Topological hierarchy in non-Hermitian three-dimensional photonic crystals

The ${C}_{n}$-symmetric topological crystalline insulator (TCI) emerges as a well-established platform known for exhibiting confined edge and corner states protected by spatial rotational symmetry in Hermitian systems. However, the exploration of non-Hermitian photonic TCIs remains relatively underdeveloped. In this study, we introduce a topological hierarchy (TH) within an alternating gain-loss photonic crystal, encompassing non-Hermitian two-dimensional surface, one-dimensional (1D) hinge, and zero-dimensional corner modes. Our methodology demonstrates a complete TH by manipulating vertical and in-plane couplings. Our analysis shows that both the real and imaginary parts of the energy bands maintain symmetry along the zero-energy line, illustrating a TH similar to that observed in Hermitian systems. Study reveals that in the non-Hermitian case, 1D hinge modes are split into two distinct directions with different real and imaginary component wave intensities. Further investigation shows that our design supports a TH and non-Hermitian higher-order topological modes in higher-orbital band gaps. This work expands the study of three-dimensional non-Hermitian photonic systems and offers a promising route to manipulate wave transmission in disentangled higher-orbital bands.