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

TL;DR

The paper develops a potential-based formalism for electrodynamics in isotropic media with weak spatial dispersion under the electric quadrupole–magnetic dipole approximation, introducing an operator-based constitutive framework and a modified Lorenz gauge that uncouples generalized wave equations for scalar and vector potentials. Time-harmonic solutions are expressed as combinations of Helmholtz-type fields, with dyadic Green's functions providing explicit field representations and allowing extension to general constitutive operators. A variational derivation yields a complete boundary-condition set, including six interface conditions and quadrupole-specific constraints, ensuring consistent solutions for both normal and oblique incidence. The authors demonstrate that quadrupole effects must be included in the Poynting vector and that longitudinal components (propagating and evanescent) play essential roles in energy balance and boundary-value problems, with concrete results for plane-wave reflection at planar interfaces. The framework generalizes to more complex media and can inform design and analysis of metamaterials and quadrupolar fluids where spatial dispersion is non-negligible.

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.

Figures (4)

• Figure 1: (a): Dispersion relations in the media with weak spatial dispersion for transverse ($k_2$) and longitudinal ($k_1$) components of electromagnetic wave for different values of the length scale parameters. Horizontal dotted lines correspond to the asymptotic cutoff frequencies $\omega_c = v/l_i$. (b): Dependence of effective refractive indices $\hat{n}_i$ on wavelength $\lambda_i=2\pi/k_i$.
• Figure 2: Plane wave incidence on a planar interface. Propagation directions defined by wave vectors $\textbf{k}_{ij}$ correspond to transverse waves (black lines) and to the longitudinal components of reflected and transmitted waves (blue lines); $\textbf{e}_{ij}$ -- polarization unit vectors, $\hat{\phi}_j$ -- potentials of evanescent waves. Angle of incidence is $\theta_{20}$.
• Figure 3: (a): Dependence of reflectance coefficient $R$ (a) and effective relative refractive index $\hat{m}$ (b) on frequency in the solution of normal incidence problem for the media with weak spatial dispersion.
• Figure 4: Solution of the oblique incidence problem. (a): Dependence of reflectance on indecent angle, (b): Dependence of Brewster angle on normalized frequency, (c): Dependence of reflection and transmission angles of transverse ($\theta_{21}$, $\theta_{22}$) and longitudinal ($\theta_{11}$, $\theta_{12}$) waves for normalized frequency value $\omega/\omega_c = 0.55$ (solid lines) and $\omega/\omega_c = 0.85$ (dashed lines), (d): Dependence of normalized surface energy of evanescent waves on incedence angle in medium 1 (solid lines) and in medium 2 (dashed lines). Classical solutions in absence of weak spatial dispersion are presented in all plots by dotted lines.