Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A solvable microscopic model for the propagation of light in a dielectric medium

Richard Dengler

TL;DR

Using a regularized discrete dipole approximation on a cubic lattice, the authors obtain an exact long-wavelength solution for light propagation in a dielectric slab. They derive the electromagnetic momentum density $\bar{\Pi}^{\mathrm{em}}=\tfrac{\epsilon_0}{2}|E^{\mathrm{avg}}_2|^2 n\hat{k}$ and show the total momentum per energy is $\bar{\Pi}^{\mathrm{tot}}/\bar{u}^{\mathrm{tot}}=\tfrac{1}{2}\left(n+\frac{1}{n}+\frac{M(n^2-1)^2}{n}\right)$, where the Madelung constant $M\approx 0.1716$ encodes microscopic structure. The mechanical momentum arises from both the Lorentz and Coulomb forces, with $\bar{\Pi_3^{\mathrm{lorentz}}}=\tfrac{\epsilon_0}{4}(n^3-n)|E^{\mathrm{avg}}|^2$ and $\bar{\Pi_3^{\mathrm{coulomb}}}=\tfrac{\epsilon_0}{4}|E^{\mathrm{avg}}|^2 M (n^2-1)^2 n$, consistent with Peierls' continuum results in the appropriate limit. The work clarifies how the conventional momentum value emerges and under which conditions microscopic details matter, guiding interpretation of experiments on light momentum in dielectrics and suggesting precise tests for intrinsic momentum transfer in finite-width signals and oblique incidence.

Abstract

Maxwell's equations resemble Schrödinger's equation in that an exact solution for a well-defined model delivers all physically relevant details. Solvable microscopic electrodynamic models, however, are rare. An exception is the discrete dipole approximation (DDA), which models a medium as a lattice of point dipoles. We use a regularized DDA variant to examine mechanical and electromagnetic momentum of light signals in such a medium in detail. The results agree in essential parts with that of the theory of R. Peierls from 1976.

A solvable microscopic model for the propagation of light in a dielectric medium

TL;DR

Using a regularized discrete dipole approximation on a cubic lattice, the authors obtain an exact long-wavelength solution for light propagation in a dielectric slab. They derive the electromagnetic momentum density and show the total momentum per energy is , where the Madelung constant encodes microscopic structure. The mechanical momentum arises from both the Lorentz and Coulomb forces, with and , consistent with Peierls' continuum results in the appropriate limit. The work clarifies how the conventional momentum value emerges and under which conditions microscopic details matter, guiding interpretation of experiments on light momentum in dielectrics and suggesting precise tests for intrinsic momentum transfer in finite-width signals and oblique incidence.

Abstract

Maxwell's equations resemble Schrödinger's equation in that an exact solution for a well-defined model delivers all physically relevant details. Solvable microscopic electrodynamic models, however, are rare. An exception is the discrete dipole approximation (DDA), which models a medium as a lattice of point dipoles. We use a regularized DDA variant to examine mechanical and electromagnetic momentum of light signals in such a medium in detail. The results agree in essential parts with that of the theory of R. Peierls from 1976.

Paper Structure

This paper contains 26 sections, 47 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The model
  3. Basic idea of the discrete dipole approximation
  4. Classical atoms
  5. Electrostatics
  6. Charge and current density
  7. Average electric field
  8. Formal solution
  9. Dispersion relation
  10. Electric and magnetic field
  11. Densities of energy and momentum
  12. Energy density
  13. Electromagnetic momentum density
  14. Momentum density from stress tensor
  15. Mechanical momentum from Lorentz force
  16. ...and 11 more sections

Figures (2)

  • Figure 1: A light signal in a solid. The signal (the gray block) moves to the right, slightly accelerates atoms at its right front to the right, and slightly decelerates atoms at its left front, thus leaving behind a trace of atoms shifted to the right, but again at rest. Atoms within the signal are moving to the right and contribute to the signal momentum.
  • Figure 2: A long wave train of finite width reflected at a mirror. The standing wave in the hatched triangle transfers momentum to the mirror and to the boundaries with the propagating waves. The mismatch of the stress tensors leads to a deposition of mechanical momentum at the boundaries.