Realization of complex conjugate media using non-PT-symmetric photonic crystals

While parity-time (PT)-symmetric systems can exhibit real spectra in the exact PT-symmetry regime, the PT-symmetry is actually not a necessary condition for the real spectra. Here we show that non-PT-symmetric photonic crystals carrying Dirac-like cone dispersions can always exhibit real spectra as long as the average non-Hermiticity strength within the unit cell for the eigenstates is zero. By building a non-Hermitian Hamiltonian model, we find that the real spectra of the non-PT-symmetric system can be explained using the concept of pseudo-Hermiticity. We demonstrate using effective medium theories that in the long-wavelength limit, such non-PT-symmetric photonic crystals behave like the so-called complex conjugate medium (CCM) whose refractive index is real but whose permittivity and permeability are complex numbers. The real refractive index for this effective CCM is guaranteed by the real spectrum of the photonic crystals and the complex permittivity and permeability comes from the non-PT-symmetric loss-gain distributions. We show some interesting phenomena associated with the CCM, such as the lasing effect.