Bulk-edge correspondence in open photonic systems

We study the bulk-edge correspondence in topological photonic crystals with open boundary. If boundary is open, edge states become leaky inside the light cone, but still exhibit a chiral and gapless property taking into account the blurring of their band structure due to the leakage. The so-called bulk-edge correspondence is thus verified. On the other hand, in closed boundaries, edge states exhibit the well-defined band structure without the blurring and show clearly the bulk-edge correspondence. To demonstrate these results, we employ the transfer-matrix formalism and derive reflection matrices of semi-infinite systems. Optical density of states for the system with open boundaries is available via the Krein-Friedel-Lloyd formula for the reflection matrices. The leaky photonic band structure of the edge states is obtained by following the peaks and widths of the density of states as a function of momentum parallel to the boundary. Our derivation of the leaky band structure does not rely on possible effective non-hermitian hamiltonians but relies on a first-principles calculation of the Maxwell equation.