Fig. 1. Schematic illustration of ML-based meta-atom design. The training dataset contains regular (expressed with the equation in the blue box) and irregular meta-atom shapes with a fixed height of 320 nm. The optical response of the complete training dataset is shown in the blue box, with the dots denoting the mean of the specific moments and shaded bands representing their corresponding standard deviation. The desired optical response at a specified operating wavelength (red box) is then fed to the developed IDM that predicts the shape producing the required response (shown in the magenta box). The meta-atoms with the optimized shapes are then fabricated and placed in an array with a large enough spacing to minimize the coupling effects between the neighboring meta-atoms in the white light spectroscopy-based experiments, shown in the orange color box.

Fig. 2. ML-driven meta-atom design. The scattering spectra of the designed meta-atoms for supporting (a) ED, (b) MD, and (c) MQ Mie-type resonant modes, with their contribution reaching 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. The desired designed wavelengths are shown with dashed blue color boxes, and the rest of the spectrum for each case is masked with a shaded region. Total scattering cross-section comparison of each ML-driven meta-atom when it is surrounded by air (dashed blue line) and the corresponding meta-atom on a glass substrate (solid red line) for (d) ED, (e) MD, and (f) MQ supporting meta-atoms. The presence of a substrate introduces an additional asymmetry to the system, leading to spectral shift and broadening of the resonant peaks. The electric field distribution of the inverse designed meta-atoms in the $x\text{-}y$ plane when the meta-atom was surrounded by air (blue box) and on a glass substrate (red box) at their corresponding desired wavelengths of (g) ${\lambda }_{\mathrm{ED}}=900\mathrm{nm}$, (h) ${\lambda }_{\mathrm{MD}}=780\mathrm{nm}$, and (i) ${\lambda }_{\mathrm{MQ}}=570\mathrm{nm}$, with black arrows representing the magnetic fields.

Fig. 3. ML-based meta-atom design results based on the QNM expansion theory. The QNM-based total (black line) and eigenmode (colored lines) scattering cross-sections of the predicted shapes correspond to (a) ED, (b) MD, and (c) MQ meta-atoms. The solid line represents the dominant QNM, whereas the dashed lines denote the background modes. The negative values of the scattering cross-section are attributed to the energy exchange between the dominant and the background QNM fields. The normalized field distributions of the first two dominant and relevant modal contributions in the close vicinity of (d) ${\lambda }_{\mathrm{ED}}$, (e) ${\lambda }_{\mathrm{MD}}$, and (f) ${\lambda }_{\mathrm{MQ}}$. The superposition of these QNMs with the other background modes yields the excitation of the desired resonant response at the designed operating wavelength, as shown in blue, red, and green color boxes.

Fig. 4. Optical response of the predicted meta-atoms arranged in a periodic array. Numerically calculated linear-optical zeroth diffracted order transmittance spectra for arrays of embedded meta-atoms corresponding to (a) ED, (b) MD, and (c) MQ resonant modes with a lattice constant of $p=1\mu \mathrm{m}$. The inset demonstrates the field distribution of each meta-atom at its design wavelength, showing an excellent agreement with its isolated counterpart. (d)–(f) The corresponding retrieved optical scattering cross-section spectra for the case of an isolated meta-atom on the glass substrate (red curve) and periodic configuration of meta-atoms on the same substrate (blue dots). The two spectra demonstrate similar behavior, suggesting that the chosen lattice constant efficiently suppressed the optical crosstalk between the meta-atoms. The 3D distribution of the electric field within an array of meta-atoms corresponding to (g) ED, (h) MD, and (i) MQ Mie-type resonances. It is evident that in the close vicinity of each meta-atom, the electric field intensity is suppressed, yielding minimal interactions between the meta-atoms when placed in an array.

Fig. 5. Experimental verification of ML-based Mie-tronics. (a) The experimental workflow of ML-based multipole engineering method. (b) The steps taken to fabricate the predicted shapes by the developed IDM. Using a multi-step process, a 320-nm ${\mathrm{TiO}}_2$ film and 80 nm of PMMA were sequentially deposited on a ${\mathrm{SiO}}_2$ substrate and then patterned with EBL. Subsequent steps included Cr evaporation, performing lift-off, etching, and Cr layer removal, resulting in the desired nanostructures on the substrate. (c) Tilted SEM images of the fabricated samples corresponding to meta-atoms supporting ED (blue box), MD (red box), and MQ (green box) resonant modes and their corresponding AFM height measurements. The scale bar represents 300 nm. (d) The schematic illustration of the experimental setup used to characterize the optical response of the fabricated samples. The incident white light is first coupled to an optical fiber and then polarized and focused on the sample on an XYZ stage. The transmitted wave is then collected by an achromatic objective and sent to a wide-range optical spectrum analyzer for spectral measurements. (e) The measured scattering cross-section spectra of the fabricated samples for meta-atoms hosting ED (blue dots), MD (red dots), and MQ (green dots). Each experimental curve represents an average of 10 independent measurements, and the color bands represent the statistical standard error calculated with the propagation error theory. The experimental measurements were repeated 10 times under identical conditions to confirm their repeatability.

Fig. 6. Application of ML-based Mie-tronics for the VUV generation. (a) The schematic of the THG in the meta-atom designed using the ML approach. The generated harmonics consist of homogeneous and inhomogeneous components. The inhomogeneous component is phase-locked with the pump and, as a result, experiences the refractive index of the material at the pump wavelength of 570 nm (corresponding to the transparent wavelength range of the ${\mathrm{TiO}}_2$). By contrast, the homogeneous component experiences the refractive index and absorption coefficient of the material at 190 nm, corresponding to the opaque part of the ${\mathrm{TiO}}_2$ spectrum. (b) Theoretically calculated nonlinear spectra of the ML-based meta-atom supporting MQ resonant mode for PL (purple squares) and HOM-TH (magenta circles) components. (c) Field distribution of the MQ meta-atom at the pump wavelength for both HOM-TH and INHOM-TH components. (d) TH power as a function of input pump power for both INHOM-TH (purple squares) and HOM-TH (magenta circles) components of inverse designed meta-atom and INHOM-TH component of ${\mathrm{TiO}}_2$ slab (blue stars) in log-log scale with a shaded region indicating the maximum intensity used to avoid laser-induced damage,³⁶ and dashed lines indicating the fitted cubic power dependence.