Introduction

With the rapid development of ultrafast lasers, the ability to sculpt spatiotemporal wavepackets has become increasingly important for both unveiling fundamental physics and fulfilling demanding industrial applications1,2. Fully customizable spatiotemporal light holds immense potential in diverse fields such as laser machining3, spatiotemporal edge detection4, nonlinear optics5,6, optical fiber propagation7, quantum optics8, and various light-matter interaction studies9,10. Traditionally, manipulating optical light fields has involved separate interventions in the spatial and temporal domains. For spatial light, holographic methods and programmable devices like liquid crystal spatial light modulators (LC-SLMs) have become prevalent tools for tailoring laser modes11,12, generating structured beams like nondiffracting beams13, and even achieving arbitrary spatial profiles14. Pulse shaping in the time domain, on the other hand, typically occurs in the spectral domain using a 4-f setup and involves strategically placing diffractive elements or 1D LC-SLMs in the focal plane15,16.

In 2020, the demonstration of photonic cyclones carrying transverse orbital angular momentum (OAM) opened a new avenue for manipulating spatiotemporal structured light17. Utilizing a 2D spatial light modulator (SLM) within a conventional pulse shaper, scientists have spearheaded the study of spatiotemporally coupled light fields, revealing their significant physical implications and promising applications. These include space-time wavepackets with intricate spatiotemporal correlations18,19, spatiotemporal optical vortices with transverse OAM17, and toroidal vortices of light20.

Researchers recently developed a 3D wavepacket generation method combining pulse shaping with multi-plane light conversion (MPLC)21. While this technology enables multi-dimensional light sculpting, it struggles with producing tight spatiotemporal structures like optical vortices due to its approach of modulating the spatial light profile at each time delay. However, relying solely on a conventional pulse shaper consisting of a grating, a cylindrical lens, and a 2D SLM as a spatiotemporal phase modulator presents a significant obstacle to efficiently and precisely manipulating spatiotemporal wavepackets on demand.

This paper introduces a novel method for shaping the spatiotemporal complex amplitude distribution of the light pulses, enabling the custom shaping of arbitrary 2D spatiotemporal optical fields. We achieve this by incorporating a computer-generated hologram (CGH) into a conventional pulse shaper. The capabilities of this device are showcased through the generation of diverse and unprecedented spatiotemporal light structures, including fundamental and higher-order spatiotemporal Bessel wavepackets, spatiotemporal crystal-like and quasi-crystal-like structures, and spatiotemporal top-hat wavepackets, highlighting its potential for applications in various fields.

Results

Figure 1a illustrates the schematic of the spatiotemporal (ST) holographic setup and the associated interferometric characterization system. The input mode-locked laser centered at 1030 nm is divided into two parts by the beam splitter (BS1). One part is compressed by a pair of gratings, serving as the probe pulse. The other one is modulated into the desired wavepacket using the ST complex amplitude modulator. The probe pulse and the generated wavepacket are recombined at the CCD camera with a small tilt angle. By changing their relative time delay, the three-dimensional information of the generated wavepacket is recorded and later reconstructed22.

Fig. 1: Illustration for spatiotemporal hologram experimental setup and its working principal.
figure 1

a Realization of spatiotemporal complex amplitude modulation with redistribution of energy into other diffraction order. BS beam splitter, M mirror, G grating, CL cylindrical lens; b One dimensional spatial linear grating with variable phase depth loaded onto the SLM; c Simulated process of redistribution of energy caused by variable modulated depth, resulting in the appearance of higher diffraction order and the variation of the 0th order between 0 and 1; d, e Experimental results for the 0th order intensity variations in c and simulation results for the 0th order retarded phase caused by different modulation depth. The extinction ratio is around 125:1.

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The ST holographic setup consists of a grating, a cylindrical lens, and a phase-only LC-SLM. They are separated by the focal length of the cylindrical lens, forming a 4-f system. With such a configuration, LC-SLM encoded with a properly designed digital hologram can simultaneously modulate the spatial-spectral phase \(\varphi \left(x,\omega \right)\) and the amplitude \(A\left(x,\omega \right)\) of the incoming optical field in its spatial-spectral plane. The encoding technique utilizes a one-dimensional blazed phase grating with different modulation depths in order to diffract light into the first order with different light intensity23. Here, we use the technique conversely so that the undesired light is diffracted into the first order while the desired light is preserved in the zeroth order. The modulation depth of blazed grating phase can be engineered so that it controls the intensity of the zeroth order light. When the depth is 2\(\pi\), all the light energy will be diffracted. When the depth is 0, none of the light will be diffracted, and, therefore, all the energy will be preserved in the zeroth order. The change of grating phase depth can continuously tune the light intensity of the zeroth order from 0 to 100%. The results as shown in Fig. 1 and more information are given in the supplementary note 1.

To measure the power of the zeroth order light, we place a small iris after the ST complex amplitude modulator so that higher-order diffracted light is fully blocked by the iris. Figure 1b shows the phase patterns of such blazed phase grating when the modulation depth of the grating phase is at \({\phi }_{{\rm{depth}}}=0.8\pi,1.2\pi,2\pi\), respectively. Figure 1c shows the theoretical energy distribution from a 1D blazed phase grating when \({\phi }_{{\rm{depth}}}\) continuously varies from 0 to 2π. Figure 1d shows both theoretical and experimental intensity of the zeroth order light when \({\phi }_{{\rm{depth}}}\) varies. The experimental data obtained by the setup shown in Fig. 1a agrees well with the theoretical curve, indicating the possibility of precise control of the ST complex optical field. The measured extinction ratio of the light intensity (Imax/Imin) is about 125:1 in this figure. The overall efficiency of our devices thus depends on (1) the reflectance from the diffraction grating, (2) the reflectivity of the SLM device, and (3) the intensity pattern of the complex amplitude modulation (see more details in Supplementary Note 2). Moreover, the bandwidth and the spatial complexity of our devices mainly depend on the operating wavelength and spatial resolution of the SLM (see more details in Supplementary Note 3).

It is noteworthy that the intensity modulation of the light field in the zeroth order also accompanies an additional retardance phase that should be properly compensated. Their relationship is shown in Fig. 1e. The cause of this retardance phase is due to the change of the overall phase in the blazed phase grating.

The results we discussed above use blazed phase grating with spatially uniform modulation depth \({\phi }_{{{\rm{depth}}}}\). To achieve arbitrary ST complex amplitude modulation of the light field \(\Psi \left(x,\omega \right)=A\left(x,\omega \right)\exp \left[i\varphi \left(x,\omega \right)\right]\), the digital hologram loaded on SLM is (see more details in Supplementary Note 1)

$${\varphi }_{{{\rm{SLM}}}}(x,\; \omega )={\phi }_{\rm{depth}}(x,\; \omega )\cdot \frac{\rm{Mod}(x,\; \varLambda )}{\varLambda }+{\phi }_{r}(x,\; \omega )+\varphi (x,\; \omega ).$$
(1)

In this expression, the first term represents the blazed phase grating with a period of Λ and a localized modulation depth of \({\phi }_{{{\rm{depth}}}}\left(x,\omega \right)\). \({\phi }_{{{\rm{depth}}}}\left(x,\omega \right)\) is related to the amplitude modulation factor \(A\left(x,\omega \right)\) given by \({\phi }_{{{\rm{depth}}}}=2\pi {{{\rm{sinc}}}}^{-1}\left(A\right)\). Figure 1d shows the experimental measured results for the relationship between A and \({\phi }_{{{\rm{depth}}}}\); the second term \({\phi }_{r}\left(x,\omega \right)\) is the phase compensation term for the phase retardance; and the last term \(\varphi \left(x,\omega \right)\) is the applied spatial-spectral phase modulation. More information including the completely complex modulation of the spatiotemporal hologram are discussed in Supplementary Note 4.

This encoding technique for ST complex amplitude modulation using programmable phase elements offers us a much wider range of applications as it can arbitrarily sculpt ST optical fields. In the process, the subtle interplay between dispersion and diffraction plays a crucial role in generating the ST optical field. Assuming paraxial approximation with group velocity dispersion, the evolution of the optical field \({E}_{0}\left({x}^{{\prime} },{\varOmega }^{{\prime} }\right)\) is determined by the second-order diffractive term and group delay dispersion (GDD) term. The spatiotemporal light field can be then expressed by \({E}_{0}\left({x}^{{\prime} },t ^{\prime} \right)=F\left\{G\left({x}^{{\prime} },{\varOmega }^{{\prime} }\right){e}^{i{\varphi }_{{{\rm{SLM}}}}}\right\}\), where \(G({x}^{{\prime} },\varOmega ^{\prime} )\) is the spatial-spectral field of the input laser and the operator \(F\left\{\cdot \right\}\) stands for 1-D Fourier transform from \(\varOmega ^{\prime}\) to t′. The evolution of the modulated spatiotemporal field can be then expressed by24

$$E\left(X,\; T{{{;}}} \;{\mbox{L}},\; {\mbox{GDD}}\right)= \frac{{w}_{b}\cdot {w}_{\varOmega }}{L}{e}^{i{k}_{0}\left(L+\frac{{X}^{2}}{L}\right)}\iint {E}_{0}\left({x}^{{\prime} },\, {\varOmega }^{{\prime} }\right){e}^{i\frac{{GDD}\cdot {w}_{\varOmega }^{2}}{2}{{\varOmega }^{{\prime} }}^{2}}{e}^{i{k}_{0}{w}_{b}^{2}\left(\frac{1}{2L}-\frac{1}{2f}\right){{x}^{{\prime} }}^{2}}\\ {e}^{i{\varOmega }^{{\prime} }T\cdot {w}_{\varOmega }}{e}^{-i{k}_{0}\frac{{x}^{{\prime} }X}{L}{w}_{b}}d{x}^{{\prime} }d{\varOmega }^{{\prime} },$$
(2)

where \({w}_{\varOmega }\) and \({w}_{b}\) are the spectral and spatial width of the input field, respectively, \({x}^{{\prime} }\)/\({\Omega }^{{\prime} }\) are normalized spatial and spectral coordinates, \(f\) is the focal length from the applied phase in the spatial direction. The balance requires \({{\rm{GDD}}}\cdot {{{\rm{w}}}}_{\Omega }^{2}={k}_{0}{w}_{b}^{2}\big(\frac{1}{L}-\frac{1}{f}\big)\). An extra focusing or defocusing phase can determine the sign of GDD. By defining \(\cot \gamma={{\rm{GDD}}}\cdot {w}_{\varOmega }^{2}={k}_{0}{w}_{b}^{2}\big(\frac{1}{L}-\frac{1}{f}\big)\) and using the following substitution \(x=\frac{\left(X\cdot \cos \gamma \right)}{{w}_{b}} \, \cdot \big(1+\frac{L}{f-L}\big)\), \(t=-T\cdot {w}_{\Omega }\cdot \sin \gamma\), Eq. (2) can be simplified to the following expression

$$E\left(X,T{{{;}}}\cot \gamma \right)={{\rm{C}}}\iint {E}_{0}\left({x}^{{\prime} },\, {\Omega }^{{\prime} }\right){e}^{i \, \frac{{{\Omega }^{{\prime} }}^{2}+{{x}^{{\prime} }}^{2}}{2}\cot \gamma }{e}^{-i{\Omega }^{{\prime} }t\csc \gamma }{e}^{-{ix}{x}^{{\prime} }\csc \gamma }d{x}^{{\prime} }d{\Omega }^{{\prime} }$$
(3)

where \({{\rm{C}}}\) is the field constant. Equation (3) is called fractional Fourier transform when \(0 \, < \, \gamma \, < \, \pi /2\) and becomes a complete Fourier transform when \(\gamma=\pi /2\). Therefore, a proper selection of the focusing/defocusing phase, GDD, and propagation distance \(L\) can achieve such “balance” between dispersion and diffraction, leading to an effective modulation of the complex field in the reciprocal plane (Fourier space).

To demonstrate the ST complex amplitude modulation capacity, we first use the device to generate Bessel spatiotemporal wavepackets (BeSTWP) and higher-order Bessel spatiotemporal optical vortex (BeSTOV) wavepackets. Compared with previous work24, the generation of these wavepackets relies upon controlling the complex amplitude in the reciprocal plane (Fourier space) where these wavepackets have an annular ring distribution and a spiral phase structure. Figure 2a–d presents the generation of the fundamental BeSTWP based on an annular ring mask in the spatial-spectral domain. The applied phase mask is shown in Fig. 2a. The desired hologram is generated using a blazed grating to preserve the light field located on the ring mask while diffracting the light field elsewhere. The spatial width \({w}_{b}\) and spectral width \({w}_{\Omega }\) of the ring are 90 μm and 1 THz, respectively. At the propagating length L = 1.25 m, the calculated γ is around 87.7o, indicating the ST field is close to the full Fourier transform form. Figure 2e–l shows higher-order BeSTOV results by applying additional ST spiral phases. BeSTOVs carrying transverse OAM with topological charges from −2 to +2 are generated and they are confirmed by the measurement.

Fig. 2: Demonstration of spatiotemporal complex amplitude modulation with the generation of the family of BeSTWP.
figure 2

a The phase mask of the fundamental BeSTWP; b–d 3D reconstructed intensity distribution with normalized iso-value 0.02 and 2D slice intensity and reconstructed phase; e–h spatiotemporal intensity profile for higher order (l from −2 to 2) BeSTWP with normalized iso-value 0.04; i–l spatiotemporal phase profile for higher order (l from −2 to 2) BeSTWP.

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The generations of the fundamental BeSTWP and BeSTOV have proven the capability of the ST holographic setup. As two-dimensional localized optical fields, BeSTWP and BeSTOV wavepackets are radially symmetric. There also exists another kind of localized state called an optical lattice with translational and rotational symmetries. Using the ST holographic technique, spatiotemporal optical lattices can be realized in the spatiotemporal domain by selecting proper pinhole patterns in the spatial-spectral domain. Figure 3a–e illustrates several configurations to generate ST crystal-like and quasi-crystal-like structures in simulation. Each pinhole is uniformly located on a circle and the number of pinholes q represents q-fold rotational symmetry. The second two rows of the figure show the spatiotemporal intensity and phase profile of the simulated wavepacket. When the number of pinholes q = 4 and q = 6, the spatiotemporal optical time crystals have translational symmetry shown in Fig. 3f, h. Especially, the spatiotemporal optical quasi-time crystal with rotational symmetry is generated at q = 5 in Fig. 3g. It is worth noting that the five-fold rotation is preserved even with a gradient phase shown in Fig. 3i, j. The transverse spiral phase can be also preserved in the spatiotemporal domain together with the five-fold rotational symmetry, forming a ST quasi-crystal structure carrying transverse OAM.

Fig. 3: Simulations of spatiotemporal optical time crystals and spatiotemporal optical time quasi-crystals.
figure 3

a–c Applied intensity masks with number of pinholes \(q=\) 4, \(q=\) 5 and \(q=\) 6 in the spatial-spectral domain for generating spatiotemporal optical time crystals; d, e Applied intensity mask with \(q=\) 5 and an extra graidient transverse phase \(l=-1\) and \(+1\) for generating spatiotemporal optical time quasi-crystals. f–o Spatiotemporal intensity and phase profile for the simulated spatiotemporal optical time crystals and quasi-crystals.

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Following the simulations shown in Figs. 3 and 4 presents experimentally generated periodic and aperiodic spatiotemporal crystal structure light fields. Figure 4a, c shows the reconstructed spatiotemporal crystals with 4-fold rotational symmetry and 6-fold rotational symmetry, respectively. Figure 4b shows spatiotemporal quasi-crystals with 5-fold rotational symmetry. Figure 4d, e shows the effect of the transverse spiral phase on the spatiotemporal quasi-crystals. The extra transverse OAM results in zero intensity in the center of spatiotemporal quasi-crystals and a phase singularity in the spatiotemporal domain. The spatiotemporal quasi-crystal structure maintains 5-fold rotational symmetry as predicted in Fig. 3.

Fig. 4: Experimental results of spatiotemporal optical time crystals and spatiotemporal optical time quasi-crystals.
figure 4

a–e 3D Reconstructions of spatiotemporal optical time crystals (\(q=\) 4, 6), spatiotemporal optical time quasi-crystals (\(q=5\)) and spatiotemporal optical time quasi-crystals; f–h, k–m Cross-sectioned spatiotemporal intensity and phase profiles for spatiotemporal optical time crystals; i, j, n, o Spatiotemporal intensity and phase profiles for spatiotemporal optical time quasi-crystals.

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Using the spatiotemporal holographic technique, we can dynamically generate two-dimensional spatiotemporal wavepackets of arbitrary shape. As discussed above, all results are products of diffraction and dispersion in dynamic equilibrium. However, especially in optical materials processing, static and higher power beam shaping is more important. Hence, we use an extra lens and de-chirped phase on the SLM to compensate for diffraction and dispersion to form spatiotemporal flat-top wavepackets in full Fourier transform. Unlike flat-top beams in the spatial domain and the flat-top pulses only in the time domain, the spatiotemporal flat-top wavepackets have both homogenous intensity distribution in \(x-t\) domain. Figure 5a shows the 3D intensity map and projections view of the reconstructed spatiotemporal flat-top pulses. Figure 5b shows the accumulated intensity in the X and T-direction, respectively. The spatial width (full-width at half-maximum(FWHM)) is 1 mm and the FWHM temporal width is around 2 ps. Just as flat-top beams and flat-top pulses play an important role in the field of optical processing and optical computing, spatiotemporal flat-top pulses with uniform intensity in both the \(x-t\) domain synthesized with this spatiotemporal holographic technique have broad potential applications in ultrafast laser machining as well as generating uniformly distributed electron wavepackets.

Fig. 5: Generation of spatiotemporal flat-top wavepacket.
figure 5

a 3D intensity isosurface of the spatiotemporal flat-top wavepacket with normalized iso-value 0.3 and its projections; b Accumulated intensity profile of spatiotemporal flat-top wavepackets and its cross lines along X = 0 and T = 0 axes.

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To conclude, we introduce a novel spatiotemporal hologram that enables control over the complex field (both amplitude and phase) of optical spatiotemporal wavepackets. Both theoretical and experimental results demonstrate the capability of this technique for generating complex spatiotemporal wavepackets with arbitrary amplitude-phase profiles. Examples include fundamental and higher-order spatiotemporal Bessel wavepackets, spatiotemporal crystal and quasi-crystal structures, and spatiotemporal flat-top wavepackets. This manipulation of spatiotemporal complex amplitude expands the range of achievable optical properties in spatiotemporal wavepackets and promises broad applications in ultrafast processing, ultrafast imaging, optical communication, and exploring novel light-matter interactions.

Methods

Parameters for spatiotemporal holographic pulse shaper

The laser source for seeding the system is a home-built Yb:fiber laser with an all-normal-dispersion (ANDi) configuration. The laser spectrum centered at 1030 nm has a 3 dB bandwidth of 10 nm. At the output, the laser beam is expanded to a beam diameter of 2 mm to accommodate the long propagation distance in this experiment.

In the spatiotemporal pulse shaper, the diffraction grating has a grating density of 1200 lines/mm. The incident angle for the grating is about 45°. The first-order diffracted light is collimated by a cylindrical lens with a focal length of 100 mm. The spatial-spectral (\(x-\omega\)) component of the light is then projected to a liquid crystal spatial light modulator with a 3840 × 2160 resolution and a pixel size of 3.74 μm (Model: Holoeye GAEA-2.1-NIR-069). The spatiotemporal hologram can be then defined by the following expression,

$${\varphi }_{{\rm{SLM}}}\left(x,\, \omega \right)={\phi }_{{{\rm{depth}}}}\left(x,\, \omega \right)\cdot \frac{{{\rm{Mod}}}\left(x,\, \varLambda \right)}{\varLambda }+{\phi }_{r}\left(x,\, \omega \right)+\varphi \left(x,\, \omega \right),$$

where \({\phi }_{{{\rm{depth}}}}\left(x,\omega \right)\) is the modulation depth for the blazed grating phase that is related to the amplitude modulation factor \(A\left(x,\omega \right)\) given by \({\phi }_{{{\rm{depth}}}}=2\pi {{{\rm{sinc}}}}^{-1}\left(A\right)\); \(\varLambda\) is the period of the blazed phase grating. We set it to 119.68 μm so that each period of the grating covers 32 pixels in the x-direction; \({\phi }_{r}\left(x,\omega \right)\) is the phase compensation term for the phase retardance; and the last term \(\varphi \left(x,\omega \right)\) is the applied spatial-spectral phase modulation.

The final component of the spatiotemporal holographic pulse shaper is the iris that is put about 55 cm after the pulse shaper (from the diffraction grating). The iris can fully block first-order and higher-order light so that the transmitted zeroth-order light can have the spatiotemporal complex amplitude modulation applied by the shaper.