Robust unfolding of MeV x-ray spectra from filter stack spectrometer data Open Access

Measurement of MeV x-ray spectra is important in several areas, including radiography,^1–5 medical physics,^6–9 particle and nuclear physics,^10,11 astronomy,¹² and inertial confinement fusion.^13,14 Many diagnostics have been fielded for such measurements, including gas Cherenkov detectors,^13,15,16 gamma reaction history diagnostics,^14,17 photo-nuclear activation measurements,^5,18 Compton spectrometers,¹⁹ scintillator attenuation spectrometers,²⁰ step filter spectrometers,²¹ and filter stack spectrometers.^22–26 These diagnostics all have different strengths and weaknesses in their precision, accuracy, complexity, time resolution, and their robustness to electromagnetic pulses (EMPs) and background particles.

The filter stack spectrometer (FSS) offers the ability to measure MeV x-ray spectra while being a simple, portable, and passive diagnostic insensitive to EMP. The FSS consists of alternating layers of filters and detectors, often image plates (IPs), where the signal measured on each detector depends on the x-ray spectrum and the filter stack design. The number of detectors in a FSS may range from a few to dozens depending on the application. The primary challenge in using an FSS is in solving the inverse problem to extract, or “unfold”, a spectrum from the detector signals. A simplified sketch of an FSS is presented in Fig. 1, and the operation of an FSS can be modeled mathematically as ∑s_iR_ij = p_j. Here, s is a vector containing the counts in the x-ray spectrum, R is the response matrix for the FSS, p is a vector containing the measured photostimulated luminescence (PSL) values, i is the indexing energy bins, and j is the indexing image plate number. The response matrix describes the amount of signal measured in each detector for an x-ray of a given energy and can be modeled using transport codes, such as Monte Carlo N-Particle Transport²⁷ (MCNP^®) or Geant4.²⁸ 

Finding a good solution to the inverse problem for FSS data is a major challenge for several reasons. First, the response matrix is often ill-conditioned^7,29 such that there is no unique solution and many local minima exist within spectral phase space that minimizes the difference between the measured PSL values and the forward-modeled PSL values of a candidate spectrum. Second, a physical constraint is needed to select between different potential solutions. This physical constraint, however, can bias the unfold if not chosen carefully. Third, MeV x-ray spectra are particularly difficult to unfold as photons have a minimum in attenuation in the MeV range (a few MeV for high-Z elements and tens to hundreds of MeV for low-Z elements³⁰) such that there is often not a one-to-one relation between photon energy and attenuation. This symmetry can be broken by using high-Z filter materials to generate secondary electrons, which also increases the sensitivity of the detector to MeV x-rays.²⁵ In addition to these challenges, solving the inverse problem is a high-risk step of data analysis, where small limitations in an unfolding algorithm or FSS design can lead to the extraction of an incorrect spectrum without a straightforward method to validate or reject the unfolded spectrum. This necessitates the use of an algorithm that is well characterized and capable of robustly unfolding a variety of spectra.

Several approaches have been taken to solve the inverse problem for FSS, analogous problems for similar diagnostics²⁰ and other particles.^31,32 These methods include expectation maximization,²⁹ Bayesian optimization,^25,33 singular value decomposition,²⁰ maximum entropy principle,³¹ machine learning,^34,35 and perturbative minimization.^7,36 Methods relying upon inverting the response matrix, such as expectation maximization, often reliably unfold x-ray spectra in the keV range, where response matrices may be well-conditioned, but often fail for MeV x-ray spectra where response matrices become ill-conditioned. Iterative searches, such as perturbative minimization (PM), take a different approach by taking small steps toward local minima while searching for a global minimum. These methods, however, can get trapped in local minima near the initial guess spectrum and, consequently, do not perform a sufficiently broad search through spectral phase space to ensure a robust unfold. For both methods, similar techniques have been applied to overcome these challenges, including using physics models to make assumptions of spectral shape in initial guesses or through fitting, or requiring arbitrary termination of the inversion procedure to avoid convergence to nonphysical spectra. However, assumptions of spectral shape or artificial termination of inversion procedures increases the risk of biasing the obtained spectrum and, in turn, provides less confidence in the unfolded spectral features.

In this work, we present an inversion method capable of extracting the primary features of MeV x-ray spectra from synthetically generated FSS data with no a priori assumptions of spectral shape or requiring arbitrary termination of the algorithm. The only assumption made by our inversion method is that the spectrum should reside near a smooth curve. Our inversion method builds upon the perturbative minimization (PM) algorithm, which has previously been shown to be an effective method for unfolding x-ray transmission data in a limited regime, in which the x-ray mass attenuation coefficient of the filter material increases monotonically with x-ray energy.^7,36 Typical FSS designs for measuring MeV x-rays, however, are well outside this regime, as high-Z materials have a minimum in attenuation as a function of photon energy at a few MeV and secondary electron production in filter materials can become a dominant term in the FSS response.

Our inversion method improves upon the PM algorithm by including regular smoothing of the candidate spectrum and by adding stochasticity to the search in two ways: (1) we randomize the order in which energy bins are perturbed in the PM algorithm, and (2) we add random noise to the candidate spectrum prior to each smoothing. The regular smoothing serves as a physical constraint to enforce a solution near a smooth curve in a similar manner to the approach of D’Agostini,³³ while the added stochasticity produces a more robust inversion as it allows the inversion method to sample a larger portion of spectral phase space. With these improvements to the PM algorithm, we demonstrate using synthetically generated data that our inversion method can successfully recover the large-scale features of x-ray spectra with endpoint energies in 10 s of MeV. For the primary features of x-ray spectra, the inversion method recovers the number of x-rays to within 10% for energy bins with widths of several MeV. Notably, our inversion method requires neither a physics model for an initial guess or for fitting, nor does it require a user-selected termination to avoid convergence to nonphysical solutions.

The rest of this paper is organized as follows: In Sec. II, we provide a detailed description of our inversion method based on the random PM algorithm. In Sec. III, we test our inversion method with synthetic data and vary free parameters to understand their impacts on the unfolded spectra. In Sec. IV, we unfold experimental FSS data obtained at the Texas Petawatt (TPW) Laser Facility at the University of Texas at Austin and the OMEGA Extended Performance (EP) laser facility at the Laboratory for Laser Energetics. In Sec. V, we discuss the need for high quality FSS data with small random errors in order to achieve inversions that provide quantitative measures of MeV x-ray spectra. In Sec. VI, we summarize the major findings of this paper and provide an outlook for future directions.

Our inversion method is based upon perturbative minimization^7,36 (PM), which is among the simplest implementations of unfolding. In this algorithm, perturbations to an initial spectrum are made by increasing and decreasing the counts in the first energy bin by a factor of 1/2^k, beginning with k = 1. The perturbed spectra are then forward-modeled to generate the expected measured signals, and the spectrum with the lowest signal error is retained. This process is repeated for each spectral bin for increasing energy, before repeating the whole process for the next k until a user-defined k_max is reached. The strengths of the PM algorithm are that it is extremely simple, it provides a more systematic search of spectral phase space as compared to other algorithms that converge more rapidly, and candidate spectra remain positive for an initial guess that is positive. Nevertheless, in its original form, the PM algorithm has several limitations similar to other iterative algorithms. These limitations include being deterministic and thereby dependent on the initial guess, a tendency to get quickly trapped in nearby local minima, and a tendency to overfit data and predict nonphysical spectra that are extremely jagged.

We overcome these limitations with two key additions to the PM algorithm. First, we smooth the spectra every N iteration of the PM algorithm using a Savitzky–Golay (SG) filter,³⁷ where N is a user-selected parameter. Second, we introduce stochasticity to the search by (1) randomizing the order that the spectral points are ordered and (2) randomly perturbing the candidate spectrum prior to each smoothing by multiplying the counts in each spectral bin to a quantity that is a function of a random number drawn from a normal distribution.

With these two additions, our inversion method does not attempt to rapidly converge toward a minimum like many existing algorithms. Instead, the search in spectral phase space is intentionally prolonged by the added stochastic perturbations, while the regular smoothing acts as a constraint to help prevent the candidate spectrum from drifting too far from a physical solution space. The stochastic perturbations allow for a greater sampling of spectral phase space as it not only helps candidate spectra escape local minima but also makes the search non-deterministic so that the search will explore a larger variety of paths through spectral phase space. As a further advantage, a non-deterministic search allows for the inversion method to generate many unfolded spectra from a single set of FSS data, which can be used to infer the variance in the unfolded spectrum arising from the inversion method itself.

Next, we provide a detailed description of our inversion method based upon the random PM algorithm, which is summarized in the flow diagram in Fig. 2.

To begin, our inverse method requires the following inputs: the response matrix, the PSL values and associated error bars, an initial guess spectrum, and the selection of several parameters. Typically, the response matrices are generated using particle transport codes, such as MCNP²⁷ or Geant4.²⁸ If secondary electron production is negligible, as is the case for low-Z FSS designs or for keV x-ray spectra, then x-ray mass attenuation coefficient tables³⁰ in combination with image plate calibration data^38–42 may be sufficient for calculating the response. The PSL values and associated error bars are both used for the unfolding, as the error bars are used to calculate the quality of fit a candidate spectrum provides. The initial guess spectrum used in this work is a flat line (i.e., all unity); however, a physics-informed initial guess can also be used. The most important parameters that need to be selected, which will be discussed in detail in Sec. III B, include the energy range of the unfolded spectrum, number of spectral points, smoothing frequency M, search breadth N, and perturbation size σ.

After repeating the random PM algorithm a user-defined M times, the resulting spectrum is compared to the best scoring candidate spectrum thus far, using the same forward-model-based metric. The spectrum yielding the lowest error is kept as the candidate spectrum, while the other spectrum is discarded. The candidate spectrum is then randomly perturbed by multiplying each spectral point by a multiplicative factor, $(1+x)sgn(x)$⁠, where x is a randomly generated number from a normal distribution centered at 0 and with a user-selected width σ. We refer to this width as the “perturbation size.” The probability density functions of the multiplicative factors for three different perturbation sizes are shown in Fig. 3. Note that σ = 1 can lead to substantial perturbations of a candidate spectrum, which is needed to robustly explore spectral phase space. After randomizing of the spectrum, the spectrum is smoothed using an SG filter. The SG smoothing is performed with the spectral counts in log-space and with a Hanning window to avoid edge effects.

The above-mentioned process repeats itself until the candidate spectrum is not updated (i.e., improved) after a user-defined N loops of perturbing, smoothing, and iterating, after which the search terminates, and the final candidate spectrum is output as the unfolded spectrum. In practice, this inversion routine can be performed multiple times, each generating a different spectrum since the search is stochastic. Furthermore, the process can also be repeated multiple times for different parameters to determine sensitivities to user-defined parameters. We include two animated examples to illustrate the unfolding process in the supplementary material. For typical parameter choices, our inversion method takes a few minutes on a laptop (Intel^® Core™ i9-11950H with 128 GB RAM) to extract a spectrum.

To assess the quality of the inversion method, we unfold synthetically generated data with known ground truths. Synthetic PSL values were generated by using a response matrix as a forward model to convert a given x-ray spectrum into measured PSL values. As an input into the inversion method, errors in the PSL values were assumed to be proportional to the square root of the PSL values. For this section, we consider a FSS that was fielded at TPW. The FSS consists of seven layers of 15-mm thick nylon-6, followed by seven layers of 15-mm thick aluminum, and finally by six layers of lead, which were two layers each of 1.5, 2.5, and 3.5 mm. An SR-IP was located after each layer of the filter material. A response matrix for the TPW FSS was generated using MCNP^®, which is provided in Fig. 4. The FSS was fielded outside the measurement chamber, so a 1.6-cm thick piece of Al was included in front of the FSS in the MCNP^® calculations. Further details on the MCNP^® calculations will be provided elsewhere. In all, we unfold three types of spectra: (1) a Findlay function⁴³ with an endpoint energy of 35 MeV, (2) Gaussian functions, and (3) sums of Findlay and Gaussian functions. Each spectrum was unfolded a total of 100 times, where the initial guess spectrum was a flat line with each spectral bin count equal to unity. For these unfolds, the parameters used were as follows: searches were performed for 15, 20, and 30 spectral points; smoothing frequency M = 10; search breadth N = 25; perturbation size σ = 1; and with an LSE metric. A detailed discussion on these parameters can be found in Sec. III B.

The first synthetic spectrum we unfold is a Findlay function, which is an approximation of a bremsstrahlung spectrum often observed in laser or electron-beam interactions with solid targets.^25,43 To provide some insights into how random measurement errors may impact the unfold, we multiplied the forward-modeled PSL values by 0, 0.3%, and 1% error prior to each unfold in Figs. 5(a)–5(c), respectively. Note that these errors were sampled from a Gaussian distribution and do not need to match the assumed error model used as an input for the inversion method. In each plot, the black solid line is the ground truth spectrum, the red dashed line is the initial guess, and each of the 100 unfolds is plotted as a dotted line.

The unfolded spectra agree qualitatively with the ground truth spectrum for all three levels of added random error. The primary region where there is a substantial disagreement is at the cutoff of the synthetic spectra, around 35 MeV. Such a feature is difficult for our inversion method to recover for two reasons: First, the x-ray count where the cutoff occurs is already four orders of magnitude lower than the peak value so that the PSL contributions for that portion of the spectrum become small as compared to the differences between the synthetic and unfolded PSLs, making it difficult to resolve. Second, the cutoff is a relatively sharp feature that becomes difficult to recover when applying regular smoothing with the SG filter.

When comparing the three cases, the variance in the unfolded spectra increases substantially as the random error is increased. However, this increase in variance does not necessarily imply that the inversion method is extracting an incorrect spectrum, as the “correct” spectrum that corresponds to the PSL values with the random error added is no longer the ground truth spectrum. Such an approach of adding random error to the ground truth PSL values is often taken to test the robustness of a complete diagnostic system in combination with the inversion method;^20,25 however, it is likely not appropriate for testing just the inversion method alone, as the FSS design likely plays a more prominent role in mitigating and robustness against random error. Nevertheless, going forward, we add 0.3% error to synthetic PSL data so that the forward and inverse models are not identical. Furthermore, 0.3% error is a reasonable estimate of the statistical uncertainty for PSL values across different image plates for a single scan.²⁵ 

Since our inversion method does not require a physics-informed initial guess, it is important to demonstrate that the method will recover a variety of spectral shapes. To this end, we unfold Gaussian functions with different means and widths in Figs. 5(e)–5(g) and the sum of Findlay and Gaussian functions in Figs. 5(i)–5(k). For all cases, the inversion method is able to recover the key features of each spectrum; however, as expected, there are substantially larger discrepancies between unfolded spectra and synthetic spectra where the x-ray counts are small. The inversion method is also unable to recover the sharpest features of the spectra, typically broadening sharp features. This limitation is expected and arises from a combination of the regular smoothing in the inversion method, the finite number of spectral bins being considered in the unfold, and the spectral resolution of the filter stack design.

These quantitative comparisons indicate that the inversion method is capable of recovering the primary features of a given spectrum to within 10% if the underlying PSL data have random errors of 0.3%. This discrepancy will increase when considering smaller energy bins relative to the extent of a given spectrum than those used here and, conversely, decrease when considering larger energy bins. In regions of a spectrum where the x-ray count becomes diminishingly small as compared to the random and fitting errors, the inversion method is unable to recover the spectral shape at all, leading to large discrepancies.

Next, we varied different user-selected parameters to determine how these choices can impact the unfolded spectrum. These include the energy range for the unfold, the number of spectral points, the size of random perturbations applied to the spectrum, the frequency of the smoothing (M), the number of smoothing loops performed before giving up (N), and the choice between least squares error and least absolute errors when comparing modeled PSL to measured PSL values. Note that many of these parameters do not impact the unfold independently of one another, so all parameters but the parameter being varied are fixed at those used in Sec. III A. We do not include error to the ground truth PSLs to avoid complicating the interpretation of the parameter scans. We present the plots of the unfolded spectra in the supplementary material and provide a summary of the results here.

When choosing a maximum energy E_max for the unfolded spectrum, it is typically not an issue for E_max to be too large. Meanwhile, too small of an E_max will often yield a poor unfold. If E_max is slightly too low, then the tail of the spectrum will be slightly high as it redistributes the missing higher energy photons into the tail of the distribution. If E_max is far too low, then this redistribution of higher energy photons can greatly disrupt the unfold. The minimum energy E_min should be selected to be above the cutoff x-ray energy of the first image plate, based on the response matrix. Otherwise, the inverted spectrum can include an arbitrarily large number of low energy photons in a nonphysical manner.

If too few spectral points are included, the unfolded spectrum typically recovers the general shape of the ground truth spectrum but misses much of the details. Meanwhile, too many spectral points can lead to artificially jagged spectrum, as the spectral point spacing becomes smaller than the energy resolution that can be recovered based on the inversion method and FSS design limitations. Furthermore, using too many spectral points can counterintuitively make smaller spectral features more difficult to discover, as the probability of randomly perturbing spectral points into the shape of, for example, a bump on a spectrum becomes less likely as the number of spectral points is increased. To address these issues, we will typically perform an inversion multiple times for increasing numbers of energy bins, for which the resulting spectrum from one inversion will be used as the initial guess of the subsequent inversion. In Sec. III A, we performed inversions for 15, 20, and 30 spectral points.

Smoothing the candidate spectrum too frequently (i.e., only performing the random PM algorithm a few times between smoothing) can be problematic in two ways: First, it will not allow the algorithm to search away from a smooth curve such that the final inverted spectrum may be overly smooth. Second, if combined with large perturbations, there may be an insufficient number of iterations of the random PM algorithm to recover from the perturbations and then to find an improved spectrum. Meanwhile, not smoothing the candidate spectrum frequently enough may allow the user to unfold more complex spectra but runs the risk of finding nonphysical solutions too far away from a smooth curve. In Sec. III A, we used M = 10.

The search breadth is controlled by the number of times N that the inversion method will perform the set of perturbation, smoothing, and random PM algorithm iterations without finding an improved spectrum before discontinuing the search. In principle, a larger search breadth should provide a better final solution, but it also increases the compute time required for the inversion method. In Sec. III A, we used N = 25.

The size of perturbations to the candidate spectrum prior to the smoothing procedure is dictated by the width σ of the normal distribution from which the perturbations are randomly selected. Increasing the perturbation size not only tends to increase the inversion method’s ability to find complexity within the spectrum, but also increases the amount of noise within the unfolded spectrum. A small perturbation size would be expected to perform better for simple spectra, while a larger perturbation size would be expected to perform better for complex spectra. It may be possible to reduce the errors associated with large perturbation sizes by systematically reducing the perturbation size during the search to mimic simulated annealing,⁴⁴ but such a strategy was not explored as part of this work. In Sec. III A, we used σ = 1.

Both LSE and LAE, Eqs. (1) and (2), respectively, are metrics that can be used to compare the forward-modeled PSL values for a candidate spectrum to the measured PSL values. As expected, the LSE metric better recovers details in the x-ray spectra when the random errors in the experiments are small and the response matrix is accurately capturing the FSS response. Meanwhile, LAE provides a more robust unfold when faced with larger random errors in the experimental data or if discrepancies exist between the modeled response matrix and the true FSS response. In Sec. III A, we used LSE as our metric, as our random errors were small (0.3%) and the response matrix was used for both generating the synthetic PSL data and performing the inversion.

As a demonstration of unfolding experimental FSS data, we unfold laser-driven x-ray FSS measurements taken at the TPW and OMEGA EP laser facilities. We do not present an extensive set of unfolded experimental data here, as we do not have ground truth spectra for these data to use as a comparison. Consequently, although the unfolded spectra appear to be reasonable, there is no method to directly check the validity of the unfolded spectra. Rather than serving a benchmark for the inversion methodology, these spectral inversions are intended to provide confidence that the inversion method does not break down in an obvious manner when faced with experimental data. For the experimental unfolds, the initial guess spectra were again flat spectra and errors in the PSL values were assumed to be proportional to the square root of the PSL values. Response matrices were generated using MCNP^® for the filter stacks used in the experiments. For the experimental unfolds, we used the same parameters as for the synthetic data in Sec. III A (searches were performed for 15, 20, and 30 spectral points; smoothing frequency M = 10; search breadth N = 25; perturbation size σ = 1), except for using the LAE metric of Eq. (2) for the TPW data, due to larger uncertainties in those experimental PSL values as compared to the OMEGA EP data.

The experimental TPW data were obtained with the FSS described earlier in Sec. III A. The data are from a 120 J, 140 fs laser pulse incident on a 1-mm thick Ta target. This interaction is expected to generate x-rays with a bremsstrahlung-like spectrum with an endpoint energy in the 10 s of MeV, as predicted by MCNP^® calculations for experimentally measured electron spectra from thin 5-μm thick Ta targets injected into 1-mm thick Ta converters. Such spectral shapes are also consistent with simulations in the literature^45,46 and with comparable MCNP^® calculations.⁴ We find that the inversion method yields such a spectrum in Fig. 6(a), again starting with an initial guess of a flat spectrum. The spectrum was unfolded 10 times, each represented by the open circles and dashed lines. The variation in the unfolded spectra represents the uncertainty associated with the inversion method itself. The average of these ten unfolded spectra are given by the black solid line. To help guide the eye, we computed the relative accumulative PSL as a function of the spectral energy and plotted it on the right axis as a red dotted line. The relative accumulative PSL was calculated as the ratio of the sum of the accumulative PSL at a given energy to the total PSL measured by the filter stack. As this dotted line approaches unity, the portion of the spectrum above that energy contributes almost nothing to the PSL values and, accordingly, cannot be unfolded with confidence. For the TPW data, this occurs at ∼20 MeV.

While the unfolded PSL values exhibit the same qualitative trends as the experimental PSL values in Fig. 6, the discrepancies between the two sets of PSL values are large when considering the need for PSL errors to be less than a few percent (at least for the TPW FSS design) to achieve an unfolded spectrum that can be analyzed quantitatively. The variation in PSL values between different unfolded spectra is also small relative to the discrepancy between experimental and averaged unfolded PSL values, as shown in the supplementary material. The consistency in unfolded PSL values, coupled with the tendency for inversion methods to over-fit data, suggests that the source of discrepancy is likely limitations in the experimental diagnostic, procedures, or modeling of the experimental diagnostic. Potential sources include physical differences between image plates, spatial asymmetry in image plate scanners, or physics that the MCNP^® calculations may not be capturing.

The OMEGA EP data were obtained using the Standard MeV filter stack in the bremsstrahlung spectrometer (BMXS) diagnostic.²² The filter stack design, which is provided in the supplementary material, consists of many filters of different thicknesses and materials and was designed to measure x-ray spectra up to ∼1 MeV. The data are from a 479 J, 700 fs laser pulse incident on a 500-nm thick CD film target. The BMXS diagnostic was fielded at 90° relative to the laser axis. Given the combination of a low-Z target and the angular position⁵ of the BMXS, a colder x-ray spectrum was expected as compared to the TPW shot.

The unfolded OMEGA EP spectrum in Fig. 7, as expected, generated a much colder spectrum with an upper bound of about 1 MeV. This is consistent with the measured PSL values, as no signal was measured in image plates 11–15, indicating that most of the x-rays were below 1 MeV. As compared with the TPW data, the experimental and unfolded PSL values have much better agreement. The better agreement arises, in part, due to having fewer PSL values that the inversion method needs to match. However, the larger dynamic range of the PSL measurements also provides more robustness against random errors and modeling the relevant transport physics accurately is simpler for x rays below 1 MeV as compared to x rays up to 25 MeV.

Achieving quantitative inversions of MeV x-ray spectra from FSS data requires the minimization of relative measurement errors between image plates of a given measurement. As discussed by Laso Garcia et al.,²⁵ although systematic errors from scan-to-scan may be at least a few percent,⁴⁷ this type of systematic error is not as significant a concern as it only leads to a change in the amplitude of the spectrum, which can be more easily cross-calibrated with other diagnostics. On the other hand, errors that change the relative signals between image plates in the FSS data can dramatically change the “correct” underlying x-ray spectral shape such that even a perfect inversion method will still yield a spectrum that differs greatly from the true spectrum. These errors may arise from random measurement errors, such as counting statistics or background noise, or from systematic measurement errors, such as variations in phosphor layer thicknesses in image plates. In either case, FSS diagnostics must be designed to be robust against such errors, through methods such as improved shielding and collimation, including a larger number of image plates, obtaining measurements over a larger dynamic range, calibrating individual image plates, or some combination thereof.

Obtaining MeV x-ray spectra with sufficient accuracy to allow for quantitative analysis using filter stack spectrometers has been complicated by the challenges associated with the inversion process to extract spectra from FSS measurements. To overcome these challenges, many approaches include the use of physics-informed initial guesses or fitting of the data, which often leaves strong imprints on the extracted spectrum. In this work, we presented an inversion method capable of providing a quantitative determination of MeV x-ray spectra from FSS data without needing a physics model that may bias the obtained spectrum. Our inversion method enforces a search near a smooth curve through regular smoothing, while adding stochasticity to the search to help prevent the inversion from being trapped at local minima and to provide a more complete sampling of spectral phase space. We applied our inversion method to unfold several sets of synthetic FSS data with known ground truths, as well as experimental data for a $>1$ MeV x-ray spectrum and for a primarily $<1$ MeV x-ray spectrum.

Future work includes benchmarking the inversion method with various FSS designs at x-ray sources that have either been carefully characterized with other diagnostics, like the Microtron at Los Alamos National Laboratory,¹⁹ or monoenergetic sources, such as the Thomson scattering source at the Berkeley Lab Laser Accelerator Center⁴⁸ or the High Intensity Gamma-ray Source at the Duke Free Electron Laser Laboratory.⁴⁹ While such measurements at known experimental x-ray sources are crucial for quantifying the quality of unfolded spectra, discrepancies in the FSS-obtained spectra will depend on the limitations of both the inversion method and the FSS design. Disentangling these two contributing factors will be a major challenge in the optimization of both the inversion method and the design of FSS.

Other future directions include extending the inversion method to other particle species, as well as to multiple particle species or multiple datasets. Applying the inversion method to other particle species can be done, in principle, by simply changing the response matrix. For multiple particle species or multiple datasets, the simplest approach is to concatenate the spectra, response matrices, and measurement data and then to apply the same inversion method.

The supplementary material contains spectral plots from our parameter sensitivity study, a plot of containing the variation in the unfolded PSL values for the TPW experimental data, the OMEGA EP Standard MeV filter stack details, and two short gifs of the inversion method for a Findlay function and the sum of Findlay and Gaussian functions.

We thank the facility staff and scientists at the OMEGA EP laser facility and the Texas Petawatt Laser Facility for their support in obtaining the experimental data presented here. This work was supported by LaserNetUS via the DOE, Office of Science, Fusion Energy Sciences, under Contract No. DE-SC0021125. This work was supported, in part, by the U.S. Department of Energy through the Los Alamos National Laboratory. This work was supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under Project No. 20220018DR. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001).

The authors have no conflicts to disclose.

C.-S. Wong: Conceptualization (equal); Investigation (equal); Methodology (lead); Software (lead); Writing – original draft (lead); Writing – review & editing (lead). J. Strehlow: Conceptualization (equal); Investigation (equal); Methodology (supporting); Software (equal); Writing – original draft (supporting); Writing – review & editing (equal). D. P. Broughton: Conceptualization (supporting); Investigation (equal); Methodology (equal); Software (equal); Writing – review & editing (equal). S. V. Luedtke: Conceptualization (supporting); Investigation (supporting); Methodology (equal); Software (equal); Writing – review & editing (equal). C.-K. Huang: Conceptualization (supporting); Investigation (supporting); Methodology (supporting); Writing – review & editing (supporting). A. Bogale: Investigation (supporting); Methodology (supporting). R. Fitzgarrald: Investigation (supporting). R. Nedbailo: Investigation (supporting). J. L. Schmidt: Investigation (supporting). T. R. Schmidt: Investigation (supporting); Methodology (supporting). J. Twardowski: Investigation (supporting). A. Van Pelt: Investigation (supporting). M. Alvarado Alvarez: Investigation (supporting); Methodology (supporting). A. Junghans: Investigation (supporting). L. T. Mix: Investigation (supporting). R. E. Reinovsky: Funding acquisition (equal); Investigation (supporting). D. R. Rusby: Investigation (supporting); Writing – review & editing (equal). Z. Wang: Investigation (supporting); Writing – review & editing (supporting). B. Wolfe: Investigation (supporting). B. J. Albright: Funding acquisition (equal); Investigation (supporting); Supervision (supporting). S. H. Batha: Funding acquisition (equal); Investigation (supporting); Supervision (supporting); Writing – review & editing (equal). S. Palaniyappan: Conceptualization (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

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