The measurement of X-rays and their spectra that are produced in an experiment is a useful diagnostic for understanding the mechanisms and dynamics at work. High-energydensity experiments regularly produce high-energy photons that need to be measured to gain a full understanding of what happened during the interaction of interest. For example, these include betatron X-rays used for imaging purposes 1 or as a diagnostic for experimental dynamics 2 , nonlinear-Compton-scattered photons measured to find evidence of a quantum electrodynamic (QED) plasma 3 , bremsstrahlung Xrays used to radiograph dense objects 4 , and X-rays that provide information on the plasma parameters for experiments at the National Ignition Facility 5 .

These spectral measurements are often made with a filter array, using Ross filter pairs or filters with thicknesses calculated to produce an accurate measurement of the spectrum 6,7 . Common arrangements include square grids and filter wheels, kept compact enough to maximize signal through as many filters as possible. They are often paired either with an X-ray charge-coupled device (CCD) detector 8 , a single image plate 9 or a stack [10][11][12] , or a scintillator screen that is imaged by a separate camera 13 . These arrays are typically designed assuming a) Electronic mail: rfitzgar@umich.edu b) Electronic mail: yongm@umich.edu a uniform spectrum across the entire area rather than taking angular variation into account 8,9,13,14 . Alternative approaches include single photon counting 7,15 , slit-grating spectrometers that rely on diffraction through a transmission grating 7 , and transmission crystal spectrometers 16 .

For an array of filters that utilizes different materials, we may assume that they attenuate photon energies according to the equation

where for a given element i, T i is the filtered transmission, T 0,i is the unfiltered transmission, µ i is the energy-dependent mass attenuation coefficient in cm 2 /g as listed by the National Institute of Standards and Technology (NIST) 17 , ρ i is the mass density in g cm -3 , and ∆x i is the thickness of the filter in cm. For a set of varied filters, this creates a pattern of intensities, from which the X-ray spectrum is calculated. Rather than assuming a spatially uniform spectrum, a filter pack can be designed to account for angular variation in the X-ray spectrum. The filter pack that we have created for this purpose relies upon an array of identical columns, spread out to cover different angles in a chosen direction. Each column has a sufficient number of varied filters to recover the spectrum at its position. The gaps between the columns allow us to account for nonuniformities in the beam and 'flatten out' the unattenuated signal. We chose to use a step filter array rather than a sequential filter stack because using an array of image plates would have significantly slowed down the acquisition time, and the X-ray CCD provides a more accurate photon flux measurement. This also allows us to remove hard hits whereas using a material with a faster response time like a scintillator would have to be calibrated for the optical system and could introduce more sources of error. More information on the specific design elements is included in section II.

By making separate measurements of the spectrum using each column, angularly-dependent variations can be recorded during a single shot. This is particularly relevant to highrepetition-rate experiments as well as to experiments that experience high shot-to-shot variability. In this paper, we will demonstrate the successful application of our filter pack method to the measurement of angularly-dependent critical energies of streaked betatron X-rays from a laser wakefield accelerator.

Laser wakefield acceleration is a useful mechanism for producing X-rays in a tabletop-scale setup [18][19][20][21][22][23] . In this scheme, a short, high-intensity laser pulse travels through a gas target, ionizing the gas and producing a plasma. As it propagates, it expels electrons from the axis of propagation due to the ponderomotive force. This creates a region of positive charge behind the pulse that attracts electrons from the background plasma. For electrons with sufficiently high velocities, they can be trapped in this 'bubble' regime and accelerated in the high acceleration gradients to energies approaching ∼ 10 GeV [24][25][26][27][28][29] .

As the electrons travel, they are constantly interacting with the electric fields set up by the sheath of the bubble around them. This leads to an oscillatory motion, producing betatron radiation emitted in a forward cone as the electrons reach relativistic velocities 8,30 . These X-rays follow a synchrotron-like spectrum with a characteristic critical energy 15,22,23 , and they have characteristics such as small source sizes, short pulse lengths, and high brightness that make them useful for imaging and ultrafast pump-probe applications 1,13,31 .

During the time of propagation, the electrons gradually move from an accelerating region of the bubble to a decelerating region. Instead of gaining energy monotonically, their energy may peak at some value before decreasing as they reach the dephasing length of the accelerator. These dynamics are difficult to measure within a single shot because typical measurements are integrated over time whereas the X-ray emission spectrum would change as the electrons are accelerated 32,33 .

To overcome this difficulty, we proposed 34 and demonstrated 35 that a transverse density gradient in our gas jet would cause the laser to veer towards the region of lower density, leading the wakefield and the electrons along a curved trajectory. As the electrons traveled, they constantly emitted X-rays that gained a horizontal spread corresponding to their time of emittance. This 'streaking' is similar to that seen in a streak camera, though in this case we are measuring the motion of the X-ray beam over time as the wakefield curves through the density gradient, and not electrons experiencing a voltage change 36 . Thus, at the measurement plane, the streaked X-rays created a relationship between spatial position and temporal emittance. By tracking how the X-ray critical energy changes with emission angle, we gained insights into how the electron energy changed with time, according to the equation

where ω β is the betatron oscillation frequency of the electrons, K is the wiggler strength parameter, and γ is the Lorentz factor of the electrons 37 . More specific relationships between the electron dynamics and the emitted X-ray spectrum at different laser and plasma conditions can be obtained using equation 1 in reference 34 and equation 3 in reference 35.

In the case with no curved wakefield, there are limitations to the fit of the synchrotron spectrum: because the X-ray emission spectrum changes during propagation, the final measured spectrum is the sum of many different synchrotron-like spectra added together, skewing the final shape 32 . By contrast, for the case with a curved wakefield, due to the dispersion of the X-rays as they propagated, the spectrum at each angle should produce a better fit to a synchrotron-like spectrum because it lacks the summation of different spectra. In addition, we neglected the contribution from the synchrotron motion due to the curved wakefield because the number of photons from this process is at least an order of magnitude less than the number produced from the betatron oscillations 38 .

In our experiment described in reference 35, we measured the critical energy of the X-ray spectrum along with its dependence on angle in order to compare our results with those demonstrated in simulations 34 . The change in energy with angle corresponded to changes in time and revealed information about the electron dynamics during propagation, such as dephasing, using a single-shot, noninvasive approach 39,40 .

The experiment was conducted at the Advanced Laser Light Source at the Institut National de la Recherche Scientifique. The setup is shown in figure 1b. Each laser pulse had an energy of about 3.2 J, a wavelength of 800 nm, and a pulse duration at full-width-half-maximum (FWHM) of 22 fs. These were focused by an off-axis parabola (OAP) to a focal spot of 15 µm on a gas jet engineered to produce a transverse density gradient. Behind the jet, a magnet deflected electrons onto a Lanex screen for measuring the electron energy spectrum. The X-rays continued forward, leaving the vacuum chamber through a beryllium window, and passing through the filter pack to be recorded by a Princeton Instruments SCX 4300 Xray CCD camera coupled to a Gd 2 O 2 S : Tb phosphor screen. The total chip size of the CCD is 50 mm × 50 mm, and each pixel is 24 µm × 24 µm. At a distance of 134.5 cm from the source, the camera's acceptance angle was therefore 37 mrad.

The filter pack is composed of 25 identical columns with 11 filters of either aluminum or copper. Each filter is 1 mm × 2 mm, with a 1 mm gap separating each column. Figure 1a shows the overall transmission of each filter element with the consideration of the quantum efficiency of the CCD as well as the attenuation of the materials such as the Be windows and the air in the beam path. The filters in each column were sufficient to find the critical energy of the X-ray spectrum at that angular position, with an angular resolution of 1.7 mrad, so each column provided data for a different angular position within a single shot.

The design assumed spectral uniformity in the vertical direction, so it only measured angular dependence in the horizontal direction. This corresponded with the direction that the laser curved toward while it traveled through the plasma. To define the angular position, we assumed that the X-rays without streaking were well-aligned with the center of the beryllium window, so this became the position of 0 divergence. Positions to the left or right were assigned an angular value in mrad.

The scaling of the synchrotron-like spectrum for on-axis photons can be expressed as

where ω is the frequency, Ω is the solid angle, ω c is the critical frequency corresponding to the characteristic critical energy E c = hω c , and K 2/3 is a modified Bessel function of the second kind 19,23,41 . The critical energy is the parameter of interest in the analysis because it provides a convenient method of comparing the spectra at different positions to understand how they differ.

To retrieve the spectrum from the transmitted intensities through the filters at a particular angle, according to equation 1, we had to measure the signal through each of the 11 filters in the column. Due to the finite size of the camera chip, we were able to pull data from only 21 of the 25 total columns. These gave us 21 data points for finding and comparing the critical energy at different angles, according to equation 3. Two of the filters fell off the array during the experiment, but the impact on the spectrum reconstruction was negligible.

To account for the presence of background signal on the camera, a piece of lead with a thickness of 3.5 mm was added at the bottom of the filter pack (not pictured). This provided a background reference for what the camera was reading in the absence of X-rays. This region was averaged and this value was subtracted from each pixel in the image. The background was less than 5% of the maximum signal.

To accurately compare the measured critical energies at different angular positions, the spatial fluctuations in the unattenuated signal between the filters had to be accounted for. These nonuniformities could skew the measured energy at one location compared to another based on the varying relative inten- sity rather than changing energies.

The process of 'flattening out' the signal is illustrated in figure 2. A picture of the physical filter pack is included in Figure 2a. Figure 2b shows the raw image with the filter pack in front of the camera. The varying unattenuated signal is evident in the gaps between the filters. These gaps were used to measure this signal, and then the griddata function in MAT-LAB was used to perform a 2-D interpolation to recover the entire unattenuated beam, as shown in figure 2c. With the beam recovered for every point, this was divided out of the original image. The resulting flattened image which shows the normalized X-ray transmission of the filters is shown in figure 2d.

With the unattenuated signal flattened out, the filter signals were measured from the image. This was done by taking an average of the signal over each rectangular region, within the visible edges of each filter 42 .

Once the signals had been retrieved for each filter in a column, the critical energy of the spectrum at that angular position could be calculated. This process began by guessing a value for the critical energy and calculating the spectrum using equation 3. From this spectrum, the calculated signal for a given filter is given by 42

where Y c is the calculated signal, S(E) is the spectrum from equation 3, Q(E) is the quantum efficiency of the camera, and T(E) is the transmission through the specific filter material, as calculated in equation 1. For the PI SCX 4300 camera and each filter material and thickness, the effective transmission

With the guessed critical energy, the calculated signal was found for each filter in the column. To measure how closely this fit with the real signals, the residuals, ∆, were calculated using a Monte Carlo minimization technique where ✚ ✚ as

where Y c is the calculated signal, Y is the measured signal, N is the number of filters, and i counts the filters in a column from 1 up to 11, or fewer for the columns not fully visible 42 . This uncertainty was propagated throughout the rest of the analysis. The uncertainty in the filter thicknesses, which is about 1.3 µm for each filter, is not included in this analysis. We estimate the error this would contribute would only be approximately 0.064% of the measured critical energy, which would not significantly affect our final results.

This process was repeated for different guesses of critical energy across a range of values in order to find the energy that minimized ∆, indicative of the best agreement between measured and calculated signals 43 . An example of the guessed energy range is shown in figure 3a, where the plot dips down to a clear minimum at 26.6 keV. The error bounds (red lines in figure 3a) were determined by taking the standard deviations of the residuals for each filter in the whole column. Figure 3b shows the comparison between the measured signal and the best fitted signal corresponding to the minimum residual at 26.6 keV, which indicates good agreement.

After repeating this for each column in the image, the results were plotted in figure 3c to show how the critical energy varied with angle and therefore with time. The residual in the final determination of the critical energy was chosen to be the width at which the lower error curve in figure 3a equaled the minimum of the upper error curve, plotted in black. In this region, the lower error curve had smaller residual values than the lowest value that the upper error curve achieved. The pixel value for the approximate center of each column was converted to angle using the fact that the pixel number, N p , corresponding to X-rays passing through the center of the beryllium window was 1057, the CCD was 134.5 cm from the source, and the pixel size was 24 µm per side, so θ = 1.8 × 10 -5 (N p -1057), assuming the small-angle approximation is applicable.

As we have described above, the spectrum of the betatron radiation emitted by a single electron with a fixed energy is synchrotron-like, as seen in equation 3, with a critical energy of E c = hω c . For a typical laser wakefield accelerator, however, the betatron spectrum differs from a synchrotron spectrum for two reasons. First, the measured spectrum is integrated over time as the electron beam evolves. This issue and its resolution via betatron streaking is discussed in section I. The second reason is that a real electron beam always has an energy spread. Thus, using a single electron energy γ (normalized to m e c 2 ) to fit the synchrotron spectrum is inaccurate. We analyze the effect of the finite energy spread in the following.

An example of the electron energy spread effect on the betatron spectrum is shown in figure 4. We use an electron beam with a Gaussian energy distribution which has a peak energy of γ peak = 1000 and a relative energy spread of ∆γ/γ peak = 50%, as shown in figure 4a. The corresponding photon spectra for each value of electron energy is given in figure 4b. Note that here we calculate the critical photon energy as a function of γ only, i.e., with fixed plasma density n p and betatron oscillation amplitude r β . The amplitudes of the photon spectra are simply weighted by the electron spectrum, dN e dγ , in figure 4a. Moreover, here for each γ value, the photon spectrum is angularly integrated with the shape of (ω/ω c ) ∞ 2ω/ω c K 5/3 (ξ )dξ to take account of all photons emitted.

By integrating the photon spectra in figure 4b with respect to γ, we obtain the overall photon spectrum for the whole electron beam, as shown by the blue line in figure 4c. For comparison, the photon spectrum for a single electron with energy γ peak , corresponding to the dashed line in figure 4b, is plotted as the orange line in figure 4c. It is clear that the integrated spectrum is nearly identical to a synchrotron spectrum emitted by a single electron by comparing the peak photon energies of 2.9 keV and 3.0 keV respectively, which have a 3.3% relative error.

Figures 4d and 4e depict the critical energies and relative errors of an electron beam as a function of either the relative electron energy spread or the peak electron energy spread, respectively, in comparison with the single electron case. The orange curves in figure 4d show that as the relative electron energy spread increases, the relative error also increases; this coincides with the critical energy increasing, as the blue curve illustrates. Nevertheless, the relative error in peak photon energy remains less than 4% and the relative error in critical photon energy remains less than 17%, even when the electron energy spread reaches 100%. The orange curve with diamonds in figure 4e shows that even for a large energy spread of 50%, the relative error in the reconstructed peak photon energy stays low, approximately 2.7%, while the peak electron energy is varied from 200 to 2200. Similarly, the relative error in the reconstructed critical energy stays below 7%, as shown by the orange curve with asterisks. The blue curves show that the peak energy of the betatron spectrum increases with peak electron energy similarly for both the electron beam and the single electron. Thus, from these low errors even at large energy spread, we may conclude that a gaussian electron energy spread has negligible influence on the betatron spectrum reconstruction, relative to other experimental errors.

In conclusion, we have demonstrated that we were able to use our filter pack to successfully measure different critical energies of the betatron spectra at different angles, all during a single shot. We could measure the change in critical energy as electrons propagated in a curving wakefield. The transverse density gradient in the gas jet caused a streaking effect of the betatron X-rays across a CCD camera, linking the spatial position on the detector with the time of emission so the evolution of the critical energy was clearly visible. The analysis showed a critical energy that clearly matched the data and provided good agreement with the calculated signal. This is valid for an electron beam with an energy spread, as the effects of this spread on the spectral reconstruction have been shown to be negligible.

These measurements were made possible by the filter pack design that included filters of different materials and thicknesses that were sufficient for reconstructing the spectrum. We believe this approach could be applied to experiments that require angularly-resolved spectral information on every laser shot without the errors introduced by high shot-to-shot variability. cil of Canada (Grant No. RGPIN-2021-04373). The authors would also like to thank Joël Maltais, Stéphane Payeur, Claude Sirois, François Légaré from ALLS for technical support.