Rapid imaging is a fundamental technique for documenting and comprehending natural phenomena. For capturing the ultrafast dynamics inherent in various processes, the duration of the illumination or exposure must be minimized. Traditional detectors, such as charge-coupled devices (CCDs) and complementary metal-oxide-semiconductor (CMOS) sensors, are constrained by their limited response and data readout capabilities, hampering imaging at durations shorter than a picosecond [1,2].

Ultrafast laser technology, capable of providing attosecond-duration illumination in extreme ultraviolet (XUV) and X-ray spectra, marks a revolution in ultrafast imaging. Currently, free electron laser (FEL) facilities produce X-ray laser pulses with hundred-attosecond-scale durations [3]. Moreover, shorter pulses with 50-attosecond durations have been reported using tabletop high-order harmonic generation (HHG) [4–6]. These pulses, often encompassing only a few or even a single electromagnetic (EM) wave cycle, enable unprecedented examinations of ultrafast electron dynamics in molecular and solid-state systems.

As pulse widths approach the wave period, there exists a trade-off governed by the uncertainty principle: high temporal resolution can only be achieved at the expense of monochromaticity. For HHG-generated attosecond lasers, the typical spectral bandwidth can extend beyond Δλ/λc≈100%, where Δλ represents the full width of the spectrum and λc signifies the central wavelength. Consequently, much research has been conducted using attosecond spectroscopy, capitalizing on its expansive spectral bandwidth and unparalleled temporal resolution.

Since its initial demonstration in 1999 [7], coherent diffraction imaging (CDI) has gained wide acceptance as a potent imaging technique in various fields, including biology [8–15] and material science [16–21]. Owing to its lensless nature, CDI dominates XUV/X-ray illumination imaging, particularly when high-quality optics are unavailable. Consequently, CDI is particularly well suited for attosecond imaging, as the wavelengths associated with attosecond pulses typically fall within the XUV and X-ray spectra. However, the expansive spectral bandwidth of attosecond pulses complicates CDI, as the resulting diffraction patterns can be convoluted by varying wavelengths. This complexity calls for the development of innovative image reconstruction algorithms.

Several solutions for broadband CDI have been proposed, including polychromatic phase retrieval algorithms [22], numerical monochromatization [23,24], and holographic methods [25]. They have enabled ultrafast CDI with femtosecond to attosecond temporal resolutions and a full spectral bandwidth on the order of Δλ/λc≈30% [23], due to limited pattern modeling as discussed in Section 2.B. Additionally, it has been demonstrated that CDI with a bandwidth approaching Δλ/λc≈100% can be achieved through ptychography [26–28] and Fourier transform imaging spectroscopy (FTIS) [29]. However, the ptychographic technique necessitates numerous position scans with approximately 94% overlap for each individual scan [28], while FTIS needs multiple delay scans.

In this work, we introduce gradient monochromatization with Fourier transformation based pattern mapping (GM-FTM), a novel method for ultra-broadband attosecond imaging, where FTM achieves high-quality pattern modeling as discussed in Section 2.B, and GM provides efficient monochromatic pattern reconstruction as discussed in Section 2.C. The bandwidth of the imaging light source was significantly extended, providing substantially higher photon efficiency and shorter temporal resolution. GM-FTM is computationally efficient and applicable to both continuous and discrete spectra, verified in simulations and experiments. To the best of our knowledge, this is the first one identified potentially applicable to snapshot CDI using sub-100-attosecond pulses, which will foster attosecond imaging with superior temporal and spatial resolution.

Furthermore, we present a high-fidelity spectrum reconstruction method using FTM, suggesting a new type of grating-free spectrometer.

In the context of CDI with broadband illumination, the resultant diffraction pattern comprises the incoherent summation of monochromatic patterns (MPs) originating from disparate frequency components. In the far field, this can be formally expressed as [23] b(u)=∫eλPλ|FT[λλ0O(λλ0x)]|2dλ,(1)where u and x denote the coordinates in the frequency and spatial domain, respectively, λ denotes wavelength, b(u) denotes the recorded broadband diffraction pattern (BP), eλ represents the quantum efficiency of the detector (e.g., CCD or CMOS) as a function of wavelength, FT signifies Fourier transformation, O(x) is the sample transmission function, and Pλ denotes the illumination spectrum. Constants can be experimentally measured and unified into a single normalization factor aλ=eλPλ.

Subsequently, the diffraction pattern corresponding to a single wavelength component can be articulated as m(u)=|FT[O′(x)]|2,(Aλm)(u)=λ2λ02|FT[O′(λλ0x)]|2,(2)where m(u) represents the diffraction pattern of the shortest wavelength and Aλ denotes the matrix transforming m(u) to the pattern corresponding to λ. This allows us to formulate the problem of monochromatization as a set of linear equations: Am(u)=b(u),A=∫aλAλdλ,(3)or in discrete form as Am=b,A=a0I+a1A1+a2A2+⋯,(4)where A is the matrix transforming MP to BP, m refers to the diffraction pattern corresponding to the shortest wavelength, b denotes the blurred diffraction pattern resulting from broadband illumination, ai=aλΔλ is the relative intensity for each wavelength, and Ai is the transformation matrix that maps the pattern from the shortest wavelength to other wavelengths in the spectrum. Thus, the MP m can be solved with the knowledge of the spectrum and transformation matrices Ai. We propose GM-FTM for MP m resolution in Eq. (4) and subsequently use traditional CDI algorithms to reconstruct the image.

To initiate the discourse on the GM-FTM solver, the explicit representation of transformation matrices Ai has to be determined. We take a diffraction experiment with a beam composed of two wavelengths λ0 and 2λ0 as shown in Fig. 1 for explanatory purpose. The actual simulation and experiments deploy more general sources. For demonstration, a 150×150 Einstein image oversampled to 300×300 [Fig. 1(b)] was used. Proper configuration results in Fraunhofer diffraction for λ0 [Fig. 2(c)], while 2λ0, with doubled oversampling ratio [Fig. 1(d)], produces a finer pattern [Fig. 1(e)], but with high frequency truncated due to the limited size of detectors [Fig. 1(f)]. The objective is to derive transformation matrix A2λ0, which maps the pattern from λ0 to 2λ0 with the original image unknown, specifically, from Fig. 1(c) to Fig. 1(f) with Figs. 1(b), 1(d), and 1(e) unknown.

We propose the transformation matrix derivation utilizing FTM as delineated in Fig. 2. The 300×300 diffraction pattern [Fig. 2(a)] of the Einstein image for λ0 was used in the demonstration. In FTM, the diffraction pattern corresponding to 2λ0 is obtained through Fig. 2 (a)→ (b)→ (c)→ (d)→ (e). First, the oversampling ratio is doubled through zero-padding around the autocorrelation [30] of the original pattern [Figs. 2(b) and 2(c)]. The finer 600×600 diffraction pattern [Fig. 2(d)] was obtained through inverse Fourier transformation (IFT) of Fig. 2(c), which contains extraneous information. This is attributed to the inclusion of zero-padding around the object, a procedure executed to augment the resolution of the diffraction pattern. However, owing to sensor size limitations, a reduced resolution is necessitated when transitioning to 2λ0, resulting in a 300×300 pattern [Fig. 2(e)] with truncated high spatial frequencies. This transformation is also achievable in Ref. [23] via scaling directly through Fig. 2 (a)→ (f) based on pixel area overlaps. Despite its simplicity, this scaling method involves approximations that might impact the pattern precision, resulting in the bandwidth limit at 30%. FTM, in contrast, is more sophisticated but lossless. The insets of Figs. 2(e) and 2(f) dictate finer details achieved by FTM. Furthermore, FTM is beneficial in terms of reasonable autocorrelation [Fig. 2(g)] compared to the scaling method [Fig. 2(h)] and successful image reconstruction [Fig. 2(i)] with a hybrid input-output (HIO) algorithm [31] compared to the failed one for the scaling method [Fig. 2(j)].

Given that the oversampling criteria are satisfied, this manipulation enables us to derive diffraction patterns for longer wavelengths based on those of shorter wavelengths, rather than the reverse. This is pivotal in solving the monochromatization problem. A theoretical benefit of this is the maximal preservation of resolution.

We represent matrix Ai in its operator form A^i and present its transpose in Eq. (5). Comprehensive details on the derivation and subsequent discussion are available in Appendix A.$$\begin{gathered} A^i(m)=CROPi{IFT{PADi[FT(m)]}}, \\ A^iT(m)=IFT{CROPi{FT[PADi(m)]}}. \end{gathered}$$(5)

Consider a broadband pattern obtained by Eq. (4). When transitioning from shorter to longer wavelengths using A^i, the diffraction pattern inevitably stretches, affecting higher spatial frequencies more substantially than lower ones. If the degree of this shift reaches a significant threshold, the sample feature with the corresponding frequency is no longer reconstructable by the traditional CDI algorithm. To formalize this, we introduce the criterion αc, defined as αc=Δλλc·lf,(6)where Δλ and λc denote the bandwidth and central wavelength of the illumination light, respectively; l and f represent the dimensions and spatial frequency of the feature, respectively. The feature can be reconstructed when αc<1. For instance, consider a sample containing a feature with 20 repeating units at a certain frequency. With lf=20, it is imperative that the bandwidth does not exceed 5% to reconstruct this feature. Similarly, the spectrum should be discretized at a step size Δλ<λc/lf in GM-FTM for a successful reconstruction of certain features.

Although the matrix A is large and dense as indicated in Appendix A, causing the inversion to be impractical, the operations Am=∑iaiA^im and ATm=∑iaiA^iTm are relatively straightforward. Consequently, we employ an iterative algorithm, specifically the gradient descent method, to find the vector m. The equation Am=b is affine, rendering the problem convex. This ensures that the gradient descent method will reliably converge to the correct solution. Assuming that in the n-th iteration we have the monochromatic pattern mn, the residual and its gradient for Am=b can be formulated as $$\begin{gathered} ϵn=|Δbn|2=|b−Amn|2, \\ ∇ϵn=2∇(Δbn)·Δbn=−2ATΔbn. \end{gathered}$$(7)

The GM-FTM can be expressed as m0=0,mn+1=mn−α∇ϵn,(8)where α represents the step size, optimally set to 0.7 for the most rapid convergence.

To accelerate the iterative process, momentum can be incorporated: v0=0,vn+1=k(vn−β∇ϵn),mn+1=mn+vn+1,(9)where the dimensionless parameters k=0.6 and β=1 for a stable and reliable reconstruction, used in both simulations and experiments. These optimization tactics are further compared in Appendix B.

After each iteration, we enforce the non-negativity of m to ensure that the result is physically meaningful. This constraint mitigates instability induced by noise, allowing image reconstruction from a single noisy pattern. The pattern can be resolved after approximately 50 iterations. It should be emphasized that the solution m is not derived by subtracting contributions from other wavelengths; rather, it is obtained by fitting. Thus, any wavelengths contribute to m regardless of the wavelength it corresponds to.

The algorithm is executed on a GPU and was tested with an RTX 3060Ti. The computation necessitates approximately 2 s and utilizes 250 MB of GPU memory to analyze a 512×512 diffraction pattern with 50 sampling points. The spectral range for this analysis spans from λ0 to 2λ0, demonstrating significant computational efficiency.

The algorithm is applicable to both continuous and discrete spectral illumination. Furthermore, GM-FTM accommodates large step sizes in the discretization of a continuous spectrum without compromising image quality, as elaborated in Appendix B. With fewer spectrum sampling points the reconstruction can be completed almost instantaneously, enabling real-time imaging.

Hitherto, the discussion has focused on monochromatizing the diffraction pattern assuming that the spectrum of the incident light is known. In this context, within Eq. (4), the parameters ai are known quantities, and the vector m is the variable to be determined. However, one may alternatively approach this equation from a different standpoint, wherein the ai parameters are unknowns to be solved for, and the vector m is a known quantity. The resolution of this inverse problem can serve as the basis for a spectrometer. First, we define an m×n matrix N, composed of the column vectors of monochromatic patterns at different wavelength and their corresponding n dimensional weight vector a: N=(m,A1m,A2m,⋯),a=(a0,a1,a2,⋯)T.(10)

The matrix N can be directly calculated by FTM with the monochromatic pattern m known. It can be found that Na=∑iaiAim=Am.(11)

The target is to solve a in the following equation: Na=b.(12)

Because the dimension of a is relatively limited, in contrast to that of m, it becomes feasible to directly compute a through matrix inversion: a=(NTN)−1NTb.(13)

Considering the spectrum’s non-negativity, a constraint is additionally incorporated in order to suppress noise: Minimize|b−Na|2s.t.ai≥0.(14)

To solve this convex optimization problem, the Lagrangian was constructed as follows: L=|b−Na|2+λTa,(15)where λ is the Lagrangian multiplier sharing the same dimension as a. The gradient algorithm [Algorithm 1] was deployed to find the Lagrange saddle point of the dual problem, where ← represents value assignment and kλ is a properly selected step size.Algorithm 1.

Spectrum Reconstruction

Simulations with two model sources and an experimental IAP source were deployed to assess GM-FTM as depicted in Fig. 3(a). A continuous spectrum following a Gaussian distribution characterized by (μ,σ)=(3.5,0.88) [Fig. 3(a)], resulting in a full width of Δλ/λc≈140%, was used to simulate CDI utilizing isolated pulses. The MNIST database image expanded to 256×256 pixels is the basis for simulation [Fig. 3(b)], with a corresponding Fraunhofer diffraction pattern at λ=1 shown in Fig. 3(c). BP was generated by summing the patterns weighted by the spectrum. After the generation, noise was further considered for snapshot imaging: the BP was first scaled down to prevent pixel overexposure, then followed by integerization. Poisson’s noise was subsequently introduced to each pixel through the equation Poisson(Signal)+Poisson(20)−20, where 20 represents the nominal detector background. This yielded a noisy 16-bit “PNG” image that closely resembled outputs from a 16-bit CCD/CMOS detector as shown in Fig. 3(d1). The detailed analysis on the impact of noises exists in Appendix C. To reconstruct the image, the MP was first derived using GM-FTM [Fig. 3(d2)], followed by a CDI algorithm containing 500 iterations of relaxed averaged alternating reflections (RAARs) [32] and shrink-wrap techniques [33] at 20-iteration intervals. The image is successfully reconstructed as shown in Fig. 3(d3), despite the insufficient high-spatial-frequency signal contributed by short wavelength components only. To evaluate the imaging quality, we contrasted the reconstructed image and the ground truth, then computed the peak-signal-to-noise ratio (PSNR) and structural similarity (SSIM) as shown in Fig. 3(d4). A PSNR value higher than 24 dB and an SSIM score close to 0.9 indicate a relatively successful reconstruction. To reveal the capability of GM-FTM for CDI utilizing attosecond pulse trains, a comb-like spectrum containing odd harmonics from the 5th to 37th order, exhibiting a plateau and a cutoff [Fig. 3(a)], was employed in another simulation. The resulting images as shown in Figs. 3(e1)–3(e4) suggest that GM-FTM allows for zeros in the spectrum. Furthermore, we incorporated the isolated attosecond pulses (IAPs) reported in Ref. [6] for a real case simulation, featuring a continuous spectrum with a full width of Δλ/λc=100% [Fig. 3(a)]. The results are depicted in Figs. 3(f1)–3(f4). Distinct high-spatial-frequency behavior of the solved MPs in these three cases can be attributed from the different noise magnification effects by monochromatization using different spectra. The artifacts of the rectangular borders in Figs. 3(e2) and 3(f2) are due to the sharp rising edge of the spectrum. Inside those borders, the signal is dominant, while outside the border the signal is completely compromised by noise due to the low intensity. In spite of that, due to the denoising tactic in GM, the low-level noise has a limited impact, resulting in the high image qualities. These conclude that GM-FTM is a promising approach for snapshot CDI with both continuous and discrete spectra, as evidenced by the successful reconstructions in all three cases. The PSNR and SSIM in Figs. 3(d4) and 3(f4) indicate that under the same circumstances, the narrower the bandwidth, the higher the image quality.

We conducted experiments to substantiate the efficacy of our methodology. The diffraction pattern was captured using a 16-bit CMOS sensor (ORCA-Flash4.0 LT3) featuring 2048×2048 pixels and a pixel size of 6.5 μm. The quantum efficiency of the sensor, provided by the manufacturer, is depicted as a dashed curve in Fig. 4. The sensor was operated without cooling, yielding an average noise level of 110/65,535 for each pixel. The beam was focused onto the sensor’s plane using an achromatic lens with a focal length of f=100mm to achieve an optimal Fraunhofer diffraction pattern. The sample was positioned approximately 35 mm upstream from the sensor.

Two distinct laser sources and targets were used for experimental validation. The first source was a femtosecond laser, spanning the spectral range of around 500–900 nm. It was generated by a Ti:sapphire laser subjected to spectral broadening via the multiple plate compression method [34]. The pulse width was measured to be approximately 8 fs (full width at half maximum). To prevent damages to the detector, the beam was attenuated before injection into the system. The CMOS exposure time was set to 30 ms for pattern collection with no pixel overexposed. The second source was a distinct supercontinuum laser (YSL Photonics SC-5), characterized by a substantially broader spectrum. Considering that the CMOS quantum efficiency beyond 1000 nm was not calibrated, a low-pass filter was implemented as a precautionary measure to exclusively permit wavelengths within the spectrum of around 480–1000 nm. To obtain a better spatial coherence, the beam went through a 75 μm pinhole before injection into the system. The CMOS exposure time was set to 3 ms. In both cases, the spectra were measured at the sample plain as depicted in Fig. 4.

A USAF 1951 resolution target, comprising only groups 6 and 7, was exposed to the first source. The target’s maximum pitch is 228.1lp/mm. The microscopic image of the target is shown in Appendix D. CDI was also performed using a He-Ne laser for comparative evaluation. The corresponding experimental data and reconstructed images are presented in Fig. 5. Upon illumination by a He-Ne laser, the resulting diffraction patterns manifest highly coherent modulations with irregularly shaped fine structures and streaks as shown in Fig. 5(a). A high-quality image can be obtained by CDI reconstructions, specifically, 1000 HIO iterations with shrink-wrap techniques in 20 iteration intervals, then 20 iterations of RAAR, and finally two iterations of error reduction [35] as shown in Fig. 5(b). However, the broadband spectrum of the femtosecond laser rendered a diffraction pattern of 50% bandwidth obfuscating the coherence as shown in Fig. 5(c), where the fine structures and streaks are completely overwhelmed by the radial trails. The direct CDI reconstruction fails as shown in Fig. 5(d). We then performed GM-FTM + CDI reconstruction on the broadband pattern. Most fine structures of the diffraction pattern were recovered precisely, resulting in a high-quality image reconstruction at the shortest wavelength (480 nm) [Figs. 5(e1) and 5(e2)]. The results were contrasted to that from He-Ne CDI reconstruction [Fig. 5(b)] and PSNR/SSIM were computed for quality evaluation as shown in Fig. 5(e3). A PSNR value of 23.349 dB and an SSIM score of 0.817 indicate a high degree of similarity between them. In principle, under the same experimental setup, a wavelength of 480 nm represents higher resolution in CDI compared to 632 nm from He-Ne lasers. However, due to the ill-posed nature of the problem, the noise is amplified in the monochromatization process, especially in the outer part of the pattern where the signal is mostly compromised. This effect, as a result, slightly reduces the imaging resolution and quality. Nonetheless, the final resolution of Fig. 5(e2) is close to that of Fig. 5(b) as shown in Fig. 6, indicated by the similar width of each bar.

To illustrate the advances we have made, the imaging was additionally performed using published methods for comparison, as discussed in Appendix E.

A specialized target engraved with the letters “ATTO-XIOPM” was exposed to the second source. The microscopic image of the target is shown in Appendix D. The resulting experimental data and reconstructed images are displayed in Fig. 7. The bandwidth of this source reaches ∼60%, causing the blurring effect to be even more significant. CDI reconstruction was first performed using a He-Ne laser as shown in Figs. 7(a) and 7(b) and then using a supercontinuum source as shown in Figs. 7(c) and 7(d). The circular pattern arising from the character “O” and the vertically distributed dots from the horizontal line in the pattern are completely invisible in the broadband pattern [Fig. 7(c)]. Nonetheless, GM-FTM recovered the original monochromatic pattern with the expected features as shown in Fig. 7(e1). The CDI algorithms were deployed on the patterns as described previously in this section as shown in Fig. 7(e2). With an even broader spectrum, we draw the same conclusion from PSNR and SSIM analysis as shown in Fig. 7(e3): the GM-FTM enables CDI of ultra-broadband illumination with high quality and resolution.

An additional experiment was conducted to demonstrate the potential use of this method as a spectrometer, the results of which are illustrated in Fig. 8. To enhance spectral resolution, a grid sample was utilized as depicted in Fig. 8(a). The sample was first illuminated by a He-Ne laser to obtain the monochromatic pattern [Fig. 8(b)], and then illuminated by a coherent beam to obtain the broadband pattern [Fig. 8(c)] for spectrum measurement. The spectrum reconstructed from the two patterns using the proposed method was compared to the one measured from a commercial spectrometer as shown in Fig. 8(d). The reconstructed and measured spectra align closely, validating the efficacy of the technique.

We introduced a new method to address the challenges inherent in ultra-broadband CDI. It facilitates high-resolution imaging across both spatial and temporal dimensions, validated by simulations under varying conditions, including octave-span supercontinuum illumination and a several-octave comb-like spectrum. An experimental demonstration was conducted using two distinct ultra-broadband supercontinuum light sources and in both cases high-quality object images were successfully reconstructed. The experimental setup pushed the spectral-bandwidth-to-central-wavelength ratio, Δλ/λc, to ∼60%, limited only by the available bandwidth. In simulations, we managed to extend this ratio to ∼140%. Reconstruction requires only approximately 2 s using standard office computing hardware, making it ideal for real-time imaging.

This work facilitates the advancement of ultrafast imaging into the attosecond era, enabling temporal resolutions into ∼50 as scale while preserving high spatial resolution. Furthermore, it has enhanced compatibility with feeble sources such as HHG-generated attosecond pulses by harnessing each detected photon across the whole spectral range. Moreover, our method holds potential for particle pulse illumination, enabling attosecond and sub-angstrom resolutions, which are particularly valuable [36] for ultrafast atomic-level imaging.

Additionally, we proposed a new method to accurately reconstruct broadband spectra, offering straightforward and reliable spectral measurement without high-quality gratings. It could be particularly advantageous within the realm of X-rays where those gratings are difficult to fabricate.