Integrating machine learning (ML) with optical metamaterials holds promise for accelerating advancements in optical technologies by facilitating the development of smarter and more efficient devices.1^–4 Leveraging the capabilities of artificial intelligence and engineered material systems, the synergy of these distinct fields is poised to expand the design spaces, optimize structures, address real-world applications, and solve practical challenges. However, despite the substantial advancements in metamaterials over the last two decades, their design process is primarily constrained by human intuition and is limited to simple topologies such as spheres, cylinders, cubes, or their combinations (meta-molecules), whereas the designs relying on complex, irregular, or pseudo-random shapes are likely to uncover whole new design spaces.5 Consequently, these limitations present a prime opportunity for ML to make a meaningful impact and significant contribution to metamaterials and the field of Mie-tronics.6^,7 Following their remarkable contributions to other fields of research and technology, ML-based approaches have recently emerged as a powerful tool in photonics for optimizing and analyzing various subwavelength structures and systems. In this perspective, inverse design methods have already facilitated the so-called “design by specification” techniques that produce irregularly shaped integrated optical devices with desired functionalities.8 However, although such performance-driven approaches represent an essential step toward engineering optical meta-devices at nanoscales, the physical interpretation of the mechanisms enabling their operation is not immediately intuitive.

In particular, the underlying physics behind nanophotonic structures’ high-dimensional design space and their optical responses’ intricate interplay can be harnessed as critical advantages in the ML design processes to explore novel and more efficient meta-devices. In this perspective, the first step in utilizing such an overlooked physics within the high-dimensional design space of nanophotonic structures is to implement the underlying information of how light interacts with the individual building blocks of these platforms, meta-atoms, through their resonant optical responses.9^,10 Although only a few recent numerical ML-based studies have focused on the resonant response of meta-atoms in terms of their multipoles (or resonant cavity modes), they were limited to regular and relatively simple geometries that typically support only low-order multipolar resonances.1^,4^,5^,11^–13 However, suppose physics-based ML could be applied to design meta-atoms with a specified multipolar response within a spectrum of interest. In that case, it may facilitate entirely new regimes of light–matter interactions relying on higher-order electric and magnetic multipolar resonances that have been largely overlooked. In particular, recently, achieving high-order multipolar responses has gained significant attention, as these local resonant modes support high-quality factors ($Q$-factor), which, contrary to their non-local counterparts, can also provide an opportunity to manipulate the wavefront of the incoming light beam14 and can lead to new mechanisms for nonlinear wavelength conversion enabled by magnetic resonances, high field confinement, and considerable mode overlap of electromagnetic fields at different frequencies essential for efficient harmonic generation.15^–18

Recently, we proposed a theoretical ML-based approach that facilitates forward and inverse design of individual complex meta-atom topologies and fast prediction of the electromagnetic field distribution within these optimized nanostructures, enabling the desired low and high multipolar (resonant) responses.19 Here, we report an experimental validation of the effectiveness of this ML approach in generating uniquely shaped meta-atoms with tailored higher-order resonances. We use the forward prediction model (FPM) to predict the scattering response of arbitrarily shaped individual meta-atoms and an inverse design model (IDM) to design meta-atoms possessing the desired optical responses at a given operating wavelength within a second. In contrast to previous ML-based approaches for meta-structure design, here, we optimize the response of the individual meta-atoms from the physics of light–matter interaction. This new capability, once validated, would allow the development of an extensive library of meta-atoms with a wide range of desired electromagnetic properties that can be used for custom designing of versatile nanophotonic devices and systems without any constraints at the time of simulations. For experimental validation purposes, using the developed ML toolkit, we design three different shapes meta-atoms possessing electric dipole (ED), magnetic dipole (MD), and magnetic quadrupole (MQ) Mie-type resonances at three different operating wavelengths within the visible spectral range, as shown in Fig. 1. We then study the underlying physical mechanisms of the resonances with the framework of quasi-normal mode (QNM) expansion theory and show good agreement between the two approaches. Although in the original ML design, we considered isolated meta-atoms surrounded by air, in our experimental studies, such meta-atoms are usually fabricated on a substrate and arranged periodically. Therefore, we investigated the effects of the glass substrate and that of the various lattice constants. Subsequently, we fabricated the designed titanium dioxide (${\mathrm{TiO}}_2$)-based meta-atoms on top of a glass substrate using electron beam lithography (EBL). We characterized their optical response using a white light spectroscopy setup. The experimental results agreed with the theoretical predictions based on multipole decomposition and QNM expansion theories, confirming the validity of the developed IDM to design meta-atoms with desired responses within the visible range. To further demonstrate the application of the presented work, we implemented the designed meta-atom supporting MQ resonant mode to generate third harmonic (TH) frequencies within the vacuum ultraviolet (VUV) spectrum. To the best of our knowledge, this is the first experimental realization of an ML-based fast optimization of arbitrary-shaped dielectric meta-atoms in the optical regime, which not only serves as a compelling confirmation of the developed design approach but also will likely open new avenues for innovative applications in optical material design and advanced light manipulation technologies.

Although the theoretical details of our proposed ML model are discussed in our previous work,19 here, we focus on the experimental validation of the ML predictions within the context of Mie scattering and study the effect of different inevitable experimental parameters. Therefore, we only briefly revisit the core aspects and the dataset used in training our models, setting the foundation for the experimental validation and the findings presented below. In our previous study, we developed three ML approaches to efficiently approximate multipolar resonances of meta-atoms and inverse design their shapes to achieve the desired response avoiding the time-consuming conventional numerical methods such as the finite-element method (FEM) or finite difference time domain for complex structures and predict three-dimensional (3D) field distribution within the meta-atoms. Specifically, our method utilized a densely connected convolutional network (DenseNet) encoder architecture, significantly expediting the process compared with conventional simulation software (see Appendix A). The developed ML approach involved compressing meta-atom shape information into a reduced dimension and employing fully connected dense layers to predict multipolar resonances at specific wavelengths. Our dataset consisted of 36,300 unique combinations of wavelengths and meta-atoms with a fixed height of 320 nm, shown in the blue box in Fig. 1 (see Sec. I and Fig. S1 in the Supplementary Material for more details on the implemented geometries). To derive the multipolar response of the meta-atoms used for training, we conducted numerical simulations using FEM COMSOL Multiphysics, obtained the corresponding induced currents within each meta-atom using the calculated polarization distribution ($\mathit{J}=\partial \mathit{P}/\partial t$) and then retrieved their multipolar contributions (see details in Appendix B and Appendix C). We found that the inverse design of meta-atoms is likely complicated by the potential non-existence of specific multipolar resonances and the possibility of multiple meta-atoms having similar resonances. Given these additional challenges, traditional models such as DenseNet were not directly applicable. Thus, we implemented a tandem inverse design model (TIDM), effectively balancing accuracy and computational speed.20 In particular, the TIDM incorporates a DenseNet decoder for creating meta-atom shapes and a pre-trained FPM for resonance validation. From this perspective, the mean-squared error (MSE) metric guided the design process, ensuring that the decoder converged to the designs with the desired resonances. This process involved using FPM to estimate the IDM-designed meta-atoms moments and an MSE loss to quantify the discrepancies between predicted and desired moments. Utilizing gradient descent, the meta-atom design was iteratively adjusted to minimize the loss function, thereby refining the design quality

The design of meta-atoms capable of generating distinct optical responses is essential for a range of applications, including the enhancement of optical nonlinearity,16 the creation of Fano resonances,21 and the directional scattering of light,22 to name a few. This section focuses on numerically designing meta-atoms that exhibit EDs, MDs, and MQs at distinct operating wavelengths. We employed specific loss functions for each design example, expressed as $L_{\mathrm{ED}}=-{\sigma }_{\mathrm{ED}}/{\sigma }_{\mathrm{Tot}}$, $L_{\mathrm{MD}}=-{\sigma }_{\mathrm{MD}}/{\sigma }_{\mathrm{Tot}}$, and $L_{\mathrm{MQ}}=-{\sigma }_{\mathrm{MQ}}/{\sigma }_{\mathrm{Tot}}$, wherein ${\sigma }_{\alpha }$ represents the scattering cross-section of a particular resonant mode (ED, MD, MQ, or total). These loss functions were used to iteratively refine the resonant features within the input layer until they converged to their minimum, resulting in the optimal design parameters with maximal ratiometric peaks within the scattering spectra. Upon achieving this convergence, the finalized design of the meta-atom was derived from these optimized parameters, ensuring that the desired resonant characteristics were met (see red and magenta boxes in Fig. 1). Using this method, we designed the meta-atoms with the ED, MD, and MQ Mie-type resonances with the contribution of 67%, 61%, and 50% of the total response at the operating wavelengths of ${\lambda }_{\mathrm{ED}}=900\mathrm{nm}$, ${\lambda }_{\mathrm{MD}}=780\mathrm{nm}$, and ${\lambda }_{\mathrm{MQ}}=570\mathrm{nm}$, respectively, as shown in Figs. 2(a)–2(c). It is noteworthy to mention that the main objective of the developed IDM was to maximize the meta-atoms performance precisely at the desired wavelengths of ${\lambda }_{\mathrm{ED}}$, ${\lambda }_{\mathrm{MD}}$, and ${\lambda }_{\mathrm{MQ}}$ (ratiometric peaks), whereas the spectral response of the meta-atoms at other wavelengths (highlighted with the blue-shaded region) was not of interest.

Although our previous ML model and its corresponding training dataset were obtained for the case when the meta-atom is located within a homogenous background medium (air with $n_b=1$), in the realistic experimental setup, the meta-atoms are fabricated on a substrate, which, in general, might modify the optical response of the meta-atoms in the uniform air background, designed using the developed IDM.23 Because in our experiments, the ${\mathrm{TiO}}_2$ nanostructures were fabricated on a glass substrate with the refractive index of $n_s=1.45$ in the operating wavelength range of interest (see Sec. II in the Supplementary Material and Fig. S2 for the ellipsometry measurements of the glass substrate), we compare the optical response of the designed meta-atoms with and without the presence of substrate as shown in Figs. 2(d)–2(f). As can be seen from these results, compared with the homogenous (air) background (blue dashed curve), the corresponding optical response of the meta-atom on a substrate (red color curve) is not only shifted toward longer wavelengths but also broadened. We found that the meta-atom with the dominant MD is more sensitive to the substrate’s presence than other meta-atoms, which can be attributed to its particular 3D field distribution and how it extends to the surrounding medium. The reduction of the resonant mode amplitudes is due to the decrease of the refractive index contrast of the meta-atom when it is placed on a glass substrate ($\mathrm{\Delta }n\approx 0.69$) compared with the case when it is surrounded by air ($\mathrm{\Delta }n\approx 1.14$).

It should be noted that the standard Mie theory applies to scenarios, in which the meta-atom is surrounded by a homogeneous medium, whereas including the substrate leads to asymmetry in the surrounding environment. Although several recent studies have shown how multipole moments are modified in the presence of a substrate, the generalized approach requires the origin of the coordinates to be shifted from the center of the meta-atom to the surface of the substrate, which subsequently leads to a different definition of multipole moments.23^,24 Therefore, to avoid such an issue and keep consistent definitions of the multipole moments for the cases with and without the substrate, we investigated the electric and magnetic field distributions at their corresponding resonance wavelengths, as shown in Figs. 2(g)–2(i). It can be seen that the electric and magnetic (black arrows) field distributions of the asymmetric surrounding medium (red boxes) for each of the meta-atoms closely resemble their symmetric environment counterpart (blue boxes). We note that due to the non-negligible contribution of other Mie-type modes at the design wavelengths, the field distributions within the meta-atoms are the results of beating all the modes rather than the mode profile of the dominant resonant mode, as shown in the blue boxes of Figs. 2(g)–2(i) for a given plane ($x\text{-}y$). In particular, although the field distribution of the asymmetric MD meta-atom on a substrate differs from its symmetric counterpart, the scattering spectra stay primarily unchanged (see Sec. III in the Supplementary Material and Fig. S3 for more results and estimation of multipole moments in the presence of the substrate). Therefore, the predicted spectral responses of the meta-atoms obtained from the IDM in the symmetric (air) case give an accurate prediction for the asymmetric case. This result suggests that the developed model can be used in realistic nanophotonic applications that usually rely on the complex distributions of multiple meta-atoms on a substrate.

To understand the underlying physical mechanisms of light scattering by the meta-atoms designed using the IDM, we have studied the optical response of these meta-atoms using the QNM expansion theory.25^,26 In particular, QNMs of dielectric resonators are the solutions to Maxwell’s wave equation in the absence of a source as $[\begin{array}{cc}0& -i{\mu }_0^{-1}\nabla \times \\ -i{ϵ}^{-1}\nabla \times & 0\end{array}][\begin{array}{c}{\tilde{\mathit{H}}}_m(\mathit{r})\\ {\tilde{\mathit{E}}}_m(\mathit{r})\end{array}]={\tilde{\omega }}_m\left[\begin{array}{c}{\tilde{\mathit{H}}}_m(\mathit{r})\\ {\tilde{\mathit{E}}}_m(\mathit{r})\end{array}\right]$ wherein ${\tilde{\mathit{E}}}_m(\mathit{r})$ and ${\tilde{\mathit{H}}}_m(\mathit{r})$ are the spatial distributions of the electric and magnetic modes ($m$) inside the resonator, ${\tilde{\omega }}_m={\omega }_m-\frac{i}{2{\tau }_m}$ is the complex frequency, with its real part being the resonant frequency of the meta-atom and the imaginary part being related to the mode lifetime. By solving the eigenvalue problem, the optical response of the meta-atom, under the excitation field of ${\mathit{E}}_b(\mathit{r},\omega )$, can be expressed in terms of the modal expansion as $[{\mathit{E}}_s(\mathit{r},\omega ),{\mathit{H}}_s(\mathit{r},\omega )]{|}_{{\mathit{E}}_b}=\sum _m{\alpha }_m(\omega )[{\tilde{\mathit{E}}}_m(\mathit{r},{\tilde{\omega }}_m),{\mathit{H}}_m(\mathit{r},{\tilde{\omega }}_m)]$ with ${\alpha }_m$ denoting the excitation coefficients, as outlined in Appendix D. Therefore, the scattering cross-section of a loss-free meta-atom can be retrieved from its modal fields as ${\sigma }_{\text{sct}}(\omega )=-\frac{\omega }{2S_0}{\int }_{V_p}\mathrm{Im}\{({ϵ}_p-{ϵ}_b)({\mathit{E}}_s(\mathit{r},\omega )+{\mathit{E}}_b(\mathit{r},\omega ))·{\mathit{E}}_b^{*}(\mathit{r},\omega )\}{\mathrm{d}}^3\mathit{r}$. To gauge the light–matter interaction within the QNM, we first retrieved the dominant eigenmodes in the spectral region of $500\mathrm{nm}<\lambda <900\mathrm{nm}$ and then derived the corresponding scattering cross-section of each meta-atom, as shown in Figs. 3(a)–3(c).

Figure 3 shows that the emergence/suppression of the multipoles radiated by each eigenmode aligns perfectly with the maxima and minima of the total scattering cross-section spectrum obtained from the IDM. Note that for precise reconstruction of the scattering cross-section, it is essential to consider background modes (shown with dashed lines), regardless of their resonances falling beyond the examined spectral range. Therefore, it is possible to view the emergence of these Mie-type resonant modes as a result of interaction between the dominant QNMs (shown with solid lines) and the rest of the modes (dashed lines), contributing to the background scattering. Since the QNMs do not obey the conventional conjugate inner product rules as opposed to the orthogonal modes in Hermitian systems, when the incident field impinges on the resonator, energy exchange takes place between the dominant and the background QNM fields.27^,28 Such energy exchange results, in some cases, in the negative contribution of scattering cross-section, as shown in Figs. 3(b) and 3(c). In particular, such positive and negative interferences of different QNMs, regardless of their resonances falling beyond the examined spectral range, lead to the emergence/suppression of various resonant modes. To better understand how background interference affects the desired response, we show the normalized electric field distributions of each meta-atom’s first two dominant QNMs in the $x\text{-}y$ plane in Figs. 3(d)–3(f). From these figures, it is clear that adding the QNM and background (also referred to as mode beating or interference) field distribution results in the total field we found within the meta-atoms. Comparing the results collated in Figs. 2 and 3, it is evident that the results obtained from the ML-based Mie-tronics and QNM framework agree well with each other, suggesting the validity of our numerical results for the case of isolated meta-atoms.

While experimental characterization of a single meta-atom is possible, typically, they are arranged in periodic arrays to form metasurfaces. However, due to near- and sometimes far-neighbor coupling, the optical response of the periodic array of the meta-atoms may significantly differ from that of an isolated case.29 Therefore, to validate the results obtained for an isolated meta-atom, we first need to find the periodicity that would be large enough to minimize such neighbor-to-neighbor coupling. To reduce the optical coupling between meta-atoms, we performed the numerical simulations using COMSOL Multiphysics and selected a lattice constant of the array to be $p=1\mu \mathrm{m}$, which exceeds both the largest dimensions of the meta-atoms and the operational wavelength range, thereby reducing the likelihood of interaction effects that could dominate the optical response (for more details see Sec. IV in the Supplementary Material and Fig. S4.). To calculate the transmittance (zeroth diffracted order), we used normal-incidence plane-wave excitation and evaluated the power flow through the sample. The calculated transmittance spectra are plotted in Figs. 4(a)–4(c), which show the minima at the spectral positions where the Mie-type resonant modes were predicted [Figs. 2(a)–2(c)]. To further clarify the existence of the desired modes within the designed meta-atoms in the periodic array, we plotted the electric field distributions within the meta-atom in the $x\text{-}y$ plane in the inset of Figs. 4(a)–4(c) at the corresponding designed operating wavelength. Comparing the calculated field distributions with those obtained from the isolated meta-atom on the glass substrate [Figs. 2(d)–2(f)] demonstrates a good agreement in the optical response for both cases at the specified wavelength.

Based on the calculated transmittance spectra, the emergence of the resonant minima within the periodic array can be approximately linked to the extinction cross-section of an isolated meta-atom through ${\sigma }_{\text{ext}}(\lambda )\approx \alpha (1-T(\lambda ))$, wherein $\alpha$ denotes the fitting parameter related to the periodicity of each array.30 Therefore, the extinction cross-section, which is equivalent to the scattering cross-section in the operating range of interest ($k_{{\mathrm{TiO}}_2}=0$ at $500\mathrm{nm}<\lambda <900\mathrm{nm}$), can be calculated as shown in Figs. 4(d)–4(f) with blue color dots. For better comparison, we have also plotted the scattering cross-section of an isolated meta-atom on top of the glass substrate with a red color. This figure shows that the calculated scattering cross-sections indicate a good agreement between the optical responses of isolated meta-atoms and those within the array, with the existence of minima in the scattering spectra at wavelengths corresponding to the regions where the transmittance approaches $T=1$. The observed agreement can be explained by considering the relation between an isolated meta-atom’s scattered electric field and the array configuration’s transmission coefficient. According to the multipole expansion theory, the scattered field in the far-field can be expressed as the superposition of various multipoles [up to the electric octupole (EO) term] as31$$\begin{gathered} {\mathit{E}}_{\text{sct}}(\mathit{n})=\frac{k_0^2\exp (i{k}_{0}r)}{4\pi {ϵ}_{0}r}([\mathit{n}\times [\mathit{D}\times \mathit{n}]]+\frac{1}{c}[\mathit{m}\times \mathit{n}]+\frac{i{k}_0}{6}[\mathit{n}\times [\mathit{n}\times \hat{Q}\mathit{n}]] \\ +\frac{i{k}_0}{2c}[\mathit{n}\times \widehat{M}\mathit{n}]+\frac{k_0^2}{6}[\mathit{n}\times [\mathit{n}\times {\widehat{O}}^{(e)}(\mathit{n}\mathit{n})]]), \end{gathered}$$(1)where $\mathit{D}$ corresponds to the exact total electric dipole, $\mathit{m}$ is the exact MD moment, and $\widehat{Q}$, ${\widehat{O}}^{(e)}$, and $\hat{M}$, represent the electric quadrupole (EQ), EO, and MQ tensors, respectively; $\mathit{n}=\mathit{r}/r$ is the unit vector directed from the meta-atom’s center towards an observation point, and $c$ and $k_0$ are the speed of light and wavenumber in vacuum. The transmission coefficient of an array of identical meta-atoms can be expressed in terms of the parameters of Eq. (1) via $t=1+i|\mathrm{\Xi }|[A_p\exp (i{\phi }_p)+A_m\exp (i{\phi }_m)-A_Q\exp (i{\phi }_Q)-A_M\exp (i{\phi }_M)-A_O\exp (i{\phi }_O)]$, wherein $A_p$, $A_m$, $A_Q$, $A_M$, and $A_O$ represent the magnitude of the ED, MD, EQ, MQ, and EO moments. At the same time, ${\phi }_i$ represents their corresponding phases, and $|\mathrm{\Xi }|$ is a constant related to the amplitude of the incident wave and the periodicity of the array.31 The scattering cross-section, ${\sigma }_{\text{sct}}\propto 1-T=1-|t{|}^2$, can then be expressed in terms of the $A_i$ and ${\phi }_i$ as $\sigma \propto |\mathrm{\Xi }{|}^2|A_p\exp (i{\phi }_p)+A_m\exp (i{\phi }_m)-A_Q\exp (i{\phi }_Q)-A_M\exp (i{\phi }_M)-A_O\exp (i{\phi }_O){|}^2$, which justifies why these two responses agree with each other. Therefore, for sufficiently large periodicity, such that the mode coupling between neighboring meta-atoms is negligible, the optical response of the predicted meta-atoms can be approximately retrieved from their transmittance spectra, as shown in Figs. 4(d)–4(f).

Figures 4(g)–4(i) show the 3D field distributions of a $3\times 3$ array of three different meta-atoms on a glass substrate. In particular, for the chosen periodicity of $p=1\mu \mathrm{m}$, the field distributions in the close vicinity of each meta-atom show minimal interference with those of the neighboring meta-atoms, leading to negligible cross-talks within the array and maximal field confinement inside the scatterers. Therefore, in contrast to the previously reported works that suggested that arranging the meta-atoms in the array leads to the sharpening of resonant features,32 the optical response of the meta-atoms designed using the developed IDM is not altered by the mode coupling and remains unchanged in the presence of the substrate and the periodic arrangement. These results justify that such an array of meta-atoms can be used in experimental validation of the accuracy of the IDM.19 Finally, in a typical experimental setup, instead of the strictly normal incidence case, which is usually considered in theoretical studies, a range of different $\mathit{k}$-vectors are presented in the beam impinging on the sample. Therefore, to investigate the effect of such an oblique incidence on the optical response of the meta-atoms, we numerically studied the scattering cross-section of the meta-atoms on the substrate as a function of incident angle and operating wavelength, and its results are shown in Fig. S5 of Sec. V in the Supplementary Material. These numerical studies indicate that the designed meta-atoms also possess reasonable tolerance against the illumination angle, such that both white and dark light spectroscopies can be used to characterize their optical response experimentally.

The primary focus of this work is to leverage ML within the context of physics-based nanophotonic experiments to assess the robustness of TIDM-predicted results under experimental conditions and to apply these insights to enhance nonlinear conversion efficiency in the ultraviolet (UV) regime using a single isolated meta-atom. Specifically, this involves generating optimized design starting points, as illustrated in Fig. 5(a), within the developed model; examining the influence of experimental factors; and, contingent upon favorable results, proceeding with fabrication. The ${\mathrm{TiO}}_2$ meta-atom arrays were fabricated on a glass substrate using electron-beam lithography in combination with inductively coupled plasma etching, as schematically shown in Fig. 5(b) (see Appendix E for more details). We have fabricated several arrays for each meta-atom with the lattice constant of $p=1\mu \mathrm{m}$. This choice of lattice constants will ensure both strong far-field signatures in transmittance measurements and allow for studying the particle resonances with only a small influence from grating effects, as shown in Fig. 5(c) for ED, MD, and MQ meta-atoms. The fabricated samples were measured using a home-built system consisting of a stabilized fiber-coupled light source and a wide-range optical spectrum analyzer for spectral measurements. To ensure that the effect of the illumination angle is minimized, the incident white light was only weakly focused on the sample with the incident angle of $|{\theta }_i|<15\mathrm{deg}$ (see Sec. V in the Supplementary Material for the effect of incident angles on ${\sigma }_{\mathrm{Tot}}$). In addition, a polarizer was used to ensure horizontal and vertical linear polarization of the incident beam on the sample, as shown in Fig. 5(d). The transmission spectrum was then measured by dividing the power transmitted through the array by that transmitted through the glass substrate as $T(\lambda )=I(\lambda )/I_0(\lambda )$ with $I(\lambda )$ denoting the transmitted wavelength-dependent intensity arriving at the detector when the sample is inserted into the beam path, and $I_0(\lambda )$ is the transmitted wavelength-dependent intensity arriving at the detector when the reference sample (the plain glass substrate) is inserted into the beam path. We note that to ensure the repeatability and validity of our results, we performed ten measurements for each meta-atom and reported their mean value and the statistical standard error calculated with the propagation error theory outlined in Appendix F.

Following measurement of the transmittance spectra, the scattering cross-section of each meta-atom [${\sigma }_{\mathrm{sct}}(\lambda )\propto 1-T(\lambda )$] is calculated, and its corresponding results are demonstrated in Fig. 5(e). The experimental results show that the measured scattering responses of the fabricated samples agree well with the theoretical predictions from the developed IDM from both an overall spectral shape and the full width at half-maximum of the resonant features standpoint. More specifically, at the spectral positions of excited resonant modes, the measured transmittance decreases, which can be interpreted as destructive interference in the forward direction of the incident and resonantly induced scattered waves originating from the designed resonant modes and other contributing higher-order moments. Figure 5 shows that although the presence of the substrate introduces an additional asymmetry to the system that might affect the optical response of the scatterers, this induced structural asymmetry, as well as the periodic arrangement of the meta-atoms, does not affect the suppression/emergence of various scattering modes predicted by the IDM. We note that when the incident light illuminates the fabricated array of meta-atoms close to normal incidence (${\theta }_i<\pm 15\mathrm{deg}$), the induced polarization within the scatterer acts as a source and radiates back to free space as the scattered wave. Then, the interference of these scattering fields gives rise to the discrete diffraction orders, with their in-plane wavevector expressed as $k_{\parallel }=|\frac{2\pi m}{p_x}{\hat{e}}_x+\frac{2\pi n}{p_y}{\widehat{e}}_y|$ and diffraction angles as $\sin ({\theta }_{mn})=k_{\parallel }/k_0=\lambda \sqrt{(\frac{m}{p_x}{)}^2+(\frac{n}{p_y}{)}^2}$, wherein $m$ and $n$ indicate the diffracted modes and $p_x$ and $p_y$ represent periodicity along $x$ and $y$ directions, respectively. Because of the chosen lattice periodicity and the desired wavelength spectra, although the maximum propagating modes are $-1<m,n<+1$, their contribution to the optical response plays a negligible role. Thus, the measured spectra can be mainly attributed to the fundamental mode $(m,n)=(\mathrm{0,0})$ (see Sec. VI in the Supplementary Material and Fig. S6 for more details).

However, despite the general agreement between the theory and experiment, some notable deviations are evident within the measured spectra. The first discrepancy is related to the spectral shift of $\sim 55\mathrm{nm}$, which is attributed to two reasons: (i) difference in theoretically considered and experimental refractive indices and (ii) not perfectly vertical side walls of the fabricated meta-atoms. In particular, the previously developed IDM was trained based on the assumption that the refractive index follows the values reported by Sarkar et al.33 However, our ellipsometry measurements of the fabricated thin film revealed a discrepancy of $\approx 5\%$ in the values of refractive indices, which leads to the spectral shift of contributing resonant modes. In addition, the scanning electron microscopy (SEM) and atomic force microscopy (AFM) images of the fabricated meta-atoms [shown in Fig. 5(c)] indicate an inclined side wall, which is attributed to the etching process and possible effects of the utilized Cr wet etchant during the Cr removal phase (the relative contributions of the predicted modeled versus realized refractive indices and the non-verticality of the side walls are discussed in detail in the Sec. VII in the Supplementary Material). Any additional differences between the measured results and the spectra predicted using the IDM are likely due to the small size differences between the designed IDM geometry and fabricated samples. We also note that due to the limitation in our measurement capability, such as the low power of the illumination source and low efficiency of the detectors within the spectral range of 500 to 550 nm, the measured scattering spectra are accompanied by fluctuations in this range, which can be mitigated with more precise measurement tools and other characterization techniques. We note that measuring the angle-dependent far-field scattering and correlating the profiles with known multipole emission patterns would indeed offer a direct verification of the individual multipolar contributions. However, such measurements require advanced angular-resolved scattering setups, which are beyond the capabilities of our current experimental apparatus. However, we emphasize that relying on the overall scattering behavior and comparing it to theoretical predictions has been widely adopted in prior studies and demonstrated to effectively capture key physical phenomena, such as anapole excitations, excitation of magnetic resonances, and enhancing directional scattering.10 Therefore, although angle-resolved measurements could provide an additional layer of verification, we believe that our current approach offers a robust, practical, and effective alternative for validating the predicted results by our developed ML model. Therefore, the ML approach allows the optimization of the meta-atom shapes beyond regularly shaped structures such as cubes, cylinders, and combinations while optimizing well-defined physical properties of the complex meta-atom shapes such as their multipole response. To the best of our knowledge, the results presented in this paper offer the first experimental verification of ML ability in predicting the optical response of arbitrarily shaped individual meta-atoms optimized to maximize the contribution of particular higher-order multipole moments. These results can facilitate novel applications requiring boosted light–matter interaction with specific field distribution, such as in nonlinear harmonic generation, optical switching, waveguiding, and optical communication.7^,10

Although UV light is crucial for a variety of basic science and applied fields, ranging from spectroscopy quantum optics to imaging and disinfection, the limited availability and the cost of suitable materials for operating within this spectrum are the main bottleneck to its broader utilization.34 Traditionally, the generation of light at UV, particularly VUV wavelengths, requires bulky and expensive optical systems. The design and implementation of ultra-compact, coherent sources of VUV light that are of paramount importance for many applications, including photolithography, chemical, biological, and medical devices, are still challenging.35 Nonlinear optics-based nanophotonic platforms that rely on second or third-harmonic generation (THG) to achieve frequency-upconverted coherent radiation at UV wavelengths emerge as promising solutions.34^–38 In particular, among the various UV-transparent optical materials such as niobium pentoxide, tantalum pentoxide, aluminum nitride, silicon nitride, hafnium oxide, and zinc oxide,39${\mathrm{TiO}}_2$, is of utmost importance owing to its relatively high refractive index, the wide bandgap of $\sim 3.2\mathrm{eV}$, and significant third-order nonlinear susceptibility of $\sim {10}^{-20}{\mathrm{m}}^2{\mathrm{V}}^{-1}$.40 We note that although the VUV applications of ${\mathrm{TiO}}_2$ may appear to be limited by the substantial absorption, we show the largely overlooked phenomenon of phase-locking, initially discussed in pioneering work on nonlinear optics41^,42 enables THG in ${\mathrm{TiO}}_2$ resonant meta-atoms supporting the MQ resonant mode at ${\lambda }_{\mathrm{MQ}}=570\mathrm{nm}$, as schematically shown in Fig. 6(a).

Let us briefly review the phenomenon of THG in the general case of a nonlinear medium that is transparent at the pump wavelength and absorbing at the TH wavelength. As the pump pulse crosses the interface of linear and nonlinear media, three TH components are generated. One is reflected back into the linear medium, whereas the other two are transmitted into the nonlinear medium.41^,42 These two transmitted components are the homogeneous (HOM-TH) and inhomogeneous (INHOM-TH) solutions of the nonlinear wave equation. In particular, the HOM-TH component propagates with a group velocity dictated by the dispersion characteristics of the material at the TH wavelength [${\mathit{k}}_{\mathrm{H}}=\frac{3\omega }{c}n(3\omega )$]. By contrast, the INHOM-TH component co-propagates with the pump pulse with its wavenumber determined by the refractive index of the medium corresponding to the fundamental wavelength [$k_{\mathrm{IH}}=\frac{3\omega }{c}n(\omega )$]. Because ${\mathrm{TiO}}_2$ is transparent at the pump wavelength of 570 nm, the TH wavelength (190 nm corresponding to the VUV spectral range) falls into the highly absorbing part of the spectrum for this material. Therefore, the HOM-TH wave experiences significant absorption of $\mathrm{Im}[n(3\omega )]\sim 1$, whereas its inhomogeneous counterpart experiences a nearly lossless propagation, as its associated wavenumber is proportional to the refractive index at 570 nm. As a result, the finite VUV generation is expected from the ${\mathrm{TiO}}_2$ isolated meta-atoms, as schematically shown in Fig. 6(a) inset. To demonstrate this numerically, in Fig. 6(b), we have plotted the TH conversion efficiency [$\eta =P_{\mathrm{THG}}(\lambda )/P_{\text{in}}$] in the VUV spectrum for both INHOM-TH (purple squares) and HOM-TH (magenta circles), respectively.

Figure 6(b) shows that HOM-TH and INHOM-TH (PL) components peak around 190 nm. However, the efficiency of the HOM-TH component is about 15 times lower than that of the INHOM-TH, as expected from the properties of these two components of the TH.41^,42 Due to the difference in the material properties that INHOM-TH and HOM-TH components experience as they propagate inside the medium, their corresponding field distributions within the meta-atom also differ, as shown in Fig. 6(c). In particular, the field distribution of the INHOM-TH component is similar to the field distribution at the pump wavelength shown in Fig. 2(i), whereas the HOM-TH has a different distribution. We note that the VUV THG for the case of isolated meta-atoms was calculated at the intensity of $I=1\mathrm{GW}/{\mathrm{cm}}^2$. By contrast, the TH power as a function of incident pump intensity (power shown in the top axis) is shown in Fig. 6(d) for both INHOM-TH and HOM-TH components in the log-log scale. To avoid laser-induced damage, we limited the incident pump intensity to below the reported ${\mathrm{TiO}}_2$ damage threshold ($\approx 30\mathrm{GW}/{\mathrm{cm}}^2$)36 as illustrated by the shaded region. Figure 6(d) confirms the possibility of predicted THG in the opaque wavelength range of ${\mathrm{TiO}}_2$ for the INHOM-TH component. Moreover, due to strong field enhancement facilitated by the ML-based designed meta-atoms, the THG signal in the VUV region for the INHOM-TH component is $\sim 200$ times stronger than the thin film ${\mathrm{TiO}}_2$ slab (blue stars). We note that for this study, we focused exclusively on the nonlinear INHOM-TH component of the slab due to its primary role in the nonlinear response, aiming to demonstrate how ML-driven meta-atoms enhance the VUV generation while excluding the less significant HOM-TH component for conciseness. Moreover, by fitting the spectra of both components, we show that they obey the expected cubic dependence ($E_{\mathrm{THG}}\propto E_{\mathrm{inc}}^3$), as shown with dashed lines in Fig. 6(d). These results show the feasibility of utilizing ML-based designs to not only design the meta-atoms at a particular wavelength in the visible band (corresponding to the transparency range of ${\mathrm{TiO}}_2$) but also to generate the TH using the same isolated meta-atoms in the opaque VUV region. In this work, we did not aim at maximizing the efficiency of the THG, which can be done by re-designing the meta-atoms such that they support resonances at the TH wavelength.16 Despite this, our approach yielded an almost 200-fold increase in the TH signal compared with an unstructured film, suggesting that the proposed ML approach may not only enable visible light applications of ${\mathrm{TiO}}_2$ nanostructures but also enable efficient shorter wavelength generation in the extreme ultraviolet. We note that our study aimed at (i) applying a well-established model to maximize a particular resonant response (multipole) by optimizing the shapes of the meta-atoms, (ii) validating its predictions experimentally, and (iii) achieving nonlinear harmonic generation with minimal effort toward maximizing spatial field overlap or inducing resonant responses at the generated harmonics. Importantly, the goal of this work was not to develop a new model or to identify the best existing model for nanophotonic design but rather to demonstrate these specific objectives using a proven approach. Finally, we note that while the results presented in Fig. 6 are numerical, they can also be experimentally validated using a setup similar to one we have previously developed for characterizing the nonlinear response of metasurfaces.43 As an example, a simplified experimental setup could include a Ti:sapphire laser (100 fs output pulse width, 1 kHz repetition rate, Coherent Libra system) and an ultrafast optical parametric amplifier (TOPAS-C) as the tunable light source, covering a spectral range of 260 to 2600 nm. The incident laser light could be focused using a 75 mm calcium fluoride (${\mathrm{CaF}}_2$) plano-convex lens, and the transmitted fundamental frequency and TH signals could be collected using a high numerical aperture (NA) objective. To isolate the TH signal, the fundamental light would be either attenuated by absorptive neutral density (ND) filters or short/band pass UV filters, and the TH would then be coupled into a SuperGamut UV-VIS-NIR spectrometer (BaySpec, Inc). To minimize the absorption of VUV light by oxygen in ambient air, the setup would need to be purged with nitrogen during measurements. It is important to note that this is a simplified setup, and additional considerations may need to be considered to ensure accurate experimental measurements. The specific configuration and components may be subject to change depending on practical constraints and the requirements of the experiment.

Recently, the integration of ML with the field of Mie-tronics has unveiled numerous promising avenues, significantly impacting the design of meta-atoms with desired responses and the development of metasurfaces with tailored light–matter interactions on demand. Surprisingly, despite these substantial theoretical advancements, the experimental verification of ML-based photonics platforms has been predominantly confined to metasurfaces and optical waveguides, with little effort on the experimental demonstration of ML for the design of the individual meta-atoms—the fundamental building blocks of these platforms. The proposed toolkit for the individual meta-atom design can be used to develop a library of building blocks for designing nanophotonic devices and functionalities on demand. Although our results offer the first steps toward forming such a library, they provide a compelling experimental confirmation of the predictions of the developed physics-guided ML model for all-dielectric meta-atoms hosting a particular, desired Mie-type resonance. In particular, we first used ML to design three meta-atoms with specific ED, MD, and MQ resonant modes at specific desired operating wavelengths. To verify the results obtained from the developed ML model, we characterized the optical response of the predicted meta-atoms via the QNM framework. We showed a good agreement between the results of the two theories. We then explored how the predicted optical response of the designed meta-atoms alters upon the inclusion of glass substrate once they are arranged in a periodic array with various lattice constants. The experimental results obtained from the fabricated meta-atoms show good agreement with the predicted results of the developed IDM. To further demonstrate the application of the presented work, the designed meta-atoms were implemented to generate TH within the VUV spectrum that enhances the TH signal 200 times more than an unstructured ${\mathrm{TiO}}_2$ thin film. Our findings not only serve as the first experimental verification of ML-based Mie-tronics but also might be a solid step toward the achievement of a fast, versatile, and accurate approach to design meta-atoms with tailored optical response via ML and will likely facilitate uncovering new regimes of linear and nonlinear light-matter interaction at the nanoscale as well as a versatile toolkit for nanophotonic design.

An FPM using dense convolutional network (DenseNet) architecture was developed to predict the multipolar resonances of specific meta-atoms as functions of wavelength. In particular, the meta-atom shapes were entered into six down-conversion dense blocks before reaching the bottleneck layer. The FPM output comprised multipolar resonance arrays connected to the bottleneck layer through three consecutive fully connected layers. The advantage of DenseNet architecture is that each layer within the model can receive inputs from all previous layers while forwarding its feature maps to all subsequent layers. This ML process takes only seconds, offering a significant speed advantage compared with direct FEM simulations. We also note that the performance of the meta-atom design using IDM is closely tied to the variety of examples implemented in the training dataset. In particular, during the training phase, we constrained the size and height range of the meta-atoms, focusing on studying the effect of different geometric cross-sections on the induced multipolar moments within the meta-atoms. Within this framework, the optimal meta-atom design for the targeted multipolar resonances might be achieved at other sizes or heights beyond the limited set that the model was trained on. Nevertheless, by incorporating a broader range of meta-atom examples with diverse shapes and sizes, the training dataset can be enlarged, enhancing the performance of the implemented ML models.

The numerical simulations were carried out using the FEM implemented in the commercial software COMSOL multiphysics. In particular, we used the wave optics module to solve Maxwell’s equations in the frequency domain and proper boundary conditions. For the isolated meta-atom case, we used a spherical air-filled domain and a radius of $4\lambda$ as the background medium. At the same time, perfectly matched layers of thickness $0.6\lambda$ were positioned outside of the background medium to act as absorbers and avoid back-scattering. The tetrahedral mesh was chosen to ensure the results’ accuracy and facilitate numerical convergence. The meta-atoms were studied under the plane wave illumination along the $z$-axis with an electric field pointing along the $x$-axis. We changed the spherical simulation domain to rectangular geometry for the periodic array simulations. We applied periodic boundary conditions in the $x$ and $y$ directions and the perfectly matched layer (PML) in the $z$ direction to avoid undesired reflections. Periodic ports were used along the $z$ direction to launch a plane wave and capture the transmitted wave with the same polarization as the isolated case. We utilized the undepleted pump approximation for our nonlinear simulations. This involved a two-step process to compute the intensity of the radiated nonlinear signal. Initially, linear scattering at the pump wavelength was simulated to derive the induced nonlinear polarization within the meta-atom. This induced polarization was then used as a source in the electromagnetic simulation at the harmonic wavelength to determine the TH field produced.

According to the multipole expansion theory, the scattering cross-section is given by $${\sigma }_{\text{s}\mathrm{ct}}\approx \frac{k_0^4}{12\pi {ϵ}_0^2{\eta }_0{I}_0}|D{|}^2+\frac{k_0^4{\mu }_0}{12\pi {ϵ}_0{\eta }_0{I}_0}|m{|}^2+\frac{k_0^6}{1440\pi {ϵ}_0^2{\eta }_0{I}_0}\sum _{i,j}|Q_{ij}{|}^2+\frac{k_0^6{\mu }_0}{160\pi {ϵ}_0{\eta }_0{I}_0}\sum _{i,j}|M_{ij}{|}^2+\frac{k_0^8}{3780\pi {ϵ}_0^2{\eta }_0{I}_0}\sum _{i,j,l}|O_{ijl}^{(e)}{|}^2+\frac{k_0^8{\mu }_0}{3780\pi {ϵ}_0{\eta }_0{I}_0}\sum _{i,j,l}|O_{ijl}^{(m)}{|}^2,$$(2)where $I_0$ corresponds to the maximum beam intensity in a focal plane, ${\eta }_0$, ${ϵ}_0$, and ${\mu }_0$ are the impedance, permittivity, and permeability of free space, respectively, and $i,j$, and $l$ represent the different components of each tensor. The elements of Eq. (2) can be derived via a volumetric integral over each meta-atom volume as $$\begin{gathered} D=\frac{i}{\omega }\int j_0(k_0{r}^{\prime })J(r^{\prime })\mathrm{d}{r}^{\prime }+\frac{i{k}_d^2}{2\omega }\int \frac{j_2(k_d{r}^{\prime })}{(k_d{r}^{\prime }{)}^2}[3(r^{\prime }·J)r^{\prime }-r^{\prime 2}J]\mathrm{d}{r}^{\prime }, \\ m=\frac{3}{2}\int \frac{j_1(k_d{r}^{\prime })}{k_d{r}^{\prime }}[r^{\prime }\times J]\mathrm{d}{r}^{\prime }, \\ \hat{M}=5\int \frac{j_2(k_d{r}^{\prime })}{(k_d{r}^{\prime }{)}^2}([r^{\prime }\times J]\otimes r^{\prime }+r^{\prime }\otimes [r^{\prime }\times J])\mathrm{d}{r}^{\prime }, \\ \widehat{Q}=\frac{3i}{\omega }\int \frac{j_1(k_d{r}^{\prime })}{k_d{r}^{\prime }}[3(r^{\prime }\otimes J+J\otimes r^{\prime })-2(r^{\prime }·J)\overline{\overline{I}}]\mathrm{d}{r}^{\prime }+\frac{i6k_d^2}{\omega }\int \frac{j_3(k_d{r}^{\prime })}{(k_d{r}^{\prime }{)}^3}[5(r^{\prime }·J)r^{\prime }\otimes r^{\prime } \\ -r^{\prime 2}(J\otimes r^{\prime }+r^{\prime }\otimes J)-(J·r^{\prime })r^{\prime 2}\overline{\overline{I}}]\mathrm{d}{r}^{\prime }, \\ {\hat{O}}^{(e)}=\frac{15i}{\omega }\int \frac{j_2(k_d{r}^{\prime })}{(k_d{r}^{\prime }{)}^2}(J\otimes r^{\prime }\otimes r^{\prime }+r^{\prime }\otimes J\otimes r^{\prime }+r^{\prime }\otimes r^{\prime }\otimes J-\widehat{A})\mathrm{d}{r}^{\prime }, \\ {\widehat{O}}^{(m)}=\frac{105}{4}\int \frac{j_3(k_d{r}^{\prime })}{(k_d{r}^{\prime }{)}^3}([r^{\prime }\times J]\otimes r^{\prime }\otimes r^{\prime }+r^{\prime }\otimes [r^{\prime }\times J]\otimes r^{\prime }+r^{\prime }\otimes r^{\prime }\otimes [r^{\prime }\times J]-\widehat{B})\mathrm{d}{r}^{\prime }, \end{gathered}$$(3)wherein $j_n(x)$ is the $n$’th order spherical Bessel function, $k_d$ is the wave number in the surrounding medium, $\overline{\overline{I}}$ is the $3\times 3$ unit tensor, and the operators of $·$, $\times$, and $\otimes$ represent the scalar, vector, and tensor products, respectively. It should be noted that $\hat{A}$ and $\widehat{B}$ are auxiliary tensors whose components are obtained according to $A_{ijl}={\delta }_{ij}{V}_l+{\delta }_{il}{V}_j+{\delta }_{jl}{V}_i$, and $B_{ijl}={\delta }_{ij}{V}_l^{\prime }+{\delta }_{il}{V}_j^{\prime }+{\delta }_{jl}{V}_i^{\prime }$, in which $V=0.2[2(r^{\prime }·J)\otimes r^{\prime }+r^{\prime 2}J]$ and $V^{\prime }=0.2[r^{\prime }\times J]r^{\prime 2}$. Because the meta-atoms within the training dataset had arbitrary irregular shapes, we extended our analysis beyond the first four multipole moments (ED, MD, EQ, and MQ), which are typically considered for regular shapes.23^,44^–47

In QNM analysis, we determined the eigenmodes of the meta-atoms using COMSOL. Subsequently, the excitation coefficients, internal fields, and induced currents at specific spectral positions were computed using the “Modal Analysis of Nanoresonators (MAN)” freeware, as described in Ref. 26. Afterward, we calculated the total scattering cross-section of the meta-atom as a superposition of each contributing mode cross-section, ${\sigma }_{\text{sct}}=\sum _m{\sigma }_{\text{sct}}^{(m)}$, and identified the QNMs with significant contributions to the system’s scattering behavior, as shown in Figs. 3(a)–3(c). We note that in the range of $500\mathrm{nm}<\lambda <900\mathrm{nm}$, far from the ${\mathrm{TiO}}_2$ band gap, the dispersion of the real part of the refractive index and its imaginary part values are minimal. Hence, for simplicity and ease of material implementation in MAN freeware, we assumed a dispersionless/lossless refractive index for the meta-atom in our calculations.

A 320-nm ${\mathrm{TiO}}_2$ film was deposited on the ${\mathrm{SiO}}_2$ substrate with a deposition rate of $0.65Å/\mathrm{s}$. Afterwards, 80-nm polymethyl methacrylate (PMMA) film was spin-coated on the deposited film and baked at 180°C for an hour. The desired meta-atom shapes were transferred to the PMMA resist by performing EBL (Raith E-line, 30 kV) and developed in methyl isobutyl ketone/isopropyl alcohol (MIBK/IPA) solution for 30 s at 0°C. The sample was transferred into an e-beam evaporator and directly coated with 22 nm chromium (Cr) films with a deposition rate of $0.5Å/\mathrm{s}$. After immersing the sample in Remover PG for 8 h, the PMMA was removed, and the nanostructures were transferred to Cr. The ${\mathrm{TiO}}_2$ under the Cr-covered nanostructure was then etched by an inductively coupled plasma (Oxford ICP180), and the remaining Cr film was removed by immersing the sample into the chromium etchant for 10 min.

A stabilized fiber-coupled light source (Thorlabs SLS201) was used as the white light source to illuminate the meta-atoms from 300 to 2600 nm. A Glan-Thompson polarizer (Thorlabs GTH10) was implemented to polarize the light beam along the desired direction, and the incident beam was weakly focused on the sample with the illumination beam wavevector distribution within the range of angles $|{\theta }_i|<15\mathrm{deg}$. The transmitted light was then collected by a 10× infinity-corrected objective with the NA = 0.25 (Olympus Plan Achromat Objective) and partially guided to a camera and $100\mu \mathrm{m}$ pinhole with a beam splitter. The setup used the pinhole to avoid undesired responses from the bare substrate and improve the signal-to-noise ratio. The transmitted light was then coupled to a large diameter ($600\mu \mathrm{m}$) multimode fiber via a 200-mm lens (Thorlabs LB1945) and sent to a wide-range optical spectrum analyzer (AQ6374 OSA) to perform transmittance measurements. The scattering spectra were then measured as $\sigma (\lambda )=1-I(\lambda )/I_0(\lambda )$ with $I(\lambda )$ denoting the transmitted wavelength-dependent intensity arriving at the detector when the sample was inserted into the beam path, and $I_0(\lambda )$ is the transmitted wavelength-dependent intensity arriving at the detector when the reference substrate was inserted into the beam path. To ensure the repeatability of the experimental results, we have performed ten measurements for each sample and then reported their average as the final response together with their standard deviation as $\sigma =\sqrt{\frac{1}{N-1}{\mathrm{\Sigma }}_i(M_i-\overline{M}{)}^2}$, with $M_i$ and $\overline{M}$ denoting each measurement and their average, respectively.

Hooman Barati Sedeh received a BSc degree in electrical engineering–communications from Iran University of Science and Technology, Tehran, Iran, in 2019 and his master’s degree in electrical engineering–electromagnetics at Northeastern University, Boston, Massachusetts, USA, in 2021. Hooman Barati Sedeh is currently a second-year PhD candidate, supervised by Professor Litchinitser in the Department of Electrical Engineering at Duke University. His research interest focuses on the theoretical and experimental studies of light–matter interaction with subwavelength meta-atoms for various applications, including nonlinear optics, chiroptical responses, and scattering manipulation.

Natalia M. Litchinitser is a professor of electrical and computer engineering and a professor of physics at Duke University. Her research focuses on linear and nonlinear optics in engineered nanostructures, metamaterials, topological photonics, and the engineering of the light beams themselves. She earned her PhD in electrical engineering from the Illinois Institute of Technology and a master’s degree in physics from Moscow State University in Russia. She completed her postdoctoral training at the Institute of Optics, University of Rochester, in 2000. Dr. Litchinitser previously was a professor of electrical engineering at the University at Buffalo, The State University of New York, a member of Technical Staff at Bell Laboratories, Lucent Technologies, and a senior member of Technical Staff at Tyco Submarine Systems. She has authored seven invited book chapters and over 250 journal and conference research papers. She is a fellow of the American Physical Society (APS), a fellow of Optica (formerly the Optical Society of America), a senior member of the IEEE and SPIE, and served as a co-chair of CLEO Fundamental Science and SPIE Nanoscience and Engineering Applications conferences in 2021–2022.

Biographies of the other authors are not available.