Extended symmetries in geometrical optics

We examine additional symmetries of specific refraction index profiles that are used in the well-known phenomena of perfect imaging and cloaking. In the considered cases, the translation generator and the angular momentum are conserved. We express the ray trajectory parameters through the integrals of motion and observe the existence of a photon state with maximal angular momentum, which can be used as an optical resonator. Application in plasmons and the role of polarization are discussed, and the spin Hall effect in an extended symmetry profile is predicted.

Figure 1

Refraction index profiles.

Figure 2

Ray trajectories for different values of $\mathbf{T},\mathbf{L},\kappa$. Dashed and red circles with radius $2r_0$ are given by the refraction profile (1). The red circle is the photon trajectory with maximal angular momentum of the photon in $\kappa >0$ case. In $\kappa >0$ case, symmetric trajectories with $\pm a_0$ intersect while in $\kappa <0$ case they do not. The straight lines are trajectories without angular momentum $L=0$. The small blue circles are trajectories in $L\to \infty$ limit provided that $\kappa <0$.

Figure 3

Basic ray trajectories for photon with different polarizations. $\varphi$ is the trajectory plane rotation angle.