhomogeneous Helmholtz equation. Our iterative method is based on the Born series, which we modified to achieve convergence for scattering media of arbitrary size and scattering strength. Compared to pseudospectral time-domain simulations, our modified Born approach is two orders of magnitude faster and nine orders of magnitude more accurate in benchmark tests in 1, 2, and 3-dimensional systems.

Calculating coherent light-wave propagation in large heterogeneous media

“…condition (18) reduces to α i > max x |∆(x)|. This reproduces the convergence condition found by both Osnabrugge et al 17 and Krüger et al 18 .…”

“…The quantity in the rounded brackets can thus be seen to be a diagonal matrix in the two Fourier space indices. As mentioned above, in a pair of recent papers 17,18 , it was shown that this Born series (5) can be made into a convergent series, the only proviso being that the system exhibits solely dissipation (no gain), and that the permittivity is nowhere singular. To achieve this convergence the authors used two ingenious steps.…”

“…Furthermore, biological tissues contain chiral organic molecules such as glucose, a property that is linked to the electro-magnetic coupling and cannot be modeled by anisotropic permittivity. In this paper we investigate the application of the modified Born series 17,18 to solve Maxwell's equations in large heterogeneous electromagnetic media, characterized by arbitrary linear constitutive relations.…”

“…However, a severe limitation is that the basic form of the series only converges in the limit of weak scattering. A modified Born series for scalar waves, which converges for any value of the scattering strength, was recently proposed by Osnabrugge et al 17 , and generalized to vector waves by Krüger et al 18 . Yet, as it stands, this approach is limited to media with an isotropic permittivity and without magnetic properties.…”

Stress redistribution of dynamic loading incident with arbitrary waveform through a circular cavity

“…More recently, Wang et al used the Schwartz alternating method and complex variable theory proposed highly accurate analytical solutions for stress and displacement induced by twin tunneling in semi‐infinite ground. Osnabrugge et al proposed an iterative method for numerically solving nonhomogeneous Helmholtz equations, which can achieve convergence of media of arbitrary size and scattering intensity in scattering problems.…”