Geometric optics of Bloch waves in a chiral and dissipative medium

We present a geometric optics theory for the transport of quantum particles (or classical waves) in a chiral and dissipative periodic crystal subject to slowly varying perturbations in space and time. Taking account of some properties of particles and media neglected in previous theory, we find important additional terms in the equations of motion of particles. The (energy) current density field, which traces the geometric optics rays, is not only governed by the Bloch band energy dispersion but also involves there additional fields. These are the angular momentum of the particle, the dissipation dipole density, and various geometric gauge fields in the extended phase space spanned by space time and its reciprocal, momentum, and frequency. For simplicity, the theory is presented using light propagation in photonic crystals.

Figure 1

Plot of the trajectories of a photon in a 2D ($\mathit{xy}$) hexagonal photonic crystal obtained from the geometric optics approach. The length unit is $\left|G\right|{}^{-1}=a/2\pi$, where $a$ is the lattice constant. $\lambda =0.5$, ${\eta }_0=0.008$, $d{\eta }_1/\mathit{dx}=4\times {10}^{-6}$ with ${\eta }_1(t=0)=0$. The inital wave vector of the photon has $\mathbf{p}=(\Delta /3,0)$ from the BZ corner $\mathbf{K}$. (Solid line) Real trajectory of the photon. (Dashed line) Trajectory without $\partial {}_{\mathbf{R}}\times \rho \mathbf{m}$. (Dotted line) Trajectory without both $\partial {}_{\mathbf{R}}\times \rho \mathbf{m}$ and ${\Omega }_{\mathbf{pR}}\stackrel{·}{\mathbf{R}}$.