Potential-based formalism for electrodynamics of media with weak spatial dispersion

Abstract

In this work, we develop a potential-based formalism for Maxwell's equations in isotropic media with weak spatial dispersion within the electric quadrupole-magnetic dipole approximation. We introduce an operator form of the constitutive relations along with a modified Lorenz gauge condition, which enables the derivation of decoupled generalized wave equations for electromagnetic potentials. For time-harmonic processes, we derive the representation of general solution for these equations as a combination of solutions to Helmholtz-type equations, whose parameters are determined by both standard and hyper-susceptibilities of the medium. We show that the proposed approach can be extended to more general constitutive relations and it provides a convenient framework for solving various applied problems. Specifically, using a derived closed-form solution for the problem of plane wave incidence on a planar interface, we demonstrate that a correct definition of the Poynting vector within the multipole theory must incorporate quadrupole effects -- an aspect overlooked in some previous works that has led to inconsistent results. We further establish the necessity of accounting for both propagated and evanescent longitudinal components in reflected and transmitted waves. The presence of these components, which follow directly from the general solution for electromagnetic potentials, is essential for satisfying all classical and additional boundary conditions in media with quadrupolar response (e.g., in metamaterials or quadrupolar liquid mixtures). The complete set of these boundary conditions is derived based on the least action principle, ensuring variational consistency with the field equations and generalizing previously known formulations of multipole theory.