THESIS
2019

xvi, 144 pages : color illustrations ; 30 cm

**Abstract**
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 real spectrum, which is protected by...[

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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 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 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
characteristic feature of non-Hermitian systems and can lead to many exotic phenomena, such as unidirectional invisibility, 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 square lattice. We formulate a non-Hermitian Hamiltonian model for the PC when non-H 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 Dirac-like
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 semi-infinite 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.

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