Analysis of 3D inhomogeneous optical systems via the Lorentzian formalism

The application of Lorentz geometry to the study of homogeneous media nonimaging optical systems was introduced successfully in [1], see also [2], based on the analogy between lightlike cones and the cone of edge rays. Recently, we applied this technique to the study of 2D inhomogeneous optical systems [3]. In the present paper, we generalize previous studies to stationary 3D inhomogeneous optical systems. We obtain and provide some solutions to the systems of nonlinear partial differential equations in cartesian and spheric coordinates in three dimensions. We study optical systems with exponential refraction index. The solutions provide the geometrical vector flux J at any point of these 3D optical systems. From this Lorentzian formalism, we have obtained irradiance patterns of these stationary inhomogeneous optical systems and we have compared it with irradiance patterns obtained by raytrace simulations. Finally a study of the physical interpretation of eigenvalues of the Gram matrix of the Lorentzian metric G is presented.