非稳腔模式计算中Colins公式适用性的分析

Availability analysis of Collins formula inmode calculation of unstable resonators

Ｃｏｌｉｎｓ公式将几何光学和波动光学联系在一起，能以简单的形式描述相干光在复杂光学系统中的传输，但忽略了光学系统中一些光学元件的衍射损耗。当Ｃｏｌｉｎｓ公式用于激光谐振腔的本征模式计算时，其适用程度与被忽略腔镜的有效菲涅尔数有关。本文用数值方法求解了两组谐振腔本征积分方程，一组应用Ｃｏｌｉｎｓ公司推导，另一组用菲涅尔－惠更斯衍射积分推导，结果表明：对稳定腔Ｃｏｌｉｎｓ公式基本适用；对非稳腔，当被忽略腔镜的有效菲涅尔数＞１时，原则上可以用，当被忽略腔镜的有效菲涅尔数＜１时不适用；对有源非稳腔基本适用。

When Collins formula is used to calculate the eigenmode of a laser resonator, its availability has to do with the efficient Frensel number of the mirror which size is ignorde. Two sets of integral eigenequations are derived, one from Collins formula, another from FresnelHuygens diffraction integration. Their numerical solutions point out that Collins formula is available for stable resonators and the unstable resonator in which the efficient Fresnel number of the mirror ignored is bigger than one, but it is unavailable for loaded unstable resonators and the unstable resonator in which the efficient Fresnel number of the mirror ignored is smaller than one.