Theory and numerical modelling of parity-time symmetric structures for photonics

This thesis presents the study of a relatively new class of photonic structures in-voking Parity-Time (PT)-symmetry. PT-symmetric structures in photonics, as a realisation of PT-symmetric Quantum Mechanics problems, are constructed by a judicious design of refractive index modulation which requires the real part of the refractive index to be an even function and the imaginary part of the refractive index to be an odd function in space. PT-symmetric structures in the form of Bragg gratings, coupled resonators and chain resonators are the main configurations studied in this thesis. These PT-symmetric structures feature a spontaneous symmetry breaking at which interesting wave behaviour such as an asymmetric response depending on the direction of the incident wave, unidirectional invisibility, simultaneous coherent-perfect absorber lasing and localised termination modes are observed; these behaviours are presented in this thesis. Theoretical and numerical studies of these PT-symmetric structures are undertaken which assume realistic material parameters,including material dispersion and material non-linearity. Moreover,in this thesis, potential applications of these PT-symmetric structures are explored. The first part of the thesis considers PT-symmetric Bragg grating structures which are formed by introducing a PT-symmetric refractive index modulation into a Bragg grating structure. If gain/loss dispersion is considered, it is shown that dispersion limits the PT-symmetric operation to just a single frequency. As such spontaneous symmetry breaking can only be achieved by varying the gain/loss parameter. Nevertheless, it is shown that by switching the gain/loss in the system, a switching operation can be achieved by using the PT-Bragg grating at a single frequency. Subsequently,anon-linear PT-Bragg grating is investigated by using a time-domain numerical method, namely the Transmission-Line modelling (TLM) method. For the present work a TLM code is developed from scratch in order to ensure full-flexibility when modelling a dispersive and non-linear material. Using the TLM solver, it is demonstrated that gain/loss saturation is an important material property which should be considered as it may impact the practical applications of a PT-symmetry-based device. In the context of a non-linear PT-Bragg grating (NPTBG), the gain/loss saturation affects the interplay between the PT-symmetric opearation and the Kerr non-linear effect. It is further shown that gain/loss saturation plays a crucial role in securing a stable operation of non-linear PT-based devices. For practical applications, it is demonstrated that a non-linear PT-symmetric Bragg grating offers an additional degree of freedom in their operation,by modulating the gain/loss and the intensity of the input signal,compared to a passive structure which can only be manipulated by the input signal intensity. Two applications based on the interplay of PT-symmetric behaviour and Kerr non-linearity are demonstrated,namely a memory device and a logic-gate device. The second part of the thesis studies PT-symmetric resonator structures as a coupled system and as a periodic chain system. For these studies, a semi-analytical method based on the Boundary Integral Equation (BIE) method is developed and used together with a two-dimensional TLM method. The impact of realistic material parameter on the spectral properties of the structure is again investigated. It is shown that the PT-symmetric behaviour can be observed at a single frequency. Moreover, it is shown that PT-symmetry-like behaviour is observed but with complex eigenfrequencies due to the radiation losses; this is a deviation of the strict definition of a PT-symmetric structure with balanced gain and loss. Lowering lasing threshold by increasing loss in the system is demonstrated; this occurs due to induced early symmetry breaking. The final part of the thesis studies the spectral properties of an infinite and finite chain of PT-symmetric resonators. It is shown that the type of modulation along the PT-chain affects the position of the breaking point of the PT-structure. For a finite PT-chain structure, and for a particular type of refractive index modulation, early PT-symmetry breaking is observed and shown to cause the presence of termination states which are localised at the edge of the finite-chain resulting in localised lasing and dissipative modes at each end of the chain.