Saturation Laser Model Research Articles

Analytical equivalent transformation method for nonlinear stochastic dynamics with multiple noises in high dimensions

In many practical applications, stochastic problems are high dimensionality and multiple noises. The study focuses on developing an analytical approach for processing multi-noise systems in high dimensions, which overcomes the challenge solving stochastic dynamics problems. Applying an analytical equivalent transformation and introducing a new partial differentiation, the author demonstrate a convenient method for obtaining the Fokker–Planck equation corresponding to Langevin dynamics equations for high-dimensional and multi-noise systems. This methodology is applied to analyze the cubic laser model subjected to white noises and the saturation laser model subjected to bounded sine Wiener noise. The results verify the correction, efficiency, and superiority of the method of processing multiple noises in high dimensions. This approach provides a novel framework for solving other practical dynamics problems for high-dimensional and multi-noise systems.

STATISTICAL FLUCTUATION IN A SATURATION LASER MODEL WITH CROSS-CORRELATIONS BETWEEN THE REAL AND IMAGINARY PARTS OF QUANTUM NOISE

We study the effects of cross-correlations between the real and imaginary parts of quantum noise on the intensity fluctuation of a saturation laser model. By virtue of the locked phase method,we derived an approximate Fokker–Planck equation and analytic expressions of the stationary probability distribution function (SPD) of the laser system. Based on the SPD, the mean, the normalized variance, and the normalized skewness of the steady-state laser intensity are calculated numerically. The results indicate that the correlation strength of the cross-correlations between the real and imaginary parts of quantum noise increases the intensity fluctuations.

STATISTICAL PROPERTIES OF INTENSITY FLUCTUATION OF SATURATION LASER MODEL DRIVEN BY CROSS-CORRELATED ADDITIVE AND MULTIPLICATIVE NOISES

Statistical properties of the intensity fluctuation of a saturation laser model driven by cross-correlation additive and multiplicative noises are investigated. Using the Novikov theorem and the projection operator method, we obtain the analytic expressions of the stationary probability distribution Pst(I), the relaxation time Tc, and the normalized variance λ2(0) of the system. By numerical computation, we discussed the effects of the cross-correlation strength λ, the cross-correlation time τ, the quantum noise intensity D, and the pump noise intensity Q for the fluctuation of the laser intensity. Above the threshold, λ weakens the stationary probability distribution, speeds up the startup velocity of the laser system from start status to steady work, and attenuates the stability of laser intensity output; however, τ strengthens the stationary probability distribution and strengths the stability of laser intensity output; when λ < 0, τ speeds up the startup; on the contrast, when λ > 0, τ slows down the startup. D and Q make the relaxation time exhibit extremum structure, that is, the startup time possesses the least values. At the threshold, τ cannot generate the effects for the saturation laser system, λ expedites the startup velocity and weakens the stability of laser intensity output. Below threshold, the effects of λ and τ not only relate to λ and τ, but also relate to other parameters of the system.

Stationary Properties and Stochastic Resonance for a Saturation Laser Model with Cross-correlation Between Quantum Noise Terms

The stationary properties of a saturation laser model with cross-correlation between the real and imaginary parts of the quantum noise are investigated theoretically. Using the Novikov theorem and the Sargent technique, we obtain the analytic expressions of the stationary probability density distribution, the mean, the variance and the skewness of the saturation laser model. The cross-correlation coefficient λ and other parameters can make the stationary probability density distribution P st (I) generate interesting two-extrema structure, one-extremum structure, or no-extremum structure. It is clearly found that a first- order-like-transition is induced by the coupling strength |λ| of the complex quantum noise terms in the saturation laser model. When the laser system is operated above the threshold, the mean 〈I〉 becomes larger and the output of the laser intensity increases; however the coupling strength |λ| attenuates the output of the laser intensity. When the laser is operated near and below the threshold, the mean 〈I〉 becomes smaller, the output of the laser intensity decreases, and |λ| still attenuates the output of the laser intensity. When a periodic signal is added to a saturation laser model with cross-correlation between quantum noise terms, the interesting stochastic resonance phenomena occur at λ=0. The noise intensity Q decreases the values of the resonance peak, however, the amplitude of the periodic signal B enhances the values of the resonance peak.

Effects of cross-correlation between the real and imaginary parts of the quantum noise in a two-mode saturation laser system

A two-mode saturation laser model with cross-correlation between the real and imaginary parts of the quantum noise is considered. The laser intensity Langevin equation and corresponding Fokker–Planck equation are derived by the phase-locking method. The effects of the cross-correlation strength λ between the real and imaginary parts of quantum noise and the cavity decay constant K on the steady-state intensity distribution Q ( I 1 , I 2 ) , the mean light intensity 〈 I 〉 , the normalization autocorrelation λ 11 ( 0 ) and cross correlation λ 12 ( 0 ) are studied by numerical calculation. The results show that as λ increases the Q s ( I 1 , I 2 ) show two extrema, and λ almost does not affect the 〈 I 〉 , λ 11 ( 0 ) and λ 12 ( 0 ) when the laser system is operated far above threshold. Nevertheless, when the laser system is operated at and below threshold, λ makes the curves of Q s ( I 1 , I 2 ) have the higher peak and drop faster. Furthermore, it enhances the deviation of λ 11 ( 0 ) and λ 12 ( 0 ) and lessens the mean light intensity 〈 I 〉 when the laser system is operated at and below threshold.

Intensity correlation function and relaxation time of a single-mode laser driven by quadratic pump noise and quantum noise

We study the effects of correlations between real and imaginary parts of pump noise on the correlation function and relaxation time of laser intensity in a saturation laser model driven by quadratic pump noise and quantum noise. Using the phase–locking method and the projection–operator method, we derived the stationary probability density, the relaxation time Tc and the intensity correlation function C(s) of the laser intensity, and then performing numerical computations, we find the following: (i) Tc monotonously increases as the correlation strength between the real and imaginary parts of quadratic pump noise λp increases and (ii) C(s) decreases exponentially as decay time s increases and C(s) increases as λp increases. The correlations between the real and imaginary parts of the quadratic pump noise slow down the rate of fluctuation decay of the laser intensity.

Effects of correlations between the real and imaginary parts of quantum noise on intensity fluctuation for a saturation laser model

This paper studies the effects of cross-correlations between the real and imaginary parts of quantum noise on the laser intensity in a saturation laser model. It derives the analytic expressions of the intensity correlation function C(τ) and the associated relaxation time T(C) in the case of a stable locked phase resulting from the cross-correlation λq between the real and imaginary parts of quantum noise. Based on numerical computations it finds that the presence of cross correlations between the real and imaginary parts of quantum noise slow down the decay of intensity fluctuation, i.e., it causes the increase of intensity fluctuation.

Intensity correlation function and associated relaxation time of a saturation laser model with cross-correlated noises for the case of nonzero correlation time

Considering a saturation laser model with cross-correlated additive and multiplicative noises, we derive the stationary probability distribution function Pst(I), the stationary intensity correlation function C(s) and the associated relaxation time Tc by virtute of the Novikov theorem and Stratonovich's decoupling ansatz. Based on numerical computation, we discuss the effects of the correlation strength λ, the cross-correlation time τ, and the noises intensity. The correlation strength λ attenuates the probability distribution and increases the intensity fluctuation of the laser system. The correlation strength λ and the noise intensity D and Q speed up the relaxation of the system from an arbitrary state to the steady state and enhance the stability of the laser intensity output. For the case of negative correlation (λ < 0), the cross-correlation τ enhances the probability distribution, decreases the intensity fluctuation, slows down the relaxation of the system from an arbitrary state to the steady state, and decreases the stability of the laser intensity output; by contrast, the effects of the cross-correlation time τ on Pst(x), C(s), and Tc are entirely opposite for (λ > 0).

Statistical fluctuations in a saturation laser model with correlated noises

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.

EFFECTS OF CORRELATED NOISES IN A SATURATION LASER MODEL

The effects of correlations between additive and multiplicative noises in a saturation laser model are investigated. The 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. Our results indicate that: (i) For the laser being operated above threshold, the correlation strength λ reduces the intensity fluctuation; (ii) For the laser being operated near threshold and below threshold, the correlation strength λ enhances the intensity fluctuation.

Intensity Correlation Function and Associated Relaxation Timeof a Saturation Laser Model with Correlated Noises

We investigate the intensity correlation function C(s) and itsassociated relaxation time Tc for a saturation model of single-modelaser with correlated noises. The expressions of C(s) and Tc arederived by means of the projection operator method, and effects ofcorrelations between an additive noise and a multiplicative noise arediscussed by numerical calculation. Based on the calculated results, itis found that the correlation strength λ between the additivenoise and the multiplicative noise can enhance the fluctuation decay ofthe laser intensity.

Modelling the effects of backscattering in a bidirectional ring laser with gain saturation

Abstract The statistical properties of an inhomogeneously broadened two-mode ring laser in the presence of backscattering are investigated through a more accurate gain saturation laser model. The laser intensity auto-correlation time T c and effective eigenvalue λ eff are calculated for the in-phase backcoupling. Compared with the experimental measurements, excellent agreement is obtained in both T c and λ eff using the same parameters. However, only one of the curves can be fitted by the third-order laser theory. It is suggested that by fitting the measured data of both T c and λ eff simultaneously, the cavity decay constant and the backscattering coefficient can be determined experimentally.

Saturation effects on intensity fluctuations of a laser with multiplicative white noise.

The saturation effects in a single-mode laser with multiplicative white noise are investigated through a full saturation laser model. The corresponding Fokker-Planck equation is transformed to a radial Schr\odinger equation and the intensity correlation time and effective eigenvalues are calculated. Compared with experimental measurements, excellent agreement is obtained and the cavity decay rate is also determined. Relatively large errors occur in the third-order theory and the measured data.