Frequency Shift Algorithm: Application to a Frequency-Domain Multiplexing Readout of X-ray Transition-Edge Sensor Microcalorimeters
D. Vaccaro, H. Akamatsu, J. van der Kuur, P. van der Hulst, A.C.T. Nieuwenhuizen, P. van Winden, L. Gottardi, R. den Hartog, M.P. Bruijn, M. D'Andrea, J.R. Gao, J.W.A. den Herder, R.W.M. Hoogeveen, B. Jackson, A.J. van der Linden, K. Nagayoshi, K. Ravensberg, M.L. Ridder, E. Taralli, M. de Wit
FFrequency Shift Algorithm: Application to a FDM Readout of X-ray TES Microcalorimeters
Frequency Shift Algorithm: Application to a Frequency-DomainMultiplexing Readout of X-ray Transition-Edge Sensor Microcalorimeters
D. Vaccaro, a) H. Akamatsu, J. van der Kuur, P. van der Hulst, A.C.T. Nieuwenhuizen, P. van Winden, L. Gottardi, R. den Hartog, M.P. Bruijn, M. D’Andrea, J.R. Gao,
1, 3
J.W.A. den Herder, R.W.M.Hoogeveen, B. Jackson, A.J. van der Linden, K. Nagayoshi, K. Ravensberg, M.L. Ridder, E. Taralli, andM. de Wit NWO-I/SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht,The Netherlands NWO-I/SRON Netherlands Institute for Space Research, Landleven 12, 9747 AD Groningen,The Netherlands Optics Group, Department of Imaging Physics, Delft University of Technology, Delft, 2628 CJ,the Netherlands (Dated: 12 February 2021)
In the frequency-domain multiplexing (FDM) scheme, transition-edge sensors (TES) are individually coupled to super-conducting LC filters and AC biased at MHz frequencies through a common readout line. To make efficient use of theavailable readout bandwidth and to minimize the effect of non-linearities, the LC resonators are usually designed tobe on a regular grid. The lithographic processes however pose a limit on the accuracy of the effective filter resonancefrequencies. Off-resonance bias carriers could be used to suppress the impact of intermodulation distortions, whichnonetheless would significantly affect the effective bias circuit and the detector spectral performance. In this paper wepresent a frequency shift algorithm (FSA) to allow off-resonance readout of TES’s while preserving the on-resonancebias circuit and spectral performance, demonstrating its application to the FDM readout of a X-ray TES microcalorime-ter array. We discuss the benefits in terms of mitigation of the impact of intermodulation distortions at the cost ofincreased bias voltage and the scalability of the algorithm to multi-pixel FDM readout. We show that with FSA, inmulti-pixel and frequencies shifted on-grid, the line noises due to intermodulation distortion are placed away from thesensitive region in the TES response and the X-ray performance is consistent with the single-pixel, on-resonance level.
The following paper has been accepted for publication in
Review of Scientific Instruments . I. INTRODUCTION
Transition-edge sensors (TES) are employed as very sen-sitive photon detectors in a wide number of experimentalapplications, either ground-based, balloon-borne or satellitetelescopes, as bolometers or microcalorimeters for mm-waveradiation up to gamma rays, such as LiteBIRD , SPICA-SAFARI , Athena X-IFU , LSPE-SWIPE , QUBIC .To comply with the stringent thermal requirements to op-erate large arrays of TES’s at sub-Kelvin temperatures and toreduce wiring complexity and cost, their readout is performedusing a multiplexing scheme, in which the signals of the detec-tors are modulated with a set of orthogonal carriers in a lim-ited information bandwidth through a common transmissionline. The sum of the signals is fed to a cold superconductingquantum interference device (SQUID) amplifier stage, thenthe output is retrieved back to warm, digital electronics, wheredemodulation of the individual TES signals is performed.To readout arrays of TES-based X-ray microcalorime-ters, we have been developing a frequency-domain multi-plexing (FDM) readout technology with base-band feedback a) Electronic mail: [email protected] (BBFB). In the FDM scheme (see Figure 1), the detectors arevoltage biased with sinusoidal carriers at different frequenciesin the MHz range. A tuned, high-Q LC bandpass filter is putin series with each detector, with the double function of lim-iting the information bandwidth and allowing only one carrierto provide the AC bias. The TES’s signals are then summed atthe input of a SQUID and demodulated at room temperatureby digital electronics.The condition of independence between the carriers used tocodify each pixel’s information sets a minimum ratio betweenthe carrier’s frequency spacing and the individual pixel band-width, which in turn is set by the detector speed and stabil-ity conditions under negative electro-thermal feedback . Tomake use of the available readout bandwidth in an efficientway, the LC filters are designed to fall on a regular grid, sothat the bias carriers are integer multiples of a frequency spac-ing that is constant between each subsequent carrier. Due tounavoidable fabrication tolerances and stray inductance fromwire bonding, in practice resonators do not match with the ex-pected frequency, with a typical spread of ≈ .Non-linearities in the readout chain, due e . g . to SQUIDamplifiers or DACs, can generate spurious signals appearingas line noises in the carrier sidebands. With a high num-ber of multiplexed detectors, such intermodulation line noises(IMLN) are likely to fall within the sensitive region of the de-tector response, degrading the detector performance. The fre-quencies of IMLN’s are defined by linear combinations of thebias frequencies, hence choosing the latter on a regular grid,so that also the IMLNs’ frequencies would conveniently be on a r X i v : . [ a s t r o - ph . I M ] F e b requency Shift Algorithm: Application to a FDM Readout of X-ray TES Microcalorimeters 2 FIG. 1. Diagram showing our FDM readout scheme with base-band feedback. AC bias carriers are generated by the FPGA and converted toanalog via the AC bias DAC from our custom digital electronics (DEMUX) board at room temperature. SQUIDs are controlled via a customanalog front-end electronics (FEE). The SQUID output signal is first pre-amplified via a low noise amplifier in the FEE, then fed via an ADCto the DEMUX board, where demodulation is performed to retrieve the individual TES’s signals. These signals are re-converted to analogvia the Feedback DAC and used to null the error signal at the SQUID summing point just around each bias carrier. Such base-band feedbackscheme allows an increase of the available dynamic range, so that a large number of pixels can be accommodated in one readout chain. a regular grid outside of the TES sensitive band, would avoidsuch performance degradation. Given the high Q of the res-onators however, off-resonance operation effectively changesthe bias circuit, since the reactive impedance of the LC filterbecomes no longer negligible compared to the TES resistance.In a previous work we showed that the spectral per-formance of one X-ray TES microcalorimeter, spoiled byIMLN’s, could be recovered with off-resonance operationby employing an early version of a frequency shift algo-rithm (FSA). The original concept of FSA contemplated thetuning of the LC resonance frequency by adding an activeimpedance in TES bias circuit via a feedback on the modu-lated bias signal. That version had the drawback of requiringparameter fine tuning for each pixel and frequency shift, thusnot being scalable. In this paper we report on a new imple-mentation of FSA, designed to allow scalability to multi-pixeloperation. The technical aspects of such algorithm are dis-cussed in a companion paper . The operation scheme is out-lined at the beginning of Section III.The paper is structured as follows. We first describe ourexperimental setup in Section II, then in Section III the FSAworking principle and requirements for the application on anX-ray microcalorimeter readout are discussed. In Section IVthe reproducibility of the bias circuit in off-resonance readoutis tested and finally in Section V the results of X-ray experi-ments with FSA in single-pixel and multi-pixel configurationsare reported. II. EXPERIMENTAL SETUP
For our experiments we used a 5 × et al. ). Each TES consists ofa 100 × µ m Ti/Au bilayer deposited on a SiN membraneand coupled to a 240 × µ m gold absorber via two centralpillars and four corner stems, as visible in Figure 2. Thesedevices have T C (cid:39)
72 mK and R N (cid:39)
75 m Ω .To perform the FDM readout the TES’s are coupled to cus-tom superconducting LC filters and transformers, to achieveimpedance matching between the TES and LC filter. For thissetup, LC filter and transformer chips result in a 2.5 µ H ef-fective inductance. The detectors are voltage-biased through a750 m Ω resistor and a capacitive divider with 1:26 ratio, for aneffective shunt resistance of 1.4 m Ω . Remanent magnetic fieldis cancelled by means of superconducting Helmholz coils.The cryogenic components are mounted on a oxygen-freehigh-conductivity (OFHC) copper holder, as depicted in Fig-ure 3, enclosed in a Nb shell and an outer Cryoperm shield. A Fe source is hosted on the superconducting magnetic shield(Nb shell) to exploit the Mn-K α line complex for measuringthe X-ray spectral performance of the detectors at 5.9 keV,with a count rate of ≈ µ W of cool-ing power at 110 mK. For the measurements reported inthis paper, the base temperature is 60 mK. The temperatureof the setup is monitored via a Germanium thermistor an-requency Shift Algorithm: Application to a FDM Readout of X-ray TES Microcalorimeters 3
FIG. 2. Zoomed view through the Cu collimator of the SRON R1a5 × chored to the Cu holder. Mechanical vibrations are dampedby suspending the setup to the mixing chamber with Kevlarwires. Thermal anchoring is performed by means of OFHCCu braids. A 2-stage VTT SQUID (Model J3 as Front-EndSQUID and model F5 as Amplifier SQUID) is employed astrans-impedance amplifier to pre-amplify the TES signals atcryogenic stage, which are demodulated and re-modulated forbase-band feedback by an FPGA board, at a sampling rate of40 MSPS. An RC snubber is placed in parallel to the SQUIDinput coil to lower the Q-factor of the 10’s of MHz resonancecreated by the parallel between the SQUID input inductanceand parasitic capacitance.Temperature drifts in the cooling system sensed by thedetectors are corrected in the data analysis pipeline exploit-ing the TES baseline current and the pulse height informa-tion. The energy non-linearity is calibrated out by using thezero energy information, the Mn-K α and Mn-K β (6.5 keV)lines. The energy of the X-ray event is evaluated by meansof the optimal filtering technique. The spectral performanceis then assessed by fitting the Mn-K α model to the col-lected events by using the Cash-statistics in the maximum-likelihood method. The collected events are typically 3-4thousand in order to obtain sufficient statistics, with a typicalstatistical error on the energy resolution of 0.2 eV.Considering the typical energy resolution of the 23 con-nected pixels, measured at R/R N typically in the range 0.15 to0.25 and frequencies distributed to use the full 1-5 MHz read-out bandwidth, the average single-pixel X-ray spectral perfor-mance is at a level of 2.7 eV at 5.9 keV. FIG. 3. Cryogenic components of our “XFDM" setup mounted onthe Cu sample holder: (a) SRON TES array, (b) transformer chip,(c) LC filter chip, (d) VTT 6-series array front-end SQUID, (e) RCsnubber in parallel to SQUID input coil and (f) SMD shunt resistor.
III. FSA FOR X-RAY TES MICROCALORIMETERS
In this paper we focus on the application of FSA to the read-out of X-ray TES microcalorimeters. We point out howeverthat the algorithm is designed to be implementable also in dif-ferent FDM readouts, such as for TES bolometer arrays as theone of SPICA-SAFARI . A. FSA working principle
The basic idea of FSA is that each carrier is shifted from theresonator center frequency and the carrier amplitude is com-pensated for the signal transfer loss due to the off-resonanceoperation of the LC filter. Such compensation is achieved bymeans of a proportional-integral controller, implemented inthe FPGA. Similarly to the BBFB scheme, the FSA controllerworks in base-band to avoid stability issues.requency Shift Algorithm: Application to a FDM Readout of X-ray TES Microcalorimeters 4In off-resonance operation, the imaginary impedance of theLC filter causes a phase difference between the TES currentand bias voltage. The bias voltage is known and the TES cur-rent is measured via the BBFB, whose outputs are the demod-ulated in-phase ( I ) and quadrature ( Q ) orthogonal componentsof the TES current. Since the phase difference is correlatedwith the quadrature component, a convenient target for thecontroller is the Q signal. The FSA controls the Q output ofthe BBFB via a feedback scheme, explained in Figure 4. Inthis " Q -nuller" scheme, the quadrature component of the TEScurrent is the input of the controller, which uses it as an errorsignal with respect to a Q = f PI , calculated by theratio of the integrating and proportional transfer functions. Inaddition, a digital low-pass filter with cut-off frequency f LPF is present in the loop to prevent interference between the Q -feedback signals of neighbor pixels, fundamental for multi-pixel operation. SS FIG. 4. Block diagram for the FSA controller . The quadratureBBFB output is the controller input, used as error signal with respectto the Q = K i and a proportional controller withgain K p . The ratio between the integrating and proportional transferfunctions defines the crossover frequency f PI . The complex vectorgiven by the bias signal U bias (real) and the controller signal U ctrl (imaginary) is modulated ( i.e. rotated by an angle ω c t ) and the realpart of the resulting output (the AC carrier) is fed to the LC filter.The SQUID output is then demodulated, with only the Q -componentof the BBFB output being used in the controller loop. To suppressthe out-of-band Q -signal, a second order low-pass filter is employed. Substantially, the FSA allows to handle the LC filter spreadby moving the issue from the cold components to the warm,digital readout electronics, which are more feasible to controlthe generation of intermodulation products, whose effect onperformance is discussed as follows.
B. Effect of intermodulation distortions
The TES is sensitive to a IMLN when its frequency is closeto the inverse of the effective time constant under electro-thermal feedback (thermal bandwidth). In this circumstance, the TES current is modulated and the energy signal dependson the phase at which the X-ray photon hits the detector, withthe effect of severely degrading the measured energy resolu-tion.Ideally, it would be desirable to tune the carriers back tothe original 100 kHz grid. Considering a 3 σ scatter of the LCresonances around the design values, this would require a shiftof the bias frequencies of the order of 10 kHz. A limitation ofFSA is that the controller is expected to become unstable ifthe applied shift from resonance is larger than the resonatorbandwidth. For the considered setup, the bandwidth of theRLC resonator with the TES biased at typical working point isof the order of 1 kHz. Given this boundary condition, shiftingback the carriers to the 100 kHz grid is an unrealistic goal.In practice, such extreme frequency shifts are not neces-sary. FDM for microcalorimeters requires TES’s with “slow"effective response time, so that a larger inductance can be em-ployed to shrink down the electrical bandwidth ≈ R/L to pre-vent carrier leakage from neighbouring pixels , while ensur-ing stable TES operation. For this reason, our detectors are de-signed to have a thermal bandwidth of approximately 300 Hz,“slow" enough to prevent carrier leakage at 100 kHz separa-tion. This characteristic can be also exploited to effectivelyuse frequency shifts much smaller than 10 kHz.The idea is then to place the bias carriers on a regular grid,so that the IMLN’s would fall exactly at the same frequency(and higher harmonics) in the TES response, sufficiently farfrom the thermal bandwidth. Therefore, we want to assesswhat is the nearest frequency grid (taking into account DACresolution) for which no impact to the detector performancefrom intermodulation distortions is expected.To do so, we measure the energy resolution of one pixelwith an external tone applied to mimic the effect of an IMLN(Figure 5, left panel). The amplitude of the tone is kept con-stant to a level of 3 orders of magnitude higher than the TESresponse. The frequency f tone is changed from values withinthe TES thermal bandwidth to the tail of the demodulated sig-nal cut off by the decimation filter. In the right panel of Fig-ure 5 the measured performance degradation with respect tothe typical value for the investigated pixel is shown, scaled indB units.We observe that the X-ray performance severely deterio-rates when the tone frequency falls into the thermal band-width and is recovered by shifting the tone frequency out-side the thermal bandwidth, with no significant degradationfor f tone = . f tone = FIG. 5.
Left panel : X-ray pulse spectrum and TES noise spectrum with applied external tone that mimics the effect of an IMLN.
Right panel :measured degradation of energy resolution with respect to the typical value for the considered pixel as a function of tone frequency f tone , whichis defined from the difference to the LC resonator frequency. The X-ray pulse spectrum is rescaled and overplotted in dashed red to show thethermal bandwidth, evaluated at the 3 dB point. we expect the 0.6 dB degradation at f tone = C. Impact on AC bias DAC
The TES current is maintained to its on-resonance valuewhen changing the bias frequency at the expense of an in-crease in the voltage fed by the bias DAC. A requirementfor practical application is that such increase in DAC outputshould be acceptable from the point of view of (a) dynamicrange and (b) noise contribution to the energy resolution.To assess the impact on the bias DAC output, we measureits raw RMS voltage and rescale it to the full-range peak-to-peak value. We first characterize the increment in voltage asa function of the frequency shift, as shown in Figure 7, leftpanel. The measurement is done for one pixel in three differ-ent bias points in the typical operation range. For lower TESresistance values, the Q -factor of the resonance is higher andso is the increase in the additional impedances when movingaway from resonance (0 frequency shift). For this reason, theincrement in DAC signal is larger for lower bias points.Since for this setup the average shift to place the carriers on a 1 kHz grid is approximately 250 Hz, from this measurementwe expect a DAC voltage increase of ≈ N ≈ . ≈
250 Hz.We conclude that the average frequency shift to place thecarriers on grid allows to make a fair prediction of the increasein DAC voltage, which can be exploited to set a safety marginin the used dynamic range to avoid DAC saturation.By design, the controller provides a signal amplificationthat compensates for the LC filter attenuation: however in thisscheme also noise is passed through the feedback loop andamplified, thus increasing the effective DAC noise contribu-tion to the spectral performance.To make a quantitative evaluation of the impact on the DACnoise contribution to the energy resolution, we refer to theestimated energy resolution budget for the FDM readout ofX-ray TES microcalorimeters in Athena X-IFU , where thetotal energy resolution of one pixel is considered as the rootsum square of several contributions, among which 2 eV comefrom the detector noise in set point and 0.23 eV from AC biasDAC noise.Considering only these two contributions, the pixel energyresolution is affected as follows: (cid:113) ∆ E TES + ∆ E DAC = (cid:113) ( ) + ( .
23 eV ) ≈ .
01 eV . requency Shift Algorithm: Application to a FDM Readout of X-ray TES Microcalorimeters 6 FIG. 6. TES noise spectra for 9 pixels (3 each for low-, mid- and high-frequency regime in 100 kHz separation) acquired in a 20-pixelmultiplexing measurement. FSA is applied on all the pixels using a 1 kHz grid configuration. As a result, the bias frequencies are shifted suchthat all the IMLN’s appear at integer multiples of 1 kHz.FIG. 7.
Le f t panel : AC bias DAC voltage normalized to on-resonance value as a function of frequency shift, for 3 different bias points in thetypical R/R N region. The resonance frequency used for reference is measured with the TES biased at 20% of its transition (black points). Theasymmetry of the parabola for the red and blue points is due to the slight change in resonance frequency at different bias points. Right panel :voltage fed by the bias DAC with FSA active normalized to the on-resonance value as a function of active pixels. The normalised DAC voltagefor one pixel active is close to 1 since for that pixel the shift towards its grid frequency is almost zero. requency Shift Algorithm: Application to a FDM Readout of X-ray TES Microcalorimeters 7When FSA is active and frequency shift is applied, the DACnoise is amplified along with the signal. Calling ε the increasein signal fed by the bias DAC, the estimated spectral perfor-mance then becomes ∆ E FSA ≡ (cid:113) ∆ E TES + ( ε · ∆ E DAC ) . To avoid underestimations, since we are not taking into ac-count other effects such as the increase of DAC non-linearity,we assume the worst possible case: all the pixels need a700 Hz shift (approximately the maximum observed shift fora 1 kHz grid operation with this setup). Considering the bluecurve from Figure 7, this means a voltage increase of roughly40%. With ε = .
40 we then get ∆ E FSA (cid:39) .
03 eV.Therefore, even making a substantial overestimation, weexpect the increase in DAC noise to not contribute signifi-cantly to the root sum square spectral performance, in par-ticular compared to the degradation observed due to (1) off-resonance operation without FSA and (2) the presence ofIMLN’s within the thermal bandwidth, as shown in previoussubsection.One more observation is that, to perform these measure-ments, the system could handle stable FSA operation withmore than 20 pixels active. This is a fundamental step forwardwith respect to past FSA implementations and a crucial pointfor practical FDM application on X-ray TES microcalorime-ters. IV. BIAS CIRCUIT REPRODUCIBILITY
The basic functionality of FSA is to preserve the electricalbias circuit when shifting the carrier from the resonance fre-quency. In particular, the controller must react if the measuredTES current is not in-phase with the bias voltage. Since evenon resonance a phase angle can be present, due to unavoidabledelays in the signal path, the phase of the carrier is tuned tocompensate for such static offset. This is done with an auto-mated routine that scans for the phase angle that minimizes thedemodulated Q − signal with the TES in normal state. In thisway, the controller correctly keeps the TES current in phasewith the bias voltage when frequency shift is applied.To verify the conservation of the bias circuit, we measurethe TES IV curve with FSA under frequency shift and com-pare it with the original IV curve on-resonance. Early testsshowed that the IV curves diverged for increasing frequencyshifts. This was understood to be the effect of non-zero ad-mittance of the other resonators in parallel with the measuredpixel. On resonance, the TES impedance is much lower thanthe parallel impedance of other resonators and the effect isnegligible. However, when applying a frequency shift theimpedance of the LCR series gets higher and the contributionof the parallel impedance becomes significant.The compensation for such effect is made in the firmware,by multiplying the bias signal for a scaling factor, essentiallythe parallel admittance converted to adimensional units. Thisscaling factor is given by numerical parameters depending onthe setup (LC resonances, inductance value, SQUID coil ra-tio, bias and feedback transfer function, conversion factors FIG. 8. Comparison of the IV curves on-resonance and shifted withFSA active. In the bottom panel, the residuals of the shifted curveswith respect to the on-resonance reference, zoomed in the supercon-ducting transition region are plotted scaled in percent. Some points,recorded in coincidence with an X-ray hit, are removed from the plotfor sake of clarity. of warm electronics) and is readily calculated for each pixel.In addition, we developed an iterative procedure that empiri-cally optimizes such scaling factor, which we use to verify thegoodness of the calculation. This approach allows us to cor-rectly compensate for the leakage of current though the paral-lel resonators.As previously stated, the electrical bandwidth at typical biaspoints for our pixels is of the order of 1 kHz. Within thisboundary ( ± Q -current oscillation due to weak-link effect. Since nosignificant impact on spectral performance is expected fromsuch small effect for these devices, we set as a requirementthat the controller should maintain the electrical circuit withinless than 1% deviation from reference. To verify this, we mea-sure how the TES current deviates from its on-resonance valuewhen frequency shift is applied. This is done by measuring theTES current (1) as a function of bias voltage (IV curve) withfixed frequency shift and (2) as a function of frequency shift(current scan) with fixed bias voltage.We verified that, by properly compensating the parallel ad-mittance effect, we are able to (1) reproduce the IV curveat shifted frequency and (2) keep the TES current constantat the typical working point within our boundary conditions( ± <
1% deviation). In Figure 8 the shifted IVcurves and in Figure 9 the TES current scan are representa-tively reported for one pixel.requency Shift Algorithm: Application to a FDM Readout of X-ray TES Microcalorimeters 8
FIG. 9. TES current change on typical bias point with FSA activeacross ± V. X-RAY SPECTRAL PERFORMANCE
As previously anticipated, the controller should be fastenough to follow the rapid decrease of the TES current whenan X-ray photon hits the detector. The parameters to set areessentially three: the integrator gain, the crossover frequency f PI (defined in Section III) and the low-pass filters cut-off fre-quency f LPF .The procedure used to set these parameters is the follow-ing: to allow the controller to correctly compensate the TEScurrent during an X-ray event, we increase the integrator gainuntil the TES response is stable and the X-ray pulse shows nosigns of oscillation. We find that the integrator gain shouldbe increased approximately in proportion with the TES biasfrequency. In practice, we use 3 sequential values for 3 cor-responding regimes, which we define as low- ( ≈ ≈ ≈ f PI covers the electri-cal bandwidth of the pixel, with a factor ≈ f LPF of the low-pass filter is then set to ≈ × f PI , value for which the out-of-band Q-feedback signal issufficiently suppressed to prevent interference with 100 kHzneighbour channels in multi-pixel operation, but with enoughmargin to avoid attenuation of the in-band TES response. Wefind that stable operation for all the pixels can be achieved byusing the same values of f PI and f LPF .Having fixed such parameters, we first perform X-ray mea-surements in single-pixel with 200 Hz frequency shift for sev-eral channels, confirming that the energy resolution does notdegrade under FSA operation. In this way we confirm that thecontroller is able to compensate for fast changes in TES cur- rent, while maintaining stable operation. In Table I the com-parison between the typical single pixel performance and themeasured performance with FSA is representatively reportedfor one pixel per each frequency regime.
Ch nr. Freq (MHz) ∆ E SP (eV) ∆ E SP FSA (eV) ± ± ± ± ± ± We then perform multiplexing experiments to verify thatthe energy resolution is not degraded in multi-pixel configura-tion. We first use three neighbouring channels in the 100 kHzseparation, namely Ch1, Ch2 and Ch3, whose single-pixel av-erage performance is at level of ≈ ± ≈
130 Hz, 210 Hz and420 Hz, respectively. We measure a summed average energyresolution of 2.55 ± f PI and f LPF are fixed, no furtherfine tuning is required for the different pixels and/or frequencyshifts. With respect to the previous version of FSA, these aremajor points of improvement that consent to scale the appli-cability of FSA up to an entire readout chain.To show FDM results is outside the focus for this paper,nonetheless we mention that we verified that in ≥ Ch nr. Freq (MHz) ∆ E SP (eV) ∆ E FSA (eV) ± ± ± ± ± ± ± ± ± ± ± ± requency Shift Algorithm: Application to a FDM Readout of X-ray TES Microcalorimeters 9 FIG. 10. Measured spectral resolutions for Ch1-2-3 in single pixel, on resonance and without FSA (left panels), and in multi-pixel at 100 kHzseparation, with FSA active and bias frequencies shifted on a 1 kHz grid (right panels). requency Shift Algorithm: Application to a FDM Readout of X-ray TES Microcalorimeters 10
FIG. 11. Summed energy spectrum from 20 pixels measured in a 22-pixel multiplexing experiment (2 outliers excluded in the analysis) withFSA. multiplexing we can obtain stable operation of the system andsummed spectral resolution at a level better than 3 eV, wherethe typical average energy resolution measured in single pixelwith this array is at a level of 2.7-2.8 eV, with one outlier ata level of 4.4 eV and two at a level of 3.5 eV. In Figure 11and Figure 12 we report, respectively, the summed histogramand individual energy spectra of 20 pixels obtained in a 22-pixel multiplexing experiment (the 4.4 eV outlier not beingactive), with a measured summed energy resolution at 5.9 keVof 2.91 ± ± , will be reported separatelyin a paper currently under preparation. VI. SUMMARY
We have developed a frequency shift algorithm to performFDM readout of TES detectors in off-resonance operation. Inthis scheme, by shifting the carriers on a regular grid, linenoises due to intermodulation distortion can be placed suf-ficiently far away from the sensitive region on the TES re-sponse, avoiding degradation in spectral performance. By ap-plying the FSA up to 20 pixels in multiplexing, we evaluatedthat the increase in bias DAC output is acceptable in termsof DAC dynamic range and noise contribution to the energyresolution.After finding and fixing the optimal FSA parameters for ourexperimental setup, we have verified that the on-resonancebias circuit can be reproduced with acceptable margin whenthe bias frequency is shifted.Finally, we have measured the X-ray spectral performancein multi-pixel configuration in 100 kHz neighboring pixelswith off-resonance, on-grid bias frequencies, observing no significant degradation from single-pixel energy resolutions.This is achieved without the necessity of fine-tuning the con-troller parameters for the different pixels and frequency shifts.Having shown scalability towards and stability in multi-pixel operation, we consider the FSA mature enough to be im-plemented on the FDM readout of X-ray TES microcalorime-ters, to finalize the demonstration of our FDM technology.
ACKNOWLEDGEMENTS
SRON is financially supported by the Nederlandse Organ-isatie voor Wetenschappelijk Onderzoek.The SRON R1a TES array used for the measurements re-ported in this paper was developed in the framework of theESA/CTP grant ITT AO/1-7947/14/NL/BW.
DATA AVAILABILITY
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