# Single photon performance characterisation of a Generation I Large Area
Picosecond PhotoDetector

R. J. Foster11footnotetext: Corresponding author. A. Scarff  and M. Malek

###### Abstract

The single photoelectron performance characteristics of a Large Area
Picosecond PhotoDetector (LAPPD) were studied using a picosecond optical laser
source. We verify that the LAPPD is capable of achieving transit time spreads
of less than $70\text{\,}\mathrm{ps}$ and spatial resolutions on the order of
millimetres when illuminated with single photons. Isolation of the single
photoelectron peak is easily possible in the pulse height distribution and the
mean single photoelectron gain is measured to be greater than $10^{6}$ with
gains above $10^{7}$ possible in certain voltage configurations. We judge the
performance of the LAPPD sufficient for application in large-scale water or
scintillator-based neutrino experiments, with additional work required to
develop the digital signal processing algorithms necessary for processing
photon hits in high photon occupancy environments.

##  1 Introduction

The Large Area Picosecond PhotoDetector (LAPPDTM) is a novel microchannel
plate (MCP) photosensor that offers single photon timing resolution of tens of
picoseconds as well as millimetre-scale spatial resolution [1]. This
performance is achieved in conjunction with a large photosensitive surface,
resulting in a relatively low cost per unit area — an attractive proposition
for many applications ranging from high energy physics to medical imaging. One
area in which LAPPDs are of particular interest is in future neutrino
experiments [2] such as DUNE [3], WATCHMAN [4], and THEIA [5]. At Fermilab,
the ANNIE experiment [6] has already successfully deployed LAPPDs inside a
water Cherenkov detector [7]. In these experiments, LAPPDs are expected to
greatly improve the vertex reconstruction ability and the fast timing is
expected to enable new techniques such as Cherenkov-scintillation separation
[8, 9] in neutrino detectors utilising novel hybrid detection media such as
water-based liquid scintillator [10].

The LAPPD was developed by the LAPPD collaboration, who sought to develop high
performance, large area photosensors in order to extract as much information
as possible from events at the Fermilab TeVatron [11]. The LAPPD is
manufactured by Incom, Inc., a Massachusetts-based company specialised in the
fabrication of glass optical devices. Traditional MCP photosensors are
fabricated using lead glass as both the substrate and the secondary emissive
surface [12]. However, there have long been concerns about limits on the total
charge that can be extracted from the plate, as well as the complex and
expensive fabrication process that results in high costs per unit of
photosensitive area. Conversely, the LAPPD is constructed from inexpensive but
robust borosilicate glass which is then later functionalised by atomic layer
deposition of a secondary emissive coating [13]. This not only enables larger
photosensitive areas by reducing the manufacturing cost, but also allows for
the electrical properties of the resistive and emissive layers to be tuned
depending on the application [14]. Since the secondary emissive layer is now
decoupled from the substrate, materials with higher secondary emission yields,
such as $\mathrm{MgO}$ or $\mathrm{Al_{2}O_{3}}$ [15], can be used to increase
the achievable gain. In addition, the use of borosilicate glass is
advantageous as low radioactivity glass samples can be chosen [16] which have
far lower [40]K content than lead glass. Since the presence of [40]K within
the glass substrate is a dominant source of background events in MCPs [17],
this dramatically reduces the overall background rate of the MCP.

In this work we aim to verify the single photoelectron performance
characteristics of the LAPPD, in particular the manufacturer specifications of
transit time spreads below $100\text{\,}\mathrm{ps}$ and gains greater than
$10^{6}$.

##  2 LAPPD 104

Figure 1: The LAPPD used in this work shown in its carry case. The SMA
connectors for signal readout can be seen on the printed circuit boards
attached onto either side of the LAPPD.

The device characterised in this work was LAPPD 104, a Generation I LAPPD with
stripline anode readout, as shown in figure 1. LAPPD 104 was manufactured and
vacuum sealed in June 2021 and delivered to the University of Sheffield in
October 2021. LAPPD 104 has a $5\text{\,}\mathrm{mm}$ thick fused-silica glass
window. On the underside of this window the multi-alkali $\mathrm{K_{2}NaSb}$
photocathode is deposited, with an average quantum efficiency of 24.3% at
$365\text{\,}\mathrm{nm}$. The active area of the detector is approximately
$$195\text{\,}\mathrm{mm}$\times$195\text{\,}\mathrm{mm}$$.

LAPPDs contain two MCPs arranged in a chevron pair. The microchannel pore
diameter is $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ with a length to
diameter ratio of 60:1. For this device, the secondary emissive layer applied
to the MCPs is made from $\mathrm{MgO}$. Five high voltage supplies are
required to power the LAPPD - these are applied to the photocathode, and the
top and bottom of each of the MCPs in the pair. In these measurements the
voltage between the anode and the bottom of the bottom MCP, and the voltage
applied to the gap between the two MCPs, is fixed at $200\text{\,}\mathrm{V}$.
The voltage applied to the photocathode, and the voltage applied across the
MCPs themselves, is adjusted as part of the characterisation measurements to
determine the optimal operating conditions. The high voltage configuration is
shown in figure 2.

Charge is collected onto an anode comprising 28 silver striplines which are
routed onto a printed circuit board into SMA connectors at both ends of the
stripline. Due to channel constraints on the digitising electronics, double-
ended readout of eight striplines was used for the measurements presented
here. Any unused channels were terminated at the SMA connector using
$50\text{\,}\mathrm{\SIUnitSymbolOhm}$ terminators.

Figure 2: The high voltage configuration for LAPPD 104 during the
characterisation measurements. The photocathode voltage and MCP voltage are
varied during the tests whilst the inter-MCP and anode-MCP voltage are fixed
at $200\text{\,}\mathrm{V}$. The voltage of both MCPs is kept identical to
ensure stable operation.

##  3 Test stand setup

The characterisation measurements were performed using an optical laser test
stand. A schematic of the setup is shown in figure 3. The LAPPD is mounted
onto a vertical plate inside of a large dark box. It is illuminated using an
NKT photonics PILAS DX PIL040-FC picosecond pulsed-diode laser. The pulse
width of the laser is less than $45\text{\,}\mathrm{ps}$, short enough such
that it is possible to make an estimate of the transit time spread of the
LAPPD. The emission wavelength of the laser is $405\text{\,}\mathrm{nm}$
(spectral width less than $5\text{\,}\mathrm{nm}$) which is closely matched
with the peak quantum efficiency of the LAPPD bialkali photocathode. The laser
pulse is guided to the LAPPD through a single-mode optical fibre into a fibre
collimator coupled to a pellicle beamsplitter, then into neutral density
filters, and finally through a plano-convex lens to focus the light onto the
photocathode surface. The total optical density of the neutral density filters
was increased up to a value of 4.0, which was sufficient to attenuate the
laser light such that the LAPPD only responded to between 6-8% of laser
pulses. The result is that the vast majority (>95%) of laser pulses which
elicit a response in the LAPPD produce a single photoelectron. The optical
apparatus is attached to an X-Y gantry allowing the laser to be scanned across
the LAPPD photocathode. In this case, the laser was centred in line with a
single strip and remained there for all measurements taken.

Figure 3: Schematic of the laser test stand setup and associated high voltage
and signal digitisation electronics.

The five required negative polarity high voltage inputs are provided by a CAEN
V6533N VME high voltage power supply. The high voltage is controlled by a
LabVIEW interface allowing for independent control of every channel. Signal
digitisation is performed by a CAEN V1742B switched-capacitor array VME
digitiser with a sampling rate of
$5\text{\,}\mathrm{G}\mathrm{S}\mathrm{/}\mathrm{s}$. CAEN WaveDump is used to
transfer the digitised signals to a PC over a CONET optical fibre link.

The laser is internally triggered using the laser controller unit at a
repetition rate of $100\text{\,}\mathrm{Hz}$. Upon generating a signal
ordering the laser to fire, this signal is also propagated to the digitiser
“fast trigger” input to begin an acquisition. A total of 16 signal channels
are digitised at a time, corresponding to each end of 8 LAPPD striplines. The
trigger signal is also digitised, enabling digital signal processing
techniques to be used post data acquisition to form precise relative
timestamps between trigger and LAPPD signals. The trigger signal jitter is
<$3\text{\,}\mathrm{ps}$ RMS according to manufacturer specifications. As a
result of the cable delays and delays within the triggering circuitry of the
digitiser, there is an arbitrary delay between the trigger signal and the
arrival of the laser light at the photocathode.

##  4 Signal processing

The binary waveform files are decoded and pre-processed before entering the
analysis chain for a given measurement. Firstly, the waveform is baseline-
subtracted to remove the arbitrary DC offset by taking the median value of all
samples in the waveform before the arrival of the trigger signal. This value
is subtracted from every sample in the waveform. All values are then converted
from ADC counts into millivolts. The digitiser has 12 bits of resolution and a
full-scale range of $1\text{\,}\mathrm{V}$ and so each ADC count below
baseline corresponded to $0.24\text{\,}\mathrm{mV}$. We intentionally do not
apply noise-reducing waveform pre-processing measures such as low-pass
filtering to ensure that we preserve as much of the high frequency components
in the fast rising LAPPD pulses as possible.

For each event, a region of interest is defined around the likely location of
the LAPPD pulse in response to the laser. The location of this region of
interest was determined by measuring the average delay between the arrival of
the trigger pulse from the laser and the LAPPD pulse using an oscilloscope. A
$10\text{\,}\mathrm{ns}$ window defines the region of interest. The integral
of the waveform across the window is calculated using trapezoidal integration,
which will later be used in the gain measurement. The maximum amplitude within
the region of interest is also calculated and if this value is above
$4\text{\,}\mathrm{mV}$ then a software-implemented constant fraction
discriminator is applied to determine the arrival time of the LAPPD pulse. The
arrival time is linearly interpolated in the likely event that the threshold
crossing occurs between samples, enabling timestamping precision better than
the sampling period of the digitiser. The constant fraction discriminator is
also applied to timestamp the trigger pulse.

##  5 Gain

The gain is measured per event by taking the numerical integration of the
LAPPD pulse to calculate a charge, taking the termination resistance to equal
$50\text{\,}\mathrm{\SIUnitSymbolOhm}$. The ratio of the calculated charge to
the elementary charge gives the dimensionless value that is then collected
into a histogram as is shown in figure 5. Since most laser pulses do not
result in a response from the LAPPD, a large peak representing the charge
pedestal can be seen as the first peak in the gain distribution centred around
zero gain. The distribution is fitted with the sum of three Gaussians, the
first representing the pedestal, the second representing the single
photoelectron peak, and the third representing the two photoelectron peak. The
fitting is constrained such that the mean of the two photoelectron Gaussian
must be exactly double that of the single photoelectron peak. The mean of the
single photoelectron Gaussian gives the single photoelectron gain. An example
of this fitting can be seen in figure 4.

As expected, the mean single photoelectron gain increases with both MCP and
photocathode voltage. It is clear that a gain in the mid $10^{6}$ range is
easily achievable even at relatively low operating voltages, with gains
approaching $10^{7}$ possible as the voltages are increased. The gain is more
sensitive to the MCP voltage, as the photocathode voltage will only affect the
energy of the photoelectron at the initial collision, whereas increasing the
MCP voltage increases the energy of all secondary electrons in the
multiplication process.

Figure 4: An example of an LAPPD gain distribution fitted with the summation
(filled in purple) of three individual Gaussian fits (shown in dashed lines).
The first Gaussian represents the charge pedestal, the second represents the
single photoelectron charge, and the third is the two photoelectron charge
which is largely suppressed due to the low intensity of the incident laser
light.

Figure 5: Left: The LAPPD gain distribution with $825\text{\,}\mathrm{V}$
applied to the MCPs and varying voltages applied to the photocathode (PC).
Right: The LAPPD gain distribution with $200\text{\,}\mathrm{V}$ applied to
the photocathode and varying voltages applied to the MCPs.

A microchannel will take a non-negligible time to recharge after performing
electron multiplication. Since the laser light is highly collimated, it is
repeatedly striking the same subset of microchannels. If the same microchannel
is struck again before it is able to fully replenish the lost charge, it is
expected that the observed gain for that even will be reduced. By gradually
increasing the laser repetition rate, this recharge rate can be investigated.
Figure 6 shows the gain distributions from $100\text{\,}\mathrm{Hz}$ to
$1\text{\,}\mathrm{MHz}$, demonstrating a reduction in the mean gain of a
factor of approximately 3 in the most extreme case. There is no change in the
gain distribution from $100\text{\,}\mathrm{Hz}$ to $1\text{\,}\mathrm{kHz}$,
indicating that the recharge rate is sub-millisecond, although the diameter of
the laser spot is certainly larger than the diameter of an individual
microchannel and as such we are likely probing the recharge rate of a small
collection of pores rather than a single microchannel.

Figure 6: The LAPPD gain distribution as a function of the laser repetition
rate, zoomed into the area around the single photoelectron peak to more
clearly highlight the differences in the distributions. The laser remains in
the same position such that the same microchannels are repeatedly illuminated.

##  6 Transit time spread

The transit time spread represents the fundamental timing resolution of a
photodetector. Both the LAPPD pulse and the laser trigger pulse timestamps are
calculated using a constant fraction discriminator. The time delta between
these timestamps is calculated and histogrammed, forming a Gaussian
distribution. The mean of this distribution is given by a combination the
LAPPD’s overall transit time, as well as any arbitrary delays resulting from
cable lengths and the triggering circuitry of the digitiser, both of which
have not been characterised in this study. The standard deviation of the
distribution gives the transit time spread. In general, the distribution will
show a tail which comprises late pulses resulting from the deflection of the
photoelectron off the first MCP before it enters a pore and begins the
amplification process. The Gaussian fit only covers the “core” distribution
and excludes this tail.

Figure 7: The distributions of the time delta between the laser pulse and the
response observed on the LAPPD for varying photocathode voltages. Figure 8:
The measured transit time spread as a function of the photocathode voltage
with and without correcting for the laser pulse width and trigger jitter.

From figure 7 it is clear to see that increasing the photocathode voltage
greatly narrows the observed time delta distribution, resulting in a smaller
measured transit time spread. The overall transit time also decreases, but
without suitable characterisation of the aforementioned arbitrary cable and
triggering delays it is not possible to calculate an absolute value for the
transit time in the current setup. Figure 8 shows the value of the transit
time spread as measured by the standard deviation of the Gaussian fit to the
time delta distribution across the range of photocathode voltages. Increasing
the photocathode voltage from an initial value of $10\text{\,}\mathrm{V}$ to
$200\text{\,}\mathrm{V}$ results in a reduction of the measured transit time
spread from $188\text{\,}\mathrm{ps}$ to $68.4\text{\,}\mathrm{ps}$. Adjusting
the MCP voltage has very little effect on the resulting transit time spread.
These values are inclusive of the uncertainties arising from the non-trivial
pulse width of the laser relative to the transit time spread of the LAPPD. We
can attempt to correct for this by subtracting in quadrature a conservatively
small estimate of the laser pulse width of around $30\text{\,}\mathrm{ps}$, as
well as the laser trigger jitter of $3\text{\,}\mathrm{ps}$. After this
correction, the best transit time spread value we measure is
$61.5\text{\,}\mathrm{ps}$ at $200\text{\,}\mathrm{V}$ on the photocathode.

##  7 Position resolution

The photon position is calculated differently in each dimension using the
stripline anode readout, shown in figure 9. In the direction parallel to the
striplines, the photon position is inferred through the difference in the time
of arrival at each end of the stripline. Position resolution in this direction
is ultimately dependent on the timing resolution that is achieved. In the
direction perpendicular to the striplines, the photon position is inferred
using the relative charge deposition on neighbouring striplines. The electron
cloud will spread as it exits the lower MCP and induce charge in several
striplines; the centroid of this charge distribution gives the reconstructed
photon position in this dimension. The position resolution depends on the
density of readout elements (i.e. more striplines will improve position
resolution at the cost of increasing readout channel density) as well as the
gain and the charge spread (inducing charge in more striplines allows for a
greater number of striplines to be used during the centroiding process, but
requires higher gain to ensure that each stripline observes a charge above the
noise threshold).

Figure 9: The stripline anode readout and position reconstruction approach on
the Generation I LAPPD. The orange dot shows the incident photon location.

Figure 10: Left: The difference in the pulse time of arrival at each end of
the stripline as the laser is scanned along the length of the stripline in
$10\text{\,}\mathrm{mm}$ increments. Right: The mean of each distribution
against the laser position. The linear fit gives a stripline pulse propagation
velocity of $0.59c$. Figure 11: The result of centroiding the charge
deposition on 8 striplines for each laser pulse. Three different laser
positions are shown, each separated by $10\text{\,}\mathrm{mm}$ in the
direction perpendicular to the striplines.

To measure the position resolution in the direction parallel to the
striplines, the laser was scanned along the length of a stripline in
$10\text{\,}\mathrm{mm}$ increments. For each laser pulse, the difference in
the time of arrival at each end of the stripline was measured, where constant
fraction discrimination with a threshold of 25% pulse height was used to
timestamp each waveform. The differential timing distributions for each laser
position are shown in figure 10. With an average differential pulse time of
arrival resolution of approximately $50\text{\,}\mathrm{ps}$, this results in
a positional resolution in the direction parallel to the striplines of
$4.4\text{\,}\mathrm{mm}$, which is confirmed when performing the position
reconstruction and fitting Gaussian functions to the subsequent position
distributions.

Figure 11 shows the result using the charge centroiding approach to infer the
photon position in the direction perpendicular to the stripline direction. The
laser is fired at three points, separated by $10\text{\,}\mathrm{mm}$ in the
direction perpendicular to the stripline and the centroid of the charge
deposition across 8 striplines is calculated for each laser pulse. The
averaged standard deviation of these distributions, calculated by fitting a
Gaussian function to each distribution, across several laser positions gives a
position resolution in this dimension of $2.0\text{\,}\mathrm{mm}$.

##  8 Discussion

We find good agreement between the single photoelectron performance expected
from the LAPPD as detailed in earlier characterisation work by Incom [1], and
the performance observed on our laser test stand. In the case of the transit
time spread measurement, the optimal value actually exceeds that measured by
Incom, Inc. as part of their standard pre-shipping test and measurement
procedure. For these measurements, the optimal measured value is likely
limited by the digitising electronics and signal processing rather than the
LAPPD itself, and so these measurements do not represent the maximal
performance of the photosensor. When measuring timing performance in the tens
of picoseconds, effects such as the cell-to-cell timing offset of the
switched-capacitor array digitiser begin to impact the measurement. We did not
modify the factory digitiser calibration performed by CAEN in this work,
although it has been demonstrated that through extensive calibration with
signal injection from a precision pulse generator that it is possible to
reduce these timing uncertainties to below $1\text{\,}\mathrm{ps}$ [18].

The LAPPD has significant potential for application in high energy physics
experiments. In large-scale water Cherenkov or liquid scintillator neutrino
experiments, the position and timing resolution offered by the LAPPD would
greatly improve the vertex reconstruction relative to standard photomultiplier
tubes which would be hugely beneficial for suppressing backgrounds. A number
of next-generation neutrino experiments are considering instrumenting their
detectors with LAPPDs as the single photon performance could unlock a number
of proposed goals. One such goal is the separation of Cherenkov and
scintillation light in large-scale water-based liquid scintillator detectors
such as the proposed Theia experiment. Depending on the scintillator used, the
LAPPD would be able to consistently resolve the nanosecond-scale time
difference between the emission of the prompt Cherenkov and delayed
scintillation light. This would allow for the retention of directional
information of the particle by measurement of the Cherenkov ring whilst still
benefiting from improved energy and vertex reconstruction arising from the
increased light yield from the scintillator. Cherenkov-scintillation
separation has been demonstrated in water-based liquid scintillator using fast
photomultiplier tubes [8] and LAPPDs [9] in small-scale or benchtop
experiments, but has also been demonstrated in large-scale pure liquid
scintillator detectors [19]. The use of LAPPDs in these larger scale
experiments would likely allow for improved separation of the Cherenkov and
scintillation components as a result of the much better temporal resolution
relative to photomultiplier tubes.

There are challenges that must be addressed, however, before LAPPDs are likely
to be deployed in any large-scale experiments. There remains uncertainty about
the performance capabilities of the stripline anode readout regarding the
ability to disambiguate individual photon hits in high photon occupancy
environments. Previous work [20] has shown that deconvolutional approaches
show promise in determining the individual hit times and locations on the
LAPPD when many photons strike the photocathode in close proximity, however
this method has only been tested in simulation. The High Rate Picosecond
PhotoDetector (HRPPD), recently developed by Incom, has the potential to
address this issue with a $32\times 32$ pixel readout array directly coupled
to a smaller $10\times 10$ ${\mathrm{cm}}^{2}$ photosensitive area which can
perform photon counting with millimetre-scale positional resolution without
the algorithmic challenges associated with stripline readout. For experiments
which do not require millimetre-scale resolution, the Generation II LAPPD
utilises a capacitively-coupled pixellated readout with which it may be
possible to count photons on individual pixels. There have been attempts to
develop algorithms for disambiguating photon hits on the stripline anode
LAPPD, although none of these have been tested on experimental data so far.
The ANNIE experiment at Fermilab has successfully deployed LAPPDs and a water-
based liquid scintillator subvolume recently [7]. Further results are expected
in the future which should give an indication of the performance of the
stripline anode in high photon density environments.

## Acknowledgments

The authors wish to thank Mark Popecki, Cole Hamel, and Stephen Clarke at
Incom for their openness and patience in answering questions regarding the
LAPPD and for their support in operating the device. In addition, the authors
wish to thank Stephen Wilson and Patrick Stowell at the University of
Sheffield for designing the LabVIEW high voltage interface and providing
feedback on the draft manuscript, respectively. This work was supported by the
Science and Technology Facilities Council [grant number ST/S006400/1].

## References

  * [1] A. Lyashenko, B. Adams, M. Aviles, T. Cremer, C. Ertley, M. Foley et al., _Performance of Large Area Picosecond Photo-Detectors (LAPPD™)_ , _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 958 (2020) 162834.
  * [2] J.R. Klein et al., _Future Advances in Photon-Based Neutrino Detectors: A SNOWMASS White Paper_ , 2203.07479.
  * [3] F. Capozzi, S.W. Li, G. Zhu and J.F. Beacom, _DUNE as the Next-Generation Solar Neutrino Experiment_ , _Physical Review Letters_ 123 (2019) .
  * [4] M. Askins, M. Bergevin, A. Bernstein, S. Dazeley, S.T. Dye, T. Handler et al., _The Physics and Nuclear Nonproliferation Goals of WATCHMAN: A WAter CHerenkov Monitor for ANtineutrinos_ , 2015.  10.48550/ARXIV.1502.01132.
  * [5] M. Askins, Z. Bagdasarian, N. Barros, E.W. Beier, E. Blucher, R. Bonventre et al., _Theia: An advanced optical neutrino detector_ , .
  * [6] A.R. Back, J.F. Beacom, M. Bergevin, E. Catano-Mur, S. Dazeley, E. Drakopoulou et al., _Accelerator Neutrino Neutron Interaction Experiment (ANNIE): Preliminary Results and Physics Phase Proposal_ , 2017.  10.48550/ARXIV.1707.08222.
  * [7] M. Ascencio-Sosa, Z. Bagdasarian, J. Beacom, M. Bergevin, M. Breisch, G. Caceres Vera et al., _Deployment of Water-based Liquid Scintillator in the Accelerator Neutrino Neutron Interaction Experiment_ , _Journal of Instrumentation_ 19 (2024) P05070.
  * [8] J. Caravaca, F.B. Descamps, B.J. Land, J. Wallig, M. Yeh and G.D. Orebi Gann, _Experiment to demonstrate separation of cherenkov and scintillation signals_ , _Physical Review C_ 95 (2017) .
  * [9] T. Kaptanoglu, E.J. Callaghan, M. Yeh and G.D. Orebi Gann, _Cherenkov and scintillation separation in water-based liquid scintillator using an LAPPD™_ , _The European Physical Journal C_ 82 (2022) .
  * [10] J. Caravaca, B.J. Land, M. Yeh and G.D. Orebi Gann, _Characterization of water-based liquid scintillator for cherenkov and scintillation separation_ , _The European Physical Journal C_ 80 (2020) .
  * [11] B.W. Adams, K. Attenkofer, M. Bogdan, K. Byrum, A. Elagin, J.W. Elam et al., _A Brief Technical History of the Large-Area Picosecond Photodetector (LAPPD) Collaboration_ , 2016.
  * [12] J. Ladislas Wiza, _Microchannel plate detectors_ , _Nuclear Instruments and Methods_ 162 (1979) 587–601.
  * [13] M.J. Minot, D.C. Bennis, J.L. Bond, C.A. Craven, A. O’Mahony, J.M. Renaud et al., _Pilot production & commercialization of LAPPD™_, _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 787 (2015) 78–84.
  * [14] J.W. Elam, A.U. Mane, J.A. Libera, J.N. Hryn, O.H. Siegmund, J. McPhate et al., _(Invited) Synthesis, Characterization, and Application of Tunable Resistance Coatings Prepared by Atomic Layer Deposition_ , _ECS Transactions_ 58 (2013) 249–261.
  * [15] Z. Insepov, V. Ivanov and H. Frisch, _Comparison of candidate secondary electron emission materials_ , _Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms_ 268 (2010) 3315–3320.
  * [16] M.A. Popecki, B. Adams, C.A. Craven, T. Cremer, M.R. Foley, A. Lyashenko et al., _Microchannel plate fabrication using glass capillary arrays with atomic layer deposition films for resistance and gain_ , _Journal of Geophysical Research: Space Physics_ 121 (2016) 7449–7460.
  * [17] O. Siegmund, J. Vallerga and B. Wargelin, _Background events in microchannel plates_ , _IEEE Transactions on Nuclear Science_ 35 (1988) 524–528.
  * [18] D. Stricker-Shaver, S. Ritt and B.J. Pichler, _Novel calibration method for switched capacitor arrays enables time measurements with sub-picosecond resolution_ , _IEEE Transactions on Nuclear Science_ 61 (2014) 3607–3617.
  * [19] Borexino Collaboration collaboration, _First directional measurement of sub-mev solar neutrinos with borexino_ , _Phys. Rev. Lett._ 128 (2022) 091803.
  * [20] G.R. Jocher, M.J. Wetstein, B. Adams, K. Nishimura and S.M. Usman, _Multiple-photon disambiguation on stripline-anode micro-channel plates_ , _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 822 (2016) 25–33.


# Ghost-imaging-enhanced non-invasive spectral characterization of stochastic
x-ray free-electron-laser pulses

Kai Li<EMAIL_ADDRESS>Department of Physics, The University of Chicago,
Chicago, IL 60637 USA  Chemical Sciences and Engineering Division, Argonne
National Laboratory, Lemont, IL 60439 USA  Joakim Laksman  European XFEL,
Holzkoppel 4, 22869 Schenefeld Germany  Tommaso Mazza  European XFEL,
Holzkoppel 4, 22869 Schenefeld Germany  Gilles Doumy  Chemical Sciences and
Engineering Division, Argonne National Laboratory, Lemont, IL 60439 USA
Dimitris Koulentianos  Chemical Sciences and Engineering Division, Argonne
National Laboratory, Lemont, IL 60439 USA  Alessandra Picchiotti  The Hamburg
Centre for Ultrafast Imaging, Hamburg University, Luruper Chaussee 149, 22761,
Hamburg, Germany  Svitovar Serkez  European XFEL, Holzkoppel 4, 22869
Schenefeld Germany  Nina Rohringer  Center for Free-Electron Laser Science
CFEL, Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg,
Germany  Department of Physics, Universität Hamburg, 20355 Hamburg, Germany
Markus Ilchen  Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607
Hamburg, Germany  Michael Meyer  European XFEL, Holzkoppel 4, 22869 Schenefeld
Germany  Linda Young<EMAIL_ADDRESS>Department of Physics, The University of
Chicago, Chicago, IL 60637 USA  Chemical Sciences and Engineering Division,
Argonne National Laboratory, Lemont, IL 60439 USA  James Franck Institute, The
University of Chicago, Chicago, IL 60637 USA

###### Abstract

High-intensity ultrashort X-ray free-electron laser (XFEL) pulses are
revolutionizing the study of fundamental nonlinear x-ray matter interactions
and coupled electronic and nuclear dynamics. To fully exploit the potential of
this powerful tool for advanced x-ray spectroscopies, a noninvasive spectral
characterization of incident stochastic XFEL pulses with high resolution is a
key requirement. Here we present a new methodology that combines high-
acceptance angle-resolved photoelectron time-of-flight spectroscopy and ghost
imaging to dramatically enhance the quality of spectral characterization of
x-ray free-electron laser pulses. Implementation of this non-invasive high-
resolution x-ray diagnostic can greatly benefit the ultrafast x-ray
spectroscopy community by functioning as a transparent beamsplitter for
applications such as transient absorption spectroscopy as well as covariance-
based x-ray nonlinear spectroscopies where the shot-to-shot fluctuations
inherent to a SASE XFEL pulse are a powerful asset.

##  I Introduction

X-ray free-electron lasers, with brilliance ten orders of magnitude higher
than synchrotrons, continuous tunability over the soft and hard x-ray regimes
and sub-femtosecond pulse duration duris2020tunable , have emerged as a
powerful tool both to explore fundamental nonlinear x-ray interactions in
isolated atomic and molecular systems young2010femtosecond ; Hoener-2010-PRL ;
Doumy-2011-PRL ; Kanter-2011-PRL ; Rudenko-2017-Nature ; Mazza-2020-PRX , and,
to follow photoinduced electronic and nuclear dynamics on their intrinsic
femtosecond timescales via pump/probe techniques young2018roadmap ;
Wolf-2017-NatComm . For the latter objective, core-level x-ray transient
absorption (XTAS) with ultrafast x-ray pulses has become a workhorse - it
projects core electronic states onto unoccupied valence/Rydberg states,
thereby capturing the evolution of valence electronic motion following an
excitation pulse. However, realization of XTAS is challenging at XFELs where
the x-ray pulses with bandwidth $\Delta E/E\sim 1\%$, typically produced by
self-amplified spontaneous emission (SASE), have spiky temporal and spectral
profiles that vary stochastically on a shot-by-shot basis kim1986analysis ;
andruszkow2000first ; milton2001exponential ; Geloni-2010-NJP ;
hartmann2018attosecond . The traditional approach for XTAS with XFELs is to
monochromatize the SASE beamLemke-2013-JPCA ; Higley-2016-RSI  and scan the
monochromatic beam ($\Delta E/E\sim 0.01\%$) across the desired spectral
range. This makes inefficent use of the full XFEL beam, imposes limits on time
resolution via the uncertainty principle, and, by reducing the pulse
intensities, hampers realization of nonlinear x-ray spectroscopies. An
alternative approach is to monitor incident and transmitted intensity to
obtain an absorption spectrum, $I_{T}(\omega)/I_{0}(\omega)$, across the
entire SASE bandwidth. With this approach one may realize novel experimental
techniques employing correlation analysis that take advantage of the intrinsic
stochastic nature of XFELs pulses Kimberg2016SD ; ratner2019pump ;
cavaletto2021high ; kayser2019core . By using pulses with uncorrelated
fluctuations one can leverage the noise such that each repetition of the
experiment, i.e. each XFEL shot, represents a new measurement under different
conditions. As an example, spectral ghost imaging has been applied to obtain
an absorption spectrum with energy resolution better than the averaged SASE
bandwidth driver2020attosecond ; li2021time . In general, the characterization
of the incident pulses is essential to this class of novel covariance
spectroscopies as previously demonstrated in UV regime Tollerud2019PNAS .

Several diagnostic tools have demonstrated well-resolved spectral measurements
on a single-shot basis without compromising the quality of the x-ray beam. A
commonality is the use of optical elements to split the incident x-ray beam
into reference and sample beams. Beamsplitters for hard x-rays use crystal
Bragg diffraction Zhu-2012-APL ; makita2015high  while diffraction gratings
are used for soft x-raysengel2020parallel ; Brenner2019OE . An alternative is
to use photoionization of a dilute target gas and measure the kinetic energy
of ejected photoelectrons to retrieve the incident photon spectrum via the
photoelectric effect viefhaus2013variable ; laksman2019commissioning ;
Walter:ys5104 . Indeed, the use of an array of 16 electron time-of-flight
spectrometers (eTOFs) radially distributed about the propagating x-ray beam
(colloquially named the ”cookie-box”) has enabled the measurement of the
position, polarization, and central energy of an x-ray photon beam as
demonstrated at the PETRA-P04 beamline viefhaus2013variable . At XFELs, while
it is straightforward to measure the central photon energy with the cookie-box
laksman2019commissioning  as has been demonstrated for two-color x-ray pulses
serkez2020opportunities  and to obtain simultaneous polarization diagnostics
lutman2016polarization ; hartmann2016circular , it is more challenging to
obtain single-shot spectra with an energy resolution comparable to a grating
spectrometer.

Here we use a ghost-imaging algorithm to improve the energy resolution of the
raw cookie-box measurements. Thousands of SASE spectra were measured
simultaneously by a cookie-box and a grating spectrometer and ghost imaging
was applied to compute the response matrix of the cookie-box. The response
matrix was then used to reconstruct the x-ray spectrum with energy resolution
improved from $\sim 1$ eV to 0.5 eV at a central energy of 910 eV for a
resolution of $\Delta E/E\sim 1/2000$ under the present conditions. This
response matrix derived from ghost imaging also provides predictive power for
the spectral profile of yet-to-be-measured XFEL pulses.

##  II Results

###  II.1 Spectral ghost imaging

Ghost imaging is an experimental technique which uses statistical fluctuations
of an incident beam to extract information about an object using a beam
replica that has not physically interacted with the object Padgett2017PTRS .
It can be used in the spatialpelliccia2016experimental ; yu2016fourier ,
temporalratner2019pump  and spectraldriver2020attosecond ; li2021time
domains. Traditional ghost imaging requires a beam splitter to separate the
incident beam into two replicas, the object beam and the reference beam. The
object beam interacts with the sample and a low-resolution detector is used to
measure the signal whose intensity is proportional to the interaction and the
incident beam. The reference beam is directly measured by a high-resolution
detector to extract knowledge of the incident beam. The incident light source
varies shot-by-shot and numerous measurements are carried out to calculate the
correlation function between the two signals from the object and the reference
beams. The correlation function of the measurements is analyzed to extract
information of the sample. The advantage of ghost imaging is that the object
beam does not necessarily need to be strong – thus protecting the samples from
radiation damage. In addition, due to the fluctuations of the light source and
correlation analysis, ghost imaging is robust to noise and background signals.

Ghost imaging essentially maps the high-resolution signal onto the low-
resolution one, making it an ideal tool to calibrate devices with high
resolution. The correlation function generated by ghost imaging contains
information on the response of a device to the different incident signals.
This extracted information can be further used to correct defects or
discrimination present in a device. The ghost imaging calibration method
reconstructs a high quality signal that achieves resolution beyond the low-
resolution instrumental limit. The stochastic nature of a SASE XFEL makes it
well-suited for ghost imaging in the temporal and spectral domains. Here,
ghost imaging is used to calibrate the eToFs of the cookie-box and obtain a
response matrix, which is then applied to reconstruct a more accurate incident
x-ray spectrum. One challenge for applying the ghost imaging method in the
x-ray regime is the requirement of a beamsplitter. Although x-ray
beamsplitters are available as mentioned above, the non-invasive gas-target
measurement is suitable to replace the function of the beamsplitter.

###  II.2 Experimental procedure

Figure 1:  Schematic of the experimental setup. SASE XFEL pulses first
interact with dilute neon gas in the cookie-box chamber where the kinetic
energies of $1s$ photoelectrons are measured by the array of eToFs. The
transmitted x-ray pulse is then focused on the VLS grating by a spherical
mirror and dispersed on a YAG:Ce crystal. The induced fluorescence is recorded
by a charge-coupled device (CCD) as a 2D image from which the single-shot
”reference” spectrum is extracted.

The energy spectrum of the incident x-ray beam was characterized non-
invasively by photoionization of dilute neon gas at the center of an array of
16-eToFs, i.e. the cookie-boxlaksman2019commissioning  as shown in Fig. 1. The
arrival times of Ne $1s$ photoelectrons were measured by the eToFs located in
the plane perpendicular to the beam propagation direction (see Supplementary
Note 1). SIMION simulations were carried out to establish a traditional
calibration between the electron time-of-flight and kinetic energy, $E_{k}$,
given the drift tube length and retardation voltages. The incident photon
energy was derived by adding the Ne $1s$ binding energy, 870 eV, to the
measured $E_{k}$. The spectrum obtained by cookie-box for several random shots
using this traditional method is shown in Fig. 1 as the object measurement.
Under the present experimental conditions, the energy resolution achievable by
the cookie-box was around 1 eV, which is not comparable to the high-resolution
grating spectrometer measurement where 0.2 eV FWHM ($\Delta E/E$) can be
readily achieved.

After passing through cookie-box, the same FEL beam was characterized by a
spectrometer based on a VLS grating and a Ce:YAG screen as shown in Fig. 1 as
the reference measurement. The cookie-box contains very dilute gas which does
not attenuate or otherwise alter the x-ray beam. Thus, ideally the same
spectrum would be obtained from the electron (cookie-box) and photon (grating
spectrometer) measurements. However, the measurement of a single random shot
shown in Fig. 3a reveals differences. The grating spectrometer resolution is
much higher than the resolution of the cookie-box, thus creating a large
deviation between the two spectra. The use of ghost imaging to retrieve a
response matrix which is then used to improve the performance of the cookie-
box measurements is demonstrated in the following.

###  II.3 Principle of reconstruction

Theoretically the photoelectron signal $c$ (after normalization to the gas
density) is proportional to the incident photon spectrum $s$ as measured by
the spectrometer

$c=As$ (1)

where $A$ relates the cookie-box signals to the incident photon spectrum, is
an $(m\times n)$ matrix with the cookie-box ToF points $m=137$ and the
spectrograph pixels $n=1900$ in the region of interest between 895 and 920 eV.
This equation resembles the basic equation in ghost imaging and is usually
used to obtain sample information by solving for $A$. However, in order to
predict the incident spectrum based on cookie-box measurements, we formally
write equation (1) as

$s=Rc$ (2)

where the response matrix $R$ is the formal inverse of matrix $A$. $R$ maps
the low-resolution cookie-box measurements to high-resolution grating
spectrometer measurements. In other words, $R$ is a calibration matrix
contains information of the characteristics of eToFs. After retrieving the
response matrix $R$, according to equation (2), it can be used to generate
high-resolution spectrum with the intrinsic defects and broadening of cookie-
box removed.

###  II.4 Ghost imaging reconstructed spectrum

To solve equation (2), we take advantage of the $N$ independent measurements
obtained. Each shot gives a realization of $s_{i}$ and $c_{j}$ in equation
$s_{i}=\sum_{j=1}^{m}R_{ij}c_{j}$ with $m$ unknown variables $R_{ij}$.
Combining all measurements gives $N$ independent linear equations which can be
solved to uniquely determine the unknown variables if $N>m$. Instead of
directly solving these equations, the response matrix elements are determined
by least square regression, i.e. by minimizing the quantity $|s-Rc|^{2}$.
Single-photon Ne $1s$ ionization exhibits a dipole angular distribution
pattern due to the linear (horizontal) polarization of the x-rays and the
spherical $1s$ electron orbital. To increase the signals, we combined six
eToFs near the polarization direction which have strong 1s peaks, to form the
cookie-box measurement vector $c$ with dimension $m=6\times 137=822$.

Figure 2:  The cookie-box response matrix computed with six eToFs and all the
shots $N=15337$.

The calculated cookie-box response matrix using all shots ($N=15337$) is shown
in Fig. 2. Compared with the traditional calibration function, which just maps
ToFs onto kinetic energy, here we retrieved a matrix whose values represent
the sensitivity of the cookie-box to photons of different energy. As expected,
there are six different calibration lines connecting the eToFs to spectrograph
pixels. The linewidth represents the instrumental broadening. One eToF does
not work well and gives relatively small signals. We tried different
regression optimizers and got essentially the same response matrix,
demonstrating the robustness of our method. As discussed below, the response
matrix can be used to obtain a better spectrum. Note that one can quickly
obtain the traditional calibration lines of eToFs, by fitting the lines within
a nonconverged response matrix obtained by using only 1500 shots.

Figure 3:  Single-shot electron-derived and photon spectra before (a) and
after (b) ghost-imaging reconstruction. a Raw electron-based spectrum from a
single eToF (blue) and grating-based photon spectrum (red). b Ghost-imaging-
reconstructed electron-based spectrum (blue) and the grating-based spectrum
after convolution with a Gaussian ($e^{-x^{2}/(2\sigma^{2})}$ with
$\sigma=0.2$ eV).

It is reasonable to assume the response matrix of the cookie-box does not
change for a given photon energy, gas target, photoelectron energy range and
cookie-box configuration (fixed retardation, bias voltage…); thus $R$ can be
used to predict the spectra of new shots. Higher resolution electron spectra
$s_{r}$ can be reconstructed by multiplying the response matrix $R$ by the
cookie-box measurement $c$ according to equation (2). As illustrated in Fig.
3, the peak profile and intensities of the cookie-box data (a) are changed
after multiplying the matrix with the cookie-box measurement (b).

The photon spectrum in Fig. 3 b was convolved with a Gaussian function with
$\sigma$=0.2 eV to compare with the ghost-imaging-reconstructed spectrum. The
$\sigma$ was derived by considering the number of data points in the cookie-
box versus the spectrometer measurement. There are $1900/137\approx 14$
spectrograph pixels between two eToF points; multiplying by the 0.013 eV/pixel
dispersion of the spectrometer gives 0.2 eV which we take to be $\sigma$. One
function of reconstruction is to remove instrumental broadening. Thus one
observes the higher resolution of the reconstructed spectrum, which matches
well with the convolved grating spectrometer measurement. This also indicates
that in our case the resolution of reconstructed spectrum is limited by the
number of data points within the Ne 1s photoelectron peak of cookie-box
signal.

Figure 4:  Improvement in spectral resolution from the ghost imaging
algorithm. $\Delta\sigma_{e-p}$ is the standard deviation of the difference
between cookie-box and spectrometer measurements. Original (blue) and after
the ghost-imaging reconstruction (red). Data from six eToFs and all the shots
were used to converge the regression and determine the response matrix $R$.
The results of 1000 shots are shown here.

To quantify the performance of the reconstruction, we calculated the standard
deviation of the difference signal between the electron-derived and the photon
spectra $s_{i}$

$\Delta\sigma_{e-p}=\sqrt{\frac{\sum_{i=1}^{n}|(c_{i}-s_{i})-(\bar{c}-\bar{s})|^{2}}{n}}$
(3)

where $\bar{s}$ and $\bar{c}$ are the mean value of spectrometer and cookie-
box measurement, respectively, $n$ is the number of spectrometer pixels.
Depending on the situation, the value of $c_{i}$ is either interpolated
cookie-box data of one eToF or the ghost imaging reconstructed spectrum. As
shown in Fig. 4 the deviation $\Delta\sigma_{e-p}$ of reconstructed spectrum
drops to half of the original value, which indicates the improvement after
reconstruction. In addition, the smaller fluctuation of the deviations means
that the reconstructed spectrum is stable and more reliable than the raw
electron-derived spectrum. The significantly better matching of spectrum after
ghost-imaging is further confirmed by a good correlation between reconstructed
and photon spectrum i.e. averaging 0.72 Pearson correlation coefficient across
the spectrum (see Supplementary Note 2).

###  II.5 Predictive power and performance analysis

Figure 5:  Number of learning shots required for robust predictive capability.
The standard deviation for learning and prediction shots as a function of
number of shots used in the regression. The inserted plot at fewer number of
learning shots (1433) shows a marked jump in standard deviation from learning
to prediction shots, while the standard deviation remains relatively constant
when more learning shots (7568) are used.

One of the most interesting aspects of the ghost imaging method is its
predictive power for future shots. This requires numerous ”learning” shots to
obtain a converged response matrix. Data from six eToFs were used and the
response matrix learned from different number of shots is then used to predict
the spectra for 100 new shots that were not used in the regression. As shown
in Fig. 5, when fewer shots are used, the deviation of the learning shots is
small because the regression is under-determined. Meanwhile, the deviation for
the new shots is large, indicating a poor predictive power of an unconverged
response matrix. As the information from more shots are included in the
regression process, the deviation of learning shots rises, whereas the
deviation for the new shots decreases indicating the gain of predictive power.
The regression converges when the deviation of learning and prediction meet
around 8000 shots (roughly 10 times the number of unknown variables i.e.
cookie-box vector elements). It is important to note that as more shots are
used the error bar for the prediction, which measures the fluctuation of the
deviation, also decreases, which means the prediction becomes more stable.

Figure 6:  Improved reconstruction as a function number of eTOFs used. The
reconstructed spectrum and Gaussian convolved spectrum using one eToF (a) and
six eToFs (b). All the shots are used to get a converged result. The inserted
plot shows the decrease in the standard deviation when more eToFs are used.

Combining electron spectra from several eToFs increases the signal intensities
and suppresses the noise. However, the eToF signals cannot be added on top of
each other directly, due to the different calibrations of each eTOF. As
mentioned before, we put the signals from different eToFs together to form a
larger vector and differences of eToFs are automatically taken into account
when calculating the response matrix. The upper panel and the lower panel of
Fig. 6 show the spectrum of a random shot where one eToF with strongest signal
and six eToFs are used, respectively. Compared with the Gaussian convolved
spectrometer measurement clearly indicates the advantages of using six eToFs.
The inset plot shows that the normalized deviation drops gradually from 1 for
one eToF to 0.87 with six eToFs. The data from different eToFs complement with
each other and improve the correlation with the spectrometer measurement thus
resulting in the better overall reconstructed spectrum.

Our analysis indicates that the performance of the ghost imaging
reconstruction depends on experimentally controllable parameters, number of
shots used, number of eTOFs used. Ghost imaging is based on the correlation
between the object measurement and the reference measurement, i.e. the
sensitivity of the cookie-box signals to the fluctuations of the incident
spectrum as measured by a grating spectrometer. Obtaining better correlation
function and reconstruction, i.e. response matrix and spectrum with higher
resolution, requires more data points within the Ne 1s photoelectron peak as
well as high signal-to-noise ratio. More data points in the eTOF spectrum can
be obtained by increasing the retardation voltages to slow the electrons,
using larger drift length tubes, or, more simply by increasing the digitizer
sampling rate which is presently 2 GHz. In addition, increased detection
sensitivity can be readily achieved by using more eToFs or by increasing the
gas density to produce more photoelectrons and a higher signal-to-noise ratio
(see discussion in Supplementary Note 3).

##  III Discussion

Figure 7:  Effect of averaging on the deviation between ghost-imaging-
reconstructed and grating-based spectra. (a) Upper panel shows the averaged
spectrum for a group of 1000 shots. The spectrum is divided into regions 1-4,
going from low to high photon energies. Lower panel shows corresponding
spectral percentage deviation for a single 1000-shot group (dashed) and
averaged over 7 1000-shot groups (solid). (b) The percentage deviation as a
function of number of averaged shots for spectral regions 1-4.

Despite the considerable improvement to achieve resolution $\Delta E/E\sim
1/2000$, the ghost-imaging reconstruction of a single-shot XFEL spectrum under
the current experimental conditions can not characterize completely the SASE
structure containing inter-spike spacings down to $\sim 0.1$ eV, corresponding
to a required resolution of $\Delta E/E\sim 1/10000$. For this particular case
using the Ne $1s$ photoelectron, the width of the final state Ne $1s^{-1}$ of
0.27 eV represents another barrier. However, the ability to reconstruct an
averaged spectral profile with high fidelity is of considerable interest.
High-precision incident spectra obtained by averaging are extremely useful for
transient absorption measurements, e.g. when a dispersive spectrometer is
placed after the sample to extract spectral features below the SASE bandwidth
as was done previously in the XUV spectral range for strong-field induced
modifications of the lineshape of a doubly-excited state in He Ott2019PRL .
This averaging method is very common for pump/probe transient absorption
experiments, e.g. with broadband soft x-ray high-harmonic generation (HHG)
sources where single-shot spectra sequentially taken with pump-on and pump-off
configurations are averaged to obtain spectral transients Attar-2017-Science ;
Pertot2017Science .

The effect of shot averaging for ghost-imaging reconstruction is shown in Fig.
7 where the deviation between the ghost-imaging-reconstructed spectrum and the
grating spectrum is shown. These deviations were evaluated as follows:
measurements were randomly selected to form 7 groups of spectra with $X$ shots
in each group. The spectra within each group were then ensemble averaged. Fig.
7a shows the result for $X$=1000 shots with the lower panel displaying the
normalized deviations ($\Delta s/s$) for photon energies, $i$:

$|\Delta
s/s|_{i}=\frac{\sum_{k=1}^{X}\left|c_{i}^{k}-s_{i}^{k}\right|/s_{i}^{k}}{X}$
(4)

where the upper index $k$ denotes the shot number. The deviation for one group
is shown as a gray-dashed line, and that for the average of 7 groups is shown
as a black-solid line.

The deviations show a spectral dependence as is clear from the lower panel of
Fig. 7a . The data were divided into 4 regions separated at 901.75, 907.5,
913.25 eV as depicted in Fig. 7a. The percentage deviations ($\Delta s/s$) for
each region are shown in Fig. 7b as a function of the number of shots
averaged. The $\Delta s/s$ decreases from around 28% for single-shot to 3% and
1% for 100 and 1000 shots, respectively. It is clear that the ghost imaging
reconstructed spectrum matches significantly better with the grating spectrum
after averaging hundreds of shots. This demonstrates the accuracy of the
calculated response matrix. If the single-shot deviation comes from random
noise, the deviation after averaging $X$ shots will be proportional to
$1/\sqrt{X}$. The log-log plot inset in Fig. 7b clearly shows a negative power
relationship. A curve fit of the data in region (2) yields a function of
$1/X^{0.45}$, which is close to the expected $1/\sqrt{X}$. We note that the
deviation is somewhat larger in region (1) which corresponds to the low-
photon-energy, low-photoelectron-kinetic-energy region. This may due to the
small cookie-box signals (corresponding to a low signal-to-noise ratio) for
electrons with lower kinetic energy. Region (1) also has a relatively small
0.53 Pearson coefficient (see further discussion in Supplementary Note 2).
Averaging to achieve a $\sim 1\%$ precision after $\sim 1000$ shots (requiring
only 1 ms at MHz repetition rates using the non-invasive photoelectron
spectroscopy scheme) can foster the realization of transient absorption
measurements at FELs Ott2019PRL .

Returning to the single-shot spectral profile measurements, the ghost imaging
reconstruction demonstrated here is easy to apply for better calibration and
resolution improvement of other instruments designed for XFEL diagnostics such
as the newly inaugurated angle-resolved photoelectron spectroscopy (ARPES)
instrument at LCLS Walter:ys5104 . After a training period using multiple
eTOFs in conjunction with a high-resolution spectrograph to obtain a converged
response matrix, spectral profiles of new shots with enhanced resolution can
be obtained at MHz repetition rates. From an XFEL-machine perspective, the
enhanced energy resolution for an ARPES-based x-ray photon diagnostic, which
has already demonstrated characterization of spatial, temporal and
polarization properties, is a sought-after breakthrough since it enables a
deeper understanding of the machine operation and allows for a fast-feedback
on SASE-formation characteristics.

From a scientific perspective, the high-resolution, non-invasive single-shot
incident x-ray spectrum characterization achieved here is anticipated to
benefit standard photon-in/photon-out transient absorption spectroscopy at
XFELs enabling exploration of nonlinear effects arising from propagation
through dense absorbing media Li2020PRA . It represents a step forward for all
techniques that require incident spectral characterization such as s-TrueCARS
cavaletto2021high  and others that take advantage of the intrinsic stochastic
nature of XFEL pulses Kimberg2016SD ; ratner2019pump ; kayser2019core .
Moreover, an incident pulse spectral characterization should enable a greater
depth of understanding for processes where the SASE-pulse structure can play
an important role, such as the recently-discovered transient resonance
phenomena in core-hole dynamics in gaseous media which rely on mapping out
narrow energy levels of highly elusive states of matter Mazza-2020-PRX , SASE-
FEL studies of chiral dynamics using photoelectron circular dichroism which
have been compromised by averaging over subtle dynamics due to the large
bandwidth Ilchen2021CommChem  and resonance-enhanced scattering for single-
particle imaging Ho-2020-NatComm . In summary, the use of a multiple
photoelectron spectrometer array as a transparent high-resolution beamsplitter
as demonstrated in this work is expected to be a huge asset for both the
technological and scientific development of x-ray spectroscopies at XFELs.

##  IV Methods

###  IV.1 XFEL photon delivery

Our experiment was performed on the SQS (Small Quantum Systems) branch of the
SASE3 beamline at the European XFEL decking2020mhz ; tschentscher2017photon
with SASE soft x-rays at 910 eV central photon energy. The averaged FWHM
bandwidth of the XFEL pulses was 9 eV and the standard deviation of pulse
energy fluctuation was 9% for an average pulse energy of 3.8 mJ as measured
with an x-ray gas monitor detector (XGM). The XFEL ran at a 10 Hz repetition
rate over a 25 minutes data acquisition time to obtain $N=15337$ shots.

###  IV.2 Electron measurement - cookie-box photoelectron spectrometer array

The cookie-box is located far from any x-ray focus points and the SASE beam
spot size was estimated to be $\sim 5$ mm in diameter, ensuring the
photoionization remains within the linear regime. The number of photoelectrons
generated is proportional to the product of the gas density and the
photoionization cross section. The base background gas pressure in the chamber
is $1\times 10^{-8}$ mbar and the pressure of gas injected was adjustable from
$1\times 10^{-7}$ mbar to $1\times 10^{-5}$ mbar. The gas density was set to
$2.5\times 10^{-7}$ during our experiment. The retardation voltage of 30 V
slowed photoelectrons with 40 eV initial kinetic energy to 10 eV. The total
time-of-flight for Ne $1s$ photoelectrons from the interaction region to the
detector is $\sim 60$ ns (see ToF signals in Supplementary Note 1). The
detector signals from the microchannel plate (MCP) stack were recorded every
0.5 ns. The electrons were slowed in order to create a $1s$ photoelectron peak
with more ToF sampling points while keeping the signal well above the
background.

###  IV.3 Photon measurement - grating spectrograph

The FEL beam (3.8 mJ/pulse) was attenuated prior to the spherical mirror with
Kr gas (transmission = 35.5%); combined with the grating efficiency of 36%,
0.49 mJ was incident on the screen. The VLS grating has a groove density of
150 lines/mm, with the imaging screen located at 99 mm distance from the
grating, in the focus of the spherical premirror. The emitted fluorescence
from the Ce:YAG screen was detected by a camera to produce the photon spectrum
serkez2020opportunities . The spectral range recorded on the YAG screen was
from 895.5 eV to 919.8 eV and was spread over 1900 pixels. The estimated
resolving power for the spectrometer, based on independent measurements, is
$E/\Delta E=10000$ Gerasimova_prep . This allows to resolve the single SASE
spikes of the XFEL pulses, which show a minimum spacing of $\sigma=90$ meV in
the present measurements.

##  V Data Availability

The experimental data were collected during beamtime 2935 at the European
XFEL. The metadata are available at
[https://in.xfel.eu/metadata/doi/10.22003/XFEL.EU-DATA-002935-00].

## Acknowledgments

This work was supported by the U.S. Department of Energy, Office of Science,
Basic Energy Science, Chemical Sciences, Geosciences and Biosciences Division
under contract number DE-AC02-06CH11357. We thank Chuck Kurtz for assistance
with Figure 1. We thank Natalia Gerasimova for assistance with the grating
spectrometer setup and operation. We acknowledge European XFEL in Schenefeld,
Germany for provision of the x-ray free-electron laser beam time at the SASE 3
undulator and thank the staff for their assistance.

## Author Contributions

K.L. and L.Y. conceived the use of ghost-imaging reconstruction for high-
resolution spectral characterization with multiple photoelectron
spectrometers. T.M., J.L., M.M., K.L., L.Y. planned and designed the
experiment. T.M. and J.L. set up the experimental configuration and performed
data collection together with K.L., D.K., G.D., A.P., M.M., M.I. and L.Y..
S.S. and N.R. provided discussions on spectral averaging analysis. K.L.
performed data analysis and together with L.Y. interpretation. K.L. and L.Y.
wrote the paper. All authors discussed the results and contributed to the
final manuscript.

## References

  * (1) Duris, J. _et al._ Tunable isolated attosecond x-ray pulses with gigawatt peak power from a free-electron laser.  _Nature Photonics_ 14, 30–36 (2020).
  * (2) Young, L. _et al._ Femtosecond electronic response of atoms to ultra-intense x-rays.  _Nature_ 466, 56–61 (2010).
  * (3) Hoener, M. _et al._ Ultraintense x-ray induced ionization, dissociation, and frustrated absorption in molecular nitrogen.  _Phys. Rev. Lett._ 104, 253002 (2010).
  * (4) Doumy, G. _et al._ Nonlinear atomic response to intense ultrashort x rays.  _Phys. Rev. Lett._ 106, 083002 (2011).
  * (5) Kanter, E. P. _et al._ Unveiling and driving hidden resonances with high-fluence, high-intensity x-ray pulses.  _Phys. Rev. Lett._ 107, 233001 (2011).
  * (6) Rudenko, A. _et al._ Femtosecond response of polyatomic molecules to ultra-intense hard x-rays.  _Nature_ 546, 129 (2017).
  * (7) Mazza, T. _et al._ Mapping resonance structures in transient core-ionized atoms.  _Phys. Rev. X_ 10, 041056 (2020).
  * (8) Young, L. _et al._ Roadmap of ultrafast x-ray atomic and molecular physics.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ 51, 032003 (2018).
  * (9) Wolf, T. J. A. _et al._ Probing ultrafast $\pi\pi*/n\pi*$ internal conversion in organic chromophores via k-edge resonant absorption.  _Nat. Commun._ 8, 29 (2017).
  * (10) Kim, K.-J.  An analysis of self-amplified spontaneous emission.  _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 250, 396–403 (1986).
  * (11) Andruszkow, J. _et al._ First observation of self-amplified spontaneous emission in a free-electron laser at 109 nm wavelength.  _Physical Review Letters_ 85, 3825 (2000).
  * (12) Milton, S. _et al._ Exponential gain and saturation of a self-amplified spontaneous emission free-electron laser.  _Science_ 292, 2037–2041 (2001).
  * (13) Geloni, G. _et al._ Coherence properties of the european xfel.  _New Journal of Physics_ 12, 035021 (2010).
  * (14) Hartmann, N. _et al._ Attosecond time–energy structure of x-ray free-electron laser pulses.  _Nature Photonics_ 12, 215–220 (2018).
  * (15) Lemke, H. T. _et al._ Femtosecond x-ray absorption spectroscopy at a hard x-ray free electron laser: Application to spin crossover dynamics.  _The Journal of Physical Chemistry A_ 117, 735–740 (2013).
  * (16) Higley, D. J. _et al._ Femtosecond x-ray magnetic circular dichroism absorption spectroscopy at an x-ray free electron laser.  _Review of Scientific Instruments_ 87, 033110 (2016).
  * (17) Kimberg, V. & Rohringer, N.  Stochastic stimulated electronic x-ray raman spectroscopy.  _Structural Dynamics_ 3, 034101 (2016).
  * (18) Ratner, D., Cryan, J., Lane, T., Li, S. & Stupakov, G.  Pump-probe ghost imaging with sase fels.  _Physical Review X_ 9, 011045 (2019).
  * (19) Cavaletto, S. M., Keefer, D. & Mukamel, S.  High temporal and spectral resolution of stimulated x-ray raman signals with stochastic free-electron-laser pulses.  _Physical Review X_ 11, 011029 (2021).
  * (20) Kayser, Y. _et al._ Core-level nonlinear spectroscopy triggered by stochastic x-ray pulses.  _Nature communications_ 10, 1–10 (2019).
  * (21) Driver, T. _et al._ Attosecond transient absorption spooktroscopy: a ghost imaging approach to ultrafast absorption spectroscopy.  _Physical Chemistry Chemical Physics_ 22, 2704–2712 (2020).
  * (22) Li, S. _et al._ Time-resolved pump–probe spectroscopy with spectral domain ghost imaging.  _Faraday Discussions_ 228, 488–501 (2021).
  * (23) Tollerud, J. O. _et al._ Femtosecond covariance spectroscopy.  _Proceedings of the National Academy of Sciences_ 116, 5383–5386 (2019).
  * (24) Zhu, D. _et al._ A single-shot transmissive spectrometer for hard x-ray free electron lasers.  _Applied Physics Letters_ 101, 034103 (2012).
  * (25) Makita, M. _et al._ High-resolution single-shot spectral monitoring of hard x-ray free-electron laser radiation.  _Optica_ 2, 912–916 (2015).
  * (26) Engel, R. Y. _et al._ Parallel broadband femtosecond reflection spectroscopy at a soft x-ray free-electron laser.  _Applied Sciences_ 10, 6947 (2020).
  * (27) Brenner, G. _et al._ Normalized single-shot x-ray absorption spectroscopy at a free-electron laser.  _Opt. Lett._ 44, 2157–2160 (2019).
  * (28) Viefhaus, J. _et al._ The variable polarization xuv beamline p04 at petra iii: Optics, mechanics and their performance.  _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 710, 151–154 (2013).
  * (29) Laksman, J. _et al._ Commissioning of a photoelectron spectrometer for soft x-ray photon diagnostics at the european xfel.  _Journal of synchrotron radiation_ 26, 1010–1016 (2019).
  * (30) Walter, P. _et al._ Multi-resolution electron spectrometer array for future free-electron laser experiments.  _Journal of Synchrotron Radiation_ 28, 1364–1376 (2021).
  * (31) Serkez, S. _et al._ Opportunities for two-color experiments in the soft x-ray regime at the european xfel.  _Applied Sciences_ 10, 2728 (2020).
  * (32) Lutman, A. A. _et al._ Polarization control in an x-ray free-electron laser.  _Nature Photonics_ 10, 468–472 (2016).
  * (33) Hartmann, G. _et al._ Circular dichroism measurements at an x-ray free-electron laser with polarization control.  _Review of scientific instruments_ 87, 083113 (2016).
  * (34) Padgett, M. J. & Boyd, R. W.  An introduction to ghost imaging: quantum and classical.  _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 375, 20160233 (2017).
  * (35) Pelliccia, D., Rack, A., Scheel, M., Cantelli, V. & Paganin, D. M.  Experimental x-ray ghost imaging.  _Physical review letters_ 117, 113902 (2016).
  * (36) Yu, H. _et al._ Fourier-transform ghost imaging with hard x rays.  _Physical review letters_ 117, 113901 (2016).
  * (37) Ott, C. _et al._ Strong-field extreme-ultraviolet dressing of atomic double excitation.  _Phys. Rev. Lett._ 123, 163201 (2019).
  * (38) Attar, A. R. _et al._ Femtosecond x-ray spectroscopy of an electrocyclic ring-opening reaction.  _Science_ 356, 54–59 (2017).
  * (39) Pertot, Y. _et al._ Time-resolved x-ray absorption spectroscopy with a water window high-harmonic source.  _Science_ 355, 264–267 (2017).
  * (40) Li, K., Labeye, M., Ho, P. J., Gaarde, M. B. & Young, L.  Resonant propagation of x rays from the linear to the nonlinear regime.  _Physical Review A_ 102, 053113 (2020).
  * (41) Ilchen, M. _et al._ Site-specific interrogation of an ionic chiral fragment during photolysis using an x-ray free-electron laser.  _Communications Chemistry_ 4, 119 (2021).
  * (42) Ho, P. J. _et al._ The role of transient resonances for ultra-fast imaging of single sucrose nanoclusters.  _Nature Communications_ 11, 167 (2020).
  * (43) Decking, W. _et al._ A mhz-repetition-rate hard x-ray free-electron laser driven by a superconducting linear accelerator.  _Nature Photonics_ 14, 391–397 (2020).
  * (44) Tschentscher, T. _et al._ Photon beam transport and scientific instruments at the european xfel.  _Applied Sciences_ 7, 592 (2017).
  * (45) Gerasimova, N. & et al.  Manuscript under preparation.