Double-freeform lens design for angular-spatial control of light fields

Precise simultaneous control of both angular and spatial light-field distributions remains a longstanding challenge in optical design, often requiring complex multi-element configurations. In this work, we propose a compact single-lens solution that achieves unified angular-spatial modulation through the co-optimization of double freeform surfaces. The problem is formulated as an extended caustic design that enforces prescribed irradiance patterns on two distinct receptive planes, where the dual-plane constraint implicitly defines the directional characteristics of the light field while preserving spatial accuracy. This framework eliminates the need for auxiliary optical components while delivering performance comparable to that of conventional multi-lens systems. Comprehensive numerical simulations verify the method's effectiveness, demonstrating accurate and stable control of both angular and spatial light-field properties. The proposed approach establishes a practical foundation for compact, high-performance optical systems and provides a promising route toward integrated angular-spatial light-field engineering.