Looking for optimal materials for whispering gallery modes applications at the 2 $\mu$m window

The diverse applications of whispering gallery modes in spherical microresonators are strongly related to the sphere size and material composition. Their design should therefore be optimized to ensure that parameters such as the quality factor and the free spectral range are maximized. Because of the imminent capacity crisis of the optical communication systems operating at the 1550 nm wavelength regime, it is time to explore optical communications at the 2 $\mu\mbox{m}$ wavelength window. In this work, we analytically investigate key resonator parameters - quality factor and free spectral range - as a function of wavelength, aiming to establish a methodology to help identify optimal materials for whispering gallery mode sensors, with special attention at the 2 $\mu\mbox{m}$ wavelength window. Specifically, we examine three materials: fused silica, AsSe chalcogenide glass and calcium fluoride, and we perform a comparison between them in order to identify the region in the parameter space of resonant wavelengths and sphere radius, $(\lambda_R,R$), where the WGM resonators are optimal at wavelengths $1.8 \mu\mbox{m}<\lambda<2.1 \mu\mbox{m}$.