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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Detuned inhomogeneously broadened laser linewidth

Jacques Arnaud

Published Aug 1, 1997• 5 citations
Publisher

Abstract

Essentially closed-form formulae are given for the linewidth of inhomogeneously broadened lasers, with arbitrary cavity detuning and pumping statistics (Poissonian or quiet pumps). The so-called `bad-cavity' mode of operation is treated. The general formula is obtained from a semiclassical theory first derived from a linearized symmetrically ordered quantum theory in the limit of a large number of atoms. This semiclassical theory is most easily explained in terms of independent quantum jumps performed by emitting and absorbing atoms. The effect of inhomogeneous broadening on linewidth is found to be negligible for some pumping schemes, but when the atoms are independently pumped the linewidth is proportional (at some constant power level) to the gain-medium spectral width. We consider, in particular, an intermediate situation in which the rate at which atoms are promoted from the lower to the upper level (pump rate) is proportional to the lower-state population for each atom class. For that particular model, pump fluctuations do not affect the linewidth, even when the cavity is detuned from the centre of the atomic transition frequency distribution.

DOI
10.1088/1355-5111/9/4/003

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Essentially closed-form formulae are given for the linewidth of inhomogeneously broadened lasers, with arbitrary cavity detuning and pumping statistics (Poissonian or quiet pumps). The so-called `bad-cavity' mode of operation is treated. The general formula is obtained from a semiclassical theory first derived from a linearized symmetrically ordered quantum theory in the limit of a large number of atoms. This semiclassical theory is most easily explained in terms of independent quantum jumps performed by emitting and absorbing atoms. The effect of inhomogeneous broadening on linewidth is found to be negligible for some pumping schemes, but when the atoms are independently pumped the linewidth is proportional (at some constant power level) to the gain-medium spectral width. We consider, in particular, an intermediate situation in which the rate at which atoms are promoted from the lower to the upper level (pump rate) is proportional to the lower-state population for each atom class. For that particular model, pump fluctuations do not affect the linewidth, even when the cavity is detuned from the centre of the atomic transition frequency distribution.
We derive a general expression for the linewidth of inhomogeneously broadened traveling-wave gas lasers, without any assumption on the relative magnitude of the atomic and field decay constants, and for several types of atomic pumping statistics, ranging from Poissonian to regular. For small inhomogeneous broadening, an important linewidth narrowing occurs in the bad-cavity limit. This effect disappears, however, in the large inhomogeneous broadening limit (Doppler limit). \textcopyright{} 1996 The American Physical Society.
Background: Theoretical studies of superradiant lasing on optical clock transitions predict a superb frequency accuracy and precision closely tied to the bare atomic linewidth. Such a superradiant laser is also robust against cavity fluctuations when the spectral width of the lasing mode is much larger than that of the atomic medium. Recent predictions suggest that this unique feature persists even for a hot and thus strongly broadened ensemble, provided the effective atom number is large enough. Methods: Here we use a second-order cumulant expansion approach to study the power, linewidth and lineshifts of such a superradiant laser as a function of the inhomogeneous width of the ensemble including variations of the spatial atom-field coupling within the resonator. Results: We present conditions on the atom numbers, the pump and coupling strengths required to reach the buildup of collective atomic coherence as well as scaling and limitations for the achievable laser linewidth. Conclusions: We show how sufficiently large numbers of atoms subject to strong optical pumping can induce synchronization of the atomic dipoles over a large bandwidth. This generates collective stimulated emission of light into the cavity mode leading to narrow-band laser emission at the average of the atomic frequency distribution. The linewidth is orders of magnitudes smaller than that of the cavity as well as the inhomogeneous gain broadening and exhibits reduced sensitivity to cavity frequency noise.
We derive the quantum-limited linewidth of a bad-cavity laser, where the cavity-loss rate can be comparable to or larger than the gain bandwidth. We allow for incomplete inversion and arbitrary mirror losses and show how the position-dependent gain, refractive index, and populations co-determine the laser linewidth. The result obtained factorizes in parts that have straightforward physical interpretations. In the bad-cavity regime the laser linewidth is shown to be reduced as compared to the Schawlow-Townes expression as a result of anomalous dispersion associated with the optical gain. Saturation is shown to lead to a power-independent contribution to the laser linewidth; the magnitude of this contribution depends critically on the assumed pumping and decay rates.
We solve the problem of a laser with variable pumping statistics for any relative magnitude of the atomic- and cavity-decay constants, and obtain a different regime of sub-Poissonian light generation. We show that, even for Poissonian pumping, the noise in the amplitude quadrature outside the cavity can be reduced up to 50% below the shot-noise level when the polarization but not the populations can be adiabatically eliminated. Maximum noise reduction in this case is obtained when the lower level decays much faster than the upper one and occurs at a frequency given by the geometrical mean of the decay rates of the field and the lower-level population. Furthermore, the full consideration of atomic memory effects leads to a generalization of previous results on regularly pumped lasers. We find that, for regular pumping, maximum amplitude-noise reduction (up to complete quieting) still occurs at zero frequency in all cases. However, a long-living polarizaton leads to an increase in amplitude noise and may even eliminate the dip at zero frequency, at the same time leading to a significant quenching of the laser linewidth in the bad-cavity limit.
The domain of the third-order theory of the single-mode laser is usually defined by the condition nB/A ≪ 1, where n is the photon number in the cavity, and A and B are the gain and saturation coefficients. When the laser cavity is detuned from line center by Δω ≡ ηγ, where γ is the natural linewidth of the atomic transition, A and B change in the prescribed manner given by the Lamb theory, and it is usually taken for granted that the theory is valid both for small and large detunings η. By examining the relationships between various parameters we show that the limits on the domain of the third-order laser theory effectively limit the allowed detuning in many common lasers to one natural linewidth (η ≈ 1).
A recent SSS-matrix-based theory of the quantum-limited linewidth, which is applicable to general lasers, including spatially nonuniform laser cavities operating above threshold, is analyzed in various limits. For broadband gain, a simple interpretation of the Petermann and bad-cavity factors is presented in terms of geometric relations between the zeros and poles of the SSS matrix. When there is substantial dispersion, on the frequency scale of the cavity lifetime, the theory yields a generalization of the bad-cavity factor, which was previously derived for spatially uniform one-dimensional lasers. This effect can lead to sub--Schawlow-Townes linewidths in lasers with very narrow gain widths. We derive a formula for the linewidth in terms of the lasing mode functions, which has accuracy comparable to the previous formula involving the residue of the lasing pole. These results for the quantum-limited linewidth are valid even in the regime of strong line pulling and spatial hole burning, where the linewidth cannot be factorized into independent Petermann and bad-cavity factors.