Quantised helicity in optical media

TL;DR

The paper reframes optical helicity as the generator of Heaviside–Larmor duality in media by introducing an AC-type duality transform that acts on both electromagnetic potentials and the material degrees of freedom. By incorporating matter explicitly through a Hopfield-type Lagrangian, the authors derive a medium helicity density $h=\frac{1}{2}(\mathbf{A}\cdot\mathbf{B}-\mathbf{C}\cdot\mathbf{D})+\boldsymbol{\Pi}^P\cdot\mathbf{M}-\boldsymbol{\Pi}^M\cdot\mathbf{P}$ with flux $\mathbf{v}=\frac{1}{2}(\mathbf{E}\times\mathbf{A}+\mathbf{H}\times\mathbf{C})$, ensuring the correct symmetry and Noether conservation in dual-symmetric media. Upon diagonalising the coupled light–matter system, helicity becomes the difference between left- and right-handed polaritons, with three polariton branches and a dispersion relation $k^2=\omega_i^2 n^2(\omega_i)$; in dual-symmetric media each polariton carries helicity equal to the total energy density over its frequency, while in general media helicity is not conserved and can oscillate between helicity eigenstates, analogous to neutrino oscillations. The framework yields a consistent, general description applicable to inhomogeneous, lossy, chiral, and nonreciprocal media and clarifies the relationship between helicity, polaritons, and optical chirality, while outlining extensions to more complex media and reservoirs.

Abstract

We present a new approach to the definition of optical helicity in a medium. Our approach resolves the problem that duality transformations which simultaneously combine $\mathbf{E}$ with $\mathbf{H}$ and $\mathbf{D}$ with $\mathbf{B}$ are incompatible with linear constitutive relations. We find that the helicity density in a medium, as the conserved quantity associated with duality transforms, must contain an explicit contribution associated with the polarisation and magnetisation of the matter, and that it can be expressed naturally in terms of the elementary polarised excitations of the system. In media for which the helicity is conserved, each circular excitation carries a well-defined helicity. However, in a medium for which the helicity is not conserved, we find that the time-varying helicity can be viewed in terms of oscillations between different helicity eigenstates, analogous to neutrino oscillations. Here we explicitly study the helicity in homogeneous and lossless media but we believe that, differently to other choices, this helicity is readily generalisable to media that may be inhomogeneous, lossy, chiral or nonreciprocal.

Figures (3)

• Figure 1: Illustration comparing two ways of defining the duality transform in a medium. (a) The duality transform defined in equation \ref{['eq:dualityTransformDB']} implies a simple rotation between the $\mathbf{D}$ and $\mathbf{B}$ fields; (b) The duality transforms defined in equations \ref{['eq:dualityTransformEpsilonMu']} and \ref{['eq:dualityTransformRescaling']} are equivalent to a rotation and scaling, and alter both the relative lengths of $\mathbf{D}'$ and $\mathbf{B'}$ and the angle between them. It is clear that the transformations \ref{['eq:dualityTransformEpsilonMu']} and \ref{['eq:dualityTransformRescaling']} are both incompatible with those in equation \ref{['eq:dualityTransformDB']}.
• Figure 2: The polariton branches, i.e. solutions to the dispersion relation in Eq. \ref{['eq:dispersion']}, with $\alpha = \omega_0/2$. The sign of $\omega_i$ has been chosen to ensure that the wave-vector and group velocity direction coincide.
• Figure 3: Some illustrative examples of the expectation value of $\hat{h}$ for an excitation in a single polariton mode, with a range of non-dual-symmetric model parameters. The corresponding dispersion curves for the polariton modes are shown below each graph. Figure 3a shows a typical case, with the helicity approaching unity when the branches are photon-like. Note that $\langle\hat{h}\rangle$ is not bounded between $\pm 1$, and the middle branch's contribution changes sign as $k$ varies. Figure 3b shows a case with $\omega_E=\omega_M$, but $\alpha\neq\beta$, where the asymptotic behaviour of the matter-like branches does not approach $0$ or $1$ for large $k$. Figure 3c shows a nonmagnetic material, which can be described in this formalism by letting $\omega_M \rightarrow \infty$. One of the branches then becomes a flat band at a very high frequency (not shown), and the system effectively reduces to two branches.