Interaction of semiconductor metasurfaces with short laser pulses: From nonlinear-optical response toward spatiotemporal shaping

Optical metasurfaces can be considered as the metamaterial versions of natural two-dimensional materials, interfaces, and optical surfaces.^1,2 They consist of a single or a few layers of nanoscale building blocks, often called meta-atoms, arranged in a two-dimensional fashion. By careful engineering of the meta-atoms and their arrangement, metasurfaces can provide a broad range of optical functionalities, including wavefront shaping,^3–6 polarization control,^7,8 and spectral filtering.⁹ 

Importantly, the metasurface platform also allows for combining multiple functionalities in a single device.^7,10,11 Additionally, metasurfaces offer interesting possibilities for dynamically switching their response,¹² e.g., by mechanical deformation,¹³ by laser-induced modification,¹⁴ or by hybridizing them with functional materials, which change their optical properties in response to an externally controlled stimulus.^15,16 Oftentimes, the meta-atoms will be designed to act as optical nanoantennas, confining light in nanoscale volumes, thereby enhancing light-matter interactions, such as absorption or spontaneous emission.¹⁷ 

While metasurfaces have originally been pioneered in plasmonics, we have witnessed a recent surge toward all-dielectric and semiconductor implementations of optical metasurfaces. This is, in part, granted by the lower losses and additional opportunities to excite multipolar Mie-type resonances in high-refractive-index dielectric building blocks.^18,19 Particularly attractive in this respect are semiconductor materials, such as silicon or gallium arsenide. For photon energies below their fundamental electronic bandgap, their optical response is characterized by a high real part and a negligible imaginary part of their refractive index. Additionally, many semiconductors exhibit strong second-order (for materials with noninversion-symmetric lattice structure, e.g., zincblende) and/or third-order nonlinear-optical response, supporting effects such as harmonic generation, frequency mixing, nonlinear absorption, Kerr effect, and optical rectification. Moreover, the possibility to excite free carriers offers additional avenues for dynamically changing the semiconductor optical response at ultrafast time scales. Nanostructuring the semiconductor materials in metasurfaces composed of resonant meta-atoms allows for enhancing and tailoring all the mentioned effects.^20,21 Drawing from the nanoantenna functionality of the meta-atoms semiconductor Mie-resonators can confine the light inside their volume, allowing for a large overlap of the near-fields of the resonant modes with the semiconductor material.²² Thereby, nonlinear optical effects can be boosted.^23–25 

The orientation of the local near-fields can also be engineered by the nanostructure design of the metasurface, which, e.g., enables breaking of conventional polarization selection rules of nonlinear processes for the effective nonlinear metasurface response.^26,27 Nanoantenna-mediated directional effects can also be exploited and custom-tailored by design.²⁷ Moreover, nanostructuring can provide light escape routes that are absent in a bulk semiconductor wafer, and introduces recombination centers for free carriers, reducing their lifetimes.²⁸ 

The outlined potential has been clearly consolidated by a collection of recent works. In this Perspective, we will review the achieved progress in this field in a compact fashion. Instead of attempting a comprehensive review, we will concentrate on a few selected works to illustrate the principles. Extrapolating from the existing demonstrations, we postulate that some of the unique future potential of metasurfaces lies particularly in their interaction with short intense laser pulses. Combining their enhanced and custom-tailored nonlinear and ultrafast optical response with their capabilities for spatial and spectral control of light fields may enable the realization of planar solid-state devices, offering response features and functionalities inaccessible by natural materials or existing technologies. Examples, which will be discussed in more detail, include ultrafast beam scanners and ultrafast spatial multiplexing.

Furthermore, we will discuss the potential of semiconductor metasurfaces to solve an important open challenge in photonics, namely, the deterministic generation of complex light fields, which vary both in space and on an ultrafast time scale in an arbitrary, user-defined fashion. An artist’s impression of a metasurface that interacts with a preshaped pulse to synthesize a complex spatiotemporal light field is shown in Fig. 1. While some avenues towards and the potential of custom-tailored metamaterials for engineering spatiotemporal effects were pointed out previously,^29,30 in Sec. IV of this Perspective, we will discuss and develop this idea for the case of resonant semiconductor metasurfaces.

Here, $P NL$ is the nonlinear polarization in the material with the nonlinear susceptibility tensors $χ ^ ( n )$⁠, and $E$ is the pump electric field. Most prominently, the nonlinear polarization can be the source of frequency components that were not present in the excitation field. Among the most commonly exploited effects is the process of second harmonic generation (SHG): $P ( 2 ω ) = ε 0 χ ^ ( 2 ) ⋅ E ω E ω$⁠; here, the dot denotes the inner tensor product. The nonlinear polarization results in the far-field radiation $I 2 ω$ at the doubled frequency of the driving laser frequency $ω$⁠. This effect can, e.g., extend the spectral range of the existing light sources. SHG can only be observed in noncentrosymmetric materials or at material boundaries. Traditionally, to achieve a reasonable conversion efficiency $η 2 ω = I 2 ω / I ω$⁠, a sizable thickness of nonlinear material and involved phase-matching schemes are required. Efficient conversion at the nanoscale would help alleviate phase-matching issues but remains a challenging task. Previous successful efforts to boost frequency conversion at the nanoscale utilized plasmonic resonators.^32–34 Many materials used for all-dielectric nanophotonics, such as silicon, germanium, titanium dioxide, and many others are centrosymmetric. This property makes the generation of second-order nonlinear effects such as SHG limited to surfaces, and its detection very challenging. Although SHG-enhancement by all-dielectric nanoresonators has been theorized for a while,⁴⁰ it could not be experimentally demonstrated until relevant platforms for noncentrosymmetric III-V-semiconductor-based metasurfaces were established.^22,25,41 Early manifestations of size-dependent SHG from nanostructured semiconductors and dielectrics were provided in GaP⁴² and $LiNbO 3$ nanowires.^43,44 First demonstrations of SHG by magnetic Mie-type resonances in AlGaAs and GaAs nanoparticles, respectively, were reported later.^25,26 Liu et al.²⁶ revealed four orders of magnitude enhancement with respect to unstructured GaAs and a conversion efficiency of $η 2 ω ≈2× 10 − 5$⁠. This figure was improved over the years,^39,45–47 peaking at $η 2 ω ≈ 10 − 4$ in AlGaAs nanostructures.²⁷ 

The zincblende crystal structure of most III-V’s defines a $χ ^ ( 2 )$ tensor where $χ i j k ( 2 ) ≠0$ only for $i≠j≠k$⁠. As a consequence, SHG emitted from nanostructures based on III-Vs has certain polarization and scattering properties. For a bulk GaAs wafer with two out of three crystallographic axes lying within the wafer’s surface (⁠ $x$ and $y$⁠), no SHG will be observed under normal incidence. The only nonlinear polarization will be generated along $z$⁠: $P z =2 χ z y x ( 2 ) E x E y$⁠, and it will not create a radiating field along the original direction, $z$⁠. However, engraving an array of Mie-resonant nanoparticles out of such a wafer does allow efficient far-field SHG for two reasons.⁴⁸ First, if the period of the array is larger than the SHG wavelength, the diffraction pattern will utilize the $P z$ polarization, giving rise to detectable diffraction orders. Second, nanostructures can support local fields along $z$⁠, which can couple to the far field efficiently, e.g., by Mie-type resonances, creating SH polarization along $x$ or $y$⁠, that can enable SH emission at close-to-normal direction, which is otherwise very weak.⁴⁹ 

Other implications of the nontrivial III-V tensor structure are certain polarization properties of the SHG radiation itself,⁴⁵ which can be utilized to unambiguously verify that the main contribution to the SHG signal is rather the one of the bulk than the one of the surface. On the practical side, the structure of the tensor allows nonlinear generation of optical vector beams characterized by azimuthally or radially polarized light.²⁷ 

Coupling between periodic arrangements of resonant nanoparticles yields a more complicated spectral response, beyond the simple Mie theory. One of the advantages of such coupling is the emergence of dark modes; these modes do not couple to free-space beams, unless a defect is introduced that breaks the symmetry of the nanoparticle.⁵⁰ The defect can couple the dark mode to a bright mode, causing a Fano-type interference⁵¹ that shows up as a prominent narrow-band dip or peak in transmittance or reflectance spectra.^52,53 Dark resonances support local fields that are orders of magnitude larger than the incoming beam. A recent study of a GaAs-based Fano-resonant metasurface showed nontrivial spectral shaping of SHG and multifold efficiency enhancement induced by high field localization and enhancement inside the broken-symmetry resonators.⁴⁷ 

The success of III-V-based nanostructures in achieving comparatively high SHG-conversion efficiency lies, in part, in their high nonlinear susceptibility values, one of the highest available. A downside is the relatively small bandgap energy, translating into higher absorption coefficients for the harmonic generation radiation, impairing the conversion efficiency in certain spectral ranges.²⁶ Other material platforms have been utilized for efficient SHG at shorter wavelengths, with notable examples being perovskites (such as $BaTiO 3$^54–56), silicon carbide,⁵⁷ and selenium.⁵⁸ Nanocrystals can also create interfaces where the symmetry is broken; for instance, in Ref. 59, SHG from a nanocrystalline silicon nanoparticle was enhanced by two orders of magnitude with respect to an unstructured silicon film using Mie resonances.

Third order effects such as four-wave mixing, self-phase modulation, third harmonic generation (THG), and two-photon absorption can also be observed in centrosymmetric semiconductors. As one of the most well-understood materials in nature, silicon has historically been the most popular source for all-dielectric nanophotonics structures. Unsurprisingly, the first manifestations of nonlinearities enhanced by magnetic Mie-types resonances were found in silicon nanoparticles,²³ where THG was found to be enhanced by two orders of magnitude with respect to a thick silicon wafer. The conversion efficiency has been found to be on the order of $η 3 ω ≈ 10 − 7$⁠, on par with the best-performing plasmonic nanostructures. Over time, conversion efficiencies from the IR to visible and UV have been successfully optimized by choices of geometry, materials, polarization, spectral range, and other parameters.^{23,35,36,38,60–62} An overview of the current conversion efficiency figures obtained with the semiconductor metasurfaces is given in Fig. 2.

Studies of THG from single nanoparticles can shed light on the role of different nanoparticle resonances in their nonlinear response. Magnetic resonances usually produce more efficient THG than electric ones, as experimentally and numerically verified in Ref. 63, where single silicon nanodisks were excited by a tunable femtosecond laser source. Having certain symmetry properties, these modes can be selectively excited by structured light beams, as probed by enhanced THG.⁶⁴ Higher-order modes, such as anapole modes,⁶⁵ have been utilized to aid efficient frequency conversion even further.^36,62 Here, amorphous germanium was used instead of silicon, since it has a larger refractive index of $n≈4$⁠, and higher nonlinear susceptibility in the chosen spectral range. As a result, an enhancement of about 4 orders of magnitude was found with respect to an unstructured film, on top of another three orders with respect to a silicon wafer.³⁶ 

If several nanoparticles are brought together so that the distance between them is smaller than or on the order of a wavelength, they are often referred to as oligomers.⁶⁶ Pairs of nanoparticles, or dimers, were shown to create hot spots of enhanced local fields,⁶⁷ which were then used to tailor the THG in the far-field.⁶⁸ Changing the distance between the nanoparticles within oligomers can cause changes in coupling and hybridization of the individual nanoparticle modes, which can be used to tune the nonlinear response of nanoparticles, too. In Ref. 69, oligomers consisting of three nanoparticles (trimers) excited at their magnetic dipole mode showed prominently different THG spectra for different sets of nanoparticle diameters and interparticle spacing. Magnetic Fano resonances excited in subwavelength quadrumers bring about additional enhancement of THG.³⁸ 

Quasi-infinite arrays of nanoparticles that have periods of less than a free-space wavelength can be used to further increase the nonlinear response by high-quality factor (high-Q) collective modes. In Ref. 35, metasurfaces with Q-factors of up to 500 have been used to enhance the THG by 5 orders of magnitude with respect to a silicon film of the same thickness. A similar approach has been taken in Ref. 70 with a different structure possessing dark modes with a small net dipole moment to enhance THG by a factor of 300 with respect to a thick silicon substrate. Although high-Q metasurfaces represent a promising route to enhanced optical nonlinearities, both papers expressed concerns regarding wasting most of the bandwidth of the femtosecond pulse that is much wider than the bandwidth of the resonance. The time-bandwidth limit is a major obstacle to efficient interactions of femtosecond laser pulses with high-Q cavities, and the full advantage of the field enhancement in a high-Q cavity remains an open question.

Some other functionalities of Mie-resonant nanostructures in the nonlinear regime are: generation of UV light through THG,⁶⁰ manipulation of the nonlinear wavefront so as to form efficient, direction-specific nonlinear diffraction and other types of wavefront engineering,^71,72 phase-matching-free SHG in $LiNbO 3$ waveguides,⁷³ enhanced nonlinearities by complementary structures,⁷⁴ and possibilities in probing optical coupling of all-dielectric nanostructures to optical waveguides.⁷⁵ 

Second and third harmonic generation processes are enabled by consolidation of two or three fundamental photons to a higher-energy photon so that the overall energy is conserved: $ℏ ω n =nℏω$⁠, where $n$ is the process order. For a general frequency mixing process, a nonlinear susceptibility can unify any number of photons having an arbitrary set of frequencies: $∑ i = 1 N ℏ ω i = ∑ j = 1 M ℏ ω j$⁠. First, manifestations of four-wave mixing were provided in germanium nanodisks,⁷⁶ where effective third-order susceptibilities as high as $2.8× 10 − 16 m 2 / V 2$ were found. A more sophisticated technique utilized a bi-color pump experiment, where 11 new frequencies have been generated in III-V-based semiconductor metasurfaces;³⁹ see Fig. 3(a) for a typical frequency mixing spectrum spanning the entire visible range and beyond.⁴

Equation (1) is a Taylor series with $E/ E at$ as the expansion parameter, where $E$ is the incident field strength and $E at ≈5× 10 11 V / m$ is the characteristic atomic electric field strength. Under the condition of $E≪ E at$⁠, nonlinear polarization is much smaller than the linear one, the nonparabolicity of potentials for electrons is small, $P ( n ) ≫ P ( n + 1 )$⁠, and the regime is called perturbative. The closer the laser intensities are to the atomic scale, the more the Taylor series fails to describe the real polarization of materials, as the high-order terms become comparable to each other. In GaAs, for instance, one can estimate the critical intensity as that when $P ( 2 ) ≈ P ( 3 )$⁠, or, conversely, $E crit ≈ χ ( 2 ) / χ ( 3 ) ≈ 10 9 V / m$⁠, which corresponds to intensities of approximately $1 TW / cm 2$⁠. This so-called nonperturbative regime has become accessible not only through the advent of femtosecond laser sources that can easily reach such intensities, but also with resonant nanostructures⁷⁷ that can funnel light to form hot spots of much higher field intensity.

The most straightforward manifestation of nonperturbative nonlinearities is the process of high-harmonic generation (HHG). In it, electrons become field-ionized but recombine with their host atom almost instantly, generating optical harmonics in the extreme UV. While best known for gases, this process can also be observed in solid-state media.⁷⁸ The intensity requirements for this process are high, and one of the approaches to significantly enhance HHG is to use localized resonances. Semiconductor metasurfaces with Mie-type resonances have been successfully used to observe harmonics up to the fourth in noncentrosymmetric materials³⁹ and odd harmonics up to the 11th in silicon-based metasurfaces.⁷⁹ Even a moderate local field enhancement, when coupled to a high-order nonlinear term, may generate orders of magnitude larger high-harmonic response. For instance, the 9th harmonic from the metasurface can be detected at intensities as low as $50 GW / cm 2$ with the signal two orders of magnitude above the noise level, whereas any detectable 9th harmonic from an unpatterned film of the same thickness shows up only at $200 GW / cm 2$⁠. Importantly, at some intensities, the 7th and 9th harmonics equalize in intensity, providing foundations for tailored HHG-sources and possibly diverging series in Eq. (1).

One of the most straightforward results of Eq. (1) is that, if pumped by a narrowband laser centered at frequency $ω$⁠, the resulting harmonics will have spectra centered at $nω$⁠, where $n$ is the order of the nonlinear process. On the other hand, this rule may fail in dynamically evolving systems, such as rapidly generated plasmas in gases and semiconductors.⁸⁰ In these systems, generation of free carriers leads to blue shifting of the emitted fundamental and harmonic generation, a process sometimes dubbed “photon acceleration.”⁸¹ However, typically intensities of $10 15 W / cm 2$ are needed to observe considerable blue-shifts of the emitted photons. At much lower intensities of up to $30 GW / cm 2$⁠, photon acceleration has been recently demonstrated in semiconductor metasurfaces by observing the spectrum of the THG radiation as a function of the pump intensity.⁸² In the experiment, the generated photons had a variable carrier frequency from $3ω$ to $3.1ω$⁠, tuned by the intensity of the mid-infrared pump. A simple coupled-mode theory model with time-dependent coefficients provided an explanation of the observed spectra, showing that Eq. (1) is not sufficient to describe the nonlinear polarization of semiconductor materials under strong electromagnetic fields.

Apart from pure enhancement of nonlinear processes, the capability of inhomogeneous metasurfaces to control the phase of an incident light field as a function of in-plane position can also be used for nonlinear wavefront control, where a desired spatial phase and intensity profile is imprinted onto the nonlinearly generated wave. So far, this was demonstrated for third harmonic waves using silicon metasurfaces. Using metasurfaces composed of silicon nanopillars, creating a linear-phase gradient at the TH wavelength, Wang et al. have demonstrated THG at a designed angle,⁷¹ as shown in Fig. 3(b). The generation of third-harmonic focused vortex beams was also accomplished. Nonlinear holographic metasurfaces were recently demonstrated by Gao et al.⁷² using a metasurface composed of C-shaped Si nanoantennas; see Fig. 3(c). The design is such that the incident laser is enhanced by the fundamental metasurface resonance, whereas the generated THG signals are redistributed to the air gap region by higher-order resonance. This way, the absorption loss of silicon at the short third-harmonic wavelength is reduced, allowing for the generation of cyan and blue THG holograms for illumination with a near-infrared laser with a conversion efficiency of up to the order of $10 − 6$⁠.

In the future, we expect that nonlinear wavefront shaping metasurfaces will be extended to other functionalities and different nonlinear processes. Moving to materials with a noninversion-symmetric crystal structure, such as GaAs or GaP, will allow for wavefront shaping of SHG, further enhancing conversion efficiency. Extension of the concept to nonlinear mixing processes, optical rectification, and higher-order processes is also an interesting option.

The presence of free carriers in Mie-resonant semiconductor nanoparticles not only changes the back- or forward-scattering of light but can also tailor the scattering pattern in general, as shown by several theoretical efforts.^84–86 

The rate of relaxation to the initial state strongly depends on the constituent material of the metasurface. Crystalline materials with an indirect bandgap, such as silicon, are poor candidates for ultrafast metasurfaces, as relaxation times can be as long as hundreds of picoseconds.⁹⁰ In Ref. 85, silicon nanoparticles were pumped at high fluences of around $40 mJ / cm 2$ so as to achieve large estimated free carrier density of about $5× 10 20 cm − 3$⁠. This led to dominance of the Auger recombination process, making it as fast as 2.5 ps.

Another approach to shorten the lifetime of free carriers in a semiconductor is to increase the probability of monomolecular recombination through inhomogeneities of the crystal structure. Naturally, metasurfaces come at an increased relative surface area due to nanostructuring, making surface recombination occur more frequently than in the bulk source material. This effect was shown to dominate the relaxation in GaAs-based metasurfaces, leading to relaxation of the magnetic dipolar Mie-mode back to its initial state in only 6 ps and $1/e$ relaxation time of 2.5 ps.²⁸ 

Although not experimentally demonstrated at the time of submission of this Perspective, to our knowledge, one can easily imagine that the ultrafast and spectrally tunable response of semiconductor metasurfaces can be used for ultrafast wavefront shaping. Spatially inhomogeneous metasurfaces could be designed such that the phase profile, which they imprint onto an incident probe beam, changes in a desired manner in the presence of an optical pump beam. Alternatively, spatially homogeneous metasurfaces could be combined with a spatially inhomogeneous pump beam illumination. Important potential applications are ultrafast beam deflectors, which could find applications in beam scanners e.g., for self-driving vehicles. Another interesting application scenario is ultrafast spatial multiplexing for optical communication. As mentioned in Sec. III A, for GaAs metasurfaces recovery times of the dominant component to the ultrafast all-optical reflectance modulation below 10 ps were observed in pump-probe experiments,²⁸ which would imply possible modulation frequencies exceeding 0.1 THz. However, in the performed experiments, the repetition rate of the pump laser was much lower than this, such that the time-averaged pump fluence could be kept low. To fully exploit the dynamics offered by the semiconductor metasurfaces for ultrafast wavefront shaping at such high frequencies, much higher average pump fluences would be required, which would also pose a problem for the damage threshold of the structures.

Full spatiotemporal control of light fields is required to unlock control over light-matter interactions in its most general form. For example, spatiotemporal shaping is required to efficiently couple light with single quantum systems⁹¹ or control of spatial light distributions in combination with temporal focusing.^92,93 Both temporal^94–96 and spatial⁹⁷ degrees of freedom of the applied light field were shown to matter when it comes to the reliable preparation, manipulation, and readout of quantum states or light-induced chemical reactions. Full spatiotemporal control of light fields is furthermore needed to maximize the amount of information that can be contained in a single short laser pulse. We envision that in optical communications, it can pave the way to ultrahigh bit rates by providing a route for simultaneous spatial⁹⁸ and temporal⁹⁹ multiplexing. Furthermore, the spatiotemporal mode structure of a photon can also be used for encoding quantum state superpositions.¹⁰⁰ Thus, the possibility to custom-tailor light fields spatially as well as at an ultrafast time scale opens important new avenues in many areas of research and technology including quantum optics and optical communications.

Currently, even for rather simple cases, complicated and bulky optical setups, usually including spatial light modulators (SLMs), are required to control both spatial and temporal degrees of freedom of light fields. Moreover, the creation of light fields, whose temporal and spatial dependencies are inseparable, was rarely considered. Graphically speaking, if the separable case is represented by a static image whose brightness changes as a function of time, the inseparable case can be compared with a motion picture. The ability to shape both degrees of freedom in a desired way, including nontrivial inseparable cases, by a compact optical element opens fundamentally new possibilities for the mentioned fields and beyond.

However, while excellent control over the spatial properties (spatial distribution, including polarization) of the output light field was demonstrated using plasmonic and semiconductor metasurfaces, the full control of a light field also requires control over its temporal structure at a femtosecond time scale. While most wavefront shaping metasurfaces will actually exhibit intrinsic spatiotemporal couplings, these are rarely mentioned and remain rather unstudied and uncontrolled. An optically resonant metasurface can precisely control the phase and/or amplitude change acquired by light passing through (or being reflected from) it in a manner that depends both on the wavelength of light and on the position on the metasurface, thus shaping the wavefront of this beam in a spatial and spectral manner. As opposed to classical means of wavefront shaping, metasurfaces are designed through modification of their structure and local resonance properties. Their phase and/or amplitude change $exp{−α(x,y,ω)−iϕ(x,y,ω)}$ can be tailored and will modify the local spectrum according to $E out (x,y,ω)=exp{−α(x,y,ω)−iϕ(x,y,ω)} E in (x,y,ω)$⁠, where $E in / out (x,y,ω)$ are the local spectral amplitudes of the in-going and out-going laser pulse, i.e., the inverse Fourier transforms of their local temporal wave-forms $E in / out (x,y,t)$⁠.

The typical minimum bandwidth of the resonant features of dielectric metasurfaces in the order of a few tens of nanometers give a lower limit on the bandwidth of a pulse, which can be modified by such a structure in a significant way. This results in a maximum transform limited pulse duration of typically less than a picosecond. Metasurfaces are thus naturally suited to modify femtosecond laser pulses.

Metasurfaces leverage the degrees of freedom that are not accessible to conventional spectral pulse shapers.^101–104 Spectral pulse shapers use a spatial light modulator (SLM) in combination with two prisms or gratings to achieve a reprogrammable spectral resolution down to line-by-line pulse shaping.¹⁰⁵ Arrays of dielectric metasurfaces placed in a setup consisting of a pair of diffraction gratings and a pair of parabolic mirrors were also recently used to demonstrate pulse shaping of femtosecond pulses in the near-infrared spectral range by manipulating the phase and amplitude of the frequency components of the input pulse.¹⁰⁶ Various pulse-shaping operations, including splitting, compression, chirping, and higher-order distortion were performed. However, independent of the implementation (with an SLM or with a metasurface), pulse-shaping approaches that rely on spatially dispersing the frequency components of the input pulse in order to manipulate them selectively require bulky optical setups. Chirped mirrors,¹⁰⁷ in contrast, demonstrate that pulses can be reshaped directly, if a sufficiently dispersive system can be created by nanostructuring of the pulse’s propagation environment.

Metasurfaces utilize a nanostructured surface, composed of dispersive nanoresonators, to generate locally fixed pulse modifications. They can be created in a spatially nonuniform manner down to the scale of a single nanoresonator,⁷ giving access to truly planar polarization-spacetime pulse control. The principle feasibility of wavelength selective metasurfaces has been demonstrated for both plasmonic¹⁰⁸ and dielectric metasurfaces.¹⁰⁹ 

This, however, requires careful control over the resonance properties of the metasurfaces. For high wavelength selectivity, high Q-factors with bandwidths well below that of the input pulses are needed. These can, e.g., be tailored based on Fano resonances^5,9,52 or bound states in the continuum⁵¹ while dynamic effects have to be taken into account.

In the following, we discuss and develop some of the possibilities offered by dispersion-engineered dielectric metasurfaces.

For spatially homogeneous plasmonic metasurfaces and nanoantennas,^111,112 femtosecond pulse manipulation was theoretically analyzed. Particularly, Ref. 112 suggests that a plasmonic metasurface is capable of stretching and compressing of ultrashort $≈10–30 fs$ pulses at $λ=810 fs$⁠, as well as reshaping their polarization, with control over the characteristics by engineering the resonances of the plasmonic metasurface.

However, experimental demonstrations of the suggested effects are lacking and plasmonic metasurfaces are generally impractical for pulse-shaping applications. On the one hand, they usually exhibit low transmittance at resonance, making pulse shaping highly inefficient. On the other hand, plasmonic resonances are broadened by their intrinsic absorption, limiting the number of frequency channels which could be independently addressed without having to spatially disperse the beam.

Thus, resonant, highly transmissive semiconductor metasurfaces or stacks thereof may deliver the missing key to experimental research in this direction. In Ref. 110, the recompression of a chirped 120-fs laser pulse at $λ=800 nm$ using semiconductor Huygens metasurfaces was theoretically studied; see Figs. 5(a) and 5(b). Furthermore, it is worth mentioning that apart from the usual geometrical parameters of metasurfaces, like the size and shape of the meta-atoms and the pitch, positional disorder was recently also found to strongly influence the metasurface dispersion. In particular, for silicon Huygens’ metasurfaces, it was demonstrated that depending on the degree of positional disorder and the spectral detuning of the two electric and magnetic dipole resonances, the phase angle of the transmission coefficient exhibits a phase transition from normal to anomalous dispersion.¹¹³ 

One of the benefits of metasurfaces for pulse shaping is their ability to independently tailor the geometry and thus shape the pulse in spatial superpixels on a single surface. This would allow for a spatial multiplexing of pulse-shaping experiments.

If the spatially multiplexed metasurface was placed in the image plane of an optical system, then each superpixel’s response could be mapped on a section of an optical sample under investigation. This would, e.g., allow for single-shot pump-probe experiments, where each superpixel encodes a different delay.

Another application would be the characterization of femtosecond optical pulses. As these pulses cannot be measured directly using optical detectors, experimentalists typically resort to reconstruction of so-called parametrized nonlinear process spectra,^115,116 of which FROG,^117,118 iFROG,¹¹⁹ d-scan,^120,121 and MIIPS^122,123 are probably the most well-known. In these schemes a pulse is modified with a series of linear filter operations $H δ (ω)=exp{−α(ω;δ)−iϕ(ω;δ)}$⁠, which depend on a parameter $δ$⁠. The so-modified optical pulse is then converted using a nonlinear-optical element, e.g., a suitable broadband SHG-crystal, and the resulting nonlinear field is analyzed using a spectrometer. From the resulting series of $δ$-dependent nonlinear spectra, the complex laser pulse can be reconstructed. Experiments typically involve the scanning of an optical element and the successive recording of individual nonlinear spectra.

Spatially multiplexed nonlinear-optical surfaces could be used to do these experiments with a single laser pulse, where a the filter set $H δ (ω)$ is implemented in spatial superpixels of the metasurface. This single-element approach would allow for single-shot temporal reconstruction of optical pulses with a single all-planar optical element, composed of a spatially multiplexed metasurface, a thin $χ ( 2 )$-crystal, and an optical grating. Novel reconstruction algorithms¹¹⁶ further allow for the relaxation of design constraints on the metasurface.

The grand challenge of spatiotemporal metasurfaces would of course be the creation of an arbitrary spatiospectral filter. Such filters could be exploited to mix spatial, spectral, and polarization modes of light. They could be used to add or subtract spatiotemporal couplings, such as, e.g., spatial chirp, angular dispersion, pulse front tilt, and the ultrafast lighthouse effect.^124,125

Today, spatiotemporal engineering is rarely utilized, for a lack of a simple ways to create them. Some exceptions are attosecond science, where spatiotemporal coupling serves to generate isolated attosecond pulses,¹²⁶ or laser-materials-processing schemes based on spatiotemporal focusing.¹²⁷ All of these schemes are, however, based on the availability of discrete optical elements, which induce specific types of spatiotemporal couplings.

Fully spatiotemporal control may be achieved using spatially variant, optically resonant metasurfaces. Metasurfaces are in general dispersive structures.²⁹ Spatiotemporal metasurfaces could be utilized to exploit the interplay of particularly shaped incident pulses with dispersion-tailored wavefront shaping metasurfaces toward the generation of arbitrary ultrafast spatiotemporal light fields [see Fig. 6(a)]. Importantly, the spatial and spectral dependencies could be inseparable, i.e., $E(x,y,t)≠E(x,y)×E(t)$⁠, which should allow for generating ultrafast holographic “movie sequences.”

It was already demonstrated that multiresonant metasurfaces can be used to exert independent wavefront control over disjunct bands of the optical spectrum. The result is a class of metasurfaces, which have an entirely different functionality for various wavelengths. Such components were suggested, e.g., for multicolor holography,^128,129 (de)multiplexing and multiplexing of orbital angular momentum¹³⁰ or wavelength-division multiplexing systems, and entangled states of light,¹³¹ advanced aberration correction,^68,132 and optical components for (higher) harmonics, just to name a few. The appeal of semiconductor metamaterials for this application stems from the extremely unfavorable design-space scaling experienced by plasmonic metamaterials, which are much more broadband. If completely independent spectral control over $N$ wavelength bands is to be attained, and for each spectral band $M$ distinctly different responses are to be used, one would have to design $M N$ different classes of superpixels. This means that any conceivable design space would be exhausted quickly. To the best of our knowledge, no plasmonic system with more than $N=2$ independent channels and $M=2$ independent values of phase shifts has ever been demonstrated.¹⁰⁸ As semiconductor nanoresonators typically have much narrower bandwidths, one may break this scaling, by placing multiple, independent nanoresonators with distinct resonance wavelengths in each superpixel, addressing the response for each wavelength independently. This results in much more favorable $M⋅N$ design space scaling law.⁶ 

So far, spatiotemporal metasurfaces were theoretically suggested for decomposing the temporal spectrum of a broadband wave in two spatial dimensions, where each frequency is mapped onto a specific point in a plane behind the metasurface with application potential for real-time 2D spectrum analysis.¹¹⁴ However, it was not specified how these metasurfaces could be implemented. Metasurfaces exhibiting spacetime gradients are theoretically considered in Ref. 133. However, while theoretical research has recently started taking an interest in spatiotemporal metasurfaces, experimental studies in this direction are extremely sparse. As one of few examples,¹³⁴ realized fast beam steering based on frequency arrayed optics with metasurfaces. The prospect of creating arbitrary spatiotemporal light fields is of fundamental scientific interest by itself. However, this capability also offers interesting application perspectives. For example, the metasurface dispersion and input-beam pulse shape could be used to control the frequency content and temporal sequence with which light is focused in different ways or positions, which may be useful for stimulated emission depletion (STED) superresolution microscopy⁹⁷ or STED-inspired direct laser writing.¹³⁵ Usually, such schemes rely on superimposing the foci of various lasers. Metasurfaces could make such setups much simpler, robust, and cost-effective.

Furthermore, spatiospectral inseparability is an interesting prospect for quantum hyperentanglement schemes¹³⁶ and may help create such states of light. Spatiotemporally sculpted light fields may also be of interest for laser material processing, where recent research has already demonstrated that both pulse shaping^137,138 and beam shaping¹³⁹ can have a dramatic effect on the process outcome.

While a range of nonlinear, ultrafast, and dispersive effects were demonstrated in semiconductor metasurfaces during the last few years, much more remains to be explored regarding the interaction of semiconductor metasurfaces with short laser pulses. On the one hand, various nonlinear optical effects have not yet been fully exploited for frequency conversion and ultrafast switching in metasurfaces, such as optical rectification and the Franz-Keldysh effect, respectively, as well as interactions in the strong-field and single-cycle regimes. On the other hand, the vast majority of nonlinear and ultrafast metasurfaces realized today was spatially homogeneous. Combining the already observed strong nonlinear and ultrafast optical response with the established spatial control offered by spatially variant metasurfaces will allow the creation of sources of sculpted nonlinear and quantum light fields and for ultrafast wavefront control. Furthermore, semiconductor metasurfaces with a tailored, spatially variant dispersion offer unprecedented opportunities for synthesizing arbitrary spatiotemporal light fields, which could open a completely new chapter in the study of light-matter interactions.

I.S. gratefully acknowledges financial support by the Thuringian State Government within its ProExcellence initiative (⁠ $ACP 2020$⁠) and by the German Research Foundation (No. STA 1426/2-1). M.S. acknowledges support from the Russian Science Foundation (Grant No. 18-12-00475). F.E. gratefully acknowledges support by the Germany Federal Ministry for Education and Science BMBF (Grant-ID: 13XP5053A).