Lin Psang Dain
National Cheng Kung University, Department of Mechanical Engineering, Tainan, Taiwan.
J Opt Soc Am A Opt Image Sci Vis. 2011 Oct 1;28(10):1995-2005. doi: 10.1364/JOSAA.28.001995.
The second-order derivative matrix of a scalar function with respect to a variable vector is called a Hessian matrix, which is a square matrix. Our research group previously presented a method for determination of the first-order derivatives (i.e., the Jacobian matrix) of a skew ray with respect to the variable vector of an optical system. This paper extends our previous methodology to determine the second-order derivatives (i.e., the Hessian matrix) of a skew ray with respect to the variable vector of its source ray when this ray is reflected/refracted by spherical boundary surfaces. The traditional finite-difference methods using ray-tracing data to compute the Hessian matrix suffer from various cumulative rounding and truncation errors. The proposed method uses differential geometry, giving it an inherently greater accuracy. The proposed Hessian matrix methodology has potential use in optimization methods where the merit function is defined as ray aberrations. It also can be used to investigate the shape of the wavefront for a ray traveling through an optical system.
标量函数关于变量向量的二阶导数矩阵称为海森矩阵，它是一个方阵。我们的研究小组之前提出了一种确定倾斜光线关于光学系统变量向量的一阶导数（即雅可比矩阵）的方法。本文将我们之前的方法进行扩展，以确定倾斜光线在被球面边界表面反射/折射时关于其源光线变量向量的二阶导数（即海森矩阵）。使用光线追迹数据来计算海森矩阵的传统有限差分方法存在各种累积舍入和截断误差。所提出的方法使用微分几何，使其具有更高的固有精度。所提出的海森矩阵方法在将品质因数定义为光线像差的优化方法中具有潜在用途。它还可用于研究光线穿过光学系统时的波前形状。
