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Journal of Optics B Quantum and Semiclassical Optics

Theory of the fundamental linewidth of a two-mode laser

Aurangzeb Khurram Hafiz, Rupamanjari Ghosh

Published Apr 27, 2004• 0 citations
Publisher

Abstract

Starting from the field master equation for a laser oscillating simultaneously in two modes, we probe the dependence of its fundamental phase-diffusion linewidth on the degree of coupling between the two modes. We find that the heterodyned intrinsic linewidth shows the usual decrease when the output power of each mode increases with an increase in the gain, while the nonlinear self-saturation and cross-coupling coefficients are held constant. The linewidth also decreases with increase in the cross-coupling between the modes for constant gain when the output power of each mode is kept constant by decreasing the self-saturation coefficient. The linewidth, however, increases with increasing cross-coupling when (a) the gain and the self-saturation coefficient are held constant and hence the output power in each mode decreases and (b) the gain is increased keeping the output power and the self-saturation of each mode constant.

DOI
10.1088/1464-4266/6/7/003

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Starting from the field master equation for a laser oscillating simultaneously in two modes, we probe the dependence of its fundamental phase-diffusion linewidth on the degree of coupling between the two modes. We find that the heterodyned intrinsic linewidth shows the usual decrease when the output power of each mode increases with an increase in the gain, while the nonlinear self-saturation and cross-coupling coefficients are held constant. The linewidth also decreases with increase in the cross-coupling between the modes for constant gain when the output power of each mode is kept constant by decreasing the self-saturation coefficient. The linewidth, however, increases with increasing cross-coupling when (a) the gain and the self-saturation coefficient are held constant and hence the output power in each mode decreases and (b) the gain is increased keeping the output power and the self-saturation of each mode constant.
The problem of the quantum-limited or intrinsic linewidth of a good-cavity laser is revisited. Starting from the Scully-Lamb master equation, we present a fully analytical treatment to determine the correlation function and the spectrum of the cavity field at steady state. For this purpose, we develop an analytical approximation method that implicitly incorporates the microscopic fluctuations of both the phase and intensity of the field, and, in addition, takes full account of the saturation of the nonlinear gain. Our main result is a simple formula for the quantum-limited linewidth that is valid from near to far above threshold and also includes the presence of thermal photons. Close to the threshold, the linewidth is twice as large as predicted by the standard phase-diffusion treatment neglecting intensity fluctuations, and even 50% above threshold the increase is still considerable. In general, quantum fluctuations of the intensity are present and continue to influence the linewidth as long as the photon-number distribution is not strictly Poissonian. This inherent relationship is displayed by a formula relating the linewidth and the Mandel Q parameter. More than 100% above treshold the linewidth is found to be smaller than predicted by the standard treatment, since the simple phase-diffusion model increasingly overestimates the rate of phase fluctuations by neglecting gain saturation. In the limit of a very large mean photon number the expected perfectly coherent classical field is obtained.
We present a theory of the homogeneously broadened laser that is based on the simple boundary conditions that govern the connection of intracavity traveling fields and the leaking vacuum field at the cavity mirrors. The approach is uniformly valid for arbitrary output-mirror transmissions. We compute the intrinsic linewidth of the laser both below and above threshold and for arbitrary mirror outcoupling. In the limit of complete output-mirror transmission, for which any semblance of the cavity to a lasing cavity is completely untenable and the laser is severely below threshold, we show that the laser linewidth reduces to the natural linewidth of spontaneous emission by atoms in free space. Above threshold, in contrast with most previously obtained results, the fundamental linewidth exhibits a power-independent contribution that arises from gain saturation that is spatially nonuniform.
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 study the linewidth of a one-photon laser through the approach of the two-time correlation function. Our numerical calculations cover the regions of below, near, and above threshold. Well below threshold, the laser linewidth decreases linearly with increasing linear gain. Near and above threshold, in comparison with the approximate linewidth given by the phase-diffusion coefficient at the mean photon number (which is inversely proportional to the laser intensity---the Schawlow-Townes formula), we find that the laser intensity fluctuations increase the laser linewidth and that the Schawlow-Townes formula is valid only when the mean photon number is large and the photon-number variance is much smaller than the square of the mean photon number. Near threshold, the laser linewidth can be twice as large as that given by the phase-diffusion coefficient at the mean photon number (i.e., twice as large as that given by the Schawlow-Townes formula).
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.
Abstract The Schawlow–Townes limit established spontaneous emission as the fundamental quantum mechanism defining the minimum linewidth of an ideal laser oscillator. According to this description, random phase perturbations introduced by spontaneous photons become progressively less significant as the intracavity photon population increases, resulting in a continuous narrowing of the emission linewidth with increasing optical power. Although this relation remains a cornerstone of laser physics, it represents an ideal limit in which other sources of phase instability are absent. Practical laser systems frequently deviate from this behaviour: after an initial power-dependent narrowing regime, the linewidth often approaches a residual value that cannot be removed by further increasing the optical power. Such saturation suggests that linewidth should be treated not only as a photon-statistical quantity, but also as a property of the complete laser system, including the active medium. In this work, we develop a phenomenological framework in which laser linewidth is described as the combined result of spontaneous-emission noise, pump-induced fluctuations, and medium induced phase perturbations. The proposed approach preserves the Schawlow–Townes limit as the fundamental quantum contribution while introducing an active-medium-dependent term associated with the finite spectral response and microscopic fluctuations of the gain medium. The physical origin of this residual contribution is analysed using optical line-shape theory, considering homogeneous broadening, inhomogeneous broadening and the role of local environments in solid-state gain media. Er³+-doped systems are considered as representative rare earth active media, since their shielded 4f–4f transitions remain sufficiently narrow to support coherent emission while still retaining measurable signatures of crystal-field splitting, host disorder and spectral broadening. The model provides a physical interpretation for linewidth saturation and highlights that the ultimate optimisation of laser coherence requires not only improved cavity design and pump stabilisation but also control of the microscopic stability of the active medium. This viewpoint reframes the linewidth floor as a system-level property emerging from the interaction between quantum fluctuations, excitation dynamics and the gain medium.