Bergstein L
Appl Opt. 1968 Mar 1;7(3):495-504. doi: 10.1364/AO.7.000495.
A method is developed for the approximate determination of the normal modes of stable and unstable optical resonators and the associated resonant frequencies and power losses. The method is based on replacing the finite integration limits in the integral equation for the normal modes by infinite limits and, subsequently, finding a differential equation whose solutions coincide with or approximate the solutions of this integral equation. When the end reflectors of the resonator are conical surfaces, a differential equation is found which corresponds exactly to the integral equation with infinite limits. Moreover, the equivalent differential equation is found to be of the same form as the wave equation for a monochromatic transverse electric wave propagating in an inhomogeneous medium of infinite extent with the inhomogeneity being transverse to the direction of propagation, showing the correspondence between the modes of a homogeneously filled conical resonator and the eigenmodes of an infinite inhomogeneous mdium. For the stable, low loss (convergent) region the solutions of the differential equation are readily found. For the unstable, high loss (divergent) region the solutions are found by using the principle of analytic continuation. The specific example of parabolic end reflectors is treated in more detail, and solutions for the eigenvalues and eigenfunctions are given for the case of infinite strip and circular geometries.
本文提出了一种近似确定稳定和不稳定光学谐振器的正常模式以及相关谐振频率和功率损耗的方法。该方法基于用无限积分限取代正常模式积分方程中的有限积分限,随后找到一个微分方程,其解与该积分方程的解一致或近似。当谐振器的端面反射镜为圆锥面时,可找到一个与具有无限积分限的积分方程精确对应的微分方程。此外,发现等效微分方程与单色横向电波在无限延伸的非均匀介质中传播的波动方程具有相同形式,其中非均匀性垂直于传播方向,这表明均匀填充圆锥谐振器的模式与无限非均匀介质的本征模式之间存在对应关系。对于稳定、低损耗(收敛)区域,微分方程的解很容易得到。对于不稳定、高损耗(发散)区域,通过使用解析延拓原理来求解。文中更详细地讨论了抛物面端面反射镜的具体示例,并给出了无限长条形和圆形几何形状情况下的特征值和特征函数的解。