论文标题
关于激光动作中的参数共振
On parametric resonance in the laser action
论文作者
论文摘要
我们考虑了固态激光器的自以为是的半经典麦克斯韦（Schrödinger）系统，该系统由麦克斯韦方程组成，耦合到$ n \ sim 10^{20} $Schrödinger方程，用于活性分子。该系统包含时间周期性抽水和较弱的耗散。我们介绍了相应的Poincaré地图$ P $，并考虑适当的固定状态$ y^0 $的差异$ dp（y^0）$。我们推测{\ it稳定激光动作}是由于{\ it parametric resonance}（pr）引起的，这意味着相应的乘数的最大绝对值大于一个。乘数定义为$ dp（y^0）$的特征值。公关使固定状态$ y^0 $高度不稳定，我们假设该不稳定维护{\ it IT相干激光辐射}。 我们证明，如果耗散足够小，频谱规范$ \，dp（y^0）$相对于单位圆$ |μ| = 1 $对称。麦克斯韦 - 布洛克系统获得了更详细的结果。我们通过连续近似计算相应的庞加莱地图$ p $。计算乘数的关键作用是通过二阶近似值的$ n $正术语的总和为总电流的总和。这一事实可以解释为所有活性分子中分子电流的{\ it同步，这与{\ it刺激发射}在激光作用中的作用相符。总和的计算取决于概率论点，这是我们方法的主要新颖性之一。其他主要新颖性是i）“ hopf表示”中的微分$ dp（y^0）$的计算，ii）差异的块结构，iii）通过对缓慢旋转平均的新估计值的“旋转波近似”的正当性。
We consider the selfconsistent semiclassical Maxwell--Schrödinger system for the solid state laser which consists of the Maxwell equations coupled to $N\sim 10^{20}$ Schrödinger equations for active molecules. The system contains time-periodic pumping and a weak dissipation. We introduce the corresponding Poincaré map $P$ and consider the differential $DP(Y^0)$ at suitable stationary state $Y^0$. We conjecture that the {\it stable laser action} is due to the {\it parametric resonance} (PR) which means that the maximal absolute value of the corresponding multipliers is greater than one. The multipliers are defined as eigenvalues of $DP(Y^0)$. The PR makes the stationary state $Y^0$ highly unstable, and we suppose that this instability maintains the {\it coherent laser radiation}. We prove that the spectrum Spec$\,DP(Y^0)$ is approximately symmetric with respect to the unit circle $|μ|=1$ if the dissipation is sufficiently small. More detailed results are obtained for the Maxwell--Bloch system. We calculate the corresponding Poincaré map $P$ by successive approximations. The key role in calculation of the multipliers is played by the sum of $N$ positive terms arising in the second-order approximation for the total current. This fact can be interpreted as the {\it synchronization of molecular currents} in all active molecules, which is provisionally in line with the role of {\it stimulated emission} in the laser action. The calculation of the sum relies on probabilistic arguments which is one of main novelties of our approach. Other main novelties are i) the calculation of the differential $DP(Y^0)$ in the "Hopf representation", ii) the block structure of the differential, and iii) the justification of the "rotating wave approximation" by a new estimate for the averaging of slow rotations.
