Manipulating light trace in a gradient-refractive-index medium: a Lagrangian optics method

Light propagation obeys Fermat's principle, and an important inference of Fermat's principle is the optical Lagrange equation, from which the light trace can be determined with a given refractive index. Here, we consider the inverse problem of how to derive the refractive index distribution of a planar geometric optical system once the trace is predetermined. Based on the optical Lagrange equation, we propose a dynamic equation model which associates the refractive index with the light trace. With the consideration of a certain trace, we illustrate the process of solving the partial differential equation of refractive index through first integral method. By setting the distribution function of a gradient-refractive-index (GRIN) medium, one can control the light traveling along a desirable curve, adjust the incoming and outgoing rays, and also use the trace to paint geometrics. This method develops the Lagrangian optics in the application of ray dynamic system design, such as lens, beam splitter, metasurface and optical waveguide. It provides a theoretical guidance to manipulate the ray in a GRIN medium.