Analytical study of mode degeneracy in non-Hermitian photonic crystals with TM-like polarization

We present an analytical study of the mode degeneracy in non-Hermitian photonic crystals (PC) with $C_{4v}$ symmetry, from the perspective of the coupled-wave-theory (CWT). The wave couplings and leakages within the non-Hermitian PCs are depicted, and the condition of accidental triple degeneracy is derived which leads to a Dirac-cone like dispersion. We prove that, similar to the real Dirac-cone, the Dirac-cone like band in non-Hermitian PC possesses good linearity and isotropy in the vicinity of the $\Gamma$ point. Moreover, some topological transit of the band may occur when the parameters adiabatically evolve. However, the Berry phase remains zero at the $\Gamma$ point, indicated the Dirac-like-cone dosen't obey Dirac equation and it is only Dirac-cone.