Capps D M
J Opt Soc Am A Opt Image Sci Vis. 2019 Apr 1;36(4):563-577. doi: 10.1364/JOSAA.36.000563.
The goal of optical simulation is to determine the performance characteristics of an optical system from a knowledge of its physical construction and how it affects light sent through it. To produce meaningful results efficiently, two simulation approaches are available for passing light through a system, geometrical raytracing and wave optics. Within the wave optics realm, there are many techniques for determining the optical fields within a system, both numerical and analytical. A few of the numerical techniques are finite-difference, finite-element, and FFT-based; analytical techniques include modal expansions, coupled wave theory, series expansions, and Green function propagators. A propagator is a function that gives the light fields at any specified location if they are known at a source location; this is possible because the light fields, electric and magnetic, satisfy a differential equation, in the case of time harmonic fields, the Helmholtz equation. The propagator is a transfer function for the fields and often takes the form of an integral, in which case, the integrand is a product of the transfer function with the source field distribution, and the integration is performed over the source field coordinates. The integrand transfer function, also known as a Green function or propagation kernel, is a solution of the Helmholtz equation. An approximation is often used in finding a solution to the Helmholtz equation, called the paraxial approximation, in which the second derivative in the propagation direction is dropped. If no approximation is made, and all second derivatives are kept, the solution is nonparaxial. In the present paper, a Green function for the propagator of the Helmholtz equation over two-dimensional domains is derived, differing in functional form from previous work on two-dimensional propagation. An angular spectrum integral is evaluated and the resulting Green function, the propagator kernel, is a nonparaxial analytic solution of the Helmholtz equation. The propagator could be applied directly to the electric and magnetic field components; instead, it is applied to the Hertz vector components. The Hertz vector is a potential function, similar to the vector potential, defined such that the electric and magnetic fields are found by taking derivatives of it. An advantage of the Hertz vector is that only it needs be propagated, versus two, electric and magnetic, vectors. In this paper, the derived propagator is applied to Hertz vector components defined by Legendre polynomial expansions, and derivatives are taken of the propagated Hertz vector components to calculate the associated electric and magnetic fields. The Green function propagator and all field quantities produced by its application are closed form analytic expressions.
光学模拟的目标是，根据光学系统的物理结构及其对通过该系统的光的影响，来确定该光学系统的性能特征。为了高效地得出有意义的结果，有两种模拟方法可用于使光通过系统，即几何光线追踪和波动光学。在波动光学领域内，有许多确定系统内光场的技术，包括数值技术和解析技术。一些数值技术有有限差分法、有限元法和基于快速傅里叶变换（FFT）的方法；解析技术包括模态展开、耦合波理论、级数展开和格林函数传播子。传播子是一个函数，如果已知光源位置处的光场，那么它能给出任意指定位置处的光场；这是可行的，因为光场（电场和磁场）满足一个微分方程，对于时谐场而言，就是亥姆霍兹方程。传播子是光场的传递函数，通常采用积分形式，在这种情况下，被积函数是传递函数与源场分布的乘积，积分是在源场坐标上进行的。被积函数传递函数，也称为格林函数或传播核，是亥姆霍兹方程的一个解。在求解亥姆霍兹方程时，经常使用一种近似，称为傍轴近似，即在传播方向上舍弃二阶导数。如果不做近似，保留所有二阶导数，那么解就是非傍轴的。在本文中，推导了亥姆霍兹方程在二维域上传播子的格林函数，其函数形式与之前关于二维传播的工作不同。对一个角谱积分进行了求值，得到的格林函数即传播核，是亥姆霍兹方程的一个非傍轴解析解。该传播子可以直接应用于电场和磁场分量；相反，它应用于赫兹矢量分量。赫兹矢量是一个势函数，类似于矢量势，其定义使得通过对它求导来得到电场和磁场。赫兹矢量的一个优点是，只需要传播它，而不是传播电场和磁场这两个矢量。在本文中，将推导得到的传播子应用于由勒让德多项式展开定义的赫兹矢量分量，并对传播后的赫兹矢量分量求导以计算相关的电场和磁场。格林函数传播子及其应用所产生的所有场量都是封闭形式的解析表达式。
