AI-based Metasurface Lens Design

Introduction

Conventional optical imaging systems are bulky and complex, requiring multiple elements to correct aberrations. Optical metasurfaces, planar structures that are capable of manipulating light at subwavelength scales, are compact alternatives to conventional refractive optical elements. Their miniature volume is suitable for technologies like AR/VR displays and wearables. However, existing metalenses face significant challenges from monochromatic (e.g., coma) and chromatic aberrations, limiting their applicability. Here we present an end-to-end AI-based computational method that parametrizes the profile of metalenses and optimizes it based on customized loss functions. This innovation enables wide-angle imaging with corrected aberrations while retaining a single-layer form factor, overcoming the key limitations of existing metalenses and advancing their potential for miniaturized imaging systems.

Background

What Is Metasurface?

A metasurface is an two-dimensional artificially engineered material composed of subwavelength-scaled patterns. Metasurfaces manipulate electromagnetic waves, e.g. light waves, through specific boundary conditions imposed at their interfaces, unlike conventional materials which manipulate EM waves through its bulk properties, such as refractive index. Metasurface's unique approach enables precise control over wavefronts, allowing for innovative applications such as planar lenses (metalenses) and holograms. Their thin profile and design flexibility are important in integrated photonics and advanced imaging systems.

File:Fig-metasurface.png

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

Metasurface Optic

Transfer function of conventional thin refractive lenses has amplitude and phase. Phase change, of an incident plane wave propagating through the lens, will leads to a change in the wave vector. That is, the incident plane wave will change its propagating direction. In a metalens, the phase is induced via the response of nanostructures. By applying local, gradient phase shifts to incoming waves, metasurfaces generalize the conventional laws of reflection and refraction, called generalized Snell's law. 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. Strel ratio (SR) is a quantitative metric that describes the quality of the PSF. After normalizing the PSF and a perfect diffration-limited airy disk by the total energy, SR is the ratio of the peak intensity of the PSF to the peak intensity of the airy disk. At normal incident angle, the PSF is spherical-aberration-free and outperform the conventional lenses. As the incident angle increasing, the metalens shows an severe coma aberration, which limits the field of view (FOV) of metalens to great extent.

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.

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, applying the appropriate transfer function 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 comprehensive 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.

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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