Electromagnetic energy, momentum, and angular momentum in an inhomogeneous linear dielectric

Abstract

In a previous work, Optics Communications 284 (2011) 2460-2465, we considered a dielectric medium with an anti-reflection coating and a spatially uniform index of refraction illuminated at normal incidence by a quasimonochromatic field. Using the continuity equations for the electromagnetic energy density and the Gordon momentum density, we constructed a traceless, symmetric energy-momentum tensor for the closed system. In this work, we relax the condition of a uniform index of refraction and consider a dielectric medium with a spatially varying index of refraction that is independent of time, which essentially represents a mechanically rigid dielectric medium due to external constraints. Using continuity equations for energy density and for Gordon momentum density, we construct a symmetric energy-momentum matrix, whose four-divergence is equal to a generalized Helmholtz force density four-vector. Assuming that the energy-momentum matrix has tensor transformation properties under a symmetry group of space-time coordinate transformations, we derive the global conservation laws for the total energy, momentum, and angular momentum.

Continuum electrodynamics and the Abraham-Minkowski momentum controversy

“…The consequent Abraham-Minkowski controversy refers to the century-long inability of scientists to conclusively identify the role of electromagnetic momentum for the propagation of light in linear media [8][9][10][11][12]. To be sure, the total momentum [12][13][14],…”

“…Here we consider the case of an arbitrarily large homogeneous simple linear dielectric draped with a gradient index antireflection coating. The gradient of the index of refraction ∇n can be made sufficiently small that reflections can be neglected [14]. It is in this limit of an unimpeded continuous flow of electromagnetic radiation in which there are no sources or sinks of energy or momentum that the energy and momentum densities can be expressed through continuity equations and a total energy-momentum tensor for a thermodynamically closed system.…”

“…is conserved. Although the mathematics, ∂ β T iβ M = 0, tell us that the Minkowski momentum is conserved, it is well-known that the Minkowski momentum is a factor of n greater than the incident momentum [12][13][14]. Then the continuity equation, Eq.…”

Electromagnetic momentum and the energy–momentum tensor in a linear medium with magnetic and dielectric properties

“…Given the arbitrariness of the field momentum and the model dependence of the material momentum that go into a construction of the total momentum, the only convincing way to determine the total momentum is by the application of conservation principles. Then conservation of the total energy is all that is needed to prove that the total momentum [13,[17][18][19] …”

“…Then an inhomogeneity in the index of refraction, ∇n = 0, is associated with work done by the field on the material or work done on the field by the interaction with the material [18]. Obviously, the gradient of the index of refraction in our anti-reflection coating must be sufficiently small that the work done is perturbative.…”