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A mathematical description of the spin Hall effect of light in inhomogeneous media

Sam C. Collingbourne, Marius A. Oancea, Jan Sbierski

Abstract

We study Gaussian wave packet solutions for Maxwell's equations in an isotropic, inhomogeneous medium and derive a system of ordinary differential equations that captures the leading-order correction to geodesic motion. The dynamical quantities in this system are the energy centroid, the linear and angular momentum, and the quadrupole moment. Furthermore, the system is closed to first order in the inverse frequency. As an immediate consequence, the energy centroids of Gaussian wave packets with opposite circular polarisations generally propagate in different directions, thereby providing a mathematical proof of the spin Hall effect of light in an inhomogeneous medium.

A mathematical description of the spin Hall effect of light in inhomogeneous media

Abstract

We study Gaussian wave packet solutions for Maxwell's equations in an isotropic, inhomogeneous medium and derive a system of ordinary differential equations that captures the leading-order correction to geodesic motion. The dynamical quantities in this system are the energy centroid, the linear and angular momentum, and the quadrupole moment. Furthermore, the system is closed to first order in the inverse frequency. As an immediate consequence, the energy centroids of Gaussian wave packets with opposite circular polarisations generally propagate in different directions, thereby providing a mathematical proof of the spin Hall effect of light in an inhomogeneous medium.

Paper Structure

This paper contains 26 sections, 30 theorems, 307 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Discussion of main results
  3. Overview of the proof
  4. Outline of the paper
  5. Preliminaries
  6. Notation and conventions
  7. Maxwell's equations in an inhomogeneous medium
  8. Energy, momentum, and multipole moments
  9. Optical geometry
  10. Main results
  11. The approximate solutions
  12. Construction of the Gaussian beam approximation
  13. Construction of the phase function
  14. Construction theorem
  15. Conservation laws
  16. ...and 11 more sections

Key Result

Theorem 3.10

Let $\mathcal{K}$-supported Gaussian beam initial data of order $2$ as in DefGBID be given and consider the corresponding solution $({E}, {H})$ to Maxwell's equations EqMax. Construct the null geodesic $\gamma$ and time-dependent purely imaginary and invertible matrix $A_{ij}(t)$ as defined above. L Here, $\mathfrak{n}$ and its derivatives are all evaluated at $\mathbb{X}(t)$. The constant implici

Figures (1)

  • Figure 1: A sketch of the spin Hall effect of light in an inhomogeneous medium. The geometric optics geodesic is represented by the green ray, which is independent of the frequency and of the polarisation. When higher-order spin Hall corrections to the geometric optics approximation are included, we obtain frequency- and polarisation-dependent rays. This leads to the spin Hall effect of light, where the propagation of the energy centroids of wave packets with opposite circular polarisation is then represented by the blue and red rays. On the right part of the figure, the spin Hall ray separation is also represented in terms of the shifted energy densities of the two wave packets with opposite circular polarisations.

Theorems & Definitions (67)

  • Definition 3.1
  • Theorem 3.10
  • Proposition 3.12
  • proof
  • Proposition 3.15
  • proof
  • Remark 3.18: On the form of the ODE system \ref{['EqMainThm']}
  • Remark 3.20: Null geodesic motion at leading order
  • Definition 3.25: A class of circularly polarised initial data
  • Proposition 4.5
  • ...and 57 more