Enhanced Light-Matter Interactions in Dielectric Nanostructures via Machine Learning Approach

A key concept underlying the specific functionalities of metasurfaces, i.e. arrays of subwavelength nanoparticles, is the use of constituent components to shape the wavefront of the light, on-demand. Metasurfaces are versatile and novel platforms to manipulate the scattering, colour, phase or the intensity of the light. Currently, one of the typical approaches for designing a metasurface is to optimize one or two variables, among a vast number of fixed parameters, such as various materials' properties and coupling effects, as well as the geometrical parameters. Ideally, it would require a multi-dimensional space optimization through direct numerical simulations. Recently, an alternative approach became quite popular allowing to reduce the computational cost significantly based on a deep-learning-assisted method. In this paper, we utilize a deep-learning approach for obtaining high-quality factor (high-Q) resonances with desired characteristics, such as linewidth, amplitude and spectral position. We exploit such high-Q resonances for the enhanced light-matter interaction in nonlinear optical metasurfaces and optomechanical vibrations, simultaneously. We demonstrate that optimized metasurfaces lead up to 400+ folds enhancement of the third harmonic generation (THG); at the same time, they also contribute to 100+ folds enhancement in optomechanical vibrations. This approach can be further used to realize structures with unconventional scattering responses.


Introduction
Metasurfaces are thin and flat arrays of subwavelength nanoparticles, enabling control over the polarization, phase, amplitude, and dispersion of light. 1 They can be used for light emission, detection, modulation, control and/or amplification at the nanoscale. In recent years, metasurfaces have been a subject of undergoing intense study as their optical properties can be adapted to a diverse set of applications, including superlenses, tunable images, holograms, etc. 1-4 High-refractive-index dielectric metasurfaces provide a powerful platform for controlling light that can go beyond plasmonics, as they cause negligible losses as compared with plasmonic metasurfaces. Dielectric materials offer unique ability to efficiently manipulate light at the nanoscale based on the simultaneous excitation and control over the optically induced electric and magnetic Mie-type resonances.
Resonant dielectric metasurfaces with high-quality factor (high-Q) resonances, in particular, are of significant interests due to their possibilities to strongly enhance the electromagnetic near-fields and boost the light-matter interactions at the nanoscale. In other words, they allow to enhance the response of metasurfaces to an external electromagnetic-field at a particular frequency. Moreover, high-Q metasurfaces can increase the storage time of photons and, thus, light-matter interactions within the subwavelength resonators. It will facilitate various nanophotonics applications, such as enhanced nonlinear photon generations, optical sensing, optoacoustic vibrations as well as narrowband filtering. In the last decade, high-Q metasurfaces were mainly associated with Fano resonances (FRs) featuring asymmetric spectral line profiles. [5][6][7][8] In FRs, the asymmetry originates from a close interaction of a discrete (localized) state with a continuum of propagation modes [9][10][11] . Some examples are trapped modes in arrays of dielectric nanodisks with asymmetric holes [12][13][14] , ring and disk cavities 15 , Dolmen structures 16,17 , and aggregated nanoparticles [18][19][20][21] . Recently, it has been shown that different approaches for generating high-Q resonances are based on the bound-state-in-thecontinuum (BIC), i.e., a localized state with zero linewidth that is embedded in the continuum. 13,22 Indeed, optical BICs provide a unique opportunity to manipulate the light-matter interaction within the radiative continuum because of their ultrahigh-Q origin and associated giant enhancement of the electromagnetic near-field. 23 On the other hand, designing metasurfaces with high-Q resonances are usually achieved via continuous parameters tuning, with limited control on the linewidth, amplitude, and spectral positions. Currently, one of the typical approaches for designing metasurfaces with high-Q resonance is based on direct optimization of one or two parameters via brute-force simulations 24-33 . This is a timeconsuming task accompanied by a random success on the output parameters of the desired resonances 34 . Recently, deep learning approaches, based on the artificial neural networks (ANNs), have emerged as a revolutionary and robust methodology in nanophotonics [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49] . Indeed, applying the deep learning algorithms to the nanophotonic inverse design can introduce remarkable design flexibility that can go far beyond that of the conventional methods. The inverse design approach works based one the training process, that enables fast prediction of complex optical properties of nanostructures with intricate architectures.
As a (non-unique) example, we have targeted toroidal dipoles, due to their promising applications in the formation of anapole states and electromagnetic energy localization. We investigate the nonradiating toroidal dipole (TD) supported by two parallel silicon bars, as the building blocks of the metasurface. The reason behind choosing such a geometry is the reasonable number of parameters to be optimized, as a proof of concept. The parameters include the length and width of the bars, as well as the gap between them. At the same time, this TD corresponds to the symmetry-protected BIC. Subsequently, via taking this TD-BIC model, we demonstrate the deep-learning-assisted inverse design of arbitrary high-Q resonances with different line width, amplitude and spectral position. We employ a multi-layer perceptron (MLP) variant of the ANN as our model [50][51][52][53] . An MLP consists of multiple layers of perceptrons (MLP), including an input and an output layer with several hidden layers. In this work, each artificial neuron in one layer connects with a certain weight to every neuron in the following layer, that are adapted during the learning state. Once learned the weight values remain fixed and the model can be used to infer the target metasurface design parameters. The hidden layers establish a nonlinear mapping between the input and output through training from the given dataset, and then be able to predict the response of the system, or inversely determine the design parameters for the desired performance.
The proposed deep-learning-assisted inverse approach is used to obtain a bi-functional metasurface dealing with photon and phonons, simultaneously. [54][55][56][57][58][59][60][61][62][63] Photon-photon conversions, so-called nonlinear nanophotonics is at the heart of modern macroscopic optics, including lasers, sensors, imaging and information technology. On the other hand, photon-phonon conversations are the stateof-the-art solution for precision mass sensors, micro-manipulation, and sensing biochemical materials, with transformative implications in the fields of health and security. A combined photon-phonon conversion can be used for non-ionizing and non-invasive imaging 62 . Here, by using the deep-learningassisted inverse approach we design and fabricate a single opto-acoustic metasurface that enhances third-harmonic generation intensity for 400+ folds and opto-mechanical mode excitations for 100+ folds, concurrently, all through a designed high-Q resonance, associated with a strong electric nearfield enhancement. The inverse design approach proposed in this paper is extendable to other characteristics and applications of metasurfaces and significantly circumvents the time-consuming, case-by-case numerical models in conventional electromagnetic nanostructure designs.

Results and Discussions:
To obtain the initial high-Q resonances, we have defined the building blocks of metasurfaces to be two identical silicon nanobars with width , length and the offset 0 , which is the distance between the center of the two bars fabricated on a glass substrate, as shown in Fig. 1a. It is worth mentioning that there is a large variety of other geometries, demonstrated earlier for generating high-Q resonances. 24-33, 35-37 However, discussions on advantages and disadvantages of various geometries are beyond the scope of the current study. We rather concentrate on customizing of the generated high-Q resonances. The thickness of bars is fixed at 150 nm and the periodicity of unit cells is fixed at D=900 nm in both x and y directions. As can be seen in Fig. 1b, this geometry supports a strong Fano resonance in the transmission spectrum, when the incident light is polarized along y-axis. This Fano resonance is formed by the interference and coupling between a "bright" electric dipole resonance and a "dark" toroidal dipole mode . Fig. 1b shows the corresponding spherical multipolar decomposition of the metasurface. As can be seen, the optical response is dominated by electric dipole (ED) excitation with a small contribution from the magnetic quadrupole (MQ) resonance. Such a pronounced ED feature was further investigated by performing the Cartesian multipolar analysis (Fig.  1b). It is worth noting that the ED response is mainly due to the strong excitation of toroidal dipole mode with an in-plane ED mode , which is polarized along the same direction as the optical pump.
Interestingly, due to the C2 symmetry of our sub-diffractive system, the toroidal dipole (TD) and MQ do not contribute to the far-field radiation along the z-direction. The far-field optical response is dominated by the ED mode . The non-radiating TD mode is a symmetry-protected BIC, where the ED mode plays a role to open a leaky channel and transform this ideal BIC into quasi-BIC with a finite Q-factor. Unlike the MD-BIC studied before where geometrical asymmetry is introduced to open a leaky channel, here the leaky channel, i.e., the excitation of ED mode , can be formed directly by properly choosing the structural sizes of the symmetric nanobars. Owing to the non-radiating nature of the dominant resonance -TD mode, a clear enhancement of the stored electric energy inside the nanobars is observed, as shown in the bottom of Fig. 1c. Fig. 1d gives the calculated electric near-field distributions. A pronounced poloidal current distribution can be observed from the two nanobars, indicating strong TD excitation. The small portion of MQ excitation shown in Fig. 1d, is due to the uncompensated circulating magnetic field in such flat geometry, formed by two anti-parallel magnetic dipole moments at the nodes of the poloidal current distribution. A comparison between the electric near-fields between y-polarized pump and x-polarized pump incidence can be seen in Fig. S1 of the Supporting Information. We observed significant near-field enhancement inside the nanostructure for y-polarized pump incidence as compared to the case for x-polarized pump incidence. A further investigation on the band structure and the corresponding mode profiles can be seen in Fig. S2 and S3 of Section I in the Supporting Information. As mentioned earlier, achieving scalable metasurfaces with several interdependent characteristics, including quality factor and spectral position, is the main target of this manuscript. There is a significant demand in the photonics community to achieve both conditions, rather than solely obtaining a high-Q resonance. In this respect, we employ the deep learning approach to inversely design our metasurfaces with simultaneously controlling and optimizing on Q-factor, amplitude and spectral position. For this task, we use the open-source neural-network library Keras 64 written in Python to implement our method. For the inverse design of nanophotonic structures using a deep learning approach, one challenge is that the same far-field electromagnetic response can correspond to different designs, i.e., several different structures can give the same responses. This nonuniqueness of the response-to-design mapping will induce conflicting examples within the training set and may prevent from converging. To avoid this issue, we use the Tandem Network (TN) approach 65,66 , as shown in Fig. 2. The TN architecture consists an inverse design network connected to a forward model network. The forward network learns the mapping from the structural parameters to the optical responses and is trained separately first. After the forward network is trained, it is placed after the inverse-design model network and its network weights keep fixed during the training of the inverse-design model network. The inverse-design network learns mapping from the optical responses to the structural parameters. When training the inverse-design network, its weights are updated to minimize the loss objective: = 1 ∑ ( − ) 2 with and being the values predicted by the neural network and the ground truth of the response (forward model network) or the structural parameters (inverse-design model network). Trained in this way, the inverse-design network is not constrained to produce a prespecified design. Rather it is free to infer any design that results in the desired forward behaviour.
In our case, we specify three structural parameters of the nanobars to learn: width , length and the offset distance 0 . By randomly specifying them, we first use the rigorous coupled-wave analysis (RCWA) 67 to generate 25,000 training examples, where we obtain the transmission spectra of the metasurfaces covering wavelength range 1400 nm to 1600 nm. RCWA is a frequency-domain modal method based on the decomposition of the periodic structure and the pseudo-periodic solution of Maxwell's equations in terms of their Fourier expansions 68 . It has been widely used for modelling light responses from periodic optical structures due to its fast convergence and accurate far-field calculations 69 . It is quite suitable for modelling the electromagnetic responses of metasurfaces and generate massive training data, especially when considering the inverse design, based on neural networks 35,70 . It is worth mentioning that for multilayer structures, the transfer matrix method can be one more choice 71 .
The forward model network is designed to have four fully-connected layers with each layer having 400-600-400-200 dimensions, respectively. We set the learning parameters batch size as 256, and use a learning rate of 0.001 and decay of 1 × 10 −6 . We first train the forward model network and evaluate it to see how well it can predict the given transmission spectra. Figure 3a shows the learning curves for the training and validation loss. It can be seen that both the training loss and validation loss decrease significantly after 10,000 epochs of training and become less than 0.005 after 30,000 epochs of training. This indicates that the trained network can estimate an appropriate spectrum, which is similar to the spectrum calculated analytically. As a test example, we use RCWA to simulate the transmission spectrum for an individual Si nanobar metasurface with parameters [ , , 0 ] being [300, 700, 300] nm, as shown by black dashed curve of Fig. 3(b). Then, we input these structural parameters to the network, and predict the corresponding output, which is shown in the blue curve of Fig. 3(b). As can be seen, our forward network can predict the transmission spectrum from our metasurface accurately. We have also tested our forward network by specifying different structural parameter sets corresponding to different transmission spectra (see Fig. S4 in the Supporting Information), which all verify the effectiveness of our forward network.
Next, we train the inverse-design model network by fixing the weights in the pre-trained forward model network. Since the forward model network is differentiable we are able to train the inversedesign model network with a loss placed after the forward model network. As mentioned above, this will overcome the issue of non-uniqueness in the inverse spectrum of electromagnetic waves, as the design by the neural network is not required to be identical to the design parameters that produced the training samples, only that the spectrum inferred by the forward model network match the target spectrum. The loss can be further lowered when the generated design and the real design have similar responses after training. The inverse-design model network has five fully-connected layers with 600, 600, 400, 200, 200 dimensions, respectively. For inverse design of our metasurfaces, the transmission spectrum is considered as the input of the tandem network. The design parameters are predicted from the intermediate layer of the whole network. The training process is shown in Fig. 3(c). As can be seen, by using TN approach, the learning of inverse design has converged effectively. We then test our inverse-design network by using a Fano formula to define the transmission spectrum 72,73 : (1) where 0 and Γ are the resonance frequency and linewidth, respectively. 0 and 0 are constant factors and fixed at 1 in the rest of this paper. is a dimensionless factor that describes the ratio between the resonant and non-resonant transition amplitudes in the spectrum.
Here, our target is a Fano resonance with a peak value of 100% and dip value of 0%. Therefore, the resonance frequency and linewidth of the transmission resonances can be obtained via: We first design a target Fano-resonance at 0 = 1500 , with resonance linewidth ∆ = 5 , and = 0.5, as shown by the black dashed curve of Fig. 3d. Here, we input the resonance frequency linewidth Γ by the defined resonance wavelength linewidth ∆ by Γ = ∆ 0 / 0 . We then use the network to predict the structural parameters [ , , 0 ] of the required metasurface as [316,580,189] nm. The transmission spectrum of the predicted metasurface is shown in the dashed red curve of Fig.  3(d). It matches well with the desired Fano-shape curve based on Equations 1 and 2. The deep network provides a powerful approach to inversely design nanophotonics structures. Once the training process is finished, the inverse design calculation takes only around 0.05 second in our case (only using CPU in a normal desktop with 64-bit Operating System: Processor: Intel(R) Core(TM) i7-4770 CPU @ 3.40 GHz, RAM: 16.0 GB), which is both promising and much more effective, as compared to conventional electromagnetic solvers. For nanophotonics applications, as mentioned above, it is a requirement to obtain scalable metasurfaces with controllable characteristics. In the following, using the TN approach, we further test the inverse design of Fano resonances with different spectral positions, linewidths or amplitudes.
According to Equations 1 and 2, we first specify target Fano resonances at 0 = 1450 nm, 1500 nm, and 1550 nm, respectively. Subsequently, we keep the linewidth ∆ = 15 nm and = 0.8, as shown in the black dashed curve of Figs. 4a-c. We then use the trained neural network to predict the structural parameters of the metasurfaces that can provide such spectra. The transmission spectra of the designed metasurfaces are shown in the red curves of Fig. 4a-c. They match and satisfy the design goal well. By varying the value of parameter or the linewidth of the resonance ∆ , the neural network can easily predict the metasurface design for the required Fano resonance with different characteristics, as shown in Fig. 4d-i. This provides a powerful method to control the near-field and electric energy confinement at the nanoscale (see Fig. S5 in the Supporting Information). So far, based on the deep learning approach, we have optimized the design to obtain high-Q resonances via altering various parameters, s including the linewidth, spectral position and amplitude, simultaneously. Such a high-Q resonance can significantly facilitate nanostructures to enhance lightmatter interactions for various applications such as nonlinear optics, optomechanics, etc. As an example, here, we investigate the third-harmonic generation (THG) from three designed metasurfaces with resonances at 1450 nm, 1500 nm and 1550 nm respectively, as shown in Fig. 4a-c. Here, we have taken into account that the spectral full-width-at-half-maximum (FWHM) of our experimentally used laser is around 15 nm for the wavelength range 1400 nm to 1600 nm. Thus we consider the resonances with this linewidth to maximize the nonlinear signal generation 13 .
The scanning electron microscope (SEM) image of one of the fabricated metasurfaces with designed resonance at 1500 nm is shown in Fig. 5a. We first measured the linear transmission spectra of the three metasurfaces under plane wave normal incidence with the electric field polarized along the yaxis, as shown in Fig. 5b. Pronounced asymmetric Fano resonances are observed around the desired spectral positions. Subsequently, we perform the TH spectroscopy measurement. A femtosecond laser beam with 200 fs pulse width and 80 MHz repetition rate was focused by an aspheric lens with focal length being 5 cm to a beam waist of 20 µm. The pump polarization was adjusted along the y-axis, in order to excite the designed TD BIC state, and tuned ranging from 1400 nm to 1600 nm, with maximum mean power in the sample plane up to around 66 mW, leading to a maximum peak intensity value of I 0 ≈ 0.66 GW/cm 2 . An objective with a numerical aperture (NA) NA=0.7 was used to collect the transmitted TH emission power in the forward direction (see Fig. S6 in Section IV. Experimental setup for nonlinear measurements of the Supporting Information). The experimentally measured TH signals from the three designed metasurfaces are shown in Fig. 5c. As can be seen, the TH signals are significantly enhanced around the resonances, while no THG enhancement are observed when the laser beam is polarized along x-axis (see Fig. S7 in the Supporting Information). By comparison, we observed 400-fold enhancement of the TH signal at the resonance position under y-polarized pump as compared to the case under x-polarized pump. Similarly, by performing the nonlinear multipolar analysis, the TH signal is predominant by the TD excitation with small portions of MQ and EQ excitations, which exhibit the same symmetry with respect to C 2 , This further leads to a stronger TH emission in the first-order dirraction compared to the zero-order diffraction, due to the absence of coupling to these modes and the normal outgoing waves (see Fig. S8 in the Supporting Information). Another advantage of high-Q resonances is that the particular electric field distributions can also facilitate the optomechanical vibration process [74][75][76][77][78] . In other words, by designing a high-Q resonance through machine learning, one can engineer the vibrational modes of nanoparticles based on the optomechanical processes, too. Here, by considering the designed metasurface shown in Fig. 2d, we numerically investigate the optomechanical vibrations and mechanical mode excitations. The mechanical vibration is modelled via the following equation: where is the displacement field characterizing the mechanical vibration, the constant , , and represent mass density, stiffness tensor and decay time of the silicon material, respectively. ( , ) is the driving force induced by the electromagnetic field here.
The optical pump in our analysis is a single y-polarized laser pulse with pulse duration 200 , peak intensity I 0 = 50 GW/cm 2 , at time 0 = 0 . It is worth mentioning that the radiation force is derived from the induced optical near-field profiles. Subsequently, we analyze the vibrations by measuring the changes in the width and length along the center of nanobars (see Fig. 7a). The vibration displacements along x and y directions, i.e. and , are shown in Fig. 7b. The corresponding spectral density is computed by the Fourier transfer analysis of this time-dependent vibrational amplitude and plotted in Fig. 7c showing that the coherent phonon oscillation frequency for this nanobar is around 12 GHz for and 9.5 GHz for . For comparison, Fig. 7d-f shows the corresponding optomechanical vibrations when the pump is polarized along x-axis. The amplitude of optomechanical vibration is around 100-fold stronger from the metasurface under y-polarized pump incidence, which corresponds to the excitation of the designed TD BIC state. In order to demonstrate the importance of high-Q resonance on opto-acoustic modes, we further calculate the spectral densities for different pump wavelengths, as shown in the 3-dimensional map in Fig. 8. As can be seen, a significant peak with the frequency of 12 GHz for (9.5 GHz for ) appears for wavelength at 1500 nm, due to the excitation of TD BIC state at the optical pump. The excitation strength of optomechanical mode decreases dramatically when the optical pump is away from the TD BIC state. We then estimate the feedback of the mechanical vibration on the resonant optical response. Based on the transient vibration shown in Fig. 7b, by assuming deform around 50pm displacement in x direction or 25-pm displacement in y direction, the maximum sensitivity of the scattering response in such metasurfaces is around 0.055% pm −1 (see Fig. S9 in Supporting Information). Utilizing the high-quality TD BIC state, up to 4.5% modulation of the transmission near the resonance can be expected when using a pulse laser with peak intensity 50 GW/cm 2 through the radiation force on the silicon nanostructures. These results suggest new opportunities for optomechnical applications such as light modulation and nanosensing with nanostructures.

Conclusion
To summarize, utilizing machine learning approach, we have demonstrated the inverse design of highquality Fano resonant metasurfaces composed of two nanobars with scalable characteristics including the spectral position, line width and amplitude of the transmission. The Fano resonance is originated from the TD-BIC state, featuring a strong near-field enhancement and intense electric energy localized inside the nanobars. We further employ these metasurfaces to simultaneously enhance photonphoton and photon-phonon interactions and achieving 400+ folds THG enhancement and 100+ folds enhancement for optomechanical vibrations. Our proposed scalable metasurfaces suggest new opportunities to control and enhance light-matter interactions, showing promising applications for realizing optoacoustic nonlinear metasurfaces.