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.     

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.     

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.     

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.     

Stochastic Resonance in Linear Region of a Single-Mode Laser： Effects of Amplitude Modulation of Signal

A single-mode laser noise model driven by quadratic colored pump noise and amplitude modulation signal is proposed. The real and imaginary parts of the pump noise are assumed to be cross-correlation. The effect of cross-correlation of noise and amplitude modulation of signal on laser statistical properties is studied by using the linearized approximation. The analytic expression of signal-to-noise ratio (SNR) is calculated. It is found that the phenomena of stochastic resonance (SR) respectively exist in the curves of the SNR versus the noise cross-correlation coefficient λ and the SNR versus the pump parameter a, as well as the SNR versus the signal frequency ω in our model. It is shown that there are three different typies of SR in the model: the conventional form of SR, the SR in the broad sense, and the bona fide SR.     

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.     

Stochastic resonance in a single-mode laser driven by quadratic pump noise and amplitude-modulated signal

This paper investigates the phenomenon of stochastic resonance in a single-mode laser driven by quadratic pump noise and amplitude-modulated signal. A new linear approximation approach is advanced to calculate the signal-to-noise ratio. In the linear approximation only the drift term is linearized, the multiplicative noise term is unchangeable. It is found that there appears not only the standard form of stochastic resonance but also the broad sense of stochastic resonance, especially stochastic multiresonance appears in the curve of signal-to-noise ratio as a function of coupling strength λ between the real and imaginary parts of the pump noise.     

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.     

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 Noise on Time Evolution of Intensity Correlation Function

Using the linear approximation method, we have studied how the correlation function C（t） of the laser intensity changes with time in the loss-noise model of the single-mode laser driven by the colored pump noise with signal modulation and the quantum noise with cross-correlation between the real and imaginary parts. We have found that when the pump noise self-correlation time T changes, （i） in the case of r 〈〈 1, the C（t） vs. t curve experiences a changing process from the monotonous descending to monotonous rise, and finally to the appearance of a maximum; （ii） in the case of r 〉〉 1, the curve only exhibits periodically surging with descending envelope. When r 〈〈 i and T does not change, with the increase of the pump noise intensity P, the curve experiences a repeated changing process, that is, from the monotonous descending to the appearance of a maximum, then to monotonous rise, and finally to the appearance of a maximum again. With the increase of the quantum noise intensity O,, the curve experiences a changing process from the monotonous rise to the appearance of a maximum, and finally to the monotonous descending. The increase of the quantum noise with cross-correlation between the real and imaginary parts will lead to the fall of the whole curve, but not affect the form of the time evolution of C（t）.     

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.     

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.     

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 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 driven by time-modulated correlated coloured noise sources in a single-mode laser

This paper investigates the phenomenon of stochastic resonance in a single-mode laser driven by time-modulated correlated coloured noise sources. The power spectrum and signal-to-noise ratio R of the laser intensity are calculated by the linear approximation. The effects caused by noise self-correlation time τ₁, τ₂ and cross-correlated time τ₃ for stochastic resonance are analysed in two ways: τ₁, τ₂ and τ₃ are taken to be the independent variables and the parameters respectively. The effects of the gain coefficient Γ and loss coefficient K on the stochastic resonance are also discussed. It is found that besides the presence of the standard form and the broad sense of stochastic resonance, the number of extrema in the curve of R versus K is reduced with the increase of the gain coefficient Γ.     

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。     

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     

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 ${C(\tau)}$ and the associated relaxation time ${T({C})}$ in the case of a stable locked phase resulting from the cross-correlation ${\lambda_q}$ 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.     

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.