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.     

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.     

加性信号调制下指数形式关联噪声驱动的单模激光的光强关联时间

采用线性化近似,计算了加性信号调制下的由具有指数关联的两白噪声驱动的单模激光增益模型的光强关联时间.发现两噪声间关联程度对光强关联时间随噪声强度的变化曲线有很大的影响,两噪声间关联程度取不同值时,光强关联时间随噪声强度的变化曲线中将出现极大值(即出现共振) 或极小值(即出现抑制) .     

Effect on Intensity Correlation Time by Input Signal in a Single-Mode Laser with Bias Signal Modulation

The effect on intensity correlation time T by input signal is studied for gain-noise model of a single-mode laser driven by colored pump noise and colored quantum noise with colored cross-correlation with a bias signal modulation in this paper. By using the linear approximation method, we detect that there exists maximum （i.e., resonance） in the curve of the intensity correlation time T upon bias current io when the noise correlation coefficient λ is positive; and there exists minimum （i.e., suppression） in the T-io curve when λ is negative. And whenλ is zero, T increases monotonously with increasing io. Furthermore, the curve of T upon the signal frequency Ω is also studied. Our study shows that no matter what the value of λ is, there exists minimum （i.e., suppression） in the T-Ω curve.     

Effect on Intensity Correlation Time by Input Signal in a Single-Mode Laser with Bias Signal Modulation

The effect on intensity correlation time T by input signal is studied for gain-noise model of a single-mode laser driven by colored pump noise and colored quantum noise with colored cross-correlation with a bias signal modulation in this paper. By using the linear approximation method, we detect that there exists maximum (i.e., resonance) in the curve of the intensity correlation time T upon bias-current i0 when the noise correlation coefficient λ is positive; and there exists minimum (i.e., suppression) in the T-i0 curve when λ is negative. And when λ is zero, T increases monotonously with increasing i0. Furthermore, the curve of T upon the signal frequency Ω is also studied. Our study shows that no matter what the value ofλ is, there exists minimum (i.e., suppression) in the T-Ω curve.     

Stationary Intensity Distribution of Single-Mode Laser Driven by Additive and Multiplicative Colored Noises with Colored Cross-Correlation

Applying the approximate Fokker-Planck equation we derived, we obtain the analytic expression of thestationary laser intensity distribution Pst(Ⅰ) by studying the single-mode laser cubic model subject to colored cross-correlation additive and multiplicative noise, each of which is colored. Based on it, we discuss the effects on the stationarylaser intensity distribution Pst(Ⅰ) by cross-correlation between noises and “color“ of noises (non-Markovian effect) whenthe laser system is above the threshold. In detail, we analyze two cases: One is that the three correlation-times (i.e.the self-correlation and cross-correlation times of the additive and multiplicative noise) are chosen to be the same value(τ1 = τ2 = τ3 = τ). For this case, the effect of noise cross-correlation is investigated emphatically, and we detect thatonly when λ≠ 0 can the noise-induced transition occur in the Pst(Ⅰ) curve, and only when τ≠ 0 and λ≠ 0, can the“reentrant noise-induced transition“ occur. The other case is that the three correlation times are not the same value,τ1 ≠τ2 ≠τ3. For this case, we find that the noise-induced transition occurring in the Pst (Ⅰ) curve is entirely differentwhen the values of τ1, τ2, and τ3 are changed respectively. In particular, when τ2 (self-correlation time of additivenoise) is changing, the ratio of the two maximums of the Pst(Ⅰ) curve R exhibits an interesting phenomenon, “reentrantnoise-induced transition“, which demonstrates the effect of noise “color“ (non-Markovian effect).     

Intensity Correlation Time of a Single-Mode Laser Driven by Two Coloured Noises with Coloured Cross-Correlation with Bias Signal Modulation

The intensity correlation time T is studied by employing a gain-noise model of a single-mode laser driven by coloured pump noise τ1 and coloured quantum noise τ2 with coloured cross-correlation τ3 with a bias signal modulation. By using the linear approximation method, we detect that a maximum （i.e. resonance） exists in the curves of the intensity correlation time T versus the noise intensities D and Q when the noise correlation coefficient A is positive; and a minimum （i.e. suppression） exists in the T - D and T - Q curves when A is negative. When A is zero, T increases monotonously with increase of D and decreases monotonically with increase of Q. Furthermore, the curve of T versus the pump noise self-correlation time τ1 is also studied. Our study shows that, no matter what the value of A is, there exist one maximum and one minimum in the T - τ1 curve.     

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.     

Dynamical Properties of a Single-Mode Laser with Both Colored Noise and Injected Signal

The colored noise in a single-mode laser with injected signal is investigated by the method of two-dimensional unified colored noise approximation. The normalized correlation function λ₂(0), the intensity correlation time T_c and effective eigenvalue λ_eff are calculated. The effects of the injected signal I₀, colored noise strength P^ˊ and the noise correlation time τ on the dynamical properties of the single-mode laser are 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.     

关联噪声对单模激光动力学性质的影响

应用线性近似方法,计算了具有指数形式关联的两白噪声驱动下的单模激光光强的关联函数、功率谱及关联时间。根据计算结果讨论了噪声间的关联程度和噪声间互联时间对以上各量的影响,并和噪声间具有δ函数关联形式的情况进行了比较     

单模激光系统输入信号后的稳态平均光强相对涨落

研究了具有实虚部间关联的量子噪声和抽运噪声驱动的单模激光系统输入信号后的统计性质,采用线性化近似方法计算了系统的稳态平均光强相对涨落,分析了量子噪声实虚部间关联系数、量子噪声强度、抽运噪声强度、输入信号振幅和频率、净增益等对稳态平均光强相对涨落的影响,发现在量子噪声实虚部间弱关联、小噪声、远离阚值、信号振幅不大和频率较高的条件下激光场的统计涨落较小。     

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.     

Fluctuations of Single-Mode Laser Driven by Two Different Kinds of Colored Noise

A single-mode laser with coupling between additive and multiplicative noise terms is investigated when the multiplicative noise and the coupling between two noise terms are colored fluctuations with finite correlation times T1 and τ2. Combining the unified colored noise approximation (UCNA) and the functional analysis, the stationary probability distribution (SPD) and the variance of the laser intensity is derived. It is found that the colored nature of multiplicative noise and the coupling strength between two noise terms can affect both the structure and the height of the SPD, while the colored nature of the coupling between two noise terms can only affect the height of the SPD. The multiplicative noise can enhance the intensity fluctuations while the additive noise can reduce the fluctuations in a laser system. Numerical simulations are presented and consistent to the analytical results.     

周期脉冲序列调制色噪声作用下单模激光的光强关联时间

采用周期矩形脉冲信号直接调制色噪声作用下的单模激光增益模型,运用线性近似的方法计算得到了模型输出光强的自关联函数和关联时间(Tc),并讨论了光强关联时间随噪声强度和调制脉冲信号的变化关系.研究结果发现:噪声关联程度λ＜0时,光强关联时间Tc随噪声强度Q、D及脉冲信号的振幅A的变化曲线均出现了随机共振现象,系统的涨落达到最小,而在λ≥0时,Tc单调变化;在-1＜λ＜1范围内,Tc随噪声关联时间τ和信号的脉冲宽度θ的变化曲线也均出现了随机共振现象,且随λ的减小,共振现象越明显;Tc随信号周期T的变化却出现了抑制现象,λ越小,抑制作用越强.     

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.     

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 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 realand 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)-a0 and λ3(0)-a0, (here a0 is the net gain coefficient) go up as λq increases. However the entire curves of λ2(0)-a0 and λ3(0)-a0are abruptly suppressed when λq = 1, in similarity to phase transition of stationary intensity distribution.     

交叉关联程度受时间周期调制情况下的单模激光光强关联函数

采用线性化近似方法计算了单模激光系统的光强关联函数,讨论了光强关联函数随时间的演化关系,发现光强关联函数呈现周期振荡的现象.关联函数振荡的幅度和周期不受量子噪音强度、抽运噪音强度和交叉关联程度的影响,输入信号频率会改变关联函数振荡的周期和幅度,但不改变振荡的平衡位置,量子噪音强度会改变振荡极值的位置,四个参量都改变关联函数衰减的初始值,但关联函数对时间周期调制频率不敏感.     

Extinction Time of a Metapopulation Driven by Colored Correlated Noises

The simplified incidence function model which is driven by the colored correlated noises is employed to investigate the extinction time of a metapopulation perturbed by environments. The approximate Fokker-Planck Equation and the mean first passage time which denotes the extinction time （Tex） are obtained by virtue of the Novikov theorem and the Fox approach. After introducing a noise intensity ratio and a dimensionless parameter R = D /α （D and a are the multiplicative and additive colored noise intensities respectively）, and then performing numerical computations, the results indicate that： （i） The absolute value of correlation strength A and its correlation time τ3 play opposite roles on the Tex; （ii） For the case of 0 〈λ〈 1,α and its correlation time τ2 play opposite roles on the Tex in which R〉 1 is the best condition, and there is one-peak structure on the Tex - D plot; （iii） For the case of-1 〈 λ≤ 0, D and its correlation time τ1 play opposite roles on the Tex in which R 〈 1 is the best condition and there is one-peak structure on the Tex - τ2 plot.