Lin Psang Dain
National Cheng Kung University, Department of Mechanical Engineering, Tainan, Taiwan.
Appl Opt. 2012 Feb 1;51(4):486-93. doi: 10.1364/AO.51.000486.
The first-order derivative matrix of a function with respect to a variable vector is referred to as the Jacobian matrix in mathematics. Current commercial software packages for the analysis and design of optical systems use a finite difference (FD) approximation methodology to estimate the Jacobian matrix of the wavefront aberration with respect to all of the independent system variables in a single raytracing pass such that the change of the wavefront aberration can be determined simply by computing the product of the developed Jacobian matrix and the corresponding changes in the system variables. The proposed method provides an ideal basis for automatic optical system design applications in which the merit function is defined in terms of wavefront aberration. The validity of the proposed approach is demonstrated by means of two illustrative examples. It is shown that the proposed method requires fewer iterations than the traditional FD approach and yields a more reliable and precise optimization performance. However, the proposed method incurs an additional CPU overhead in computing the Jacobian matrix of the merit function. As a result, the CPU time required to complete the optimization process is longer than that required by the FD method.
在数学中，函数关于变量向量的一阶导数矩阵被称为雅可比矩阵。当前用于光学系统分析和设计的商业软件包使用有限差分（FD）近似方法，在单次光线追迹过程中估计波前像差关于所有独立系统变量的雅可比矩阵，这样通过计算已生成的雅可比矩阵与系统变量相应变化的乘积，就能简单地确定波前像差的变化。所提出的方法为基于波前像差定义品质因数的自动光学系统设计应用提供了理想基础。通过两个示例说明了所提方法的有效性。结果表明，与传统的有限差分方法相比，所提方法所需的迭代次数更少，并且能产生更可靠、精确的优化性能。然而，所提方法在计算品质因数的雅可比矩阵时会产生额外的CPU开销。因此，完成优化过程所需的CPU时间比有限差分方法更长。
