Deflectometry for measuring inhomogeneous refractive index fields in two-dimensional gradient-index elements.

We present a numerical method for calculating inhomogeneous refractive index fields in rectangular gradient-index (GRIN) elements from measured boundary positions and slopes of a collection of rays that transit the medium. The inverse problem is reduced to a set of linear algebraic equations after approximating ray trajectories from the measured boundary values and is solved using a pseudo-inverse algorithm for sparse linear equations. The ray trajectories are subsequently corrected using an iterative ray trace procedure to ensure consistency in the solution. We demonstrate our method in simulation by reconstructing a hypothetical rectangular GRIN element on a  15×15 discrete grid using 800 interrogating rays, in which RMS refractive index errors less than 0.5% of the index range (n(max)-n(min)) are achieved. Furthermore, we identify three primary sources of error and assess the importance of data redundancy and system conditioning in the reconstruction process.