Bulk-edge correspondence in open photonic systems

Abstract

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.

Keywords:

• Topological photonics
• edge states
• density of states
• open quantum systems

Acknowledgments

This work was partially supported by KAKENHI Grant No. 17K05507.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Such a boundary condition is necessary to count the DOS of the radiation modes without ambiguity. Although the PEC boundary condition is introduced just to count the DOS, the boundary condition can be approximately implemented by putting a metal wall with a high-enough absolute permittivity.

2 The PMC wall is approximately realized with a ferrite medium at microwave frequencies near a magnetic resonance.

Additional information

Funding

This work was supported by Japan Society for the Promotion of Science KAKENHI [grant number 17K05507].