Towards Optical Simulation of Topological Phenomena with Ring Resonators

Abstract

In recent years, there has been a growing interest in synthetic dimensions. Unlike physical dimensions where one is restricted with three possible dimensions, one can tailor a lattice system with several higher order dimensions even with a lower-order physical system. In this thesis, we will explore, both theoretically and experimentally, how equidistant frequency modes in a resonator—specifically a ring resonator—can be used as a synthetic dimension. In this respect, we will first explore basics of resonators by reviewing a Fabry Perot resonator and developing a coupled-mode theory to understand its transmission and reflection spectrum. We will then expand and develop a coupled-mode theory for the case of a ring resonator, and explore experimental results for a basic fiber-beam splitter based ring resonator. Next, we will modify the ring resonator with a few off the shelf fiber optic components like a Dense Division Wavelength Multiplexing filter, a fiber electro-optic modulator, and an optical amplifier, and see how they transform the transmission characteristics. Lastly, we will modulate the ring resonator at the same frequency as the free-spectral range of the resonator to couple the modes to observe a band structure in the reciprocal space, much like atoms in a periodic solid-state crystal coupled via Coulomb interactions lead to energy bands in the conjugate momentum space. For this purpose, we will develop a coupled-mode theory for a ring resonator modulated by a phase modulator, look at some theoretical results for symmetric and asymmetric band structures, and observe some experimental results for the symmetric case. Thus, we highlight the potential of fiber ring resonators as versatile platforms for optical simulation of topological systems. Further, the ability to replicate higher-dimensional phenomena using frequency modes opens new pathways for investigating exotic physical behaviors, such as non-Hermitian systems and synthetic magnetic fields, in compact and experimentally accessible setups.