Whispering Gallery modes in optical disc resonators: Mode formation and temperature sensing

Abstract

The "Whispering Gallery Mode" (WGM) phenomenon occurs when a wave travels along the inner surface of a circularly symmetric chamber by total internal reflection and, after reaching the starting point, the waves interferes. In this case, the total number of wavelengths that can fit into the chamber are amplified and the resonant frequencies characteristic of the chamber, or WGMs, are generated. This phenomenon was named after Lord Rayleigh's discovery, over a century ago, that in St Paul's Cathedral, whispering could be heard from one side of the 34-meter dome to the other. Today, this phenomenon is applied to make small but highly capable optical resonators, where light is confined to outer rims of circular discs. Interest in such disc resonators arise from their numerous applications, that include e.g. fundamental studies on nonlinear dynamics, optical metrology, and environmental sensing.

In this thesis, we investigate theoretically, through a literature review and analytically using MATLAB software, how WGMs are generated and what kind of longitudinal and transverse modes the WGM resonator can support. The longitudinal modes are analysed by calculating the wavelengths amplified in the resonator. Transverse modes are studied by plotting figures of the electric field distributions generated in the cross section of the resonator. The mode analyses are performed on a magnesium fluoride disc resonator with a diameter of 1 mm.

This thesis also examines how temperature change affects the radius and the refractive index of a WGM resonator. This was investigated using the temperature coefficients of thermal expansion and refractive index of the materials for resonators made of silica and magnesium fluoride. One interesting application of WGM resonators is the fabrication of sensitive sensors, as these resonators have particularly sharp resonances. This, and the fact that WGMs are very sensitive to changes in ambient conditions, allows WGM resonators to be used for high-accuracy measurements.

The results show that increasing the mode numbers decomposes the electric field distribution of light into smaller spherical parts in both the azimuthal and radial directions. For the longitudinal modes, it was found that the resonance wavelength decreases with increasing mode numbers and that the effect is stronger for modes with lower order numbers. The effect of temperature change on the radius and refractive index of the WGM resonator was found to depend entirely on the material. It was also found that the combined effect of radius and refractive index was stronger in a resonator made of MgF_2 than in a one made of SiO_2. However, it was concluded that an analysis of the other properties of the materials would be needed to determine which material should be preferred in a WGM-based temperature sensor.