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## A MATRIX THEORY FOR OPTICAL PASSIVE RESONATORS (IN CYLINDRICAL COORDINATES) (II)——CALCULATION FOR AXIAL SYMMETRIC STABLE RESONATORS

LI XIAN-SHU, GAO YAN-QIU, CHEN ZHI-TIAN, FENG ZENG-YE
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• #### 摘要

本文的目的是探讨应用文献[1]中提出的光学无源谐振腔的矩阵方程,解决腔中振荡横模的具体计算方法问题,并讨论其精度。为了便于与已有公式较可靠的结果相比,我们挑选了由两面对称的球面反射镜组成的、菲涅耳数N≤1的各种稳定腔作为计算对象;还分别用数学试验法和文献[1]中得出的误差上限公式,求出所得结果的误差。结果表明:对于基横模,所得结果与文献[3,6]的结果在报道的精度内很好地符合;对于包括高阶模在内的各阶横模,文献[1]中所得公式的误差上限都是正确的,但往往大大偏高。计算结果实际上往往具有高得多的精度。文中对g=0,0.5,0.8,0.9,0.95,菲涅耳数N=1的稳定腔,列出了l=0,p=0,1,2,3,4,5的各阶横模的本征值;绘出了l=0,1,2,p=0,1,2各阶横模场的相对振幅与位相分布曲线。还绘出了上述诸类稳定腔中N=1/π时TEM00,TEM01模的场分布曲线。从所得结果中得出了一些新的规律,并进行了某些讨论。本文结果表明:用这种矩阵理论,确实可以较为方便地一次求得模损耗不极近于1的、角模数l任意给定而径模数p不同的、所有各阶横模的性质,并能给出所得结果的误差上限,保证其具有相当高的精度。由于所用坐标系关系,本方法仅适用于具有理想轴对称性的谐振腔。

#### Abstract

Attempts are made to explore the way of solving the problem of calculating modes in optical passive resonators on the basis of the matrix equation in the previous papers, and to determine the superior limits of calculation errors resulting from truncating the matrix equation. For convenience of comparison between the results of ours and those obtained previously by other methods in literatures, we have calculated the characteristics of the modes in various stable resonators with two symmetric reflectors for Fresnel numbers N ≤ 1, and have determined the superior limits of the calculation errors using both the trial method of mathematics (the matrix has been truncated into various order to solve the matrix equation) and the rigorous formulae derived in the previous paper in general. It is shown that the superior limits followed from these formulae are resonable for all modes, including high order modes, but they are far larger than real errors. So that a new formula, which is more suitable for determining the superior limits of errors in our calculation than the previous one, has been adopted.Some new characteristics of the modes are obtained.It is obvious that, by means of the matrix equation, all modes in a resonator, whose losses are not very close to unity, with various radial mode numbers p and an arbitrary given angular mode number l can be caculated at a time, It has been found that a satisfactory accuracy can be achieved for the modes in resonators 0 ≤g≤0.95 in our calculation,provided the matrix equation is truncated into (N + 1) order, where N≥ 8N(1+g)1/2-l/2.

#### 施引文献

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##### 出版历程
• 收稿日期:  1982-05-10
• 刊出日期:  1983-04-05

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