Analytical wave tracing, including higher-order aberrations, is the state of the art, but at present it is only provided for single propagation or refraction. A complex optical system being a sequence of many refractive surfaces and gaps in between can only be treated by successive application of elementary wave-tracing steps, which is time-consuming for a large number of surfaces or even impossible if a system specification by surfaces is not available. Provided the ray transfer properties of a system are summarized as a nonlinear function ${\textbf{f}}$ whose Jacobian is the ABCD matrix of the system, by multiple derivative we obtain wave-tracing equations for the wavefront’s local derivatives of any desired order. The outgoing wavefront derivative of any order can be written as a sum of multinomials of derivatives of the incoming wavefront, weighted by system-dependent coefficients and by powers of the factor $\beta = {(A - B{E_2})^{- 1}}$, where ${E_2}$ is the second-order aberration of the incoming wavefront. Compared to stepwise wave tracing, this approach is extremely efficient when tracing many different wavefronts through one fixed optical system.

中文翻译：

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复杂光学系统局部高阶波前追踪方程的通用方法

分析波跟踪，包括高阶像差，是最先进的技术，但目前仅提供用于单次传播或折射。一个复杂的光学系统是由许多折射表面和其间的间隙组成的序列，只能通过连续应用基本波跟踪步骤来处理，这对于大量表面来说是耗时的，如果按表面指定系统规格，这甚至是不可能的无法使用。假设系统的光线传递特性被概括为非线性函数${\textbf{f}}$其 Jacobian 矩阵是系统的 ABCD 矩阵，通过多重导数我们得到波前任意阶次的局部导数的跟踪方程。任何阶的输出波前导数都可以写成输入波前导数多项式的总和，由系统相关系数和因子$\beta = {(A - B{E_2})^{- 的幂加权1}}$，其中${E_2}$是传入波前的二阶像差。与逐步波跟踪相比，这种方法在通过一个固定光学系统跟踪许多不同的波前时非常有效。