论文标题
通过随机速率方程式描述的增益转换激光器中光相方差的差异
Divergence of the variance of the optical phase in gain-switched semiconductor lasers described by stochastic rate equations
论文作者
论文摘要
在本文中，我们报告了对增益切换的单模半导体激光器中相扩散的理论研究。我们为电场使用随机速率方程来分析增益开关激光的相统计。它们的使用避免使用光子数率和光相数较小时使用的速率方程式获得的不稳定性。但是，我们表明，与字段方程集成时会出现一个新问题：光相的方差变得不同。由于先前的不稳定性，因此无法通过对光子数和光相的常用方程的数值整合来观察这种差异。相位方差的差异意味着随着积分时间步长的减少，该数量未达到固定值。我们得到的是，随着积分时间步长的减小，相位方差也会增加，即使对于微小的步骤也没有饱和行为的迹象。我们通过与二维布朗运动的问题进行比喻来解释差异。差异似乎并不奇怪，因为在1940年，保罗·莱维（PaulLèvy）已经证明，二维布朗运动中极角的方差是一个不同的数量。我们的结果表明，光子数和相位的随机速率方程不适合描述光子数小时的相统计。对电场的随机速率方程的模拟与Lèvy的结果一致，但给出了非物理结果，因为可以测量的数量获得无限的值。
In this paper, we report a theoretical study of the phase diffusion in a gain-switched single-mode semiconductor laser. We use stochastic rate equations for the electrical field to analyze the phase statistics of the gain-switched laser. Their use avoid the instabilities obtained with rate equations for photon number and optical phase when the photon number is small. However we show that a new problem appears when integrating with the field equations: the variance of the optical phase becomes divergent. This divergence can not be observed with the numerical integration of the commonly used equations for photon number and optical phase because of the previous instabilities. The divergence of the phase variance means that this quantity does not reach a fixed value as the integration time step is decreased. We obtain that the phase variance increases as the integration time step decreases with no sign of saturation behaviour even for tiny steps. We explain the divergence by making the analogy of our problem with the 2-dimensional Brownian motion. The fact that the divergence appears is not surprising because already in 1940 Paul Lèvy demonstrated that the variance of the polar angle in a 2-dimensional Brownian motion is a divergent quantity. Our results show that stochastic rate equations for photon number and phase are not appropriated for describing the phase statistics when the photon number is small. Simulation of the stochastic rate equations for the electrical field are consistent with Lèvy's results but gives unphysical results since an infinite value is obtained for a quantity that can be measured.
