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

TL;DR

The paper addresses designing whispering-gallery mode microspheres for the $2\,\mu\mathrm{m}$ window by analytically evaluating the quality factor $Q$ and free spectral range $FSR$ across three materials: SiO$_2$, CaF$_2$, and AsSe chalcogenide. It integrates Sellmeier dispersion, material absorption, and TE-mode eigenvalue equations to compute resonant wavelengths $\lambda_{TE,\ell}^{R,q}$ and evaluates how $Q$ and $FSR$ depend on radius $R$ and wavelength, identifying optimal $(\lambda_R, R)$ regions. Key findings show silica generally delivers the highest $Q$ in the 2 μm window for moderate to large spheres ($R\gtrsim 20\,\mu$m), while AsSe offers smaller radii advantages due to higher index and reduced radiative loss, with CaF$_2$ providing a competitive trade-off; the study derives practical design guidelines for WGM-based sensing at 2 μm. The methodology, based on analytic and numerical treatments of dispersion, absorption, and spectral properties, provides a framework for material selection and geometry optimization in WGM resonators beyond the telecom band, with implications for biosensing, environment monitoring, and photonic sensing at mid-IR wavelengths.

Abstract

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 $μ\mbox{m}$ 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 $μ\mbox{m}$ 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, $(λ_R,R$), where the WGM resonators are optimal at wavelengths $1.8 μ\mbox{m}<λ<2.1 μ\mbox{m}$.

Figures (8)

• Figure 1: (a) and (b) Refractive index, $n(\lambda)$, and (c) absorption coefficient, $\alpha(\lambda)$, for: silica (black dot-dashed line), calcium fluoride (red dashed line) and AsSe chalcogenide (blue solid line).
• Figure 2: Resonant wavelengths of the first radial mode as a function of the azimuthal number, for a fixed radius $R~=~25\,\mu$m, for the three materials under consideration.
• Figure 3: Radial TE field intensity of the fundamental radial mode in a microsphere with $R=25$$\mu$m and: (a) azimuthal mode $\ell$=150 ($\lambda_{SiO_2} = 1.4264\,\mu$m, $\lambda_{CaF_2} = 1.4080\,\mu$m, $\lambda_{AsSe} = 2.7379\,\mu$m); (b) resonant wavelength around 2 $\mu$m ($\ell_{SiO_2} = 104$, $\ell_{CaF_2} = 103$, $\ell_{AsSe} = 208$).
• Figure 4: Resonant wavelengths of the first radial mode as a function of the sphere radius, for a fixed azimuthal number $\ell=150$, for the three materials under consideration.
• Figure 5: $Q$-factor as a function of wavelength for fused silica (black dot-dashed line), calcium fluoride (red dashed line) and AsSe chalcogenide (blue solid line), for different sphere radii: (a) $R=5\,\mu$m, (b) $R=25\,\mu$m, (c) $R=50\,\mu$m, (d) $R=100\,\mu$m.