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Physical Review A

Theory of the fundamental laser linewidth. II

Philip Goldberg, Peter W. Milonni, Bala Sundaram

Published Oct 1, 1991• 17 citations
Publisher

Abstract

A recent theory of the fundamental laser linewidth developed by the authors is extended to include the effects of the Henry \ensuremath{\alpha} parameter. The usual (1+${\mathrm{\ensuremath{\alpha}}}^{2}$) enhancement of the linewidth is shown to be increased for arbitrary output coupling by spatial hole burning. A total enhancement factor is derived that multiplies the Schawlow-Townes linewidth and takes into account line broadening due to amplification of quantum noise as well as that due to phase-amplitude coupling.

DOI
10.1103/physreva.44.4556

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A recent theory of the fundamental laser linewidth developed by the authors is extended to include the effects of the Henry \ensuremath{\alpha} parameter. The usual (1+\ensuremathα2{\mathrm{\ensuremath{\alpha}}}^{2}\ensuremathα2) enhancement of the linewidth is shown to be increased for arbitrary output coupling by spatial hole burning. A total enhancement factor is derived that multiplies the Schawlow-Townes linewidth and takes into account line broadening due to amplification of quantum noise as well as that due to phase-amplitude coupling.
The theory of the laser linewidth is formulated to account for arbitrarily large output couplings and spatial hole burning. We show explicitly that the linewidth can be interpreted in terms of either spontaneous-emission noise or the amplification of vacuum field modes leaking into the cavity, depending on the ordering of operators in the correlation function determining the laser spectrum. This allows us to derive the Petermann K factor associated with ``excess spontaneous-emission noise'' in a physically transparent and mathematically simple way, without the need to introduce adjoint modes of the resonator. It also allows us to straightforwardly include spatial-hole-burning effects, which are found to increase the K factor and the linewidth in high-gain systems appreciably.
We perform a first-principles calculation of the quantum-limited laser linewidth, testing the predictions of recently developed theories of the laser linewidth based on fluctuations about the known steady-state laser solutions against traditional forms of the Schawlow-Townes linewidth. The numerical study is based on finite-difference time-domain simulations of the semiclassical Maxwell-Bloch lasing equations, augmented with Langevin force terms, and includes the effects of dispersion, losses due to the open boundary of the laser cavity, and non-linear coupling between the amplitude and phase fluctuations (α factor). We find quantitative agreement between the numerical results and the predictions of the noisy steady-state ab initio laser theory (N-SALT), both in the variation of the linewidth with output power, as well as the emergence of side-peaks due to relaxation oscillations.
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.
The problem of fundamental laser line broadening due to random spontaneous emission of radiation and amplification of thermal radiation noise is analyzed in terms of a classical fluctuating field phasor model. We derive a general expression for the intrinsic linewidth, given in terms of the spectral power of the radiation noise source, which can be classical or quantum mechanical in nature. In the case of a two-level atomic laser, we recover by the use of Einstein relations, the traditional linewidth formula of the Schalow-Townes form. In the case of the free-electron laser (FEL), using the explicit expression for the spontaneous emission, we present calculation of the laser linewidth by purely classical methods. The result agrees with the one obtained in the framework of a quantum-mechanical model. By using ``extended Einstein relations'' which are applicable to classical radiators, we show that a Schalow-Townes-type formula can also be obtained for the FEL. The theory predicts extremely narrow intrinsic linewidth ${(10}^{\mathrm{\ensuremath{-}}7}$ Hz) for cw FEL's with parameters similar to those of the FEL experiment of Elias et al.
The simple intuitive picture of phase diffusion in a traveling wave is introduced to calculate the fundamental (Schawlow-Townes) limit to the laser linewidth. Contrary to the usual phase-diffusion model, our variety is not limited to the regime of low outcoupling; it predicts an enhancement of the linewidth for large outcoupling. It is demonstrated that the results of the traveling-wave phase-diffusion model are identical to those of the theory of nonorthogonal eigenmodes. Finally, the effects of saturation on the linewidth are estimated.