Upper bound for the time derivative of entropy for a dynamical system driven by coloured cross-correlated white noises

It is shown how the cross-correlation time and strength of coloured cross-correlated white noises can set an upper bound for the time derivative of entropy in a nonequilibrium system. The value of upper bound can be calculated directly based on the Schwartz inequality principle and the Fokker--Planck equation of the dynamical system driven by coloured cross-correlated white noises. The present calculations can be used to interpret the interplay of the dissipative constant and cross-correlation time and strength of coloured cross-correlated white noises on the upper bound.     

Statistical properties of a single-mode laser driven by additive and multiplicative coloured noises with a coloured cross-correlation for different correlation times

We study a system for a single-mode laser driven by additive and multiplicative coloured noises with a coloured cross-correlation. The analytical expression of the stationary intensity distribution (SID) for the laser is derived in the case of three different correlation times. The influences of each stochastic parameter on the SID are discussed, the effects of the noise "colour" and the cross-correlation of noises on the mean intensity , the variance, λ_2(0), and the skewness,λ_3(0) of the single-mode laser are investigated. We find that there are colourful phase transitions for the SID above a threshold, and re-entrant transitions induced by the "colour" of the additive noises. Further research of the curves of -τ_i, λ_2(0)-τ_i, and λ_3(0)-τ_i (i=1,2,3) (where τ_i is the correlation time) shows that the output intensity not only increases with the additive noise correlation time τ_2 and the cross-correlation time τ_3, but also the quality of the output of laser beams is optimized.     

Effects of signal modulation and coloured cross-correlation of coloured noises on the diffusion of a harmonic oscillator

The diffusion in a harmonic oscillator driven by coloured noises $\xi(t)$ and $\eta(t)$ with coloured cross-correlation in which one of the noises is modulated by a biased periodic signal is investigated. The exact expression of diffusion coefficient d as a function of noise parameter, signal parameter, and oscillator frequency is derived. The findings in this paper are as follows. 1) The curves of d versus noise intensity D and d versus noises cross-correlation time $\tau_3$ exist as two different phases. The transition between the two phases arises from the change of the cross-correlation coefficient $\lambda$ of the two Ornstein--Uhlenbeck (O-U) noises. 2) Changing the value of $\tau_3$, the curves of d versus Q, the intensity of colored noise that is modulated by the signal, can transform from a phase having a minimum to a monotonic phase. 3) Changing the value of signal amplitude A, d versus Q curves can transform from a phase having a minimum to a monotonic phase. The above-mentioned results demonstrate that a like noise-induced transition appears in the model.     

Time dependence of entropy flux and entropy production for a dynamical system driven by noises with coloured cross-correlation

This paper shows the Fokker--Planck equation of a dynamical system driven by coloured cross-correlated white noises in the absence and presence of a small external force. Based on the Fokker--Planck equation and the definition of Shannon's information entropy, the time dependence of entropy flux and entropy production can be calculated. The present results can be used to explain the extremal behaviour of time dependence of entropy flux and entropy production in view of the dissipative parameter γ of the system, coloured cross-correlation time \tau and coloured cross-correlation strength \la.     

Stochastic resonance in a single-mode laser driven by frequency modulated signal and coloured noises

By adding frequency modulated signals to the intensity equation of gain--noise model of the single-mode laser driven by two coloured noises which are correlated, this paper uses the linear approximation method to calculate the power spectrum and signal-to-noise ratio (SNR) of the laser intensity. The results show that the SNR appears typical stochastic resonance with the variation of intensity of the pump noise and quantum noise. As the amplitude of a modulated signal has effects on the SNR, it shows suppression, monotone increasing, stochastic resonance, and multiple stochastic resonance with the variation of the frequency of a carrier signal and modulated signal.     

The statistical fluctuation of a single-mode laser system driven by coloured pump noise with signal modulation and the quantum noise with cross-correlation between its real and imaginary parts

Using the linear approximation method, this paper studies the statistical property of a single-mode laser driven by both coloured pump noise with signal modulation and the quantum noise with cross-correlation between its real and imaginary parts, and calculates the steady-state mean normalized intensity fluctuation and intensity correlation time. It analyses the influences of the modulation signal, the net gain coefficient, the noise and its correlation form on the statistical fluctuation of the laser system respectively. It is found that the coloured pump noise modulated by the signal has a great suppressing action on the statistical fluctuation of the laser system; the pump noise self-correlation time and the specific frequency of modulation signal have the result that the statistical fluctuation tends to zero. Furthermore, the `colour' correlation of pump noise has much influences on the statistical fluctuation of the laser system. Increasing the intensity of pump noise will augment the statistical fluctuation of the laser system, but the intensity of quantum noise and the coefficient of cross-correlation between its real and imaginary parts have less influence on the statistical fluctuation of the laser system. Therefore, from the conclusions of this paper the statistical property can be known and a theoretical basis for steady operation and output of the laser system can be provided.     

Intensity correlation time of a single-mode laser driven by two coloured noises with coloured cross-correlation with direct signal modulation

In this paper, the intensity correlation time T is studied for the gain-noise model of a single-mode laser driven by coloured pump noise and coloured quantum noise with coloured cross-correlation with a direct signal modulation. By using the linear approximation method, it is found that when the pump noise is modulated directly by a signal, the effects of the cross-correlation between the pump noise and the quantum noise will disappear. In addition, there exists a maximum (i.e. resonance) in the curve of the intensity correlation time $T$ versus the pump noise self-correlation time \tau¹. Furthermore, when \tau¹\le\tau², the intensity correlation time T increases monotonically with the increase of $D$ and decreases monotonically with the increase of Q, but when \tau¹>\tau², the intensity correlation time T increases monotonically with the increase of Q and decreases monotonically with the increase of D.     

Moments of the intensity of a single-mode laser driven by coloured pump noises with a cross-correlation between the real and imaginary parts of the quantum noise

By means of the unified coloured noise approximation and phase lock, we study in this paper the stationary intensity distribution of the single-mode laser cubic model driven by coloured pump noise with cross-correlation between the real and imaginary parts of the quantum noise. We present a thorough discussion of how the cross-correlation λ_q between the real and imaginary parts of the quantum noise and the self-correlation time τ of the pump noise determine the behaviours of the mean intensity 〈I〉 and variance λ_2 (0) for both below and above the threshold operation, and many new phenomena are discovered. When the laser is operated above the threshold, increasing the cross-correlation intensity λ_q makes the 〈I 〉－τ curves exhibit a "re-entrant phase transition". Whether the laser is above or below the threshold, the whole 〈I 〉－τ curve moves down as λ_q increases; however, when λ_q=1 (perfect cross-correlation), the curve abruptly runs up. A nonzero value of cross-correlation time τ (τ≠0) leads to the entire λ_2(0)－p′ curve being suppressed (here p′ is the pump noise intensity). This indicates the increasing precision of the laser beam due to the existence of the self-correlation time τ. The behaviour of the λ_2 (0)－p′ curve as a function of λ_q is similar to that of the 〈I 〉－τ curve against λ_q: that is, whether the laser is above or below the threshold, the λ_2 (0)－p′ curve moves up as λ_q increases; however, when λ_q=1, the curve suddenly moves down. Through the study in this paper, we can find a way to optimize for output laser intensity.τ     

Associated relaxation time and intensity correlation function of a bistable system driven by cross-correlation additive and multiplicative coloured noise sources

The associated relaxation time and the intensity correlation function of a bistable system driven by an additive and a multiplicative coloured noise with coloured cross-correlation are investigated. Using the Novikov theorem and the projection operator method, the analytic expressions of the stationary probability distribution P_st(x), the relaxation time T_c, and the normalized correlation function C(s) of the system are obtained. The effects of the noise intensity, the cross-correlation strength λ and the cross-correlation time τ are discussed. By numerical computation, it is found that the cross-correlation strength |λ| and the quantum noise intensity D decrease the relaxation of the system from unstable points. The cross-correlation time τ delays relaxation of the system from unstable points. The cross-correlation strength λ and the cross-correlation time τ can alter the effects of the pump noise intensity Q. Thus, the relaxation time T_c is a stochastic resonant phenomenon, and distribution curves exhibit a single-maximum structure.     

Stochastic resonance in a bistable system with coloured correlation between additive and multiplicative noise

The phenomenon of stochastic resonance (SR) in a bistable nonlinear system with coupling between additive and multiplicative noises is investigated when the correlation between two noise terms is coloured. It is found that the signal-to-noise ratio (SNR) of the system is affected not only by the coupling strength λ between two noise terms, but also by the noise correlation time τ. The SNR is changed from a single peak, to two peaks with a dip, and then to a monotonically decreasing function with noise strength. The dependence of the SR on the initial conditions is entirely caused by the coupling strength λ between two noise terms.     

Stochastic resonance in an asymmetric bistable system driven by coloured correlated multiplicative and additive noise

This paper investigates the stochastic resonance （SR） phenomenon in an asymmetric system with coupling between multiplicative and additive noise when the coupling between two noise terms is coloured. The approximate expression of signal-to-noise ratio has been obtained by applying the two-state theory and SR exhibits in the bistable system. Moreover, the potential asymmetry r and cross-correlation strength λ can weaken the SR phenomenon, while the cross-correlation time r can strengthen the SR phenomenon.     

单模激光系统随机共振的模拟研究

用数值模拟方法对具有调制信号的单模激光系统进行了研究, 分析了在阈值附近和远离阈值两种工作方式下的输出信噪比、输出信号功率、 背景噪声功率,并与线性化近似结果进行了对比. 研究发现,激光在阈值附近工作时,输出信噪比会随着量子 噪声的增大出现随机共振现象,这一现象不要求泵噪声与量子噪 声存在关联,但噪声之间的关联会进一步增大信噪比;在远离阈值时, 只有噪声间存在关联时才会发生随机共振现象.研究还发现噪声也 会影响输出信号的功率.     

Intensity Correlation Function of a Single-Mode Laser Driven by Colored Cross-Correlation Noises in Modulation of a Bias Signal

By using the linear approximation method, the intensity correlation function is calculated for a single-mode laser modulated by a bias signal and driven by colored pump and quantum noises with colored cross-correlation. We found that, when the correlation time between the two noises is very short, the behavior of the intensity correlation function versus the time, in addition to decreasing monotonously, also exhibits several cases, such as one maximum, one minimum, and two extrema. When the correlation time between the two noises is very long, the behavior of the intensity correlation function exhibits oscillation and the envelope is similar to the case of short cross-correlation time.     

Stimulated Raman spectra generated by ps-pulses in a single-mode fiber

Efficient frequency conversion of 5 ps-pulses from a cavity dumped Rh6G-dye laser by stimulated-Raman-scattering (SRS) into the first and second Stokes order has been achieved. The simulateneously produced asymmetric spectrical broadening of the pump pulses leads to double peaks in the Raman spectrum. By means of spectrally resolved cross-correlation measurements with 2 ps pulses from a second synchronized dye laser, the relative arrival time of the various spectral components is measured. Furthermore it is shown that the effect of a reduction of the pump peak power by SPM is partially compensated by the temporal broadening of the pump pulses leading to a longer interaction length for SRS.     

噪声交叉关联强度的时间周期调制对线性过阻尼系统的随机共振的影响

针对由加性、乘性噪声和周期信号共同作用的线性过阻尼系统, 在噪声交叉关联强度受到时间周期调制的情况下,利用随机平均法推导了系统响应的信噪比的解析表达式. 研究发现这类系统比噪声间互不相关或噪声交叉关联强度为常数的线性系统具有更丰富的动力学特性, 系统响应的信噪比随交叉关联调制频率的变化出现周期振荡型随机共振, 噪声的交叉关联参数导致随机共振现象的多样化.噪声交叉关联强度的时间周期调制的引入有利于提高对微弱周期信号检测的灵敏度和实现对周期信号的频率估计. 关键词： 随机共振 周期振荡型共振 噪声交叉关联强度 信噪比     

Stochastic resonance in single-mode semiconductor lasers

Owing to the considerable virtues of semiconductor lasers for applications, they have become the main optical source for fiber communication systems recently. The behavior of stochastic resonance (SR) in direct-modulated semiconductor laser systems is investigated in this article. Considering the carrier and photon noises and the cross-correlation between the two noises, the power spectrum of the photon density and the signal-to-noise ratio (SNR) of the modulated laser system were calculated using the linear approximation method. We found that the SR always appears in the dependence of the SNR upon the bias current density, and is strongly affected by the cross-correlation coefficient of the carrier and photon noises, the frequency of modulation signal and the photon lifetime in the laser cavity. Hence, it is promising to use the SR mechanism to enhance the SNR of direct-modulated semiconductor laser systems and improve the quality of optical communication.     

Stochastic resonance for signal-modulated pump noise in a single-mode laser

By adopting the gain-noise model of the single-mode laser in which with bias and periodical signals serve as inputs, combining with the effect of coloured pump noise, we use the linear approximation method to calculate the power spectrum and signal-to-noise ratio (SNR) of the laser intensity under the condition of pump noise and quantum noise cross-related in the form of δ function. It is found that with the change of pump noise correlation time, both SNR and the output power will occur stochastic resonance (SR). If the bias signal α is very small, changing the intensities of pump noise and quantum noise respectively does not lead to the appearance of SR in the SNR; while α increases to a certain number, SR appears.     

Influences of quantum noises on direct-modulated properties of 1.3-μm InGaAsP/InP laser diodes

Due to the zero dispersion point at 1.3μm in optical fibres, 1.3-μm InGaAsP/InP laser diodes have become main light sources in fibre communication systems recently. Influences of quantum noises on direct-modulated properties of single-mode 1.3-μm InGaAsP/InP laser diodes are investigated in this article. Considering the carrier and photon noises and the cross-correlation between the two noises, the power spectrum of the photon density and the signal-to-noise ratio (SNR) of the direct-modulated single-mode laser system are calculated using the linear approximation method. We find that the stochastic resonance (SR) always appears in the dependence of the SNR on the bias current density, and is strongly affected by the cross-correlation coefficient between the carrier and photon noises, the frequency of modulation signal, and the photon lifetime in the laser cavity. Hence, it is promising to use the SR mechanism to enhance the SNR of direct-modulated InGaAsP/InP laser diodes and improve the quality of optical fibre communication systems.     

Intensity Correlation Function of a Single-Mode Laser Driven by Two Colored Noises with Colored Cross-Correlation

By using the linear approximation method, the intensity correlation function and the intensity correlation time are calculated in a gain-noise model of a single-mode laser driven by colored cross-correlated pump noise and quantum noise, each of which is colored. We detect that, when the cross-correlation between both noises is negative, the behavior of the intensity correlation function C(t) versus time t, in addition to decreasing monotonously, also exhibits several other cases, such as one maximum, one minimum, and two extrema (one maximum and one minimum), i.e., some parameters of the noises can greatly change the dependence of the intensity correlation function upon time. Moreover, we find that there is a minimum Tmin in the curve of the intensity correlation time versus the pump noise intensity, and the depth and position of Train strongly depend on the quantum noise self-correlation time T2 and cross-correlation time T3.     

Improved approach for target detection and recognition on optoelectronic hybrid joint transform correlator

A conventional joint transform correlator (JTC) often generates large correlation sidelobes as well as a large correlation peak width, strong zero-order peak, and low diffraction efficiency in target detection and recognition, which make the detection ability of JTC lower. To conquer these difficulties, firstly, a joint power spectrum (JPS) subtraction technique was proposed in Fourier plane, where power spectrum of reference image and power spectrum of object image are subtracted from the JPS before inverse Fourier-transform operation, it is evident that the improved JPS removes the zero-order term. Secondly, a fringe-adjusted filter (FAF) was presented to restrain sidelobes and noises. The revised JPS is multiplied by a FAF before the inverse Fourier-transform operation to obtain the cross-correlation peak. Computer simulations showed the improved method can markedly eliminate zero-order diffraction and effectively control the sidelobes and noises compared to traditional JTC, and then enhance the detection ability for JTC. Experimental results presented the sharp correlation peak and also demonstrated this approach effectiveness.