Double-Freeform Lens Design for Angular-Spatial Control of Light Fields

Abstract

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.

Figures (13)

• Figure 1: Overview of the forward light-transport model. A collimated light beam passing through an incident-surface triangle $\triangle \mathbf{v}_1^i \mathbf{v}_1^j \mathbf{v}_1^k$ is refracted toward its corresponding exit-surface triangle $\triangle \mathbf{v}_2^i \mathbf{v}_2^j \mathbf{v}_2^k$. After the second refraction, the outgoing rays intersect receptive planes $A$ and $B$, forming imaging triangles $\triangle \mathbf{p}_A^i \mathbf{p}_A^j \mathbf{p}_A^k$ and $\triangle \mathbf{p}_B^i \mathbf{p}_B^j \mathbf{p}_B^k$, respectively. Each imaging triangle carries the same luminous flux as its corresponding incident-surface triangle. The per-pixel irradiance on each plane is obtained by accumulating the flux contributions from all overlapping imaging triangles and then converting to pixel values via gamma encoding, producing simulated images that can be directly compared with the prescribed target images $g_A$ and $g_B$.
• Figure 2: Target outgoing light field. For each ray sample (vertex index) $i$ on the double-freeform lens, we assign a pair of OT-derived target locations $(\tilde{\mathbf{p}}_A^i,\tilde{\mathbf{p}}_B^i)$ on the two receptive planes $A$ and $B$. Each pair defines a target optical-path line $\tilde{l}_{\mathrm{out}}^i$ (yellow), specifying both the desired landing locations on the two planes and the corresponding propagation direction between them; these per-ray targets provide structured guidance for the subsequent surface optimization.
• Figure 3: Side view of the OT-guided optimization process. The red points $\tilde{\mathbf{p}}_A^{i,j}$ and $\tilde{\mathbf{p}}_B^{i,j}$ denote the fixed OT-derived target intersections on planes $A$ and $B$, respectively, while the blue points $\mathbf{p}_A^{i,j}$ and $\mathbf{p}_B^{i,j}$ denote the actual intersections of the refracted rays. During optimization, the incident and exit surfaces evolve through updates of the vertex positions $\mathbf{v}_1^{i,j}$ and $\mathbf{v}_2^{i,j}$, together with the exit-surface normals $\mathbf{n}_2^{i,j}$, so that the actual ray intersections progressively approach the OT-derived targets on both receptive planes.
• Figure 4: A local example of the first two stages. The target positions are obtained by downsampling from the finest-stage, while the mesh for the finer stage is initialized by subdividing the optimized mesh from the coarser stage.
• Figure 5: Optimization progress across stages: the left panels show the energy reduction over the first 300 iterations at the coarsest and finest stages, while the right panels compare the simulated images on planes A and B before and after optimization for each stage.