Single-shot hyperspectral wavefront imaging

Domain: physics
Published: 2025-12-04
Editorial outcome: major_revision

Abstract
Single-shot hyperspectral wavefront sensing is essential for applications like spatio-spectral coupling metrology in high-power laser or fast material dispersion imaging. Under broadband illumination, traditional wavefront sensors assume an achromatic wavefront, which makes them unsuitable. We introduce a hyperspectral wavefront sensing scheme based on the Hartmann wavefront sensing principles, employing a multicore fiber as a Hartmann mask to overcome these limitations. Our system leverages the angular memory effect and limited spectral correlation width of the multicore fiber, encoding wavefront gradients into displacements and the spectral information into uncorrelated speckle patterns. This method retains the simplicity, compactness, and single-shot capability of conventional wavefront sensors, with only a slight increase in computational complexity. It also allows a tunable trade-off between spatial and spectral resolution. We demonstrate its efficacy for recording the hyperspectral wavefront cube from single-pulse acquisitions at the Apollon multi-petawatt laser facility, and for performing multispectral microscopic imaging of dispersive phase objects.

Introduction
Wavefront sensing and quantitative phase imaging are widely used for beam metrology, adaptive optics1 and biomedical imaging2. A wavefront sensor (WFS) typically measures the wavefront of a single monochromatic beam. For broadband beams, it is assumed that the wavefronts are achromatic, otherwise, the WFS can only measure a meaningless spectrally-averaged wavefront. Spectrally resolved WFS is typically required to quantify spatio-spectral coupling in high power laser metrolgy3, to study light matter interaction in plasma4, to image material dispersion5 or to measure hemoglobin concentration in living organisms6. In these applications, single-shot measurements are preferable or even required, especially to monitor fast events like in flow cytometry7, in pump probe experiments4, in light matter interaction8, or to characterize single laser pulses8,9. Assisted by a chirped pulse, ultrafast events can be acquired4,8. Also, the performances of ultrafast and high-power laser systems critically suffer from non-linear optical aberrations10. While scanning techniques can arguably characterize the spatiotemporal pulse profiles in high repetition rate systems3, they are incompatible with low repetition rate systems, such as petawatt and multi-petawatt-class lasers, due to potential drift with time and limited pulse-to-pulse reproducibility. The key challenge in this context lies in the spectral discrimination of wavefronts. In this manuscript, we distinguish between “multispectral” and “hyperspectral” techniques – referring, respectively, to approaches that resolve wavefronts over discrete spectral lines versus those spanning contiguous spectra, following the criterion established by Goetz11. This distinction is justified by the increased difficulty of sensing wavefronts across contiguous spectra compared to a discrete set of wavelengths. To manage continuous spectra, a spectrum-sampling Fabry–Perot interferometer was proposed in combination with a pair of gratings, which splits the spectral lines onto a conventional WFS, thereby enabling single-shot multispectral wavefront sensing over four spectral lines9. Although spectral sampling using a Fabry–Perot interferometer is efficient, the transmitted spectrum is sensitive to local wavefront tilt. Multispectral phase imaging could also be achieved by digital holography by encoding spectral information in the k-space12 or even without the need for a reference arm by combining a wavefront sensor with a multispectral camera13. Low-resolution (12 sampling points) and low-throughput single-shot hyperspectral wavefront sensing can be achieved using a coded aperture in combination with a phase retrieval algorithm14. More recently, hyperspectral phase maps could be rebuilt experimentally by combining a hyperspectral camera (so-called CASSI) with a phase diversity technique4. In a very recent publication, hyperspectral phase imaging could also be demonstrated by combining a diffraction grating with a micro-lens array15. So far, hyperspectral high-resolution WFS has only been investigated through numerical simulations combining a CASSI with a lateral shearing interferometer16. As a general principle, cascading existing techniques for achieving multiplexed optical sensing results in increased experimental complexity and spatial footprints, so hindering practical use and dissemination of the technique.

Here, we propose a single-shot hyperspectral wavefront sensing scheme that combines both spectral discrimination and wavefront sensing functionalities into a single mask imaged at a close distance from a camera sensor. Our approach thus builds on the same principles underlying conventional WFS17,18 in which the local wavefront tilts are encoded as local translations of the intensity pattern after propagation to the camera. This design has become one of the most widely used approaches in metrology and quantitative phase imaging19–21 because of its simplicity, compactness, single-shot capability, robustness to vibrations, and compatibility with broadband light sources. The application of this same principle to the multispectral/hyperspectral working regime has faced the challenge of spectral discrimination when dealing with broad spectra13.

Specifically, our system leverages both the so-called “angular memory effect”22 and the limited spectral correlation width23 occurring at a short distance from the output of a multicore fiber bundle (MCF). The angular memory effect encodes wavefront gradients in intensity pattern displacements24, while the limited spectral correlation width ensures distinct pattern encoding. The proposed instrument thus inherits the advantages of conventional WFS with only a modest increase in computational post-processing complexity and no need for regularization. We demonstrate the quantitative nature of our approach on well-defined optical systems. Next, we demonstrate the practical usability of our instrument by characterizing the hyperspectral wavefront cube from a single-pulse beam of multi-PW laser line at the Apollon laser facility, captured in a single-shot image acquisition. Finally, since our system relies on Hartmann deflectometry, it enables tuning the trade-off between spatial resolution and phase sensitivity24. We thus show that our implementation also enables single-shot, multispectral, quantitative phase microscopy with a high spatial resolution.

Results
Conceptual framework

Suitable optical mask for single-shot hyperspectral WFS. In the quest for instrument simplification, spectrally resolved wavefront sensing calls for a single optical element that simultaneously provides both spectral discrimination and wavefront sensitivity.

Several masks can be placed in the close vicinity of a camera that result in wavefront-sensitive intensity patterns: arrays of holes, microlens arrays17, meta-micro-lens arrays25, thin diffuser24, diffraction gratings26. A key characteristic for wavefront reconstruction is that a local tilt of the wavefront is transmitted through the mask, such that a wavefront tilt at the entrance of the mask results in a local translation of the intensity pattern at the camera plane. This principle, generalized in the context of scattering media and coined “memory effect”22, is satisfied by optical elements whose transmission matrix is diagonal in the canonical basis.

The spectral sensitivity of a mask is characterized by the spectral correlation width of the intensity patterns at the camera. All the masks listed above provide highly achromatic WFS properties. Spectral decorrelation may arise from chromatic dispersion within the mask itself or from free-space propagation over the distance d between the mask and the camera. However, axial dilation of intensity patterns due to free-space propagation27,28 contributes only marginally to spectral sensitivity when the mask is positionned at millimeter-scale distances24. To achieve hyperspectral WFS, the use of spectrally dispersive optics is therefore required. For interference effects to be wavelength-sensitive, these optics must involve a diversity of large optical path lengths. Nanometer-scale spectral sensitivity is typically achieved using complex media, such as multiply scattering samples29 or multimode fibers30. Notably, diffraction gratings are achromatic at short distances19 and thin diffusers–which we have previously shown to enable effective WFS–are only weakly dispersive24. Furthermore, complex media have been shown to be well suited for multiplexed spectral31,32 and polarimetric33 imaging, and even for multiplexed wavefront sensing in a single spectral channel34.

Therefore, optics that satisfy both spectral and angular properties required for achieving spectrally resolved wavefront sensing are not numerous. For instance, thick multiply scattering diffusers exhibit improved spectral sensitivity at the expense of degraded angular memory effect as thickness increases, drastically limiting the applicability to WFS22. In contrast, MCF exhibit both large angular memory effect35 and narrowband spectral correlation properties36. MCF exhibit a diagonal transmission matrix if fibers are perfectly monomode and uncoupled. In practice, many bundles are composed of few-modes fibers, so limiting the angular range of the memory effect. The physical mechanism behind the limited spectral correlation width in these fibers is still unclear but involves statistical dispersion in longitudinal wavenumbers in individual fibers due to variations in their diameters and potentially coupling between fibers. This dispersion is not specified by manufacturers and probably results from a lack of control in the fabrication process.

Image formation. The speckle pattern created by a scattering medium exhibiting an angular memory effect22 encodes the incident wavefront gradients into local speckle grains displacements at a short distance d24,34,37–40. For a monochromatic light beam, the relationship between a reference speckle pattern R(x, y) obtained under plane wave illumination and a speckle pattern M(x, y) obtained with unknown beam wavefront and intensity profile is:

M(x,y)=α(x,y)R[x+u(x,y),y+v(x,y)] (1)

where α(x,y) is the beam intensity and u(x,y), v(x,y) the components of the vector displacement map induced by the wavefront gradient. The spatial-phase gradient ∇ ⊥ φ is related to u(x, y) and v(x, y) according to:

u(x,y)=k0d ∂φ/∂x (2)
v(x,y)=k0d ∂φ/∂y

Measuring the displacement map (u,v) thus enables the reconstruction of the wavefront φ by a numerical integration step41–43. In addition, the propagation through a complex medium encodes spectral information into uncorrelated speckle patterns23. The possibility to unmix such statistically independent speckle patterns under broadband illumination enables spectroscopy29,30 and even hyperspectral intensity imaging31,32. In these systems, the spectral resolution is determined by the number of interfering spatial modes, which typically scales with the spectral correlation width of the scattering medium, δλmedium, defining the spectral detuning necessary for speckle decorrelation to occur44.

As argued above, to achieve single-shot spectrally resolved wavefront sensing, the Hartmann mask must be carefully chosen. Here we suggest using a MCF which demonstrated both a wide angular memory effect35 and a limited spectral correlation width36,45. The latter can be tuned by adjusting the length of the MCF (see Supplementary Fig. S4). As illustrated in Fig. 1a, b, when a broadband light beam illuminates the MCF, the speckle pattern M(x, y) is the sum, over the beam spectrum, of the monochromatic reference speckle intensity patterns R(x, y, λ), weighted by the spatially-dependent power spectrum amplitude of the beam intensity α(x, y, λ), and distorted according to the incident hyperspectral phase gradients ∂φ/∂x, ∂φ/∂y according to Eq.(2) (Fig. 1b). Therefore, because of the spectral diversity of speckles, both the spectro-spatial phases and amplitudes information of the incident field are encoded and multiplexed in the single speckle pattern image M(x, y). Since spectral information is spatially encoded in the speckle patterns captured by the camera, increasing spectral resolution requires allocating more camera pixels to each sampling point, which in turn reduces the number of spatial sampling points on the wavefront sensor. Ultimately, this trade-off between the number of spectral and spatial modes is made with the contraint that the total number of modes is limited by the number of camera pixels (see Supplementary Section S2).

Reconstruction algorithm. The inverse problem in Eq.(3), summarized in Fig. 1b, is solved doing the two following assumptions: (i) the displacements u(x,y,λ) and v(x,y,λ) are sufficiently small, relative to the speckle grain size, to justify a first-order Taylor expansion of Eq.(3). (ii) Both the beam amplitude α and wavefront gradient (i.e. (u,v)) can be considered as constant within a given neighborhood. This latter assumption is at the basis of the Lucas–Kanade optical flow algorithm46, a variant of which we use here. Over this neighborhood, Eq.(3) can be re-written in the discretized matrix formalism as

M=ReX+noise (4)

where Re is the matrix containing the reference speckle patterns and their derivatives, and X is the vector containing the amplitude and the wavefront gradients at each wavelength λi. For a discrete set of wavelength (i.e. for multispectral imaging), the matrix Eq.(4) is solved over a sliding Gaussian window based on a least square minimization by computing the Moore–Penrose pseudo-inverse X*=(ReT Re)−1 ReM. When a contiguous spectrum is considered (i.e. for hyperspectral imaging), Eq.(4) is solved over a set of patches covering the camera surface, using a truncated singular value decomposition (SVD) estimation47 where the low singular values are filtered out with a half-Gaussian function of width σSVD48. The first order Taylor expansion performed to linearize Eq.(3) is only valid over displacements of the order of one speckle grain size (Supplementary Fig. S5b). To make the resolution robust to larger displacements, we implemented an iterative multi-scale registration algorithm49. The principle of the multi-scale approach consists in adjusting the speckle grain size at each iteration by Gaussian-filtering all images (M and R), thereby ensuring the first order approximation remains valid at each step by distorting the reference images according to the estimated larger-scaled displacement maps. The filter width σscale is then progressively decreased to measure finer displacement structures while preserving robustness to noise (Supplementary Fig. 5c). The detailed description of the inversion process, including the definition of the neighborhood and wavelength sampling, can be found in the Methods section and the algorithm workflow is illustrated in Supplementary Fig. S6.

After solving Eq.(4) over a dense set of patches, a numerical integration of the displacement maps (ui, vi) is achieved in the Fourier domain. To avoid artifacts associated with periodic boundary conditions, Fourier integration is achieved after symmetrizing the gradient vector field42.

Experimental results

Optical characteristics of our system. The built-up hyperspectral wavefront sensor consists of a few-mode MCF (FIGH-100-1500N, Fujikura, Japan), a relay imaging system (×10, 0.3 NA microscope objective and a 300 mm tube lens), and a camera (PCO Panda 4.2). The spectral sensitivity results from the statistical dispersion of optical path lengths between cores, which is an inherent characteristic of MCF45. In our case, the angular correlation of the few-mode MCF, measured by tilting a collimated beam at the entrance of its input facet, exhibits a ≃30% drop for a tilt of 20 mrad (See Fig. 1c and Sec. for technical details). This limited angular memory effect reduces our ability to recognize speckles and reconstruct wavefronts. Our experiments were conducted within this range, thereby validating this lower bound of the dynamic range for our wavefront sensor implementation. This limited angular correlation is attributed to the fact that the MCF is made of few-modes fibers (≃4) and not monomode fibers. As detailed in the Supplementary Information, once interference occurs between even a few cores, the spectral correlation width is determined solely by the intrinsic properties of the fiber (see Supplementary Fig. S4). For a distance d=88 mm between the fiber image plane and the camera, we experimentally measured correlation widths on the order of 2 nm across the entire 500−800 nm spectral range (Fig. 1d). Both the angular memory effect and the resulting spectral correlation widths are well suited for hyperspectral wavefront sensing applications.

The experimental parameters must then be chosen based on the desired measurement characteristics. In particular, the required wavefront sensitivity defines the optimal distance d between the MCF and the camera. Given that our algorithm can reliably detect displacements as small as 1/50 of a pixel (see Supplementary Fig. S5b), increasing d enhances sensitivity. However, due to free-space propagation, this same distance also sets the spatial resolution of the wavefront sensor, which scales as NA⋅d, where NA is the numerical aperture of the MCF. Finally, the number resolvable spectral channels is linked to the ability to solve the inverse problem, which depends on the rank of the calibration matrix that encodes the reference patterns across the desired spectral range. According to the Marchenko-Pastur law for random matrices (See Supplementary Section S2), the distribution of singular values is governed by the ratio r=Nλ/M, where Nλ is the number of desired spectral channels and M= NA/πk0P is the number of independent spatial modes sampled within a given patch, with k0 the central wavenumber and P the patch size that should match the spatial resolution NA × d. The upper bound for the condition number κ, which reflects the invertibility and stability of the problem, is given by κ= 1+√r / 1−√r (5).

Here, we demonstrate two implementations of our instrument corresponding to two trade-offs between spatial and spectral resolutions:

1. One hyperspectral wavefront sensing modality measuring the first few Zernike wavefront modes over ≃20 spectral contiguous channels over a continuous 100-nm-wide spectrum.

2. One quantitative multispectral and high-resolution wavefront imaging modality (220×220 pixels) performed at three discrete wavelengths in a microscopic imaging configuration.

Quantitative hyperspectral wavefront sensing. To verify the instrument’s quantitativity, a first setup was implemented to perform hyperspectral wavefront sensing over a continuous spectrum using simple test optical systems. The test optical systems under consideration were: three wedge prisms made of NBK7 material (with deviation angles of 2∘, 4∘, and 6∘ at 633 nm), and two afocal lens-pair systems (f=50 mm) in NBK7 and SF11 materials. A supercontinnum laser coupled with a computer-controled filter box was used as a light source. The output plane of the optical system under test was imaged onto the MCF facet using a 1:3 demagnifying telescope. The distance d≃90 mm between the fiber output image plane and the camera was finely calibrated over the whole 100-nm-wide spectrum of interest, so taking into account chromatic aberrations of relay optics. For numerical reconstruction, the patch size at the camera was qualitatively adjusted to match the diffraction pattern of a single fiber after free space propagation, resulting in the reconstruction of phase and intensity maps of resolution 16×16-patches across a 100 nm spectral range with a spectral resolution of 5 nm. Next, the wavefronts were reconstructed both from the single broadband multiplexed images and from sequential images acquisition while scanning the spectrum. Throughout the article, we use the following terminology: multiplexed acquisition refers to a single-shot measurement of the spectrally resolved wavefront data cube, while sequential acquisition refers to a series of quasi-monochromatic wavefront measurements obtained by scanning the spectrum. More details about the experiment and data processing are given in the Methods section. An illustration of the reconstructed multiplexed and sequential wavefronts is shown in Fig. 2b for the 2∘ prism at 640 nm, while Fig. 2d similarly provides an illustration for the SF11 afocal system at 560 nm. The wavefront accuracy evaluated by computing the root mean square error (RMSE) between the expected and measured wavefronts over the whole spectrum reconstructed from the prisms experiments, is 1.3 nm (i.e. λ/450 at 600 nm) for the sequential reconstruction and 9.1 nm (i.e. λ/65 at 600 nm) for the multiplexed reconstruction.

Finally, we extracted the tip/tilt and defocus components from the wavefronts, for the prisms and afocal systems, respectively, and measured the differential chromatic contribution, taking 600 nm as an arbitrary reference wavelength. These results are shown in Fig. 2a for the prisms, and in Fig. 2c for the afocal systems. Results obtained from multiplexed measurements exhibit excellent agreement with the ones obtained from sequential data. These experimental results were further compared with values expected from numerical simulations (based on the geometrical specifications of optics and the tabulated refractive indices of the materials). The theoretical spectral defocus induced by the afocal systems were computed within the frame of the thin lens approximation. All measurements show excellent agreement with one another and together with theoretical expectations and demonstrate the quantitative nature of our hyperspectral wavefront sensing approach.

Single pulse reconstruction at the Apollon laser facility. Next, we installed our single-shot hyperspectral wavefront sensor in the Long Focus Area (LFA) at the Apollon multi-PW laser facility50 (Saint-Aubin, France). The full footprint of our wavefront sensor does not exceed 100 cm2 and could be directly integrated into the diagnostics setup of the Apollon laser in a few hours only. Wavefront measurements were carried out through the 1 PW beam-line, using a test laser beam having 50 mJ pulse energies at 10 Hz repetition rate (activating 2 amplification stages out of 5). Since following the very same optical path as in the full-energy configuration (nominal pulse-energy of 15 J at a rate of 1 shot/min), the laser beam was carrying the main static residual spatio-temporal coupling aberrations of the system. The 140 mm diameter laser beam was demagnified down to a beam diameter of 0.7 mm at the input facet of the MCF with achromatic telescope relay systems. The hyperspectral wavefront sensor was calibrated in situ over the spectral range 650−1000 nm with a resolution of 1 nm. More details about the system and its calibration can be found in the Methods section.

The multiplexed speckle image is shown in Fig. 3a. The beam exhibits a central hole resulting from the sampling mirror of the current LFA configuration (see Methods). From this single image, the ≃40-nm-wide spectrum of the laser is retrieved (Fig. 3b) together with hyperspectral intensity beam profiles (Fig. 3c) showing very good agreement with the real Apollon beam characteristics. The hyperspectral optical path differences (OPD) are also retrieved (see illustration of the OPD in Fig. 3d for the wavelength 825 nm). Before integration of wavefront gradients, the missing information in the central zero-intensity region was filled in using a median filter. The measured wavefronts (Fig. 3d) mostly present low order aberrations. The differential chromatic aberrations between 785 nm and 825 nm, shown in Fig. 3e, mostly exhibits chromatic defocus but negligible spectral tilt, so demonstrating the excellent alignement of the Apollon laser system (up to a few μrad at the WFS plane). The residual spectral tilt and defocus are plotted in Fig. 3f, g. The chromatic defocus is measured to be about 50 nm PtV (λ/16) over the FWHM-bandwidth of the Apollon pulses, corresponding to a ±zR/8 longitudinal depth of focus, with zR the Rayleigh length of the focused beam. This value is also in good agreement with the theoretically expected longitudinal chromatism of the Apollon beam transport system employing in its front-end part relay imaging telescopes based on simple singlet fused silica lenses for an aperture up to 18 mm in diameter51.

To characterize laser pulses in the temporal domain, measuring the spectral phase is required, at least within a sub-aperture of the near-field beam50. In the Apollon laser system, the spectral phase is routinely measured and optimized for the central portion of the beam. This enables pulse compression nearly to the Fourier transform limit (FTL). The resulting focus at the focal plane and the electric field time traces at the center of the focus and on one lobe are presented in Fig. 3h, i assuming a flat spectral phase at the center of the focus and preserving the missing hole in the amplitude pupil data. The power spectrum was apodized with a Tukey window with α = 0.35. The measurements realized in this study are in agreement with previous multi-shot characterization campaigns and indicate negligible peak intensity losses in the order of 15% compared to an ideal focus. In contrast to multi-shot acquisition systems, our instrument now enables the characterization of potential non-linear spatio-temporal coupling at the single-pulse level.

Multispectral quantitative phase microscopy. Single-shot multispectral imaging is also of interest for microscopy applications. We thus implemented our system in high-spatial resolution working regime by reducing the distance d down to ≃6 mm between the image of the fiber-bundle output and the camera. The distances d were accurately measured over the full field of view (Supplementary Fig. S7). The sample was illuminated in transmission mode with a custom-built source based on three dielectric bandpass filters (480/20 nm, 580/10 nm and 690/10 nm), and imaged onto the PCF using a ×20 magnification and 0.4 NA microscope objective. The PSF sampling factor by the MCF is dictated by the 4.5 μm mean nearest-neighbor distance between fiber cores and varies in the range between 2.7 (at 480 nm) and 3.8 (at 690 nm). Eq.(4) was solved using a Moore–Penrose pseudo-inversion over a sliding Gaussian window with standard deviation σGauss = 4 pixels. The sample was prepared, consisting in 5.2 μm PMMA beads immersed in an aqueous solution. Sucrose was added to the solution to reduce the refractive index mismatch between water and PMMA. A chromophore (Blue Brillant FCF) was also added in order to introduce a controled amount of spectral dispersion52. All details regarding the optical system, the sample preparation and the algorithm can be found in the Methods section. Then, multiplexed and sequential multispectral wavefront imaging of the sample was carried out.

The single-shot multispectral intensities and OPDs are shown in Fig. 4a. Noteworthy, the imaging system between the sample and the WFS exhibits a slight chromatic aberrations since the red channel (690 nm) appears unfocused. Based on both the measured intensity and OPDs, numerical refocusing is performed (Supplementary Fig. S8). Comparison of OPD profiles obtained by sequential and multiplexed measurements can then be done and confronted to theoretical expectations based on the solution composition and the specified geometric characteristics of the beads (Fig. 4b). The only missing parameters, the exact sucrose and chromophore concentrations, were fitted from experimental data. The dispersion of the sample clearly appears in the images shown in Fig. 4c where the difference in OPD at 480 nm and 690 nm is displayed. Finally, the peak OPDs are plotted in Fig. 4d both from sequential and multiplexed data, demonstrating very good agreement between experimental results and the spectral dispersion expected from theoretical modeling. The scattering observed in the measurements of refractive indices is due to the dispersion in the sizes of the micro-beads (Supplementary Fig. S9).

Methods
Optical design of the wavefront sensor

The output facet of a 1-cm-long multicore few-mode fiber (FIGH-100-1500N, Fujikura, Japan) is imaged at a distance d from a camera (PCO Panda 4.2) with a microscope objective (×10, NA=0.3) on a translation stage and a tube lens (f = 300 mm and f = 200 mm for the hyperspectral system and multispectral system respectively), as illustrated in Fig. 1a. The fraction of transmitted light energy through the MCF is 95%.

Light sources

For hyperspectral WFS, the optical system is illuminated using narrow linewidth collimated beams generated by broadband laser sources and filtered using computer-driven filter boxes. For the prism/afocal experiment, the supercontinuum source (Leukos, Electro Vis) and the filter box (Leukos, Bepop) provided a minimum bandwidth of ≃5 nm in the spectral range between 500 nm and 800 nm. At the Apollon laser, a Ti:Sa laser (Rainbow, Femtolasers Produktions GmbH, Austria) covering the spectral range 650−1000 nm was used for calibration, in association with a custom-built monochromator. The monochromator consisted of a pair of blazed gratings, an afocal telescope and a slit mounted on a motorized translation stage. After the monochromator, the beams were spatially filtered using single mode fibers and re-collimated before being sent on the input facet of the MCF for the test optics. The reference speckle patterns Ri were then sequentially recorded over the given spectral ranges by steps of 1 nm. Broadband illumination could be obtained by tuning the linewidth up to 100 nm for the commercial filter box, and simply by removing the slit from the custom-built monochromator.

For the multispectral wavefront imaging experiment, a white halogen lamp, spatially filtered by a pinhole and collimated, was split into three optical paths. The three illuminating wavelengths were obtained by inserting off-the-shelf available bandpass filters (480/20 nm, 580/10 nm and 690/10 nm), placed in each path before recombination using beam-splitters.

Spectral calibration of the hyperspectral WFS

Multicore fibers (MCFs) exhibit a limited spectral correlation width due to the inherent statistical dispersion of optical path lengths between cores45. The spectral correlation width of the MCF was measured over the full spectrum by measuring the speckle contrast C as a function of the central wavelength of the 5-nm-wide laser line. The contrast is directly related to the number of independent speckles Nλ present in the image as: Nλ=1/C223,59. The speckle contrast decreases as the number of independent spectral speckles increases. We thus conventionally define the spectral correlation width of the MCF as δλmedium = σlaser/Nλ where σlaser is the linewidth of the laser used. We initially purchased a 50-cm-long MCF which we shortened down to 1 cm to tune its spectral correlation width in accordance with the requirements of our applications (see Supplementary Fig. S4). As shown in Fig. 1d, we measured correlation widths of the order of 2 nm over the full spectrum in between 500 nm and 800 nm. The spectral width of the MCF ultimately determines the maximum spectral resolution of our hyperspectral WFS.

Angular memory effect & calibration of the distance d

Multicore fibers (MCFs) exhibit an angular memory effect. The angular correlation, limited by the number of modes in individual fibers, was measured experimentally. The tilt angle of a collimated beam was scanned at the input facet of the MCF. The tilt angle was controlled by translating the output of a monomode fiber in the focal plane of a fixed collimating lens (f = 30 mm) and the output of the lens was then conjugated with the input facet of the MCF using a G = 1:6 magnification afocal telescope. The output speckle measured at a distance d on the camera were then digitally cross-correlated to measure the angular memory effect. The results shown in Fig. 1c exhibit a ≃30% drop in correlation after a 20 mrad tilt, over the full spectrum of interest. This limited angular memory effect reduces our ability to recognize speckles and reconstruct wavefronts, thereby restricting the number of accessible spectro-spatial modes.

The described system also allowed an accurate measurement of the distance d between the output facet image and the camera, accounting for potential chromatic defocus in the relay imaging system. The distance d was measured using Eq.(2), relying on tilt-angle conservation between the input and the output facets of the MCF, and within the range of the angular memory effect. Considering the magnification Gin and Gout of the relay imaging system before and after the MCF, respectively, the distance d is measured according to:

d = f cam (Gin δucam / Gout δusource) (6)

where ⟨δucam⟩ is the mean displacement measured over the camera and δusource the known displacement amount of the monomode fiber used as a point source. For the hyperspectral experiments, an average distance d across the field of view was measured. In the multispectral experiments, to achieve metrology-grade phase imaging, the distance d was estimated individually for each camera pixel. The resulting d-maps obtained for each wavelength were fitted using low-order (N=3) 2D polynomials, as shown in Supplementary Fig. S7.

Description of the hyperspectral wavefront sensing experiments

The metrological measurements on the prisms and afocal systems where carried out by imaging the test optics with afocal imaging telescope with de-magnification 1:3, onto the MCF. Special care was taken to align the test beams with the calibrating beams at the entrance of the MCF. The imaged field of view corresponded to an area of 2.4×2.4 mm2 of the test optics. The distance d between the output facet image plane and the camera was accurately measured, and was observed to vary from 88 mm to 97 mm over the spectrum of interest (550–650 nm). This relatively large mean distance was intentionally chosen to enhance sensitivity to small wavefront distortions, such as those produced by chromatic optical elements that simulate spatio-temporal coupling in typical ultrashort laser facility systems.

Description of the Apollon laser beam metrology

For hyperspectral WFS acquisitions, the 140 mm diameter Apollon laser beam was sampled after the focal plane of the end-chain focusing optic (F=6 m) and collimated to 7 mm diameter beam using afocal telescopes within the interaction chamber of the LFA. Relay imaging systems, composed of achromatic lenses both in vacuum and in air, were employed to transfer the image plane (defined at the entrance of the experimental chamber) to the input of the hyperspectral WFS. The overall demagnification factor of the imaging system was 1: 200, reducing the beam diameter to 0.7 mm at the input facet of the MCF. The corresponding field of view at the MCF facet is 0.8×0.8 mm2.

Description of the multispectral wavefront microscope

Poly(methyl methacrylate) beads of 5.2 μm (PMMA-R-5.2, microparticles GmbH) were deposited on glass microscope slides. The dry beads were then covered with a solution consisting in blue brilliant absorber FCF in water (E133 in water, Vahine, Avignon, France) mixed with sucrose (2 g/ml) and then sealed with a cover glass. A description of the model used to describe our sample refractive index can be found in Supplementary Materials. The beads sample was imaged onto the input plane of the wavefront sensor by a microscope imaging system, consisting of a ×20, 0.4 NA microscope objective and a 180 mm tube lens. The resulting field of view was 120×120 μm2. The distances d between the output facet image and the camera were accurately measured over the full field of view and varied in the range between 4 mm and 8 mm (see Supplementary Fig. S7) after taking into account the magnification after the MCF.

Data processing

For multispectral reconstruction involving a limited number of wavelengths with spectral shifts exceeding several δλmedium, Eq.(4) can be addressed using the Moore–Penrose pseudoinverse: X*=(ReT Re)−1 ReM applied over a neighborhood defined by a Gaussian sliding windows. The full width at half maximum of the gaussian sliding window is selected to ensure that the condition number of the matrix Re remains below 10. Typically, for reconstruction using three discrete wavelengths, this size corresponds to the average distance between two speckle grains observed at the camera plane.

For hyperspectral wavefront reconstruction over a continuous (and contiguous) spectrum, the references are captured for a discrete set of wavelengths having a step smaller than the spectral correlation width of the MCF. We found that good performances were obtained for a spectral step equal to δλmedium/3. We solved Eq.(4) for a dense set of square patches covering the full camera surface. To solve Eq.(4) for a given patch, the corresponding matrix Re is inverted through a truncated singular value decomposition (SVD)47. The truncation aims at canceling the contribution of low singular values that dominates after inversion. They are thus filtered out by multiplying inverted singular values by a (half-)Gaussian function of width σSVD48. In practice, the value of σSVD is set empirically in order to reduce the oscillation-like artefacts appearing during the matrix inversion60. Furthermore, to speed up the processing time, a sparse SVD routine is used. The number of singular values conserved after SVD filtering (the σSVD value) will, in fine, drive the maximum spectral resolution of the system. Three singular values are needed per wavelength-band λi±δλi/2 to reconstruct all the information: one singular value for the amplitude αi and two for the displacements (ui, vi). With our optical instrument, we used σSVD = 30 nm for a full spectral range spanning over 100 nm.

Both for multispectral and hyperspectral phase imaging, the first order Taylor expansion performed to linearize Eq.(3) is only valid over displacements of the order of one speckle grain size. For larger displacements, we implemented an iterative multi-scale version49. The principle of the multi-scale approach consists in adjusting the speckle grain size at each iteration by Gaussian-filtering all images (M and R). At each iteration, the reference images are distorted according to the estimated large-scaled displacement maps. The filter width σscale is then progressively decreased to measure finer displacement structures.

For the images acquired with the Apollon laser facility, an initial registration step was achieved by patches by digital image cross-correlation between the acquired multiplexed speckle and a synthetic broadbrand reference (i.e. the sum of the measured reference intensities) to compensate a displacement offset between the reference speckles and the multiplexed ones. This offset was due to a residual non co-linearity error between the Apollon laser beam and the calibrating beam.

A summary of the parameters used for the reconstructions, along with the resulting imaging characteristics, is provided in Supplementary Table S1.

Image processing for multispectral microscopy

Additional post-processing was necessary to evaluate the dispersion of the PMMA beads sample in a Blue Brillant solution. Due to chromatic aberrations in the microscope imaging system, all three channels could not be simultaneously focused. Furthermore, airflow fluctuations in the experimental room introduced low-frequency background variations in the reconstructed optical path delays, with amplitudes comparable to the spectral dispersion. Defocus was corrected using the angular spectrum method within the Fresnel approximation. Assuming that the objects are transparent, the optimal numerical propagation distance was determined by minimizing the standard deviation of intensity images. The slowly-varying wavefront background was estimated by segmenting the image. At the bead positions, an interpolation was performed. This measured background was then filtered using a Gaussian low-pass filter with a full-width half-maximum of 5 pixels, and then subtracted from the optical path delay images. The complete reconstruction workflow is illustrated in Supplementary Fig. S8. Finally the peak values of OPDs were measured at the center of each bead by averaging over a disk with a radius of 10 pixels (=0.6 μm).

Data availability

The data generated in the quantitative dispersion microscopy experiments and their related post-processing algorithms have been deposited in the Zenodo repository under accession code: https://doi.org/10.5281/zenodo.17523847.

Code availability

The post-processing algorithms used in the quantitative dispersion microscopy experiments have been deposited in the Zenodo repository under accession code: https://doi.org/10.5281/zenodo.17523847.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "See attached review.", "reviewer_2": "See the attached pdf for formatted comments.\nThe authors have submitted their work “Single-Shot Hyperspectral Wavefront Imaging” for consideration to be published in\nNature Communications. The article comprises a very interesting merging of computational microscopy-like topics, classical\nwavefront sensing, and ultrashort pulse characterization, and I appreciate the unique nature of these topics being joined\ntogether to achieve such a result. It is also very interesting that they successfully did measurements on different systems\nusing their device in different configurations.\nRecommendation:\nI think that this work, despite its high interest to the field and potential novelty, requires significantly improved explanations\nand more thorough (or rearranged/improved/clarified) presentation of the methods and results. There is still potential to be\npublished in Nature Communications, and I strongly propose keeping the paper for another round of review after major\nrevisions.\nOn one hand, the results are convincing in a general sense, but on the other hand, as you will see below, I have a few\nspecific issues with some of the explanations. Additionally, on a more general note, I find that the framing of the work is\nstrongly from the perspective of the authors and therefore many of the explanations are either too superficial or too technical\ndepending on what is assumed of the reader, i.e. not the best considering the actual implementation and the final result. The\nproblem is approached from the perspective of a scattering media in general, but the implementation is actually very specific,\nand I think that in retrospect the writing could be changed to better explain the specific implementation.\nLastly, as is a comment of mine on the majority of works prepared for such journals, too much of the important or even crucial\ninformation is saved for the methods section. The methods are almost 4 pages! and some of the content there is absolutely\ncrucial for understanding the very initial motivation for why an MCF can act as the proper technology for their device, for\nexample. As a reviewer and potential reader, the paper would be easier to understand and therefore useful if more of the\nmethods text was in the main body at the appropriate place. I apologize in advance if this is contrary to editorial advice.\nAt the end of my comments I mention some other results/papers that are related intimately to the work in various ways,\nwhere a discussion would be illuminating.\nComments:\n• Your implementation relies very heavily on “angular memory” and “spectral decorrelation” but these are not well defined\nenough in this article, nor is it properly or significantly enough motivated, in this reviewer’s opinion, why a multi-core fiber\nhas these properties or conversely why it is necessarily to introduce these properties so generally.\no You write “The speckle pattern created by a scattering medium exhibiting an angular memory effect [15] encodes the\nincident wavefront gradients into local speckle grains displacements at a short distance d [17-22].”\n  In the case of a multi-core fiber, isn’t this trivial? Phase delays at the input will be transferred 1-to-1 to the output, so that,\nat least considering interferences between near cores after the fiber exit and wavefront slopes that aren’t too large,\nthe speckles will move according to the slope of the encoded phase delays.\n  For me your explanations are actually obfuscating what is going on in the true implementation. Angular memory is not\ntrivial at all in other types of scattering media, sure, but in MCFs it seems guaranteed to a certain point, so it seems over-\ncomplicated to introduce it so generally. Maybe I am wrong here. Did you try other scattering media first that did not have\nso strong angular memory?\n  It is unfortunate that interesting and important details are only in the methods section 5.4. Having these details next to\nFigure 1c and introducing it in a more pedagogical way would help the reader.\n  Put another way, could the strength of the “angular memory” and therefore the correctness of Eq. 2 or 3, be derived by only\nknowing the layout and dimensions of the MCF? (properties which you fully know) Or does the random path difference\nbetween the cores also come in to play here? Is this what’s in Ref. [31]?\no You write “The spectral decorrelation results from the random dispersion of delays between cores, which is an inherent\ncharacteristic of MCF [33].”\n  Does this mean that each core has a slightly different path length than the others, and that this is random? Is “dispersion of\ndelays” really a good way to say this? It seems like in [33] that’s what it is since they have such a long fiber (I would still not\nuse “dispersion” since it could be confused with the spectral property of materials), but with your 1 cm long fiber, is it really a\n“delay” or rather a phase effect? If you are looking at the speckle of each individual color, delay turns into phase anyway\nsince in the context of each color it is not a pulse anymore. It makes sense to me that a random phase offset between the\nmany cores produces speckle, but your referencing and explanations are lacking.\n  When you write “and a precisely controlled spectral bandwidth [32,33]” in the main text, should it rather be “spectral\nbandwidth over which there is spectral decorrelation” ? And what does “precisely controlled” mean? Should it not just be\n“sufficiently large” ?\n  I do not find methods section 5.3 as helpful in this case, unfortunately (as was 5.4 for the angular memory topic).\n  Just like with angular memory, I also wonder if the spectral decorrelation could be derived from known MCF parameters.\n  This aspect is where I understand the least, but also where the article is the least convincing.\n• You use different notation in Fig. 1b(iii) from the equations in the paper and this could lead to confusion. Put another way,\nEq. 2 could be expanded to also include the derivatives to match the notation of the figure.\n• Before equation 3 you write “This speckle is the incoherent sum.” Why is it “incoherent?” You are working with intensity\nimages, so it’s just a normal sum, right?\n• Are the speckle pattern and reference images in Fig. 1b real raw data from one of the measurements?\no It is a shame how few raw reference images or speckle patterns are shown in the paper and supplemental document. Even\njust from a reproduction point of view, there should be more raw data. Those few included, Fig. 3a and Fig. S2a are difficult to\nsee and, to be honest, not very useful or instructive.\n• It is actually difficult to see the speckle in Fig. 3a since it is rather small.\n• Why not scale the curves for the second plot in Fig. 3i? We cannot see any detail here.\n• You write “In contrast to multi-shot acquisition systems, our instrument now enables the characterization of potential non-\nlinear spatio-temporal coupling at the single-pulse level.”\no Why emphasize non-linear here? Other devices can measure non-linear or arbitrary spatio-temporal effects as well, just\nnot necessarily in a single shot.\n• I find that you do not necessarily rigorously introduce your terminology “multiplexed” and “sequential” and that it may be\nmisleading. “sequential” means that you are doing non-hyperspectral measurements at different colors one after another,\nand “multiplexed” just means you are doing the measurements that are the main result of this work. There may be either a\nbetter way to word this, or you could more implicitly define the terms when they are first used.\n• In the methods section 5.1 you write “multicore few-mode fiber” but everywhere else you emphasize multicore single-mode\nfibers. Is this an important aspect of your specific experimental implementation? Is this a typo?\n• You do not cite the recent work IMPALA “Simple few-shot method for spectrally resolving the wavefront of an ultrashort\nlaser pulse” (https://doi.org/10.1364/OL.502000) nor their previous works, which also work with a “speckle”. However, it\nseems like they lack spectral decorrelation and therefore have to perform multiple shots. It would be really nice to have a\ndiscussion on this point by you, the experts.\n• You do note cite two other works that used Coded aperture snapshot spectral imaging (CASSI), which may be tangentially\nrelated to scattering media due to the nature of the coded aperture. Both of these results also sense the wavefront of an\nultrashort pulse in a single shot and could reasonably be used for phase microscopy as well. A discussion of your work in that\ncontext would be important.\no “Single-shot compressed optical field topography” LSA (2022) (https://doi.org/10.1038/s41377-022-00935-0)\no “Hyperspectral compressive wavefront sensing” HPLSE (2023) (https://www.doi.org/10.1017/hpl.2022.35)\n• More context and comparison to your previous work in Ref. [20] would also be appreciated, if appropriate.", "reviewer_3": "The paper from Blochet et al. reports a novel optical scheme to measure the wavefront phase across a large\nspectral domain. The idea is to use the spectral decorrelation of the speckle arising after a multicore fiber and its speckle distortion to\nrecover hyperspectral wavefront cubes through an unmixing algorithm. The paper is very well written, with clear\nexplanations and is nicely illustrated in two cases where hyperspectral wavefront cubes are acquired in a single shot for a\nhigh-power fs pulsed laser and to recover microscopic phase images. The paper is of very high scientific level both from the\nexperimental implementation and the computational unmixing algorithm that is implemented.\nBeside this overall very good impression of the work I have few points that I would like the author to address in a revised\nversion of their manuscript.\n1- Could the authors provide more insight and explanations on how the spatial resolution and spectral resolution are linked.\nThis is clear from the paper that they are related to the distance d but the underlying physics is not sufficiently explained nor\na table/curve are provided to show the evolution of the spatial and spectral resolution with this key d parameter. I think it\nwould be valuable to the paper if discussion could be more developed on this point.\nFor instance, d=88mm the spectral decorrelation width is 3nm, at d=6mm I could not find the spectral decorrelation width.\nHow the spatial and spectral resolutions scale with d?\n- The multicore fiber (MCF) is used here as a thin diffuser, in a previous implementation [17] the authors used a thin diffuser\nto perform wavefront sensing. In this case the angular memory effect encodes wavefront gradients in intensity pattern\ndisplacements but should exhibit also spectral decorrelation. This is not clear to me what are the advantages of using a MCF\nover a thin diffuser for distinct spectral pattern encoding. Is it because the spectral decorrelation is stronger with MCF as\ncompared to thin diffuser? It would be nice to include a discussion to clarify this comparison between a thin diffuser and a\nMCF.\n- I wonder if a combination of a thin diffuser and a grating would perform the same spatio-spectral encoding avoiding the\ncomplexity of the MCF fiber core fixed spacing, could the author comment on that?"}, "author_response": "Response to reviewers\nWe thank the reviewers for their insightful reading of our manuscript and for their constructive comments\nand criticisms. We now addressed all their concerns and do believe that the manuscript has now substantially\nimproved. The reviewers will find below our point by point answers in addition to the red-lined version of\nthe updated article.\n• Reviewers comments appear in black ink\n• Our answers are appended in blue ink\n• Changes to the manuscript are detailed in red ink, in bold for additions\nReviewer #1\nIn their paper \"Single-Shot Hyperspectral Wavefront Imaging\", Baptiste Blochet and colleagues\npropose a\nsetup for spectrally-resolved phase measurements in the context of laser wavefronts and phase contrast\nimaging. Their method relies on the incoherent superposition of speckle patterns from different wavelengths.\nThis is an extension of the \"Wavefront sensing with a thin diffuser\" concept Marc Guillon's group introduced\nin 2017 (Ref. 17), where a key insight is that each speckle pattern itself can be modeled via local linear\ndisplacements under the small angle approximation. The current work goes beyond this initial concept by\nrecognizing that speckle patterns have a relatively short spectral correlation width and thus, wavefronts from\ndifferent wavelengths can be reconstructed using a linear inverse problem as opposed to a non-linear one.\nThe paper is well-presented, and the methodology is, in general, sound.\nWe thank the reviewer for their positive general appreciation.\nI will begin my review with a more general contextualization of this work. As discussed in their opening\nstatement, hyperspectral phase measurements and spatio-spectral laser diagnostics have received recent\ninterest within the research community. A multitude of multi-shot and single-shot methods have been\nproposed and demonstrated, thus making it central to the manuscript's perceived impact how their work goes\nbeyond the state-of-the-art. The authors justify the relevance of their contribution as follows: \"All\nhyperspectral techniques developed thus far have abandoned the principles of Hartmann-based WFS,\nconsequently forfeiting several of its many advantages that made it highly popular for metrology and\nquantitative phase imaging [12–14]: simplicity, compactness, single-shot capability, robustness to vibrations,\nand compatibility with broadband light sources.\" I regret to say that this blanket statement is as stated\nmisleading or even false, hinging on either implicit or quite specific definitions that are not as universal as\nthe statement implies:\nThe summarizing sentence at the basis of the main reviewer’s comments was written for the sake of\nconciseness. It aimed at helping contextualizing the presented work and must be understood in the light of\nthe whole article.\nNevertheless, we understand that the current phrasing may be prone to misinterpretations and we agree to\ncomply with the reviewer’s request to rephrase it in a more accurate way. We now replaced the highlighted\nsentence by a more extended paragraph trying to avoid the possible misinterpretations raised by the reviewer.\nActions taken:\nThe introduction has now been substantially modified to provide more accurate contextual material to the\nreader.\n• The sentenced hightlighted by the reviewer was now removed.\nll hyperspectral techniques developed thus far have abandoned the principles of Hartmann-based WFS,\nconsequently forfeiting several of its many advantages that made it highly popular for metrology and\nquantitative phase imaging [16–18]: simplicity, compactness, single-shot capability, robustness to vibrations,\nand compatibility with broadband light sources.\n• some more detailed background was added, including a article published after the first submission of\nour manuscirpt.\nIn contrast to hyperspectral phase imaging, multispectral recording only considers a discrete set of\nwavelengths. Also, the performances of ultrafast and high-power laser systems critically suffer from\nnon-linear optical aberrations [10]. While scanning techniques can arguably characterize the\nspatiotemporal pulse profiles in high repetition rate systems [3], they are incompatible with low\nrepetition rate systems, such as PW and multi-PW-class lasers, due to potential drift with time and\nlimited pulse-to-pulse reproducibility. The key challenge in this context lies in the spectral\ndiscrimination of wavefronts. In this manuscript, we distinguish between “multispectral” and\n“hyperspectral” techniques – referring, respectively, to approaches that resolve wavefronts over\ndiscrete spectral lines versus those spanning contiguous spectra, following the criterion established by\nA. Goetz [11]. This distinction is justified by the increased difficulty of sensing wavefronts across\ncontiguous spectra compared to a discrete set of wavelengths. To manage continuous spectra, a\nspectrum-sampling Fabry-Perot interferometer was proposed in combination with a pair of gratings,\nwhich splits the spectral lines onto a conventional WFS, thereby enabling single-shot multispectral\nwavefront sensing over four spectral lines [9]. Although spectral sampling using a Fabry-Perot\ninterferometer is efficient, the transmitted spectrum is sensitive to local wavefront tilt. Multispectral\nphase imaging can could also be achieved by digital holography by encoding spectral information in the k-\nspace [12] or even without the need for a reference arm by combining a wavefront sensor with a\nmultispectral camera [13]. More recently, hyperspectral phase maps could be rebuilt experimentally over a\ncontinuous spectrum thanks to a phase retrieval algorithm applied to two defocused images acquired with\nhyperspectral cameras (so-called CASSI) by combining a hyperspectral camera (so-called CASSI) with a\nphase diversity technique [4]. In a very recent publication, hyperspectral phase imaging could also be\ndemonstrated by combining a diffraction grating with a micro-lens array [14]. recent So far,\nhyperspectral high-resolution WFS has only been investigated through numerical simulations have\npaved the way to associating a CASSI imager with a high resolution WFS based on quadriwave combining a\nCASSI with a lateral shearing interferometer [15]. As a general principle, cascading existing techniques\nfor achieving multiplexed optical sensing results in increased experimental complexity and spatial\nfootprints, so hindering practical use and dissemination of the technique.\n• A paragraph was added explaining better the specific improvement carried out in our work as\ncompared to the state of the art\nHere, we propose a single-shot hyperspectral wavefront sensing scheme based that combines both spectral\ndiscrimination and wavefront sensing functionalities into a single mask imaged at a close distance from\na camera sensor. Our approach thus builds on the same principles underlying conventional WFS [19,20]\nin which the local wavefront tilts are encoded as local translations of the intensity pattern after\npropagation to the camera. This design has become one of the most widely used approaches in\nmetrology and quantitative phase imaging [16–18] because of its simplicity, compactness, single-shot\ncapability, robustness to vibrations, and compatibility with broadband light sources. The application\nof this same principle to the multispectral/hyperspectral working regime has faced the challenge of\nspectral discrimination when dealing with broad spectra [13].\n• Let's start at the beginning of the sentence: What does \"hyperspectral techniques\" mean? The authors\nargue that \"hyperspectral\" methods are different from \"multispectral\" in that the latter \"only\nconsiders a discrete set of wavelengths\". I would argue that among practitioners, the distinction more\noften seen as one of number of channels used. Typically, one sees multispectral used for 4-10\nchannels and hyperspectral for >10 channels, but the boundaries are fluid. I would argue that three\nchannels in the optical range are usually not called multispectral, because of the ubiquitous RGB use\ncase based on Bayer filters. Personally, I would say their laser wavefront measurements are\nhyperspectral and the phase contrast measurements are RGB. But as the authors appear quite\ninsistent on their interpretation of multi-/hyperspectral, I would like to point out that their definition\nwould mean their own technique is actually multispectral, too, since it is limited by the correlation\nwidth. Even for the higher resolution case, their purported “continuous” coverage is still discretely\nsampled and subsequently interpolated by the low-rank (SVD) approach. In essence, each spectral\nchannel is reconstructed from a finite set of reference wavelengths and then smoothed together, such\nthat one obtains a synthetic “continuous” map only by merging neighboring channels via truncation.\nConsequently, the actual resolution is inherently limited by the correlation width and the rank cut-off,\nleaving the method closer to a “multispectral” interpolation scheme than a true hyperspectral one\n—if the latter really exists at all.\nIn this comment, the reviewer raises several points.\nThe reviewer first raises an objection regarding the definition we employ for hyperspectral and multispectral\nin our manuscript. We agree with the reviewer that our definition, based on continuous vs. discrete spectra\nwas unclear. The distinction we now make between hyperspectral and multispectral is based on the\ndefinition proposed by A. Goetz in the literature (Goetz, 2009) and that was subsequently adopted by other\nauthors in the optics and metrology community. A. Goetz makes a distinction between techniques providing\nthe spectrum for a discrete set of wavelengths and those providing data over contiguous spectra. The\nconfusion in our first definition came from the fact that the characterization over contiguous spectra appears\nas specifically relevant when dealing continuous spectra.\nThe definition of hyperspectal proposed by A. Goetz points out much better the specific difficulty we address\nin our work, a noncontiguous, discrete set of wavelengths being much simpler to manage than a contiguous\nset. Except in new ref. [14] that we now cite but published after our manuscript submission, all current\nwavefront sensing techniques have developed solutions to sub-sample the spectra of beams with continuous\nspectrum supports such as spectral filters or Fabry-Perot interferometers. In contrast, the technique we\npropose can directly manage spectra with broad and continuous supports, taking measurements on\ncontiguous spectral bands. This ability raises fundamental questions about the splitting of information\nbetween spatial and spectral modes. As suggested by reviewer 3, we have now introduced an analytical\nmodel, numerical simulations and new analysis of our experimental data to account for the scaling laws of\nthe spectro-spatial sampling achieved by our approach, and in particular, the number of independent spectral\nchannels.\nThe distinction we make in our manuscript about the discrete or continuous/contiguous nature of the\nspectrum support of is thus a key element to appreciate the advance provided by our technique in the field of\nhyperspectral metrology with respect to alternative solutions published in the literature.\nActions taken:\n• We now rephrased the sentence in the introduction by clearly explicating the definition of\nmultispectral and hyperspectral we use in our manuscript to avoid any ambiguity.\nIn contrast to hyperspectral phase imaging, multispectral recording only considers a discrete set of\nwavelengths. Also, the performances of ultrafast and high-power laser systems critically suffer from\nnon-linear optical aberrations [10]. While scanning techniques can arguably characterize the\nspatiotemporal pulse profiles in high repetition rate systems [3], they are incompatible with low\nrepetition rate systems, such as PW and multi-PW-class lasers, due to potential drift with time and\nlimited pulse-to-pulse reproducibility. The key challenge in this context lies in the spectral\ndiscrimination of wavefronts. In this manuscript, we distinguish between “multispectral” and\n“hyperspectral” techniques – referring, respectively, to approaches that resolve wavefronts over\ndiscrete spectral lines versus those spanning contiguous spectra, following the criterion established by\nA. Goetz [11]. This distinction is justified by the increased difficulty of sensing wavefronts across\ncontiguous spectra compared to a discrete set of wavelengths. To manage continuous spectra, a\nspectrum-sampling Fabry-Perot interferometer was proposed in combination with a pair of gratings,\nwhich splits the spectral lines onto a conventional WFS, thereby enabling single-shot multispectral\nwavefront sensing over four spectral lines [9]. Although spectral sampling using a Fabry-Perot\ninterferometer is efficient, the transmitted spectrum is sensitive to local wavefront tilt. Multispectral\nphase imaging can could also be achieved by digital holography by encoding spectral information in the k-\nspace [12] or even without the need for a reference arm by combining a wavefront sensor with a\nmultispectral camera [13]. More recently, hyperspectral phase maps could be rebuilt experimentally over a\ncontinuous spectrum thanks to a phase retrieval algorithm applied to two defocused images acquired with\nhyperspectral cameras (so-called CASSI) by combining a hyperspectral camera (so-called CASSI) with a\nphase diversity technique [4]. In a very recent publication, hyperspectral phase imaging could also be\ndemonstrated by combining a diffraction grating with a micro-lens array [14]. recent So far,\nhyperspectral high-resolution WFS has only been investigated through numerical simulations have\npaved the way to associating a CASSI imager with a high resolution WFS based on quadriwave combining a\nCASSI with a lateral shearing interferometer [15].\nAs a second comment, the reviewer argues that, provided the distinction we make between multispectral and\nhyperspectral, our numerical filtering approach of the spectrally-resolved transmission matrix makes our\nsystem a “multi-spectral” technique, rather than a “hyperspectral” one. We believe that the definition of\nhyperspectral we now rely on makes this point clearer.\nWe acknowledge with the reviewer that the spectral resolution of our instrument is limited, as any\nspectrometer is. In our case, the spectral resolution is ultimately limited by the spectral correlation width of\nthe MCF like a spectral resolution of spectrometer is typically limited by the number of lines of its grating.\nThe numerical SVD filtering process we carry can further limit the spectral resolution if too few modes are\nkept but cannot fake improved spectral resolution: a too high modal cut-off only results in more noisy\nspectral data. The numerical modal cut-off must be chosen according to the intrinsic spectral correlation\nwidth of the MCF used. Nowaydays, no spectrometer is based on the use of spectral filters sub-sampling the\nbeam spectrum, because the intrinsic spectral discrimination offered by gratings is much simpler to\nimplement experimentally (in addition to better discrimination performances). Similarly, our technical\napproach suggests a similar improvement in the simplification for achieving spectral information retrieval\nover contiguous spectra thanks to the spectral correlation width of the MCF in smart combination with the\ncorrespondingly suited numerical data processing. The number of independent spectral channels is then\ndriven by the intrinsic spectral correlation width of the MCF itself and not by the filter of the singular value\ndecomposition, whose role is just to make the numerical problem more robust to noise.\nAction taken:\nAs also requested by reviewer 3, we now derived some additional material about the physical scaling laws\ndriving the spectral and the spatial resolution of our instrument. These details are in the new supplementary\nsection 2.\nS2.1.2 Scaling of N with physical parameters\nλ\nThe number of independent spectral channels Nλ allowed by the hyperspectral wavefront sensor is\ndriven by two factors: the statistical distribution of optical path differences Ln introduced by the\nindividual monomode fibers, and the spectral dispersion generated by diffraction through free-space\npropagation until the camera plane.\nHere, for the sake of simple analytical modeling, we assume a statistical distribution of optical path\nlength differences given by a centered normal distribution ρ(L ) = N (0, ∆L). In this case, N can be\nn λ\nqualitatively estimated as the width of the two-wavelength mutual coherence function of speckle\npatterns generated by the fibers\n(1)\nwhere δk is the spectral detuning. The width of this correlation function is qualitatively equal to 1/∆L,\nmeaning that the average number of independent spectral channels is\n(2)\nNoteworthy, Nλ does not depend on the number of interfering fibers since this equation is obtained by\nstatistical averaging relying on the central-limit theorem. However, the singular value decomposition\nof the experimental calibration speckles demonstrate that with our MCF, we have Nλ ≃ 10. This value\nis the maximum value but Nλ can be lowered like in the case of multispectral microscopy where we\nonly considered 3 color channels.\nIn addition, free space propagation between the output of the MCF and the camera plane introduces\nsome extra\nspectral dispersion because of differential diffraction angles. In the frame of Fresnel approximation,\nthe diffraction angle scales as 1/λ so resulting in a homothetic dilation of the intensity pattern scaling\nas λd [1, 2]. For a spectral shift inducing an axial elongation larger or equal to the axial dimension of\nthe speckle grain, the speckles correlation is lost. The spectral detuning induced by diffraction is thus\nThird, the reviewer mentions the use of ubiquitous color cameras wherein color sensitivity is provided by\nstandard RGB Bayer filters. Having been developed to adapt human color vision, usual RGB filters\nequipping standard color cameras cannot pretend to provide high-grade metrology performances, unless used\nwith a set of three corresponding narrow-band excitation lines for illumination (complemented with a Fabry-\nPerot interferometer sharpening the spectral channels). Indeed, the spectral transmission properties of usual\nfilters are not sharp, resulting in (1) inhomogeneous transmission in the transparent window and (2) low\noptical densities outside resulting in significant spectral channel coupling. Therefore, the use of a standard\nRGB camera cannot provide any quantitative dispersion measurements for continuous broad white spectrum\nwithout significant additional instrumental and numerical efforts.\nFinally, we point out that the “multispectral” microscopy experiment, in our work, was carried out using\nthree spectral lines because of the illumination source available but the spectral sensitivity of the instrument\nis driven by the MCF itself. For high imaging at high spatial resolution, the considered macro-pixels for\nmatrix inversion must be small, so reducing the number of measure points. The result is a broadening of the\nstatistical distribution of the condition number of the matrix over the image surface, but not of its average\nvalue. This statistical broadening of the condition number results in an increased sensitivity to noise for a\ngiven spectral rank. For this reason, we illustrate quantitative multispectral wavefront imaging with a set of\nthree wavelengths to mitigate this effect. Details about the scaling law of the standard deviation of the\ncondition number of the system with the number of of the number of rebuilt color channels is now\nthoroughly detailed in new supplementary section 2. The derived scaling law, together with our numerical\nsimulations (Figs. S1 and S2) and experimental characterizations (Fig. S3), demonstrates that a high-\nresolution multispectral wavefront imager can be implemented for up to 10 wavelengths, even under\nbroadband illumination with a bandwidth of 40 nm.\nAction taken:\n• We added to the Methods the fact that the number of spectral channels used were limited by the\navailability of spectral filters\nThe three illuminating wavelengths were obtained by inserting off-the-shelf available bandpass filters\n(480/20 nm, 580/10 nm and 690/10 nm), placed in each path before recombination using beam-splitters.\n• We now added a supplementary section 2.2 explaining how the condition number of the matrix to be\ninverted evolves with the number of spectral channels and the number of measure points in the\nmacro-pixel. This models is in excellent agreement with experimental data.\nS2.2 Analytical and numerical modeling\nS2.2.1 Analytical model\nThe ability to resolve the inverse problem is given by the rank of the calibration matrix containing the\nreference patterns over the spectrum of interest. For each macro-pixel, the number of rows of this\nmatrix is M and the number of columns must be at least Nλ. Provided that we always have M ≫ Nλ in\nour experiments, the maximum rank of this matrix is Nλ. We can thus consider that the calibration\nmatrix is of dimensions M × Nλ without loss of generality. The calibration matrix is thus a matrix of M\n× Nλ random intensity values whose each column vector can be normalized. For the sake of\nsimplification, we consider normalizing column vectors after removing their average value. As a\nconsequence, the so-obtained matrix is composed of random independent and identically distributed\nvectors lying on the unit sphere. The singular values distribution of such a matrix is given by the\nMarchenko-Pastur (MP) law and the probability density functions of singular values s is given by\n(6)\nwith r = Nλ/M and where s± = 1 ± √r are the extrema values of the distribution. Consequently, the\nmaximum value of the condition number of such a matrix is\n(7)\nThe system must be designed so as to minimize this upper bound.\nS2.2.2 Numerical model\nFor modeling the experiment, we considered the wavefield generated by a square array of point\nsources having random optical path differences and placed at a distance from a two-dimensional\npixilated sensor array. The resulting wave-field at the detector is given by Fourier transform :\n(8)\nwhere p is the separation distance between the sampling points of the intensity sensor. The numerical\nmodel and the MP model are compared in Figs. S1 and S2. In Fig. S1, the probability density function\nof singular values are shown for the two typical experimental configurations at play in the manuscript\ndemonstrating that the MP model accurately describes our physical system. In particular, this\ntheoretical distribution is bounded by two extrema values s± = 1 ± √r, so providing an estimate of the\nmaximum bound value for the condition number of the reference matrix. In Fig. S2, the histograms of\nthe reciprocal condition numbers obtained from numerical simulations are compared to its theoretical\nlower bound (vertical red dashed lines), again demonstrating the good agreement\nbetween our optical system and the theoretical model.\nS2.2.3 Experimental validation\nWe experimentally measured the probability density function (PDF) of the singular values of the\nreference matrix used in the laser metrology experiment. The results, shown in Fig. S3, demonstrate\nexcellent agreement with both the theoretical model and the numerical simulation.\n• With this broader context let's now discuss if research has \"abandoned the principles of Hartmann-\nbased WFS\" and given its overlap with the present work, let's include multi-spectral methods here:\nThere are several works in the literature that directly used multi-spectral imaging of Shack-Hartmann\nsensors [Dorrer as cited, Kim Opt. Express, 29(13):19506–19514 (2021)]. But what are even the\n\"principles of Hartmann-based WFS\"? Hartmann just used a pinhole mask. The authors appear to\nhave a quite specific definition in mind where the local intensity pattern following a linear\nrelationship to a patch's local wavefront, an approximation usually valid under homogenous\nillumination and small angles. But is this a meaningful distinction? Rouzé, Bastien, et al. [Optics\nExpress 29.4 5193-5204 (2021)] showed that Shack-Hartmann patterns can be equally well\nunderstood under the interference framework, showing large analogy between Shack-Hartmann and\nquadriwave lateral shearing (QWLS) interferometry. Indeed, the authors of Ref.11 have also\npresented a hyperspectral use of Shack-Hartmann sensors instead of QWLS interferometry at\nconferences. But more importantly, one could easily say that QWLS is more similar to Hartmann\ntype sensing than their method is. We could also argue that it is the essence of Hartmann sensors is\nthat they are easy to calibrate and interpret by eye (and so forth). I find the definition via the\nmemory effect somewhat contrived, because this can basically encompass any optical element. Which\nimmediately brings us to the next point.\nWe thank the reviewer for their comment. First, the reviewer raises the question regarding the definition of\nwhat we called a “Hartmann-based WFS”. As pointed out by the reviewer themselves and supported by their\nreference to Rouzé, Bastien et al., the principle of Shack-Hartmann sensors and QWLS interferometry can be\nunified under a same theoretical framework as belonging to the family of wavefront sensing techniques that\nuse a mask in the close vicinity of the camera sensor. From a practical point of view, the main advantage of\nthese approaches is that they encode wavefront distortions into the deformation vector field of an intensity\npattern. Thanks to its simple and compact design, this approach is especially robust. It is also one of the few\nfamilies of methods capable of performing single-shot wavefront measurements without requiring a\nreference arm.\nFrom a physical point of view, the very structuring principle of this family of techniques is that a wavefront\ntilt results in a shift of the intensity pattern (eg. a focal spot, an interference fringe or a speckle grain) at the\ncamera sensor. Mathematically, this property requires that the transmission matrix of the mask commutes\nwith the tilt transform matrix. This property is only satisfied by so-called “thin” optical elements, that is,\noptical elements whose transmission matrix is diagonal in the canonical position basis. This property was\ngeneralized in the context of complex and scattering media and coined “memory effect”.\nUnlike suggested by the reviewer, not all optical elements satisfy this property, all the more when providing\nspectral discrimination. For instance, a pair of gratings, such as used for multispectral wavefront sensing,\ndoes not satisfy this property (such as used in Kim et al. Opt Express (2021) cited by the reviewer). As\nanother example a Fabry-Perot interferometer, involves several reflections on two axially-separated mirrors.\nIts spectrally-resolved transmission matrix is thus not tilt invariant, so making the “thin optics”\napproximation only valid in a limited extent. Finally, a multiple scattering sample optics also provides high\nspectral sensitivity (Ref. [31], R. French et al. Opt. Lett. 2017) but exhibits limited “memory effect”,\nunsuitable for wavefront sensing. We tested such an implementation but results were of too poor quality to be\nshown in the manuscript.\nFor all these reasons, we consider that our reference to the very general principle of “memory effect” as a\nfederating definition of the family of wavefront sensor what we call the “Hartmann-based mask” techniques\nis enlightening since referring to a very fundamental physical property of optics.\nHowever, we agree that the expression “memory effect” is coming to the specific community of complex\nmedia and we now rephrased this sentence into a full paragraph making clear this structuring common point\nof WFS techniques relying on a single mask.\nAs discussed here, it is far from straightforward to identify an optical element that simultaneously satisfies\nall the necessary requirements (namely, a linear relationship between pattern displacement and phase\ngradient over a wide angular range, along with a narrow spectral correlation width to enable effective\nspectral unmixing) for extending what we refer to as the Hartmann-based wavefront sensing principle to\nhyperspectral wavefront imaging. In the newly added Section 2.1 of the main manuscript (Suitable Optical\nMask for Single-Shot Hyperspectral WFS), we explicitly highlight why a multicore fiber fulfills both key\ncriteria.\nActions taken:\n We now removed the sentence criticized by the reviewer and replaced it by a full paragraph was\nadded to more explicitely refer our work to the state of the art, highlighting the specific improvement\nthat we carried out. Therein, we include the reference to the article of Rouzé, Bastien, et al. [Optics\nExpress 29.4 5193-5204 (2021)] suggested by the reviewer (new reference 20).\nll hyperspectral techniques developed thus far have abandoned the principles of Hartmann-based WFS,\nconsequently forfeiting several of its many advantages that made it highly popular for metrology and\nquantitative phase imaging [16–18]: simplicity, compactness, single-shot capability, robustness to vibrations,\nand compatibility with broadband light sources.\nHere, we propose a single-shot hyperspectral wavefront sensing scheme based that combines both spectral\ndiscrimination and wavefront sensing functionalities into a single mask imaged at a close distance from\na camera sensor. Our approach thus builds on the same principles underlying conventional WFS [19,20]\nin which the local wavefront tilts are encoded as local translations of the intensity pattern after\npropagation to the camera. This design has become one of the most widely used approaches in\nmetrology and quantitative phase imaging [16–18] because of its simplicity, compactness, single-shot\ncapability, robustness to vibrations, and compatibility with broadband light sources. The application\nof this same principle to the multispectral/hyperspectral working regime has faced the challenge of\nspectral discrimination when dealing with broad spectra [13].\n We now added the reference Kim et al. Opt. Express (2021) suggested by the reviewer (new\nreference [9]) as a typical illustration of spectrum sub-sampling of a continuous spectrum. This\nreference indeed accurately emphasize the contrast with hyperspectral WFS over contiguous spectra\nthat we carry out in our work.\nTo manage continuous spectra, a spectrum-sampling Fabry-Perot interferometer was proposed in\ncombination with a pair of gratings, which splits the spectral lines onto a conventional WFS, thereby\nenabling single-shot multispectral wavefront sensing over four spectral lines [9].\n We added a new section 2.1 Suitable Optical Mask for Single-Shot Hyperspectral WFS explaining\nwhy a multicore fiber fulfills the criteria to expend Hartmann-based wavefront sensing to\nhyperspectral wavefront sensing.\nIt includes an explanation of what the memory effect is and why this concept, created in the\ncommunity of complex media, is especially suited for grasping the principle of the WFS discussed in\nthe manuscript.\nIt also includes a comment about thick multiple scattering samples, which are perfect counter-\nexamples of masks that cannot be used to perform deflectometry-based WFS because not having a\nmemory effect, although they have been demonstrated to perform efficient hyperspectral imaging\n(without wavefront sensing).\n2.1 Suitable Optical Mask for Single-Shot Hyperspectral WFS\nIn the quest for instrument simplification, spectrally-resolved wavefront sensing calls for a single\noptical element that simultaneously provides both spectral discrimination and wavefront sensitivity.\nSeveral masks can be placed in the close vicinity of a camera that result in wavefront-sensitive\nintensity patterns: arrays of holes, microlens arrays [19], meta-micro-lens arrays [24], thin diffuser\n[23], diffraction gratings [25]. A key characteristic for wavefront reconstruction is that a local tilt of\nthe wavefront is transmitted through the mask, such that a wavefront tilt at the entrance of the mask\nresults in a local translation of the intensity pattern at the camera plane. This principle, generalized in\nthe context of scattering media and coined “memory effect” [21], is satisfied by optical elements whose\ntransmission matrix is diagonal in the canonical basis.\nThe spectral sensitivity of a mask is characterized by the spectral correlation width of the intensity\npatterns at the camera. All the masks listed above provide remarkably achromatic WFS properties.\nSpectral decorrelation may arise from chromatic dispersion within the mask itself or from free-space\npropagation over the distance d between the mask and the camera. However, axial dilation of intensity\npatterns due to free-space propagation [26, 27] contributes only marginally to spectral sensitivity when\nthe mask is positionned at millimeter-scale distances [23]. To achieve hyperspectral WFS, the use of\nspectrally dispersive optics is therefore required. For interference effects to be wavelength-sensitive,\nthese optics must involve a diversity of large optical path lengths. Nanometer-scale spectral sensitivity\nis typically achieved using complex media, such as multiply scattering samples [28] or multimode\nfibers [29]. Notably, diffraction gratings are achromatic at short distances [16] and thin diffusers –\nwhich we have previously shown to enable effective WFS – are only weakly dispersive [23].\nFurthermore, complex media have been shown to be well suited for multiplexed spectral [30, 31]\nand polarimetric [32] imaging, and even for multiplexed wavefront sensing in a single spectral\nchannel [33].\nTherefore, optics that satisfy both spectral and angular properties required for achieving spectrally-\nresolved wavefront sensing are not numerous. For instance, thick multiply scattering diffusers exhibit\nimproved spectral sensitivity at the expense of degraded angular memory effect as thickness increases,\ndrastically limiting the applicability to WFS [21]. In contrast, MCF exhibit both large angular memory\neffect [34] and narrowband spectral correlation properties [35]. MCF exhibit a diagonal transmission\nmatrix if fibers are perfectly monomode and uncoupled. In practice, many bundles are composed of\nfew-modes fibers, so limiting the angular range of the memory effect. The physical mechanism behind\nthe limited spectral correlation width in these fibers is still unclear but involves statistical dispersion in\nlongitudinal wavenumbers in individual fibers due to variations in their diameters and potentially\ncoupling between fibers. This dispersion is not specified by manufacturers and probably results from a\nlack of control in the fabrication process.\n• The authors continue to state that not using \"Hartmann-based WFS\" will be \"forfeiting [advantages\nlike] simplicity, compactness, single-shot capability, robustness to vibrations, and compatibility with\nbroadband light sources\". In this framing, they imply that their method indeed possesses all these\nabilities while others don’t. But this is not the case. There are several works that would tick all these\nboxes except for single-shot capability. Single-shot devices tend to be more complex and less\ncompact, but there are still examples for them in the literature. The question is whether their method\ntruly addresses all these issues? I am not sure to which level the terms simplicity and robustness\napply (imaging onto a millimeter-sized fiber, speckle pattern depends on accurate fiber calibration\nunder current environment conditions, reconstruction is quite slow and involved, etc.). Furthermore, the\ncompatibility with broadband light sources is not entirely clear. The method faces the same\nspectral vs. spatial resolution trade-offs as every other method, evidenced by the fact that their phase\ncontrast demonstration did not use a broadband light source, but discrete line sources.\nAs underlined in the title of our manuscript, our work specifically focuses on spectrally discriminating\n“single-shot” wavefront sensing. Single-shot wavefront sensing and imaging is especially critical for laser\nmetrology and also represents an interesting advantage in other applications dealing with real time imaging\nsuch as microscopy of living biological samples. As pointed out by the reviewer themselves, much less\ntechniques were published in the single-shot regime, and all investigated approaches resulted in more\ncomplex instrumental systems. Existing single-shot techniques are either too complex to be implemented in\nlarge-scale PW systems worldwide (new ref. [10]: Danson et al.), requiring important modifications of the\nexisting laser diagnostics platforms and dedicated bulky setups, or too invasive, requiring the installation of\nlarge-size equipment, which is especially critical in vacuum chambers.\nIn contrast to formerly published techniques, our approach specifically relies on encoding both spectral and\nwavefront information with a single optical element: the multicore fiber. This MCF just needs to be re-\nimaged onto the camera sensor with a relay telescope. The MCF provides the two optical properties required\nfor hyperspectral wavefront sensing which have always been found in two different optical elements so far:\n(i) the spectral discrimination ability and (ii) the “memory effect” converting a local wavefront tilt into a\nmere intensity pattern translation. The full footprint of our \"device\" does not exceed 100 cm² and could be\ndirectly integrated into the diagnostics setup of the Apollon laser in a few hours only. With our approach,\nthe complexity of the system is reduced through a proper data post-processing algorithm based on standard\nlinear matrix algebra and fundamental physics of complex media.\nProvided these elements, we think that the reviewer will agree that our new approach combining the dual\noptical requirement of hyper-spectral wavefront sensing into a single optical element represents a significant\nsimplification over previously developed instruments.\nActions taken:\n• In the introduction, we now emphasize more the importance of single-shot WFS to avoid any\npossible misunderstanding and to clearly distinguish from the literature treating from color-blind\nWFS.\nIn these applications, single-shot measurements are preferable or even required, especially to monitor fast\nevents like in flow cytometry [7], in pump probe experiments [4], in light matter interaction [8], or to\ncharacterize single laser pulses [8, 9]. Assisted by a chirped pulse, ultrafast events can be acquired [4, 8].\nAlso, the performances of ultrafast and high-power laser systems critically suffer from non-linear\noptical aberrations [10]. While scanning techniques can arguably characterize the spatiotemporal\npulse profiles in high repetition rate systems [3], they are incompatible with low repetition rate\nsystems, such as PW and multi-PW-class lasers, due to potential drift with time and limited pulse-to-\npulse reproducibility. The key challenge in this context lies in the spectral discrimination of\nwavefronts.\n• We added a sentence about the size and the installation complexity\nNext, we installed our single-shot hyperspectral wavefront sensor in the Long Focus Area (LFA) at the\nApollon multi-PW laser facility. The full footprint of our wavefront sensor does not exceed 100 cm² and\ncould be directly integrated into the diagnostics setup of the Apollon laser in a few hours only.\nIn fact, given the existence of many other spatio-temporal laser diagnostics, the significance of this work\nhinges on its implied interdisciplinary applications beyond laser diagnostics. However, that breadth feels\nartificially emphasized: the multi-spectral phase contrast application relies on a custom three-color source;\nunder these circumstances phase contrast imaging can easily be replicated with a minimal setup such as\nstandard wavefront sensor or interferometer plus an RGB camera. Without the custom source, the\nspeckles would probably be too blurred out. Consequently, the purported general-purpose nature of the\ntechnique is not convincingly established. As it is, the paper is interesting for an audience from an optics\nbackground, but even in this context the method is not groundbreaking given the numerous alternative\nmethods for laser characterization available.\nBelow, we answer point by point to the reviewer’s concerns that are enumerated in this paragraph.\n• There are “numerous alternative methods” and our technique is “not groundbreaking given the\nnumerous alternative methods for laser characterization available”\nThe reviewer acknowledges that the examples of single-shot laser metrology experiments published in the\nliterature are not numerous: “there are still examples for them in the literature.”; in contrast to their\nconcluding statement. We interpret the reviewer acknowledgment as a recognition that such approaches are\nstill emerging, and we welcome the opportunity to position our contribution within this developing context.\nFollowing our response to the former paragraph, we now believe that the context of single-shot WFS is now\nclear.\n• Our manuscript “artificially emphasize” on the “interdisciplinary applications beyond laser\ndiagnostics”\nSingle-shot wavefront sensing and imaging is highly critical for characterizing non-linear dispersive\naberrations in laser metrology but also represents a significant advantage in microscopy for addressing fast\nmonitoring of living/moving biological samples. In this context our technique provides both modalities\nwithin a single instrument and the versatility of our approach, allowing to mitigate the balance between\nspatial and spectral resolution is an advantage that no other technique can claim. Based on these two\narguments, we feel that the reviewers’ comment about the “artificial emphasis” as unfair.\n• The microscopy experiment could be “easily be replicated with a minimal setup such as standard\nwavefront sensor or interferometer plus an RGB camera”\nAs extensively answered above, we do not see how standard RGB cameras could provide metrology-grade\nperformances without significant calibration and proper control of the illumination spectrum.\n• For the microscopy experiment, “Without the custom source, the speckles would probably be too\nblurred out.”\nThe reviewer raises here an important point. The reviewer is right in saying that the contrast of the image\ndegrades while the number of spectral channels increases. However, this decrease in speckle contrast is not\nspecific to the microscopy experiment and the 10 modes resolved in the laser/optics metrology experiment\ndemonstrates that more than 3 channels can be resolved by our instrument. As now thoroughly detailed in\nnew supplementary section 2, the spectral correlation width (i.e. the spectral resolution) of our instrument is\ndriven by the spectral correlation width of the MCF. The number of resolvable spectral channels depends on\nthe balance desired between the spatial resolution and the signal to noise ratio. Resolving more spectral\nchannels qualitatively requires having more independent measure points, and thus a larger macro-pixel. In\nthe microscopy experiment, we arbitrarily limited the number of spectral channels to 3 because of the\navailability of light sources but theoretical and numerical developments carried out in the new supplementary\nFig. S2 demonstrate that 10 wavelengths could be also resolved, at the expense of a ~30% decrease in the\nsignal to noise ratio in the spectral/wavefront reconstruction. We emphasize that for other configurations, our\nnew theoretical developments in supplementary section S2 very accurately match experimental data.\nAction taken:\n• The balance between spatial resolution, spectral resolution and the conditioning of the inverse\nproblem is now extensively developed in Supplementary Section S2, demonstrating in Fig. S2 that\nthe system would remain well conditioned for 10 wavelengths in the microscopy configuration. Fig.\nS2 shows the probability density function of the condition number obtained for 10 wavelengths from\nnumerical simulations in the microscopy configuration.\n• Exploiting our experimental data for multispectral experiments in the microscopy configuration, we\nalso demonstrate in Fig. S3 that the system remains well conditioned for more than 3 wavelengths.\nNoteworthy, the obtained experimental histograms obtained for singular values perfectly match our\ntheoretical model.\nS2.2.3 Experimental validation\nWe experimentally measured the probability density function (PDF) of the singular values of the\nreference matrix used in the laser metrology experiment. The results, shown in Fig. S3, demonstrate\nexcellent agreement with both the theoretical model and the numerical simulation.\nI would also have some hesitation regarding the method’s ease of use that I hope the authors can clarify.\nFor instance, how do you know in which plane you are actually imaging the laser in? It seems difficult for me to\nfigure out the exact plane when you have to image onto a 1.4 mm fiber entrance, which will be a\ndemagnification of >100 of the APOLLON beam. And how stable are the patterns really; is the setup\ntemperature dependent, how difficult are calibrations, is the reconstruction really unique or does the\nprocedure sometimes fail to converge (it does not appear to me that there are global convergence\nguarantees)?\nThe reviewer raises some reasonable doubts about the precision requirements of the imaging plane on the\ndiffuser. This is indeed true due to the large downscaling factor of 1:100. However, we should note that the\nrequired imaging precision also scales inversely with the square of the size of the Apollon beam. For a 140\nmm beam, a few meters of propagation would have an absolutely insignificant impact on the wavefront of\nthe beam or the low spatial order spatiotemporal coupling effects studied with our setup.\nFinally, we would like to clarify the apparent complexity of the proposed hyperspectral wavefront imager.\nWhile the system may initially seem intricate, its complexity is comparable to that of other metrology-grade\nimaging systems and stems mainly from two key areas: calibration and reconstruction.\nAs with any metrology imaging system, proper calibration is essential. In our case, two forms of calibration\nare required:\nSpectral calibration: In hyperspectral computational imaging systems, where spectral information is\nencoded in spatial patterns measured by a camera, calibration is needed to associate each wavelength with its\ncorresponding pattern. This can be achieved either by:\n Scanning the wavelengths individually and recording the resulting patterns at the detector, as\ndemonstrated in [Video-rate hyperspectral camera based on a CMOS-compatible random array of\nFabry-Perot filters, Nature Photonics], or\n Predicting the patterns based on the known system layout, as done in some CASSI-based\napproaches. However, this method is more sensitive to alignment errors, system aberrations, and\nnon-linear dispersion introduced by prisms or other dispersive elements, which can lead to\ninaccuracies in the predicted spectral encoding.\nPhase calibration: For nanometer-level wavefront measurements, a flat wavefront measurement is required\nas reference; either via a flat reference arm (in interferometry) or flat illumination (in wavefront sensing).\nSince our system measures the spatial phase for each wavelength, both spectral and phase calibrations are\nnecessary. In practice, this amounts to generating flat illumination across the spectral range. We do not\nbelieve this is significantly more complex than the calibration procedures required for conventional\nhyperspectral imaging systems.\nOur reconstruction task involves solving a linear inverse problem (Eq. 4), derived under reasonable\nassumptions relevant to wavefront sensing: namely, that the phase gradients are small, and that both the\nphase gradients and amplitude vary smoothly over local spatial regions\nIn the revised supplementary materials, we now provide experimental evidence, numerical simulations,\nand an analytical model demonstrating that this inverse problem is well-posed. Unlike other approaches\nsuch as CASSI, which could rely on sparsity or continuity constraints, our formulation does not require\nadditional assumptions. This ensures stable reconstructions without excessive noise amplification.\nThe reconstruction algorithm leverages standard and well-established algorithms from the computer vision\ntoolkit:\n Moore-Penrose pseudoinverse: Used to recover spectrally resolved amplitude and phase gradients.\n Iterative optical flow algorithms: Based on image interpolation, these are adapted to recover large\nphase gradients.\n Optional low-rank projection: A singular value truncation step can be applied to reduce noise,\nthough it is not strictly required.\nGiven that we aim to reconstruct a high-dimensional space (spatial phase × spectrum), longer computation\ntimes are expected compared to standard imaging systems. However, we emphasize that several steps, such\nas the pseudoinverse matrix computation and the low-rank projection, can be precomputed and reused\nacross experiments, provided the system configuration and calibration remain unchanged. These steps are\ntherefore one-time costs, not per-experiment overheads. The most computationally intensive operations in\nour current pipeline are the image interpolations, for which we use spline interpolation to ensure high\naccuracy. Faster alternatives, such as FFT-based interpolation or GPU acceleration, could significantly\nreduce runtime. Although these optimizations have not yet been implemented, they remain promising\ndirections for improving computational efficiency in future iterations of the system\nAction taken:\nWe added in the supplementary materials (fig SI.5) raw data of reference images and multiplexed image\nzoomed on one patch used for reconstruction\nWe also added a detailed algorithm workflow illustrating the key steps of the reconstruction process (fig\nSI.6)\nFig. S5: a. Example of raw data collected from a single patch, corresponding to the hyperspectral\nwavefront sensing experiment shown in Figure 2. b. Illustration of the validity region for the first-\norder Taylor approximation shown for different speckle grain sizes obtained via Gaussian filtering\nwith standard deviation σ. A single, uniform displacement is applied to all spectral components. c.\nSingular values of the matrix R = [R, ∂ R, ∂ R], computed over one patch for images filtered with\ni x i y i\nvarying Gaussian widths σ\nFig. S6: a. Workflow of the modified iterative optical flow algorithm. b. Example of raw data from a\nhyperspectral wavefront sensing experiment. c. Evolution of the absolute displacements (u, v),\naveraged over the full spectral range, for a single patch. d. Normalized root mean square error\ncomputed for one patch as a function of the iteration. e. Displacement fields (u, v) stitched together for\nthe lens metrology experiment shown in Figure 2c,d. f. Reconstructed beam amplitude after stitching.\ng. Retrieved optical path delays obtained via numerical integration of the stitched displacement fields\n(u, v).\nTo summarize, we respectfully emphasize that our approach advances the current state of the art by\nbringing together several key aspects:\n Single-shot operation: This is a cornerstone for high-power laser metrology and real-time biological\nimaging, where dynamic samples or chromatic aberrations make scanning-based or multi-shot\ntechniques impractical.\n Contiguous spectral performance: Our system could operate over a contiguous spectral range,\nrather than relying on discrete or filtered bands, as is the case in approaches such as Kim et al., Opt.\nExpress 29(13):19506–19514 (2021).\n Improved metrology performance: Hartmann-type wavefront sensing approach offers superior\noptical path difference (OPD) accuracy among single-shot, reference-less methods, reaching values\nup to λ/600 in the monochromatic regime. In comparison, state-of-the-art phase diversity methods\ntypically achieve accuracy on the order of λ/120, even with Cramer-Rao-optimized dual-plane\nimplementations [e.g., High-accuracy wavefront sensing by phase diversity technique with\nbisymmetric defocuses, Sci. Rep., 2017]. In addition, our approach enables robust detection of\nstructured phase features, including optical vortices [Wu et al.].\n Quantified accuracy in the hyperspectral regime: In contrast to other recent single-shot\napproaches, such as Compressed Optical Field Topography [Kim et al., Opt. Express, 2021] and the\nmethod of Dopp et al., Nat. Photonics, 25th June 2025, which also perform the previously mentioned\nkey aspects, our method explicitly quantifies wavefront accuracy in the hyperspectral regime. We\ndemonstrate a solid performance of λ/60 over a 100 nm spectral range.\n Well-posed inverse problem: As mentioned in our response, we provide both analytical and\nnumerical evidence that the inverse problem associated with our method is well-posed. In contrast,\nphase retrieval methods often lack guaranteed convergence, and compressive sensing approaches\n(e.g., CASSI) depend on specific sparsity or structural assumptions about the object to ensure\naccurate reconstruction.\n Versatility across domains: The proposed architecture is tunable and broadly adaptable, suitable for\napplications ranging from laser metrology to broadband biological microscopy, without the need for\nsubstantial hardware modifications.\n Conceptual elegance: The approach is built on a compact, physically intuitive design that leverages\nthe angular memory effect and the spectral decorrelation in a multicore fiber bundle. This principle\nenables hyperspectral wavefront sensing with a new and effective trade-off between spatial resolution,\nspectral coverage, and measurement sensitivity.\nWe believe that this combination of practicality, accuracy, and conceptual clarity makes our method a strong\nand original contribution to the field. By leveraging a novel imaging concept, the approach transfers\nhardware complexity into software, and its successful application in a demanding high-power laser\nenvironment highlights its operational relevance and practical potential.\nReviewer #2\nThe authors have submitted their work “Single-Shot Hyperspectral Wavefront Imaging” for consideration to\nbe published in Nature Communications. The article comprises a very interesting merging of computational\nmicroscopy-like topics, classical wavefront sensing, and ultrashort pulse characterization, and I appreciate\nthe unique nature of these topics being joined together to achieve such a result. It is also very interesting that\nthey successfully did measurements on different systems using their device in different configurations.\nRecommendation:\nI think that this work, despite its high interest to the field and potential novelty, requires significantly\nimproved explanations and more thorough (or rearranged/improved/clarified) presentation of the methods\nand results. There is still potential to be published in Nature Communications, and I strongly propose\nkeeping the paper for another round of review after major revisions.\nOn one hand, the results are convincing in a general sense, but on the other hand, as you will see below, I\nhave a few specific issues with some of the explanations. Additionally, on a more general note, I find that the\nframing of the work is strongly from the perspective of the authors and therefore many of the explanations\nare either too superficial or too technical depending on what is assumed of the reader, i.e. not the best\nconsidering the actual implementation and the final result. The problem is approached from the perspective\nof a scattering media in general, but the implementation is actually very specific, and I think that in\nretrospect the writing could be changed to better explain the specific implementation.\nWe thank the reviewer for their compliments and for pointing out the broad potential impact of the approach\nwe propose. We highly appreciate its constructive comments and suggestions for making our article more\naccessible to physicists with various backgrounds. We now revised the manuscript according to their\nsuggestions.\nLastly, as is a comment of mine on the majority of works prepared for such journals, too much of the\nimportant or even crucial information is saved for the methods section. The methods are almost 4 pages! and\nsome of the content there is absolutely crucial for understanding the very initial motivation for why an MCF\ncan act as the proper technology for their device, for example.\nAs a reviewer and potential reader, the paper would be easier to understand and therefore useful if more of\nthe methods text was in the main body at the appropriate place. I apologize in advance if this is contrary to\neditorial advice.\nAs the reviewer encourage us to, our intention was to make the manuscript accessible to as broad a\nreadership as possible by focusing on the fundamental physics principles in the main body, while placing\nmore detailed information in the methods and supplementary materials. We appreciate the reviewer's\nvaluable feedback and are happy to modify our manuscript to include the missing details that the reviewer\nconsiders as essential into the main text for a comprehensive understanding.\nAt the end of my comments I mention some other results/papers that are related intimately to the work in\nvarious ways, where a discussion would be illuminating.\nWe thank the reviewer for their suggestion. Our detailed answer is appended below.\nComments:\n• Your implementation relies very heavily on “angular memory” and “spectral decorrelation” but these\nare not well defined enough in this article, nor is it properly or significantly enough motivated, in\nthis reviewer’s opinion, why a multi-core fiber has these properties or conversely why it is\nnecessarily to introduce these properties so generally.\nWe agree with the reviewer that most of sentences referring to a “spectral decorrelation” could be phrased in\na more precise manner. In all concerned places, we now refer to the “spectral correlation width” that relies on\na clear mathematical definition.\nAlso, we understand that, as pointed out by the reviewer 1, the reason for choosing a multicore fiber was\nlacking some more detailed explanations in relation with the angular memory effect and the spectral\ncharacteristics. The physical reasons for the limited spectral correlation width of multicore fibers are still\nunclear but can be related to a differences in individual core diameters that may result in different\nlongitudinal wave-numbers, and thus, in different optical path lengths.\nActions taken :\n• we replaced most of the expression “spectral decorrelation” by “limited spectral correlation width”\nor “spectral sensitivity” in most of the occurrences in the manuscript.\nAbstract\nOur system leverages the angular memory effect and limited decorrelation from spectral correlation\nwidth of the multicore fiber, encoding wavefront gradients into displacements and the spectral\ninformation into uncorrelated patterns.\nLast paragraph of the introduction\nSpecifically, our system leverages both the so-called “angular memory effect” [18] and the limited\nspectral correlation width spectral decorrelation [19] occurring at a short distance from the output\nof a multicore single-mode fiber bundle (MCF). The angular memory effect encodes wavefront\ngradients in intensity pattern displacements [20], while the limited spectral correlation width\nspectral decorrelation ensures distinct pattern encoding.\n2nd paragraph of Section 2.2\nIn addition, the spectral decorrelation that arises after propagation through a complex medium\nencodes spectral information into uncorrelated speckle patterns [19].\n1st paragraph of Section 3.1\nMCFs exhibit both angular memory effects and a limited spectral correlation width spectral\ndecorrelation simultaneously. The spectral decorrelation sensitivity results from the random\nstatistical dispersion of delays optical path lengths between cores, which is an inherent\ncharacteristic of MCF [36].\n1st line of Section 5.3\nMulticore single-mode fibers (MCFs) exhibit a limited spectral correlation width spectral\ndecorrelation due to the inherent random statistical dispersion of delays optical path lengths\nbetween cores [36].\n• An introductory paragraph was added as a section 2.1 at the beginning of the first part to explain the\nchoice of a MCF as a mask for hyperspectral WFS. In this paragraph, we now explicitly define to the\n“spectral correlation width” as the correlation product of intensity patterns and we provide\ninformation about the possible physical reason for the limited spectral correlation width of MCFs.\n2.1 Suitable optical mask for single-shot hyperspectral WFS\nIn the quest for instrument simplification, spectrally-resolved wavefront sensing calls for a single\noptical element that simultaneously provides both spectral discrimination and wavefront sensitivity.\nSeveral masks can be placed in the close vicinity of a camera that result in wavefront-sensitive\nintensity patterns: arrays of holes, microlens arrays [19], meta-micro-lens arrays [24], thin diffuser\n[23], diffraction gratings [25]. A key characteristic for wavefront reconstruction is that a local tilt of\nthe wavefront is transmitted through the mask, such that a wavefront tilt at the entrance of the mask\nresults in a local translation of the intensity pattern at the camera plane. This principle, generalized in\nthe context of scattering media and coined “memory effect” [21], is satisfied by optical elements whose\ntransmission matrix is diagonal in the canonical basis.\nThe spectral sensitivity of a mask is characterized by the spectral correlation width of the\nintensity patterns at the camera. All the masks listed above provide remarkably achromatic WFS\nproperties. Spectral decorrelation may arise from chromatic dispersion within the mask itself or from\nfree-space propagation over the distance d between the mask and the camera. However, axial dilation\nof intensity patterns due to free-space propagation [26, 27] contributes only marginally to spectral\nsensitivity when the mask is positionned at millimeter-scale distances [23]. To achieve hyperspectral\nWFS, the use of spectrally dispersive optics is therefore required. For interference effects to be\nwavelength-sensitive, these optics must involve a diversity of large optical path lengths. Nanometer-\nscale spectral sensitivity is typically achieved using complex media, such as multiply scattering samples\n[28] or multimode fibers [29]. Notably, diffraction gratings are achromatic at short distances [16] and\nthin diffusers – which we have previously shown to enable effective WFS – are only weakly dispersive\n[23]. Furthermore, complex media have been shown to be well suited for multiplexed spectral [30, 31]\nand polarimetric [32] imaging, and even for multiplexed wavefront sensing in a single spectral\nchannel [33].\nTherefore, optics that satisfy both spectral and angular properties required for achieving spectrally-\nresolved wavefront sensing are not numerous. For instance, thick multiply scattering diffusers exhibit\nimproved spectral sensitivity at the expense of degraded angular memory effect as thickness increases,\ndrastically limiting the applicability to WFS [21]. In contrast, MCF exhibit both large angular memory\neffect [34] and narrowband spectral correlation properties [35]. MCF exhibit a diagonal transmission\nmatrix if fibers are perfectly monomode and uncoupled. In practice, many bundles are composed of\nfew-modes fibers, so limiting the angular range of the memory effect. The physical mechanism behind\nthe limited spectral correlation width in these fibers is still unclear but involves statistical dispersion in\nlongitudinal wavenumbers in individual fibers due to variations in their diameters and potentially\ncoupling between fibers. This dispersion is not specified by manufacturers and probably results\nfrom a lack of control in the fabrication process.\n• You write “The speckle pattern created by a scattering medium exhibiting an angular memory\neffect [15] encodes the incident wavefront gradients into local speckle grains displacements at a\nshort distance d [17-22].”\nIn the case of a multi-core fiber, isn’t this trivial? Phase delays at the input will be transferred 1-to-1 to the\noutput, so that, at least considering interferences between near cores after the fiber exit and wavefront slopes\nthat aren’t too large, the speckles will move according to the slope of the encoded phase delays. For me your\nexplanations are actually obfuscating what is going on in the true implementation. Angular memory is not\ntrivial at all in other types of scattering media, sure, but in MCFs it seems guaranteed to a certain point, so it\nseems over-complicated to introduce it so generally. Maybe I am wrong here. Did you try other scattering\nmedia first that did not have so strong angular memory?\nIf we interpret correctly the reviewer’s point: they are challenging our choice to refer to the general principle\ncalled “memory effect” while the phase transmission through a bundle of monomode fiber seems obvious.\nThis objection is related to the former reviewer’s remark that our explanations and motivations for using a\nMCF were too weak in the former version of the manuscript.\nThe memory effect is a generalization of a phenomenon that is indeed trivial for several optics like lenses or\nbundles of monomode fibers. This generalization was achieved in the context of complex and scattering\nmedia. Here, the ability to relate a local wavefront tilt to the local translation of an intensity pattern is\nintrinsically linked to the fact that the transmission matrix of the optical element is diagonal in the canonical\nbasis.\nInitially, our first attempt was achieved using scattering media which both exhibit angular memory effect and\nlimited spectral correlation width. However, having both a large angular memory effect and a narrow spectral\ncorrelation width with such media is highly challenging because the angular and spectral properties exhibit in\nopposite requirement directions with the sample thickness. We thus gave up this approach and turned\ntowards MCF.\nMoreover, most of MCF, like ours, are actually made of few-modes fibers and not single-mode fibers. For\nthis reason, the angular memory effect exhibits a limited extent that must be measured and compared to the\ndynamic of wavefront distortions at play. Because this memory effect must be characterized, we need this\nconcept of “memory effect” introduced in the context of complex and scattering media.\nActions taken:\nWe removed confusing references to single-modes fibers since our bundle is composed of few modes fibers.\nLast paragraph of the introduction\nSpecifically, our system leverages both the so-called “angular memory effect” [18] and the limited\nspectral correlation width spectral decorrelation [19] occurring at a short distance from the output\nof a multicore single-mode fiber bundle (MCF). The angular memory effect encodes wavefront\ngradients in intensity pattern displacements [20], while the limited spectral correlation width\nspectral decorrelation ensures distinct pattern encoding.\n1st line of Section 5.3\nMulticore single-mode fibers (MCFs) exhibit a limited spectral correlation width spectral\ndecorrelation due to the inherent random statistical dispersion of delays optical path lengths\nbetween cores [36].\n1st line of Section 5.4\nMulticore single-mode fibers (MCFs) exhibit an angular memory effect.\nWe added a detailed explanation regarding few-modes MCF\nNew section 2.1:\nMCF exhibit both large angular memory effect [34] and narrowband spectral correlation\nproperties [35]. MCF exhibit a diagonal transmission matrix if fibers are perfectly monomode and\nuncoupled. In practice, many bundles are composed of few-modes fibers, so limiting the angular\nrange of the memory effect. The physical mechanism behind the limited spectral correlation width\nin these fibers is still unclear but involves statistical dispersion in longitudinal wavenumbers in\nindividual fibers due to variations in their diameters and potentially coupling between fibers. This\ndispersion is not specified by manufacturers and probably results from a lack of control in the\nfabrication process.\nWe now make it clear that what we are using is a few-mode MCF\n1st line of Section 3.1:\nThe built-up hyperspectral wavefront sensor consists of a few-mode MCF (FIGH-100-1500N,\nFujikura, Japan)\nWe link the limited memory effect and the number of modes in individual fibers\nsection 3.1:\nThis limited angular memory effect reduces our ability to recognize speckles and reconstruct\nwavefronts. Our experiments were conducted within this range, thereby validating this lower bound\nof the dynamic range for our wavefront sensor implementation. This limited angular correlation is\nattributed to the fact that the MCF is made of few-modes fibers (≃ 4) and not monomode fibers.\n• It is unfortunate that interesting and important details are only in the methods section 5.4. Having\nthese details next to Figure 1c and introducing it in a more pedagogical way would help the reader.\nWe thank the reviewer for their suggestion. We agree that the calibration protocol is of the highest\nimportance for metrology-grade experiments, and we believe that the accurate calibration and the thorough\nvalidation of our instrument is indeed one of the main strength of our work.\nSection 5.4 aims at describing the technical details about the calibration protocol for measuring the distance\nbetween the image of the output facet of the MCF and the camera. Although we agree that this calibration\nprotocol is of high importance, we believe that introducing such sharp technical details in the main body will\nhinder the readability.\nAction taken:\nWe now moved up the comments of Sec. 5.4 about the angular memory effect of our few-modes MCF and\nalso now explicitly refer to Sec. 5.4 for additional technical details about the experiment.\nIn our case, the angular correlation of the fiber bundle few-mode MCF, measured by tilting a\ncollimated beam at the entrance of its input facet, exhibits a ≃ 30% drop for a tilt of 20 mrad (See\nFig. 1c and Sec. 5.4 for technical details). This limited angular memory effect reduces our ability\nto recognize speckles and reconstruct wavefronts. Our experiments were conducted within this\nrange, thereby validating this lower bound of the dynamic range for our wavefront sensor\nimplementation. This limited angular correlation is attributed to the fact that the MCF is made of\nfew-modes fibers (≃ 4) and not monomode fibers.\n• Put another way, could the strength of the “angular memory” and therefore the correctness of Eq. 2\nor 3, be derived by only knowing the layout and dimensions of the MCF? (properties which you\nfully know) Or does the random path difference between the cores also come in to play here? Is this\nwhat’s in Ref. [31]?\nAs a short answer: No, unfortunately, the range of the memory effect cannot be deduced from the usual\nspecifications provided by the manufacturer such as the NA. The memory effect mostly depends on the\nnumber of spatial modes in individual fibers of the MCF as well as from potential inter-fiber coupling in the\nbundle. These characteristics are not specified by the manufacturers we know.\nThe former version of the manuscript was confusing about this point because it was citing a sentence from\n(former) Ref. [31] (Porat et al. Opt. Express 2016) which claims that the memory effect depends on the NA\nof the fibers. We now hold this statement as highly confusing. For an ideal MCF made of single-mode\nfibers, one would indeed expect a large memory effect, indeed primarily constrained by the fiber’s NA, as\nthe reviewer rightly suggests. However, in practice, the fibers are not perfectly single-mode and may support\na small number of higher-order modes. As the illumination angle increases, these higher-order modes\nbecome increasingly excited, leading to a degradation in angular correlation (see figure 1.c). Additionally,\ncore-to-core coupling introduces further limitations on the angular memory range, as discussed in\n[https://doi.org/10.1364/OE.23.030532].\nIn our MCF, we estimated that the number of spatial modes are of the order of 3, exhibiting Hermite-\nGaussian like modes with indices (0,0), (0,1) and (1,0). Some fibers exhibit slightly more complex modes.\nFor this reason, we estimate the average number of modes in the fibers to approximately 4.\nActions taken:\nwe removed two occurrences of the confusing statement found in the literature and make it clear that the\nangular memory effect is chiefly driven by the number of spatial modes in individual fibers:\n7th line of Section 3.1\nThe angular correlation is given by the numerical aperture (NA) of the cores [23].\nFirst line of Section 5.4\nMulticore single-mode fibers (MCFs) exhibit an angular memory effect. The angular correlation,\nlimited by the numerical aperture (NA) of the cores [23] number of modes in individual fibers, was\nmeasured experimentally.\nThe paragraph at the end of the new section 2.1 gives more details about the MCF:\nIn contrast, MCF exhibit both large angular memory effect [34] and narrowband spectral\ncorrelation properties [35]. MCF exhibit a diagonal transmission matrix if fibers are perfectly\nmonomode and uncoupled. In practice, many bundles are composed of few-modes fibers, so\nlimiting the angular range of the memory effect. The physical mechanism behind the limited\nspectral correlation width in these fibers is still unclear but involves statistical dispersion in\nlongitudinal wavenumbers in individual fibers due to variations in their diameters and potentially\ncoupling between fibers. This dispersion is not specified by manufacturers and probably results\nfrom a lack of control in the fabrication process.\n• You write “The spectral decorrelation results from the random dispersion of delays between cores,\nwhich is an inherent characteristic of MCF [33].”\nDoes this mean that each core has a slightly different path length than the others, and that this is random? Is\n“dispersion of delays” really a good way to say this? It seems like in [33] that’s what it is since they have\nsuch a long fiber (I would still not use “dispersion” since it could be confused with the spectral property of\nmaterials), but with your 1 cm long fiber, is it really a “delay” or rather a phase effect? If you are looking at\nthe speckle of each individual color, delay turns into phase anyway since in the context of each color it is not\na pulse anymore. It makes sense to me that a random phase offset between the many cores produces speckle,\nbut your referencing and explanations are lacking.\nWe agree with the reviewer that the word “delay” is not the most appropriate one for clarity and that “optical\npath length” is better.\nRegarding the word dispersion, we agree that it may be confusing with the intrinsic spectral dispersion\nproperties of materials but we used this word in the context of statistical physics. To describe the width of the\nprobability density function of the stochastic values of optical path lengths, we think that there cannot be\nbetter wording than “statistical dispersion” which is now replacing the expression “random dispersion”\nwhich was less idiomatic and more prone to confusion. We believe that it is now more clear that “dispersion”\nmust be understood in the context of statistics.\nRegarding the physical mechanism at play in this statistical dispersion in optical path lengths, it is still\nunknown, to the best of our knowledge. The scaling of this statistical dispersion with the fiber length is also\nunclear. Finally, our 1cm long fiber cannot be compared with the one used in former Ref [33] Accanto et al.\nNeuron (2023) since not only the manufacturers are different (Schott vs. Fujikura) but also the fabrication\ntechnologies.\nActions taken\nWe now make it more clear where the spectral correlation properties of MCFs may originate, although this\nphenomenon has not been studied, to the best of our knowledge:\nNew section 2.1, last paragraph\nThe physical mechanism behind the limited spectral correlation width in these fibers is still\nunclear but involves statistical dispersion in longitudinal wavenumbers in individual fibers due to\nvariations in their diameters and potentially coupling between fibers. This dispersion is not\nspecified by manufacturers and probably results from a lack of control in the fabrication process.\nWe replaced the expression “random dispersion” by “statistical dispersion” to make it clear that we a\nreferring to the width of a stochastic process and not the the chromatic dispersion of a material. We also\nreplaced the word “delays” by “optical path lengths”\n1st paragraph of Section 3.1\nMCFs exhibit both angular memory effects and a limited spectral correlation width spectral\ndecorrelation simultaneously. The spectral decorrelation sensitivity results from the random\nstatistical dispersion of delays optical path lengths between cores, which is an inherent\ncharacteristic of MCF [36].\n1st line of Section 5.3\nMulticore single-mode fibers (MCFs) exhibit a limited spectral correlation width spectral\ndecorrelation due to the inherent random statistical dispersion of delays optical path lengths\nbetween cores [36].\n• When you write “and a precisely controlled spectral bandwidth [32,33]” in the main text, should it\nrather be “spectral bandwidth over which there is spectral decorrelation” ? And what does “precisely\ncontrolled” mean? Should it not just be “sufficiently large” ?\nI do not find methods section 5.3 as helpful in this case, unfortunately (as was 5.4 for the angular memory\ntopic).\nThe reviewer indeed points out an improper phrasing which we now changed. In this sentence, we indeed\nmeant that the MCF is suitable for hyperspectral WFS thanks to its limited spectral correlation bandwidth.\nNevertheless, the spectral correlation width of the MCF can be roughly tuned by adjusting its length, which\nwe actually did. We now added this technical detail both in the main text and in Sect. 5.3. Detailed\ninformation about the scaling of spectral bandwidth with fiber length is now added to Supplementary\ndocument in section S3, Fig. S4.\nAction taken:\nWe now changed the sentence highlighted by the reviewer\nAs argued above, to achieve hyperspectral single-shot spectrally-resolved wavefront sensing, the\nHartmann mask must be carefully chosen.\nHere we suggest using a MCF which demonstrated both a wide angular memory effect [34] and a\nlimited precisely controlled spectral correlation bandwidth [35, 44]. The latter can be tuned by\nadjusting the length of the MCF (see Supp. Fig. S4).\nIn Methods section 5.3, we now added this sentence to explain how the spectral correlation width can be\ncontrolled:\nWe initially purchased a 50 cm-long MCF which we shortened down to 1 cm to tune its spectral\ncorrelation width in accordance with the requirements of our applications (see Supp. Fig. S4).\nWe added a figure in the supplementary materials (Supp. Fig. S4) showing the evolution of the spectral width\nwith the fiber length (50cm, 8cm and 1cm)\nS3 Additional experimental characterization of the multicore fiber bundles\nFig. S4: Caption: Left. Spectral correlation widths measured for different bundle lengths over the visible\nspectrum. Right. Spectral correlation width for an 8cm MCF as a function of the MCF image and the\ncamera distance d.\n• Just like with angular memory, I also wonder if the spectral decorrelation could be derived from\nknown MCF parameters.\nUnfortunately, the answer is no again. Neither the range of the memory effect or the spectral correlation\nwidth of the MCF can be deduced from the usual specifications provided by the manufacturers. As explained\nabove, even though little is known about the physical mechanism at play in this phenomenon, we can just\nrely on the fact that the spectral correlation width decreases with the MCF length.\nActions taken\nIn the introductory paragraph, we now make it more clear where the spectral correlation properties of MCFs\nmay originate, although this characteristic is not provided by the manufacturers:\nThe physical mechanism behind the limited spectral correlation width in these fibers is still\nunclear but involves statistical dispersion in longitudinal wavenumbers in individual fibers due to\nvariations in their diameters and potentially coupling between fibers. This dispersion is not\nspecified by manufacturers and probably results from a lack of control in the fabrication process.\nWe also added the following sentences to the main manuscript:\n1st paragraph of Section 3.1\nMCFs exhibit both angular memory effects and a limited spectral correlation width spectral\ndecorrelation simultaneously. The spectral decorrelation sensitivity results from the random\nstatistical dispersion of delays optical path lengths between cores, which is an inherent\ncharacteristic of MCF [36].\n1st line of Section 5.3\nMulticore single-mode fibers (MCFs) exhibit a limited spectral correlation width spectral\ndecorrelation due to the inherent random statistical dispersion of delays optical path lengths\nbetween cores [36].\nIn Methods section 5.3, we now added this sentence to explain how the spectral correlation width can be\ncontrolled, in addition to precisions about the method used to measure the spectral correlation width of the\nfiber:\nWe initially purchased a 50 cm-long MCF which we shortened down to 1 cm to tune its spectral\ncorrelation width in accordance with the requirements of our applications (see Supp. Fig. S4).\nThis aspect is where I understand the least, but also where the article is the least convincing.\nWe believe that we now satisfactorily answered both reviewer’s point regarding the angular memory effect\nand the limited spectral correlation width. We thank the reviewer for highlighting this lack of clarity in the\nmanuscript.\n• You use different notation in Fig. 1b(iii) from the equations in the paper and this could lead to\nconfusion. Put another way, Eq. 2 could be expanded to also include the derivatives to match the\nnotation of the figure.\nActions taken:\nWe now re-wrote Eq. 2 in the same explicit way as in Fig. 1b(iii)\n(2)\n• Before equation 3 you write “This speckle is the incoherent sum.” Why is it “incoherent?”\nYou are working with intensity images, so it’s just a normal sum, right?\nWe used the word “incoherent” to explain that what we obtain is a sum of intensities and not an interfering\nphenomenon. The word incoherent then refer to the mutual coherence of wavefields at different wavelengths.\nWe decided to change this phrasing as considered by confusing by the reviewer.\nActions taken:\nWe removed all three occurrences of “incoherent” in the manuscript and made it clear that we are referring to\na sum of “pattern intensities”.\nCaption of Fig. 1\nb. A multiplexed image (I) consists of an incoherent sum of locally translated reference intensity\npatterns (ii) according to the model (iii).\nSection 5.8\n(i.e. the incoherent sum of the measured reference intensities)\n• Are the speckle pattern and reference images in Fig. 1b real raw data from one of the measurements?\nIt is a shame how few raw reference images or speckle patterns are shown in the paper and supplemental\ndocument. Even just from a reproduction point of view, there should be more raw data. Those few included,\nFig. 3a and Fig. S2a are difficult to see and, to be honest, not very useful or instructive.\n• It is actually difficult to see the speckle in Fig. 3a since it is rather small.\nAction taken:\nWe added in the supplementary materials (fig SI.5) raw data of reference images and multiplexed image\nzoomed on one patch used for reconstruction\nWe also added a detailed algorithm workflow illustrating the key steps of the reconstruction process (fig\nSI.6)\nFig. S5: a. Example of raw data collected from a single patch, corresponding to the hyperspectral\nwavefront sensing experiment shown in Figure 2. b. Illustration of the validity region for the first-\norder Taylor approximation shown for different speckle grain sizes obtained via Gaussian filtering\nwith standard deviation σ. A single, uniform displacement is applied to all spectral components. c.\nSingular values of the matrix R = [R, ∂ R, ∂ R], computed over one patch for images filtered with\ni x i y i\nvarying Gaussian widths σ\nFig. S6: a. Workflow of the modified iterative optical flow algorithm. b. Example of raw data from a\nhyperspectral wavefront sensing experiment. c. Evolution of the absolute displacements (u, v),\naveraged over the full spectral range, for a single patch. d. Normalized root mean square error\ncomputed for one patch as a function of the iteration. e. Displacement fields (u, v) stitched together for\nthe lens metrology experiment shown in Figure 2c,d. f. Reconstructed beam amplitude after stitching.\ng. Retrieved optical path delays obtained via numerical integration of the stitched displacement fields\n(u, v).\n• Why not scale the curves for the second plot in Fig. 3i? We cannot see any detail here.\nWe showed the two curves on a same scale to allow qualitative visual comparison. Nevertheless, we agree\nthat the curve was a bit small and we now increased its size.\nAction taken:\nThe plots in Fig.3i have been enlarged to facilitate better readability.\n• You write “In contrast to multi-shot acquisition systems, our instrument now enables the\ncharacterization of potential non-linear spatio-temporal coupling at the single-pulse level.”\nWhy emphasize non-linear here? Other devices can measure non-linear or arbitrary spatio-temporal effects\nas well, just not necessarily in a single shot.\nReviewer (2) is right to underline this important point, which is not straightforward except for people expert\nin the specific field of ultra-high intensity laser technology. Non-linearities do exist and have a non-\nnegligible impact on practically all types of ultrafast laser systems. For those that operate in the high\nrepetition rate regime (>1 Hz), scanning, multi-shot spatiotemporal characterization techniques can\neffectively provide complete information about the spatiotemporal couplings, including the non-linear\neffects. However, PW and multi-PW-class systems operate at low repetition rates (providing few pulses per\nminute or even lower, down to only a few shots per hour) and are prone to induce especially large non-linear\nbehaviors. For these large-scale systems, the full impact of the non-linear effects on the spatiotemporal\ncoupling processes can only be investigated at full power operation mode and thus at a very low repetition\nrate.\nThe limitations of existing scanning techniques are twofold:\n1. The long acquisition time and potential drift during spectral scanning\nThe number of shots required (typically in the range of 100 or more) leads to long acquisition times during\nwhich the laser system is not necessarily perfectly stable. Typical instabilities (pointing, spectral phase,\nspectral bandwidth, wavefront, and energy fluctuations, to mention only a few) even if achromatic, could\nthen be falsely interpreted by the scanning measurement as chromatic ones and as non-existent\nspatiotemporal coupling effects. As the most trivial example, we can think of a simple sequential wavefront\nmeasurement at different narrowband spectral lines. If during this measurement the global pointing of the\nlaser drifts or if its wavefront fluctuates due to air turbulence in the system, these instabilities would be\nsimply seen as chromatic variations of the wavefront, leading to wrong conclusions.\n2. The limited shot-to-shot reproducibility due to the chaotic nature of non-linear phenomena that are\nsensitive to small variations in working conditions\nNonlinearities in a high-intensity laser can strongly fluctuate on a shot-to-shot basis. The total accumulated\nnon-linear effect, such as the Kerr effect in both the temporal and spatial domains, can be drastically\nimpacted by the instabilities of the pump sources of each amplification stage, even if the output energy of the\nsystem is globally stable. The laser system will reach more or less the same saturation level at the last\namplifier, but the saturation of intermediate amplifiers can vary significantly. Such shot-to-shot instabilities\nand their consequent shot-to-shot spatiotemporal instabilities can, of course, only be measured by single-shot\ntechniques since scanning methods will average them out, probably down to zero.\nActions taken:\nAdditional sentences have been included to highlight the critical role of single-shot acquisition in the\nmeasurement of nonlinear aberrations.\nIn these applications, single-shot measurements are preferable or even required, especially to\nmonitor fast events like in flow cytometry [7], in pump probe experiments [4], in light matter\ninteraction [8], or to characterize single laser pulses [8, 9]. Assisted by a chirped pulse, ultrafast\nevents can be acquired [4, 8]. Also, the performances of ultrafast and high-power laser systems\ncritically suffer from non-linear optical aberrations [10]. While scanning techniques can\narguably characterize the spatiotemporal pulse profiles in high repetition rate systems [3], they\nare incompatible with low repetition rate systems, such as PW and multi-PW-class lasers, due\nto potential drift with time and limited pulse-to-pulse reproducibility. The key challenge in this\ncontext lies in the spectral discrimination of wavefronts.\nAnd\nIn contrast to hyperspectral phase imaging, multispectral recording only considers a discrete set of\nwavelengths. Also, the performances of ultrafast and high-power laser systems critically suffer\nfrom non-linear optical aberrations [10]. While scanning techniques can arguably characterize\nthe spatiotemporal pulse profiles in high repetition rate systems [3], they are incompatible with\nlow repetition rate systems, such as PW and multi-PW-class lasers, due to potential drift with\ntime and limited pulse-to-pulse reproducibility. The key challenge in this context lies in the spectral\ndiscrimination of wavefronts.\n• I find that you do not necessarily rigorously introduce your terminology “multiplexed” and\n“sequential” and that it may be misleading. “sequential” means that you are doing non-hyperspectral\nmeasurements at different colors one after another, and “multiplexed” just means you are doing the\nmeasurements that are the main result of this work. There may be either a better way to word this, or\nyou could more implicitly define the terms when they are first used.\nAction taken:\nNext, the wavefronts were reconstructed both from the single broadband multiplexed images and from\nsequential images acquisition while scanning the spectrum. Throughout the article, we use the following\nterminology: multiplexed acquisition refers to a single-shot measurement of the spectrally resolved\nwavefront datacube, while sequential acquisition refers to a series of quasi-monochromatic wavefront\nmeasurements obtained by scanning the spectrum.\n• In the methods section 5.1 you write “multicore few-mode fiber” but everywhere else you emphasize\nmulticore single-mode fibers. Is this an important aspect of your specific experimental\nimplementation? Is this a typo?\nThis comment relates to the reviewer’s earlier remarks regarding the angular memory effect of multicore\nfibers (MCFs). To avoid potential confusion, we have removed the term “single-mode” from the manuscript.\nActions taken: We removed improper reference to single-modes fibers since our bundle is composed of few\nmodes fibers.\nLast paragraph of the introduction\nSpecifically, our system leverages both the so-called “angular memory effect” [18] and the limited\nspectral correlation width spectral decorrelation [19] occurring at a short distance from the output\nof a multicore single-mode fiber bundle (MCF). The angular memory effect encodes wavefront\ngradients in intensity pattern displacements [20], while the limited spectral correlation width\nspectral decorrelation ensures distinct pattern encoding.\n1st line of Section 5.3\nMulticore single-mode fibers (MCFs) exhibit a limited spectral correlation width spectral\ndecorrelation due to the inherent random statistical dispersion of delays optical path lengths\nbetween cores [36].\n1st line of Section 5.4\nMulticore single-mode fibers (MCFs) exhibit an angular memory effect.\nwe added a detailed explanation regarding few-modes MCF\nNew section 2.1:\nMCF exhibit both large angular memory effect [23] and narrowband spectral correlation\nproperties [24]. These optical elements exhibit a diagonal transmission matrix if fibers are\nperfectly monomode and uncoupled. In practice, many bundles are composed of few-modes fibers,\nso limiting the angular range of the memory effect.\nwe now make it clear that what we are using is a few-mode MCF\n1st line of Section 3.1:\nThe built-up hyperspectral wavefront sensor consists of a few-mode MCF (FIGH-100-1500N,\nFujikura, Japan)\nwe link the limited memory effect and the number of modes in individual fibers\nsection 3.1:\nThis limited angular correlation is attributed to the fact that the MCF is made of few-modes fibers\n(≃ 4) and not monomode fibers.\n• You do not cite the recent work IMPALA “Simple few-shot method for spectrally resolving the\nwavefront of an ultrashort laser pulse” (https://doi.org/10.1364/OL.502000) nor their previous\nworks, which also work with a “speckle”. However, it seems like they lack spectral decorrelation and\ntherefore have to perform multiple shots. It would be really nice to have a discussion on this point by\nyou, the experts.\nThe IMPALA approach relies on an optimized distribution of 12 pinholes placed in the pupil plane to\ngenerate interference patterns that encode the hyperspectral wavefront cube. This configuration enables the\nretrieval of spectrally resolved wavefronts, spatially sampled at the pinhole positions.\nSpectral information is directly accessible in the Fourier plane, where the different spectral components\nappear as elongated streaks, arising from the spectrally dependent spatial dilation of the interference patterns\nduring propagation. The number and positions of the pinholes are carefully chosen to ensure that these\nspectral streaks remain distinguishable in the Fourier domain. The number of pinholes is optimized to 12,\nwhich appears to represent a practical upper bound for spatial resolution in a single-shot configuration. For\ncomparison, our system provides a spatial sampling of 16×16 pixels in a single shot in the hyperspectral\nexperiments. In IMPALA, higher spatial resolution requires sequential acquisition through rotation of the\npinhole mask. The spectral resolution is determined by the extent of the streaks and is governed by the\ngeometry of the mask and the optical design. Ultimately, their approach achieves a spectral resolution\ncomparable to ours : on the order of 10 nm. Phase retrieval in the IMPALA method is performed via an\niterative algorithm, which is likely well-constrained given the low number of spatial sampling points (12\npinhole) in the Fourier domain.\nAdvantages of the IMPALA approach include a relatively simple calibration procedure and a tunable\nspectral resolution. Drawbacks include very limited spatial resolution and lower optical throughput due to\nthe small number of sampling apertures.\nActions taken: We added the IMPALA reference that was lacking in our introduction\nMultispectral phase imaging can could also be achieved by digital holography by encoding spectral\ninformation in the k-space [12] or even without the need for a reference arm by combining a\nwavefront sensor with a multispectral camera [13]. Low-resolution (12 sampling points) and low-\nthroughput single-shot hyperspectral wavefront sensing can be achieved using a coded\naperture in combination with a phase retrieval algorithm. [14].\n• You do note cite two other works that used Coded aperture snapshot spectral imaging (CASSI),\nwhich may be tangentially related to scattering media due to the nature of the coded aperture. Both\nof these results also sense the wavefront of an ultrashort pulse in a single shot and could reasonably\nbe used for phase microscopy as well. A discussion of your work in that context would be important.\n◦ “Single-shot compressed optical field topography” LSA (2022)\n(https://doi.org/10.1038/s41377-022-00935-0)\n◦ “Hyperspectral compressive wavefront sensing” HPLSE (2023)\n(https://www.doi.org/10.1017/hpl.2022.35)\nWe did cite the last two papers mentioned by the reviewer in references [4] and [11] in the original version of\nthe manuscript; however, we acknowledge that a proper comparison with these works was missing. We now\naddress this by summarizing the key distinguishing features of our approach relative to alternative methods:\n Single-shot operation: This is a cornerstone for high-power laser metrology and real-time biological\nimaging, where dynamic samples or chromatic aberrations make scanning-based or multi-shot\ntechniques impractical.\n Broadband spectral performance: Our system could operate over a continuous spectral range,\nrather than relying on discrete or filtered bands, as is the case in approaches such as Kim et al., Opt.\nExpress 29(13):19506–19514 (2021).\n Improved metrology performance: Hartmann-type wavefront sensing approach offers superior\noptical path difference (OPD) accuracy among single-shot, reference-less methods, reaching values\nup to λ/600 in the monochromatic regime. In comparison, state-of-the-art phase diversity methods\ntypically achieve accuracy on the order of λ/120, even with Cramer-Rao-optimized dual-plane\nimplementations [e.g., High-accuracy wavefront sensing by phase diversity technique with\nbisymmetric defocuses, Sci. Rep., 2017]. In addition, our approach enables robust detection of\nstructured phase features, including optical vortices [Wu et al.].\n Quantified accuracy in the hyperspectral regime: In contrast to other recent single-shot\napproaches, such as Compressed Optical Field Topography [Kim et al., Opt. Express, 2021] and the\nmethod of Dopp et al., Nat. Photonics, 25th June 2025, which also perform the previously mentioned\nkey aspects, our method explicitly quantifies wavefront accuracy in the hyperspectral regime. We\ndemonstrate a solid performance of λ/60 over a 100 nm spectral range.\n Well-posed inverse problem: As mentioned in our response, we provide both analytical and\nnumerical evidence that the inverse problem associated with our method is well-posed. In contrast,\nphase retrieval methods often lack guaranteed convergence, and compressive sensing approaches\n(e.g., CASSI) depend on specific sparsity or structural assumptions about the object to ensure\naccurate reconstruction.\n Versatility across domains: The proposed architecture is tunable and broadly adaptable, suitable for\napplications ranging from laser metrology to broadband biological microscopy, without the need for\nsubstantial hardware modifications.\n Conceptual elegance: The approach is built on a compact, physically intuitive design that leverages\nthe angular memory effect and the spectral decorrelation in a multicore MCF. This principle enables\nhyperspectral wavefront sensing with a new and effective trade-off between spatial resolution,\nspectral coverage, and measurement sensitivity.\nWe believe that this combination of practicality, accuracy, and conceptual clarity makes our method a strong\nand original contribution to the field.\nAction taken: We have rephrased our introduction to better highlight the main advantages of our system\ncompared to existing approaches in the literature; namely: single-shot operation, high spatial resolution, the\nuse of a single optical element to encode the spectral wavefront datacube, and compatibility with a\ncontiguous spectral range.\nTo manage continuous spectra, a spectrum-sampling Fabry-Perot interferometer was proposed\nin combination with a pair of gratings, which splits the spectral lines onto a conventional WFS,\nthereby enabling single-shot multispectral wavefront sensing over four spectral lines [9].\nAlthough spectral sampling using a Fabry-Perot interferometer is efficient, the transmitted\nspectrum is sensitive to local wavefront tilt. Multispectral phase imaging can could also be\nachieved by digital holography by encoding spectral information in the k-space [12] or even without\nthe need for a reference arm by combining a wavefront sensor with a multispectral camera [13].\nLow-resolution (12 sampling points) and low-throughput single-shot hyperspectral wavefront\nsensing can be achieved using a coded aperture in combination with a phase retrieval\nalgorithm. [14]. More recently, hyperspectral phase maps could be rebuilt experimentally over a\ncontinuous spectrum thanks to a phase retrieval algorithm applied to two defocused images\nacquired with hyperspectral cameras (so-called CASSI) by combining a hyperspectral camera\n(so-called CASSI) with a phase diversity technique [4]. In a very recent publication,\nhyperspectral phase imaging could also be demonstrated by combining a diffraction grating\nwith a micro-lens array [15]. So far, hyperspectral high-resolution WFS has only been\ninvestigated through numerical simulations have paved the way to associating a CASSI imager\nwith a high resolution WFS based on quadriwave combining a CASSI with a lateral shearing\ninterferometer [16]. As a general principle, cascading existing techniques for achieving\nmultiplexed optical sensing results in increased experimental complexity and spatial footprints,\nso hindering practical use and dissemination of the technique\n• More context and comparison to your previous work in Ref. [20] would also be appreciated, if\nappropriate.\nWe thank the reviewer for the opportunity to clarify the relationship between our current work and our\nprevious publication [Ref. 20, Wu et al., 2024]. While both studies investigate the use of complex media for\nmultiplexed wavefront sensing, the two works are largely unrelated in terms of scientific context, optical\nphysics and algorithm.\nBrief summary of Wu et al., Optica 2024\nThe context and challenge. Our former publication addressed the challenge of measuring several\nwavefronts to image beyond an isoplanatic patch in the context of fluorescence microscopy.\nThe optical element. To achieve this, we used a thin diffuser, which yields non-periodic intensity patterns\nhaving remarkable achromatic properties, as studied in Ref. [17] (Berto et al. Opt. Lett. 42(24), 5117 (2017)).\nBeing almost achromatic, this kind of thin diffuser is highly unsuitable for hyperspectral wavefront sensing.\nThe algorithm. The algorithm developed in this former context enabled the unmixing of wavefronts\noriginating from up to 5 discrete fluorescent point sources separated by large angles. The unmixing algorithm\nsolely relied on the non-periodic nature of the diffuser (in addition to large angular memory effect).\nBrief summary of the achievements reported in our current manuscript\nThe context and challenge. Our current manuscript addresses the challenge of single-shot hyperspectral\nwavefront sensing for laser metrology and label-free microscopy imaging.\nThe optical element. Identifying an optical element suitable for hyperspectral wavefront sensing far from\nstraightforward. The optical element should exhibit a short spectral correlation width to exhibit different\nspeckle patterns for different information channels while maintaining the “angular memory” effect, at play in\nWFS to encode wavefront information in patterns distortions. This point is now explained in detail in a new\nsection.\nThe algorithm. While Ref. [20] was limited to a discrete set of well separated (angle) signals, the current\nimplementation achieves multiplexed wavefront sensing over a contiguous (spectral) data cube. This new\ncapability – critical for a wider range of applications, including hyperspectral microscopy and laser\nmetrology involving broadband sources – is enabled thanks to a rigorous theoretical characterization of the\ninverse problem, supported both by analytical derivations and numerical simulations.\nNew theoretical ground. We demonstrate that, under appropriate physical and mathematical conditions, the\ninverse problem is well-posed, which had not been done before. Furthermore, as suggested by the valuable\nfeedback of Reviewer 3, this manuscript introduces a complete analytical framework of complex media\nsuitable for hyperspectral wavefront sensing.\nConsequently, despite the difficulty to explicitly relate to our previous work (beyond the citation already\npresent in the first version of the manuscript) provided that the context, the optical element and the algorithm\ndiffer substantially, we found it relevant to clarify that our work lies in the general and broader framework of\ninformation multiplexing using complex media – an approach that has enabled the multiplexing of spectral,\npolarimetric and wavefront information.\nActions taken:\nOur former work, already cited in the introduction is now cited in the new section 2.1 explaining in details\nthe properties required for the multiplexing optical element to enable hyperspectral WFS. We now added the\nfollowing sentence (new ref. [33]):\nFurthermore, complex media have been shown to be well suited for multiplexed spectral [30, 31]\nand polarimetric [32] imaging, and even for multiplexed wavefront sensing in a single spectral\nchannel [33].\nReviewer #3\nThe paper from Blochet et al. reports a novel optical scheme to measure the wavefront phase across a large\nspectral domain. The idea is to use the spectral decorrelation of the speckle arising after a multicore fiber and\nits speckle distortion to recover hyperspectral wavefront cubes through an unmixing algorithm. The paper is\nvery well written, with clear explanations and is nicely illustrated in two cases where hyperspectral wavefront\ncubes are acquired in a single shot for a high-power fs pulsed laser and to recover microscopic phase images.\nThe paper is of very high scientific level both from the experimental implementation and the computational\nunmixing algorithm that is implemented.\nWe thank the reviewer for highlighting the clarity and the scientific significance of our manuscript.\nBeside this overall very good impression of the work I have few points that I would like the author to address\nin a revised version of their manuscript.\n1- Could the authors provide more insight and explanations on how the spatial resolution and spectral\nresolution are linked. This is clear from the paper that they are related to the distance d but the underlying\nphysics is not sufficiently explained nor a table/curve are provided to show the evolution of the spatial and\nspectral resolution with this key d parameter. I think it would be valuable to the paper if discussion could be\nmore developed on this point.\nThe reviewer raises here the very fundamental question of the scaling of the number and the distribution of\nspatio-spectral modes as a function of design parameters. This question is difficult and calls an analytical\nmodel. However, we agree with the reviewer that the scaling law of the instrument with optical parameters is\nof very high importance for designing the instrument and that such additional material is highly valuable to\nthe manuscript impact. For this reason, to answer this highly significant and valuable question, we added\nsome supplementary materials wherein we now derive an analytical model supported by numerical\nsimulations. This new theoretical study now relates the sought-for performances and trade-off of the\ninstrument to the optical and physical parameters of the instrument. In a summarizing paragraph, we also\nnow provide a step-by-step “tutorial” for designing the instrument.\nIn more details, starting from the target performances of the instrument in terms of\n• spectral resolution,\n• spatial resolution,\n• number of spectral channels,\n• wavefront sensitivity,\nWe provide the requirements regarding\n• the spectral dispersion of the MCF\n• the magnification of the relay telescope between the bundle output and the camera\n• the separation distance between the image of the output facet and the camera surface\n• the size of the macro-pixel to consider for numerical solving of the inverse problem\nOur theoretical analysis reveals that the performances and trade-offs of the instrument are driven by three\ndimension-less parameters, namely:\n• the number of sought-for spectral channels\n• the number of interfering fibers at the camera sensor,\n• the number of independent measure points in each macro-pixels at the camera,\nWe do agree with the reviewer that this fundamental physical study leading to a practical design protocol is a\nvery valuable add-on to our manuscript.\nActions taken:\n• We now added the new supplementary section S2\nS2 Scaling of the number of spatial vs. spectral modes with physical parameters\nS2.1 Parameters at play\nS2.1.1 Notations\nThe hyperspectral wavefront sensor can be modeled as an array of monomode fibers imprinting\nrandom optical path differences Ln to an impinging beam, and placed at a distance from in a pixelated\ncamera sensor. The physical parameters at play are then:\n• the separation distance between the monomode fibers a\n• the intensity sampling period p (related to the pixel size through the intermediate telescope\nmagnification)\n• the separation distance d between the fiber bundle and the camera sensor\n• the numerical aperture of the monomode fibers NA\n• the size P of the macro-pixel used for matrix inversion\n• the statistical distribution of the optical path differences ρ(Ln)\n• the average wavenumber k0 and the spectral width ∆k of interest\nThese physical parameters drive the speckle generation and its properties at the camera plane. In fine,\nthe question of the inverse problem can be simplified by considering a system governed by three\ndimension-less parameters:\n• the number of independent spectral modes Nλ\n• the number of interfering fibers N at the camera plane\n• the number of independent measure points M at the camera\nS2.1.2 Scaling of Nλ with physical parameters\nThe number of independent spectral channels Nλ allowed by the hyperspectral wavefront sensor is\ndriven by two factors: the statistical distribution of optical path differences Ln introduced by the\nindividual monomode fibers, and the spectral dispersion generated by diffraction through free-space\npropagation until the camera plane.\nHere, for the sake of simple analytical modeling, we assume a statistical distribution of optical path\nlength differences given by a centered normal distribution ρ(L ) = N (0, ∆L). In this case, Nλ can be\nn\nqualitatively estimated as the width of the two-wavelength mutual coherence function of speckle\npatterns generated by the fibers\n(1)\nwhere δk is the spectral detuning. The width of this correlation function is qualitatively equal to 1/∆L,\nmeaning that the average number of independent spectral channels is\n(2)\nNoteworthy, Nλ does not depend on the number of interfering fibers since this equation is obtained by\nstatistical averaging relying on the central-limit theorem. However, the singular value decomposition\nof the experimental calibration speckles demonstrate that with our MCF, we have Nλ ≃ 10. This value\nis the maximum value but Nλ can be lowered like in the case of multispectral microscopy where we\nonly considered 3 color channels.\nIn addition, free space propagation between the output of the MCF and the camera plane introduces\nsome extra\nspectral dispersion because of differential diffraction angles. In the frame of Fresnel approximation,\nthe diffraction angle scales as 1/λ so resulting in a homothetic dilation of the intensity pattern scaling\nas λd [1, 2]. For a spectral shift inducing an axial elongation larger or equal to the axial dimension of\nthe speckle grain, the speckles correlation is lost. The spectral detuning induced by diffraction is thus\nS2.1.3 Scaling of N with physical parameters\nThe number of interfering fibers N at the camera plane can be simply expressed as\n(4)\nwhich yields, N ≃ 26 × 26 = 676 for the hyperspectral metrology experiments and N ≃ 5 × 5 = 25 for\nthe multispectral microscopy experiments.\nS2.1.4 Scaling of M with physical parameters\nThe matrix inversion performed to retrieve the hyperspectral amplitude and wavefront data requires\ndefining a macro-pixel at the camera. In practice, the size of the macro-pixel P was chosen to\nqualitatively match the spatial spread NA × d of the beam due to propagation between the fiber bundle\nand the camera. For the hyperspectral and multispectral experiments, the beam spread are\nrespectively 65 μm and 12 μm; and the macro-pixel dimensions are P = 50 μm and P = 6 μm,\nrespectively.\nThe number of independent measure-points M used for inversion qualitatively equals the number of\nspeckle grains in the analysis patch. We then get\n(5)\nFor the hyperspectral optics and laser metrology experiments, M is estimated to M ≃ 33×33 = 1089\nand M ≃ 25×25 = 625, respectively. For the multispectral microscopy configuration, M ≃ 4 × 4 = 16.\nS2.1.5 Experimental values\nS2.2 Analytical and numerical modeling\nS2.2.1 Analytical model\nThe ability to resolve the inverse problem is given by the rank of the calibration matrix containing the\nreference patterns over the spectrum of interest. For each macro-pixel, the number of rows of this\nmatrix is M and the number of columns must be at least Nλ. Provided that we always have M ≫ Nλ in\nour experiments, the maximum rank of this matrix is Nλ. We can thus consider that the calibration\nmatrix is of dimensions M × Nλ without loss of generality. The calibration matrix is thus a matrix of M\n× Nλ random intensity values whose each column vector can be normalized. For the sake of\nsimplification, we consider normalizing column vectors after removing their average value. As a\nconsequence, the so-obtained matrix is composed of random independent and identically distributed\nvectors lying on the unit sphere. The singular values distribution of such a matrix is given by the\nMarchenko-Pastur (MP) law and the probability density functions of singular values s is given by\n(6)\nwith r = Nλ/M and where s± = 1 ± √r are the extrema values of the distribution. Consequently, the\nmaximum value of the condition number of such a matrix is\n(7)\nThe system must be designed so as to minimize this upper bound.\nS2.2.2 Numerical model\nFor modeling the experiment, we considered the wavefield generated by a square array of point\nsources having random optical path differences and placed at a distance from a two-dimensional\npixilated sensor array. The resulting wave-field at the detector is given by Fourier transform :\n(8)\nwhere p is the separation distance between the sampling points of the intensity sensor. The numerical\nmodel and the MP model are compared in Figs. S1 and S2. In Fig. S1, the probability density function\nof singular values are shown for the two typical experimental configurations at play in the manuscript\ndemonstrating that the MP model accurately describes our physical system. In particular, this\ntheoretical distribution is bounded by two extrema values s± = 1 ± √r, so providing an estimate of the\nmaximum bound value for the condition number of the reference matrix. In Fig. S2, the histograms of\nthe reciprocal condition numbers obtained from numerical simulations are compared to its theoretical\nlower bound (vertical red dashed lines), again demonstrating the good agreement\nbetween our optical system and the theoretical model.\nS2.2.3 Experimental validation\nWe experimentally measured the probability density function (PDF) of the singular values of the\nreference matrix used in the laser metrology experiment. The results, shown in Fig. S3, demonstrate\nexcellent agreement with both the theoretical model and the numerical simulation.\nS2.2.4 Discussion\nScaling of Nλ and M. The MP law predicts a singular-value distribution whose bounds are s± = 1 ± √r,\nwith r = Nλ/M ∈ [0, 1[ (for M < Nλ, the system is under-determined and cannot be inverted,\nobviously). The distributions are thus centered on the value 1 with standard deviations scaling as √r. In\nthe frame of this model, the condition number of the reference matrix is thus necessarily smaller than\nκ = (1 + √r)/(1 − √r). κ is an increasing function of r that is minimum for M ≫ Nλ and exhibits a\nvertical asymptote for M = Nλ. In this latter case M = Nλ, the lower bound of the distribution is 0, so\npotentially making the smaller singular value arbitrarily small. However, the statistical nature of the\nprocess makes that the reference matrix may be, sometimes, well-enough conditioned to be inverted.\nTypically, the MP model predicts for instance that taking M = 4Nλ qualitatively ensures condition\nnumbers smaller than 3, and that M = 2Nλ yields condition numbers smaller than ≃ 6.\nFor a given number of camera pixels, the number of spatial modes scales as 1/M , since M is the\nnumber of measure points used (and even “required” for optimized systems) for the system to be\ninvertible in each macro-pixel. Moreover, setting a maximum condition number κ, the number of\nspectral modes Nλ is also fixed and scales as M . Consequently, it turns out that the total number of\nmodes (given by the product between Nλ multiplied by the number of spatial modes) qualitatively\nequals the number of camera pixels.\nThe role of N , the number of interfering fibers. The number of interfering fibers is not so important to\nmake a hyperspectral spectrometer. As a smaller limit, a simple pair of point sources can be used as a\nspectrometer. The spectral sensitivity will depend on how far the measure point is with respect to the\nco-phased position. Nevertheless, the random nature of large number of fibers makes the statistical\nMP model work better in this case.\nLimit of the models. In the analytical and numerical models considered here, we removed the average\nvalue of speckles to match the MP-model validity requirements. However, in experiment, we normalize\nthe column vectors with non-zero average values. Since all vectors of the calibration matrix\nconcentrate near the mean direction, i.e. close to the vector 1. As a result, a spike is observed in the\nspectrum due to larger leading singular value (≃ 2), so resulting in an increased condition number as\ncompared to the MP model. The distribution of singular values is no longer supported on a single bulk\ninterval and the remaining singular values follow a deformed MP distribution. The reduced contrast\nwithin the limited quantized dynamic of the detection camera, will thus result in a increased noise level\nas compared to the MP model analytically derived here. Nevertheless, even though the true κ value is\nnot accurately estimated with the MP mode, Nλ is still expected to scale as M.\nS2.3 Conclusion\nIn a nutshell the hyperspectral/multi-spectral wavefront sensor must be designed according to the\nfollowing constraint order:\n1. Fiber bundle parameters:\n• The statistical distribution of optical path lengths in the fiber bundle drives the spectral\nresolution.\n• The number of mono-mode fibers in the bundle is chosen so as to have at least two interfering\nfibers per macro-pixel.\n2. Optical design parameters:\n• The magnification between the output facet of the fiber-bundle and the camera is set so that\nthe camera pixel is small enough to properly resolve the speckle grains.\n• The wavefront sensitivity of the instrument is tuned by adjusting the separation distance d\nbetween the output of the fiber bundle and the camera.\n• The spectral resolution correction due to free-space propagation is estimated.\n3. Numerical parameters:\n• The size of the macro-pixel, driving M , is chosen equal to the diffraction spatial spread of one\nmono-mode fiber: d × NA.\n• The number of spectral modes Nλ is tuned so that the calibration matrix is “well-enough”\nconditioned in every macro-pixel.\nIn the last step, the criterion to estimate that the condition number is small enough depends on the\nexperimental requirements about reconstruction-noise sensitivity.\n• We also added a detailed explanation regarding the parameters scaling in section 3.1:\nThe distance d must be tuned so as to control the trade-off between spatial resolution vs. wavefront-tilt\nsensitivity. Increasing d decreases the spatial resolution but improve the sensitivity (according to eq (2)).\nAdditionally, a lower spatial resolution allows for a larger neighborhood to be considered when solving\nEq. (4) , so enabling the use of more pixels to reconstruct additional wavelengths.\nFor a distance d = 88 mm from the fiber image plane, we measured correlation widths of the order of 2 nm\nall over the spectrum 500 − 800 nm (Fig. 1d). Both the angular memory effect and the spectral correlation\nwidths are suitable for hyperspectral wavefront sensing applications.\nAs detailed in the Supplementary Information, once interference occurs between even a few cores, the\nspectral correlation width is determined solely by the intrinsic properties of the fiber (see Section S3\nand Supp. Fig. S4). For a distance d = 88 mm between the fiber image plane and the camera, we\nexperimentally measured correlation widths on the order of 2 nm across the entire 500 − 800 nm\nspectral range (Fig. 1d). Both the angular memory effect and the resulting spectral correlation widths\nare well suited for hyperspectral wavefront sensing applications. The experimental parameters must\nthen be chosen based on the desired measurement characteristics. In particular, the wavefront\nsensitivity defines the optimal distance d between the MCF and the camera. Given that our algorithm\ncan reliably detect displacements as small as 1/50 of a pixel (see Section Fig SI), increasing d enhances\nsensitivity. However, due to free-space propagation, this same distance also sets the spatial resolution\nof the wavefront sensor, which scales as NA · d. The ability to solve the inverse problem depends on the\nrank of the calibration matrix that encodes the reference patterns across the desired spectral range.\nAccording to the Marchenko–Pastur law for random matrices, the distribution of singular values is\ngoverned by the ratio r = N /M ,where N is the number of desired spectral channels and\nλ λ\nis the number of independent spatial modes sampled within a given patch with k0 the\ncentral wavenumber and P the patch size that should match the spatial resolution NA.d. The upper\nbound for the condition number κ, which reflects the invertibility and stability of the problem, is given\nby\n(5)\nas detailed in SI Section 2.2.1. To ensure a well-conditioned system, the number of spectral channels\nNλ must be chosen such that κ remains sufficiently small, depending on the desired reconstruction\nfidelity and tolerance to noise.\nFor instance, d=88mm the spectral decorrelation width is 3nm, at d=6mm I could not find the spectral\ndecorrelation width. How the spatial and spectral resolutions scale with d?\nAs detailed in the added supplementary materials, the number of interfering fibers is not critical for enabling\nhyperspectral spectrometry, as long as at least a few fibers interfere coherently. This point is illustrated in the\nfigure below, where we measure the speckle contrast under broadband illumination (5 nm bandwidth) as a\nfunction of the distance from the fiber output. The contrast rapidly reaches a plateau, indicating that a\nminimal number of interfering cores is sufficient to generate stable speckle patterns for spectral\ndiscrimination.\nAction taken\nWe added additional calibrations of MCF in SI as a function of the MCF length and the distance between the\nimaged MCF output fiber and the camera.\nS3 Additional experimental characterization of the multicore fiber bundles\nCaption: Left. Spectral correlation widths measured for different bundle lengths over the visible\nspectrum. Right. Spectral correlation width for a 8cm MCF as a function of the MCF image and the\ncamera distance d.\n- The multicore fiber (MCF) is used here as a thin diffuser, in a previous implementation [17] the authors\nused a thin diffuser to perform wavefront sensing. In this case the angular memory effect encodes wavefront\ngradients in intensity pattern displacements but should exhibit also spectral decorrelation. This is not clear to\nme what are the advantages of using a MCF over a thin diffuser for distinct spectral pattern encoding. Is it\nbecause the spectral decorrelation is stronger with MCF as compared to thin diffuser? It would be nice to\ninclude a discussion to clarify this comparison between a thin diffuser and a MCF.\nWe thank the reviewer for this insightful question, which points to a key design choice in our\nsystem. In response, we have added a dedicated discussion in Section 2.1 of the revised manuscript\n(Suitable Optical Mask for Single-Shot Hyperspectral WFS) to motivate and clarify the use of a\nmulticore fiber over other speckle-generating masks, such as thin diffusers.\nTo enable simultaneous wavefront and spectral imaging, the optical mask must exhibit both a\nbroad angular memory effect and a narrow spectral correlation width. Thin diffusers fulfill the\nfirst criterion but not the second. In such systems, spectral decorrelation arises primarily from free-\nspace propagation effects, which induce dilation of the speckle patterns. However, this effect is\nweak when the mask-to-camera distance is on the millimeter scale, as typically implemented (see\nBerto et al., Opt. Lett., 2017). Consequently, thin diffusers provide high angular memory effect\nbut poor spectral discrimination, making them unsuitable for hyperspectral wavefront sensing.\nWe also point out that even at larger propagation distances, where free-space propagation introduces\nsome spectral decorrelation (see DiffuserSpec: Spectroscopy with Scotch Tape, Waller & Boyden),\nreliable hyperspectral reconstruction remains fundamentally unfeasible. The main limitation stems\nfrom the non-dispersive nature of the thin diffuser: at first order, changes in wavelength induce\nonly spatial dilation of the speckle pattern, which, in a small local patch, reduces to a simple\ntranslation. This causes degeneracies in the inverse problem, where a pattern at one wavelength\ncan be confused with a translated version of another, making spectral unmixing non-unique and\nunreliable.\nAn alternative to thin diffusers could be the use of thicker diffusers, as they exhibit improved\nspectral sensitivity due to increased path length distribution. However, this comes at the cost of a\nsignificantly degraded angular memory effect as the thickness increases, which severely limits\ntheir applicability for wavefront sensing. For instance, we measured that achieving the desired\nspectral correlation width of ~2 nm with a thick diffuser results in an angular memory range\nreduced to approximately 1 mrad which is insufficient for practical wavefront measurements. In\ncontrast, multicore fibers (MCFs) provide a compelling solution by decoupling angular and\nspectral correlations. The angular memory effect remains high (greater than 20 mrad) regardless of\nfiber length, while the spectral correlation width narrows with increasing fiber length, enabling\nhyperspectral discrimination. MCFs therefore meet both essential requirements: a wide\nangular memory range for wavefront gradient encoding and a narrow spectral correlation width for\naccurate spectral demixing.\nWhile the precise origin of spectral decorrelation in MCFs is not yet fully understood, it likely\narises from statistical variations in longitudinal wave vectors, due to minor differences in core\ndiameters and possibly inter-core coupling. These imperfections introduce effective dispersion\nacross the fiber cores. Although not intentionally engineered, this effect proves beneficial in our\ncontext and enables nanometer-scale spectral sensitivity..\nAction taken\nWe added section 2.1 to motivate the choice of a multicore fiber versus others speckle-generating alternatives\n(thin diffuser, thick diffuser)\n2.1 Suitable Optical Mask for Single-Shot Hyperspectral WFS\nIn the quest for instrument simplification, spectrally-resolved wavefront sensing calls for a single\noptical element that simultaneously provides both spectral discrimination and wavefront sensitivity.\nSeveral masks can be placed in the close vicinity of a camera that result in wavefront-sensitive\nintensity patterns: arrays of holes, microlens arrays [19], meta-micro-lens arrays [24], thin diffuser\n[23], diffraction gratings [25]. A key characteristic for wavefront reconstruction is that a local tilt of\nthe wavefront is transmitted through the mask, such that a wavefront tilt at the entrance of the mask\nresults in a local translation of the intensity pattern at the camera plane. This principle, generalized in\nthe context of scattering media and coined “memory effect” [21], is satisfied by optical elements whose\ntransmission matrix is diagonal in the canonical basis.\nThe spectral sensitivity of a mask is characterized by the spectral correlation width of the intensity\npatterns at the camera. All the masks listed above provide remarkably achromatic WFS properties.\nSpectral decorrelation may arise from chromatic dispersion within the mask itself or from free-space\npropagation over the distance d between the mask and the camera. However, axial dilation of intensity\npatterns due to free-space propagation [26, 27] contributes only marginally to spectral sensitivity when\nthe mask is positionned at millimeter-scale distances [23]. To achieve hyperspectral WFS, the use of\nspectrally dispersive optics is therefore required. For interference effects to be wavelength-sensitive,\nthese optics must involve a diversity of large optical path lengths. Nanometer-scale spectral sensitivity\nis typically achieved using complex media, such as multiply scattering samples [28] or multimode\nfibers [29]. Notably, diffraction gratings are achromatic at short distances [16] and thin diffusers –\nwhich we have previously shown to enable effective WFS – are only weakly dispersive [23].\nFurthermore, complex media have been shown to be well suited for multiplexed spectral [30, 31] and\npolarimetric [32] imaging, and even for multiplexed wavefront sensing in a single spectral\nchannel [33].\nTherefore, optics that satisfy both spectral and angular properties required for achieving spectrally-\nresolved wavefront sensing are not numerous. For instance, thick multiply scattering diffusers exhibit\nimproved spectral sensitivity at the expense of degraded angular memory effect as thickness increases,\ndrastically limiting the applicability to WFS [21]. In contrast, MCF exhibit both large angular memory\neffect [34] and narrowband spectral correlation properties [35]. MCF exhibit a diagonal transmission\nmatrix if fibers are perfectly monomode and uncoupled. In practice, many bundles are composed of\nfew-modes fibers, so limiting the angular range of the memory effect. The physical mechanism behind\nthe limited spectral correlation width in these fibers is still unclear but involves statistical dispersion in\nlongitudinal wavenumbers in individual fibers due to variations in their diameters and potentially\ncoupling between fibers. This dispersion is not specified by manufacturers and probably results from a\nlack of control in the fabrication process.\n- I wonder if a combination of a thin diffuser and a grating would perform the same spatio-spectral encoding\navoiding the complexity of the MCF fiber core fixed spacing, could the author comment on that?\nWe thank the reviewer for their interesting question about this alternative configuraiton. However, the\nconfiguration combining a thin diffuser and a diffraction grating is not suitable for hyperspectral wavefront\nsensing over a contiguous spectrum. If the thin diffuser produces achromatic speckle patterns, the system\nwill yield spectrally shifted but structurally identical speckle fields: essentially, translated versions of the\nsame intensity distribution at different wavelengths. As a result, the forward model described by equation (4)\nin the main manuscript becomes non-injective, and the inverse problem lacks a unique solution.\nSuch a configuration can only be effective for a discrete set of well-separated wavelengths, where the\nspectral shift induced by the grating is larger than the maximum speckle displacement caused by wavefront\ndistortions. This condition ensures that the speckle patterns remain distinguishable across angle, as used in\n[Wu et al., Optica] for multiplexed angular imaging and [Kim et al., Opt. Express 29(13):19506–19514\n(2021)] for multispectral wavefront sensing.\nFor microscopy applications, the point spread function (PSF) must be adequately sampled by the Hartmann\nmask; in practice, this means that the PSF should cover a couple of fiber cores. Fortunately, the inter-core\nspacing of the multicore fiber (MCF) closely matches the scientific camera pixel size and the typical PSF\nsize found in standard microscope designs, making it well suited for this purpose. The fixed core-to-core\nspacing inherent to MCFs has not posed any significant limitation for wavefront sensing in our\nimplementation, unlike in imaging applications, where the regular bundle structure can become visible and\nmay introduce artifacts. The primary challenge when using a multicore fiber lies in its limited field of view,\nwhich requires intermediate relay optics to properly image the fiber facet onto the sensor.\nAction taken:\n• In the introduction, we now make it clear that the main contribution of our work is to combine the\ntwo optical properties required by hyperspectral WFS in a single optics for the sake of optical\nsimplicity and stability:\nAs a general principle, cascading existing techniques for achieving multiplexed optical sensing results\nin increased experimental complexity and spatial footprints, so hindering practical use and\ndissemination of the technique.\nAll hyperspectral techniques developed thus far have abandoned the principles of Hartmann-based WFS,\nconsequently forfeiting several of its many advantages that made it highly popular for metrology and\nquantitative phase imaging [17–19]: simplicity, compactness, single-shot capability, robustness to vibrations,\nand compatibility with broadband light sources.\nHere, we propose a single-shot hyperspectral wavefront sensing scheme based that combines both spectral\ndiscrimination and wavefront sensing functionalities into a single mask imaged at a close distance from\na camera sensor. Our approach thus builds on the same principles underlying conventional WFS [19,20]\nin which the local wavefront tilts are encoded as local translations of the intensity pattern after\npropagation to the camera. This design has become one of the most widely used approaches in\nmetrology and quantitative phase imaging [16–18] because of its simplicity, compactness, single-shot\ncapability, robustness to vibrations, and compatibility with broadband light sources.\n• We added a full section 2.1 to better explain the choice of a MCF as a Hartmann mask.\nResponse to reviewers\nWe thank the reviewers for their positive appreciation of our work and we thank them again for helping\nimproving our manuscript.\n• Reviewers comments appear in black ink\n• Our answers are appended in blue ink\nReviewer #1 (Remarks to the Author)\nThe authors have provided a comprehensive and detailed response to the initial reviews. I value their\neffort and rigor. However, framed as it is, I remain doubtful whether the presented technique truly\npresents a methodological breakthrough rather than an intriguing proof of concept of *a* method for\nhyperspectral wavefront sensing.\nThe authors claim you need the transmission matrix to be diagonal in the canonical (position) basis -\nwhat they call the \"memory effect.\" Their argument is:\n\"This principle, generalized in the context of scattering media and coined 'memory effect', is satisfied\nby optical elements whose transmission matrix is diagonal in the canonical basis.\"\nBut I would argue that this is unnecessarily restrictive. The actual requirement is that your forward\nmodel must be invertible within your computational constraints. Diagonality in the position basis is\nsufficient but not necessary. I believe this might be best explained in the context of two other works,\ni.e. the IMPALA concept by Smartsev et al. and the RAVEN by Howard et al. (their new reference 15).\n- IMPALA uses a structured pinhole mask to generate a known speckle pattern structure in the far field,\nwhich can then be used to retrieve the phase of multiple colors. This is arguably the simplest possible\nsetup to achieve spatio-spectral encoding; the method is however limited in resolution as interferences\nscale up combinatorially. While Smartsev used it in a multi-shot setup, there is no reason this would not\nwork in single-shot mode (except for the reduced spatial resolution). Note that the transmission\nfunction is decidedly not diagonal in position space - it's a complex speckle pattern from a sparse\nbinary mask. However, in Fourier space, each pinhole pair creates interference at a specific frequency.\nThe pseudo-random arrangement ensures these components don't overlap significantly. So the forward\nmatrix is approximately diagonal - but in the Fourier basis, not the canonical basis.\n- RAVEN presents an intentional separation between the phase encoding and the spectral encoding,\ndemonstrating that the full laser field can be recovered. One can note that RAVEN does not work\nbecause of diagonal structure in the canonical basis but because of an engineered block structure.\nArguably, the RAVEN can be interpreted as a \"thick\" optical element whose combined forward model\nis complex but engineered to be invertible.\nI acknowledge RAVEN's publication timing (after this manuscript's submission), so this is not a\ncriticism of novelty (for the purpose of laser characterization, it maybe argued that RAVEN provides\nsuperior results). Rather, RAVEN and IMPALA demonstrate that the theoretical constraints the authors\npropose are not fundamental to the problem: It appears to me that the paper conflates physical structure\n(thin element, position-space locality) with mathematical structure (invertible forward model). These\nare distinct. One could, in principle, work with complex media violating the memory effect entirely -\nthe constraint is computational tractability, not optical physics.\nThis matters because the authors frame their MCF approach as uniquely addressing fundamental\nrequirements. In reality, they've made a particular design choice - trading optical simplicity (one\nelement) for algorithmic complexity. This is not to say that their method does not work. Rather, the\nframework the authors propose does not seem to hold up entirely to scrutiny.\nWe thank the reviewer for sharing their opinion about the potential impact of our approach and for\nplacing our contribution in the context of alternative techniques.\nWe fully agree with the reviewer that using a single optical element, as in the IMPALA technique,\nsimplifies the system design. In this respect, we argue that the angular memory effect provides an\neffective compromise between optical and computational simplicity, since it relies on principles similar\nto those used in Shack–Hartmann wavefront sensing.\nWe further demonstrate that our design achieves wavefront sensing at a higher spatial resolution than\nthe current state-of-the-art IMPALA approach reported in the literature, even in its lowest-resolution\nconfiguration. Moreover, our method allows flexible tuning between spectral and spatial resolution,\nfollowing rigorous mathematical criteria derived from standard linear algebra.\nAsides, the authors argue extensively that RGB cameras cannot provide 'metrology-grade\nperformances' without 'significant additional instrumental and numerical efforts.' However, the MCF\napproach also requires extensive calibration: sequential wavelength scanning across the full spectral\nrange, precise distance measurements accounting for chromatic aberrations, and characterization of the\nangular memory effect. The objection appears to rest on spectral overlap in RGB filters, but this is\nmathematically equivalent to solving the linear system RX = M where R includes the filter responses.\nThis is no more complex than their Equation (4). So the claim of significantly reduced calibration effort\nis not really substantiated, thereby undermining the relevance of their example for high-resolution\nimaging.\nWe agree with the reviewer that developing a hyperspectral wavefront sensor based on color or\nhyperspectral cameras is certainly of interest but not simpler than our approach.\nIn conclusion, the experimental work is solid, the presentation outstanding and the authors have\nsuccessfully demonstrated hyperspectral wavefront sensing using a multicore fiber. The\ncharacterization of the fiber's properties and the reconstruction algorithm represent genuine\ncontributions. However, the theoretical framework positioning this as a uniquely suitable or\nfundamentally simpler approach does not withstand scrutiny against recent literature. Currently, the\nmanuscript presents a proof-of-concept for *a* method for single-shot hyperspectral wavefront sensing,\nbut the attempt to frame it as *the* solution to longstanding challenges in this field is not sufficiently\nsupported.\nReviewer #2 (Remarks to the Author)\nThe authors have made significant changes and improvements in line with my and the other reviewer's\ncomments and suggestions. I believe that the paper could be accepted now, and is much more readable\nand important as a result.\nReviewer #3 (Remarks to the Author)\nThe authors have adequately addressed my comments.\nI consider the paper is suitable now for publication."}]

Decision letter