Realization of complex conjugate medium by using a ring of exceptional points in non-PT-symmetric photonic crystals

We propose a method to locate exceptional points in general non-Hermitian 2-dimensional (2D) photonic crystals (PCs). We focus on systems carrying a Dirac-like cone at k=0, where a ring of exceptional points can emerge when PT-symmetric complex potentials are introduced. We then build a Hamiltonian to find the ring of exceptional points in non-PT-symmetric PCs analytically. By adjusting the loss/gain ratio of our PC, the ring of exceptional points could be made to sit on the real axis even though the PC is not PT-symmetric. We also explored the properties of the PC from an effective medium theory point of view. For frequencies near the ring of exceptional points, the PC behaves like a homogenized complex conjugate medium, whose refractive index is real while the permittivity and permeability are complex individually. The optical properties of such medium will be discussed.