# Photon Energy-Resolved Velocity Map Imaging from Spectral Domain Ghost
Imaging

Jun Wang<EMAIL_ADDRESS>Stanford PULSE Institute, SLAC National
Accelerator Laboratory, Menlo Park, CA 94025, USA  Department of Applied
Physics, Stanford University, Stanford, CA 94305, USA  Taran Driver
<EMAIL_ADDRESS>Stanford PULSE Institute, SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA  SLAC National Accelerator Laboratory,
Menlo Park, CA 94025, USA  Felix Allum  Stanford PULSE Institute, SLAC
National Accelerator Laboratory, Menlo Park, CA 94025, USA  Christina C.
Papadopoulou  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607
Hamburg, Germany  Christopher Passow  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  Günter Brenner  Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Siqi Li  SLAC National
Accelerator Laboratory, Menlo Park, CA 94025, USA  Stefan Düsterer  Deutsches
Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Atia Tul
Noor  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg,
Germany  Sonu Kumar  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85,
22607 Hamburg, Germany  Philip H. Bucksbaum  Stanford PULSE Institute, SLAC
National Accelerator Laboratory, Menlo Park, CA 94025, USA  SLAC National
Accelerator Laboratory, Menlo Park, CA 94025, USA  Benjamin Erk  Deutsches
Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Ruaridh
Forbes  Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo
Park, CA 94025, USA  SLAC National Accelerator Laboratory, Menlo Park, CA
94025, USA  James P. Cryan<EMAIL_ADDRESS>Stanford PULSE Institute,
SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA  SLAC National
Accelerator Laboratory, Menlo Park, CA 94025, USA

(Oct 2022)

###### Abstract

We present an approach that combines photon spectrum correlation analysis with
the reconstruction of three-dimensional momentum distribution from velocity
map images in an efficient, single-step procedure. We demonstrate its efficacy
with the results from the photoionization of the $2p$-shell of argon using the
FLASH free-electron laser (FEL). Distinct spectral features due to the spin-
orbit splitting of Ar${}^{+}(2p^{-1})$ are resolved, despite the large average
bandwidth of the ionizing pulses from the FEL. This demonstrates a clear
advantage over the conventional analysis method, and it will be broadly
beneficial for velocity map imaging experiments with FEL sources. The
retrieved linewidth of the binding energy spectrum approaches the resolution
limitation prescribed by the spectrometers used to collect the data. Our
approach presents a path to extend spectral-domain ghost imaging to the case
where the photoproduct observable is high-dimensional.

##  I Introduction

Modern x-ray spectroscopy provides a sensitive probe of local electronic
density in molecular systems with atomic-site specificity Siegbahn (1982);
Picón  _et al._ (2016); Erk  _et al._ (2014). As a result, x-ray free-electron
lasers (XFELs) with continuous wavelength tunability throughout the soft x-ray
region, unparalleled peak brightness, and short temporal duration, have
enabled significant advances in time-resolved measurements of molecular
dynamics Wolf  _et al._ (2017); Jay  _et al._ (2022); Brauße  _et al._ (2018);
Allum  _et al._ (2022). Oftentimes the intrinsic photon energy jitter of a
self-amplified spontaneous emission (SASE) XFEL, along with the strongly
fluctuating sub-structure, is thought to be the limiting factor for achievable
energy resolution in time-resolved spectroscopy Brauße  _et al._ (2018); Allum
_et al._ (2022); Mayer  _et al._ (2022). However, the spectral fluctuations
inherent to SASE operation can, in fact, be exploited as a notable advantage
by correlating x-ray observables with properties of the incident pulse on a
shot-to-shot basis. This is a powerful approach to improving the resolution of
experiments at XFEL facilities, beyond the bandwidth limit. In particular, the
application of the spectral-domain ghost imaging (SDGI) technique has
demonstrated sub-bandwidth resolution for spectroscopies employing XFELs
Driver  _et al._ (2020a); Li  _et al._ (2021); Kayser  _et al._ (2019); Klein
_et al._ (2022).

Thanks to their high throughput, $4\pi$ collection solid-angle, and ability to
provide angle-resolved photoproduct yields, velocity map imaging (VMI)
spectrometers Eppink and Parker (1997) have emerged as a popular instrument
for time-resolved XFEL studies of dynamics in gas phase systems Erk  _et al._
(2014); Squibb  _et al._ (2018); Eppink and Parker (1997); Rouzée  _et al._
(2011); Fushitani  _et al._ (2021); Prince  _et al._ (2016); Brauße  _et al._
(2018); Allum  _et al._ (2022). A VMI spectrometer typically measures the two-
dimensional (2D) projection of the three-dimensional (3D) momentum
distribution of charged particles. In most cases, however, the quantity of
interest is the underlying 3D distribution or another related quantity such as
the kinetic energy (KE) spectrum, access to which requires post-processing of
the 2D projection that inverts the Abel transform Garcia  _et al._ (2004);
Roberts  _et al._ (2009); Dick (2014); Montgomery Smith  _et al._ (1988);
Vrakking (2001). In trying to apply SDGI to retrieve the photon energy-
resolved KE spectrum of charged particles from a VMI data set, the momentum
projection, inherent to the VMI operation principal, could complicate the
procedure. A feature with a specific KE is mapped to a broad range of pixels
on the VMI detector, and thus features with distinct KEs will produce heavily
overlapping features in the projected distribution This is in contrast with
previous applications of SDGI, where KE is more directly related to the
experimental measurement Li  _et al._ (2021). In this work, we present a
single-step regression approach which can simultaneously reconstruct the
spectral response and the 3D momentum distribution by exploiting correlation
between the single-shot VMI image and the corresponding photon spectrum
demonstrating that the projection inherent in the VMI concept is not
detrimental to the SDGI procedure.

Figure 1: Schematic of a typical experimental layout that correlation analysis
approaches can be applied to. XFEL pulses are incident on gaseous sample. The
vital observables for our approach are the charged particles’ VMI images and
the x-ray spectra simultaneously measured with a high-resolution photon
spectrometer on a shot-to-shot basis. Throughout this work, we refer the VMI
axis as $x$ direction, and in this figure the x-ray is polarized along $z$
direction and propagates along $y$.

The primary components of a typical experiment where correlation analysis can
be applied is illustrated in Fig. 1. Briefly, a stochastic light source (here
the SASE XFEL Feldhaus (2010)) produces pulses that intercept a grating. The
reflected beam (zeroth order) is directed toward the interaction point of a
VMI spectrometer, which records the single-shot photoelectron momentum
distribution. The incident spectrum is measured from the first order
diffraction of the grating Brenner  _et al._ (2011).

In Sec. II, we describe our single-step regression approach. In Sec. III, we
demonstrate it with an experiment at the Free-electron LASer in Hamburg
(FLASH), where spectral resolution below the average bandwidth of the incident
x-ray pulses has been achieved. We point out that sub-bandwidth resolution is
a common advantage of photon spectrum correlation analysis approaches. In Sec.
IV, we describe the advantage we observe in regularizing with a single-step
reconstruction compared to other possible implementations. Our approach
promises to enhance the energy resolution of XFEL experiments using VMI,
opening up opportunities to study time-dependent phenomena through the
variation in finely resolved energy structures Inhester  _et al._ (2019);
Allum  _et al._ (2022).

##  II Method

###  II.1 Model

The primary quantity of interest in VMI measurements is the 3D momentum
distribution of charged particles, $f(\bm{p})$, which depends on the spectral
profile of the incoming pulse, according to

$f(\bm{p})=\int\chi(\bm{p},\omega)a(\omega)\mathrm{d}\omega,$ (1)

where $a(\omega)$ is the frequency spectrum of an incident pulse, and
$\chi(\bm{p},\omega)$ is the (linear) response to the spectral intensity at
frequency $\omega$. This model applies to light-matter interactions in the
linear regime, such as single-photon ionization. The image obtained in a VMI
spectrometer is given by the Abel transform of $f$ with a mapping from
momentum to position $\bm{r}=\bm{p}/\sqrt{2m\alpha}$,

$b(y,z)=\int\tilde{\chi}(\bm{r},\omega)a(\omega)\mathrm{d}x\mathrm{d}\omega~{},$
(2)

where $x$ is the direction of projection, $\alpha\propto
mt_{\mathrm{ToF}}^{-2}$ is the magnification factor depending on the time-of-
flight $t_{\mathrm{ToF}}$ and particle mass $m$, and
$\tilde{\chi}(\bm{r},\omega)=(2m\alpha)^{3/2}\chi(\bm{p},\omega)$ is the
$\omega$-dependent momentum distribution.

Figure 2: Schematic of the overall analysis procedure. Given a single-shot
spectral profile, our model composes a 3D momentum distribution based on
parameters $C$ and conducts the Abel transform, obtaining an expected 2D
distribution to be compared with the corresponding VMI image. Tuning the model
parameters arrives at an optimal point of the objective function that consists
the total discrepancy and the regularization terms. The optimal set of
parameters is eventually converted to the photon-frequency dependent momentum
distribution.

Our analysis approach is summarized graphically in Fig. 2. The spectral
response of the sample, $\chi(\bm{p},\omega)$, is extracted by fitting a model
to a combination of single-shot VMI images and photon spectra. The most likely
$\tilde{\chi}$ is the function that minimizes the difference between the
measured and predicted electron momentum distributions, subject to
regularization terms that favor sparsity and smoothness in the
$\omega$-dependent distribution $\tilde{\chi}$. We restrict our considerations
to cases with cylindrical symmetry, where the axis of symmetry ($z$), is
oriented perpendicular to the direction of projection ($x$), such as the case
shown in Fig. 1. Other configurations satisfying the same symmetry
requirements are discussed in Sec. IV.4 . Such a model is sufficient to
describe experiments using either linearly- or circularly-polarized laser
pulses acting on isotropic samples. The Abel transform is uniquely invertible
under this symmetry condition, and the quality of the reconstruction will
depend on the number of FEL shots used.

###  II.2 Implementation

In general, the inverse Abel transform is ill-conditioned, being susceptible
to experimental noise Pretzier  _et al._ (1992). Several inversion algorithms
Garcia  _et al._ (2004); Roberts  _et al._ (2009); Dick (2014); Montgomery
Smith  _et al._ (1988); Vrakking (2001) have been developed to mitigate this
issue and to robustly reconstruct the underlying 3D momentum distribution. In
our method we adopt the basis functions employed in the well-known pBasex
Garcia  _et al._ (2004) algorithm to represent $\tilde{\chi}(\bm{r},\omega)$
and combine the inversion procedure with spectral domain ghost imaging Driver
_et al._ (2020a); Li  _et al._ (2021). pBasex expands the three-dimensional
momentum distribution with a basis set that is the product of radial basis
functions and Legendre polynomials:

$\tilde{\chi}(\bm{r},\omega)=\frac{1}{2\pi}\sum_{l=0}^{N_{L}}\sum_{k=1}^{N_{K}}c_{lk}(\omega)f_{k}(r)P_{l}(\frac{z}{r})~{},$
(3)

where $P_{l}(z/r)$ is the $l$-th order Legendre polynomial, and the sum over
$l$ can be truncated at twice the highest order of light-matter interaction
$N_{L}$ (i.e. twice the number of photons involved in the ionization process).
$\\{f_{k}(r)\\}_{k=1}^{N_{K}}$ is a set of radial basis functions, and
$c_{lk}(\omega)$ are their $\omega$-dependent coefficients.

Combining Eqn. (3) with Eqn. (2), the projected 2D distribution can be written
as:

$b(y,z)=\sum_{l=0}^{N_{L}}\sum_{k=1}^{N_{K}}\int
c_{lk}(\omega)G_{lk}(y,z)a(\omega)\mathrm{d}\omega~{},$ (4)

where $G_{lk}(y,z)=\int f_{k}(r)P_{l}(z/r)\mathrm{d}x/(2\pi)$, is the
projection of each basis function perpendicular to the axis of cylindrical
symmetry. It is worth noting, that after discretization on a Cartesian grid
$\bm{R}_{q}=(y_{q},z_{q})$, the projections $G_{lk,q}=G_{lk}(y_{q},z_{q})$ are
the same integrals encountered in the standard pBasex inversion method (Eqn. 4
of Ref. Garcia  _et al._ (2004)). Being further uniformly discretized in
photon energy $\omega_{w}$, Eqn. (4) is

$\hat{B}_{i,q}=\sum_{l=0}^{N_{L}}\sum_{k=1}^{N_{K}}\sum_{w=1}^{N_{\omega}}G_{lk,q}A_{i,w}C_{lk,w}~{}.$
(5)

where for each single-shot, $i$, $\hat{B}_{i,q}$ is the expected intensity at
pixel $q$ given the spectral profile $A_{i,w}=A_{i}(\omega_{w})$ and the
spectral response of the sample encoded in the coefficients
$C_{lk,w}=C_{lk}(\omega_{w})$. Solving for the coefficients $C$ from the
expected image $\hat{B}$ is straightforward, as the tensor $H\equiv G\otimes
A$ can be reshaped into a matrix $H_{iq,lkw}=G_{lk,q}A_{i,w}$, whose pseudo-
inverse maps $\hat{B}$ to $C$. However, solving $C$ from the measured images
$B$ by application of the pseudo-inverse, is highly sensitive to experimental
noise in both $B$ and photon spectra $A$. A more robust approach is to
minimize an objective function, $h$ consisting of $h_{0}$, which quantifies
the model-measurement discrepancy, and regularization terms favoring expected
qualities of $C_{lk,w}$, such as sparsity and smoothness Li  _et al._ (2021),
which is similar to the procedure discussed in Ref. Dick (2013). In our case,
$h(C)$ is

$h(C)=h_{0}(C;A,G,B)+\lambda_{\mathrm{sp}}h_{\mathrm{sp}}(C)+\lambda_{\mathrm{sm,\omega}}h_{\mathrm{sm,\omega}}(C)+\lambda_{\mathrm{sm,r}}h_{\mathrm{sm,r}}(C)~{},$
(6)

where

$h_{0}(C;A,G,B)=\sum_{i}\sum_{q}W_{q}\left|\hat{B}_{i,q}(C;A,G)-B_{i,q}\right|^{2}$
(7)

is the Gaussian log-likelihood up to a global factor, with weights $W_{q}$
over each pixel $q$ in the VMI image. Weights $W_{q}$ can be chosen to enhance
the sensitivity of the reconstruction to certain regions of the VMI image, and
henceforth we denote $W$ to be the diagonal matrix constructed by them.
$h_{\mathrm{sp}}(C)$ and $h_{\mathrm{sm,\cdot}}(C)$ are the sparsity and
smoothness regularization terms, respectively.

We separate smoothness into two terms $h_{\mathrm{sm,\omega}}(C)$ and
$h_{\mathrm{sm,r}}(C)$, to differentiate the smoothness along the frequency
($\omega$), and radial ($r$) directions. The corresponding hyperparameters,
$\lambda_{\mathrm{sp}},\lambda_{\mathrm{sm,\omega}},\lambda_{\mathrm{sm,r}}$
control the degree to which sparsity and smoothness are enforced in the
retrieved $\tilde{\chi}$. For each direction in $d=\omega,r$, the smoothness
term quantifies the roughness with the second order difference, i.e.
$h_{\mathrm{sm,d}}(C)=\|L^{(d)}C\|_{2}^{2},~{}d=\omega,r$, with $L^{(d)}$
representing the finite-difference Laplacian operator along direction $d$.
Common choices for the form of $h_{sp}$ include the $\mathrm{L}_{1}$-norm
Tibshirani (1996) and $\mathrm{L}_{2}$-norm squared Tikhonov  _et al._ (1995).
We choose the latter for the demonstration in Sec. III. We discuss strategies
for choosing the proper values for the hyperparameters in the supplementary
material.

From the point of view of implementation, the discrepancy term,
$h_{0}(C;A,G,B)$, is quadratic in the $C$ coefficients, conducting the
summation over shot $i$ to formulate $h_{0}$ into a quadratic form prior to
the optimization significantly improves the efficiency, although at the cost
of caching a large matrix $A^{T}A\otimes GWG^{T}$ in memory.

##  III Results

Figure 3: Representative single-shot data and average spectra. (a,b) Two
single-shot photon spectra after a global background subtraction, with (d,e)
showing their corresponding VMI images. For visualization only, we have
blurred the raw images with a 3px-wide (standard deviation) gaussian kernel.
(c) Average photon spectrum, with the shade representing the standard
deviation over all FEL shots. (f) Average VMI image, where the left half is
the average raw image, and the right half is the reconstruction of average
image (see main text).

We demonstrate this reconstruction method for a dataset collected from soft
x-ray ionization of argon above the $2p$ ionization threshold Heimann  _et
al._ (1987); Avaldi  _et al._ (1994a); Jurvansuu  _et al._ (2001). The
experiment is conducted using the CAMP instrument Erk  _et al._ (2018) at
beamline BL1 of the FLASH Feldhaus (2010). Argon gas is introduced via
supersonic expansion to produce a continuous molecular beam. Following
collimation through two skimmers and an aperture, the beam is intercepted by
focused x-ray pulses produced by the FEL in the interaction region of the
spectrometer. The velocities of the photoelectrons are mapped to a position-
sensitive microchannel plate/phosphor screen detector and recorded with a CMOS
camera at $10$ Hz.

The pulses have an average bandwidth of $5$ eV full-width at half maximum
(FHWM) at $264$ eV central photon energy. The estimated pulse duration is
$100-150$ fs (FWHM) with a mean pulse energy of $6~{}\mu$J/pulse at the
sample. The incident photon spectrum is recorded on a shot-to-shot basis using
an upstream variable line spacing (VLS) spectrometer Brenner  _et al._ (2011).
The zeroth order beam from the grating is delivered to the interaction point,
while the first order is collected by a detector to image the spectrum.

Figure 4: Comparison between the conventional binning-and-averaging method
(a)(d) and our method (b)(c)(e)(f). (a) Energy distribution obtained with
pBasex applied to the cumulative VMI image, with preliminary grouping of shots
by the centre-of-mass (CoM) of spectral profile. (b-c) Photon-energy resolved
argon $2p$ photoelectron kinetic energy (KE) spectra of the $l=0$ and $l=2$
Legendre polynomial components of the angular distribution, both being
normalized to the maximum intensity in $l=0$. The grey dotted lines delineate
the photon energy range for (e-f). (d-f) Binding energy (BE) spectra of (a-c)
extracted by shifting the KE spectra at each photon energy, see main text. The
brown solid lines are the mean over the photon energy range, and the shaded
regions represent the standard deviation. The red dashed lines in (e-f) are
the sum of fitted Gaussians.

This measurement is performed with linearly polarized light from the FEL,
which implies that the resulting electron momentum distribution has reflection
symmetry about the $xy$ plane. In this case, we can limit the sum in Eqn. (3)
to only even-order Legendre polynomials Yang and Bersohn (1974). The dataset
consists of $2.8\times 10^{4}$ FEL shots, and the representative single-shot
raw VMI images and photon spectra are shown in Fig. 3, where the average VMI
image and photon spectrum are also shown. The single-shot VMI images are quite
sparse, containing 40 electrons in each image on average.

From the minimization of Eqn. (6), we obtain the most probable parameters
$C_{lk,w}$ given the dataset, which represents the fitted
$\chi(\bm{p},\omega)$. Using Eqn. (5), the average image is reconstructed by
averaging the expected intensity $\hat{B}_{i,q}$ across all FEL shots. As
shown in Fig. 3 (f), the reconstructed average image agrees well with the
average of raw images except being less noisy. The reconstructed average
image, nevertheless, is the projected momentum distribution of the mean photon
spectrum and can hardly show the underlying fine structure. With the same set
of most-probable parameters $C_{lk,w}$, a better way to reveal the fine
structure is to reconstruct the KE spectrum for each $l$-th Legendre
polynomial component and each photon energy $\omega$, which is discretized as

$I_{l}(E_{r},\omega_{w})=\sum_{k}C_{lk,w}f_{k}(\sqrt{\frac{E_{r}}{\alpha}})\sqrt{\frac{E_{r}}{4\alpha^{3}}}~{},$
(8)

with $E_{r}=\alpha r^{2}$ denotes the KE at radial grid point $r$. The
ionization from argon $2p$-shell is a single-photon process, so we visualize
$I_{0}(E,\omega)$ and $I_{2}(E,\omega)$, in Fig. 4 (b)(c) respectively.

We observe two dispersive features that correspond to the spin-orbit split
cationic states of argon, ${}^{2}P_{1/2}$ and ${}^{2}P_{3/2}$ Jurvansuu  _et
al._ (2001). For each photon energy in Fig. 4 (b), we shift the reconstructed
KE spectrum to binding energy (BE) according to
$\mathrm{BE}=\hbar\omega-\mathrm{KE}$, which is shown in Fig. 4 (e). The spin-
orbit splitting is well resolved with our approach. We use a previously
reported measurement Jurvansuu  _et al._ (2001) of this splitting, $2.12$ eV,
to calibrate the VMI and VLS energy axes. With this calibration, we extract a
line-width of $1.1$ eV FWHM from Fig. 4 (e), which is resolved beyond the
width of the averaged FEL pulse spectrum, $5$ eV FWHM. Furthermore, the
extracted line-width is below the average bandwidth of a single pulse $3.5$ eV
FWHM. Thus, with the application of spectral domain ghost imaging in this XFEL
experiment, we are able to achieve sub-bandwidth resolution.

In Fig. 4 (c) and (f) we plot the projection of the 3D momentum distribution
onto the second-order Legendre polynomial $I_{2}(E,\omega)$, which is related
to the photoemission anisotropy parameter, $\beta_{2}$ Cooper and Zare (1968).
We extract a value of $0.46\pm 0.05$ and $0.39\pm 0.05$ for the anisotropy
parameter for the ${}^{2}P_{1/2}$ and ${}^{2}P_{3/2}$ ionic states,
respectively. These values are averaged over a range of photon energies from
$261$ to $265$ eV, which is in good agreement with previous measurements in
Ref. Avaldi  _et al._ (1994b) and calculations in Ref. Lindle  _et al._
(1988). The results in Fig. 4 (b)(c)(e)(f) are obtained with a set of
$N_{K}=125$ radial basis functions and uniform pixel weight $W$.

In contrast, directly applying pBasex on the cumulative image results in a
single spectral feature with a width of $5.4$ eV FWHM. Moreover, grouping the
FEL shots by the centre-of-mass of photon spectrum does not resolve the two
spin-orbit features, and the corresponding average binding energy spectrum has
a width of $5.1$ eV (FWHM), as shown in Fig. 4 (a) and (d). In order to obtain
the resolution observed in Fig. 4 (b) and (c), we need to explicitly consider
the correlations between the single-shot spectra and the VMI images.

##  IV Discussion

###  IV.1 Resolution achieved in our demonstration experiment

To characterize the resolution of this technique, it is useful to define the
correlation length of the spectral measurement. The correlation length is a
measure of how strongly coupled the intensity fluctuations are between
neighboring pixels in the photon spectrum. Here, we define this metric
$\delta_{A}$ to be the average distance between two frequencies where the
Pearson’s correlation coefficient drops to $1/e$ Driver  _et al._ (2020b). The
correlation between spectrometer pixels arises from two sources: the
resolution of the photon spectrometer, and the intrinsic variation due to
spectral fluctuations of the source. In our experiment, the correlation length
is measured to be $\delta_{A}=22$ VLS pixels, corresponding to $0.99$ eV, as
shown in Fig. S1(a) of the supplementary material. This is much larger than
the instrumental resolution of the photon spectrometer Brenner  _et al._
(2011), and thus, we conclude that the dominant contribution to the
correlation length is the intrinsic correlation of the FEL source.

The average width of the photoemission features in the BE spectrum in Fig. 4
(e) is determined to be $\sigma_{\mathrm{BE}}=0.48~{}\mathrm{eV}$ in standard
deviation. This width is the result of the finite VMI energy resolution and
the correlation in the spectral measurement, which approximately sum in
quadrature
$\sigma^{2}_{\mathrm{BE}}\approx\sigma^{2}_{\mathrm{VMI}}+\sigma^{2}_{\phi\mathrm{Corr}}$.
The contribution from spectral correlation, $\sigma_{\phi\mathrm{Corr}}$, is
not necessarily the correlation length $\delta_{A}$, but the two quantities
are related by a constant of proportionality. As described in the supplemental
material, and shown in Fig. S1(b), $\sigma_{\phi\mathrm{Corr}}^{2}$ is
estimated to be $0.11~{}\mathrm{eV}^{2}$ based on the measured $\delta_{A}=22$
px. Having accounted for $\sigma_{\phi\mathrm{Corr}}^{2}$, we find the
remaining contribution to $\sigma^{2}_{\mathrm{BE}}$ to be
$\sigma_{\mathrm{VMI}}^{2}=0.10~{}\mathrm{eV}^{2}$. This corresponds to a VMI
energy resolution of 2.3% at $\mathrm{KE}=15~{}\mathrm{eV}$, which is
consistent with the reported value of 2% for this instrument Erk  _et al._
(2018). We note that the average bandwidth of incident x-ray pulses is not a
limiting factor for the resolution $\sigma_{\mathrm{BE}}$ of our technique.
Instead, it is determined by the kinetic energy resolution of the VMI
spectrometer and the correlation length of the photon spectrum measurement.

###  IV.2 Extension of SDGI

Spectral domain ghost imaging, in general, exploits spectral fluctuations of
the incident source to capture the sample response. The key development in the
present work is incorporating the native projection of VMI into the ghost
imaging model. This method unravels the projection of different Newton
spheres, while simultaneously correlating these unravelled shells with the
incident photon spectra. More specifically, our approach models the
photoproduct momentum space with a reduced dimensional representation (similar
to the pBasex approach) and inverts the tensor product of spectral integration
and Abel transform in a single step. Such an approach is a unique way to
extend the scope of SDGI. Although under ideal circumstances it is equivalent
to conducting the two regression steps sequentially, the single-step approach
has several advantages over either sequential approach.

In comparison to the sequential application of SDGI followed by Abel
inversion, the single-step approach represents the momentum distribution with
a polar basis set, reducing the dimensionality of the problem and coordinating
the comprehensive behavior of the momentum distribution over multiple pixels.
When applying SDGI directly to the 2D VMI data, incorporating the smoothness
term that penalizes high spatial frequencies in momentum space is inefficient
and in most cases intractable. This is because the dimension of the image
space is too large. Conceding this smoothness over momentum space, we can
implement a sequential approach by applying SDGI to each VMI pixel
independently and subsequently applying pBasex to the result. This approach
remains efficient, but it is plagued by compromised performance in noise
handling, as shown in the supplementary material Fig. S2. The key difference
between this sequential approach and the single-step is that the regression is
performed independently for each pixel in the first of the two sequential
steps.

Another feasible sequential approach is to apply single-shot Abel inversion
followed by SDGI. Although the regularization terms can remain in the same
form as in the single-step approach, in this approach the Gaussian log-
likelihood in the objective function (Eqn. 6) is replaced by,

$h_{0,\mathrm{p+S}}(C;A,G,B)=\sum_{i}\left|\sum_{w}A_{i,w}C_{lk,w}-\sum_{q}\left(G^{+}\right)_{q,lk}B_{i,q}\right|^{2}~{},$
(9)

where $G^{+}=(GWG^{T})^{-1}GW$ is the pseudo-inverse of the Abel transform,
which is conducted separately for each photon energy bin.

In our demonstration experiment, both sequential approaches show inferior
performance compared to the single-step approach, which is illustrated in the
comparison in the supplementary material. Under ideal circumstances, when
regularization becomes unnecessary to suppress experimental noise, the optimal
point of the original objective function in Eqn. (6) is

$\displaystyle C_{lk,w}^{(\lambda=0)}$ $\displaystyle=$
$\displaystyle\sum_{i,q}\left((G\otimes A)^{+}\right)_{wlk,iq}B_{i,q}$ (10a)
$\displaystyle=$
$\displaystyle\sum_{i,q}\left(A^{+}\right)_{w,i}\left(G^{+}\right)_{q,lk}B_{i,q}~{},$
(10b)

where $A^{+}=(A^{T}A)^{-1}A^{T}$, and in going from Eqn. (10a) to (10b) we
have changed the order of operations between the pseudo-inverse and tensor-
product of $A$ and $G$. Equation (10a) describes the single-step approach and
Eqn. (10b) denotes both the sequential approaches described above. Thus it can
be seen that in the absence of regularization, both sequential approaches are
equivalent to the single-step method. The regularization in Eqn. (6)
necessitates the computationally heavy step of inverting an
$N_{\omega}N_{L}N_{K}$-dimensional matrix, which is not true for the
sequential approaches. However, as described above, the single-step
outperforms the sequential approaches by virtue of being more robust to noise.
Similar results between the single-step and sequential approaches may be
achieved in other measurements, and the precise choice of method depends
critically on the properties of the measurement; e.g. signal-to-noise ratio,
size of the dataset, and number of parameters to fit.

###  IV.3 Connection to Covariance Approach

We note that there are a number of mathematical techniques to extract
correlations in large datasets. While the method presented in this work relies
on linear regression, covariance analysis is another common method for
extracting correlations Frasinski  _et al._ (1989, 2013); Frasinski (2016);
Allum  _et al._ (2022); Amiot  _et al._ (2018). The photoproduct yield can be
correlated with other intrinsic or extrinsic measurements, to extract signal
from noisy data. Both regression and covariance have been employed in ghost
imaging experiments in the spectral domain, and these two techniques are
closely related.

The connection is shown by regressing the mean-subtracted VMI images $\Delta
B_{i}\equiv B_{i}-\langle B\rangle$ on the mean-subtracted spectra $\Delta
A_{i}\equiv A_{i}-\langle A\rangle$, with $\langle\cdot\rangle$ denoting the
average across all shots. Given the standard definition of sample covariance
$\mathrm{Cov}[X,Y]=\Delta X^{T}\Delta Y/(N_{\mathrm{shot}}-1)$, the
unregularized spectral regression of $\Delta B$ on $\Delta A$ gives

$(\Delta A)^{+}\Delta B=(\Delta A^{T}\Delta A)^{-1}(\Delta A^{T}\Delta
B)=\mathrm{Cov}[A,A]^{-1}\mathrm{Cov}[A,B]~{},$ (11)

where the right hand equality demonstrates this is identical to applying the
inverse of the photon autocovariance matrix to the photon-electron covariance
matrix.

The transform from the projected momentum distribution to the coefficients $C$
is a linear operation in the electron momentum space, which is in tensor
product with the operations in the photon energy space. Therefore the
aforementioned single-step and sequential regression approaches are all
applicable to the mean-subtracted data. Comparing to the original approach
elaborated in Sec. II, regressing the mean-subtracted data is more robust to
static background in both the VMI images and photon spectra, but it may suffer
from the loss of information due to subtraction of the mean, especially when
the spectral fluctuation is limited. Regardless whether the mean is subtracted
or not, one can further constrain the parameters with additional prior
knowledge about the sample, to solve the most probable parameters in a space
with dimension lower than $N_{\omega}N_{L}N_{K}$. Although related, these
further restricted approaches require more assumptions and are less
generalisable than the main approach we present in this work. We provide an
example in the supplementary material.

###  IV.4 Applicable Apparatus Configurations

As for standard inverse Abel transform procedures, a prerequisite of our
method is the cylindrical symmetry about any axis $z$ that is perpendicular to
the projection axis of the VMI $x$, for which the coordinate system shown in
Fig. 1 is not the only configuration. Here we describe two other
configurations without exhausting all possible cases. For circularly polarized
x-rays, such as in Ilchen  _et al._ (2021), the layout in Fig. 1 is still
applicable except that the symmetry axis $z$ is along the axis of beam
propagation. For a co-axial VMI Li  _et al._ (2018) with linearly polarized
x-rays propagating along the VMI axis $x$, $z$ falls along the x-ray
polarization axis.

##  V Conclusion and Outlook

We present a regression approach which can achieve VMI measurements at
resolution better than the inherent bandwidth of a noisy photon source, by
simultaneously reconstructing the spectral response and performing the inverse
Abel transform. Our approach demonstrates a clear advantage over the
conventional binning-and-averaging method. In our experimental demonstration
on the photoionization of argon, the retrieved linewidth is dominated by the
resolution limit prescribed by the VMI resolution and the measured spectral
correlation. We anticipate that the outlined approach will be of great use in
the emerging field of time-resolved inner-shell photoelectron spectroscopy at
FELs, which promises to interrogate photoinduced nuclear and electronic
dynamics in a site-selective manner.

We demonstrate the connection between covariance and regression analysis and
we relate and compare our single-step approach to other sequential approaches.
Our method extends the scope of SDGI by adopting a reduced dimensional
representation for the properties of the photoproducts, and it can be directly
applied to any measurement where VMI images are recorded in coincidence with
the incident photon spectrum. This makes the technique broadly applicable to
many different light sources.

##  VI Data Availability

The data that support the findings of this study are available from the
authors upon reasonable request. An implementation of the proposed approach
can be found in our custom package developed for general SDGI Wang (2022).

##  VII Author Contributions

T.C.D., J.P.C and R.F. initiated the research. J.W., T.C.D. and J.P.C
conceived the method, which is implemented by J.W. The experiment was carried
out by C.C.P., C.P., G.B., S.D., A.T.N., S.K., and B.E. . All authors
contributed to collection or analysis of the experimental data and writing of
the manuscript.

##  VIII Acknowledgement

We acknowledge the Max Planck Society for funding the development and the
initial operation of the CAMP end-station within the Max Planck Advanced Study
Group at CFEL and for providing this equipment for CAMP@FLASH. The
installation of CAMP@FLASH was partially funded by the BMBF grants 05K10KT2,
05K13KT2, 05K16KT3 and 05K10KTB from FSP-302. The analysis work was supported
by the U.S. Department of Energy (DOE), Office of Science, Office of Basic
Energy Sciences (BES), Chemical Sciences, Geosciences, and Biosciences
Division (CSGB). We acknowledge DESY (Hamburg, Germany), a member of the
Helmholtz Association HGF, for the provision of experimental facilities. Parts
of this research were carried out at FLASH and the Maxwell computational
resources operated at Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany.
We thank Jordan O’Neal and Alice E. Green for fruitful discussions.

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