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| == Introduction ==
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| 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.
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| 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.
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| 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.
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| == Background ==
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| === What Is Metasurface? ===
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| 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.
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| [[File:Fig-metasurface.png|thumb|center|600 px|Figure 1: Metasurfaces with various nanostructures. (Source: Meinzer 2014)]]
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| === Metasurface Optic ===
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| 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, <math> \theta_t </math>, <math> \theta_i </math> are the reflection/refraction and incident angle, respectively; <math> n_t </math>, <math> n_i </math> are the refractive index of the corresponding material; <math> \lambda </math> is the wavelength of the light; and <math> \Phi </math> is the phase profile of the metalens.
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| <math display="block"> n_t \sin \theta_t - n_i \sin \theta_i = \frac{\lambda}{2\pi} \frac{\text{d}\Phi}{\text{d}x} </math>
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| 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.
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| [[File:Gsl.png|thumb|center|350 px|Figure 2: Generalized Snell's law. (Source: Yu 2011)]]
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| === Hyperbolic Phase Profile ===
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| Hyperbolic phase profile is commonly used for metalens, given by the equation below, where <math> \rho = \sqrt{x^2 + y^2} </math> and <math> (x,y) </math> represents the coordinate at the aperture plane; <math> f </math> is the focal length of the metalens; and <math> \lambda </math> is the wavelength of light that the metalens designed for.
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| <math display="block"> \Phi (\rho) = - \frac{2\pi}{\lambda} (\sqrt{\rho ^ 2 + f ^ 2} - f) </math>
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| In Fig. 3, the amplitude, which is fixed to 1, and the hyperbolic phase is shown. This metalens has the following specs:
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| * Radius <math> R = 100 \text{ } \mu \text{m} </math>,
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| * Focal length <math> f = 200 \text{ } \mu \text{m} </math>,
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| * Numerical aperture <math> NA = 0.89 </math>,
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| * Pitch size <math> 330 \text{ nm} </math>.
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| [[File:Hyperbolic_phase.png|thumb|center|500 px|Figure 3: The constant amplitude profile and the hyperbolic phase profile of metalens.]]
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| 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.
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| [[File:Psf_before.png|thumb|center|500 px|Figure 4: PSFs of the hyperbolic phase metalens at different incident angles and their Strel ratio SR.]]
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| == Methods ==
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| === Angular Spectrum Method ===
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| 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.
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| 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.
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| <math display="block"> I = | \mathcal{F}^{-1} \{ \mathcal{F} \{U_0 \} \cdot \mathcal{H} \} | ^ 2 </math>
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| 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 <math> \mathcal{F} \{U_0 \} </math>, applying the appropriate transfer function <math> \mathcal{H} </math> 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.
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| === Parametrization of the Phase ===
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| 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 <math> c_n </math>. This parametrization not only reduces the computational complexity of simulating the metalens but also provides flexibility in optimizing its optical performance for specific applications.
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| <math display="block"> \Phi(\rho) = \sum_n c_n(\frac{\rho}{R})^n </math>
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| 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.
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| === Optimization Algorithm ===
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| 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.
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| <math display="block"> \mathcal{L} = \sum_\theta SR(\theta) = \sum_\theta \frac{I_{real}(x(\theta),y(\theta))}{I_{ideal}(0,0)}, \quad \theta \in \{0, 10, 20, 25\} </math>
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| == Results ==
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| === Optimized Phase Profile ===
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| The phase profile after optimization is shown in Figure 5. On the right, a slice of the phase at <math> y = 0 \text{ } \mu \text{m}</math> is compared to the same slice of the hyperbolic phase.
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| [[File:Optimized_phase.png|thumb|center|550 px|Figure 5: Optimized phase profile and the slice of optimized phase profile.]]
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| === PSF Comparison ===
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| 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 <math> SR > 0.1 </math> for incident angles at <math> 10^{\circ} </math>, <math> 20^{\circ} </math>, <math> 25^{\circ} </math>. On the other hand, The <math> SR </math> 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.
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| [[File:Psf_after.png|thumb|center|500 px|Figure 6: PSFs at different incident angles of the metalens with optimized phase.]]
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| 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 <math> 10^{\circ} </math>, <math> 20^{\circ} </math>, <math> 25^{\circ} </math>. 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.
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| [[File:MTF.png|thumb|center|500 px|Figure 7: MTFs along x-axis and y-axis of the metalens of hyperbolic phase and optimized phase.]]
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| === Image Simulations ===
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| [[File:S_sim.png|thumb|center|600 px|Figure 8: ]]
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| The following images are simulated using space-varying 2D convolution algorithm. In space-varying convolution, we divide the whole FOV into <math> n </math>-by-<math> n </math>
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| 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.
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| [[File:Simulation_img.png|thumb|center|400 px|Figure 9: Simulated images from space-varying convolution of the target image with PSFs of two metalenses.]]
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| == Conclusions ==
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| == Appendix ==
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| You can write math equations as follows:
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| <math>y = x + 5 </math>
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| You can include images as follows (you will need to upload the image first using the toolbox on the left bar, using the "Upload file" link).
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| [[File:Snip 20210106183207.png|200px]]
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