信号调制色泵噪声和实虚部间关联量子噪声驱动下单模激光的随机共振现象

计算了受信号调制的色泵噪声和实虚部间关联的量子噪声驱动的单模激光损失模型的输出光强信噪比.发现信噪比R随泵噪声自关联时间τ、调制信号频率Ω和量子噪声实虚部间关联系数λq的变化均存在随机共振,这种现象扩展了“信噪比R对噪声强度的变化曲线具有极大值”的典型随机共振. 若以Ω为参数,当Ω增加时,Ｒ随τ的关系曲线经历了从同时出现共振和抑制到单峰共振,最后到单调上升的变化,呈现多种形式的随机共振．若以τ为参数,当τ增加时,Ｒ随Ω的关系曲线经历了从单调上升到同时出现共振和抑制,最后又到单调下降的变化过程．Ｒ随λq的关 关键词： 噪声 信噪比 随机共振     

Influence of the net gain on characteristic of stochastic resonance in a single-mode laser system

The phenomenon of stochastic resonance (SR) is found in a single-mode laser system driven by the colored pump noise with signal modulation and the quantum noise with cross-correlation between the real and imaginary parts. When the net gain a0 changes, it is found that, 1) the shape of the curve of the signal-tonoise ratio (SNR) versus the pump noise self-correlation time T exhibits a changing process of multiform SR, from single-peak SR to simultaneous existence of resonances and suppressions; 2) the curve of SNR versus signal frequency Ω experiences a complicated changing process from the monotonous descending to the simultaneous appearances of a maximum and a minimum, and finally to monotonous descending; 3)the curve of SNR versus cross-correlation coefficient between the real and imaginary parts of the quantum noise λq appears an acute single-peak SR. Therefore, the net gain a0 greatly influences the characteristic of SR of laser system.     

Time evolution of the intensity correlation function in a single-mode laser driven by both the coloured pump noise with signal modulation and the quantum noise with cross-correlation between the real and imaginary parts

Using the linear approximation, we have studied the time evolution of intensity correlation function C（t） in a single-mode laser driven by both the colored pump noise with signal modulation and the quantum noise with cross-correlation between its real and imaginary parts. In the case of the pump noise self-correlation time, we find that the time evolution of C（t） varies with modulation signal frequency Ω, amplitude B, and net gain ɑ0. (i) As the Ω increases, the time evolution of C（t） experiences a process changing from the monotonous descension to the descension with a flat appearing initially, and finally to the a form of damping oscillation; (ii) As the B increases, it experiences from monotonous descension to the appearance of a maximum; (iii) As the net gain ɑ0 increases, it experiences a process repeatedly changing from the monotonous descension to monotonous ascension, and to the appearance of a maximum, finally to monotonous descension again. However, in the case of, the time evolution of C（t） only exhibits a form of damping oscillation .     

单模激光系统中信噪比对净增益的随机共振

研究了受信号调制的色泵噪声和实虚部间关联的量子噪声驱动的单模激光系统的随机共振现象,发现信噪比随激光系统净增益系数存在随机共振.当泵噪声自关联时间和调制信号频率增加时,信噪比随激光系统净增益系数的变化曲线经历了从同时出现共振和抑制到单调上升的演化过程;当调制信号振幅、泵噪声强度和量子噪声强度、量子噪声实虚部间关联系数等变化时,该曲线一直同时出现共振和抑制,但共振峰和抑制谷有很大的变化.     

Influence of Signal and Noise on Statistical Fluctuation of Single-Mode Laser System

On the basis of calculating the steady-state mean normalized intensity fluctuation of a signal-mode laser system driven by both colored pump noise with signal modulation and the quantum noise with cross-correlation between its real and imaginary parts, we analyze the influence of modulation signal, noise, and its correlation form on the statistical fluctuation of the laser system. We have found that when the amplitude of modulation signal weakens and its frequency quickens, the statistical fluctuation will reduce rapidly. The statistical fluctuation of the laser system can be restrained by reducing the intensity of pump noise and quantum noise. Moreover, with prolonging of colored cross-correlation time, the statistical fluctuation of laser system experiences a repeated changing process, that is, from decreasing to augmenting,then to decreasing, and finally to augmenting again. With the decreasing of the value of cross-correlation coefficient, the statistical fluctuation will decrease too. When the cross-correlation form between the real part and imaginary part of quantum noise is zero correlation, the statistical fluctuation of laser system has a minimum. Compared with the influence of intensity of pump noise, the influence of intensity of quantum noise on the statistical fluctuation is smaller.     

Influence of Signal and Noise on Statistical Fluctuation of Single-Mode Laser System

On the basis of calculating the steady-state mean normalized intensity fluctuation of a signal-mode laser system driven by both colored pump noise with signal modulation and the quantum noise with cross-correlation between its real and imaginary parts, we analyze the influence of modulation signal, noise, and its correlation form on the statistical fluctuation of the laser system. We have found that when the amplitude of modulation signal weakens and its frequency quickens, the statistical fluctuation will reduce rapidly. The by reducing the intensity of pump noise and quantum noise. statistical fluctuation of the laser system can be restrained Moreover, with prolonging of colored cross-correlation time, the statistical fluctuation of laser system experiences a repeated changing process, that is, from decreasing to augmenting, then to decreasing, and finally to augmenting again. With the decreasing of the value of cross-correlation coe~cient, the statistical fluctuation will decrease too. When the cross-correlation form between the real part and imaginary part of quantum noise is zero correlation, the statistical fluctuation of laser system has a minimum. Compared with the influence of intensity of pump noise, the influence of intensity of quantum noise on the statistical fluctuation is smaller.     

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.     

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     

Study of the normalized intensity correlation function of a single-mode laser system with colored cross-correlated noises

A single-mode laser system with colored cross-correlated additive and multiplicative noise terms is considered. By the means of projection operator method, we study the effects of the cross-correlation time τ and the cross-correlation intensity λ between noises on the normalized intensity correlation function C(s). It is found that if λ＞ 0 (λ＜ 0), the normalized intensity correlation function C(s) increases (decreases) with increasing the cross-correlation time τ, and at large value of τ, the variation of the normalized intensity correlation function C(s) becomes small. With the increase of the net gain a0, C(s) exhibits a maximum when λ is larger. However, a minimum and a maximum appear on C(s) curves with the increase of a0 when λ becomes smaller and smaller.     

Stochastic Resonance in a Single-Mode Laser Driven by Pump Noise and Quantum Noise with Cross-Correlated Real and Imaginary Parts

We present an analytic investigation of the signal-to-noise ratio (SNR) by studying a signal modulated model of a single-mode laser system driven by pump noise and quantum noise with correlated real and imaginary parts,and find there is a maximum in the curve of the dependence of SNR upon the cross-correlation coefficient λq between the real part and the imaginary part, i.e., stochastic resonance appears in the SNR vs. λq curve. Moreover, when the SNR is at the maximum, the cross-correlation coefficient λq = O, which is coincidentally at the minimum of the mean normalized intensity fluctuation. The influences on stochastic resonance by the intensities of the pump and the quantum noise, the amplitude of the modulation signal, and the net gain of the laser are also studied. Furthermore, in order to ensure that the results obtained in this paper is reliable, the valid range for the linear approximation method is discussed.     

周期信号调制色泵噪音驱动下单模激光光强关联函数的时间演化特性

采用周期信号调制色泵噪音驱动的单模激光模型,运用线性化近似方法研究了单模激光系统光强关联函数C(t)随时间的演化关系,分析了调制信号的振幅B、频率Ω等对光强关联函数随时间演化的影响. 发现在泵噪音自关联时间τ>>1的情形下,随着调制信号频率Ω、振幅B的增加,C(t)随时间的演化为单调衰减;在τ>>1的情形下,随着调制信号频率Ω、振幅B的增加,C(t)随时间的演化均为周期性振荡衰减.     

输入信号和噪声对单模激光随机共振的影响

采用色抽运噪声和实虚部间关联的量子噪声驱动的单模激光损失模型,运用线性化近似方法计算了周期性信号加性输入时激光系统的输出光强信噪比,发现用信噪比与量子噪声实虚部间关联系数的关系曲线描述的随机共振现象.在抽运噪声自关联为短时关联情况下,当信号振幅增大和频率增快、抽运噪声色关联时间增大时,系统的随机共振加强;而噪声强度的增加会削弱系统的随机共振.在抽运噪声自关联为长时关联情况下,当信号振幅增大和量子噪声强度减弱时,系统的随机共振加强;而信号频率、抽运噪声强度、抽运噪声色关联时间的变化对系统随机共振的影响很小.     

Analyses of Valid Range for the Linear Approximation in a Single-Mode Laser

Using the linear approximation method, we calculated the steady-state mean normalized intensity fluctu-ation for a loss-noise model of a single-mode laser driven by a pump noise and a quantum noise, whose real part andimaginary part are cross-correlated. We analyzed the valid range for thelinear approximation method by studying theinfluences on the steady-state mean normalized intensity fluctuation by the cross-correlation coefficient, the intensities ofthe quantum and pump noise, the net gain, and the amplitude and frequency of the input signal, and we found that thevalid range becomes wider when the cross-correlation between the real and imaginary part of quantum noise is weaker,the noise intensities of quantum and pump are weaker, the laser system is far from the threshold and the signal hassmaller amplitude and higher frequency.     

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.     

Steady-State Analysis of a Single-Mode Laser Driven by Colored Pump Noise with Cross-Correlation Between Real and Imaginary Parts of Quantum Noise

Applying the method of the unified colored noise approximation and phase lock, we study in this paper the stationary intensity distribution of the single-mode laser driven by colored 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 λq between the real and imaginary parts of the quantum noise and the self-correlation time τ of the pump noise determine the behaviors of the stationary distribution Qst(I), the mean (I）, and the variance λ2(0) of the laser intensity. It is shown that cross-correlation intensity λq of the complex quantum noise can induce a first-order-like transition. When the pump noise is colored noise (τ≠0), improving the pump parameters monotonously will make the curves of Qst(I) exhibit reentrant phase transition. The fluctuations of laser intensity are strongly influenced by λq and τ when the laser is operated near or below threshold. Especially when τ≠0, the heights of the peaks of the curves of λ2(0)-α0 and α3(0)-α0, (here a0 is the net gain coefficient) go up as λq increases. However the entire curves λ2(0)-α0 and λ3(0)-α0 are abruptly suppressed when λq = 1, in similarity to phase transition of stationary intensity distribution.     

Steady-State Analysis of a Two-Mode Laser System with Cross-Correlation Between Real and Imaginary Parts of Quantum Noise

An inhomogeneously broadened two-mode laser system with cross-correlations between the real and imag- inary parts of quantum noise is considered. The Fokker-Planek equation of the system is derived by the phase-locking method. The steady-state probability distribution, the mean light intensity, the normalization autocorrelation function, and cross correlation function are calculated. The results indicate that： （i） The cross-correlation between the real and imaginary parts of quantum noise can cause the stationary probability distribution from one peak structure to two extrema structure when the laser system is operated above threshold; （ii） The cross-correlation between the real and imaginary parts of quantum noise enhance the light intensity fluctuation and decrease the laser output when the laser system is operated below or near threshold; （iii） The effect of the cross-correlation between the real and imaginary parts of quantum noise is very weak on the stationary properties when the laser system is operated far above threshold.     

Stochastic resonance in a gain--noise model of a single-mode laser driven by pump noise and quantum noise with cross-correlation between real and imaginary parts under direct signal modulation

Stochastic resonance (SR) is studied in a gain--noise model of a single-mode laser driven by a coloured pump noise and a quantum noise with cross-correlation between real and imaginary parts under a direct signal modulation. By using a linear approximation method, we find that the SR appears during the variation of signal-to-noise ratio (SNR) separately with the pump noise self-correlation time \tau , the noise correlation coefficient between the real part and the imaginary part of the quantum noise \lambda_q , the attenuation coefficient \gamma and the deterministic steady-state intensity I_0 . In addition, it is found that the SR can be characterized not only by the dependence of SNR on the noise variables of \tau and \lambda_q, but also by the dependence of SNR on the laser system variables of \gamma and I₀. Thus our investigation extends the characteristic quantity of SR proposed before.     

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. T3.     

白交叉关联色噪声驱动的线性振子的扩散

研究了由白交叉相关的两色噪声所驱动的线性振子的扩散,其中一个噪声受偏置的周期信号所调制.揭示了乘性色噪声的自关联时间τ1和加性色噪声的自关联时间τ2对d-Q,d-D及d-Ω曲线的影响,其中d,Q,D和Ω分别为振子的扩散系数、受信号调制的噪声强度、未受调制的噪声强度和信号的频率.发现d-Q和d-D曲线均存在一个极小值,而d-Ω曲线则在Ω=～Ω处呈现极大值,～Ω为振子频率.推导了使扩散系数d=0的条件.