• Advanced Photonics Nexus
  • Vol. 4, Issue 3, 036009 (2025)
Anton Ovcharenko1,*, Sergey Polevoy2, and Oleh Yermakov1,3,*
Author Affiliations
  • 1V. N. Karazin Kharkiv National University, Department of Computational Physics, Kharkiv, Ukraine
  • 2O. Ya. Usikov Institute for Radiophysics and Electronics of NAS of Ukraine, Radiospectroscopy Department, Kharkiv, Ukraine
  • 3Leibniz Institute of Photonic Technology, Department of Fiber Photonics, Jena, Germany
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    DOI: 10.1117/1.APN.4.3.036009 Cite this Article Set citation alerts
    Anton Ovcharenko, Sergey Polevoy, Oleh Yermakov, "Forward and inverse design of single-layer metasurface-based broadband antireflective coating for silicon solar cells," Adv. Photon. Nexus 4, 036009 (2025) Copy Citation Text show less

    Abstract

    Almost half of the solar energy that reaches a silicon solar cell is lost due to the reflection at the silicon–air interface. Antireflective coatings aim to suppress the reflection and thereby to increase the photogenerated current. The conventional few-layer dielectric antireflective coatings may significantly boost the transmission of solar light, but only in a narrow wavelength range. Using forward and inverse design optimization algorithms, we develop the designs of antireflective coatings for silicon solar cells based on single-layer silicon metasurfaces (periodic subwavelength nanostructure arrays), leading to a broadband reflection suppression in the wavelength range from 500 to 1200 nm for the incidence angles up to 60 deg. The reflection averaged over the visible and near-infrared spectra is at the record-low level of approximately 2 % and 4.4% for the normal and oblique incidence, respectively. The obtained results demonstrate the potential of machine learning–enhanced photonic nanostructures to outperform the classical antireflective coatings.

    1 Introduction

    Photovoltaics studies ways to convert sunlight directly into electricity. Optical reflection at the surface of a solar cell is one of the main loss factors when high-index semiconductors are used. More specifically, the reflection of sunlight from a flat silicon surface ranges from 35% to 50% in the visible and near-infrared spectra,1 significantly reducing the efficiency of silicon solar cells. This problem may be overcome using antireflection coatings (ARCs), which suppress reflection losses at the interface of two media and, as a consequence, improve the power conversion efficiency. The simplest ARCs include single-, double-, and triple-layer thin dielectric films.2 They allow suppression of the reflection down to <1%, but only for a limited wavelength range, typically, for spectral window widths of between 100 and 300 nm.3 Besides, multilayered systems bring technological complexity and increased costs. Hence, the standard thin-film ARCs with great performance are multilayered and narrowband. By contrast, real-world applications require ultrathin ARCs that minimize light reflection over a wide range of wavelengths, particularly from visible to near-infrared, where the irradiance of sunlight is at its maximum.

    Some previous approaches to enhance light absorption and suppress reflection include dense ultrathin clusters of nanoparticles of various shapes, among them spheres, hemispheres, disks, pyramids, pillars, etc.48 The latter approaches include photonic nanostructures, especially the metasurfaces representing the periodic arrays of subwavelength plasmonic and dielectric scatterers.610 One should notice that plasmonic metasurfaces bring additional parasitic absorption,11 whereas materials used for all-dielectric metasurfaces exhibit negligibly small absorption in the visible and near-infrared ranges.12

    The proposed metasurface-enhanced solutions demonstrate high functionality in terms of the reflection reduction with ARCs on glass and other low-index substrates. Nevertheless, the application of ARCs for the widely used silicon solar cells is still a great challenge due to the high contrast between air (n=1) and silicon (n3.55), which results in high reflectance.13 The spherical and cylindrical scatterers as well as the subwavelength nanogratings implementing the Mie resonances for the reflection suppression open a new page for antireflection design.14,15 Specifically, Huygens’ metasurfaces1618 implementing the Kerker forward-scattering effect19,20 represent a new promising platform for antireflection tasks.21

    Inverse design is becoming increasingly important for developing advanced metasurfaces, especially as the desired functionalities become more complex.22,23 Although traditional design methods, such as phase matching, rely on human intuition and pre-existing solutions, the inverse design utilizes computational algorithms to explore a wider range of possibilities. This allows for the creation of free-form metasurfaces with enhanced light steering, antireflection, and diffraction properties, exceeding the capabilities of conventional forward design approaches. For example, inverse design can be used to create efficient large-angle beam-deflecting metagratings through topology optimization.24 In addition, the inverse design allows researchers to manipulate the diffraction orders of light, enabling higher dimensional modulation designs for metasurfaces.25 Beyond single functionalities, inverse design is particularly beneficial for creating multifunctional and broadband metasurfaces and meta-lenses.26 The integration of artificial intelligence, specifically machine learning algorithms, is further enhancing the capabilities of inverse design for metasurfaces. Techniques such as surrogate modeling allow for the efficient exploration of large design spaces, whereas end-to-end design strategies co-optimize both the physical structure and postprocessing algorithms for advanced imaging systems.2729 The engineering of the metasurface-based ARCs is a time-consuming process requiring a number of numerical simulations. The development of universal inverse design approaches is in high demand and should take into account the fabrication constraints.3032

    In this work, we aim to create a single-layer metasurface-based ARC for silicon solar cells with negligible absorption losses demonstrating a record-high broadband reflection suppression. We apply forward and inverse design approaches to find the optimum design of silicon metasurface on silicon substrate aiming to minimize the reflection in the wavelength from 500 to 1200 nm. The two design approaches produce metasurfaces based on the cross-circle and free-form meta-atoms, respectively. The developed metasurfaces suppress average (for any wavelength and polarization) reflectances up to 2% under normal incidence. For oblique incidence in the angular range up to 60 deg, the average reflectance is suppressed up to 4.4%, demonstrating better reflection suppression by about an order of magnitude compared with an unstructured flat silicon surface (Fig. 1). The results obtained outperform all known ARCs for silicon solar cells and even achieve the performance level of the coated and multilayered ARCs.

    Illustration of reflectance suppression achieved by a metasurface-based silicon antireflective coating, composed of cross-circle meta-atoms, demonstrating approximately an order of magnitude improvement compared with an unstructured silicon solar cell.

    Figure 1.Illustration of reflectance suppression achieved by a metasurface-based silicon antireflective coating, composed of cross-circle meta-atoms, demonstrating approximately an order of magnitude improvement compared with an unstructured silicon solar cell.

    2 Materials and Methods

    2.1 Statement of Problem

    We aim to design an etching pattern for the silicon surface of a solar cell to help alleviate the momentum mismatch between the incident solar radiation in the free space and the transmitted wave in the high-index dielectric medium. Within this task, we consider a broad wavelength window from 500 to 1200 nm containing over 80% of the total solar irradiation.33 Mathematically, the problem is formulated as a silicon planar metasurface design that sits between two semi-infinite media on each side—air and polycrystalline silicon, which stands for the solar cell material [see Fig. 2(a) for schematics]. The refractive index dispersion for the silicon material used for the metasurface and the substrate was the same as the one used in Refs. 34 and 35. As the absorption losses of the studied materials in the visible and near-infrared ranges can be neglected,12 only the real part of the refractive index dispersion was used. The information (Fig. S1 in the Supplementary Material) gives the case of the complex refractive index taking into account the real absorption losses of crystalline silicon. Moreover, we analyze only the reflection R(θ,λ) as a function of the wavelength λ and the angle of incidence θ, even though the true goal is to maximize the transmission. The typical absorption losses of the silicon in the wavelength range under study do not exceed 1%, i.e., ε/ε<0.01,13,34,35 where ε and ε are the real and imaginary parts of the silicon permittivity, respectively.

    Schematic depiction of the studied setup: (a) ARC as a single layer of Si etched on Si. (b) Hybrid meta-atom that is a superposition of a rectangular cross and cylindrical post geometries. The geometric parameters are the circle radius (R), cross arm width and length (W and L, respectively), height (H), and unit cell period (P).

    Figure 2.Schematic depiction of the studied setup: (a) ARC as a single layer of Si etched on Si. (b) Hybrid meta-atom that is a superposition of a rectangular cross and cylindrical post geometries. The geometric parameters are the circle radius (R), cross arm width and length (W and L, respectively), height (H), and unit cell period (P).

    2.2 Definitions

    To analyze the performance of the ARC, we introduce the angle-related (Rθ) and wavelength-related (Rλ¯) average reflectance coefficients defined as Rθ¯(λ)=θ1θ2R(θ,λ)dθ(θ2θ1),Rλ¯(θ)=λ1λ2R(θ,λ)dλ(λ2λ1).

    Other important values to evaluate the functionality of the proposed ARCs are the boosting factors related to the reflection suppression (ηR) and the enhancement of the transmitted power (PT) caused by adding the ARC with respect to a flat unstructured silicon. The first type of these values shows how many times the ARC reduced the reflectance depending on wavelength or angle of incidence ηθR(λ)=Rθ¯Si(λ)Rθ¯ARC(λ),ηλR(θ)=Rλ¯Si(θ)Rλ¯ARC(θ),where RSi and RARC correspond to the average reflectances for a flat silicon and various ARCs, respectively. The second type of the boosting factor value shows how many times the ARC has increased the power of the transmitted solar radiation {PθT(λ)=λ1λ2W(λ)[1RARC(θ,λ)]dλλ1λ2W(λ)[1RSi(θ,λ)]dλPλT(θ)=θ1θ2W(λ)[1RARC(θ,λ)]dθθ1θ2W(λ)[1RSi(θ,λ)]dθ,where RSi(θ,λ) and RARC(θ,λ) are the reflectance from the bare air–silicon interface and ARC-on-silicon, respectively, and W(λ) is the spectral distribution of the surface power of the Sun.

    Finally, we also introduce the integral value providing the total reflectance averaged over all the wavelengths and incident angles (R¯) defined as R¯=θ1θ2λ1λ2R(θ,λ)dλdθ(λ2λ1)(θ2θ1).

    In our case, the ranges of the incident angles and wavelengths within the paper are [θ1,θ2]=[0,60]° and [λ1,λ2]=[500,1200]  nm.

    2.3 Optimization Approach

    Within this paper, we set out to compare both commonplace forward design and advanced inverse design approaches. Even though the optimizations we carried out with normal incident waves only, when evaluating the device performance, we calculate reflections at angles up to 60 deg as oblique reflections are crucial for the performance of a real-life solar cell. To calculate the average values of reflectance over both angle and wavelength, we will use the discretized form of average reflectance [Eq. (1)] with the 5 nm step for the wavelength and 1 deg for the angle.

    2.3.1 Cross-circle forward design

    The forward design approach is based on using the hybrid cross-circle meta-atom geometry, shown in Fig. 2(b), previously used by Ndao et al. for a metalens design in Ref. 36. This design provides the polarization-independent performance (see Fig. S2 in the Supplementary Material). It allows the electric and magnetic dipole resonances of the meta-atom to be controlled separately. The antireflection effect is based on the forward scattering of light by the lattice of meta-atoms due to the collective generalized Kerker condition.1621 The fields of the eigenmodes are shown in Fig. S3 in the Supplementary Material. The variable parameters are the cross arm width and length, W and L, respectively, the circle radius, R, meta-atom height, H, and the unit cell period, P. The design process in this case consists of simulating different combinations of these parameters within the acceptable ranges and choosing the best performing one. For simplicity, within the optimization process, we consider only the normal incident light, otherwise, the number of simulations would be prohibitively large.

    2.3.2 Free-form inverse design

    The optimization procedure of the inverse design approach utilizes the ResNet-based GLOnet algorithm35,37 modified by us to be applied to a 3D problem with the use of the Reticolo software package as a solver, which is a MATLAB-based implementation of the rigorous coupled-wave analysis (RCWA).38 In a nutshell, the process is to optimize a probability distribution in a many-dimensional latent space—the number of dimensions is equal to the number of free parameters (pixels)—defined via weights of the generator network. The network can be thought of as a more sophisticated high-level abstract representation of the target geometry. This way, when the space is sampled randomly, the generator produces predominantly high-performing designs. The training of the generator, therefore, constitutes updating said weights by backpropagating the loss defined as35L(x,g,Eff)=1Mm=1M1σexp(Eff(m)σ)x(m)g(m)γ·1Mm=1M|x(m)|(2|x(m)|),where x stands for the device image representation, in our case, pixel values of ±1 stand for air and silicon, respectively, Eff and g are its simulated efficiency and gradient with respect to the geometry, M is the batch size and m indicates a specific device from the batch, and σ and γ are the tunable hyperparameters. In these calculations, x, Eff, and g are considered mathematically independent from each other, for the purposes of differentiation. The second term in loss is so-called regularization—it is minimized when a pixel absolute value approaches 1, which biases the generator to output binary images. The tunable hyperparameter γ is necessary so that this regularization does not overwhelm the whole loss, at least not in the beginning. The gradient g=Eff/ε(x,y) is calculated using the adjoint technique,24,39,40 which involves a so-called adjoint simulation with the input and target directions swapped. Thus, normally, we have one additional simulation per each target channel. In our case, instead of simulating many of the transmitted orders, we focus on the reflection, where only the single order is propagating and the adjoint field would be equal to the forward one with the additional reflected field phase factor, alleviating the need for additional simulations.

    During the backpropagation, the weights are updated by adding the gradient of L, which is computed using the chain rule as follows: Lw=m=1MLx(m)x(m)w,where w stands for the vector of weights of the deep network. The first term, namely, Lx is calculated analytically from the loss expression, which uses Eq. (5), whereas the second one, xw, is computed using the automatic differentiation capabilities of any deep learning framework. However, it should be noted that during the GLOnet optimization process, the efficiency used for the loss function was weighted against the black body radiation spectrum to prioritize the shorter wave region of the band, which in turn will be beneficial both for the applications and average reflectance values as this region tend to show larger reflectance overall due to more complicated light–matter interactions.

    The generator is able to generate pixelized images that define the design profile in the XY cross-section. As we enforce the reflection symmetry along x- and y-axes, it actually generates a quarter of the image, which is later replicated for the simulation and gradient calculation. The layer thickness and unit cell period must be fixed beforehand. We chose these values to match those from the best designs of the parameter sweep; namely, we fixed the height of H=150  nm and the period of P=380  nm. The same as for the forward design study, only the normal incidence is considered.

    2.4 Numerical Simulation

    To conduct numerical simulations, we have used Reticolo38 for RCWA simulations during optimizations and Comsol Multiphysics41 for post-optimization finite element method (FEM) modeling. Both methods work in the frequency domain. RCWA represents the problem as a stack of layers and is analytic in the perpendicular direction. The surrounding air and silicon substrate are considered semi-infinite. The method is accurate, provided a sufficient number of Fourier terms are taken into account. Here, we take 21 terms (from 10 to +10), which our convergence study proved to be adequate.

    In the FEM simulation, the silicon substrate layer had a finite thickness of 500 nm with a 300 nm thick perfectly matching layer below. Figure S4 in the Supplementary Material compares the spectra obtained with the two methods showing a good coincidence when the maximum cell size in the FEM discretization is below 50 nm.

    3 Results

    The designs of metasurfaces acting as ARCs have been obtained using the optimization approaches described above (see Fig. S5 in the Supplementary Material for GLOnet training progress visualization). Figures 3 and 4 show optimal geometries and reflectance colormaps of the cross-circle and free-form optimized meta-atoms, respectively, as a function of incident wavelength and incident angle for both transverse electric (TE) and transverse magnetic (TM) incident polarizations. The optimum cross-circle structure corresponds to the values of 90, 60, and 300 nm for R, W, and L, respectively (rounded by up to 2 nm to the nearest tens). The average values over the wavelength range from 500 to 1200 nm for the reflectance at normal incidence are Rλ¯(θ=0°)=2.07% and Rλ¯(θ=0°)=1.96% while averaged over angles up to 60 deg and both polarizations – R¯=5.02% and R¯=4.42% for cross-circle and free-form meta-atoms, respectively. The colormaps reveal the presence of an area for lower wavelengths and higher angles where the reflection is increased, which corresponds to the emergence of the first diffraction order in reflection.

    Reflectance maps for the cross-circle geometry showed in (a) with R=90 nm, W=60 nm, L=300 nm, H=150 nm, and P=380 nm, for (b) TE and (c) TM polarized plane waves as a function of incident wavelength and incident angle.

    Figure 3.Reflectance maps for the cross-circle geometry showed in (a) with R=90  nm, W=60  nm, L=300  nm, H=150  nm, and P=380  nm, for (b) TE and (c) TM polarized plane waves as a function of incident wavelength and incident angle.

    Reflectance maps for the free-form geometry showed in (a) for (b) TE and (c) TM polarized plane waves as a function of incident wavelength and incident angle.

    Figure 4.Reflectance maps for the free-form geometry showed in (a) for (b) TE and (c) TM polarized plane waves as a function of incident wavelength and incident angle.

    Figure 5 compares the reflection spectra for bare silicon surface, single-layer (Si3N4 layer of the 80 nm thickness) antireflection coating (SLARC),14 and the two proposed designs (cross-circle and free-form meta-atom arrays) with respect to the standard solar spectrum. One can see that for the developed metasurface-based ARCs, the reflection is below 5% (marked by the horizontal dashed line in Fig. 5) for the majority of the band, which is a significant improvement over standard analogs, such as SLARC that shows comparable performance only within 200  nm wide window. Note that the position of the SLARC minimum is determined by its thickness. Here, it is chosen to minimize the average reflection, which is our metric of choice. For solar cell applications, it is obviously more favorable to coincide the maxima of solar emission and the minima of the ARC’s reflectance spectrum. The chosen wavelength range of 800 nm spectral width contains 80% of the solar energy, whereas the uniform single- or double-layer ARC may suppress the reflection efficiently within the 200 to 300 nm window. Hence, the engineering solution for the broadband single-layer ARC requires structuralization and patterning.

    Reflectance spectra of the bare silicon substrate (light gray), single-layer antireflective Si3N4 80 nm coating (black), optimal cross-circle (red), and free-form (blue) periodic 150 nm thick arrays under (a) normal incidence and (b) 30 deg oblique incidence. The dashed horizontal line marks reflectance level of 5%. The shaded region marks the AM1.5 solar spectrum.

    Figure 5.Reflectance spectra of the bare silicon substrate (light gray), single-layer antireflective Si3N4 80 nm coating (black), optimal cross-circle (red), and free-form (blue) periodic 150 nm thick arrays under (a) normal incidence and (b) 30 deg oblique incidence. The dashed horizontal line marks reflectance level of 5%. The shaded region marks the AM1.5 solar spectrum.

    Figure 6 plots the boosting factors for the reflected and transmitted power, ηR and PT, respectively, averaged over either wavelength or incident angle, for the three types of single-layer coatings considered. One can notice the multifold enhancement in the reflection suppression compared with the flat unstructured silicon surface [Figs. 6(a) and 6(b)]. One can see that the metasurface-based coatings show a broadband enhancement of capturing the reflected power, whereas the standard SLARC has only a single albeit extremely strong resonance, where the reflectance is very close to 0. The boosting of the transmitted power with metasurfaces is not multifold because the transmission of the flat silicon surface is 50% to 65% and cannot be increased more than 1.5 to 2 times. Uniform layer structures are also less sensitive to the incidence angle, unlike structured periodic layers, which exhibit more reflection under a larger angle; however, their performance is still better overall. The metasurface-based ARCs are much better at small angles of incidence, whereas their performance for an angle larger than 45 deg becomes comparable to the SLARC. However, the developed metasurface-based ARCs are significantly better almost at all wavelengths except the optimized wavelength region for the SLARC.

    (a) and (c) Reflectance boosting factor, ηR [see Eq. (2)] and (b) and (d) power transfer boosting factor PT [see Eq. (3)], averaged over the wavelength and incident angle, respectively, for the single-layer antireflective Si3N4 80 nm coating (black), optimal cross-circle (red), and free-from (blue) periodic 150-nm thick arrays.

    Figure 6.(a) and (c) Reflectance boosting factor, ηR [see Eq. (2)] and (b) and (d) power transfer boosting factor PT [see Eq. (3)], averaged over the wavelength and incident angle, respectively, for the single-layer antireflective Si3N4 80 nm coating (black), optimal cross-circle (red), and free-from (blue) periodic 150-nm thick arrays.

    We did our best to find the best designs for a few of the geometries mentioned within a reasonable time frame for our band and materials of interest. The results are summarized in Table 1. The exact parameters used to obtain the reflectance values are provided in Table S1 in the Supplementary Material. The comparison shows the free-form design to have the lowest reflection among all single-layer (uncoated) candidates under both normal and oblique incidences. However, it should be noted that the cross-circle metasurface is not far behind. The two structures can be argued to be visually similar. Indeed, as Fig. 5 shows, their reflection spectra have a similar nature. The field profiles for 580 nm incident wavelength are depicted in Fig. 7. They demonstrate that the same fundamental modes are responsible for the metasurfaces operation. Similar plots for a few other important wavelengths can be found in Fig. S6 in the Supplementary Material. The fact that the free-form design can be considered a perturbation of a cross is due to geometric constraints used during the optimization (symmetries and Gaussian filters to eliminate sharp edges and small features), as well as large wavelength bands considered.

    Best obtained reflectance (%)
    Structure typeReferenceNormal incidenceOblique incidence
    Multilayered system
    Three-layer thin-film stack372.122.95
    Coated silicon cylinder141.863.01
    Single-layered system
    Single-layer uniform coating147.839.46
    Silicon cylinder145.619.02
    Dual-sized pillar152.695.23
    Cross-circle waveguide36 (originally) as ARC—this paper2.075.02
    Free-form structureThis paper1.964.42

    Table 1. Comparison of efficiencies of different typical design approaches using single- or multilayer planar structures.

    Field profiles for the normally incident 580 nm plane wave for (a) cross-circle and (b) free-form designs.

    Figure 7.Field profiles for the normally incident 580 nm plane wave for (a) cross-circle and (b) free-form designs.

    4 Discussion and Conclusion

    One can apply a variety of known photonic resonant structures to reduce reflection from a silicon surface with various degrees of success. Generally, as a rule of thumb, the more complicated the design, i.e., the more useful design parameters there are, the wider range of performance values one can achieve. Previously, periodic arrays of simple geometries were applied as ARCs on silicon. These include silicon cylindrical subwavelength nanoparticles with and without Si3N4 coating layer14 or a dual-period square post metasurface.15

    We use GLOnets to find the optimal free-form pattern design. In fact, the original GLOnet authors, Jiaqi Jiang and Jonathan Fan, have already used it to design a multilayer thin-film ARC.37 They have managed to present a design with 1.8% reflectance over 400 to 1100 nm wavelength range, 0 to 60 deg angle range, and both polarizations. Note, however, that this value was calculated as a sum over the solid angle, rather than an average value [Eq. (1)], which we use throughout this paper. Moreover, the obtained design assumes constant real-value material refractive indices. For a fair comparison, we have carried out a few GLOnet optimizations for the three-layer dispersive dielectric thin-film structures with dispersion available from the original work.37 The best-performing device showed reduced reflection down to 2.12% in the range of 500 to 1200 nm under the normal incidence and 2.95% averaged over the angles and polarizations. Overall, the thin-film design approach shows the smallest performance decay due to the oblique incidence, compared with other candidates. This is a fairly well-known drawback of planar structures especially compared with more complicated gradual-change inhomogeneous structures, e.g., moth-eye or bullet-type meta-atoms.1

    The fabrication of the proposed silicon metasurfaces may be done with electron-beam lithography methods.17,42 However, we are generally not limited to silicon as a material for metasurfaces, especially for low-index substrates and organic solar cells. Thus, the two-photon polymerization lithography opens a plethora of possibilities for advanced manufacturing and designing allowing to implement of the nontrivial configurations of meta-atoms with high precision.43,44

    In conclusion, forward design can produce very promising results given the appropriate choice of geometric parameters, whereas when using the inverse design approach, one does not have to settle on a geometry type in advance and be reasonably confident that the solution would be close to the global optimum. We have managed to obtain both forward-designed cross-circular and free-form inverse-designed structures with the best reported antireflection properties for single-layer structures. In our setup, parametric sweep for the cross-circular geometry with steps of 15  nm in W, R, and L and 20 nm for the wavelength, using fixed values of height and period, took 6 to 7 min computed in parallel on a 16-core processor. Obtained optimal values were further manually refined on a smaller mesh with additional runs. GLOnet training time are highly dependent on its hyperparameters and can be anything from under an hour to several hours on the same hardware setup. A reasonable set of parameters results in a few hours of training time, which is several times longer than a more precise parametric sweep, but the parameter space explored is incomparably more wasted.

    Biographies of the authors are not available.

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    Anton Ovcharenko, Sergey Polevoy, Oleh Yermakov, "Forward and inverse design of single-layer metasurface-based broadband antireflective coating for silicon solar cells," Adv. Photon. Nexus 4, 036009 (2025)
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