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Acta Physica Sinica

LASER LINEWIDTH WITHOUT ROTATING WAVE APPROXIMATION

Xiong Jin, Xiangming Hu, PENG JIN-SHENG, 化中师范大学物理系，武汉 430079

Published Jan 1, 1999• 0 citations
Publisher

Abstract

In this paper the laser linewidth without rotating wave approximation is studied by keeping the ratio of the atomic natural decay rate to the radiation frequency to two orders.It is shown that while the linewidth is confined to the Schawlow-Townes formula,the non-rotating wave terms would cause the linewidth to increase since the non-rotating wave terms decrease the average photon number.The degree of increasing in linewidth depends not only on the square of the ratio of the atomic natural de-cay rate to radiation frequency,but also on the average photon number in rotating wave approxima-tion.The influence of the non-rotating wave terms on the laser linewidth is much larger than on the photon number fluctuations.

DOI
10.7498/aps.48.1864

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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).
The effect of counter-rotating terms in the interaction Hamiltonian on the photon statistics is investigated for a single-mode laser. The treatment is based on the Scully-Lamb model. A perturbation expansion is used to derive an equation of motion for the field-density matrix. The counter-rotating terms are found to lead to both transient and secular effects. With some reasonable approximations the steady-state photon-number distribution is derived and it is shown that the counter-rotating terms tend to raise the threshold of laser action and broaden the photon-number distribution. The relative size of these effects is found to be very small and depends on the square of the ratio of the atomic linewidth and the transition frequency.
We use standard quantum laser theory to examine the linewidth of lasers with a four-level atomic gain medium as a function of the spontaneous emission rate into the lasing mode and the decay rates from the upper and lower lasing levels. We find that the laser linewidth ceases to exhibit the customary decrease with increased photon number when a critical photon number, proportional to the ratio of the decay rate from the lower lasing level to the rate of spontaneous emission into the lasing mode, is reached. This critical photon number would decrease as the size of the optical cavity is decreased. In addition we show that, in general, the laser linewidth depends critically on the ratio of the overall decay rates from the upper and lower lasing levels.
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.
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.