Spectrometer optimization

The most significant drawback of the FSS is the ill-posed linear system, which results in substantial errors in spectrum measurements. Because the scintillator can function as both filter and sensor, parameters such as the layer thickness, number of layers, filter density, and scintillator density need to be finely tuned to alleviate the ill-conditioned nature of the RM. Rusby et al. [36] optimized their online FSS based on the difference in the scintillator output within the energy range of interest. The optimal scintillator should exhibit the largest difference in scintillator output. However, this optimization criterion was spectrum-dependent and the scintillator thicknesses were kept constant, limiting the optimization efficiency.

To propose a universal optimization method, we first establish a formal model for encoding an X-ray spectrum S(E) into the experimental channel Di for an N-channel FSS. The mathematical definitions and notation employed in this study are consistent with those used by Fehl et al. [45, 46]. Di can be written as follows: $D_i={\displaystyle {\int }_0^{E_{\text{MAX}}}}{R}_i\left(E\right)S\left(E\right)\text{d}E+{ϵ}_i=d_i+{ϵ}_i\left(i=\mathrm{1,...}N\right)\mathrm{,}$ (1) where Di represents the experimental channel data, $R_i\left(E\right)i{=1}^N$ denotes the response function, di represents the noise-free channel data, and ϵi represents uncertainties and noise. Equation (1) can be expressed in vector-matrix notation as follows: $D=ℝS+\text{Δ}D=d+\text{Δ}D\mathrm{.}$ (2) The errors that affect the unfolding accuracy can be divided into two parts. The first part is the perturbation ΔD superimposed on D during the data-gathering process, which includes factors such as statistical uncertainty, signal noise, digitization errors, and uncorrected signal baselines, among others. The second part is the bias between the experimentally measured or simulated RM and the actual RM, expressed as $ℝ\to ℝ+\text{Δ}ℝ$. The presence of $\text{Δ}ℝ$ can be attributed to drifts and uncertainties in the fitting parameters, such as the calibration of the light yield and light collection efficiency discussed in Sect. 2.3. Because $(\text{Δ}ℝ)S$ is analogous to ΔD [46], only ΔD is discussed here.

Solving Eq. (2) poses challenges because it is an ill-posed problem. Various unfolding algorithms can be employed to address this problem, including regularization methods, least-squares spectrum adjustment, parameter estimation, iterative unfolding methods, and the maximum entropy principle [47]. However, these unfolding algorithms often struggle to provide mathematically rigorous or realistic estimates of the error propagation relationship between the spectral unfolding error ΔS and the data perturbation ΔD. Nevertheless, it is worth noting that the error propagation relationship between ΔS and ΔD is primarily determined via the response function $ℝ$. For instance, an upper-bound measure for ΔS can be estimated from ΔD and the condition number of the RM is denoted by $Cond(ℝ)$, as described in [45]. $\left|\right|\text{Δ}S\left|\right|/\left|\right|S\left|\right|\le Cond(ℝ)\left|\right|\text{Δ}D\left|\right|/\left|\right|D\left|\right|\mathrm{,}$ (3) where $\left|\right|D\left|\right|={\left({\displaystyle {\sum }_{i=1}^N{D}_i^2}\right)}^{\frac{1}{2}}$ represents the norm of D. RM $ℝ$ can be either a square or non-square matrix [59], and the condition number $Cond(ℝ)$ can be calculated via singular value decomposition. By reducing $Cond(ℝ)$, for example, by alleviating the ill-conditioned nature of the RM, the upper-bound measure of ΔS can be lowered, resulting in improved precision in terms of the spectrum measurements. Furthermore, $Cond(ℝ)$ is determined solely by the FSS configuration. Hence, $Cond(ℝ)$ can serve as a measure of merit for FSS optimization, independent of the spectrum.

Another challenge in spectrometer optimization is the multi-parameter optimization problem. FSS typically comprises numerous filter and sensor layers, and the thickness and density (determined by the material) of each layer can affect $Cond(ℝ)$. Consequently, multiple parameters must be optimized. Owing to the practical limitations on the layer thickness for both the filter and sensor, as well as the absence of derivatives and linearity in this problem, the genetic algorithm is a suitable method for optimization.

Several factors restrict the range of the optimization parameters. In an FSS, filters and sensors closer to the X-ray source are typically thinner, whereas those farther away are thicker. This design ensures that most low-energy X-rays deposit their energy in the layers near the sources and high-energy X-rays do so in the layers farther away. Consequently, RM $ℝ$ has a smaller condition number ($Cond(ℝ)$). To address the low light-collection efficiency of thin scintillators, which can result in significant statistical errors, sensors with a 300 μm depletion layer thickness, such as PIN diodes, were used in the layers near the X-ray source. For custom production convenience, the scintillators are made of Gd₃Al₂Ga₃O₁₂ (GAGG) [48], while the filters are made of aluminum or copper. After determining the number of layers, $Cond(ℝ)$ should be minimized because it is scale-dependent. Considering that the number of layers is a tradeoff between the applicable energy range and the complexity of the signal readout system, 15 layers were selected, including six PIN diodes and nine scintillators. The overall configuration of the proposed online FSS is listed in Table 1. The layer number represents the distance from the X-ray source with the filter closer to the source in each layer. As an aluminum film of 50 μm is always used as the first layer for electromagnetic shielding, the parameters to be optimized are the filter thickness in layers 2-15 and the scintillator thickness in layers 7-15.

A key aspect for implementing the genetic algorithm is the fitness assignment method. As the population consists of 2000 individuals and there are 300 iterations, approximately 6×10⁵ $Cond(ℝ)$ calculations, i.e., RM calculations, are required. Accurately RM calculations using the Monte Carlo method [28, 37, 36] consume significant computational power. To reduce the computational power requirement, a simplified RM calculation model that utilizes the X-ray mass energy absorption coefficients and mass attenuation coefficient is proposed. First, the expected energy deposition of a single X-ray with energy E in the i-th sensor, denoted by E_{dep, i}(E), can be calculated as $E_{\text{dep}\mathrm{,}i}\left(E\right)=\left(1-e^{-{\mu }_{\text{en}\mathrm{,}i}\left(E\right)s_i}\right)\cdot {\displaystyle \prod _{j=1}^{i-1}}{e}^{-{\mu }_j\left(E\right)s_j}\cdot {\displaystyle \prod _{j=1}^i}{e}^{-{\mu }_j\left(E\right)f_j}E\mathrm{,}$ (4) where μ_en,i(E) represents the X-ray mass energy absorption coefficient of the i-th sensor, μj(E) represents the X-ray mass attenuation coefficient of the j-th sensor or filter, sj and fj denote the thickness of the j-th sensor and filter, respectively. Thus, the RM equation can be written as $R_i\left(E\right)=E_{\text{dep}\mathrm{,}i}\left(E\right)\cdot ES{C}_i\mathrm{.}$ (5) where ESCi are the energy-signal coefficients of the i-th channel. For the PIN diode, $ES{C}_i=1/3.62\times q\times {\rho }_i\mathrm{,}$ (6) where 3.62 eV is the pair creation energy of silicon [49], q is the elementary charge, and ρ is the gain of the electronics in ohm. For GAGG, $ES{C}_i={\left(LY\times LCE\times QE\right)}_i\times q\times {\rho }_i\mathrm{,}$ (7) where LY is the light yield of GAGG, LCE is the light collection efficiency, and QE is the quantum efficiency of the photon detector. If we disregard the nonlinearity of the scintillator light yield and electronics, ESCi becomes independent of th X-ray energy E. Therefore, the practical response equation to be solved can be derived using Eqs. (1) and (5) as follows: $\text{D}/\text{ESC}={\mathbb{\text{E}}}_{\text{dep}}{\text{S}}_{\text{unfold}}.$ (8) where Sunfold is the spectral unfolding and ${\mathbb{E}}_{\text{dep}}$ is the matrix notation of E_dep,i(E). An error ΔS exists between Sunfold and the real X-ray spectrum S owing to perturbation ΔD in D. In Eq. (8), ${\mathbb{E}}_{\text{dep}}$ is equivalent to RM $ℝ$, and can be determined from Eq. (4). Therefore, the optimization process for reducing $Cond\left({\mathbb{E}}_{\text{dep}}\right)$ is illustrated in Fig. 1.

Fig. 1Full-screen View

Optimization process of the proposed online FSS using the genetic algorithm

The genetic algorithm was implemented using the Sheffield Genetic Algorithm toolbox [51]. The population consisted of 2000 individuals, with each individual containing 14 filter thicknesses and nine scintillator thicknesses. These thicknesses are represented as floating-point numbers and encoded as bit strings to create chromosomes. Each individual contained 24 binary chromosomes, representing the 14 filter thicknesses and nine scintillator thicknesses. To consider practical layer thickness limitations, lower and upper limits were set for the filters and scintillators, 0.01 and 50 mm and 2 and 50 mm, respectively. Each binary chromosome uses 20 bits to ensure sufficient precision. For the iterations, all filter thicknesses were set to zero, and any thickness below 0.1 mm was also set to zero. The generation gap, mutation probability, and crossover probability were set to 0.9, 0.01, and 0.7, respectively. The termination criterion was reached when the iteration time was 300.

The minimum and average $Cond\left({\mathbb{E}}_{\text{dep}}\right)$ values during the iterations are shown in Fig. 2. The optimal $Cond\left({\mathbb{E}}_{\text{dep}}\right)$ is approximately 4×10⁴, which represents a significant reduction compared with the unoptimized online FSS configuration. For instance, the previously reported CsI FSS [37] exhibited a $Cond\left({\mathbb{E}}_{\text{dep}}\right)$ of 2×10¹² (considering 15 channels). The optimal configuration obtained during the iterations is listed in Table 2. The 0.01-mm precision is maintained, considering that the machining tolerance of the GAGG crystal is 0.02 mm.

Fig. 2Full-screen View

(Color online) Evolution of $Cond\left({\mathbb{E}}_{\text{dep}}\right)$ during the genetic algorithm iterations

It is worth noting that the optimization process using a simple RM calculation model has certain limitations. First, because the mass attenuation and mass X-ray absorption coefficients are available only within an energy range of 1 keV to 20 MeV, the optimization process considers X-rays ranging from 10 keV to 10 MeV. Second, it is well known that the simple RM calculation model does not consider secondary effects, such as Compton scattering and pair production processes, which become dominant in photon-matter interactions starting at hundreds of keV. Consequently, the simple RM calculation model results deviate from the actual values above hundreds of keV. However, it still provides a reasonable estimation of the actual RM, and we demonstrated that optimization is an effective approach for achieving a partially optimized structure. This is evident from the significant reduction in the unfolding error, as demonstrated next.

Unfolding spectrometer response

Numerical experiments were conducted to test the unfolding procedure and evaluate the accuracy of the unfolded radiation spectra. Using Eqs. (2) and (5), the expected energy deposition ${\mathbb{E}}_{\text{dep}}$, X-ray spectrum S, energy-signal coefficients ESC, and perturbation ΔD were calculated and modeled to simulate the channel data.

First, the optimized online FSS was modeled using GEANT4 [52], a simulation platform using the Monte Carlo method, to obtain an accurate ${\mathbb{E}}_{\text{dep}}$. The simulation model consists of the filters, sensors, scintillation photon detector, PIN diode carrier board, collimator, shielding, and other mechanical parts. The incident X-rays were simulated as pencil-like beams uniformly and randomly distributed within a 6-mm-diameter circle, matching the collimator diameter. The energy range of the incident X-rays was set from 10 keV to 200 MeV. For each energy bin, the X-ray energies were uniformly and randomly sampled and 10⁷ X-rays were simulated. The energy deposition curves of the X-ray beams impinging on the online FSS, i.e., ${\mathbb{E}}_{\text{dep}}$, are shown in Fig. 3.

Fig. 3Full-screen View

(Color online) Response matrix expressed as ${\mathbb{E}}_{\text{dep}}$ obtained via Monte Carlo simulations

The X-ray spectrum S is modeled as follows: $S\left(E\right)=S_{\text{Beta}}\left(E\right)+S_{\text{Brems}}\left(E\right)+S_{\text{BG}}\left(E\right)\mathrm{,}$ (9) where $S_{\text{Beta}}\left(E\right)=A{E}^{\alpha }\text{exp}\left(-\beta E\right)$ represents the on-axis betatron component, $S_{\text{Brems}}=\frac{B}{E}\left(\nu -\text{ln}\left(E\right)\right)$ represents the on-axis bremsstrahlung component, and $S_{\text{BG}}\left(E\right)=C\text{exp}\left(-\eta E\right)$ represents the background component [52, 28]. Note that E is in MeV. Parameters A, α, β, B, ν, C, and η define the shape of the curves and the fluence ratios of the three components. Specifically, they are set to 12.47, 0.768, 7.52, 4.09×10^-4, 3.88, 0.01, and 1, respectively. In this scenario, the betatron radiation critical and peak energies are 0.25 and 0.1 MeV, respectively, and the fluence ratio of the background component is 7%. This represents a typical betatron radiation spectrum with background [28]. Assuming X-ray energies ranging from 10 keV to 10 MeV and a total photon count of 10⁵, the simulated energy depositions in the online FSS channels were obtained by multiplying the expected energy deposition ${\mathbb{E}}_{\text{dep}}$ by the photon count in each energy bin, S(E)dE.

ESC can be estimated using Eqs.(6) and (7). GAGG-HL-type GAGG crystals manufactured by EPIC CRYSTAL Co., Ltd., with a light yield LY of 54000 photons/MeV were used in this study. The light-collection efficiency LCE was approximately 60% under typical packaging and optical coupling conditions [54]. The maximum emission wavelength for GAGG is approximately 530 nm [49], and the quantum efficiency QE of PIN diodes, such as the 0.3-mm Si-PIN, is approximately 90% at a wavelength of 530 nm. Because ESC may vary owing to the different scintillators, packaging, and optical couplings, it was experimentally calibrated as described in Sect. 2.3. The FEE gains (ρ) are the product of the gains of the TIAs and main amplifiers, which collectively determine the overall gain of the electronic unit. The gains of the TIAs and main amplifiers were determined individually to ensure that the signal amplitudes conformed to the dynamic ranges of each stage. A detailed discussion of ρ is provided in Sect. 3.2 and the values are listed in Table 4.

The perturbation ΔD is primarily caused by the statistical uncertainty σ(E)_st, where E represents the energy deposition in the detector, and the electronic noise σ_noise. The total perturbation σ_total is calculated as ${\sigma }_{\text{total}}=\sqrt{\sigma {\left(E\right)}_{\text{st}}^2+{\sigma }_{\text{noise}}^2}$. The statistical uncertainty in the PIN diode channels can be estimated using $\sigma {\left(E\right)}_{\text{st}}=2.355\sqrt{F\cdot E\cdot W}$, where F=0.12 is the Fano factor and W=3.62 eV is the energy required for the formation of a charge carrier pair. Statistical uncertainties in the GAGG channels were estimated based on a previous calibration experiment conducted in our laboratory using radioactive sources [54]. In particular, $\sigma {\left(E\right)}_{\text{st}}=0.09\cdot 662\times {10}^3\cdot \sqrt{\frac{662\times {10}^3}{E}}$. The electronic noise model is complex and beyond the scope of this study. The calculation method for the electronic noise in a high-speed waveform-sampling detector is described in detail in [55]. According to Cang et al. [55], when the pulse amplitude fully utilizes the ADC range and the integration time window is 150 ns, the electronic noise reaches approximately 0.12%. Considering that our time integration window is 600 ns and the pulse amplitude may vary from pulse to pulse, a 1% electronic noise error estimation is reasonable. Therefore, an electronic noise of σ_noise = 0.01×Di is applied to all channels, where Di represents the noise-free channel data.

Based on the simulated channel data, $D_{\text{sim}}$, the unfolded X-ray spectrum was obtained by solving Eq. (8) using the expectation-maximization method [56]. The convergence degree is indicated by the normalized mean absolute distance (MAD) between the fitting and simulated channel data. $MA{D}_{\text{data}}=\frac{1}{N}{\displaystyle \sum }\frac{|D_{i\mathrm{,}\text{fit}}-D_{i\mathrm{,}\text{sim}}|}{D_{i\mathrm{,}\text{sim}}}\mathrm{,}$ (10) where N denotes the number of channels. Using a constant function of unity as the initial guess for S_unfold(E)dE, the fitting errors for different numbers of iterations are plotted in Fig. 4(a). In the case of the optimized GAGG array (represented by the red open circles), the fitting error initially decreases rapidly to a low level within the first few steps and eventually reaches <1% after 50 steps. The deviation between the unfolded spectrum and the spectrum model function is also indicated by the normalized MAD between them. $MA{D}_{\text{spec}}=\frac{1}{M}{\displaystyle \sum }\frac{|S_{\text{unfold}}\left(E\right)\text{d}E-S\left(E\right)\text{d}E|}{S\left(E\right)\text{d}E}\mathrm{,}$ (11) where M denotes the number of energy bins. As shown in Fig. 4(b), the divergence can reach a minimum of approximately 16% in an energy range of 10 keV to 10 MeV when the fitting converges. In this study, 100 iteration steps were chosen, which is sufficiently large to observe the convergence properties, obtain a reasonable spectrum shape, and fit the response of the FSS, as shown in Fig. 5. It should be noted that the choice of the unfolding algorithm can significantly affect the unfolding results, and the expectation-maximization method was selected because of its ability to avoid the need for many additional constraints and its greater universality.

Fig. 4Full-screen View

(Color online) (a) Converge curve and (b) unfolding error for three kinds of online FSSs

Fig. 5Full-screen View

(Color online) Fitting data (right) plotted with the corresponding spectra (left)

For a comparative analysis, the CsI array proposed by Behm et al. [37] and the BGO array proposed by Rusby et al. [36] were also simulated following the aforementioned procedure. The channel number and noise level were kept the same as those in the GAGG array, that is, 15 channels. The thickness of the BGO crystals was 0.2 cm. The condition numbers for the CsI and BGO arrays were 2×10¹² and 1×10¹², respectively. The corresponding minimum fitting errors were 115% for the CsI array and 133% for the BGO array, respectively, as shown in Fig. 4. The unfolding error of the optimized GAGG array was significantly reduced compared to that of the CsI and BGO arrays with uniform thickness.

Quantitative calibration of the spectrometer response

An accurate RM determination is of vital importance for precise spectrum unfolding. In Eq. (5), E_dep,i(E) can be precisely obtained through Monte Carlo simulations [28, 37, 36]. However, ESC must be experimentally calibrated for scintillator channels because the light yield LY and light collection efficiency LCE can vary depending on the scintillators, packages, and optical couplings, making it challenging to model and calculate them.

The most challenging aspect of the experimental calibration campaign for LY and LCE is determining the energy deposition in the scintillators. Behm et al. [37] performed an experimental calibration by measuring the scintillation signal resulting from bremsstrahlung interactions. They theoretically calculated the bremsstrahlung X-ray beam in GEANT4 by simulating the collision of a typical electron beam with a 9-mm-thick piece of lead. The energy deposition in the CsI array was then determined using GEANT4. Rusby et al. [36] calibrated their online FSS by exposing the detector to radiation sources for an extended period of time and integrating the camera images. Energy deposition during this period was also determined via GEANT4 simulation.

In the optimized online FSS, a PIN diode with a 0.3-mm Si-PIN was also used as the scintillation photon detector. The 300-μm depletion layer of the 0.3-mm Si-PIN allows us to obtain LY × LCE × QE by comparing the full-energy peaks of the radioactive sources. One source directly irradiates the PIN diode 0.3-mm Si-PIN, while the other irradiates the GAGG scintillator [57]. The energy deposition in the scintillators, that is, the gamma ray lines of the radioactive sources, can be accurately determined, thereby providing more reliable calibration results.

The calibration experiment utilized ²⁴¹Am with a 59.5-keV line and ²²Na with a 1274.5-keV line. Charge-sensitive pre-amplifiers were used instead of TIAs to amplify the signals induced by the radioactive sources. An example of the spectrum measured during the experiment for a single channel is shown in Fig. 6. The peak positions were obtained by fitting the experimental spectrum using a Gaussian curve superimposed on a second-order polynomial background, denoted by the green lines; the background is represented by the brown lines.

Fig. 6Full-screen View

(Color online) Spectra obtained during the calibration experiment for one channel

As the readout electronics are identical for GAGG and PIN diodes, according to Eqs. (5)-(7), LY × LCE × QE can be calculated as follows: $LY\times LCE\times QE=\frac{P_{\text{GAGG}}{E}_{\text{dep,PIN}}}{3.62[\text{e}V]\cdot P_{\text{PIN}}{E}_{\text{dep,GAGG}}}\mathrm{,}$ (12) where P_GAGG and P_PIN are the fitting peak positions in the experimental spectrum, E_dep,GAGG and E_dep,PIN are the energy depositions in GAGG and PIN diodes, which are 1274.5 and 59.5 keV, respectively. The corresponding spectra are shown in Fig. 6, P_GAGG and P_PIN are 1.0646±0.0002 and 0.4314±0.0001 V, respectively; therefore, LY × LCE × QE = 31826±5 electrons/MeV.

Filter and sensor unit

GAGG scintillators were chosen because of their good mechanical characteristics, non-hygroscopic properties, high light yield (30-70 ph/keV), and fast decay times (∼100 ns). The scintillators had front faces measuring 1 cm × 1 cm with variable thicknesses. The filter and scintillator configurations are listed in Table 3. The prototype configuration was obtained using the original version of our optimization code without proper constraints on the thickness, resulting in a configuration different from that listed in Table 2. Nevertheless, the condition number of the prototype response matrix is 6×10⁵, which is close to the optimized condition number presented in Sect. 2.1 (compared with the uniform-thickness configuration), and an improved performance is expected.

The reflection layers of the scintillators are enhanced specular reflectors (ESRs), 65-μm polymers with high reflectance (>98%) manufactured by 3M. The ESRs were cut into specific shapes to allow the coupling of the scintillation light from the side of the scintillators. The scintillators were fixed to an aluminum frame and interleaved with filters, as shown in Fig. 8.

Fig. 8Full-screen View

(Color online) Filters and scintillators used in the prototype

The scintillation light output is bright when the scintillators are irradiated by laser-induced brilliant X-ray pulses, for example, >5 × 10⁷ scintillation photons (10³ MeV × 54000 photons/MeV) per channel per pulse. A non-multiplying photon detector is suitable for collecting scintillation light. To improve the light-collection efficiency and coupling stability, a PIN diode with a large sensitive area (1 cm × 1 cm) was used as the photon detector, and the PIN diodes were directly coupled with the scintillators using optical grease. As there are six PIN diode channels used as X-ray sensors, all PIN diodes are mounted on one PIN diode carrier board vertically or horizontally, as shown in Fig. 9.

Fig. 9Full-screen View

(Color online) PIN diodes and their carrier board

The sensors, filters, and PIN diodes were placed in an X-ray- and electromagnetic-shielding enclosure along with the collimator and laser sight, making up the filter and sensor unit, as shown in Fig. 10. Bias voltages and electric signals were applied or elicited using coaxial cables.

Fig. 10Full-screen View

(Color online) Filter and sensor unit

Using the calibration method described in Sect. 2.3, the LY × LCE × QE values for the nine GAGG scintillation channels were calibrated experimentally, as shown in Fig. 11. Error bars are not visible in this figure because of the low peak fitting error.

Fig. 11Full-screen View

Experimentally calibrated LY × LCE × QE for the nine GAGG scintillation channels of the prototype

Electronics unit

The PIN diode signal is fed to a TIA via alternating current (AC) coupling, and a standard high-speed amplifier with a 1.6-GHz gain bandwidth product is adopted as the TIA amplifier. A 2000-ohm resistor connects the cathode of the PIN diode to the ground, serving as the direct-current (DC) path, which can limit the current of the PIN diode and increase the system reliability. The TIA is followed by the main amplifier, which adjusts the voltage amplitude to match the dynamic range of the DAQ system. In addition to the 16-channel TIAs and main amplifiers, the FEE board includes a trigger circuit to generate an inner trigger signal for the DAQ system.

The FEE circuit gains are determined to ensure that the signal amplitude is within the dynamic range of each circuit stage. First, the energy deposition per pulse in the PIN diodes and scintillators was calculated using GEANT4. The spectrum of the X-ray source was modeled using Eq. (9) with a photon fluence of 1×10¹¹ photons/sr/pulse, and is located 1 m away from the collimator of the online FSS. Therefore, the number of X-ray photons passing through the collimating aperture is 2.8×10⁶ per pulse.

The temporal waveforms were calculated using the transient response model, which includes the GAGG scintillation light, PIN diode output current, and output voltage of each circuit stage. The PIN diode was treated as a second-order Butterworth low-pass filter (LPF) with a 40-MHz -3 dB bandwidth [59]. The transient response of the GAGG scintillator can be expressed as a single exponential decay signal with a decay time of 100 ns, and the TIA can also be treated as a second-order Butterworth LPF with a -3 dB bandwidth of $\sqrt{\text{GBP}/\left(2\pi R_{\text{F}}{C}_{\text{D}}\right)}$ [54], where GBP is the 1.6-GHz gain bandwidth product, R_F is the feedback resistance, and CD is the input capacitance of the TIA. CD is mainly attributed to the terminal capacitance of the PIN diodes and the stray capacitance of the cables, and is in the order of ∼100 pF. The main amplifier was designed as an inverting second-order voltage-controlled voltage source LPF with a 10-MHz -3 dB bandwidth. The simulated FEE output waveform obtained by the transient response model is plotted along with the experimental waveform measured in the laser-plasma experiment, as shown in Fig. 12. The simulated waveform is in good agreement with the experimental waveform, e.g., the relative difference in the peak amplitudes is less than 2% when the integrals from 130 to 700 ns are equal (with the same energy depositions), thus validating the transient response model built in this study.

Fig. 12Full-screen View

(Color online) Simulated and experimental waveforms of the FEE output

As the voltage output swing of the TIA is 0-3 V and the input dynamic range of the DAQ system is 1 V, the main amplifier gains are determined to fully utilize the dynamic range of the DAQ system, which is 0.32. Then, using the transient response model, the TIA gains are determined to ensure that the peak amplitudes of the 15-channel TIA outputs are in the order of 500 mV. The results are presented in Table 4, with the TIA and main amplifier gains listed on the left and right sides of the multiplication signs, respectively. The main purpose of this section is to describe the hardware design. As the X-ray fluence can vary significantly in different laser-plasma experiments, the FEE is designed as a plug-in module so that the gains can be determined using the described methods and can be easily changed.

To preserve the temporal waveform information and enable flexible digital signal processing, a commercial 16-channel high-speed analog-to-digital conversion module based on DRS4 chips was adopted as the DAQ system. The DAQ system provides a 1-GHz sample rate, a 1024-sample acquisition window (equivalent to 1.024 µs at 1 GHz), and a 30-MB/s transfer rate (using the USB 2.0 protocol). Given that the GAGG decay time is 100 ns, a 1.024-µs acquisition window is sufficient to capture the complete waveform. The online FSS system requires a high repetition rate of several hundred kHz. However, the DAQ data transfer rate imposes a limitation of approximately 1 kHz, calculated as 30 MB/s divided by "16 channels × 12 bits/sample × 1024 samples/channel/pulse," resulting in 1220 pulses/s. In future campaigns in which higher repetition rates may be available, upgrading the data transfer protocol would allow for a faster online FSS system.