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Shaping the structure of light with flat optical devices has driven significant advancements in our fundamental understanding of light and light-matter interactions, and enabled a broad range of applications, from image processing and microscopy to optical communication, quantum information processing, and the manipulation of microparticles. Yet, pushing the boundaries of structured light beyond the linear optical regime remains an open challenge. Nonlinear optical interactions, such as wave mixing in nonlinear flat optics, offer a powerful platform to unlock new degrees of freedom and functionalities for generating and detecting structured light. In this study, we experimentally demonstrate the non-trivial structuring of third-harmonic light enabled by the addition of total angular momentum projection in a nonlinear, isotropic flat optics element -- a single thin film of amorphous silicon. We identify the total angular momentum projection and helicity as the most critical properties for analyzing the experimental results. The theoretical model we propose, supported by numerical simulations, offers quantitative predictions for light structuring through nonlinear wave mixing under various pumping conditions, including vectorial and non-paraxial pump light. Notably, we reveal that the shape of third-harmonic light is highly sensitive to the polarization state of the pump. Our findings demonstrate that harnessing the addition of total angular momentum projection in nonlinear wave mixing can be a powerful strategy for generating and detecting precisely controlled structured light.

Recent years have witnessed significant developments in the study of nonlinear properties of various optical materials at the nanoscale. However, in most cases experimental results on harmonic generation from nanostructured materials are reported without the benefit of suitable theoretical models and appropriate comparisons to assess enhancement of conversion efficiencies compared to the intrinsic properties of a given material. Here, we report experimental observations of even and odd optical harmonics generated from a suspended subwavelength silicon film, a dielectric membrane, up to the 7th harmonic tuned deep in the UV range at 210nm, which is the current limit of our detection system, using peak power densities of order 3TW/cm^2. We explain the experimental data with a time domain, hydrodynamic-Maxwell approach broadly applicable to most materials. Our approach accounts simultaneously for surface and magnetic nonlinearities that generate even optical harmonics, as well as linear and nonlinear material dispersions beyond the third order to account for odd optical harmonics, plasma formation, and a phase locking mechanism that makes the generation of high harmonics possible deep into the UV range, where semiconductors like silicon start operating in a metallic regime.

Nonlinear metasurfaces offer a new paradigm to realize optical nonlinear devices with new and unparalleled behavior compared to nonlinear crystals, due to the interplay between photonic resonances and materials properties. The complicated interdependency between efficiency and emission directionality of the nonlinear optical signal on the existence, localization, and lifetimes of photonic resonances, as well as on the nonlinear susceptibility, makes it extremely difficult to design optimal metasurfaces using conventional materials and geometries. Inverse design using topology optimization is a powerful design tool for photonic structures, but traditional approaches developed for linear photonics are not suitable for such high dimensional nonlinear problems. Here, we use a topology optimization approach to inverse-design a fabrication-robust nonlinear metasurface that includes quantum-engineered resonant nonlinearities in semiconductor heterostructures for efficient and directional second harmonic generation. Furthermore, we also demonstrate that under practical constraints, among all the parameters, the nonlinear modal overlap emerges as the dominant parameter that enhances conversion efficiency, a finding that contrasts with intuition-driven studies that often emphasize Purcell enhancement. Our results open new opportunities for optimizing nonlinear processes in nanophotonic structures for novel light sources, quantum information applications, and communication.

Digital signal processing is the cornerstone of several modern-day technologies, yet in multiple applications it faces critical bottlenecks related to memory and speed constraints. Thanks to recent advances in metasurface design and fabrication, light-based analog computing has emerged as a viable option to partially replace or augment digital approaches. Several light-based analog computing functionalities have been demonstrated using patterned flat optical elements, with great opportunities for integration in compact nanophotonic systems. So far, however, the available operations have been restricted to the linear regime, limiting the impact of this technology to a compactification of Fourier optics systems. In this paper, we introduce nonlinear operations to the field of metasurface-based analog optical processing, demonstrating that nonlinear optical phenomena, combined with nonlocality in flat optics, can be leveraged to synthesize kernels beyond linear Fourier optics, paving the way to a broad range of new opportunities. As a practical demonstration, we report the experimental synthesis of a class of nonlinear operations that can be used to realize broadband, polarization-selective analog-domain edge detection.

Biomechanical biofeedback may enhance rehabilitation and provide clinicians with more objective task evaluation. These feedbacks often rely on expensive motion capture systems, which restricts their widespread use, leading to the development of computer vision-based methods. These methods are subject to large joint angle errors, considering the upper limb, and exclude the scapula and clavicle motion in the analysis. Our open-source approach offers a user-friendly solution for high-fidelity upper-limb kinematics using a single low-cost RGBD camera and includes semi-automatic skin marker labeling. Real-time biomechanical analysis, ranging from kinematics to muscle force estimation, was conducted on eight participants performing a hand-cycling motion to demonstrate the applicability of our approach on the upper limb. Markers were recorded by the RGBD camera and an optoelectronic camera system, considered as a reference. Muscle activity and external load were recorded using eight EMG and instrumented hand pedals, respectively. Bland-Altman analysis revealed significant agreements in the 3D markers' positions between the two motion capture methods, with errors averaging 3.3$\pm$3.9 mm. For the biomechanical analysis, the level of agreement was sensitive to whether the same marker set was used. For example, joint angle differences averaging 2.3$\pm$2.8{\deg} when using the same marker set, compared to 4.5$\pm$2.9{\deg} otherwise. Biofeedback from the RGBD camera was provided at 63 Hz. Our study introduces a novel method for using an RGBD camera as a low-cost motion capture solution, emphasizing its potential for accurate kinematic reconstruction and comprehensive upper-limb biomechanical studies.

By uncovering novel aspects of second harmonic generation in aluminum we show that there are unusual and remarkable consequences of resonant absorption, namely an unexpectedly critical role that bound electrons play for light-matter interactions across the optical spectrum, suggesting that a different basic approach is required to fully explain the physics of surfaces. We tackle an issue that is never under consideration given the generic hostile conditions to the propagation of light under resonant absorption. Unlike most noble metals, aluminum displays Lorentz-like behavior and interband transitions centered near 810 nm, thus splitting the plasmonic range in an atypical manner and setting its linear and nonlinear optical properties apart. Studies of aluminum nanostructures having complex topologies abound, as do reported inconsistencies in the linear spectral response of surface plasmons and harmonic generation. Our experimental observations of second harmonic generation from aluminum nanolayers show that bound electrons are responsible for a unique signature neither predicted nor observed previously: a hole in the second harmonic spectrum. A hydrodynamic-Maxwell theory explains these findings exceptionally well and becomes the basis for renewed studies of surface physics., Comment: 8 pages, 5 figures

Notwithstanding its long history, the study of nonlinear optics from metal surfaces is still an active field of research. For instance, in view of the presence of absorption questions remain concerning the possibility of significantly enhancing harmonic conversion efficiencies in the visible and UV ranges. While to many it may seem that metals do not easily lend themselves to that purpose, they are nevertheless crucial materials in the development of nanophotonics, and more generally, to electromagnetism at the nanoscale. Here, we report our experimental observations and numerical simulations of second and third harmonic generation from a gold nanograting, which exhibits a plasmonic resonance whose spectral position depends on incident angle. All things being equal, the enhancement of nonlinear optical processes from the UV to the near IR range manifests itself in dramatic manner: second harmonic generation conversion efficiencies increase more than three orders of magnitude compared to a flat gold mirror, while third harmonic generation conversion efficiency increases by nearly four orders of magnitude, both in excellent agreement with predictions. The clear inferences one may draw from our results are that our model describes the dynamics with unprecedented accuracy, and that much remains to be revealed in the development of nonlinear optics of metals at the nanoscale.

The predominant methods currently used to determine nonlinear optical constants like the nonlinear refractive index n2 or chi3 rely mostly on experimental, open and closed z-scan techniques and beam deflection methods. While these methods work well when the linear absorption is relatively small or negligible, the retrieval process is more complicated for a strongly scattering, dispersive or absorbing medium. The study of optics at the nanoscale in the ps or fs regimes demands the development of new theoretical tools experimental approaches, to extract and verify both linear and nonlinear optical dispersions exhibited by matter, especially when material constituents are fashioned into nanostructures of arbitrary shape. We present a practical, combined experimental and theoretical approach based on a hydrodynamic model that uses experimental results of harmonic generation conversion efficiencies to retrieve complex, nonlinear dispersion curves, not necessarily only for third order processes. We provide examples for materials that are of special interest to nanophotonics, silicon, gold, and indium tin oxide, which displays nonlocal effects and a zero-crossing of the real part of the dielectric constant. The results for silicon and gold compare well with analytical predictions based on the nonlinear oscillator model. Based on our assessment of THG conversion efficiencies in silicon, we predict chi3(w) and chi3(3w) are of order 10^(-17)(m/V)^2, in the visible and IR ranges, with respective peaks of 10^(-14) and 10^(-16)(m/V)^2 in the UV range. Similarly, gold's chi3(w) and chi3(3w) are of order 10^(-17) and 10^(-16)(m/V)^2, and predict chi3(w)~10^(-17)(m/V)^2 and chi3(3w)~10^(-18)(m/V)^2 for ITO. These results suggest that judicious exploitation of the nonlinear dispersion of ordinary semiconductors can transform device physics in spectral regions that extend well into the UV range.

Nonlinear light sources are central to a myriad of applications, driving a quest for their miniaturisation down to the nanoscale. In this quest, nonlinear metasurfaces hold a great promise, as they enhance nonlinear effects through their resonant photonic environment and high refractive index, such as in high-index dielectric metasurfaces. However, despite the sub-diffractive operation of dielectric metasurfaces at the fundamental wave, this condition is not fulfilled for the nonlinearly generated harmonic waves, thereby all nonlinear metasurfaces to date emit multiple diffractive beams. Here, we demonstrate the enhanced single-beam second- and third-harmonic generation in a metasurface of crystalline transition-metal-dichalcogenide material, offering the highest refractive index. We show that the interplay between the resonances of the metasurface allows for tuning of the unidirectional second-harmonic radiation in forward or backward direction, not possible in any bulk nonlinear crystal. Our results open new opportunities for metasurface-based nonlinear light-sources, including nonlinear mirrors and entangled-photon generation., Comment: 16 pages, 8 figures

Understanding how light interacts at the nanoscale with metals, semiconductors, or ordinary dielectrics is pivotal if one is to properly engineer nano-antennas, filters and, more generally, devices that aim to harness the effects of new physical phenomena that manifest themselves at the nanoscale. We presently report experimental results on second and third harmonic generation from 20nm- and 70nm-thick gold layers, for TE- and TM-polarized incident light pulses. We highlight and discuss for the first time the relative roles bound electrons and an intensity dependent free electron density (hot electrons) play in third harmonic generation. While planar structures are generally the simplest to fabricate, metal layers that are only a few nanometers thick and partially transparent are almost never studied. Yet, transmission offers an additional reference point for comparison, which through relatively simple experimental measurements affords the opportunity to test the accuracy of available theoretical models. Our experimental results are explained well within the context of the microscopic hydrodynamic model that we employ to simulate second and third harmonic conversion efficiencies, and to simultaneously and uniquely predict the nonlinear dispersive properties of a gold nanolayer under pulsed illumination. Using our experimental observations and our model, based solely on the measured third harmonic power conversion efficiencies we predict |chi3|~10^(-18)-10^(-17)(m/V)^2, triggered mostly by hot electrons, without resorting to the implementation of a z-scan set-up.

Epsilon-near-zero materials are exceptional candidates for studying electrodynamics and nonlinear optical processes at the nanoscale. We demonstrate that by alternating a metal and a highly doped conducting-oxide, the epsilon-near-zero regime may be accessed resulting in an anisotropic, composite nanostructure that significantly enhances nonlinear interactions. Using two independent and different computation techniques we show that the structure can enhance the local field intensity by nearly two orders of magnitudes, in large part due to the onset of the effective anisotropy. The investigation of the multilayer nanostructure using a microscopic, hydrodynamic approach also sheds light on the roles of two competing contributions that are for the most part overlooked, but that can significantly modify linear and nonlinear responses of the structure: nonlocal effects, which blueshift the resulting resonance, and the hot electron nonlinearity, which redshifts the plasma frequency as the effective mass of free electrons increases as a function of incident power density. Finally, we show that, even in absence of second order bulk nonlinearity, second order nonlinear processes are also significantly enhanced by the layered structure.

We study electromagnetic pulse propagation in an indium tin oxide nanolayer in the linear and nonlinear regimes. We use the constitutive relations to reconstruct the effective dielectric constant of the medium, and show that nonlocal effects induce additional absorption resonances and anisotropic dielectric response: longitudinal and transverse effective dielectric functions are modulated differently along the propagation direction, and display different epsilon-near-zero crossing points with a discrepancy that increases with increasing intensity. We predict that hot carriers induce a dynamic redshift of the plasma frequency and a corresponding translation of the effective nonlinear dispersion curves that can be used to predict and quantify nonlinear refractive index changes as a function of incident laser peak power density. Our results suggest that large, nonlinear refractive index changes can occur without the need for epsilon-near-zero modes to couple with plasmonic resonators. At sufficiently large laser pulse intensities, we predict the onset of optical bistability, while the presence of additional pump absorption resonances that arise from longitudinal oscillations of the free electron gas give way to corresponding resonances in the second and third harmonic spectra. A realistic propagation model is key to unraveling the basic physical mechanisms that play a fundamental role in the dynamics.

Using a hydrodynamic approach we examine bulk- and surface-induced second and third harmonic generation from semiconductor nanowire gratings having a resonant nonlinearity in the absorption region. We demonstrate resonant, broadband and highly efficient optical frequency conversion: contrary to conventional wisdom, we show that harmonic generation can take full advantage of resonant nonlinearities in a spectral range where nonlinear optical coefficients are boosted well beyond what is achievable in the transparent, long-wavelength, non-resonant regime. Using femtosecond pulses with approximately 500 MW/cm2 peak power density, we predict third harmonic conversion efficiencies of approximately 1% in a silicon nanowire array, at nearly any desired UV or visible wavelength, including the range of negative dielectric constant. We also predict surface second harmonic conversion efficiencies of order 0.01%, depending on the electronic effective mass, bistable behavior of the signals as a result of a reshaped resonance, and the onset fifth order nonlinear effects. These remarkable findings, arising from the combined effects of nonlinear resonance dispersion, field localization, and phase-locking, could significantly extend the operational spectral bandwidth of silicon photonics, and strongly suggest that neither linear absorption nor skin depth should be motivating factors to exclude either semiconductors or metals from the list of useful or practical nonlinear materials in any spectral range., Comment: 12 pages, 4 figures