Transformation optics and topological phases of chiral photonic crystals

Abstract

The topology of chiral materials is attractive because a wider parameter space can be provided by the coupling of electric and magnetic fields in their constitutive relations. Parity inversion symmetry is already broken without applying external magnetic fields due to left-handed and right-handed circular polarization of optical activity; however, the time reversal symmetries can still be preserved. Optical activities has been gradually taken into consideration when developing the topological theories of bi-anisotropic materials. In this study, the techniques of transformation optics have been used to design one-dimensional chiral photonic crystals. Transformation optics is a theory based on the invariance of Maxwell's equations under coordinate transformation. Due to the flexibility of Jacobian matrix, the trajectory of a light beam can be controlled so that some "unrealistic" phenomena in conventional view can be easily achieved, for example, illusion optics. However, coordinate transformation is not valid to predict electromagnetic properties in bi-anisotropic medium via simple anisotropic medium since the constitutive relations of the media are different. Other transformation method is also investigated such that there exists a conformal mapping between two spaces. Instead of the trajectory, optical activity attributed to chirality can be controlled owing to the flexibility of the transformation matrix. The designed chiral materials will act as a unit cell of a photonic crystal such that a chiral photonic crystal can be constructed. Topological features, Zak phases and topological edge states, of the designed chiral photonic crystals are examined. The eigenmodes, TE and TM modes, are no longer degenerated because of anisotropy and existence of chirality. Zak phases calculated by these coupled eigenstates are briefly discussed. With choosing suitable electromagnetic tensors, two independent modes can be decoupled such that the topological theories developed from simple isotropic layered photonic crystals can be totally applied.