1 Introduction

The centroid of a Gaussian peak can be determined with a statistical uncertainty of \(\sigma /\sqrt{N}\), where \(\sigma \) is the standard deviation and N is the number of counts of the peak. High precision, therefore, requires a detector with high energy resolution, \(\delta {E}_\mathrm{FWHM} = 2.355\cdot \sigma \), that can quickly capture a large number of counts. Superconducting tunnel junctions (STJs) offer higher energy resolution than semiconducting detectors and a faster response time than other cryogenic detectors  [1, 2]. In this paper, we demonstrate the use of STJs for high-precision measurements in the extreme ultraviolet (EUV) and discuss the requirements for high accuracy.

2 Experimental Setup

The detectors used in this study were \(208\,\upmu \)m \(\times \,208\,\upmu \)m Ta-Al-AlO\(_2\)-Al-Ta STJs with characteristic rise times of 6 \(\upmu \)s and decay times of 100 \(\upmu \)s fabricated at STAR Cryoelectronics  [3]. They were cooled to \(<\)100 mK in an adiabatic demagnetization refrigerator (ADR). The ADR was retrofitted with a 50 \(\upmu \)m diameter single mode optical fiber (CeramOptec UV50/55p) with the fiber optic output incident on a hot mirror (Edmund Optics #B64-453 0\(^\circ \) AOI) and heat absorbing glass (Edmund Optics #B49-092 KG-5) that reflect and absorb IR radiation, respectively  [4].

The STJs were illuminated with a pulsed frequency tripled Nd:YVO\(_4\) laser (Spectra-Physics J40-8S-40). The Nd:YVO\(_4\) crystal was temperature regulated to minimize drift in the laser wavelength. The wavelength was monitored by splitting the beam and calibrating it with a grating spectrometer relative to the 253.65 and 365.02 nm lines from a mercury vapor lamp  [5]. No systematic drift in wavelength was observed over the entire range of pump diode currents indicating that the temperature of the laser crystal did not vary significantly  [6]. The measured single photon energy for our laser was 3.4975 \(\pm \) 0.0002 eV, which added a systematic uncertainty \(\delta E_\mathrm{L} = 0.2 - 5.6\) meV to the energy calibration between 3.5 and 100 eV in our current setup. The measured energy differs from the literature value  [6] by about 2 meV, most likely due to aging of the pump diode. The laser pulse width of 12 ns was significantly shorter than the detector rise time, so that multiple photons from a single pulse were effectively observed as a single event with an energy corresponding to an integer multiple of 3.4975 eV.

3 Results and Discussion

The STJ response to the heavily attenuated laser at a repetition rate of 5 kHz shows several peaks that correspond to the absorption of different integer numbers of photons from a laser pulse (Fig. 1). For constant laser output, the response is a Poissonian distribution convolved with a Gaussian of constant width due to the energy resolution of the STJ [2]. However, small fluctuations in the laser intensity shift the Poissonian distribution over time. Therefore, the STJ response is fitted with a superposition of multiple Gaussian functions whose amplitudes and centroid positions are allowed to vary freely. The resulting least squares fit agrees well with the measured spectrum, and the fit residuals match the statistical uncertainty of the number of counts per bin (Fig. 1 bottom).

Fig. 1
figure 1

Top The detector response (black) is graphically indistinguishable from fit (gray). Bottom fit residuals (diamonds) and statistical uncertainty (black)

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Fig. 2
figure 2

Detector response (Top) and residuals (eV) (Bottom) to a linear energy calibration with pump diode current at 14 (left), 16 (center), and 18 A (right) (Color figure online)

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To calibrate the detector in the energy range up to 100 eV, we vary the laser pump diode current between 14 and 18 A (Fig. 2). The single photon peak is identified by reducing the laser intensity until only the noise and a single peak are observed. Subsequent peaks differ by integer multiples of 3.4975 eV. We extract the centroids and perform a linear energy calibration. The residuals for this calibration are compatible with zero to within statistical uncertainties, the numerical values depending on the total number of counts in the peak (Fig. 2 bottom). The accuracy of the energy calibration is determined by the uncertainty of the laser calibration \(\delta E_\mathrm{L}\) and that of the detector calibration \(\delta E_\mathrm{D}\).

We estimate the statistical limitation on the precision of a measurement by calculating the uncertainty of the centroid \(\delta E_\mathrm{C}\) as a function of total counts N per peak. We compare this uncertainty for two detectors with different leakage currents and consequently different energy resolutions of 0.9 and 1.6 eV FWHM. The low-energy response of the two detectors at a repetition rate of 5 kHz is shown in Fig. 3. For a resolution of 0.9 eV, the peaks are fully separated, while there is a small overlap at a resolution of 1.6 eV. The centroid uncertainty of the Gaussian fit is plotted as a function of counts in the peak for several runs with different acquisition times and pump diode currents to vary the number of counts in each peak (Fig. 4). The difference in the uncertainties of the two STJs is due to their different energy resolutions. Normalizing this uncertainty by the standard deviation (\(\sigma _\mathrm{STJ} = \mathrm{{FWHM}}/2.355\)) of the Gaussian for each peak, we see that the uncertainty is given by 1/\(\sqrt{N}\) as expected (Fig. 4 inset). The uncertainty is slightly larger for some of the peaks because of the spectral background and neighboring peaks. Regardless, centroids of peaks with >10\(^5\) counts can be determined with a statistical precision \(\delta E_\mathrm{C}\) approaching 1 meV. At these high statistics, the uncertainty in the detector calibration \(\delta E_\mathrm{D}\) can be as low as 2 meV for energies up to 100 eV.

Fig. 3
figure 3

Low-energy detector response with an energy resolution of 0.9 and 1.6 eV FWHM (Color figure online)

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Fig. 4
figure 4

Centroid uncertainty as a function of peak counts, N, for detectors with energy resolution of 0.9 eV (circles) and 1.6 eV FWHM (crosses). (Inset) Same data normalized by the standard deviation of the Gaussian peak compared to the expected \(1/\sqrt{N}\) (black) (Color figure online)

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In general, the accuracy of the energy calibration should be higher than the statistical uncertainty of any signal whose centroid is to be determined. A desired calibration accuracy \(\delta E_\mathrm{C}= 10\) meV can be achieved in \(\sim \)30 s at a laser repetition rate of 5 kHz, and \(\delta E_\mathrm{C} = 3\) meV in 5 min, subject to the uncertainty of the laser calibration. Alternatively, the laser can be operated continuously at a rate of a few 100 Hz to monitor drifts in the detector response over time scales of minutes.

4 Conclusions

We have used a 355 nm Nd:YVO\(_4\) laser to characterize Ta-based STJs in the EUV energy range below 100 eV. The laser was calibrated to an uncertainty of 0.2 meV using a grating spectrometer and an Hg vapor lamp. The detectors had a resolution of 0.9 and 1.6 eV FWHM and were linear below 100 eV to within statistical uncertainty. The precision of the fitted Gaussian centroids was determined to be as low as 1 meV for peaks with \(>\)10\(^5\) counts. The accuracy of the energy calibration for high statistics was as low as 2 meV for energies below 100 eV. The total uncertainty is currently limited by the STJ counting statistics at low energy, and by the laser calibration at higher energy. Both can be further improved if required.