Calorimeter with Bayesian unfolding of spectra of high-flux broadband x rays

High-intensity lasers are widely used to study the physics of the coupling of extreme electromagnetic fields with matter. Transient charge separation fields in the plasmas generated in this interaction lead to the acceleration of charged particles to energies comparable to conventional particle accelerators.¹ 

In the case of solid targets, such as foils, the high electric fields ionize the matter and accelerate the electrons to relativistic energies. The electrons traversing the target produce bremsstrahlung radiation.^2,3 Fast electrons with high enough energy can escape the target,⁴ their spectrum usually measured with magnetic spectrometers and compared with the current available scaling models.^5,6 However, electrons with lower energies rebound at the target interfaces due to sheath fields,⁷ recirculating and heating the target.⁸ Diagnosing the evolution of the electron distributions within the target is important for understanding the mechanisms by which laser energy couples to electrons and, in the case of ion acceleration, to protons and ions. X rays emitted from the target are a good probe for this purpose. Currently, target heating can be diagnosed via Kα emission⁹ and other x-ray techniques in the keV range. Photonuclear activation measurements provide thresholding information on the spectrum in the MeV range.¹⁰ Calorimetry has been used to diagnose photon emission in the MeV range.¹¹ 

In the case of gas targets, the interaction of the laser with underdense plasmas leads to laser wakefield acceleration (LWFA). This generates electron bunches of high charge (up to nC) and high energies (up to several GeV).^12,13 These electron bunches can be used to generate bright bremsstrahlung beams (N_γ ≈ 10⁹) with also a short duration (fs)¹⁴ and high energy (up to GeV) when using a high-Z radiator.^15–17 They can additionally be used to create energy tunable x-ray beams via inverse Compton scattering of the electrons with a high-intensity laser pulse.^18–23 

In all these cases, the high particle numbers coupled with the short pulse duration (fs to ps) pose a challenge for the characterization of the x-ray emission. Furthermore, the spectral distribution varies for each source in shapes (exponential-like for solid targets and Gaussian-like for inverse Compton scattering) as well in the energy range from MeV up to GeV.

Current calorimetric techniques based on structures of absorbers interlaced with sensitive detectors,^24–29 scintillator-based systems^30–33 as well as Compton-based bremsstrahlung spectrometers,^20,34 have been proposed and used in laser-plasma experiments. They rely on numerical methods to unfold the x-ray spectrum from the energy deposition depth-profile in the multi-layered detector. They require an initial and precise knowledge of the functional description of the energy spectrum (known in Bayesian analysis as the functional “priors”) or rely on Monte Carlo simulations to forward calculate the energy deposition in the detector and performing a least-squares minimization to reach the best fitting one.

Hannasch et al.³⁵ recently published an extensive study of the detection of x rays generated in a laser wakefield accelerator using the same detector structure as in our work, extending the work by Jeon et al.³⁶ There, the unfolding procedure was based on a forward calculation of the energy deposition and a posterior least-squares minimization. The technique was used to reconstruct spectra from betatron, bremsstrahlung, and inverse Compton radiation.

In this work, we discuss the use of the same type of the differential calorimeter coupled with a Bayesian unfolding. We apply this method to conventional radioactive sources, laser generated bremsstrahlung and inverse Compton radiation. We benchmark our unfolding with the one from Ref. 35. The wide applicability of the Bayesian unfolding provides a unified approach to the detection and unfolding of laser-plasma generated x rays. Furthermore, the final solution for the spectrum of x rays calculated by the unfolding procedure is not tightly constrained to the initial guess as opposed to other techniques with even different functional priors leading to the same solution. The main benefit of this effect is the possibility of observing unexpected particularities in the x-ray spectrum, an observation that is not possible when the unfolding is constrained by the functional form of the prior.

Calorimeter detectors have been used in high energy physics for decades.³⁷ Typical calorimeters are massive detectors that rely on the measurement of the energy deposition in the material to identify and determine the energy of incoming particles. In our case, we develop a calorimeter detector to determine the energy spectrum of an ultrashort x-ray bunch generated via laser–matter interaction.

The calorimeter is structured as a sampling calorimeter. Layers of materials with increasing density and thickness are used as absorbers for low energy x rays and as secondary particle shower developers for high energy x rays. The sensitive part of the detector consists of image plates (Fujitsu BAS-MS). Image Plates (IPs)³⁸ are based on photo-stimulated luminescence (PSL).³⁹ Incoming x rays or secondary particles ionize Eu²⁺ dopant ions in phosphor crystals embedded in plastic, converting them to Eu³⁺. Subsequent visible radiation in the scanner stimulates recombination of the trapped ionized electron with a Eu³⁺ ion, converting it back to excited-state Eu²⁺, which then luminescences with an intensity proportional to the stored energy. These image plates are placed behind each absorber layer. A similar structure was used by Horst et al.,²⁴ who describe a layered detector with absorbers and thermoluminescence dosimeters as readout. They used such a calorimeter to reconstruct spectra from clinical accelerators and laser–plasma interactions.

Our detector was designed to cover a wide range of cases: bremsstrahlung generated in laser–solid interactions at photon energies up to 5–10 MeV, inverse-Compton x rays (<2 MeV) generated by colliding a visible light pulse with laser-wakefield accelerated electrons, and high energy bremsstrahlung (>100 MeV) generated by the laser-wakefield accelerated electrons traversing through a high-Z target. Figure 1 summarizes the calorimeter’s layer-by-layer configuration. We chose the number of image plates (24) as a compromise between maximizing the sampling of deposited energy and minimizing scan time. Since our Fujitsu Bas 1800II scanner accommodated 12 IPs at once, a full detector turnover required two scans or about 20 min. All cases under study used the same detector; however, in the case of the solid target, the readout was limited to 16 layers, while the LWFA cases used the full stack readout.

The specific materials and thicknesses were chosen based on the expected x-ray spectrum contributions. Low energy x rays (E $<100$ keV) from general backgrounds or from betatron emission are absorbed within the detector. The first PMMA layer places a threshold on the lowest detectable energy at 7 keV. For high photon energies (E $>10$ MeV), the secondary particle shower develops. The brass and steel material layers are used to provide a maximum on the shower development followed by a reabsorption of the secondaries, resulting in a peaked energy deposition profile in later layers. This effect will be discussed in Sec. V C.

The general formulation of the problem at hand can be as follows: let us assume that our detector provides an energy deposition per layer E_dep,j; let us assume also that the detector has a response matrix P(E_dep,j, E_in), which relates the impinging energy spectrum to the energy deposited in the detector. Then, the x-ray spectrum f(E_in) and the measured one are related by

where f(E_in,i) denotes the binned energy spectrum and the sum is performed over all the energy bins.

Many different approaches have been proposed to solve the general unfolding problem.^40–42 In this report, we offer a new solution to the bremsstrahlung unfolding by combining the differential calorimeter with a Bayesian unfolding developed by D’Agostini.^43,44 This algorithm has already been successfully applied at ATLAS to reconstruct particle jets in proton–proton collisions.⁴⁵ 

The main advantage of this method is that it is not constrained to the prior functional form. This means that while a certain shape of the spectrum must be assumed as a prior, the final solution does not necessarily follow the same function. The Bayesian approach enables spectral features to appear that a chosen prior solution did not anticipate, which can, in turn, lead to new understanding.

The full details of the unfolding procedure can be found in Refs. 43 and 44. In the following, we present a summary of the algorithm while detailing the key aspects related to our system.

Let us start with the formulation of the problem as stated in Eq. (1). Consider the incoming bremsstrahlung normalized spectrum as a probability distribution. In this case, the contents of each bin f(E_in,i) represent the probability of a photon to have energy E_in,i. The problem can now be expressed as

where P(E_dep,j|E_in,i) is again the response matrix and represents the energy deposit in layer j due to a photon with energy E_i. A probabilistic interpretation of this equation is as follows: the contents of each “effect” bin, E_dep,j, are the sum of all the contents of the “cause” bins, f(E_in,i), times the probability of the cause bin i resulting in an effect bin j, P(E_dep,j|E_in,i).

From here, we can apply Bayes’ theorem to estimate the probability of a cause bin i due to an effect bin j, P(E_in,i|E_dep,j),

where f₀ represents the initial assumed prior spectrum.

Finally, the spectrum due to the observed energy deposition in the layers is calculated as

where ɛ represents the inefficiency, i.e., not every x-ray crossing the detector produces an event. This is simply calculated in the Monte Carlo simulations as the ratio of the number of interactions to the number of primary particles used.

The response matrix of the detector to incoming x rays was simulated via the Geant4 package.^46–48 The layers were recreated exactly as in the real detector, and the internal structure of the image plates was also implemented according to Ref. 49. The physics list used was the electromagnetic model Penelope, which provides the most accurate results for the low energy range⁴⁹ combined with the high energy physics list QGSP_BIC_HP to account for secondary processes at higher energies. The primary beam consists of monoenergetic photons. The beam shape and dimensions were chosen according to the beam source size and divergence estimated from the IPs in the stack for each of the cases. Nearby objects close to the beam path where scattered radiation might affect the stack were also accounted for. In the case of calibration radioactive sources, the elements included in the simulation were the source capsule, magnetic electron filter, table support, and lead collimator. For the laser–solid interaction, the elements included were a magnetic electron spectrometer, Kapton foil at the chamber flange, and collimator. Finally, for the laser-wakefield acceleration case, the magnetic electron spectrometer and Kapton window were used. No collimator was used due to the beam-like structure of the radiation. Accounting for the beam shape implies accounting for the radiation leakage due to the finite size of the detector. Figure 2(a) shows the simulation results for a matrix covering the energy range from 10 keV up to 800 MeV. For unfolding up to 10 MeV, the response matrix was sampled with 10 keV energy bins. For energies under 100 keV, the deepest layers show no energy deposit due to the high absorption of the photons in the material in front of them. For unfolding up to 800 MeV, the response matrix was sampled with 1 MeV energy bins.

Selecting a prior in any unfolding depends largely on the knowledge of the physical process taking place. For example, a radioactive source such as ⁶⁰Co will emit x rays with a very well defined energy. In this case, a prior with a Gaussian shape (to account for finite detector resolution) on a uniform background would be the most general form. In case of bremsstrahlung, an exponential shaped prior or a parameterization of the process cross section would be a good candidate.

The cases we consider are the following:

• Uniform prior. The most basic case is that of a flat spectrum. It will be used in Sec. IV to illustrate the evolution of the unfolding with the complexity of the prior.

• Bremsstrahlung exponential prior. One possible prior is a simple exponential function,

  $f0=ATγ×e−EγTγ$

  (5)

  with A being a scaling factor and T_γ being the temperature expressed in units of energy k_BT_γ, where k_B is the Boltzmann constant.
• Bremsstrahlung Findlay prior. One can use a parameterization describing the shape of a bremsstrahlung spectrum. This is based on an approximate parameterization of forward integrated bremsstrahlung emission tables as expressed in Ref. 50,

  $f0=aZ2(Eγ−1−bEe−1),$

  (6)

  where a ≈ 11 mbarn, b ≈ 0.83, E_γ is the photon energy, E_e is the electron energy, and Z is the atomic number of the target. For simplicity, we will refer to this as a Findlay prior.
• Inverse Compton scattering case. The function is a Gaussian

  $f0=A×e−(b−Eγ)22c2+d$

  (7)

  of amplitude A and width c centered at E_γ = b on top of a uniform background d.

• Calibrated radioactive sources. The prior is the same as the inverse Compton case.

In all cases, the prior is normalized so that the total integral is unity.

Following D’Agostini’s procedure, the unfolding is performed in an iterative way. The prior is used in Eq. (3) together with the response matrix. The resulting matrix is then used to calculate the corresponding energy deposit in the detector. Finally, a chi-square is evaluated to compare the measured energy deposition with the values obtained via the algorithm,

where E_dep,unf,j represents the energy deposit in layer j obtained from applying Eq. (1) to the unfolded spectrum, E_dep,exp is the measured energy deposit in the image plate, and σ_dep,exp is the statistical uncertainty of the measurement. The sum is performed over all the layers.

The convergence criterion to stop the iteration is (χ_iter − χ_iter−1)/χ_iter < 0.001. If the criterion is not fulfilled, the unfolded spectrum is smoothed and then injected as prior in the next iteration. Several smoothing procedures have been tested: a Savitzky–Golay filter implemented in the NumPy Python package and⁵¹ a running average filter and a convolution filter implemented in the Astropy package.^52,53 In general, there is no single recipe for the smoothing procedure to be used or what the length of the window should be. The window width will determine the final energy resolution of the unfolding due to its inherent broadening of the features in the data (for example in the case of Gaussian spectra). The smoothing must be tailored to each specific case.

In this section, we discuss the different uncertainty sources in the measurements and the propagation to the final unfolded spectrum.

• Statistical uncertainty in the measurement: The underlying statistic distribution on the image plate hits follows a Poisson distribution. The number of particles impinging on the detector in all the cases analyzed in this paper results in a very low relative uncertainty in the energy deposition measured at each layer. An analysis of the distributed hits on a given IP provides an estimate of the mean value λ from which an uncertainty is obtained. The relative uncertainty in the cases under discussion ranges from 0.1% to 0.4%. This is further cross-checked against the final number of photons unfolded. As discussed in Sec. V C, the total number of photons incident in the detector is N_ph ≈ 10⁸. The detector efficiency extracted from Monte Carlo simulations is 1%. The number of photons actually depositing energy in the active layers is about 10⁶. Assuming a Poisson distribution, this results in a relative uncertainty $λ/λ=0.1%$⁠.

• Systematic uncertainty in the measurement: The main source of systematic uncertainty is the scanner calibration. The calibration factor for the scanner was determined to be α = 0.29 ± 0.06 PSL/MeV. However, this uncertainty simply results in an offset in the energy deposition. The relative values of the energy deposition between layers remain unchanged. As a consequence, this only results in an offset in the total number of photons in the unfolded spectrum.

• Uncertainty in the unfolded spectrum: The uncertainty in the energy deposition is propagated to the unfolding spectrum by generating 100 distributions based on the estimated Poisson statistics. For each layer, the parameter λ is estimated from the measurement and used to generate 100 energy deposition profiles. Each one of the energy deposition profiles is unfolded separately. Then, the unfolded spectra for each of the cases are averaged together, and a standard deviation σ_unfold,bin per bin is calculated.

Furthermore, the statistical uncertainty in the unfolded spectrum is calculated as $σpoiss,bin=nbin$⁠.

The final uncertainty reported is the combination of both, $σtotal,bin=σunfold,bin2+σpoiss,bin2$⁠.

The unfolding procedure was tested with extensive Monte Carlo simulations. In this section, we will provide an example of the full unfolding while discussing key aspects such as the prior selection, convergence criteria, and uncertainty levels. We will also compare the result with commonly used unfolding methods.

Geant4 was used to simulate the bremsstrahlung generated by hot electrons produced in a laser–solid target interaction. The target was a slab made of titanium with a thickness of 700 μm. The primary particles were electrons with an exponential spectrum. The temperature of the spectrum was T_hot = 2.2 MeV, taken from scaling laws⁵ for a 100 TW laser system. The real geometry of the detector was implemented and located 1 m downstream along the laser direction. The impinging bremsstrahlung spectrum on the detector is scored as well as the energy deposition per layer. These parameters were chosen to simulate the solid target case discussed in Sec. V B.

The adaptability of the algorithm was tested by using priors of increasing complexity. The first prior type to be tested was a uniform distribution. The algorithm uses Bayes’ theorem and obtains a posterior from the prior and the energy deposit profile. The χ² value is calculated using Eq. (8). The posterior is smoothed by convolution with a uniform kernel. The smoothed posterior is then fed as the prior for the next iteration. An example of the evolution of the unfolding is shown in Fig. 3. The solution evolves toward a multiple exponential result despite the uniform prior [Fig. 3(a)]. Once the convergence condition is reached, the final calculation of the unfolded spectrum is performed. The process is repeated for 100 energy deposition profiles, which are extracted from the original energy deposition profile, assuming a Poisson distribution. Finally, the mean value of all the unfolded spectra is calculated, and the uncertainty is obtained. The final unfolded spectrum compared to the Geant4 input spectrum can be seen in Fig. 3(d). Figure 3(g) shows the simulated energy deposition and the energy deposition corresponding to the unfolded spectrum. The result of a uniform prior shows an exponential decay for low energies but remains flat for energies higher than 0.5 MeV. Of particular interest is the very low energy region of the spectrum. The algorithm is capable of adapting the solution to reproduce the self-absorption in the target itself. This effect is visible in the impinging bremsstrahlung spectrum simulated by Geant4.

Refining the prior to a more accurate parameterization naturally leads to better fitting of the unfolded spectrum to the simulated one. In Fig. 3(b), a single exponential spectrum was used as the prior. The unfolded spectrum fits better the simulated one [Fig. 3(e)]. Finally, in Fig. 3(c), a Findlay prior following Eq. (6) is used, and the best match to the simulated spectrum is obtained, Fig. 3(f). The energy deposition corresponding to the unfolded spectrum is shown in Figs. 3(g)–3(i).

We benchmark the unfolded spectrum against the models developed by Findlay⁵⁰ and Galy.⁵⁴ Assuming an exponential electron spectrum, such as the one used in the Geant4 simulations, the bremsstrahlung emission is expressed as

with a and b being the same parameters as defined in Eq. (6), N_e being a scaling factor for the electron distribution, i.e., the amplitude of the assumed electron exponential spectrum, T_e being the electron temperature, E₀ being the electron cut-off energy, Z being the atomic number, n_a being the atomic density, and l being the target thickness. A fit to this function results in N_e = (1.3 ± 0.2) × 10³ MeV⁻¹, T = 2.7 ± 0.1 MeV, and E₀ = 3.2 × 10³ MeV. The temperature value obtained is within 20% of the value input in the simulation, T_e = 2.2 MeV, with the difference due to the approximation in the cross section itself. The large value of the electron cut-off energy indicates that the detector is not able to resolve E₀. The unfolded spectrum does not show any signature of a cutoff. This is equivalent to assume that the effective cutoff extracted lies in the infinity. We test this by setting E₀ → ∞ in Eq. (9), obtaining

Fitting this function to the unfolded spectrum yields N_e = (1.3 ± 0.2) × 10³ MeV⁻¹ and T_e = 2.7 ± 0.1 MeV, the same values as the previous fit.

The sensitivity of the detector to the cut-off energy is related to the energy bandwidth. Naively, one could assume to increase the response matrix to higher energies until the cutoff is visible. However, it is easy to demonstrate that for an exponentially decreasing spectrum in the low MeV range, adding energy bins to the response matrix for a fixed detector configuration does not increase the amount of information contained in them. This was studied by calculating the energy deposition for photons in 1 MeV energy intervals. Figure 4 shows the relative energy deposition for slices of the unfolded spectrum. Most of the energy is deposited in the detector by x rays with energies <1 MeV. Subsequent slices, deposit a smaller amount, until the contribution of higher energies is negligible. Conversely, for higher x-ray energies where the secondary shower development is much more pronounced, the response matrix must be extended to cover a higher energy range. This phenomenon is used to reconstruct the high energy bremsstrahlung spectrum in the LWFA case in Sec. V C.

We benchmarked the accuracy of the reconstruction and its correlation to the energy deposition uncertainty. We used Eq. (10) to generate bremsstrahlung distributions with different temperatures. Thus, we can consider Eq. (10) to be exact without effects of target absorption or cross section approximations. We generated bremsstrahlung beams in Geant4 with temperatures ranging from 2 to 50 MeV (without target) and recorded the energy deposit in the calorimeter. We used the energy deposit profiles as the input of the unfolding while considering four cases with different relative uncertainties, ΔE_dep/E_dep = 10⁻⁴, 10⁻³, 10⁻², 10⁻¹. We unfolded each case separately and fit the spectrum to Eq. (10). The uncertainty of the extracted T_e was 5% for the first three cases and 10% for the last one. In all cases, the input T_e was within the uncertainty of the fitted value. This indicates that the unfolding can reconstruct temperatures with an accuracy of 5% with the accuracy deteriorating as the uncertainty in the measurement increases beyond that value.

We also cross-checked the Bayesian unfolding results against two commonly used unfolding methods: a χ² minimization procedure via a steep-gradient (also known as Newton’s method) and a forward calculation. We used a Findlay prior for all cases. The comparison is seen in Fig. 5. As expected, the parameterized unfolding does not reproduce the self-absorption part of the spectrum. This, in turn, introduces a deviation at higher energies with respect to the simulated one. This effect represents one of the strengths of the Bayesian unfolding: a basic parameterization of the prior can lead to more complex forms not introduced beforehand.

The technique was tested with calibrated high-activity sources to determine its accuracy and capability to unfold peaked spectra from energy deposit profiles. Calibrated radioactive sources test not only energy resolution but also enable testing the ability of the algorithm to calculate the total number of photons that crossed the detector.

A ⁶⁰Co and a ¹³⁷Cs source were used for this experiment. They are well established calibration sources and their decay scheme known in detail. ⁶⁰Co has two characteristic photopeaks at 1.173 and 1.332 MeV,⁵⁵ while ¹³⁷Cs has a characteristic gamma emission at 661 keV.⁵⁶ The activity of the sources according to the supplier was 10.00 GBq with a 10% tolerance.

The full experimental setup is shown in Fig. 6. The source was encased in a lead shielding with an aperture of 30° in the forward direction. This aperture was covered by two blocks of lead, ensuring that no radiation leaked when it was closed. The opening of this aperture was performed via a motorized stage. A filtering magnet with a magnetic field strength of 0.5 T and a length of 10 cm was used to deflect the electrons produced in the nuclear decay. Further downstream, a lead collimator with a thickness of 5 cm and a hole diameter of 1.2 cm was used to stop secondaries produced at the magnet. We placed the calorimeter 6 cm after the collimator. The calorimeter was exposed to the source for 20 min. Due to the opening and closing of the lead frontal cover of the source, the uncertainty of this time is about 10 s.

The analysis of each image plate signal was as follows: (i) the image data were converted to PSL via the scanner calibration, (ii) the data were corrected for fading following the parameterization in Ref. 38, (iii) the signal within the collimator area was integrated by selecting a circular spot with increasing diameter as a function of depth according to the angle subtended by the collimator, and (iv) the energy deposition as a function of depth in the detector was calculated. The resulting energy deposit as a function of depth is then used as the input in the unfolding algorithm. The initial guess used for the unfolding is a Gaussian function on top of a background as expressed in Eq. (7).

For each of the sources, a spectroTRACER scintillator detector was used to measure the precise spectrum at the calorimeter position. The results of the unfolding and the spectrometer are shown in Fig. 7. The spectroTRACER measurement shows the two photopeaks, the standard Compton continuum (due to the detector scintillator), and a low energy component at energies <100 keV. This low energy contribution is attributed to scattered photons in the collimator and on the other elements of the setup. The algorithm reconstructs this contribution even though it was not encoded in the prior. As expected, the unfolding does not reconstruct the Compton continuum shown by the spectroTRACER as this is a detector artifact. The final resolution of the unfolding is not enough to reconstruct both photopeaks of ⁶⁰Co. In the case of ¹³⁷Cs, the unfolded spectrum shows a single peak with some low energy contribution similar to the spectroTRACER measurement. A Gaussian fit to the ¹³⁷Cs photopeak results in a value of E_fwhm,Cs = 0.17 MeV. The fit of the ⁶⁰Co results in E_fwhm,Co = 0.31 MeV. This value is compatible with assuming the ⁶⁰Co energy resolution to be the sum of the ¹³⁷Cs result plus the distance of the two ⁶⁰Co photopeaks, ΔE = 0.16 MeV; E_fwhm = E_fwhm,Cs + ΔE = 0.33 MeV.

One of the main purposes of this detector is the determination of the bremsstrahlung spectrum emitted in a relativistic laser–solid interaction. The proof-of-concept of this system was performed at the DRACO laser at HZDR. DRACO^57,58 is a Ti:Sa laser consisting of two arms, one with PW peak power and the other with 150 TW. The 150 TW arm is capable of delivering an ultrashort pulse of 30 fs with energies up to 3 J on the target with a focal spot of $≈3μ$m, achieving intensities of 10²⁰ W/cm².

The laser irradiated the target at an angle of 45° with respect to the target normal. The calorimeter was placed 60 cm down the laser propagation axis but with an angle of 10° with respect to the laser p-polarization plane. Before the calorimeter, an electron magnetic spectrometer with a magnetic field of 0.5 T and a gap of 1 cm was used as an electron filter. The calorimeter had a direct line of sight to the target through the magnet and a Kapton window.

The analysis of two titanium targets, one of 50 μm and one of 700 μm thickness, is shown in the following. The raw data corresponding to the 700 μm target can be seen in Fig. 8. The round shape is due to a lead collimator at the electron spectrometer with the sharp vertical lines deviating from a full circle arise from the spectrometer magnet shadow. The energy deposited in each layer was integrated over an area within the collimator and then normalized by the solid angle subtended by said area in the detector plane. From the integrated signal at each layer, an energy deposition profile is obtained. The full unfolding is performed in the same way as for the simulated data.

The result of both cases is displayed in Fig. 9(a). The spectrum shows a self-absorption feature in the case of the 700 μm target, which is not present on the 50  µm one. A similar result was observed for similarly sized targets in Ref. 11. In Fig. 9(b), the energy deposit as a function of layer is shown and compared with experimental data.

The number of bremsstrahlung x rays scales linearly with the target thickness in the absence of electron recirculation. Recirculation of electrons depends on the target thickness, laser pulse duration and lifetime of the sheath fields. Recent experiments with 100 μm targets suggest the existence of a strong sheath that could drive recirculation,⁵⁹ particle-in-a-cell simulations for experimental conditions analog to the ones described here indicate that electron recirculation can exist up to hundreds of femtoseconds.⁶⁰ In our experiment, the laser pulse duration τ was 30 fs, with the distance covered by electrons at the speed of light being d = cτ = 9 µm, much smaller than any of the targets under study. Considering a sheath duration of 300 fs, the travel time is 90 μm. We expect the contribution of recirculating electrons for our particular case to not have a large influence in the bremsstrahlung generation. The ratio of x rays with energies larger than 1 MeV in this case is N_700µm/N_50µm = 12, with the ratio of thickness being d_700µm/d_50µm = 14.

Finally, a fit analog to the simulation data can be performed. The results obtained are N_e = (1.2 ± 0.3) × 10⁶ MeV⁻¹ and T_e = 0.56 ± 0.04 MeV for the case of 700 μm Ti and N_e = (2.5 ± 0.4) × 10⁶ MeV⁻¹ and T_e = 0.53 ± 0.03 MeV for the case of 50 μm Ti. It is important to notice that the angular acceptance of the detectors was low, and away from the laser axis. Therefore, the forward integrated cross section approximation is no longer suitable. A detailed analysis of the angular distribution is out of the scope of this paper.

The calorimeter was also fielded at the DRACO laser-wakefield acceleration facility. The beam was 2.5 J on target, 30 fs pulse length focused onto a 20 μm beamspot. The gas jet was a mixture of He–N₂ from a 3 mm de-Laval nozzle. This setup, operating in the Self-Truncated Ionization Scheme, can provide high energy electron beams, with a quasi-monoergetic peak with energies up to 250 MeV with a few tens MeV spread and high bunch charges in the order of hundreds of pC.⁶¹ High energy x rays generated in this facility were studied under two different generation schemes: Compton backscattered photons and bremsstrahlung generated by the electrons traversing a thick converter.

The inverse Compton beam (ICB) setup makes use of the reflection of the laser beam via a thin plasma mirror.^20,62 In our case a 25 µm Kapton foil, located after the gas jet, was used as plasma mirror as shown in Fig. 10. The thickness was chosen to reduce bremsstrahlung background from the interaction of the accelerated electron bunch with the foil. The calorimeter was placed behind a Kapton window ≈1.5 m downstream from the interaction point, outside the vacuum chamber. The electrons are deflected and their energy recorded by a magnetic spectrometer. The expected energies for the ICB lie in the range of 0.6–1.5 MeV depending on the position of the plasma mirror. However, in this case, contributions from the betatron and bremsstrahlung generated in the plasma mirror and secondaries generated in the Kapton window must be taken into account.

A shot with the plasma mirror located directly at z = 4000 μm after the gas nozzle has been analyzed. Assuming a Gaussian propagation and reflection of the laser, we estimate the laser dimensionless electric field amplitude a₀ ≈ 0.5. The electron energy spectrum for that shot is seen in Fig. 11(a). The peak electron energy is E_e,peak = 235 MeV, which corresponds to $Eγ∼4γe2ℏωLa0=1.16$ MeV. Figure 11(b) shows the reconstructed spectrum. It shows a low energy contribution from the betatron radiation generated by the electrons inside the accelerator and a peak from the inverse Compton radiation generated in front of the plasma mirror. The peak of the inverse Compton radiation lies at E_γ,peak = 1.1 MeV, very similar to the theoretical expectations. For completeness, Fig. 11(c) compares the measured energy deposition distribution with the energy distribution corresponding to the unfolded spectrum.

The wakefield-accelerated electrons were also transported through an 800 μm-thick tantalum foil located 30 cm behind the gas exit to generate bright energetic bremsstrahlung. The calorimeter was placed in the same position as in the ICB case. The recorded electron energy was used as the input in Geant4 to calculate the expected bremsstrahlung field at the detector location. The unfolded spectrum is then compared with the simulation results. Figure 12(a) shows the agreement between the simulated output and the reconstructed spectrum. The resulting energy deposition profiles from the simulation and the unfolding are depicted in Fig. 12(b) together with the experimental values. In contrast with all the cases shown before, the energy deposit increases with the depth in the detector. This effect is due to pair-production processes becoming dominant and developing a shower within the detector. Therefore, as opposed to the previous cases, the information on the spectrum is contained on the deeper layers in the detector. Extending such a configuration to higher photon energies is then achieved by simply adding high-Z or high-density layers, enabling the usage of such a detector in GeV-class laser-wakefield accelerators.

Our results can be benchmarked against the ones from Ref. 35. We compare the total number of photons and energy endpoints of the unfolded spectrum. The results are compiled in Table I. Here, we remark that the uncertainty quoted by Hannasch et al. includes the systematic effect of the IP readout scanner, while we report the statistical uncertainty separately.

It was discussed in Sec. IV how the unfolding can reconstruct electron temperatures with an uncertainty of 5%. Here, we obtain an end point energy with an uncertainty of 1%. As mentioned before, at these higher energies, secondary pair production in the stack dominates, generating a shower of secondary particles that deposit their energy in the detector. Thus, higher energy photons disproportionately deposit energy in comparison to lower energy ones as discussed in Ref. 35. This enables for the accurate unfolding of the high energy part of the spectrum.

In this paper, we have shown how a sampling calorimeter with image plates was used to determine nuclear decay radiation, bremsstrahlung, and ICS spectra from a variety of sources. We have also shown how an unfolding algorithm without a tight prior constraint can be used to unfold the spectrum in these cases. We further show how the unfolded spectrum can be significantly different than the prior one used due to background radiation or self-absorption in thick radiators. The performance of the detector in high-power laser–solid interaction is also discussed, and the results of two different targets are compared. The characterization of inverse Compton scattered photons is demonstrated with the results agreeing with simple calculations of expected parameters. Finally, the results of very high energy bremsstrahlung measurements are also shown and compared with simulations.

We have shown how a simple and inexpensive detector can be used to detect a wide range of x-ray spectra in different physical contexts. In these cases, a reasonable agreement between the final unfolded spectrum and simulations has been found.

While the method described here is robust, simple, and easy to field, its largest weakness is the low repetition rate. Each set of image plates has to be mounted, irradiated, scanned, and erased, making this a good detection method in low repetition laser systems. A scintillator based system has been designed and fielded in all the laser situations described in this paper, the results of which shall be discussed in a later publication.

The authors would like to acknowledge the DRACO crew for their support and their expert laser operation. We would also like to thank S. Eber, M. Walter, and D. Loehnert from Strahlenschutz, Analytik und Entsorgung Rossendorf e. V. for their support and operation of the calibration sources. We also thank T. Toncian for fruitful discussions on the laser–solid interaction physics. This work was supported by Bundesministerium für Bildung und Forschung through the Helmholtz Matter and Technologies program. A.H., R.Z., and M.C.D. acknowledge support from U.S. Department of Energy Grant No. DE-SC0011617, and M.C.D. acknowledges support from the Alexander von Humboldt Foundation.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.