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| == Introduction ==
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| 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.
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| == Background ==
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| === What Is Metasurface? ===
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| 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.
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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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| === Parametrization ===
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| === Optimization ===
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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} , 20 ^ {/circ} , 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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| [[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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| == 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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