Resonant wavelengths of whispering gallery modes in dispersive materials

TL;DR

This work analyzes how chromatic dispersion in bulk-fused silica microspheres shifts whispering-gallery-mode resonances. It compares exact solutions of the TE/TM characteristic equations with dispersive asymptotic expansions that incorporate $n_s(\lambda)$, assessing their accuracy via resonance positions, FSR, and $Q$-factors. The study finds sub-1% agreement between methods for large azimuthal numbers, but dispersion induces nontrivial wavelength-radius relationships that prevent simple rescaling, particularly affecting FSR while leaving $Q$ largely dispersion-insensitive. The results provide a practical framework for designing silica microspheres to operate at specific wavelengths across a broad range, with implications for sensing and photonics applications.

Abstract

In this work we compute the resonant wavelength of whispering gallery modes for bulk-fused silica microspheres including chromatic dispersion. This is done following two methods: by solving the exact characteristic equation and, on the other hand, by solving the nonlinear equations that result for a variable refractive index in the asymptotic approximations. Similar results with both methods are obtained with differences below $1\%$ . Nevertheless, important differences are found with respect to the resonant wavelengths computed with a constant index and with a variable index. We compute the free spectral range and the quality factor, and make a comparison between the variable index and the constant index cases. The differences are of significant relevance for the free spectral range, while for the quality factor, the constant case is insensitive to the chromatic dispersion. Our work could be useful as a pathway for designing microspheres for different applications.

Figures (7)

• Figure 1: (a) Refractive index, $n_s$, and (b) absorption coefficient, $\alpha$, for bulk-fused silica as a function of wavelength, calculated using Eqs. \ref{['sellmeier']} and \ref{['alpha']}, respectively.
• Figure 2: Dispersion curves for the first radial TE modes of $R~=1\,\mu$m and $R=5\,\mu$m microspheres, calculated with different approximations and $n_s(\lambda)$. The dotted line corresponds to the "exact" dispersion curve calculated for the $R=1\,\mu$m resonator and escalated for $R=5\,\mu$m.
• Figure 3: Percent relative error computed between the "exact" and the different asymptotic solutions for a silica microsphere with $R=5\,\mu$m, for TE modes, $n=1$ and $n_s(\lambda)$.
• Figure 4: Percent relative error for $R=5\,\mu\hbox{m}$ (solid lines) and $R=1\,\mu\hbox{m}$ (dotted lines), for different values of radial modes ($n=1,2,3,4$), for both TE and TM modes and $n_s(\lambda)$.
• Figure 5: $\Delta x_{1,l}$ computed with $n_s(\lambda)$ (solid lines), and with $n_s~=~1.45$ (dotted lines), for $R=1\,\mu$m and $R=5\,\mu$m and TE modes.