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Physical Review A

Power tail of an intensity distribution function of the bad-cavity laser with field and population fluctuations

Tetsuo Ogawa

Published Feb 1, 1989• 5 citations
Publisher

Abstract

We investigate theoretically the statistical properties of the bad-cavity laser with field and population fluctuations. This system is equivalent to the stochastic Toda oscillator, whose probability distribution obeys the Kramers equation with a position-dependent diffusion coefficient. An approximate probability distribution function is calculated in the stationary state by the method of the orthogonal polynomial expansion. We predict novel statistical features of the laser intensity resulting from the power tail of the intensity distribution. Diverging intensity moments and the deviation of the photon-counting statistics from the Poisson type are also given.

DOI
10.1103/physreva.39.2264

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We investigate theoretically the statistical properties of the bad-cavity laser with field and population fluctuations. This system is equivalent to the stochastic Toda oscillator, whose probability distribution obeys the Kramers equation with a position-dependent diffusion coefficient. An approximate probability distribution function is calculated in the stationary state by the method of the orthogonal polynomial expansion. We predict novel statistical features of the laser intensity resulting from the power tail of the intensity distribution. Diverging intensity moments and the deviation of the photon-counting statistics from the Poisson type are also given.
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The noise characteristics of a single-mode laser with a low-Q cavity are investigated theoretically. After the electric field is adiabatically eliminated from the Maxwell-Bloch equations, coupled Langevin equations with both additive and multiplicative white noises are examined. The equations are solved using Rice's method in a framework of quasilinear Fourier analysis. Noise spectral densities are calculated analytically to study the dependence of the relative intensity noise (RIN) on the pumping. Through an investigation of the auto- and cross-correlations of the light intensity and population noises, their variances are obtained in order to compare their properties to the good-cavity case. The stationary intensity cumulants, the photon counting coefficient, and the photon counting probability are explicitly derived and compared with the results of a Fokker-Planck analysis previously carried out for both the good- and bad-cavity cases.>
Using population rate equations, we have calculated the intrinsic intensity fluctuations expected in the output of 3-level and 4-level CW laser oscillators. Such fluctuations stem from the quantum shot noise implicit in a restriction of the atomic and photon population-operator eigenvalues to discrete integral values (in the manner of Shimoda, Takahasi, and Townes). The intrinsic fluctuations below threshold are typical of a Gaussian distribution ofe-mfield amplitudes (Bose-Einstein distribution of photons) and above threshold of an amplitude-stabilizede-mfield (Poisson distribution of photons). The noise properties of 3-level and 4-level lasers are qualitatively similar, although for a given output power the intensity fluctuations of 3-level oscillators are greater than (or equal to) those in 4-level oscillators, principally as a consequence of the increased spontaneous emission associated with the larger upper-state occupation necessary to achieve a net gain in 3-level systems.
The statistical behavior of spectral intensity fluctuations in a Raman gain random fiber laser is investigated for the first time, to the best of our knowledge. Through a fiber Fabry–Perot tunable filter, the intensity of the filtered single wavelength from random laser (RL) spectra is measured at different pump powers. The probability density function (PDF) of spectral intensity values is analyzed. Well below the lasing threshold, the intensity of a single wavelength is stable in the time domain, and the PDF exhibits a typical Gaussian behavior. However, around the threshold, strong fluctuations in intensity emerge. In this case, the statistical distribution is characterized by a power-law-tailed function consistent with a Lévy α-stable distribution. We identify the statistical evolution as the injected pump power increases, from a Gaussian to a Lévy and then back to a Gaussian distribution, which is similar to the case of RLs with a strong disorder due to the presence of photon scatterers and gain related to electronic transitions in the active medium, indicating the universal existence of such statistical properties in random lasing systems.
It is shown that, introducing quantum-mechanical degrees of freedom for the pump field, one can model optical-pumping processes of a laser with sub-Poissonian statistics. For a class of pump models containing regular pumping at one end of the range and Poissonian pumping at the other, the exact stationary moments for the pump-field-averaged laser field are calculated in the strong-saturation limit of the lasing transition. It is demonstrated that, in a photodetection experiment of the laser output, complete noise suppression in the photocurrent fluctuation spectrum is, at least in principle, achievable. Finally, an approximate Fokker-Planck equation for photon distribution of the laser field with explicit appearance of the pump-light intensity correlation function in the diffusion term is derived and solved in the stationary limit.
We study the effects of correlations between quantum and pump noises on fluctuations of the laser intensity in a saturation laser model. An approximative Fokker–Planck equation and analytic expressions of the steady-state probability distribution function (SPD) of the laser system are derived. Based on the SPD, the normalized mean, the normalized variance, and the normalized skewness of the steady-state laser intensity are calculated numerically. The results indicate that (i) the correlation strength λ of correlated noises always enhances the fluctuation of laser intensity; (ii) the correlation time τ of correlated noises strengthens the fluctuation of laser intensity for the below-threshold case but τ weakens it for the above-threshold case.