Ohno Hiroshi
J Opt Soc Am A Opt Image Sci Vis. 2020 Mar 1;37(3):411-416. doi: 10.1364/JOSAA.378829.
A method based on Hamiltonian optics for ray tracing through gradient-index (GRIN) media is proposed. The ray equation that describes light-ray paths can be written in the form of the Hamiltonian equations. Although the Hamiltonian equations can be numerically calculated using a finite-difference explicit method, deviations from the exact equations are generally inevitable at subsequent time steps. An optical Hamiltonian can be constructed of two independent terms, i.e., one term dependent on position and the other term dependent on momentum. The symplectic integrator is applicable to such a separable optical Hamiltonian system and makes the optical Hamiltonian equations form invariant at each time step of numerical calculations. Accuracies of light-ray paths calculated using the first-order symplectic ray tracing in GRIN lenses approximate those calculated on the basis of the fourth-order Runge-Kutta algorithm, which shows the promising potential of the symplectic-ray-tracing method.
提出了一种基于哈密顿光学的光线在梯度折射率（GRIN）介质中追迹的方法。描述光线路径的光线方程可以写成哈密顿方程的形式。虽然哈密顿方程可以使用有限差分显式方法进行数值计算，但在后续时间步长中通常不可避免地会偏离精确方程。光学哈密顿量可以由两个独立项构成，即一项依赖于位置，另一项依赖于动量。辛积分器适用于这种可分离的光学哈密顿系统，并使光学哈密顿方程在数值计算的每个时间步长上保持形式不变。在GRIN透镜中使用一阶辛光线追迹计算的光线路径精度近似于基于四阶龙格 - 库塔算法计算的精度，这表明了辛光线追迹方法具有广阔的应用前景。
