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Physical Review A

Quantum fluctuations and saturable absorption in mesoscale lasers

Kaushik Roy‐Choudhury, Anthony F. J. Levi

Published Apr 21, 2011• 12 citations
Publisher

Abstract

We present a quantum-mechanical treatment of fluctuations and saturable absorption in mesoscale lasers. The time evolution of the density matrix is obtained from numerical integration and field-field and intensity-intensity correlations are calculated to obtain steady-state linewidth and photon statistics. Inclusion of a saturable absorber in the otherwise homogeneous medium is shown to suppress lasing, increase fluctuations, and enhance spontaneous emission near threshold.

DOI
10.1103/physreva.83.043827

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We present a quantum-mechanical treatment of fluctuations and saturable absorption in mesoscale lasers. The time evolution of the density matrix is obtained from numerical integration and field-field and intensity-intensity correlations are calculated to obtain steady-state linewidth and photon statistics. Inclusion of a saturable absorber in the otherwise homogeneous medium is shown to suppress lasing, increase fluctuations, and enhance spontaneous emission near threshold.
Many quantum optical systems exhibit first-order phase transition analogy.1 Fluctuations of the optical field in such systems are due to spontaneous emission noise. The nature of these fluctuations for various systems has been discussed using various approaches. Fully quantum mechanical and Langevin equation treatments have been given for the dye laser2,3 and the laser with a saturable absorber (LSA) with homogeneously broadened media.2,4 For inhomogeneously broadened media, such treatments are either valid for low intensities or do not fully take into account the effect of quantum noise.5 In this contribution we present a fully quantum mechanical treatment for the LSA with an inhomogeneously broadened absorber using the Foch state representation of the density matrix.2,6
The equation of motion for the slowly varying field amplitude of the laser containing an intracavity saturable absorber (LSA) is where a is the net gain (pump parameter) and b is a saturation parameter of the laser. In our experiment the absorber saturates faster than the amplifier and the parameter b is positive. q(t) is a complex Gaussian white noise random process representing spontaneous emission noise. From Eq. (1) we obtain for the steady-state probability density of light intensity I = | E |2. From this equation various moments of light intensity can be calculated and compared with experimentally measured quantities. For b > 0 the LSA exhibits bistability and a first order phase transition occurs at a = -3b2/16. For negative values of b there is a continuous transition at a = 0. Experiments were carried out on a He—Ne laser with an intracavity Ne absorber cell. The laser was operated close to the center of the absorption line. Slow drifts of light intensity were controlled by an electronic feedback loop. The parameter b was varied by changing the current in the absorber tube. Pump parameter a was then varied by changing passive losses in the cavity by means of an adjustable knife edge. The main beam from the LSA was allowed to fall on a detector/am- plifier combination. The voltage output from the detector was digitized by a multichannel analyzer and a histogram of voltage amplitudes was built up. Since the voltage output from the detector is directly proportional to the light intensity incident on the detector, voltage histogram is a direct representative of the intensity probability distribution. From the measured histogram moments of light intensity were calculated. These results are compared with the theoretical predictions.
The intensity fluctuations in the output of a saturable optical amplifier are studied as the amplifier length is increased. Characteristics are determined by calculation of the diagonal elements of the density matrix of the optical field. Saturation affecting both upper-state and lower-state atomic populations is considered. Spontaneous emission distributed along the amplifier causes the "photon-number" distribution to approach a Bose-Einstein distribution during linear amplification if the input field is not Bose-Einstein. The onset of saturation reduces the relative fluctuations of the optical field both by introducing nonlinear gain and by reducing the spontaneous emission. Limiting solutions show that strong saturation leads to a major reduction in the fluctuations causing the photon-number distribution to approach a Poisson distribution.
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.
The dominant role of quantum fluctuations in determining the steady-state and transient response of a laser is demonstrated when there is a small number of particles in the system. In this regime, quantum fluctuations are found to suppress the lasing threshold and create a non-Poisson probability distribution for n discrete excited electronic states and s discrete photons. The correlation between n and s damps the averaged dynamic response of laser emission. Random walk calculations verify the master equation predictions and are used to connect to systems containing larger numbers of particles.
A quantum theory of the spectral width of a laser with a saturable absorber is presented. We perform a density matrix calculation yielding the off-diagonal element pn,n+1(t ) which is proportional to the average value of the electric field (E(t )) associated with the laser field having a large photon number n. The general linewidth theory is applied to any general laser which has saturable absorber features for a portion of the building of the laser field. We have found that the saturable absorber action contribution to the linewidth of such a laser can be substantial.