In this thesis, we report on the theory of non-Hermitian photonic crystals. Physical observables are required to be real numbers since they are values measurable by experiments. Traditional quantum mechanics theory requires the operators of the physical observables to be Hermitian, which guarantees that the system is energy conservative and the time evolution is unitary. However, non-conservative elements are ubiquitous, because dissipation and energy leakage to the surrounding environment are inevitable. Thus to describe realistic physical systems, a formalism of non-Hermitian quantum mechanics is required.

Among various non-Hermitian systems, the pseudo-Hermitian system has attracted much attention in recent years because it can support a purely real spectrum, which is protected by antilinear symmetries. Parity-time (PT) symmetry is a subset of these antilinear symmetries. As we change the parameters of a PT-symmetric Hamiltonian, a phase transition occurs in the eigenvalues from purely real (exact PT symmetry) to complex numbers (broken PT symmetry). The phase transition point is called as an exceptional point (EP), a non-Hermitian degenerate point at which the eigenvalues and eigenstates simultaneously coalesce. EPs are branch point singularities in the complex energy plane. At the EPs, the non-Hermitian Hamiltonian matrix is defective and hence, the spectrum is incomplete. The emergence of EPs is a peculiar feature of non-Hermitian systems and can lead to many exotic phenomena, such as coherent perfect absorption and lasing.

Photonic crystal (PC) is an ideal platform for studying non-Hermitian systems because adding gain and loss is feasible in photonics. Loss is ubiquitous in optics, manifested as material absorption and radiation leakage to the surrounding environment. Gain can be implemented through the optical or electrical pumping of external sources. In this thesis, we studied various aspects of the non-Hermitian PCs. First, we study a two-dimensional PC comprising of dielectric cylinders arranged in a lattice. We formulate a non-Hermitian Hamiltonian model for the PC when non-Hermiticity is introduced. Using this model, we develop a method to locate the EPs. Furthermore, by properly tuning the averaged non-Hermiticiy within the unit cell to zero, we found that the Hermitian bands with Diraclike cone dispersion can be modified into pseudo-Hermitian bands, which can be used to realize complex conjugate media (CCM) from the viewpoint of effective medium theory. The refractive index of CCM is a real number, while the permittivity and permeability are complex numbers. Then we study a PC composed of two semi-infinite PCs with loss and gain, respectively. The whole PC is PT-symmetric about the interface of the two semiinfinite PCs. We found that there are localized modes at the PT-symmetric interface, and that EPs appear at the turning points of the bands formed by the interface modes. When we change the parameters of the PC, the interface bands can form a closed loop or zig-zag trajectory, which is shown to be associated with the coalescence of EPs. Such dispersions with turning points cannot be found in Hermitian systems.

We believe that theoretical work about non-Hermitian PC will have an impact because it can pave the way for experimentally realizing many unique properties of non-Hermitian systems, in particular those exotic phenomena associated with EPs. At the same time, extending the PC into the non-Hermitian regime can offer us more degrees of freedom to control the electromagnetic wave propagation.