Retrieval of refractive index fields in two-dimensional gradient-index elements from external deflectometry data

In a previous work, we presented a numerical method for retrieving inhomogeneous refractive index fields in rectangular gradient-index elements from boundary positions and internal boundary slopes associated with a set of interrogating probe beams that transit the medium. The present work extends this method to external boundary beam slopes without knowledge of the refractive index along the surface of the optical element, requiring minimal additional information (outside of beam position and slope data) such as a single known index point inside the medium. The inverse problem is cast as a linear algebraic system describing the deflection of probe beams inside the optical material, and an iterative inversion algorithm is used to generate an index field that produces the boundary value data. By incorporating Snell's law into the system equation through surface values derived from tentative reconstructions of the refractive index, we show in simulation that a series of inversion cycles applied to the system equation accurately recovers the index profile used to generate the test data.