**Phase Space Approaches for Microresonators**

Optical systems such as microresonators and optical fibres often operate at scales where the wavelength is small compared to other geometrical scales, but not negligible. We have applied a variety of methods, ranging from exact approaches using boundary integral and transfer formalisms, to asymptotic and WKB approximations, to straight ray-tracing and phase space propagation methods such as DEA and the Frobenius-Perron operator, to model such systems. The methods are chosen so that the role of underlying phase space structures of the ray limit are made apparent, even when the detailed calculations are done in a full wave setting. Transfer operators offer a means of expressing exact solutions provided by the boundary integral method so that a direct analogy can be made with transport in the ray-dynamical phase-space. Such methods have proved to be an efficient means of developing near-analytical solutions for PT-symmetric arrays of disk resonators, for example. They have also provided a foundation on which to treat tunnelling, or evanescent escape, from deformed microdisk resonators: WKB approximations for such effects are achieved in terms of complex ray evolution, which can be directly motivated by saddle point approximation of transfer operator evolution. By providing a direct route from full wave calculations to ray tracing, even in the case of incoherent light, they allow us to incorporate effects such as polarisation in ray-tracing simulations, which is currently being applied to models of optical fibres with deformed core and cladding.

**Related Publications:**

- CREAGH, S.C., HAMDIN, H.B. and TANNER, G., 2013. In-out decomposition of boundary integral equations Journal of Physics A: Mathematical and Theoretical. 46(43), 435203
- CREAGH, S.C. and WHITE, M.M., 2012. Differences between emission patterns and internal modes of optical resonators Physical Review E. 85(1), 015201
- WHITE, M.M. and CREAGH, S.C., 2012. Quality factors of deformed dielectric cavities Journal of Physics A: Mathematical and Theoretical. 45(27), 275302
- CREAGH, S.C. and WHITE, M.M., 2010. Evanescent escape from the dielectric ellipse Journal of Physics A: Mathematical and General. 43(46), 465102
- CREAGH, S.C., 2007. Directional emission from weakly eccentric resonators Physical Review Letters. 98(15), 153901

**Parity-Time (PT) Symmetric Photonics**

In photonics, PT-symmetry is achieved with a judicious profile of complex refractive index that

combines of both gain and loss. PT-symmetric photonic structures can have purely real spectra, i.e. zero net-power amplification or dissipation, despite having both gain and loss in the system. However, there exists an exceptional point defined for a certain value of a system parameter for which the PT-system undergoes a spontaneous PT-symmetry breaking. Above this exceptional point the spectrum becomes complex and power grows exponentially. We have studied the spectral properties and the dynamics of various PT-symmetric structures, and also demonstrated interesting properties of PT-photonic devices, such as loss-induced invisibility and laser generation by reversing the effect of loss at threshold. These works have led to a better understanding of fundamental physical mechanisms of PT symmetry, which leads to a wide-range of applications.

**Related Publications:**

- PHANG, SENDY, VUKOVIC, ANA, GRADONI, GABRIELE, SEWELL, PHILLIP, BENSON, TREVOR M. and CREAGH, STEPHEN C., 2017. Theory and Numerical Modelling of Parity-Time Symmetric Structures in Photonics: Boundary Integral Equation for Coupled Microresonator Structures. In: Recent Trends in Computational Photonics Springer Nature.
- PHANG, SENDY, BENSON, TREVOR M., SUSANTO, HADI, CREAGH, STEPHEN C., GRADONI, GABRIELE, SEWELL, PHILLIP D. and VUKOVIC, ANA, 2017. Theory and Numerical Modelling of Parity-Time Symmetric Structures in Photonics: Introduction and Grating Structures in One Dimension. In: Recent Trends in Computational Photonics Springer Nature.
- PHANG, SENDY, VUKOVIC, ANA, CREAGH, STEPHEN C., SEWELL, PHILLIP D., GRADONI, GABRIELE and BENSON, TREVOR M., 2016. Localized Single Frequency Lasing States in a Finite Parity-Time Symmetric Resonator Chain Scientific Reports. 6(20499), 20499
- SENDY PHANG, ANA VUKOVIC, STEPHEN C. CREAGH, TREVOR M. BENSON, PHILLIP D. SEWELL and GABRIELE GRADONI, 2015. Parity-time symmetric coupled microresonators with a dispersive gain/loss Opt. Express. 23(9), 11493-11507
- PHANG, SENDY, VUKOVIC, ANA, BENSON, TREVOR M, SUSANTO, HADI and SEWELL, PHILLIP, 2015. A versatile all-optical parity-time signal processing device using a Bragg grating induced using positive and negative Kerr-nonlinearity Optical and Quantum Electronics. 47(1), 37-47
- PHANG, SENDY, VUKOVIC, ANA, SUSANTO, HADI, BENSON, TREVOR M and SEWELL, PHILLIP, 2014. Impact of Dispersive and Saturable Gain/Loss on Bistability of Nonlinear Parity-Time Bragg Gratings Optics Letters. 39(9), 2603-2606
- PHANG, SENDY, VUKOVIC, ANA, SUSANTO, HADI, BENSON, TREVOR M. and SEWELL, PHILLIP, 2013. Ultrafast optical switching using parity-time symmetric Bragg gratings JOURNAL OF THE OPTICAL SOCIETY OF AMERICA B-OPTICAL PHYSICS. 30(11), 2984-2991