Freeform gradient-index media represented by rotation-invariant orthogonal polynomials in the unit ball for asymmetric optical design

Freeform gradient-index (F-GRIN) media offer significant degrees of freedom for the design of advanced imaging systems. As manufacturing capabilities evolve toward complex refractive index distributions, a unified representation is increasingly required. In this paper, rotation-invariant orthogonal polynomials in the unit ball (ROPB) are proposed to provide a canonical 3D representation for F-GRIN. ROPB integrates spatial coordinates through rotation-invariant base functions, ensuring orthogonality and continuity in the unit ball. The index distribution and aberration properties of ROPB terms are analyzed to facilitate their application in optical design. Furthermore, the efficacy of ROPB is validated through the performance enhancement of an ultrashort throw ratio projection system. The results demonstrate that the ROPB can not only effectively guide the optical design of F-GRIN systems but also mitigates aberrations in non-rotationally symmetric systems, providing a feasible solution for the more compact and high-performance optical design.