Self-consistent Physics-based δ_η -Regularized Green's Function for 3D Poisson's Equation in Anisotropic Dielectric Media

摘要

A novel renormalization scheme has been proposed for the regularization of infinite-domain Green's function for 3D Poisson's equation in anisotropic dielectric media. The method, generalizing the ideas presented in an accompanying paper for 2D Poisson's equation, involves the following constructive steps. (i) Starting from the given governing equations and imposing the interface conditions on a fictitiously-introduced plane z = z', expressions for the (singular) Green's function G(x - x', y - y', z - z') and the field distributions have been obtained in response to a Dirac's delta function δ(x - x', y - y') δ(z - z'). (ii) The component of the dielectric displacement vector in the direction normal to the fictitious plane is then used to construct a self-consistent problem-tailored representation for Dirac's delta function, denoted by δ_η (x - x', y - y'). Thereby, δ_η(x - x', y - y') approaches δ(x - x',y-y') continuously for η tending to zero. (iii) Using the constructed representation δ_η (x - x',y - y'), rather than the standard symbolic Dirac's delta function δ(x -x',y-y'), regularized Green's function G_η (x - x', y - y', z - z') has been constructed in closed form self-consistently. Besides the fact that the resulting δ_η-regularized Green's function is relevant for its own merit, it's closed forms in spectral - as well as in spatial domain allow illuminating a plethora of intricate details involved in the regularization process. It is claimed that the proposed method applies to more complex problems including half-spaces. The results are computationally also relevant because they play a significant role in determining dominant asymptotic terms of electrodynamic dyadic Green's functions for low-frequency regime in the near field.