A Lie algebraic approach to GRIN optimization

The Lie algebraic approach provides a systematic method to determine aberration coefficients for ray propagation in inhomogeneous media. We present the underlying concepts and report analytical expressions for aberration coefficients of arbitrary rotationally symmetric gradient-index (GRIN) lenses. These explicit formulas are well-suited for the application of optimization routines that do not require repeated ray tracing at intermediate steps. We optimize a cylindrical lens, for which we recover a classical solution, and a GRIN lens with inhomogeneous terms along the optical axis.