矩阵方法在球柱镜系统计算中的应用

Application of a matrix algorithm in spherocylindrical systems

目的 :探讨一种能在一般屈光系统中应用的矩阵算法。方法 :从光学的基本定律出发 ,推导出一般屈光系统矩阵算法的普遍公式。结果和结论 :屈光系统的矩阵算法可以对一般的球柱镜光系进行运算 ,尤其解决了有一定间距的斜交叉球柱镜光系联合的计算问题 ,对复杂系统的计算具有很大的优势 。

Objective:The classical form of expression for refractive power includes three separate qualities:sphere,cylinder and axis. In calculating refractive systems or measuring refractive power,these three qualities are usually calculated separately. But the refractive power is believed to be a unit. Furthermore,the combination of the cross spherocylindrical systems is much more complicated than classical calculations yield. A matrix has been widely applied not only in engineering or technology science,but also in social science. In his research on spherical optical system calculation,Wang Guang-ji found the calculation of spherical systems became easier when using a matrix algorithm,and the matrix expression of refractive power revealed more information than classical measurement. This paper investigates the combination and calculation of the spherocylindrical systems with a new algorithm based on the derived refractive power matrix,which can be practically applied to spherocylindrical refractive systems.Methods:From the refractive effect of the spherocylindrical system,the refractive power matrix of the system can be defined using a form of the 2×2 symmetrical matrix,and the algorithm can be derived,including the conversion of the classical and matrix forms in relation to each other,and the combined spherocylindrical systems. Two examples are discussed in detail.Results:With zero distance in the two combined spherocylindrical systems,the calculated result using the matrix method conforms to the classical forms of sphere,cylinder and axis. But when the two combined systems are separated from each other by a certain distance,the result is quite different not only in the powers of the sphere and cylinder,but also in the axis of the cylinder. And strictly speaking,the qualities a12 and a21 of the power matrix are not equal,and the refractive matrix becomes asymmetrical. So the result of a combined optical system is not strictly spherocylindrical,and may include asymmetrical qualities.Conclusion:The refractive power matrix is more convenient and can be converted into classical form. When applied in the combination of two separate spherocylindrical systems,the matrix algorithm is easier and can also reflect the asymmetry of the combined systems,which conforms with true optical quality. Such a mathematical form of expression for the refractive power and its algorithm will allow common ideas and concepts to be shared with optical engineers.