Synthesizing ultrafast optical pulses with arbitrary spatiotemporal control

Abstract

The ability to control the instantaneous state of light, from high-energy pulses down to the single-photon level, is an indispensable requirement in photonics. This has, for example, facilitated spatiotemporal probing and coherent control of ultrafast light-matter interactions, and enabled capabilities such as generation of exotic states of light with complexity, or at wavelengths, that are not easily accessible. Here, by leveraging the multifunctional control of light at the nanoscale offered by metasurfaces embedded in a Fourier transform setup, we present a versatile approach to synthesize ultrafast optical transients with arbitrary control over its complete spatiotemporal evolution. Our approach, supporting an ultrawide bandwidth with simultaneously high spectral and spatial resolution, enables ready synthesis of complex states of structured space-time wave packets. We expect our results to offer unique capabilities in coherent ultrafast light-matter interactions and facilitate applications in microscopy, communications, and nonlinear optics.

Arbitrary space-time wave packets over an ultrawide bandwidth are synthesized using metasurface optics.

INTRODUCTION

From revealing molecular dynamics in the attosecond time scale to generation of high-energy pulses in the petawatt regime, ultrafast optics continues to have far-reaching impacts across a wide range of scientific disciplines (1–5). In the time domain, the high peak intensity associated with an ultrafast pulse has enabled numerous advances in nonlinear optics (6) and high-field physics (7), whereas in the frequency domain, the corresponding equally spaced spectral lines, referred to as frequency combs, have become indispensable for precision metrology and spectroscopy (8–10). Generation of ultrafast optical pulses and their use in these various application areas rely in some form on coherently controlling their temporal evolution using techniques commonly referred to as ultrafast optical pulse shaping (11, 12). For example, functions such as pulse compression or optical arbitrary waveform generation require temporal control achieved via manipulating the spectral phase and amplitude (13, 14), whereas realization of space-time wave packets requires additional control over the spatial degree of freedom (15). Here, using metasurface optics, we present a universal approach toward arbitrary controlling the full space-time evolution of ultrafast optical pulses. We demonstrate the versatility of this approach by generating spatiotemporal waveforms carrying, simultaneously, a rich set of instantaneous polarization states and time-multiplexed spatial wavefronts with different topological charges. This is made possible by the use of metasurfaces that offer multidimensional control of light at the nanoscale (16), enabling simultaneous and independent control of the phase, amplitude, polarization, and spatial wavefront of spectrally dispersed ultrafast pulses. The nanoscale control further breaks the constraint in both spectral and spatial resolution placed by the large pixel size of traditional pulse modulation techniques, enabling synthesis of arbitrary space-time wave packets over an ultrawide bandwidth. We expect our results to offer unique capabilities in probing and manipulation of ultrafast molecular (17, 18) or topological (19, 20) excitations or in generation of complex states of extreme-ultraviolet (EUV) (21, 22) or x-ray light (23, 24).

RESULTS

We describe our approach of synthesizing arbitrary spatiotemporal pulses by treating an ultrafast pulse train as a frequency comb, where the pulse train spectrum is composed of a series of equally spaced spectral lines at frequencies given by ν_j = jν_rep + ν₀, where j is an integer, ν_rep is the repetition rate of the laser, and ν₀ is the carrier envelope offset frequency (9). Assuming an input beam of Gaussian spatial and spectral distribution, the transform-limited spatiotemporal electric field ${\overrightarrow{\mathit{E}}}_{\text{in}}$ (here, the arrow accent denotes the vectorial nature of the field whereas the bold font denotes that the field is complex) of a p-polarized (E-field parallel to the x axis) input pulse train can be expressed by its Fourier series

$${\overrightarrow{\mathit{E}}}_{\text{in}}(x,y,t)={\displaystyle \sum _j}{e}^{-\frac{x^2}{{\mathrm{\sigma }}^2}}{e}^{-\frac{y^2}{{\mathrm{\sigma }}^2}}·a_j{e}^{-i{\mathrm{\omega }}_{j}t}\left[\begin{array}{c}1\\ 0\end{array}\right]$$ (1)

where σ represents the input beam waist, a_j is the spectral amplitude, and ω_j = 2πν_j. At the input of a Fourier transform (FT) pulse shaper (Fig. 1A), the various spectral lines constituting the ultrafast pulse train are first angularly dispersed by a blazed grating and focused at the Fourier plane (ξ, η) using an off-axis parabolic mirror, transforming the input electric field into ${\overrightarrow{\mathit{E}}}_{-}(\mathrm{\xi },\mathrm{\eta },t)\propto \mathcal{F}\left\{\mathrm{G}\right\{{\overrightarrow{\mathit{E}}}_{\text{in}}(x,y,t)\left\}\right\}=\sum _j{u}_j(\mathrm{\xi }-{\mathrm{\xi }}_j)u_j(\mathrm{\eta })a_j{e}^{-i{\mathrm{\omega }}_{j}t}\left[\begin{array}{c}1\\ 0\end{array}\right]$, where ${\overrightarrow{\mathit{E}}}_{-}$ is the electric field at the input of the metasurface at the Fourier plane with the individual spectral lines ω_j dispersed along the ξ axis, ℱ is the FT, 𝒢 represents the grating function, and u_j(ξ) is the one-dimensional (1D) spatial FT of the input field at ω_j (see text S1). A transmission-mode dielectric metasurface placed at the Fourier plane is designed to impart a complex masking Jones matrix M(ξ, η) to the spectrally dispersed electric-field ${\overrightarrow{\mathit{E}}}_{-}$, resulting in an electric field ${\overrightarrow{\mathit{E}}}_{+}(\mathrm{\xi },\mathrm{\eta },t)=\left[\begin{array}{c}{\mathit{M}}_p{\mathit{E}}_{-}(\mathrm{\xi },\mathrm{\eta },t)\\ {\mathit{M}}_s{\mathit{E}}_{-}(\mathrm{\xi },\mathrm{\eta },t)\end{array}\right]$ at the exit plane of the metasurface. Here, M_p and M_s are the corresponding M(ξ, η) elements that respectively determine the p- and s-polarized components of ${\overrightarrow{\mathit{E}}}_{+}$ for an input p-polarized electric-field ${\overrightarrow{\mathit{E}}}_{-}$. This combination of Fourier decomposition coupled with the multidimensional control at the nanoscale offered by metasurfaces enables us to parallelly and independently manipulate the full spatial and temporal properties of each spectral component.

Fig. 1. Synthesis of ultrafast optical pulses with arbitrary spatiotemporal control enabled by a dielectric metasurface.

(A) Schematic of the spatiotemporal Fourier transform pulse synthesizer, with a single-layer transmission-mode metasurface consisting of Q contiguous superpixels, labeled as S_J (J = 1, 2, … Q), placed at the Fourier plane. Spatially dispersed frequency components of an incoming linearly polarized optical pulse with Gaussian spatial distribution, upon interaction with the corresponding S_J, are simultaneously transformed into targeted waveforms with the requisite amplitude, phase, polarization, and wavefront distribution, and coherently recombined. Bottom left inset: Schematic perspective view of a metasurface unit cell (pitch size p) consisting of a dielectric nanopillar of height H and in-plane dimensions L₁ and L₂, placed at a rotation angle θ. Top right inset: One representative metasurface unit cell (cyan outline in S_J) acting as a quarter–wave plate. (B) Simultaneous and independent control of the instantaneous phase, amplitude, polarization, and wavefront as a function of time enabled by the pulse synthesizer.

To achieve this, we spatially divide the metasurface into Q superpixels S_J (J = 1, 2, … Q) linearly concatenated along the ξ axis, where each superpixel S_J acts as a submetasurface of its own offering a polarization-multiplexed spectral modulation ${\mathbf{{\rm Y}}}_J^{p(s)}({\mathrm{\omega }}_J)$, and spatial modulation ${\mathbf{\Gamma }}_J^{p(s)}(\mathrm{\xi }-{\mathrm{\xi }}_J,\mathrm{\eta })$, targeted to tailor the complex transmission of a small spectral subrange ∆λ_J, centered at λ_J. The complex masking function M_p(s) of the entire metasurface can, therefore, be expressed in terms of an assembly of complex masking functions of individual superpixels as

$${\mathit{M}}_{p(s)}(\mathrm{\xi },\mathrm{\eta })=\sum _{J=1}^Q {\mathbf{{\rm Y}}}_J^{p(s)}({\mathrm{\omega }}_J){\mathbf{\Gamma }}_J^{p(s)}(\mathrm{\xi }-{\mathrm{\xi }}_J,\mathrm{\eta })\mathrm{\Pi }(\mathrm{\xi }-{\mathrm{\xi }}_J)$$ (2)

where Π(ξ − ξ_J) represents a rectangular function that defines the boundary of superpixel S_J. Each superpixel is further divided into a 2D array of square pixels of pitch p_J, each consisting of a subwavelength-size dielectric nanopillar of rectangular cross section acting as a birefringent wave plate (Fig. 1A, insets). To achieve the multifunctionality required at the nanoscale, a library of birefringent nanopillars at each ω_J is generated to simultaneously provide Φ₁(ξ, η) and Φ₂(ξ, η) covering the full [0, 2π] range, where Φ₁ and Φ₂ are the respective phase shifts along the two birefringent axes of the nanopillar. Therefore, via a judicious choice of nanopillars constituting each superpixel with spatially varying in-plane dimensions (L₁ and L₂) and rotation angles (θ), independent control of spatial wavefront, phase, amplitude, and polarization for each spectral subrange ∆λ_J can be achieved. Last, the modulated spectral components at the output of the metasurface are recombined using a second pair of parabolic mirror and grating, resulting in a synthesized output pulse train with an electric field ${\overrightarrow{\mathit{E}}}_{\text{out}}$ given by

$${\overrightarrow{\mathit{E}}}_{\text{out}}(x,y,t)=\left[\begin{array}{c}{\displaystyle \sum _{J=1}^Q} {\mathit{E}}_{\text{out},J}^p(t){\mathit{E}}_{\text{out},J}^p(x,y)\\ {\displaystyle \sum _{J=1}^Q} {\mathit{E}}_{\text{out},J}^s(t){\mathit{E}}_{\text{out},J}^s(x,y)\end{array}\right]$$ (3)

where ${\mathit{E}}_{\text{out},J}^{p(s)}(t)=a_J{\mathbf{{\rm Y}}}_J^{p(s)}({\mathrm{\omega }}_J)e^{-i{\mathrm{\omega }}_{J}t}$ and ${\mathit{E}}_{\text{out},J}^{p(s)}(x,y)={\mathrm{ℱ}}^{-1}\{{\mathbf{\Gamma }}_J^{p(s)}(\mathrm{\xi },\mathrm{\eta })\mathrm{\Pi }(\mathrm{\xi })u_J(\mathrm{\eta })\}$ represent the independent spectral and spatial shaping of the p- (s-) polarized spectral components centered at ω_J, respectively (see text S1). The novelty of the approach outlined above stems from the fact that it provides parallel control over the temporal state of light via engineering the metasurface at the superpixel level while enabling simultaneous control over its spatial wavefront evolution via engineering individual nanopillars within each superpixel (Fig. 1B).

Experimentally, pulse synthesis is achieved by placing the metasurface at the Fourier plane (ξ, η) of a custom-built FT pulse shaper (see Materials and Methods). The metasurface is designed to address spectrally dispersed linearly polarized input pulses from a Ti:sapphire oscillator (pulse duration <10 fs, corresponding to a full-width at 10th-maximum bandwidth of ≈80 THz at a central wavelength of λ_c = 800 nm). We divide the metasurface into Q = 201 superpixels, each of size 115 μm by 200 μm, to match the optimized spectral resolution ∆λ_J = 1 nm of the implemented pulse synthesizer. To demonstrate our ability of simultaneous spatiotemporal control, as an illustrative example, we first design a metasurface optic to transform an input p-polarized, Gaussian wavefront ultrafast pulse to a shaped pulse exhibiting, simultaneously, a time-varying polarization state, and a vortex spatial wavefront carrying an orbital angular momentum (OAM) order of 𝓁 = −1 (Fig. 2, A and B). The control of instantaneous polarization requires each superpixel S_J to simultaneously endow complex spectral masking functions ${\mathbf{{\rm Y}}}_J^p({\mathrm{\omega }}_J)=\text{cos}(b_{\mathrm{I}}({\mathrm{\omega }}_J-{\mathrm{\omega }}_{\mathrm{c}}))e^{i\frac{b_{\text{II}}}{2}{({\mathrm{\omega }}_J-{\mathrm{\omega }}_{\mathrm{c}})}^2}$ and ${\mathbf{{\rm Y}}}_J^s({\mathrm{\omega }}_J)=i\text{sin}(b_{\mathrm{I}}({\mathrm{\omega }}_J-{\mathrm{\omega }}_{\mathrm{c}}))e^{i\frac{b_{\text{II}}}{2}{({\mathrm{\omega }}_J-{\mathrm{\omega }}_{\mathrm{c}})}^2}$ to the spectrally dispersed electric-field components ${\overrightarrow{\mathit{E}}}_{-,J}$, where b_I and b_II are the group delay and group delay dispersion, respectively, and ω_c is the angular frequency at λ_c. Concurrently, the spatial wavefront control is achieved by imparting a spiral-phase transformation ${\mathbf{\Gamma }}_J^{p(s)}(\mathrm{\xi }-{\mathrm{\xi }}_J,\mathrm{\eta })=e^{i{\mathrm{\ell }}_J{\text{tan}}^{-1}\frac{\mathrm{\eta }}{\mathrm{\xi }-{\mathrm{\xi }}_J}}$ to each superpixel S_J, where 𝓁_J = −1. These simultaneous requirements define the targeted masking functions M_p(ξ, η) and M_s(ξ, η), which can be satisfied by leveraging the library of birefringent nanopillars and placing each nanopillar at θ = π/4. The implemented phase topography is given by Φ₁₍₂₎(ω_J, ξ, η)= ${\mathrm{\phi }}_{1(2)}({\mathrm{\omega }}_J)-{\text{tan}}^{-1}\frac{\mathrm{\eta }}{\mathrm{\xi }-{\mathrm{\xi }}_J}$ for each superpixel S_J (see text S2), where ${\mathrm{\phi }}_{1(2)}({\mathrm{\omega }}_J)=\frac{b_{\text{II}}}{2}{({\mathrm{\omega }}_J-{\mathrm{\omega }}_{\mathrm{c}})}^2\pm b_{\mathrm{I}}({\mathrm{\omega }}_J-{\mathrm{\omega }}_{\mathrm{c}})$ represents the corresponding spectral phase, b_II governs the temporal length of the synthesized pulse, and b_I/b_II determines the rotation speed of the major axis of instantaneous polarization (see text S3). For the experiment, by setting b_II = 100 fs²/rad and b_I/b_II = 0.1 rad/fs, the corresponding φ₁₍₂₎ versus λ, and Φ₁₍₂₎ for eight representative superpixels (J = 34 to 41, corresponding to λ_J = 733 to 740 nm) are plotted in Fig. 2 (C and D, respectively). Each nanopillar in-plane dimension L₁₍₂₎(ω_J, ξ, η) that simultaneously satisfies the local phase requirements for each ω_J and offers the highest transmission is determined via minimization of a figure-of-merit (FOM) function (see Materials and Methods). One representative plot illustrating the dependence of FOM versus (L₁, L₂) to simultaneously endow Φ₁ = 0.90π and Φ₂ = 1.54π at one nanopillar location within a superpixel at λ_J = 734 nm (white cross, Fig. 2D) is shown in Fig. 2E (FOM minima denoted by the white circle). The FOM minimization procedure, repeated at each λ_J, yields a library of L₁₍₂₎ values that simultaneously satisfies any requisite Φ₁₍₂₎ combination covering the full [0, 2π] range (see fig. S4 for one representative library at λ_J = 734 nm). The requisite nanopillars constituting the metasurface are patterned on a layer of 650-nm-thick amorphous-Si (a-Si) on fused silica substrate using electron beam lithography (EBL) and reactive ion etching (see Materials and Methods). An optical transmission image of the fabricated metasurface, of dimensions 2.3 cm × 200 μm, taken under unpolarized white light illumination along with a higher-magnification image of eight superpixels (corresponding to J = 34 to 41; Fig. 2D) and a representative scanning electron microscope (SEM) image are shown in Fig. 2 (F and G, respectively).

Fig. 2. Metasurface design to synthesize an OAM-carrying, polarization-swept ultrafast pulse.

(A) Numerically simulated 3D electric-field representation of the targeted pulse assuming the carrier envelope phase to be zero. Gray solid line overlaid on the surface of the pulse tracks the tip of the instantaneous electric-field vector versus t. Orange and blue projections represent the p- and s-polarized electric-field components, respectively. Insets: simulated output intensity distribution of the synthesized pulse carrying an OAM order of 𝓁 = −1 for the p- and s-polarized components. Scale bars, 5 mm. (B) Poincaré sphere representation of the time-varying polarization state in (A). Q(t), U(t), and V(t) correspond to the Stokes parameters and are defined in text S6. Red, green, and blue stars mark three representative instantaneous polarization states at −27 fs, 0 fs, and +27 fs, respectively. (C) Targeted spectral phase φ₁₍₂₎ versus λ. (D) Targeted phase topographies Φ₁ and Φ₂ for eight representative superpixels within the pink-shaded spectral region in (C). Scale bar, 100 μm. (E) Figure of merit (FOM) versus nanopillar in-plane dimensions (L₁, L₂), plotted in logarithmic scale for λ_J = 734 nm. The local FOM minima (white circle) yields a combination of L₁ and L₂ (white crosses) that simultaneously satisfies the targeted phases Φ₁ = 0.90π and Φ₂ = 1.54π for one nanopillar location within the superpixel at λ_J = 734 nm (white crosses in D), while offering the highest transmittance along the two nanopillar axes. (F) Optical transmission image of the fabricated metasurface under unpolarized white light illumination. Inset: magnified image of eight superpixels corresponding to the phase maps in (D). Scale bar, 100 μm. (G) Representative SEM image of the nanopillars (perspective view at 52°) taken at position marked by the cyan box in the magnified image in (F). Scale bar, 500 nm.

The complete spatiotemporal evolution of the synthesized OAM-carrying, polarization-swept pulse, ${\overrightarrow{\mathit{E}}}_{\text{out}}(x,y,t)$, at the output of the pulse shaper is characterized by using a combination of spectral and spatial domain methods (Fig. 3). Measuring the temporal polarization evolution of the pulse requires measurement of the spectral intensity (I_p(s)(ω_J)) and phase (φ_p(s)(ω_J)) for the two orthogonal p- and s-polarized components. I_p(s)(ω_J) is measured by directing the p- (s-) polarized component of ${\overrightarrow{\mathit{E}}}_{\text{out}}$ to a grating spectrometer, whereas φ_p(s)(ω_J) is measured using a custom-built spectral interferometry (SI) setup. SI is performed by interfering the synthesized pulse ${\overrightarrow{\mathit{E}}}_{\text{out}}$ with a transform-limited reference pulse ${\overrightarrow{\mathit{E}}}_{\text{in}}$ that was partially split at the input of the pulse shaper, at a fixed time delay τ = τ₀ (see Materials and Methods). The temporally evolving polarization state of the experimentally measured pulse is characterized by measuring the temporal amplitude A_p(s)(τ) of the p- (s-) polarized component and the phase difference Ψ_p(τ) − Ψ_s(τ), which closely matches the design (Fig. 4A). The Poincaré sphere representation of the measured temporal polarization state (Fig. 4B) also closely resembles the design (Fig. 2B). The time-averaged spatial intensity distribution of the two polarization components, ${\overline{I}}_{p(s)}(x,y)$, is measured by imaging the p- (s-) polarized component of ${\overrightarrow{\mathit{E}}}_{\text{out}}$ on a complementary metal-oxide semiconductor (CMOS) sensor (insets, Fig. 4A), whereas the vortex spatial phase distribution, Ψ_p(s)(x, y, t), for the two polarization components is confirmed by spatially interfering ${\overrightarrow{\mathit{E}}}_{\text{out}}$ with ${\overrightarrow{\mathit{E}}}_{\text{in}}$ at different time delay τ. Three representative spatial interference images, each for p- and s-polarized components, taken at τ = −27 fs, 0 fs, and +27 fs (red, green, and blue stars, respectively; Fig. 4B), clearly show a single fork dislocation, confirming that, as designed, the entire synthesized pulse is carrying 𝓁_J = −1. The fine structure in experimentally measured ${\overline{I}}_{p(s)}(x,y)$ stems from the astigmatism-limited spot size and finite spectral resolution of the implemented FT pulse shaper (∆λ_J = 1 nm), nominally corresponding to ≈6250 spectral lines each illuminating a superpixel at a slightly shifted spatial location ξ_j. The effect of spatially dispersed illumination within each superpixel S_J, including overlap at boundaries with adjacent superpixels S_J±1, is numerically modeled by calculating E_+,J(ξ, η, t) accounting for all spectral lines illuminating each S_J, and propagating it through the pulse shaper yielding a far-field response E_{out, J}(x, y, t) (see text S4). The simulated far-field intensity resulting from one representative superpixel (S₁₀₁), ${\overline{I}}_{\text{out},101}(x,y)$, accounting for all 6250 spectral lines illuminating it, shows a perturbed doughnut-shape resulting from the nonuniform superpixel illumination (Fig. 4C, left). The corresponding calculated far-field phase distribution, Ψ_out,101(x, y, t = 0), shows a perturbed 2π phase winding, confirming the synthesis of a vortex wavefront carrying 𝓁_J = −1 (Fig. 4C, right). The numerically calculated ${\overline{I}}_{\text{out}}(x,y)=\sum _J{\overline{I}}_{\text{out},J}(x,y)$, representing the collective far-field response from the entire metasurface and accounting for all 201 superpixels (Fig. 4D), closely resembles the experimentally measured intensity distribution (Fig. 4A, insets). The full analytical formulation and numerical results describing the synthesis of spatiotemporal wave packets are presented in texts S1 and S4.

Fig. 3. Schematic diagram of the experimental setup.

The experimental beam (labeled Exp.) propagates through the pulse shaper and acquires a targeted spatiotemporal modulation. The reference beam (labeled Ref.) is delayed by τ with respect to Exp. beam to perform spatial and spectral interferometry (SI; gray and pink inset, respectively) on the shaped pulses and is also independently characterized by the spectral phase interferometry for direct electric-field reconstruction (SPIDER) method (green inset). BS, beam splitter; M, mirror; G1 and G2, grating; OPM, off-axis parabolic mirror; HWP, half–wave plate; QWP, quarter–wave plate; LP, linear polarizer; NC, nonlinear crystal; Spec., spectrometer. Pink inset: polarization resolved temporal pulse reconstruction using SI. Gray inset: polarization resolved time-varying spatial wavefront reconstruction using spatial interferometry. Green inset: SPIDER technique based on spectral shearing interferometry.

Fig. 4. Experimental implementation of an OAM-carrying, polarization-swept ultrafast pulse.

(A) Temporal amplitudes A_p(τ) (top), A_s(τ) (middle), and phase difference Ψ_p(τ) − Ψ_s(τ) (bottom) of the designed (gray dashed lines) and measured (solid colored lines) pulse illustrating the rich set of time-varying polarization states endowed to the pulse. The surrounding shadows represent 1 SD from 1000 measurements. Colored dashed lines in the bottom plot denote 0 (green) and π (orange), respectively. Inset camera images show the measured ${\overline{I}}_{\text{out}}^{p(s)}$. (B) The measured time-varying polarization state of the shaped pulse represented as a trace on the Poincaré sphere. Insets: representative spatial interferometry images for the p- and s-polarization, all exhibiting an OAM order of 𝓁 = −1, measured at τ = −27 fs, 0 fs, and +27 fs. (C) Simulated ${\overline{I}}_{\text{out},101}^p$ (left) and ${\mathrm{\Psi }}_{\text{out},101}^p$ (right) accounting for all 6250 spectral lines illuminating one representative superpixel (S₁₀₁). (D) Simulated collective output intensity distribution of the synthesized pulse ${\overrightarrow{\mathit{E}}}_{\text{out}}(x,y,t)$ carrying an OAM order of 𝓁 = −1, closely matching the measured intensity distribution in (A). Scale bars in (C) and (D), 5 mm.

The high spatial and spectral resolution, full-design freedom, and ready extension to other spectral ranges either linearly or nonlinearly make our approach an attractive platform for simple realization of other form of space-time wave packets over an ultrawide bandwidth. For example, these capabilities immediately lend themselves for generation of complex light states such as those exhibiting dynamic rotation and revolution (25) or having time-varying OAM (22), which either have only been proposed theoretically or are limited to the EUV regime via high-harmonic generation. To illustrate this, we synthesize two analogs of space-time wave packets: a light coil representing a transform-limited pulse with helical intensity distribution (26) and another exhibiting coherently multiplexed time-varying OAM orders. These requisite functionalities are achieved by imparting a prescribed 𝓁_J to a preassigned set of ω_J constituting the pulse, while simultaneously controlling their spectral phase φ₁₍₂₎(ω_J). We experimentally demonstrate this by fabricating a metasurface able to encode five time-multiplexed topological charges to an ultrafast pulse, and controlling its group delay dispersion b_II, implemented via superpixel masking function, ${\mathit{M}}_J=e^{i\frac{b_{\text{II}}}{2}{({\mathrm{\omega }}_J-{\mathrm{\omega }}_{\mathrm{c}})}^2}{e}^{i{\mathrm{\ell }}_J{\text{tan}}^{-1}\frac{\mathrm{\eta }}{\mathrm{\xi }-{\mathrm{\xi }}_J}}\mathrm{\Pi }(\mathrm{\xi }-{\mathrm{\xi }}_J)$, where 𝓁_J = +2, +1, 0, −1, or −2 is assigned to each S_J within five prescribed spectral subregions (Fig. 5A). An optical transmission image of the fabricated metasurface along with a higher-magnification image of five representative superpixels (J = 26, 56, 101, 146, and 171), their intensity transmission I_+,J measured using a tunable continuous-wave laser at the corresponding λ_J, and the spatial interferogram $I_J^{\text{intf}}$ measured via interference of E_+,J with a reference Gaussian beam of the same color confirm that, as designed, each S_J is able to impart the desired 𝓁_J to each ω_J (Fig. 5A). The corresponding mode purity of the OAM beams is calculated to be 0.68, 0.80, 0.75, 0.69, and 0.58, respectively, using the method outlined in (27). When transform-limited (b_II = 0 fs²/rad), coherent interference of the various 𝓁_J orders at each time-instance t results in a radially displaced, spatiotemporally localized wave packet that revolves azimuthally around a central propagation axis (Fig. 5B). Subsequently, introduction of a chirp (b_II = 180 fs²/rad) to the masking function M_J, results in temporal separation of the various ω_J, or equivalently 𝓁_J, where the pulse now exhibits a time-varying vortex spatial phase topography. In each case, the spatiotemporal evolution of the generated pulse is determined from the measured interferograms I_intf(x, y, τ), resulting from the interference of E_out(x, y, t) with a reference pulse E_in at different time delay τ. The reconstructed spatiotemporal wave packet I_meas(x, y, τ) of the synthesized light coil, exhibiting a localized intensity revolving around a central propagation axis, closely matches the numerically simulated intensity distribution, I_sim(x, y, t) (Fig. 5B). For the synthesized time-varying OAM pulse, three-representative I_intf(x, y) snapshots clearly show three distinct OAM states at different τ, closely matching the numerically simulated spatiotemporal phase evolution, Ψ_sim(x, y, t) (Fig. 5C). The retrieved mode purity for the simulated pulse, along with those of the three experimental snapshots in Fig. 5C, is shown in fig. S11. The complete spatiotemporal evolution of I_sim(x, y, t) and Ψ_sim(x, y, t), along with the corresponding I_meas(x, y) and I_intf(x, y) versus τ for the two synthesized pulses, is analytically modeled and experimentally captured (see text S5 and movies S1 and S2), illustrating the rich diversity and control over the spatiotemporal encoding of a single ultrafast pulse.

Fig. 5. Synthesis of an ultrafast light coil and a spatiotemporal pulse carrying time-varying OAM.

(A) Five OAM orders 𝓁_J = +2, +1, 0, −1, and −2 are assigned to each superpixel within a set of five spectral subregions. Top: schematic spectral intensity distribution at frequencies ω_J of the input femtosecond pulse. Middle: optical transmission image of the fabricated metasurface designed to impart the targeted OAM orders to each Δλ_J. Bottom: magnified image (left) of five representative superpixels S_J (J = 26, 56, 101, 146, and 171), along with the corresponding transmitted intensity distribution I_J (middle) and spatial interferogram $I_J^{\text{intf}}$ (right) measured at corresponding λ_J. (B) Simulated (left) and experimentally reconstructed (right) spatiotemporal wave packet for the synthesized light coil. The plots show iso-intensity profile at 40 and 50% of the peak intensity, respectively. Left (right) insets: instantaneous simulated (reconstructed) intensity distribution I_sim (I_meas) at three representative time delays τ = −25 fs, 0 fs, and +25 fs. (C) Simulated spatiotemporal wave packet carrying time-varying OAM (left). The plot shows iso-intensity profile at 10% of the peak intensity. Simulated phase profile Ψ_sim (middle) and measured spatial interferometry I_intf (right) at τ = −26 fs, 10 fs, and +46 fs, respectively. White arrows indicate fork dislocations in the interference fringes with opposite openings.

DISCUSSION

In summary, by leveraging the multidimensional control of light at the nanoscale offered by metasurfaces, we have experimentally demonstrated a versatile photonic platform able to control the complete 4D spatiotemporal evolution of ultrafast pulses over an ultrawide bandwidth. Beyond the exquisite time-varying polarization and temporally encoded spatial-wavefront evolution demonstrated here, our approach can be simply extended to other forms of structured light (28) or space-time wave packets (29, 30) via superpixel engineering, or wavelength regimes via nonlinear control (31). While the functionalities achieved in this work are static, we expect ongoing research in the field of active metasurfaces to provide, in time, capabilities required for reconfigurable control (32).

MATERIALS AND METHODS

Spatiotemporal pulse shaping setup

Input femtosecond pulses, of duration ≈ 10 fs and time-averaged power ≈ 400 mW, centered at 800 nm (full-width at 10th-maximum bandwidth of ≈ 80 THz), are generated at a repetition rate of 75 MHz from a Ti:sapphire oscillator. An ultrafast beam splitter is used to transmit 75% of the input power into the pulse shaper and reflect the remaining 25% into the reference beam (Fig. 3). At the input of the pulse shaper, the beam has a Gaussian spatial profile of diameter ≈1 cm (1/e² of the maximum intensity). The frequency components constituting the pulse are first angularly dispersed by a blazed grating (300 grooves/mm) into the first diffraction order, and then focused by an off-axis metallic parabolic mirror (reflected focal length f = 381 mm and an off-axis angle of 15°). The diffraction efficiency of the gratings differs by ≲10% between the p- and s-polarization over the spectral range from 700 to 900 nm. The grating-mirror pair is used to spatially disperse the input beam over the full length of the metasurface along the ξ-direction, following a quasi-linear function λ(ξ) (fig. S1). Using ray tracing simulations, an effective monochromatic spot size at the center wavelength of λ_c= 800 nm is estimated to be ≈34 μm, corresponding to a theoretical upper limit on the wavelength resolution of ≈0.3 nm or a spectral resolution of ∆ω_j = 140 GHz for the implemented FT pulse shaper (33). In the experiments, metasurfaces consisting of Q = 201 superpixels, each of size 115 μm by 200 μm, correspond to an implemented ∆ω_J ≈ 467 GHz at λ_c, wherein the spectral resolution is sacrificed to minimize boundary effects between adjacent superpixels (see text S4). Along the η direction, the largest astigmatism-limited spot at the edge of the spectrum is ≈0.2 mm.

The alignment of the metasurface at Fourier plane involves translation along ξ, η, and z directions; roll within the (ξ, η) plane; pitch within the (η, z) plane; and yaw within the (ξ, z) plane. A linear translation stage is used to control the z-direction translation, and a six-axis kinematic mount is further used to control the ξ and η translation, the pitch and yaw adjustment, and roll rotation. Nine reference half–wave plate metasurface superpixels are fabricated parallel to the pulse-shaping metasurfaces to facilitate alignment along the ξ direction. By monitoring the spectrum of the transmitted light through the reference metasurface superpixels (fig. S1), rigorous alignment of the ξ-position is achieved. Yaw and pitch are aligned using the back reflection from the metasurface. Note that the metasurface is intentionally placed at a small pitch angle (≈2°) to minimize any back reflection into the laser system. The remaining η and roll alignments are performed by sequentially allowing either the long or the short end of the spectrum to transmit while blocking the remaining frequencies at the Fourier plane and monitoring the intensity in the far field.

Assuming the input pulse follows a Gaussian intensity distribution with peak power at λ_c = 800 nm and accounting for diffraction loss from the grating, the peak intensity incident on the nanopillars ≤300 kW/cm², where we do not expect any substantial nonlinear effects inside the a-Si nanopillars (34). After passing through the metasurface, the beam is recombined using a second parabolic mirror and grating pair, yielding a vectorially shaped pulse of desired spatiotemporal characteristics. The complete time-varying properties of the recombined pulse exiting the system are measured using a combination of spatial and spectral domain techniques described below.

Metasurface simulation and design

The transmission amplitudes t₁, t₂ and phase shifts ψ₁, ψ₂ along the two birefringent axes of a rectangular nanopillar are simulated for wavelengths λ_J = (699 + J) nm, where J = 1,2, …,201 using rigorous coupled wave analysis (35), for all combinations of lateral nanopillar sizes L₁, L₂ ∈ [0.15p, 0.86p], where p(λ)= (0.4λ − 60 nm) is the pitch of the unit cell along both the ξ and η directions. The wavelength-dependent complex refractive indices for a-Si used in the simulations is measured using spectroscopic ellipsometry (fig. S2). We also performed Raman spectroscopy on the electron beam–exposed a-Si to ensure no morphological change occurred during exposure (fig. S3). For a given position (ξ, η) in the Fourier plane, the desired spatiotemporal shaping function M(ξ, η) (see text S1) is projected onto the two birefringent axes of the nanopillar to respectively impart the targeted complex transmission e^iΦ₁ and e^iΦ₂. Furthermore, we define the FOM as

$$\text{FOM}(L_1,L_2)={\mid e^{i{\mathrm{\Phi }}_1}-t_1{e}^{i{\mathrm{\psi }}_1}\mid }^2+{\mid e^{i{\mathrm{\Phi }}_2}-t_2{e}^{i{\mathrm{\psi }}_2}\mid }^2$$ (4)

The nanopillar dimensions L₁(ξ, η), L₂(ξ, η) for a given height H that simultaneously satisfy the criteria of required phase shifts and offer the highest transmission are identified by the local minimum in the FOM map (one representative example is shown in Fig. 2E). The FOM minimization process yields a library of L₁, L₂ values that provide the full [0, 2π] phase coverage while maintaining high transmission for each ω_J (one representative library at λ_J=35 = 734 nm is shown in fig. S4). Using the libraries, any requisite Φ₁₍₂₎ can be satisfied. For example, the simulated Φ₁₍₂₎ and $t_{1(2)}^2$ maps to achieve the design of the eight representative superpixels (Fig. 2D) are shown in fig. S5 (A to D), and the average transmission ${\overline{t}}_{1(2)}^2$ for each superpixel S_J of the entire metasurface is shown in fig. S5E.

Metasurface fabrication

A layer of 650-nm-thick a-Si is first deposited on a 500-μm-thick fused silica substrate using plasma-enhanced chemical vapor deposition. Here, an a-Si thickness of 650 nm is chosen to concurrently satisfy the requirements on achieving the desired phase depth at longer wavelengths and minimizing optical losses at shorter wavelengths. The substrate is then coated with a 300-nm-thick layer of electron beam resist and a 15-nm-thick aluminum (Al) anti-charging layer for EBL. The nanopillar designs and alignment marker patterns are exposed using a 100-keV EBL system, followed by Al-layer removal with tetramethylammonium hydroxide, and resist development at 4°C. Next, a 50-nm-thick Al₂O₃ layer is deposited by electron beam evaporation. The developed pattern in the resist layer is subsequently transferred to the Al₂O₃ layer via a lift-off process. The resulting Al₂O₃ pattern is used as a hard mask for etching a-Si nanopillars using inductively coupled plasma (ICP)–reactive ion etching, with a mixture gas of SF₆ and C₄F₈ (1:2 ratio) at 1800 W ICP power and 15 W radio frequency power, at 15°C. The metasurface device is lastly cleaned in a mixture of hydrogen peroxide and ammonium hydroxide to remove the Al₂O₃ etch mask and any etch residues.

Spatiotemporal pulse characterization

Spectral interferometry

An ultrafast pulse with a complex time-varying polarization state can be decomposed into the two-constituent p- and s-polarized components. The temporal evolution of the electric-field vector ${\overrightarrow{\mathit{E}}}_{\text{out}}(t)$ can then be expressed as ${\overrightarrow{\mathit{E}}}_{\text{out}}(t)=\left[\begin{array}{c}{A}_p(t)e^{i{\mathrm{\Psi }}_p(t)}\\ A_s(t)e^{i{\mathrm{\Psi }}_s(t)}\end{array}\right]$, where A_p(s)(t) and Ψ_p(s)(t) are the temporal amplitude and phase for the p- (s-) polarized component, respectively. To characterize ${\overrightarrow{\mathit{E}}}_{\text{out}}(t)$, the spectral phase φ_p(s)(ω) and intensity I_p(s)(ω) for the p- (s-) polarized component are measured separately using the methods described below. The time domain waveform ${\overrightarrow{\mathit{E}}}_{\text{out}}(t)$ can then be reconstructed from the frequency domain field ${\overrightarrow{\mathit{E}}}_{\text{out}}(\mathrm{\omega })=\left[\begin{array}{c}{a}_p(\mathrm{\omega })e^{i{\mathrm{\phi }}_p(\mathrm{\omega })}\\ a_s(\mathrm{\omega })e^{i{\mathrm{\phi }}_s(\mathrm{\omega })}\end{array}\right]$, through an inverse-FT operation, where $a_{p(s)}(\mathrm{\omega })=\sqrt{I_{p(s)}(\mathrm{\omega })}$.

The spectral amplitude a_p(s)(ω) of the p- (s-) polarized component is characterized by blocking the reference beam and measuring the corresponding spectral intensity I_p(s)(ω). To calculate the normalized spectral amplitudes, an input spectrum I₀(ω) was also measured by placing only the fused silica substrate at the Fourier plane.

The spectral phase φ_p(s)(ω) for the p- (s-) polarized component, each containing part of the total pulse energy, is measured using the SI technique (36). Nonlinear pulse characterization techniques such as frequency-resolved optical gating and spectral interferometry for direct electric-field reconstruction (SPIDER) require high-input powers and are also unable to unambiguously retrieve the constant phase offset [first term in the Taylor series expansion of the spectral phase, (37)], which is required here to accurately determine the instantaneous polarization state of the shaped pulse. To overcome the input-power and phase-offset measurement limitations, SI that only relies on linear optics is leveraged here using a custom-built interferometry setup to characterize the temporal properties of the synthesized pulses. SI retrieves phase information of the shaped pulse by measuring the spectral interference fringes upon interference with a known reference pulse (Fig. 3). Pulses in the experimental beam, which originate after transmission from an ultrafast beam splitter, pass through the pulse shaper and acquire the desired spatiotemporal modulation. The reflected beam, designated as the reference beam, is subsequently delayed by a fixed time τ using a retroreflector mounted on a translation stage, and recombined with the experimental beam using a second ultrafast beam splitter. The combined beams are then directed to the SI measurement setup, where the interference pattern between these two spatially overlapped and temporally sheared pulses is recorded on a spectrometer (Fig. 3, pink inset). A half–wave plate and a linear polarizer in the experimental beam are used to switch between measuring the p- and s-polarized components, separately. The recorded SI signal $I_{\text{SI}}^{p(s)}(\mathrm{\omega })$ can be expressed as

$$I_{\text{SI}}^{p(s)}(\mathrm{\omega })=I_{p(s)}(\mathrm{\omega })+I_{\text{ref}}(\mathrm{\omega })+2\sqrt{I_{p(s)}(\mathrm{\omega })}\sqrt{I_{\text{ref}}(\mathrm{\omega })}\text{cos}({\mathrm{\phi }}_{p(s)}(\mathrm{\omega })-{\mathrm{\phi }}_{\text{ref}}(\mathrm{\omega })-{\mathrm{\omega \tau }}_0)$$ (5)

where I_ref(ω) and φ_ref(ω) are the spectral intensity and phase of the reference pulse, respectively. Here, τ₀ is the mean extracted value from 1000 measurements of $I_{\text{SI}}^p$ with only the fused silica substrate present in the experimental beam path, and the phase difference (φ_p(s)(ω) − φ_ref(ω)) at this τ₀ is then extracted from the cross-term in Eq. 5 using the fringe inversion technique (38). After characterizing φ_ref(ω) using the SPIDER technique (Fig. 3, green inset, and fig. S6), the only remaining unknown φ_p(s)(ω) can be retrieved. Note that the path length difference between the SPIDER and the SI measurement setup was measured to be ≈9 cm, which corresponds to a group delay dispersion in air of ≈1.8 fs²/rad, resulting in negligible broadening of the input pulse.

Spatial interferometry

Characterizing the time-varying spatial profile of the shaped pulse involves determining the time-dependent spatial phase Ψ_p(s)(x, y, t) and amplitude A_p(s)(x, y, t) of the p- (s-) polarized component at different time delays t = τ, which is achieved by performing spatial interferometry between the experimental beam and a transform-limited reference beam that carries a Gaussian spatial profile with a quasi-flat phase front. A small tilt angle between the wavefronts of the reference and experimental beams is introduced to create fringe interference patterns. The interference fringes are recorded on a CMOS sensor for various values of τ. By counting the number of fringe dislocations and comparing their orientations, the topological charge of Ψ_p(s)(x, y, τ) at different τ can be determined. The spatial amplitude A_p(s)(x, y, t), on the other hand, is proportional to the oscillation amplitude of the cross-term between experimental and reference fields, i.e., the contrast of the bright and dark fringes. Further details on evaluating the time-varying spatial amplitude are discussed in text S5. The time-averaged spatial amplitude ${\overline{A}}_{p(s)}(x,y)$ of the p- (s-) polarized component is directly characterized by blocking the reference beam and recording the corresponding spatial intensity distribution ${\overline{I}}_{p(s)}(x,y)={\overline{A}}_{p(s)}^2(x,y)$ on the CMOS sensor.

Supplementary Materials

This PDF file includes:

Texts S1 to S6

Figs. S1 to S11

References

Other Supplementary Material for this manuscript includes the following:

Movies S1 and S2