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Physical Review A

Ultralow Raman lasing threshold and enhanced gain of whispering gallery modes in silica microspheres

Mikhail V. Jouravlev, Daniel R. Mason, Kwang S. Kim

Published Jan 19, 2012• 7 citations
Publisher

Abstract

Remarkable demonstrations of a cavity-induced giant reduction of the Raman lasing threshold in silica microspheres have been reported in Spillane et al. [Nature (London) 415, 621 (2002)], yet a complete quantum electrodynamics (QED) treatment of this process including the rigorous expressions for the electromagnetic modes of the sphere is lacking. In this paper, we employ a cavity QED approach to describe Raman lasing in microscopic spherical dielectric resonators with third-order nonlinearity to derive explicit expressions for the Raman gain and lasing threshold in terms of mode-overlapping coefficients determined from full analytic expressions of the electromagnetic eigenmodes of a dielectric sphere. We present dependencies of the Raman lasing threshold on the the order $n$ of overlapping whispering gallery modes for microspheres with diameter in the range 20--35 $\ensuremath{\mu}$m, demonstrating ultralow thresholds of between 70 and 90 $\ensuremath{\mu}$W, consistent with recent experimental results. We explain the reduction of the lasing threshold as due to an increase in the overlapping coefficients as the mode order is increased toward the regime where the modal energy is mostly confined to the surface region of the resonator. The presented theory can be easily generalized to any microcavity of regular morphology and for any combination of interacting whispering gallery modes.

DOI
10.1103/physreva.85.013825

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Remarkable demonstrations of a cavity-induced giant reduction of the Raman lasing threshold in silica microspheres have been reported in Spillane et al. [Nature (London) 415, 621 (2002)], yet a complete quantum electrodynamics (QED) treatment of this process including the rigorous expressions for the electromagnetic modes of the sphere is lacking. In this paper, we employ a cavity QED approach to describe Raman lasing in microscopic spherical dielectric resonators with third-order nonlinearity to derive explicit expressions for the Raman gain and lasing threshold in terms of mode-overlapping coefficients determined from full analytic expressions of the electromagnetic eigenmodes of a dielectric sphere. We present dependencies of the Raman lasing threshold on the the order nnn of overlapping whispering gallery modes for microspheres with diameter in the range 20--35 \ensuremathμ\ensuremath{\mu}\ensuremathμm, demonstrating ultralow thresholds of between 70 and 90 \ensuremathμ\ensuremath{\mu}\ensuremathμW, consistent with recent experimental results. We explain the reduction of the lasing threshold as due to an increase in the overlapping coefficients as the mode order is increased toward the regime where the modal energy is mostly confined to the surface region of the resonator. The presented theory can be easily generalized to any microcavity of regular morphology and for any combination of interacting whispering gallery modes.
The whispering gallery modes (WGMs) of quartz microspheres are investigated for the purpose of strong coupling between single photons and atoms in cavity quantum electrodynamics (cavity QED). Within our current understanding of the loss mechanisms of the WGMs, the saturation photon number n0{n}_{0}n0​ and critical atom number N0{N}_{0}N0​ cannot be minimized simultaneously, so that an ``optimal'' sphere size is taken to be the radius for which the geometric mean n0N0,\sqrt{{n}_{0}{N}_{0}},n0​N0​​, is minimized. While a general treatment is given for the dimensionless parameters used to characterize the atom-cavity system, detailed consideration is given to the D2{D}_{2}D2​ transition in atomic cesium at ${\ensuremath{\lambda}}_{0}=852\mathrm{nm}$ using fused-silica microspheres, for which the maximum coupling coefficient ${g}_{a}/(2\ensuremath{\pi})\ensuremath{\approx}750\mathrm{MHz}$ occurs for a sphere radius $a=3.63\ensuremath{\mu}\mathrm{m}$ corresponding to the minimum for ${n}_{0}\ensuremath{\approx}6.06\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}.$ By contrast, the minimum for ${N}_{0}\ensuremath{\approx}9.00\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ occurs for a sphere radius of $a=8.12\ensuremath{\mu}\mathrm{m},$ while the optimal sphere size for which n0N0\sqrt{{n}_{0}{N}_{0}}n0​N0​​ is minimized occurs at $a=7.83\ensuremath{\mu}\mathrm{m}.$ On an experimental front, we have fabricated fused-silica microspheres with radii $a\ensuremath{\sim}10\ensuremath{\mu}\mathrm{m}$ and consistently observed quality factors $Q>~0.8\ifmmode\times\else\texttimes\fi{}{10}^{7}.$ These results for the WGMs are compared with corresponding parameters achieved in Fabry-Perot cavities to demonstrate the significant potential of microspheres as a tool for cavity QED with strong coupling.
We investigate the scattering threshold and cavity-enhanced gain in nonlinear spheres with second- or third-order permeability. Pairs of pump-driven idler and signal modes are considered, satisfying morphology-dependent resonance conditions. The thresholds and gain coefficients of amplified and stimulated Raman scattering, parametric downconversion, and analogous parametric processes in microspheres are derived and evaluated under typical conditions. Applications may include the measurement of chemical impurity concentrations or the creation of low-threshold optical parametric amplifiers using microspheres.
Silica microspheres behave as efficient optical microresonators when small-volume whispering-gallery modes (WGMs) are used. We have realized a cryogenic set-up to work with microspheres immersed in a superfluid helium bath. Quality factors up to 109 have been obtained at 2 K. In this environment, we have been able to observe a dispersive bistable behavior of the WGM resonances, due to the weak intrinsic Kerr optical nonlinearity of silica, with a threshold power of 10 (mu) W only. This result opens the way to the realization of thresholdless microlaser, based on a rare-earth doped silica microsphere and to other Cavity-QED projects with microspheres at low temperatures.
Summary form only given, as follows: The high Q whispering gallery modes of silica microspheres are promising for quantum optics and nonlinear optics experiments involving atoms and photons stored in very small volumes. We have developed the technology of these resonators, learning how to couple light in and out through optical fibers, how to obtain and measure record Q factors, and how to probe, by near-field optical microscopy, the electric field distribution immediately outside the sphere. Whispering gallery mode lasers have also been obtained with microspheres made of silica doped with neodymium ions. We describe these experiments and discuss projects involving the coupling of whispering gallery modes to single atoms or ions to perform cavity QED experiments in the optical domain.
Silica microspheres have been demonstrated to provide a permanent resonator which combines small mode volumes with the highest observed quality factors (Q>2.10/sup 9/) corresponding to photon storage times of the order of one microsecond. These properties have allowed us to make a very low-threshold laser with Nd-doped silica microspheres (200 nW absorbed pump power, i.e. 10/sup 6/ excited ions/mode). These microlasers are now operated at 2 K to benefit from a smaller homogeneous linewidth. We then expect to reach the "thresholdless" regime with only a few excited ions. These experiments can also be considered as a first step toward Cavity QED experiments in the strong coupling regime with single atoms or molecules coupled to a whispering gallery mode.