A scalar potential formulation and translation theory for the time-harmonic Maxwell equations

TitleA scalar potential formulation and translation theory for the time-harmonic Maxwell equations
Publication TypeJournal Articles
Year of Publication2007
AuthorsGumerov NA, Duraiswami R
JournalJournal of Computational Physics
Pagination206 - 236
Date Published2007/07/01/
ISBN Number0021-9991
KeywordsDebye potentials, Electromagnetic scattering, Fast Multipole Method, Helmholtz equation, Maxwell equations, Mie scattering, T-matrix method, Translation operators

We develop a computational method based on the Debye scalar potential representation, which efficiently reduces the solution of Maxwell’s equations to the solution of two scalar Helmholtz equations. One of the key contributions of this paper is a theory for the translation of Maxwell solutions using such a representation, since the scalar potential form is not invariant with respect to translations. The translation theory is developed by introducing “conversion” operators, which enable the representation of the electric and magnetic vector fields via scalar potentials in an arbitrary reference frame. Advantages of this representation include the fact that only two Helmholtz equations need be solved, and moreover, the divergence free constraints are satisfied automatically by construction. Truncation error bounds are also presented. The availability of a translation theory and error bounds for this representation can find application in methods such as the Fast Multipole Method.For illustration of the use of the representation and translation theory we implemented an algorithm for the simulation of Mie scattering off a system of spherical objects of different sizes and dielectric properties using a variant of the T-matrix method. The resulting system was solved using an iterative method based on GMRES. The results of the computations agree well with previous computational and experimental results.