Phase spectroscopy of topological invariants in photonic crystals

We propose a method of measuring topological invariants of a photonic crystal through phase spectroscopy. We show how the Chern numbers can be deduced from the winding numbers of the reflection coefficient phase. An explicit proof of the existence of edge states in a system with a nonzero reflection phase winding number is given. The method is illustrated for one- and two-dimensional photonic crystals of nontrivial topology.

Figure 1

The color map of the squared amplitude (b) and the phase (c) of the reflection coefficient from the left edge of a semi-infinite 1D photonic crystal (a) as a function of light frequency $\omega$ and the parameter $\varkappa$. Solid curves show the real part of the left-edge state frequency. The subplots in panel (c) show the cross sections of the phase map at different frequencies. The calculation parameters are $b=1/3,\eta =0.5,n_{\mathcal{A}}/n_{\mathcal{B}}=2$, and $d_{\mathcal{A}}/d_{\mathcal{B}}=0.2$.

Figure 2

Panels (a) and (d) show two proposed schemes of measuring of topological invariants. Panels (b) and (c) show the calculated color map of the squared amplitude and the phase of the reflection coefficient in the multiport scheme for the rectangular $N_x\times N_y=15\times 15$ lattice, with $b=1/3,\kappa /J=1,\gamma /J=0.01$. Panels (e) and (f) show the squared amplitude and the phase of the reflection coefficient in the single port scheme for the same parameters except for $\kappa /J=0.01$ and $N_x\times N_y=21\times 21$.