AI-based Metasurface Lens Design

Introduction

The evolution of optical imaging systems has been driven by the persistent need to improve image quality while minimizing system complexity and size. Conventional optical systems, such as cameras and microscopes, rely on multiple refractive elements to correct optical aberrations like spherical, chromatic, and coma aberrations. This multilayered architecture, results in bulky, heavy, and mechanically complex designs, making them less suitable for compact and portable applications.

Optical metasurfaces, 2 dimentional engineered material composed of subwavelength-scale nanostructures, have emerged as promising alternatives to traditional optics. They are able to manipulate the phase, amplitude, and polarization of light with unprecedented precision. Their thin, lightweight, and compact form factors is suitable for advanced technologies, including augmented reality (AR), virtual reality (VR), wearable displays, and compact imaging devices. However, despite their potential, current metalenses are affected by intrinsic optical limitations, for example, monochromatic aberrations like coma and chromatic dispersion across a wide spectral range. These aberrations restrict their practical deployment in real-world imaging systems.

Addressing these challenges requires innovative design methodologies that go beyond conventional optimization techniques. In this work, we present a novel end-to-end AI-based computational framework for designing metalenses. Our approach parametrizes the metasurface profile and optimizes it using customized loss functions designed to specific imaging requirements. By leveraging this method, we achieve wide-angle imaging with corrected coma aberrations while maintaining a single-layer lens design. This advancement overcomes critical limitations of existing metalenses and advances their potential for miniaturized imaging systems.

Background

What Is Metasurface?

A metasurface is a two-dimensional, artificially engineered material consisting of subwavelength-scale patterns. Unlike conventional optical materials that manipulate electromagnetic (EM) waves through bulk properties like refractive index, metasurfaces control EM waves by imposing specific boundary conditions at their interfaces. This unique mechanism enables precise wavefront manipulation, allowing for advanced optical functionalities such as planar lenses (metalenses), beam shaping, and holography. The thin profile and design flexibility of metasurfaces make them highly suitable for applications in next-generation imaging systems.

File:Fig-metasurface.png

Figure 1: Metasurfaces with various nanostructures. (Source: Meinzer 2014)

Metasurface Optic

The transfer function of conventional thin refractive lenses has both amplitude and phase modulation. When a plane wave propagates through a lens, the phase change induced by the lens material alters the wave vector, resulting in a change in the wave's propagation direction. In metalenses, phase modulation occurs through the optical response of nanostructures arranged on the metasurface. By inducing local, gradient phase shifts to incoming waves, metalenses enable precise control over light propagation, extending the traditional laws of reflection and refraction into what is known as the generalized Snell's law.

The generalized Snell's law describes the altered paths of reflected and refracted waves due to the imposed phase gradient. In the equation below, ${\theta }_t$, ${\theta }_i$ are the reflection/refraction and incident angle, respectively; $n_t$, $n_i$ are the refractive index of the corresponding material; $\lambda$ is the wavelength of the light; and $\Phi$ is the phase profile of the metalens.

$$n_t\sin {\theta }_t-n_i\sin {\theta }_i=\frac{\lambda }{2\pi }\frac{\text{d}\Phi }{\text{d}x}$$

For conventional laws of reflection and refraction, the term on the right hand side of the equation is 0. The generalized Snell's law implies that we can control the refraction angle by designing the phase profile of the metalens.

File:Gsl.png

Figure 2: Generalized Snell's law. (Source: Yu 2011)

Hyperbolic Phase Profile

Hyperbolic phase profile is commonly used for metalens, given by the equation below, where $\rho =\sqrt{x^2+y^2}$ and $(x,y)$ represents the coordinate at the aperture plane; $f$ is the focal length of the metalens; and $\lambda$ is the wavelength of light that the metalens designed for.

$$\Phi (\rho )=-\frac{2\pi }{\lambda }(\sqrt{{\rho }^2+f^2}-f)$$

In Fig. 3, the amplitude, which is fixed to 1, and the hyperbolic phase is shown. This metalens has the following specs:

• Radius $R=100\mu \text{m}$,
• Focal length $f=200\mu \text{m}$,
• Numerical aperture $NA=0.89$,
• Pitch size $330\text{ nm}$.

File:Hyperbolic phase.png

Figure 3: The constant amplitude profile and the hyperbolic phase profile of metalens.

The point spread function (PSF) at the focal plane of the metalens is shown in Fig. 4. A key metric used to evaluate the PSF quality is the Strehl ratio (SR), which quantitatively measures optical performance relative to a diffraction-limited system. After normalizing both the PSF and a perfect diffraction-limited Airy disk by their total energy, the SR is defined as the ratio of the PSF's peak intensity to that of the Airy disk.

At normal incidence, the metalens produces a PSF free of spherical aberration, showing superior performance compared to conventional lenses. However, as the incident angle increases, significant coma aberration arises, leading to distortion in the PSF. This aberration severely restricts the metalens's field of view (FOV), posing a fundamental challenge to its applicability in wide-angle imaging systems.

File:Psf before.png

Figure 4: PSFs of the hyperbolic phase metalens at different incident angles and their Strel ratio SR.

Methods

Angular Spectrum Method

The Angular Spectrum Method (ASM) is a computational technique used in wave propagation analysis, particularly in optics. It models the propagation of waves by decomposing a complex wavefront into a spectrum of plane waves with spatial frequencies and angles. Each plane wave propagates independently according to its wave vector (direction of propagation), allowing the reconstruction of the wavefront at any distance from the source. ASM is especially useful for simulating wave behavior in free space or through homogeneous media.

In the project, the optical system is straight forward. There is only one metalens with given amplitude and phase profile, and the sensor is located at the focal plane of the metalens. We assume that the object is in the far field, so its spatial structure can be represented as a distribution of spatial frequencies in the optical wave's angular spectrum. The metalens, with its specific amplitude and phase profile, acts as a spatial filter, modifying the incoming wavefront accordingly.

$$I=|{ℱ}^{-1}\{ℱ\{U_0\}\cdot \mathscr{H}\}{|}^2$$

By applying ASM, we can compute how the wavefront evolves from the metalens to the sensor located at its focal plane. This involves transforming the wavefront into its angular spectrum using a Fourier transform $ℱ\{U_0\}$, applying the appropriate transfer function $\mathscr{H}$ that accounts for metalens, and then inverse Fourier transforming the result to reconstruct the wavefront at the sensor plane. This approach captures diffraction and interference effects, allowing us to analyze your system's optical performance.

Parametrization of the Phase

To model the phase profile of the metalens, we use its radial symmetry by parametrizing the phase distribution using power series in the radial coordinate. This approach simplifies the representation while preserving the essential features of the lens’s optical behavior. Specifically, the phase profile is expressed as a polynomial expansion in terms of the radial distance from the lens center, with coefficients determining the lens's focusing characteristics. In the optimization algorithm, we need to optimize the coefficients $c_n$. This parametrization not only reduces the computational complexity of simulating the metalens but also provides flexibility in optimizing its optical performance for specific applications.

$$\Phi (\rho )=\sum _n{c}_n(\frac{\rho }{R}{)}^n$$

In addition to using standard power series, we explore several other methods to parametrize the phase profile of the metalens. One notable approach is using Zernike polynomials, which are commonly employed in optical design due to their ability to model aberrations and complex phase features. However, while Zernike polynomials provide a basis for phase representation, many of their terms are not radially symmetric, making them less suitable for systems where radial symmetry is critical. This limitation motivates us to consider alternative parametrization schemes that balance modeling flexibility with computational efficiency while respecting the inherent symmetry of the metalens design.

Optimization Algorithm

To optimize the coefficients of the power series representing the phase profile of the metalens, we use PyTorch for its automatic differentiation capabilities. We define the coefficients of the power series as trainable parameters and construct a loss function based on the Strehl ratio of the PSFs at the sensor plane. The optimization process uses the Adam optimizer, which efficiently updates the coefficients using adaptive learning rates and momentum-based corrections. This setup allows us to perform gradient-based optimization.

$$ℒ=\sum _{\theta }SR(\theta )=\sum _{\theta }\frac{I_{real}(x(\theta ),y(\theta ))}{I_{ideal}(0,0)},\theta \in \{0,10,20,25\}$$

Results

Optimized Phase Profile

The phase profile after optimization is shown in Figure 5. On the right, a slice of the phase at $y=0\mu \text{m}$ is compared to the same slice of the hyperbolic phase.

File:Optimized phase.png

Figure 5: Optimized phase profile and the slice of optimized phase profile.

PSF Comparison

The PSFs at the focal plane of the metalens with optmized phase is shown in Fig. 6. We observed that the coma aberration is corrected to a great extent. With $SR>0.1$ for incident angles at $10^{\circ }$, $20^{\circ }$, $25^{\circ }$. On the other hand, The $SR$ at the normal incident angle decreases, and we observe scattering rings which indicates spherical aberration. This implies that the optimzation algorithm is looking for a balance of the tradeoff between coma aberration and spherical aberration.

File:Psf after.png

Figure 6: PSFs at different incident angles of the metalens with optimized phase.

To better compare the performance of the metalens, Fig. 7 shows the modulation transfer function (MTF) of the metalens imaging system. For the metalens with the hyperbolic phase, we observe that the contrast drops steeply for incident angles at $10^{\circ }$, $20^{\circ }$, $25^{\circ }$. While after optimization, the MTFs across all incident angles are balanced with MTF at the normal incident angle being compromised. This is in accord with the observation from the PSF.

File:MTF.png

Figure 7: MTFs along x-axis and y-axis of the metalens of hyperbolic phase and optimized phase.

Image Simulations

File:S sim.png

Figure 8:

The following images are simulated using space-varying 2D convolution algorithm. In space-varying convolution, we divide the whole FOV into $n$-by-$n$ isoplanatic patches, where we assume PSF is invariant. We implement space-invariant 2D convolution on isoplanatic patches and then merge the matches together by apply a Gaussian window. The periphery of the simulated images with hyperbolic phase metalens are a lot more blurry compared to the ones with optimized phase, which agrees with the expectation. In this simulation, we assume that the PSF across the RGB channel are the same, so color in this case is only for visual purpose.

File:Simulation img.png

Figure 9: Simulated images from space-varying convolution of the target image with PSFs of two metalenses.

Appendix

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