As the schematic principle demonstrated in Fig. 1, since a broadband pattern I_b diffracted by a microstructure can be interpreted as a linear superposition of multiple discrete channels of monochromatic diffraction patterns in the source spectrum

where the PSF(λ_i) is a spectrum propagation function from a reference diffraction field $$\sqrt{I_m}$$ (details in Supplementary S1). A broadband measurement I_b with M*N pixels can be represented as the integral of ω(λ)[PSF(λ)]² over the wavelength range, including of M*N multi-linear equations with n parameters. Practically, it is usually impossible to solve ω(λ) by ordinary noniterative methods due to its ill-posed nature. To tackle such instabilities, we perform an improved residual norm minimization tactic applied with a weighting regularization factor, known as Tikhonov regularization

Fig 1  Principle of UDI operation.

a Geometry of UDI operation. A spectral filter is placed to modulate a quasi-monochromatic or broadband illumination the from a supercontinuum source. b PSF mapping from a monochromatic diffraction at wavelength λ_m. A broadband diffraction I_b captured in-situ can be thought of as a superposition of PSFs of I_m at different wavelengths over full spectrum, each multiplied by its corresponding power spectrum weighting ω(λ). The CSS is reconstructed via adaptive Tikhonov regularization. c Monochromatization procedure consists in the inversion of the ill-posed matrix function I_b = CI_m to retrieve the CSM pattern from the solved CSS

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Importantly, since ω(λ) represents neither the probe spectrum P(λ) nor the diffracted radiation, but the final corrected spectrum for the sample's spectral transmissivity function T(λ) and the quantum efficiency of detector QE(λ), ω(λ)= P(λ) T(λ) QE(λ). Thus, the CSS represents the principal component of the final corrected power spectrum, which considers the light-matter interaction between the broadband diffractive radiation through the sample and the diffraction photons read out by the detector over the full spectrum. Thus, there is no need to correct the spectrum for the detector response or make the strong constraint of non-dispersive specimen over the entire spectrum for BCDI. Practically, the CSS is calculated just once and can be applied to various non-dispersive samples. In situations involving dispersive objects, the objects spectral transmissivity can also be obtained from the CSS matrix.

As outlined in mono CDI, the retrieval of the monochromatic pattern can be further reduced to a linear algebra problem, rewritten to a matrix form in simplicity

where m stands for the vector of the monochromatic pattern, b represents the broadband pattern, and C can be regarded as containing the spectrally dependent PSFs over the calculated CSS. Here, we adopt a specific expression to calculate C in one dimension (detailed in Supplementary S2). Note that C is fully determined by the calculated CSS and the dimension of the measured broadband pattern (Fig. 1c). For a 2D diffraction pattern with 512 × 512 pixels, C is a 4D matrix with 256⁴ values. Thanks to the sparsity of the CSS, C is also sparse, with only a few percent of non-zero values. Crucially, once the CSS is computed within the framework of a BCDI configuration, matrix C attains a unique determination, rendering it entirely independent of the specific broadband patterns employed in BCDI. Given its nature as a matrix characterized by high ill-posedness, sparsity, symmetry, and positive definiteness, the direct application of conventional noniterative methods to solve Eq. (2) is typically deemed impractical. Nonetheless, the sparsity and positive definiteness attributes of matrix C render it notably amenable to iterative solutions, particularly through the utilization of the Conjugate Gradient-based descent algorithms. In this work, we employ a normalized BiCGStab

We firstly present a broadband ptychography conducted with a bandwidth of 20% to illustrate the performance of UDI (experimental set-up detailed in Supplementary S4). Initially, we conducted a capture of coherent diffraction at 532 nm with a 3 nm full width at half maximum (FWHM) and broadband diffraction in-situ at any identical position of the USAF target (Fig. 2a). The CSS was then extracted with 87 sparse spectrum channels (green scatters in Fig. 2b). Following the extraction of the CSS, the sparse matrix C is subsequently computed (Fig. 2d), allowing us to monochromatize the broadband measurements. Additionally, we also computed the matrix C from the dense spectrum measurement (Fig. 2c). It is evident that the CSS-derived matrix C contains only 4.1 × 10^7 non-zero values sparsely, compared to the full spectrum-derived matrix C with 6.7 × 10^7 non-zero values. This sparsity is a result of the low-energy trend of spectral leakage in CSS, which causes the CSS-derived matrix C to exhibit a sparser characteristic. It should be mentioned that the matrix C is just performed only once per spectrum and can be used for varying non-dispersive samples.

Fig 2  UDI ptychography at 20% bandwidth.

a The pre-captured quasi-monochromatic diffraction pattern at 532 nm (up right) and the corresponding broadband pattern captured in-situ (bottom left) to calculate the CSS. b The broadband source spectrum (black curve) and the recovered CSS consisting of 87 sparse spectral profiles (green scatters). c. Matrix C generated from the spectrum measurement in a. d A comparison of the sparse matrix C obtained from the CSS in b. e A frame of the broadband pattern. f The corresponding CSM pattern recovered from the broadband data in e. g and h depict the reconstructed images from the broadband ptychography and the proposed UDI ptychography, respectively, after 600 iterations of mPIE

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A comparison of the broadband measurement and the corresponding narrowband pattern captured in-situ reveals that the use of broadband illumination introduces a noticeable decoherence (Fig. 2e). Afterwards, we applied the CSM approach to monochromatize the broadband measurements. The enhancement of coherence between the broadband pattern (Fig. 2e) and the CSM result (Fig. 2f) is readily apparent, revealing that the CSM in UDI is remarkably efficient with only single iteration of monochromatization calculation. Utilizing this efficiency, we monochromatized all the broadband measurements, resulting in monochromatization with notably enhanced coherence and superfast convergence. Subsequently, a comparison between the broadband ptychography and the proposed UDI ptychography were performed using 600 iterations of the mPIE

We further extended the bandwidth of the source spectrum to 41% FWHM, and repeated the UDI ptychography experiment. The CSS can still be precisely computed (green scatters in Fig. 3b) from the diffraction signals (Fig. 3a). It is noteworthy that the spectral extension of the light source results in significant diffraction aliasing (Fig. 3c), ultimately leading to the failure of broadband ptychography convergence (Fig. 3f). The existing mono CDI

Fig 3  UDI ptychography at 41% bandwidth.

a Same as Fig. 2a but narrowband filter at 633 nm and broadband filter with 41% FWHM. b The calculated CSS consisting of 136 sparse spectral profiles (red scatters). c–e A frame of the measured broadband pattern, the corresponding monochromatized pattern via published mono CDI framework

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34 (optimum 25 iterations), and the proposed CSM pattern, respectively. f–h compare the reconstructed results from the broadband ptychography, mono ptychography, and UDI ptychography, respectively

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Additionally, a BCDI application is also showcased in Fig. 4, where the experimental setup resembles that of broadband ptychography. Mentioning that all CDI procedures were performed using the RAAR algorithm with 500 iterations. We first captured a shot of coherent diffraction pattern at 532 nm using a bandpass filter with a 3 nm FWHM (Fig. 4a). The corresponding CDI result is shown in Fig. 4e. To assess the impact of decoherence, we conducted an in-situ acquisition of a broadband pattern with a bandwidth of 20% FWHM, spanning from 480 nm to 600 nm (Fig. 4b). The decoherence nature of the broadband pattern ultimately resulted in a convergence failure in BCDI (Fig. 4f). In comparison, the CSM pattern (Fig. 4c) exhibits a notably enhanced coherence. Simultaneously, the UDI method recovers the probe spectrum, as illustrated by the CSS plotted in Fig. 4c. Furthermore, the UDI reconstructions (Fig. 4g, h) reveal a remarkable improvement in the fidelity of the reconstructions. Importantly, the UDI approach seamlessly combines the inherent advantage of high coherence in monochromatic diffraction with the high photon utilization efficiency offered by full spectrum illumination. Through the enhancement of coherence, the photon utilization efficiency is boosted by two orders of magnitude, leading to a significant reduction in detector acquisition time. Specifically, for broadband illumination, the detector acquisition time is reduced to only 0.05 ms compared to 5 ms for coherent illumination. This improvement in both efficiency and coherence contributes significantly to the overall superiority of the UDI method.

Fig 4  Broadband diffractive imaging at 20% bandwidth

a The pre-captured coherent pattern at 532 nm with 5 ms exposure time. b The corresponding broadband pattern captured in-situ with 20% bandwidth (spectrum ranging from 480 nm to 600 nm) with only 0.05 ms exposure time. c The CSM pattern recovered from the broadband data in b. d A photograph of the Siemens star target, a micro pinhole with100 μm in diameter is employed to intercept the incoming light illumination, resulting in a circular planar wave approximately 100 μm that is incident on the Siemens star. e–g depict the reconstructed patterns obtained from the CDI in a, BCDI in b, and the proposed UDI in c, respectively, after 500 iterations of RAAR. h shows an average image of ten trails of UDI reconstructions

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We are further considering a more general case of broadband diffractive imaging, where the specimen is spectrally dispersive. Most of the current state-of-the-art research on BCDI relies heavily on the strong assumption that the specimen should be non-dispersive over the spectrum

Figure 5 depicts a numerical simulation of BCDI for a spectrally dispersive EUV mask using broadband HHG source with 22% FWHM bandwidth spanning from 12 nm to 15 nm (see the magenta curves in Fig. 5a). The EUV mask's multilayer structure functions as a bandpass spectral filter, selectively reflecting the spectrum centered at 13.5 nm and absorbing the remaining wavelengths

Fig 5  Broadband diffractive imaging with a spectrally chromatic EUV mask.

a Schematic setup for a BCDI application with a chromatic EUV mask. b–d show the reconstructed images after 600 iterations of CDI using three different approaches: direct reconstruction from the recorded broadband diffraction pattern, mono CDI with a pre-knowledge of the broadband HHG source, and the proposed UDI, respectively. e The calculated CSS with only 18 sparse points of the spectrum

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It should be emphasized that the UDI outperforms the existing mono CDI for two main reasons. Firstly, UDI accurately recovers the spectral information of the imaging system. In contrast, mono CDI relies heavily on precise prior spectral measurements. Due to the detector's spectral nonlinearity or the sample's spectral dispersion, there is a significant deviation between the spectrum measurement of the light source and the spectral features in the captured diffraction image, preventing accurate spectral characterization. Besides, UDI also offers comprehensive improvements in monochromatization, coherence enhancement, noise robustness, and wide-spectrum robustness. This results in superior ultra-broadband computational imaging outcomes.

We first performed a numerical investigation to evaluate how CSS affects the accuracy of broadband diffraction pattern fitting. We chose a broadband HHG source with a bandwidth of 22% FWHM as the illumination source in numerical calculation. Our analysis involved measuring the fitting error of a broadband diffraction pattern while varying the spectral sampling channels between 25 to 600 and the detector noise levels ranging from 20 dB to noisefree, as detailed in Fig. 6. The detector noise is a mixture with Gaussian noise and Poisson noise, following the detector noise model established in our previous work

Fig 6  Broadband diffraction fitting error analysis.

a Varying sampling intervals in HHG spectrum ranging from 12 nm to 15 nm. b The evolution of the broadband diffraction pattern fitting MSE with varying spectral sampling intervals in a and varying noise levels

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We characterized the performances of CSS calculation with broadband diffractions under varying levels of noise. The "HSUT" logo (inside Fig. 7g) was used to generate ideal coherent diffraction data at 13.5 nm. Subsequently, a corresponding broadband pattern was obtained by linearly superimposing 600 discrete channels of monochromatic diffraction patterns in the HHG source spectrum. To replicate real-world scenarios, the diffraction datasets were synchronized with a 16-bit camera and added with varying levels of mixed detector noises.

Fig 7  CSS accuracy analysis.

a–e show the calculated CSS from the noised broadband patterns with varying detector noises ranging from 60 dB to 20 dB with 10 dB interval, respectively. f compares the MSE between the fitted broadband patterns generated from the calculated CSSs in a–e with the ground-truth spectrum, while varying the spectral sampling numbers. g A presentive shot of coherent diffraction pattern at 13.5 nm with 40 dB detector noise. h. The broadband diffraction pattern generated from the broadband HHG source in ground-truth with 600 discrete sampling points (black scatters in Fig. 6a), while the corresponding fitted broadband pattern from the calculated CSS under a noisy condition in e is presented in i, and its corresponding fitting error is displayed in j

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Images inside Fig. 7a–e depict the broadband patterns with varying levels of noise, and the corresponding calculated CSSs are plotted simultaneously. Seeing that the calculated CSSs exhibit a strong correspondence with the spectrum within fewer than 30 spectral channels, even in the presence of substantial noise. We employed the sparsely calculated CSSs for fitting the broadband patterns, and subsequently analyzed the fitting MSEs and fitting error rates, as depicted in Figs. 7i, f, and j, respectively. The fitting MSEs consistently demonstrate minimal values across different noise levels, and the fitting error rate remains below 10%, displaying uniformity (Fig. 7j). These findings indicate that the calculated CSS effectively aligns with the primary components of the full-spectrum radiation, maintaining high compressed sparsity and robustness against noise.

We utilized the recovered CSS with a bandwidth of 22% and 29 sparse spectral channels to create the sparse matrix C (Fig. 7a). Known that C is a sparse diagonal matrix with a sparsity of 0.03%. There are four identical sets of data distributed along the diagonal of matrix C. This is due to that the scaling of the spectral PSF is the same for all four quadrants. Figure 8b, c show the matrix C created from the CSS and the spectrum measurement, respectively for comparison. The matrix C created from the CSS has a distribution similar to that from the spectrum measurement, with only small localized differences where the sparse C is slightly non-uniform due to spectral leakage in CSS. However, these artifacts are not dominant in monochromatization due to the superiority of the proposed UDI method.

Fig 8  Broadband diffraction monochromatization.

a The calculated CSS from the broadband pattern with 60 dB detector noise. b A 2D matrix C calculated from the CSS with 29 sparse spectral points in a. c Similar with b but calculated from the full spectrum with 600 sampled spectral channels for comparison. d The recovered monochromatized patterns after 3 iterations of CSM and mono, respectively

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The generation of the matrix C from the CSS allows us to recover the optimal monochromatization from a broadband measurement. Figure 8d compares the monochromatized diffraction patterns between the mono method

We have introduced a powerful UDI method for ultra-broadband diffractive imaging. Our research comprehensively addresses the key challenges of current state-of-art BCDI. By employing UDI, we successfully achieve a significant enhancement in coherence of ultra-broadband diffraction patterns. We provide a detailed explanation of the theory and design process for our UDI method in broadband diffractive imaging, which has been experimentally verified. It presents a natural sort of superiorities:

Firstly, UDI represents an advancement in ultra-broadband diffractive imaging with an unknown probe spectrum, while simultaneously recovering the spectrum information of the diffracted radiation. UDI overcomes limitations posed by constraints on spectrally non-dispersive specimens across a wide spectrum. It is inherently applicable across a broad wavelength range and eliminates the need for prior spectral knowledge, particularly crucial for applications in EUV and soft X-ray ranges where the absorption edge effects of materials are more pronounced. Figure 9.

Fig 9  UDI workflow.

The sparse matrix C is constructed as a pre-calculation process in the broadband CDI application, as indicated by the red dashed box. Subsequently, the measured broadband pattern b is monochromatized using the pre-calculated matrix C, resulting in the monochromatic pattern m, as depicted in the blue dashed box. Notably, this monochromatization process does not require any prior knowledge of the spectrum

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Secondly, the UDI achieves coherence-enhanced, superfast-solving, and noise-robust monochromatization under ultra-wide spectral illumination. It efficiently utilizes the entire flux from a broadband source and results in a significant reduction in data acquisition time. This makes UDI highly beneficial for ultra-broadband imaging, offering a nearly fourfold improvement in bandwidth compared to existing mono CDI benchmark. The monochromatization in UDI operates with high efficiency, achieving optimal results within the initial iteration, which is 30 times faster than the state-of-the-art CGLS method

Nevertheless, certain critical matters still require clarification and warrant further research. Despite the potent ultra-broadband imaging capabilities exhibited by the UDI when handling unknown probe spectrum, it is crucial to acknowledge that UDI relies on the Fresnel diffraction approximation. Consequently, its applicability may be constrained in scenarios involving multi-layer or multi-scattering samples, potentially impeding its capacity to fully leverage the entire performance in such intricate situations. Additionally, the optimization algorithm for monochromatization and the spectral quantum efficiency of the detector may also limit further spectral bandwidth extension in UDI. Our UDI experiments demonstrate ultra-broadband diffractive imaging with a relative spectral bandwidth exceeding 40% FWHM. This bandwidth is currently limited by the detector's quantum efficiency, not by the UDI algorithm itself.

As described in Supplementary S1, Eq. (2) can be treated as an ill-posed multi-variable linear regression problem, which can be solved by Tikhonov regularization, to prevent overfitting and suppress the noise signals during reconstruction

where A is a given M*N*n matrix with elements of each column of a flattened PSF(λ_i) matrix in 1D array corresponding to the i-th slice of spectrum and b is a vector of a broadband diffraction flattened in 1D array, ω is the vector of unknown spectrum coefficients for the function. Γ is the regularization coefficient that controls the weight given to minimization of the side constraint relative to minimization of the residual norm. ||.||₂ is the l₂ norm. Note that the efficiency of these estimates depends on appropriately choice of the regularization coefficient $$\mathrm{\Gamma }$$, which should be carefully selected to balance the results of robustness and resolution. In this work, we employ a generalized Cross-Validation statistic to make the balanced choice of Γ adaptively

where I is the identity matrix and the operator Tr sums elements on the main diagonal of a matrix. As a result, we can have the CSS estimates $$\hat{\mathit{\omega }}$$ from solving Eq. (4)

The algorithm of the CSS calculation is described in detail in the supplementary information in ref.

As described in Supplementary S2, C is a sparse, symmetric, and positive definite matrix. The sparsity and positive definiteness of matrix C make Eq. (3) particularly well-suited for iterative solutions using BiCGStab. This algorithm performs as an implementation of an orthogonal projection technique onto the Krylov subspace. It involves minimizing the least squares problem to achieve the desired monochromatization. Implicitly, BiCGStab solves not only the original system $$C\mathit{m}=\mathit{b}$$ but also a dual linear system $$C^T{\mathit{m}}^{\mathbf{\ast }}={\mathit{b}}^{\mathbf{\ast }}$$ with $$C^T$$.

where $${\mathcal{K}}_k\perp {\mathcal{L}}_k$$, denotes the Krylov space orthogonally

The flowchart of BiCGStab algorithm is detailed in Supplementary S3.

Step 1: Pre-capture a shot of broadband pattern and a quasi-coherent pattern in-situ, respectively.

Step 2: Calculate the CSS from the measurements in Step 1 via adaptive Tikhonov regularization.

Step 3: Calculate the sparse matrix C which contains the spectral information of CSS.

Step 4: Monochromatize the broadband patterns in broadband imaging experiments via BiCGStab along with the pre-calculated sparse matrix C in Step 3.

Step 5: Output the optimal diffraction pattern m with enhanced-coherence in broadband imaging applications.