Optical Spin-Orbit Coupling in Dirac Metamaterials

We explore hypothetical metamaterial cavities in which ``optical spin-orbit coupling'' of continuous vector electromagnetic fields can be made nearly isospectral to electron spin-orbit coupling in the hydrogen atom. 4 by 4 matrix formulations of Maxwell's equations suggest that certain simultaneous variations of $\epsilon$ and $\mu$ make Maxwell's equations very similar the Dirac equation; we call such systems ``Dirac metamaterials.'' We present solutions to Maxwell's eqns. in hydrogen-like cavities and discuss the semiclassical meaning of various perturbation terms including the $\mathbf{L}\cdot{}\mathbf{S}$ coupling term and a Darwin-like term.