Toward a Perfect Lens: Automated Design of GRIN Double-Freeform Systems for Broadband Aberration and Chromaticity Correction

Abstract In conventional optical design, the inability of a single lens to simultaneously correct high-order geometric and chromatic aberrations over a broad spectral range constitutes a persistent physical bottleneck limiting the development of ultracompact imaging systems. This fundamentally arises from severely restricted degrees of freedom in traditional lenses, which rely solely on interfacial modulation and the intrinsic dispersion of homogeneous media. Here, we propose a global inverse design framework based on gradient-index (GRIN) dual-freeform optics to overcome single-lens constraints. By introducing a wavelength-inclusive extended phase space, the complex ray-tracing evolution is rigorously transformed into an analytical algebraic reconstruction process. This framework reveals a physical mechanism for chromatic aberration compensation: exploiting the three-dimensional spatial dispersion gradient within the GRIN medium induces an anomalous negative chromatic aberration effect. This globally and precisely cancels the intrinsic positive chromatic dispersion introduced by freeform refractions. As methodological validation, we inversely designed an objective singlet lens with a demanding half field of view of 50 degrees. Quantitative evaluations demonstrate that across the entire visible spectrum, the full-field root-mean-square (RMS) spot radius is strictly controlled within 8 nm. This achieves broadband aberration-free and achromatic imaging well within the diffraction limit. This work provides a rigorous theoretical methodology and an implementable pathway for designing ultracompact imaging optics, substantially reducing the reliance on manual experience and iterative numerical optimization in advanced freeform design.