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Journal of Optics A Pure and Applied Optics

Multivariate statistics of the intensity fluctuations of a heavily saturated amplified spontaneous emission and coupled-mode optical systems

Biman Das, Shaun Giammichele

Published Nov 22, 2001• 1 citation
Publisher

Abstract

In an optical amplifier, when the gain is non-linear due to gain saturation, the intensity fluctuation statistics are significantly altered from the statistics of the negative-exponential intensity probability distribution function (IPDF) for spontaneous emission and unsaturated amplifiers. We have systematically computed bivariate moments, trivariate moments and central moments, tetravariate moments, as well as bivariate and trivariate cumulants of intensity fluctuations for non-linear coupled-mode optical systems starting from a k -variable multivariate IPDF of a system of n ( k < n ) coupled modes. The multivariate IPDF is a negative polynomial that rests on the assumption that the variation of intensity has a maximum-intensity distribution with the constraint that the total intensity is constant. The assumption holds for multimode optical systems including a thin, long, mirrorless, gas laser where under heavy gain saturation the output tends to become constant although the intensity of any mode displays rapid fluctuations about the average intensity. The multivariate IPDF reduces to the negative-exponential IPDF of spontaneous emission when n is infinite, as expected. The new results for the multivariate characteristics of the intensity fluctuations have shown characteristic signatures that depend on the number of interacting modes, n . The results are also useful for testing the multivariate IPDFs and identifying n for heavily saturated coupled-mode optical systems.

DOI
10.1088/1464-4258/4/1/310

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In an optical amplifier, when the gain is non-linear due to gain saturation, the intensity fluctuation statistics are significantly altered from the statistics of the negative-exponential intensity probability distribution function (IPDF) for spontaneous emission and unsaturated amplifiers. We have systematically computed bivariate moments, trivariate moments and central moments, tetravariate moments, as well as bivariate and trivariate cumulants of intensity fluctuations for non-linear coupled-mode optical systems starting from a k -variable multivariate IPDF of a system of n ( k < n ) coupled modes. The multivariate IPDF is a negative polynomial that rests on the assumption that the variation of intensity has a maximum-intensity distribution with the constraint that the total intensity is constant. The assumption holds for multimode optical systems including a thin, long, mirrorless, gas laser where under heavy gain saturation the output tends to become constant although the intensity of any mode displays rapid fluctuations about the average intensity. The multivariate IPDF reduces to the negative-exponential IPDF of spontaneous emission when n is infinite, as expected. The new results for the multivariate characteristics of the intensity fluctuations have shown characteristic signatures that depend on the number of interacting modes, n . The results are also useful for testing the multivariate IPDFs and identifying n for heavily saturated coupled-mode optical systems.
Spontaneous emission from incoherently excited atoms has a negative exponential intensity probability distribution function (IPDF) In an optical amplifier when the gam is non-lincar (due to gam saturation), the intensity lluctuation statistics is sigmllcanlly altered from that of spontaneous emission as is evidenced by experimental results We have derived the expressions for (i) univariate cumulants of intensity lluctuations up to tenth order from a single mode IPDF and (ii) bi\ariate moments of intensity fluctuations up to fifth order starting from a bivariate IPDF 3'he IPDFs arc negative polynomials (whose powers depend on the number of interactive modes), which rest on the assumption that the variation of intensity has a maximum entropy distribution with the constraint that the total intensity is constant This approximation holds for a heavily saturated source of amplified spontaneous emission (ASI-) and some other coupled-mode optical systems Both the normalized cumulants and bivariate moments have shown unique features that depend only on the number of interacting modes m the system The theoretical results have been compared with the experimental ones for ASH The comparison is useful to lest the model and to identify the number of interacting modes m coupled-mode optical systems Keywords Amplified spontaneous emission, coupled-mode optical systems, univariate cumulants and bivariate moments PAC^S Nos.42 50 Ar, 02 70 --c
Intensity fluctuation characteristics of amplified-spontaneous emission are found using intensity rate equations and the noise-amplifier approximation for saturation coupling between two and four competing modes. The existence of more than one mode leads to the retention of relatively large fluctuations in each mode even as the amplifier gain is saturated, an operating regime which by contrast has been shown to reduce significantly the fluctuations when only a single beam is present. The noise-amplifier approximation is found to be an adequate approximation of the effects of the spontaneous emission distributed along the length of a laser amplifier. The intensity fluctuations in all cases approach limiting distributions functions as the amplifier becomes heavily saturated. Characteristics of these limiting distributions are found for one, two, and four beams; for combinations of beams copropagating and counterpropagating; in the presence and absence of loss; for differences in gain; and for saturation of the sources due to high intensities.
The univariate and bivariate intensity statistics of amplified spontaneous emission (ASE) from continuously excited HeNe and HeXe gas discharges have been measured for the purpose of characterizing the dependence of ASE fluctuations on gain length and degree of homogeneous broadening. Comparison with theoretical results from a model for intensity fluctuations in heavily saturated multimode systems confirms that increasing the homogeneous broadening lowers the number of competing "modes" in the system. Intrinsic to this simple coupled-intensity model are unique combinations of cumulants that are independent of the number of modes and the strength of their coupling. These combinations have been measured, and they are independent of the degree of homogeneous broadening and are constant as the discharge length is increased. The normalized cumulants and cross-correlations for coprapogating and counterpropagating orthogonal, circularly polarized beams in a HeXe homogeneously broadened medium are consistently smaller in magnitude than those for orthogonal, linearly polarized beams. Comparison of results for neon and xenon discharges will distinguish whether this is the result of back-reflections or is a property of the counterpropagating beams quasiresonant with these noble-gas transitions, which differ in their upper- and lowerstate angular momenta.
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 present a theoretical analysis and an experimental study of the statistical properties of the noise accompanying an optical pulse propagating in a nonlinear semiconductor optical amplifier. Several degrees of gain saturation corresponding to different levels of optical signal-to-noise ratios (OSNRs) are examined. We employ the Heun numerical procedure to ensure proper convergence to the Stratonovich solution of the multiplicative propagation equation. This algorithm takes also into account the effect of gain saturation due to the amplified spontaneous emission noise. Moreover, the multicanonical Monte Carlo algorithm is used to efficiently calculate the probability density functions (pdfs), including the tails, of the peak of a pulse which emerges at the output of a saturated semiconductor optical amplifier. The results are compared to the corresponding pdfs obtained in a linear amplification system (where the optical noise is additive and Gaussian) having the same gain and under identical OSNR levels. We demonstrate that the pdf of the saturated amplifier is shifted toward lower power levels and is narrower, or equivalently, the mean and variance for the saturated amplifier case are smaller. Also, the difference between the two configurations increases with the degree of gain saturation. The theoretical predictions are confirmed by a series of experiments in which the pdfs at the output of the linear amplification scheme and saturated semiconductor optical amplifier are measured for an optical pulse of /spl sim/ 70-ps duration.