Optics of Conducting Materials: An Electromagnetic Potential Perspective

Abstract

Macroscopic electrodynamics with scalar and vector potentials offers a useful paradigm to study the optical properties of conducting materials, forming a non-Cartesian geometrical structure. We elaborate this viewpoint with a new electromagnetic gauge (an extension of the Lorentz gauge) and obtain the equations of motion for the potentials. We next discuss the subtle idea of the Laplace operator on the vector field in an orthogonal curvilinear geometry. Remarkably, the Laplacian on the vector field (as opposed to the scalar one) is not separable in any curvilinear coordinate frame (except the rectangular Cartesian coordinates). This nonseparability of Laplacian, as we show with spherical polar coordinates as an example, does not allow a closed-form, fully analytical, and exact mathematical solution of the field equations. We then introduce the idea of effective field equations for a simplified description of the electromagnetic properties of the material and obtain a closed-form analytical solution for the spherical polar geometry. To account for the electromagnetic responses of the material (isotropic or forming a cubic system), we use linear constitutive relations with wavevector and frequency-dependent response functions (permittivity, permeability, and the conductivity). We use the present formalism to study the scattering of an electromagnetic wave by a spherical grain (finite radius) of a model-conducting material and obtain analytically closed-form expressions for the scattering and absorption cross sections. Present theoretical predictions are consistent with recent optical experiments on metallic nanoparticles and earlier theoretical works on this subject.