Highlight Method Results Extensions Citation

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

Optics Express 2026

Yuou Sun, Bailin Deng, Juyong Zhang
University of Science and Technology of China, Cardiff University
arXiv

A single optimized double-freeform lens directs light to form two different irradiance patterns at two receptive planes, showing joint control of both angular and spatial distributions within one compact optical element.

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.

Highlight

01

Dual-Plane Irradiance Control

The optimization simultaneously matches prescribed irradiance targets on Plane A and Plane B in one unified formulation.

02

Compact Single-Lens Design

Joint angular-spatial control is achieved with a single double-freeform lens, reducing reliance on cascaded optics.

03

Robust and Extensible

The framework remains stable across diverse target pairs and extends naturally to high-contrast cases and point-source illumination.

Method

Forward Light Transport Model

The two lens surfaces are discretized into paired triangular meshes. Light propagates through both refractive interfaces and accumulates on two receptive planes, making the entire process differentiable.

Forward transport model figure

OT-Guided Target Outgoing Field

An optimal transport(OT) mapping between the target planes provides correspondence priors for each sampled ray, improving optimization stability and physical consistency.

Target outgoing field figure

Differentiable Optimization

Dual-plane image losses and physical regularization are jointly minimized to update the two freeform surfaces, yielding a compact lens that satisfies angular-spatial constraints.

OT-guided optimization pipeline figure

Results

Double-Freeform Geometry Shape

Both the incident and exit surfaces exhibit smooth freeform features that satisfy the refractive mapping and jointly produce the two prescribed patterns with angular and spatial control.

Optimized incident and exit surface geometry

Generalization Across Target Pairs

Across multiple dual-plane target combinations, the method consistently reconstructs distinct irradiance patterns, demonstrating robust angular-spatial controllability.

Additional simulation examples

Extensions

High-Contrast Dual-Plane Design

This high-contrast case demonstrates that the lens can still recover two challenging target patterns on separated receptive planes, while maintaining physically consistent ray deflection and stable image formation.

High-contrast dual-plane design result

Point-Source Extension

With a point-source illumination model and solid-angle weighting, the same framework remains effective and preserves dual-plane target fidelity.

Point source extension

Citation

@article{sun2026doublefreeform,
  title   = {Double-Freeform Lens Design for Angular-Spatial Control of Light Fields},
  author  = {Sun, Yuou and Deng, Bailin and Zhang, Juyong},
  journal = {Optics Express},
  year    = {2026}
}