Reverse ray tracing for transformation optics

Ray tracing is an important technique for predicting optical system performance. In the field of transformation optics, the Hamiltonian equations of motion for ray tracing are well known. The numerical solutions to the Hamiltonian equations of motion are affected by the complexities of the inhomogeneous and anisotropic indices of the optical device. Based on our knowledge, no previous work has been conducted on ray tracing for transformation optics with extreme inhomogeneity and anisotropicity. In this study, we present the use of 3D reverse ray tracing in transformation optics. The reverse ray tracing is derived from Fermat's principle based on a sweeping method instead of finding the full solution to ordinary differential equations. The sweeping method is employed to obtain the eikonal function. The wave vectors are then obtained from the gradient of that eikonal function map in the transformed space to acquire the illuminance. Because only the rays in the points of interest have to be traced, the reverse ray tracing provides an efficient approach to investigate the illuminance of a system. This approach is useful in any form of transformation optics where the material property tensor is a symmetric positive definite matrix. The performance and analysis of three transformation optics with inhomogeneous and anisotropic indices are explored. The ray trajectories and illuminances in these demonstration cases are successfully solved by the proposed reverse ray tracing method.