[The precise assignment of whispering gallery modes for lasing spectra emitting from cylindrical micro-cavities]

Lasers with spherical or cylindrical dielectric resonators supported by whispering gallery modes (WGM) have attracted much interest due to their microscopic size, high cavity Q factor, and low lasing threshold. Cylindrical microcavity lasers based on the gain only in the evanescent field region of whispering gallery modes have been demonstrated in our recent works. The gain was excited by the evanescent wave of longitudinal optical pumping along the optical fiber. To well understand the obtained lasing spectra, the mode assignment is required. The explicit asymptotic formulas for the position and mode-interval of whispering gallery modes were obtained from the characteristic equation of whispering gallery modes in a cylindrical micro-cavity. The formulas were used to analyze the lasing spectra emitting from cylindrical microcavies which were evanescent-wave-gain pumped. The lasing spectra were found to be transverse magnetic modes(TM), and then the spectra were mode assigned with two integers, i.e., radial quantum numbers (1) and angular momentum numbers (n). Based on the explicit asymptotic formulas, all of the spectra from five optical fibers with a diameter ranging from 215 to 328 mm were well mode assigned. In the match between experimental spectral data and the asymptotic formula, only two matched parameters (l, n) were used, and the wavelength deviation in the match was less than 0.05 nm, which indicated that the mode assignment was reliable and precise. The spectral mode-assignment of cylindrical micro-cavity is important for computing the spatial distribution of mode intensity and is crucial for the applications of frequency-shift biosensor built in cylindrical micro-cavities. The method introduced in this paper can also be used to measure the diameters and refractive indexes of cylindrical micro-cavies precisely.