The high intensity of spatiotemporal coupling (STC) ultrafast pulse focusing is a crucial tool for exploring the atomic and molecular realms [1–4] and for controlling chemical reaction processes [5,6]. As laser pulse duration continues to decrease, the required spectral bandwidth increases. Full-dimensional characterization of broadband laser beams is essential for further observing and manipulating their focusing properties. Temporal waveform characterization techniques for spatially averaged laser beams, such as frequency-resolved optical gating (FROG) [7] and dispersion scan [8], have been widely developed. However, the spatial structure of broadband laser beams also significantly influences their focusing behavior [9–11]. For instance, chromatic aberration in lenses can shift the waist positions of beams at different wavelengths [12], and diffraction gratings and prisms used for laser pulse compression may introduce spatial chirp [13]. Accurate characterization of these phenomena requires ultrafast laser characterization approaches with high spatial resolution.

Earlier techniques included spatial beam scanning methods, such as a spatially encoded arrangement for temporal analysis by dispersing a pair of light E-fields (SEA TADPOLE) [14,15], whose scanning accuracy limits spatial resolution, and delay scanning methods like INSIGHT [16] and total E-field reconstruction using a Michelson interferometer temporal scan (TERMITES) [17]. Due to the extremely short pulse durations of ultrafast pulses, direct time-domain characterization is highly challenging. However, by leveraging the Fourier transform relationship between time and frequency, spatially resolved frequency measurements can be used to characterize the spatiotemporal laser beam. Recent studies include spatially and temporally resolved intensity and phase evaluation devices, full information from a single hologram (STRIPED-FISH) [18,19], which achieves a spatial resolution of 44 μm, and compressed optical field topography (COFT) [20,21], which offers a resolution of 50 μm. With the advent of the ptychographical information multiplexing (PIM) algorithm [22], researchers can now simultaneously reconstruct the amplitude and phase information of multi-wavelength laser beams and object functions at higher spatial resolutions [23–25].

Ptychography accelerates the convergence of phase retrieval algorithms by capturing additional diffraction information from various locations. The effectiveness of this algorithm in accurately recovering complex object functions has been well established and demonstrates ultrahigh spatial resolution [26,27]. As a result, it has been progressively extended to pulse wavefront measurements. Since ptychography relies on diffraction information for phase recovery, the laser beam needs to have good temporal coherence. Initially, it was applied to wavefront diagnosis of a single narrowband laser [28–30]. The PIM algorithm [22] has expanded the technique from a single narrowband laser source to multiple wavelength channels of narrowband laser sources [31–33], facilitating the application of ptychography to broadband laser beam reconstruction. However, the discrete spectral assumption inherent in multiplexed ptychography algorithms does not align with the continuous spectral structure of ultrafast laser pulses, leading to significant crosstalk between different wavelength channels (WCs), which poses challenges on the correct convergence of the phase retrieval algorithm and deteriorates the achieved spatial resolution. For instance, one study [23] modulated a broadband LED light field using a 200 μm pinhole and applied probe replacement constraints during reconstruction. However, half of the reconstructed probes of different WCs fail to show the expected trend of increasing diameter with wavelength, particularly at the edges, where discrepancies with simulated results were significant. The reconstruction algorithm is unable to distinguish the diffraction patterns generated by different wavelength components for continuous broadband laser pulses, with significant crosstalk between different WCs. The beams modulated by the pinhole show uniform spatial structure, whereas real ultrafast laser beams, influenced by dispersion control elements, exhibit spatiotemporal coupling effects such as spatial chirp and chromatic dispersion. In a study on complex laser beams with spatial chirp [34], eight probes were reconstructed, but only the middle five probes showed relative centroid displacement with increasing wavelength, while the spatial chirp effects of the probe in the edge WCs were completely incorrect. To address the issue of high-precision spatiotemporal reconstruction of ultrafast lasers, more reliable and advanced algorithms and experimental techniques are required.

In this work, we propose a spectral modulation method that discretizes the broadband spectrum using multi-beam interference, combined with a multiplexing ptychography phase recovery algorithm to reconstruct laser beams of different WCs. Both simulation and experimental results confirm the critical role of spectral modulation in diagnosing the spatiotemporal character of an ultrashort femtosecond laser beam with high accuracy.

We employ a multi-beam interference scheme to discretize the spectrum, ensuring compatibility with the assumptions in the PIM algorithm. The schematic and simulation results are presented in Fig. 1. In the simulation, a Gaussian interference spectrum with a bandwidth of 6.84% was used, calculated as the ratio of the pulse bandwidth to the central wavelength, generating seven interference peaks through multi-beam interference. As the number of interference beams increased, the bandwidth of the modulation peaks gradually reduced, ultimately forming a comb structure in the frequency domain [Figs. 1(b)–1(e)]. A standard interferometer or suitable Fabry–Perot etalon can facilitate this spectral modulation. In the spectrum, we selected 301 wavelengths at equal frequency intervals and combined them to simulate the diffraction pattern. In the simulation, the laser pulse we used exhibits a linear spatial chirp, with the beam of each wavelength component displaced by one pixel relative to its neighboring wavelength component. During reconstruction, only the discrete WCs corresponding to the seven interference peaks were selected.

During the reconstruction process, no restrictions were imposed on the iteration, including constraints such as probe modulus enhancement [34] and probe replacement constraints [23]. Given that the chosen object function is a simple binary test target, we expect the reconstruction object functions of WCs to be similar. The selected iterative algorithm computes the average update of the object function across all WCs [25], as shown below: Oλ′(x,y)=Oλ(x,y)+α0[∑λ|Pλ(x,y)|2]max×∑λPλ*(x,y)[φλ′(x,y)−φλ(x,y)],(1)Pλ′(x,y)=Pλ(x,y)+αpOλ*(x,y)|Oλ(x,y)|max2[φλ′(x,y)−φλ(x,y)],(2)where Pλ(x,y) represents the probe of different WCs, Oλ(x,y) is the object function, φλ(x,y)=Pλ(x,y)Oλ(x,y) is the exit-wave function, and α0 and αp are the iteration constants. Additionally, we package the two-dimensional probe and object function matrices into a three-dimensional matrix based on the number of WCs, and perform the calculations using a GPU, which significantly accelerates the iteration process.

Prior to reconstruction, the initial beams of each WC were aligned without any relative displacement. However, as the iteration count increased, the beams gradually shifted to their respective positions. The difference between the centroids of the reconstructed beams and the original spatial chirp was calculated [Figs. 1(f)–1(i)]. The input spatial chirp in the simulation was 9.7 μm/nm, and the relative displacement between adjacent WCs was expected to be 162.5 μm. For the unmodulated spectrum, the reconstruction of the spatial chirp showed significant offsets, with the average displacement for each WC being 131.9 μm. After applying dual-beam interference, reconstruction accuracy improved significantly, with the average offset reduced to 38.1 μm. However, the centroid reconstruction error for the edge WC remained relatively large. Increasing the number of interference pulses to four further reduced the offset to 20.7 μm, approximately the size of three pixels. As the number of interference pulses approached infinity, the spectrum became a fully discrete spectrum consisting of multiple single-frequency components, and the spatial chirp was nearly perfectly reconstructed, with an error of only one pixel. The resolution of the reconstructed object function also reflected the accuracy of the reconstruction. As shown in the bottom row of Fig. 1, as the number of interference pulses increased, the digits on the far left became progressively clearer, and the line shape representing the smallest resolvable digit “7” also became more distinct, indicating an improvement in reconstruction resolution. After discretizing the modulated spectrum, the reconstruction algorithm was able to more clearly differentiate the spectral components in the diffraction pattern, reducing crosstalk between WCs.

In the proof of principle experiment, we employed a Mach–Zehnder interferometer to achieve a dual-pulse interference configuration, demonstrating the critical role of spectral modulation in the spatiotemporal diagnostics of ultrafast laser beams. Figure 2 illustrates our spectrally modulated STCs diagnostic device. The spatiotemporal laser beam undergoes spectral modulation via the Mach–Zehnder interferometer, with the number of interference peaks controlled by a pair of wedge mirrors. Fused silica is placed on the other arm of the interferometer to balance the effects of dispersion. The spectrum and phase at the beam center in one arm of the interferometer are then measured by a home built SHG-FROG [35]. Finally, the beam is directed into a coherent diffraction imaging module, where the test target is mounted on a two-dimensional translation stage, with a camera positioned 3.43 cm downstream the target, enabling us to capture diffraction patterns in the Fresnel region; in the reconstruction process, we employed the angular spectrum method to calculate the beam propagation in free space [36]. Notably, no lens focusing elements are used in our imaging module, as chromatic and spherical aberrations introduced by lenses can significantly affect the accuracy of the reconstruction for broadband ultrafast laser beams.

The ptychographic 2D scan was conducted with a step size of 65 μm, and an overlap rate of 98.5%. The time required for a single scan was approximately 12 min, and the 10% random step offset was employed to eliminate grid effects during reconstruction [37]. Before and after each scan, we used a spectrometer to collect modulated spectra for comparison, ensuring that the spectrum collected by the interferometer during the scanning process remained unchanged, thereby preventing interference peak drift. The diffraction object-image distance was 3.43 cm, determined through the diffraction pattern of a monochromatic 532 nm laser combined with image edge detection. To highlight the advantages of spectral modulation, we blocked one arm of the interferometer and collected data without spectral modulation, comparing each reconstruction result under identical iterative conditions. No constraints, such as probe modulus restrictions or spectral component limits for each WC, were applied during reconstruction. The initial guess for all WCs was a Gaussian beam of uniform size, with no relative displacement, which significantly differed from the actual laser beam. Finally, 150 iterations were performed using a GPU.

In the dual-pulse configuration spatiotemporal diagnostic experiment, we first diagnosed a spatiotemporal laser beam with spatial chirp introduced by a misaligned prism pair and chromatic aberration introduced by a lens group. The output parameters of the mode-locked Ti:sapphire amplifier pulse are as follows: a central wavelength of 800 nm, a spectral width of 60 nm, a repetition rate of 1 kHz, and a pulse duration of 25 fs. Initially, the beam size is shrunk by a factor of five using a lens-based telescope, and then the beam passes through a double-prism system to induce spatial chirp. The beam is stretched into an elliptical shape with a length of 4.5 mm and a width of 2.4 mm due to the spatial chirp introduced by the prism [Fig. 3(d)]. The spectral modulation of the interferometer transfers to a continuous spectrum with a bandwidth of 6.55% into a modulated spectrum consisting of multiple peaks, with a bandwidth of only 0.92% for each peak [Fig. 3(a)]. During the iterative process, no restrictions were imposed on the iteration, as was done in the simulation, including the intensity ratios of different WCs.

We used a spectrometer (FLAME-T-VIS-NIR, Ocean Optics) to collect the spatially averaged spectrum of the spatiotemporal laser beam and compared it with the reconstructed spectral components of each wavelength [Fig. 3(b)]. The results indicate that the intensities of different WCs are approximately consistent with the spectral components measured by the spectrometer. Figure 3(c) shows the reconstructed object function. The resolution at the red line position was fitted, yielding a resolution of 10.7 μm through convolution fitting of a Gaussian function and a step function. By applying the angular spectrum propagation algorithm, the theoretical resolution of our system is determined by the pixel size of the camera, which is 6.5 μm. The experimental resolution closely matches the theoretical resolution, and the use of the diffraction integral formula could further enhance the resolution. For a beam with dimensions of 4.5 mm by 2.4 mm, our reconstruction method fully utilizes the benefits of ptychography high resolution. In the absence of relative displacement for the initial beam guess, the reconstructed beam is color-coded according to the maximum relative intensity of the three WCs [Fig. 3(e)], with intensities above 1/e2 selected for display. This reconstructed optical field closely matches the beam captured by the camera after moving the test target [Fig. 3(d)]. The spatial chirp introduced by the prism significantly elongated the beam, and improper alignment of the beam-shrinking lens and prism pair resulted in a tilted beam. These phenomena were well reconstructed.

We further investigated the effects of spatial chirp and chromatic aberration on the laser beam. The reconstructed beams for different WCs of the modulated spectrum are shown in Fig. 3(g). To facilitate observation, the beam was rotated by 90°. The results clearly demonstrate a spatial chirp along the longitudinal axis. We analyzed the centroid positions of the reconstructed beam, as indicated by the red line in Fig. 3(f). A centroid displacement of approximately 750 μm is observed over a spectral span of 76.5 nm. Additionally, as the wavelength increases, the size of the reconstructed beam gradually increases, suggesting effective reconstruction of chromatic aberration caused by lens compression. This complex laser beam reconstruction further validates the powerful capabilities of spectral modulation ptychography. To assess the impact of spectral modulation, one arm of the interferometer was blocked, and diffraction data were collected without spectral modulation under the same reconstruction conditions. The reconstructed beams for each WC are shown in Fig. 3(i). Severe crosstalk between different WCs was observed, resulting in the elongation of most reconstructed beams and the absence of noticeable chromatic aberration. A 6.55% bandwidth led to a completely incorrect reconstruction, as shown by the blue line in Fig. 3(f) for the centroid positions of the reconstructed beam.

To quantitatively assess the accuracy of the reconstructed laser beam, we placed the fiber port of the spectrometer in the spatially chirped beam for a 2D scan and measured its 2D spatial spectrum with a step size of 200 μm; the beam for each WC is displayed in Fig. 3(h). Due to the large numerical aperture of the fiber port and the relatively low scanning resolution, the collected data are convolved with a lower-resolution kernel, resulting in larger displayed beam sizes. A similar direct laser beam sampling using fibers was reported by SEA TADPOLE [14,15], where the fiber’s numerical aperture limited the achievable spatial resolution. Nevertheless, the spatial chirp along the X-axis and chromatic aberration effect as the wavelength increases are still clearly visible, consistent with the reconstructed results under the dual-pulse interference configuration. We collected the maximum intensity positions of the beam at 1 nm intervals, as indicated by the yellow line in Fig. 3(f). The yellow line aligns closely with the red line, further confirming that the modulated spectrum accurately reconstructed the spatial chirp of the laser beam. In contrast, the blue line shows a significant offset from the yellow line and exhibits only a weak spatial chirp, suggesting that the unmodulated reconstruction suffers from substantial error.

For a broadband laser beam with a 60 nm spectral width, the sampling of only six WCs is insufficient. To address this, we conducted additional experiments with denser sampling by adjusting the insertion thickness of the wedge pair, which modulated the spectrum into 14 interference peaks. The wavelength interval between adjacent channels is 6.22 nm, and the bandwidth of a single modulated peak is 0.435% [Fig. 4(a)]. At this point, the delay between the two pulses is 343 fs; however, excessive delay reduces the modulation depth of the interference spectrum. The reconstruction deviation for the spatially averaged spectral components of the 14 WCs remains minimal [Fig. 4(b)]. And the reconstructed resolution is 9.4 μm [Fig. 4(c)], which is an improvement over the six WCs. Additionally, we selected three beams with wavelengths of 775.8 nm, 799 nm, and 823.9 nm and plotted them on the same graph [Fig. 4(e)], where the elongation and tilting of the beam are also clearly reconstructed.

We present all 14 beams in Fig. 4(g), where the spatial chirp and chromatic aberration effects are more clearly visible. The tail of the low-intensity interference peaks in the long-wavelength portion of the spectrum causes relatively more severe crosstalk in the edge WC. The reconstruction result without spectral modulation is shown in Fig. 4(h), where the central beam is elongated, and the edge beams exhibit distorted shapes, clearly displaying severe crosstalk. Similarly, using the centroids from Fig. 4(f), the yellow line represents the spatial sampling result obtained with the spectrometer, which is identical to the yellow line in Fig. 3(f). The red and yellow lines remain well-aligned, confirming that both the 6-mode and 14-mode reconstructions capture the spatial chirp introduced by the prism pair. In contrast, the blue line exhibits a significant deviation, particularly in the edge WC, where it reconstructs a completely incorrect spatial chirp.

After reconstructing the spectrally resolved wavefront of the ultrashort laser beam, the relative spectral phase of each WC must also be obtained to achieve spatiotemporal characterization of the laser beam. FROG could measure the spectral phase at the center of the beam, so we performed full-dimensional characterization of the laser beam after the telescope beam compression system. The modulated spectrum [Fig. 5(a)] produces 10 interference peaks with a bandwidth of 0.64%. We present the reconstructed beam of each WC on the image plane [Fig. 5(d)] and the spatial phases corresponding to regions with a relative intensity greater than 8% [Fig. 5(f)]. The chromatic aberration effect, manifested as an increase in beam radius with wavelength, remains evident. The phase of each WC exhibits a convergent phase profile, but the overall spatial phase difference is small, not exceeding 2 rad. After back-propagating the pulse beam by 0.5 m along the propagation direction, it reaches the position behind the compressed lens [Fig. 5(e)]. Here the beam radius increases significantly relative to the image plane, indicating that the wavefront is still slowly converging. The intensity distribution along each diameter direction becomes more Gaussian-like, suggesting the presence of axial astigmatism. Based on this information, we can characterize the evolution of ultrafast laser beams at any position along the propagation direction. The measured and reconstructed FROG traces [Fig. 5(b)] exhibit a small comparison error, confirming the accuracy of the reconstruction. The reconstructed 25.4 fs full-width at half-maximum pulse spectrum and phase [Fig. 5(c)] combine with the reconstructed phase information for each WC, allowing us to determine the spectrum intensity and phase at any spatial position.

We propose and demonstrate a spectral modulation method for high-precision spatiotemporal measurement of an ultrashort laser beam. This method discretizes the broadband spectrum using multi-beam interference, ensuring compatibility with the assumptions of multiplexing ptychography to reconstruct laser beams of different WCs. Simulation results show that as the number of interference pulses increases, the bandwidth of each individual interference WC decreases, further improving the resolution and accuracy of the reconstruction. In the experiment, we employed a Mach–Zehnder interferometer to generate the modulated spectrum and combined near-field diffraction with the angular spectrum propagation algorithm to achieve lensless imaging. We reconstructed the spatially chirped laser beam introduced by a prism pair and a lens group, and compared it with spatial scanning results from a spectrometer to verify the accuracy of the reconstruction. Additionally, we employed FROG to measure the phase at the center of the laser beam, completing a full-dimensional characterization. The reconstructed resolution is approximately 9.4 μm, close to the pixel size resolution limit of the angular spectrum method.

If a suitable Fabry–Perot etalon can be customized, the experimental setup would be significantly simplified, and the robustness of the optical path would be greatly enhanced. Multi-beam interference will lead to a smaller interference peak bandwidth, and therefore further improve the accuracy of the reconstruction. Considering an 800 nm femtosecond laser beam with a bandwidth of 60 nm, if the spectrum is discretized into 20 WCs, the corresponding Fabry–Perot etalon made of fused silica would need to have a thickness of approximately 44 μm or less. This presents a challenge for the precision dual-layer coating process required for the Fabry–Perot etalon, and we anticipate further developments of this device in the future.