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Parametric Surfaces

Parametric Surfaces

A parametric surface in three-dimensional space is defined through a vector-valued function\begin{equation*}\mathbf{r}(u, v) = x(u, v)\mathbf{i} + y(u, v)\mathbf{j} + z(u, v)\mathbf{k}\end{equation*}where u and v are parameters within a specified domain D in the uv-plane. The functions x(u, v), y(u, v), and z(u, v) define the coordinates of points on the surface. As u and v vary over D, the position vector r(u, v) traces a continuous surface in space. This parametric representation is essential...

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Updated: Jun 19, 2026

Demonstration of Equal-Intensity Beam Generation by Dielectric Metasurfaces

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Arbitrary engineering of spatial caustics with 3D-printed metasurfaces.

Xiaoyan Zhou^1,2,3, Hongtao Wang^4,5, Shuxi Liu¹

¹Zhejiang Key Laboratory of Micro-nano Quantum Chips and Quantum Control, School of Physics, Zhejiang University, Hangzhou, 310058, China.

Nature Communications

|May 2, 2024

Summary

Researchers developed 3D-printed metasurfaces to control optical caustics, enabling customizable curved trajectories and in-plane patterns for structured light applications.

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Area of Science:

Optics and Photonics
Metamaterials
Structured Light

Background:

Optical caustics, envelopes of rays, appear in systems from nano-scale to astronomical scales.
Structured light with complex-amplitude distributions has applications in particle manipulation, imaging, and communication.
Engineering custom caustic fields with controllable propagation and intensity profiles remains a significant challenge.

Purpose of the Study:

To introduce a novel method for shaping caustic fields with customizable curved trajectories in free space.
To demonstrate the preservation or morphing of in-plane caustic patterns during propagation.
To provide a compact and scalable solution for generating tailored caustic structured light.

Main Methods:

Utilized 3D-printed metasurfaces incorporating a 'compensation phase' to engineer caustic fields.
Employed two-photon polymerization lithography for fast-prototyping and cost-effective large-scale fabrication.
Designed ultra-thin optical elements with sub-millimeter dimensions.

Main Results:

Successfully shaped caustic fields with curved propagation trajectories using the compensation phase.
Demonstrated the ability to maintain or transform in-plane caustic patterns during propagation.
Fabricated compact metasurface optical elements suitable for various applications.

Conclusions:

The compensation phase in 3D-printed metasurfaces offers a powerful tool for controlling optical caustics.
This approach enables unprecedented flexibility in designing structured light with tailored propagation and intensity.
The developed metasurfaces provide a compact and scalable solution for beam shaping, advanced microscopy, and light-matter interaction studies.