Identifying modes of large whispering-gallery mode resonators from the spectrum and emission pattern

Identifying the mode numbers in whispering-gallery mode resonators (WGMRs) is important for tailoring them to experimental needs. Here we report on a novel experimental mode analysis technique based on the combination of frequency analysis and far-field imaging for high mode numbers of large WGMRs. The radial mode numbers q and the angular mode numbers p=$\ell$-m are identified and labeled via far-field imaging. The polar mode numbers $\ell$ are determined unambiguously by fitting the frequency differences between individual whispering gallery modes (WGMs). This allows for the accurate determination of the geometry and the refractive index at different temperatures of the WGMR. For future applications in classical and quantum optics, this mode analysis enables one to control the narrow-band phase-matching conditions in nonlinear processes such as second-harmonic generation or parametric down-conversion.