Optical Calibration of a Frequency-Domain-Multiplexed X-Ray TES Microcalorimeter Array via Photon-Number-Resolved Statistics

1. Introduction

High-resolution X-ray spectroscopy is a key tool for investigating the hot and energetic Universe [1], including the dynamics [2], chemical enrichment [3], and thermodynamic state [4] of diffuse astrophysical plasmas. The X-ray Integral Field Unit (X-IFU) [5] on board NewAthena [6] is being developed to provide spatially resolved spectroscopy with less than 4 eV energy resolution over a broad soft-X-ray energy range. Achieving this performance requires large arrays of superconducting microcalorimeters with stable, reproducible, and accurately calibrated responses. Transition-edge sensors (TESs) are among the leading detector technologies for this purpose. A TES microcalorimeter [7] is operated within the narrow superconducting-to-normal transition, where a small temperature increase caused by photon absorption produces a measurable change in resistance. Under voltage bias, this resistance change is converted into a current pulse whose amplitude is related to the deposited energy. Negative electrothermal feedback stabilizes the detector operating point and improves the linearity and recovery of the response, enabling high-resolution spectroscopy. For TES arrays, multiplexed readout is essential to reduce the number of cryogenic wires and the associated thermal load. In frequency domain multiplexing (FDM) [8], each pixel is AC-biased through a dedicated superconducting LC resonator at a distinct carrier frequency. Photon-induced changes in each pixel’s TES impedance modulate its carrier. The carriers from all pixels are combined into a single signal, amplified by one shared two-stage SQUID, and digitally demodulated back into the individual pixel channels. This FDM readout architecture enables the characterization and operation of individual pixels in large TES arrays while keeping power consumption low, since several tens of pixels can be multiplexed through the same amplification chain [9].

A remaining challenge is the calibration and characterization of TES microcalorimeters at low energies. Conventional fluorescent X-ray calibration lines below 1 keV can be broader than the intrinsic detector resolution. Compact, lab-based monochromators are limited by the achievable photon flux, while synchrotron beamline monochromators avoid this limitation but require access to a large external facility. Electron-beam ion traps (EBITs) can likewise be coupled to TES microcalorimeters, as demonstrated for higher-energy transitions and dedicated atomic-physics studies [10,11], but similarly require large and dedicated infrastructure and provide low intensity at low energies. Besides, measurements below approximately 100–200 eV are challenging because the infrared-blocking filters, required in the cryostat to limit the thermal load on the detector, can significantly reduce the transmission of such low-energy X-rays.

Optical photons provide an attractive alternative [12,13]. When a coherent optical pulse is sufficiently attenuated, the number of absorbed photons follows Poisson statistics, generating a comb of peaks separated by the known optical photon energy, a technique recently used to cross-validate TES energy calibration against an independent X-ray source on a single detector [13,14]. In this work, we extend this approach to an Athena-like TES pixel operated within a frequency domain multiplexed (FDM) readout chain, demonstrating full-energy-scale Poissonian linearity on a single detector up to approximately 400 eV and exploring the possibility of calibrating multiple pixels illuminated by the same fiber and read out simultaneously under FDM.

2. Materials and Methods

2.1. Experimental Setup

2.1.1. The TES Array

The measurements were performed on an Athena-like transition-edge sensor array fabricated at SRON, where TES pixels are coupled to large Au/Bi absorbers. The nominal pixel pitch is 317 $\mu$m, with absorber gaps of 5 $\mu$m and an absorber stack consisting of 1.0 $\mu$m Au and 5.5 $\mu$m Bi. The TES bilayer is based on Ti/Au, with a target sheet resistance of approximately 15 m$\Omega$ [15]. In this work, we used high-ohmic 105 × 15 $\mu$m² TES geometry (Figure 1 on the left) developed to match the SRON FDM readout [16]. The 8 × 9 pixel array can be read out from one side of the chip, as shown schematically in the center of Figure 1, while a microscope image of the mounted detector assembly is shown on the right. The expected energy resolution for this array is 1.6 eV - 2.1 eV at 6 keV. The TES chip is wire-bonded to the cryogenic readout circuitry, including the micro-strip line interconnect chip and LC-filter stages, and is thermally anchored to the cold stage of the dilution refrigerator.

For the optical calibration measurements, six pixels of the array were biased and read out simultaneously through the FDM system, each coupled to its own dedicated LC resonator and amplified by the SQUID readout chain. Pixel 36 was aligned with the optical fiber and was the intended illumination target. The detector was operated at a bath temperature of approximately 58 mK during the optical pulse measurements. The fiber was aligned to pixel 36 using a dedicated cryogenic mechanical interface, consisting of a custom aluminum superconducting shield and a copper alignment bracket mounted on the detector backbone (Figure 2 left). During assembly, the copper bracket held the cleaved fiber ferrule above the TES array while the shield remained open enough to inspect the relative fiber–absorber position optically. The ferrule’s insertion depth into the copper block set its height above the TES chip, while the bracket allowed transverse (X and Y) adjustment until the fiber axis was centered on the target pixel’s absorber. Once aligned, the bare fiber was passed through a slit in the removable half of the three-part aluminum shield (Figure 2 right), which was then closed around the bracket and TES chip; only afterward was the chimney screwed onto the removable half to complete the feedthrough. This sequence avoids threading the fiber through the narrow chimney bore while it is already positioned, limiting mechanical stress on it during assembly. Once closed, the aluminum shield provides a light-tight enclosure that reduces stray illumination of pixels or internal components, and becoming superconducting below about 1 K, also provides a magnetic shielding of the TES array during operation.

2.1.2. Optical Setup

An acousto-optic modulator (AOM) generated light pulses with various averaged photon numbers. A schematic of the optical bench is shown in Figure 3. The light source was a 405 nm laser diode. At this wavelength, each photon carries an energy of approximately 3.06 eV, providing a natural few-eV calibration spacing when photon-number peaks are resolved in the TES pulse-height spectrum.

The laser beam was first collimated and then sent through the AOM, driven by a waveform generator producing pulses with widths ranging from 200 ns up to a few $\mu$s, repeated every 20 ms to fully comply with the detector dead time. The drive signal excites acoustic waves in the piezoelectric crystal, creating a time-dependent diffraction grating that deflects the beam into a first-order component only while the drive is on. Selecting this component makes the AOM act as a fast optical shutter, with the same drive signal providing an external electrical trigger synchronized to each optical pulse. This scheme is essential for reaching the lowest photon numbers, since direct modulation of our laser driver cannot produce electrical pulses of sufficiently short duration. After the AOM, the diffracted beam was isolated using a mirror pair and an iris aperture, then coupled into an optical fiber and guided into the cryostat. The optical intensity was attenuated to reach the low photon-number regime required for photon-number-resolved measurements. Inside the cryostat, the fiber delivered the optical pulses to the TES array and was positioned onto the selected pixel using the cryogenic alignment interface described above. For the acquisitions requiring the highest mean photon numbers, the AOM stage was bypassed: the waveform generator directly modulated the laser diode current, and its output fiber was connected directly to a feedthrough on top of the cryostat.

2.2. Preliminary Analysis and Software Optimization

The optical pulse datasets were analyzed using a custom Python pipeline developed for externally triggered TES measurements. For each acquisition, the software first averaged an initial subset of traces to identify the approximate pulse position and to define the pulse window. Samples outside this window were used to estimate the local baseline of each event. A robust baseline subtraction was applied trace by trace before amplitude reconstruction. The baseline was calculated from the pre- and post-pulse regions after rejecting the largest deviations from the local median, in order to reduce the effect of spurious pulses or glitches occurring outside the triggered window. This step ensures that the reconstructed pulse height is referred to a common zero level for all events. The pulse amplitude was reconstructed using an optimal filter in the frequency domain. In the analysis presented here, a fixed common pulse template was used for all datasets and is shown in Figure 4. The template was aligned vertically to the baseline and to the detected pulse position, normalized, and Fourier-transformed. For each event, the Fourier transform of the baseline-subtracted trace was compared with the external template and weighted by the inverse noise power spectral density. The optimal-filter amplitude was computed as

$$A={\displaystyle \frac{\mathrm{Re}\left[{\sum }_{f}D\left(f\right)S^{*}\left(f\right)/J\left(f\right)\right]}{{\sum }_f{\left|S\left(f\right)\right|}^2/J\left(f\right)}},$$

(1)

where $D\left(f\right)$ is the Fourier transform of the measured trace, $S\left(f\right)$ is the Fourier transform of the external template, and $J\left(f\right)$ is the noise power spectral density. This procedure provides a single pulse-height estimator for each triggered event while suppressing frequency components dominated by noise.

After optimal filtering, a baseline-drift correction was applied to compensate for slow variations in detector response during long acquisitions. The correction was obtained by evaluating the correlation between the local pre-pulse baseline and the reconstructed pulse amplitude. A linear trend was fitted to the densest region of this distribution, thereby avoiding outliers, and the corresponding correction was applied to all events. The corrected pulse amplitudes were then histogrammed to reconstruct the photon-number spectrum. The spectra were fitted with a multi-Gaussian model in pulse-height units. The zero-photon component was treated independently, while the photon peaks were constrained to follow a common spacing. The fitted peak areas were used to extract the photon-number probabilities and compare them with the Poisson distribution expected for an attenuated coherent optical source. Finally, the fitted spacing between adjacent photon-number peaks was used to calibrate the pulse-height axis in energy, using the known photon energy of the 405 nm laser.

3. Results

Optical pulse acquisitions were performed for different laser settings to probe the pixel response over a broad range of absorbed optical energies. The reconstructed spectra show a clear comb-like structure, with adjacent peaks separated by the energy of a single 405 nm photon. Representative energy-calibrated spectra for a subset of the optical pulse acquisitions analyzed in this work are shown in Figure 5 (left panels), together with the corresponding photon-number probability distribution (right panels).

For low energy optical pulses, only the first few photon-number peaks are significantly populated, while increasing the laser current or pulse duration shifts the Poisson distribution toward larger photon numbers. The fitted mean photon number ranges from approximately $\mu =1.06$ to $\mu =101.80$ in the datasets shown. In all cases, the extracted photon-number probabilities follow the expected Poissonian statistics of an attenuated coherent optical source, with residuals from the Poisson fit remaining < 0.002 in absolute probability over the full range, confirming that the observed peak structure reflects photon-number statistics rather than readout artefacts. The multi-Gaussian fits reproduce the measured spectra over a wide dynamic range, with a fitted energy resolution of approximately 2.1 to 2.5 eV FWHM for the datasets in Figure 5, which is consistent with the expected energy resolution of the TES array. This shows that the TES pixel resolves individual photon-number states from the single-photon regime up to significantly higher absorbed energies, with peaks observed up to approximately 400 eV, corresponding to more than 130 absorbed 405 nm photons. This upper bound reflects a practical limitation of the current optical setup, rather than a fundamental limit of the detector itself. The degradation in energy resolution at higher photon numbers may be due to the filling in of valleys between adjacent peaks, as observed in other experiments [12,17], possibly caused by extra noise from laser light incident on parts of the chip near the aligned pixel.

The linearity of the detector response was evaluated directly in terms of photon-number statistics, rather than through the energy-scale fit alone, using the mean number of photons independently sent to the detector for each acquisition, ${\mu }_{\mathrm{sent}}$. This quantity was obtained from power-meter measurements of the laser output, taken at pulse widths and diode currents matching each acquisition, with the power meter positioned immediately before the fiber entered the cryostat (see Figure 3), so the derived detection efficiency describes the full transmission from the cryostat input to the TES pixel. Two Poisson references are compared against each measured distribution. The first, $q\left(n\right)=\mathrm{Poisson}(n;{\mu }_{\mathrm{meas}})$ (yellow bars in Figure 6a), uses each dataset’s own fitted mean and checks only whether the measured distribution has the expected Poissonian shape, a self-Poisson comparison, simpler and less stringent than the full POVM (Positive Operator-Valued Measure) reconstruction used by Brida et al. [18]. The second, $l\left(n\right)=\mathrm{Poisson}(n;\widehat{\eta }{\mu }_{\mathrm{sent}})$ (blue bars in Figure 6a), uses a single, globally estimated detection efficiency, $\widehat{\eta }={\sum }_j{\mu }_{\mathrm{meas},j}/{\sum }_j{\mu }_{\mathrm{sent},j}=2.02\times {10}^{-5}$, and is the actual test of detector linearity: because $\widehat{\eta }$ is fixed across all datasets, agreement with $l\left(n\right)$ shows that one efficiency predicts the detected mean across nearly two orders of magnitude in sent photon number. The low value of $\widehat{\eta }$ mainly reflects splice losses inside the cryostat, from the 200 $\mu$m core delivery fiber down to a 5 $\mu$m and finally a 3 $\mu$m core fiber illuminating the TES.

Despite this low overall efficiency, the measured photon-number distributions remain in good agreement with the Poisson distribution expected for an attenuated coherent source across the optical pulse datasets, both when compared to each dataset’s own fitted mean and to the distribution predicted from ${\mu }_{\mathrm{sent}}$ and the detection efficiency, consistent with the TES response remaining Poissonian and linear from $\mu \approx 1$ up to $\mu \approx 102$. Figure 6b shows the absolute residuals, $|p_{\mathrm{meas}}\left(n\right)-q\left(n\right)|$ and $|p_{\mathrm{meas}}\left(n\right)-l\left(n\right)|$, between the measured distribution and each of the two models. As a result of using a single, fixed efficiency across all datasets, the residuals from the linear model are systematically larger than those from the self-Poisson comparison. Figure 6c shows the fidelity between the measured distribution and the linear model, $F={\sum }_n\sqrt{p_{\mathrm{meas}}\left(n\right)l\left(n\right)}$, as a function of the mean sent photon number. Figure 6d gives a complementary and more direct linearity check: the fitted Gaussian centroid energies (points, color-coded by photon number N) are compared with the expected photon energies, $E_N=N{E}_{\gamma }$, where $E_{\gamma }$ is the 405 nm photon energy, together with a linear fit (blue line). When a given photon number appeared in more than one dataset, the centroid with the highest statistics was retained. Unlike panels (a) to (c), which test the linearity of the full photon-number probability distribution against an independently known sent flux, panel (d) tests only whether the fitted peak positions themselves scale linearly with photon number, independent of any power-meter calibration. The measured centroids follow the theoretical photon-energy scale with excellent linearity over the full investigated range ($E_{\mathrm{theory}}=1.0025E_{\mathrm{meas}}-0.0439$, $R^2=0.99999$, RMSE $=0.322$ eV).

Figure 7 summarizes the nine optical pulse settings included in the stacked energy-calibrated histogram. Each coloured contribution corresponds to one acquisition condition, making visible how the different pulse widths and laser powers populate complementary photon-number ranges. The stacked representation is used here as a compact overview of the full calibration data set: the same photon-energy spacing is preserved across the acquisitions, while the logarithmic scale shows that the optical-pulse method remains usable from the single-photon regime to the highest absorbed energies investigated.

4. Conclusions

This work demonstrated fiber-delivered 405 nm optical photons as a compact calibration and characterization method for an Athena-like TES microcalorimeter array fabricated at SRON and operated with the SRON FDM readout. Externally triggered pulses were processed with a custom optimal-filter pipeline based on an external pulse template, robust baseline subtraction, baseline-drift correction, and multi-Gaussian reconstruction of the photon-number spectra. Across the optical pulse datasets, the reconstructed photon-number distributions remain in good agreement with Poisson statistics, both against each dataset’s own fitted mean and against the mean sent photon number scaled by a single global detection efficiency. This confirms that the detector response remains Poissonian and linear from the single-photon regime up to $\mu \approx 101.80$, with the fitted peak spacing consistent with the 405 nm photon energy throughout and an energy resolution in the few-eV range. These results establish fiber-coupled optical photons as a flexible tool for the low-energy characterization of Athena-like TES arrays. Future work will focus on improving the optical confinement onto selected absorbers and comparing the optical calibration with independent X-ray calibration measurements.