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Here we employ the quasi-bound state in the continuum (quasi-BIC) resonance in all-dielectric metasurfaces for efficient nonlinear processes in consideration of the backward frequency conversion. We theoretically study the second-harmonic generation (SHG) from symmetry-broken AlGaAs metasurfaces and reveal the efficiency enhancement empowered by high-$Q$ quasi-BIC resonances. By introducing the correction term of nonlinear polarization at the fundamental wave field to the conventional undepleted approximation, we uncover the effect of backward frequency conversion on the nonlinear conversation efficiency. The SHG efficiency as $2.45\times10^{-2}$ with the developed depleted model, shows a $14.3\%$ decrease compared with $2.86\times10^{-2}$ in conventional undepleted approximation, under the incident intensity of 10 MW/cm$^{2}$. Our results are of significant importance for designing efficient nonlinear metasurfaces supporting high-$Q$ resonances.

Deep neural networks (DNNs) frequently present behaviorally irregular patterns, significantly limiting their practical potentials and theoretical validity in travel behavior modeling. This study proposes strong and weak behavioral regularities as novel metrics to evaluate the monotonicity of individual demand functions (known as the "law of demand"), and further designs a constrained optimization framework with six gradient regularizers to enhance DNNs' behavioral regularity. The proposed framework is applied to travel survey data from Chicago and London to examine the trade-off between predictive power and behavioral regularity for large vs. small sample scenarios and in-domain vs. out-of-domain generalizations. The results demonstrate that, unlike models with strong behavioral foundations such as the multinomial logit, the benchmark DNNs cannot guarantee behavioral regularity. However, gradient regularization (GR) increases DNNs' behavioral regularity by around 6 percentage points (pp) while retaining their relatively high predictive power. In the small sample scenario, GR is more effective than in the large sample scenario, simultaneously improving behavioral regularity by about 20 pp and log-likelihood by around 1.7%. Comparing with the in-domain generalization of DNNs, GR works more effectively in out-of-domain generalization: it drastically improves the behavioral regularity of poorly performing benchmark DNNs by around 65 pp, indicating the criticality of behavioral regularization for enhancing model transferability and application in forecasting. Moreover, the proposed framework is applicable to other NN-based choice models such as TasteNets. Future studies could use behavioral regularity as a metric along with log-likelihood in evaluating travel demand models, and investigate other methods to further enhance behavioral regularity when adopting complex machine learning models.

Here we demonstrate the efficient photon-pair generation via spontaneous parametric down conversion from a semiconductor metasurface supporting dual quasi-bound states in the continuum (quasi-BICs). In a simple metasurface design composed of AlGaAs ellipse nano-cyclinders, the two high-$Q$ quasi-BIC resonances that coincide with the generated signal and idler frequencies significantly boost the local electric field. This leads to a substantial enhancement in the reverse classical nonlinear process of sum frequency generation and subsequently the remarkable high generation rate of photon pairs under the quantum-classical correspondence principle. Within a narrowband wavelength regime around the quasi-BIC resonances, the rate of pair production is enhanced up to $\sim10^{4}$ Hz, two orders of magnitude larger than that in the Mie resonant AlGaAs nanoantennas. Moreover, the photon pair emission is mainly concentrated in the normal direction with respect to the metasurface, and shows tunable rate with the $Q$ factor by engineering the rotation angle of nano-cylinders. The presented work enables nanoscale sources of high-quality entangled photons which will find applications in advanced quantum imaging and communications.

The miniaturization of nonlinear light sources is central to the integrated photonic platform, driving a quest for high-efficiency frequency generation and mixing at the nanoscale. In this quest, the high-quality ($Q$) resonant dielectric nanostructures hold great promise, as they enhance nonlinear effects through the resonantly local electromagnetic fields overlapping the chosen nonlinear materials. Here, we propose a method for the enhanced sum-frequency generation (SFG) from etcheless lithium niobate (LiNbO$_{3}$) by utilizing the dual quasi-bound states in the continuum (quasi-BICs) in a one-dimensional resonant grating waveguide structure. Two high-$Q$ guided mode resonances corresponding to the dual quasi-BICs are respectively excited by two near-infrared input beams, generating a strong visible SFG signal with a remarkably high conversion efficiency of $3.66\times10^{-2}$ (five orders of magnitude higher than that of LiNbO$_{3}$ films of the same thickness) and a small full-width at half-maximum less than 0.2 nm. The SFG efficiency can be tuned via adjusting the grating geometry parameter or choosing the input beam polarization combination. Furthermore, the generated SFG signal can be maintained at a fixed wavelength without the appreciable loss of efficiency by selectively exciting the angle-dependent quasi-BICs, even if the wavelengths of input beams are tuned within a broad spectral range. Our results provide a simple but robust paradigm of high-efficiency frequency conversion on an easy-fabricated platform, which may find applications in nonlinear light sources and quantum photonics.