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Physical Review A

Lasing threshold for whispering-gallery-mode microsphere lasers

Ying Wu, P. T. Leung

Published Jul 1, 1999• 82 citations
Publisher

Abstract

The analytical expression for the lasing threshold for whispering-gallery-mode microsphere lasers under the strong-coupling condition is obtained by solving a quasi-three-level quantized model. It explicitly displays the dependence of the lasing threshold on the cavity radius, the homogeneous linewidth, the quality factors and linewidths of the relevant cavity modes.

DOI
10.1103/physreva.60.630

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A quantum Langevin theory of whispering-gallery-mode microsphere laser theory is developed. The linear and nonlinear analysis are made for laser operation below and above the threshold. In these analysis, corresponding to the specific property of microsphere, the effect of inversion fluctuation is treated. The coherence functions of laser field are calculated, and the intensity, the amplitude fluctuation and the linewidth of the field are obtained, which are connected with the enhancement factor of whispering-gallery-mode microsphere. It is shown that the strong couple and strong pumping are useful for the amplification of intensity and the decrease of linewidth below the threshold. It is also shown that, for the laser action above threshold, the variances of photon number and the linewidth of internal field are related to the enhancement factor and the square of the enhancement factor, respectively.

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Scientific Reports•Jul 6, 2021•13 citations
The analytical expression for the lasing threshold for whispering-gallery-mode microsphere lasers under the strong-coupling condition is obtained by solving a quasi-three-level quantized model. It explicitly displays the dependence of the lasing threshold on the cavity radius, the homogeneous linewidth, the quality factors and linewidths of the relevant cavity modes.
A quantum Langevin theory of whispering-gallery-mode microsphere laser theory is developed. The linear and nonlinear analysis are made for laser operation below and above the threshold. In these analysis, corresponding to the specific property of microsphere, the effect of inversion fluctuation is treated. The coherence functions of laser field are calculated, and the intensity, the amplitude fluctuation and the linewidth of the field are obtained, which are connected with the enhancement factor of whispering-gallery-mode microsphere. It is shown that the strong couple and strong pumping are useful for the amplification of intensity and the decrease of linewidth below the threshold. It is also shown that, for the laser action above threshold, the variances of photon number and the linewidth of internal field are related to the enhancement factor and the square of the enhancement factor, respectively.
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 $\ensuremath{\mu}$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.
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.
In this article, we propose a simple approach to deal with the fluorescence of a quasi-three-level system in $\ensuremath{\Lambda}$ configuration in which one of the ``levels'' is actually a band with possibly a large width. By solving this quasi-three-level molecule model, we obtain an analytical expression for the microcavity-modified fluorescence decay rate suitable for arbitrary cavities with any geometrical shape under a strong coupling condition. Applying the general result to a microsphere, we express explicitly the fluorescence decay rate's dependence on the cavity radius, emission linewidth, and the quantities characterizing the involved cavity modes, and find it in remarkable agreement with the recent fluorescence experiment by Barnes et al. [Phys. Rev. Lett. 76, 3931 (1996)]. In addition we briefly discuss the suitability of the approach for other microcavity-modified optical processes such as lasing and Raman scattering.
The reduced density matrix method is used to calculate the quantum-statistical properties of the radiation of a quantum-dot laser operating on the whispering gallery mode of a dielectric microsphere. It is shown that under the conditions of strong coupling between the quantum dot and an electromagnetic field the radiation of such a laser can be in a nonclassical (sub-Poissonian) state. The laser scheme considered is characterized by an extremely low lasing threshold and a small number of saturation photons, as result of which lasing is possible with close to zero population inversion of the working levels, if g ≫ P ≫ γ ≫ Γ, where g is the field-matter interaction constant, P is the pumping rate, γ is the loss rate of the resonator, and Γ is the spontaneous emission rate. The largest squeezing inside the resonator-microsphere (the Fano factor F = 0.75) obtains for g ≫ P ≫ γ ≫ Γ, and the greatest squeezing in the fluctuation spectrum outside the resonator [V(ω = 0) ≈ 0.25] occurs for g ∼ P ∼ γ ≫ Γ, and in this case a substantial deviation of the photon number statistics of the radiation leaving the resonator from the Poissonian statistics is observed.
Microlasers, relying on the strong coupling between active particles and optical microcavity, exhibit fundamental differences from conventional lasers, such as multi-threshold/thresholdless behavior and nonclassical photon emission. As light sources, microlasers possess extensive applications in precision measurement, quantum information processing, and biochemical sensing. Here we propose a whispering-gallery-mode microlaser scheme, where ultracold alkaline-earth metal atoms, i.e., gain medium, are tightly confined in a two-color evanescent lattice that is in the ring shape and formed around a microsphere. To suppress the influence of the lattice-induced ac Stark shift on the moderately-narrow-linewidth laser transition, the red-detuned trapping beams operate at a magic wavelength while the wavelength of the blue-detuned trapping beam is set close to the other magic wavelength. The tiny mode volume and high quality factor of the microsphere ensure the strong atom-microcavity coupling in the bad-cavity regime. As a result, both saturation photon and critical atom numbers, which characterize the laser performance, are substantially reduced below unity. We explore the lasing action of the coupled system by using the Monte Carlo approach. Our scheme may be potentially generalized to the microlasers based on the forbidden clock transitions, holding the prospect for microscale active optical clocks in precision measurement and frequency metrology.