Design and performance of a gain calibration system for the POLARBEAR-2a receiver system at the Simons Array cosmic microwave background experiment

Abstract. We present an advanced system for calibrating the detector gain responsivity with a chopped thermal source for POLARBEAR-2a, which is the first receiver system of a cosmic microwave background (CMB) polarimetry experiment: the Simons Array. Intensity-to-polarization leakage due to calibration errors between detectors can be a significant source of systematic error for a polarization-sensitive experiment. To suppress this systematic uncertainty, POLARBEAR-2a calibrates the detector gain responsivities by observing a chopped thermal source before and after each period of science observations. The system includes a high-temperature ceramic heater that emits blackbody radiation covering a wide frequency range and an optical chopper to modulate the radiation signal. We discuss the experimental requirements of gain calibration and system design to calibrate POLARBEAR-2a. We evaluate the performance of our system during the early commissioning of the receiver system. This calibration system is promising for the future generation of CMB ground-based polarization observations.


Introduction
The precise measurement of cosmic microwave background (CMB) polarization, particularly the spatially odd-parity polarization pattern called "B-mode," is crucial for exploring the early universe.The B-modes at the degree scale reveal the primordial gravitational waves from inflation in the early universe, 1,2 and their precise measurement leads to quantitative studies such as the energy scale when it occurs.In addition, the B-modes at the subdegree scale are sensitive to the weak gravitational lensing effect.Precise measurements of the lensing B-modes provide unique information on the large-scale structure of the universe, which also allows us to constrain the sum of the neutrino masses. 3,4itigating the systematic errors in the observations is crucial for achieving precise measurements of CMB polarization.One of the major systematic errors is due to the uncertainty of the detector gains (responsivities).In CMB polarization observations, pairs of two orthogonal detectors are commonly used.If the gains of the two detectors are incorrectly calibrated, fake polarization signals are generated.This systematic error can be mitigated by improving the precision of the gain calibration.
Various gain calibration methods have been developed for CMB polarization observations. 5-10Calibration periods are important because gain fluctuations occur at every time scale owing to a fluctuation of the observation apparatus and atmosphere.The actual gain calibration strategy is determined by considering the performance and utility of the methods.The detector gain is finally calibrated by the temperature of the blackbody spectrum that best fits the CMB power spectrum.However, this requires observing a specific sky area for a sufficient duration.Hence, additional methods are required to track daily gain changes.Observations of celestial objects, such as planets, can provide constant sources but are not always performed owing to the daily changes in their positions and the sky conditions.Injecting a stable optical signal from a thermal source into the detectors is useful for calibrating the relative gain change in the time range from hours to days.
This study aims to mitigate the uncertainty of detector gains for the Simons Array (SA) experiments [11][12][13] through the use of a reference thermal source.The SA is a ground-based CMB polarization experiment in the Atacama Desert, Chile, at an altitude of 5200 m.It is an upgraded version of the POLARBEAR (PB-1) experiment. 5,14,15The projected constraint at the SA on the tensor-to-scalar ratio r is expected to be σðrÞ ¼ 0.006 at r ¼ 0.1, and the sensitivity to the sum of the neutrino masses is expected to be 40 meV (68% C.L.). 13,16OLARBEAR-2a (PB-2a) is the first of three-receiver systems in the SA. 13,16,17The PB-2a focal plane contains 7588 transition edge sensor (TES) bolometers 18 connected to a broadband antenna and simultaneously observes the 90 and 150 GHz bands for foreground subtraction.
We have developed a gain calibrator, a "stimulator," for PB-2a.The stimulator is designed to simultaneously calibrate the gains of all TES bolometers in two observation frequency bands, using blackbody radiation from the thermal source covering a wide range of observation frequencies.The same type of calibrator was used in some CMB polarization experiments, 5,19,20 and the design of the PB-2a stimulator is based on PB-1 equipment (Appendix A) in many respects.In this study, we detail its construction, operation, and validation of the stimulator.
The remainder of this paper is organized as follows.We introduce the general TES gain calibration in the CMB polarization observation in Sec. 2 and discuss the gain calibration requirements for PB-2a in Sec. 3 and Appendix B. Sections 4 and 5 explain the design and operation of the stimulator.Sections 6 and 7 summarize the performance evaluation during the early commissioning.In Appendix C, we discuss a case where the stimulator is used with a polarization modulator for future applications.

Formulation of Gain Calibration
The gain of a detector is defined as a factor that connects the detector's electrical output to the sky optical power.In CMB experiments, TES types of bolometers are widely used.TES bolometers read the incident optical power as a current value as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 1 1 4 ; 9 8 IðωÞ ¼ R det ðωÞP opt ðωÞ; (1 where ω is the angular frequency of signal fluctuation and a bolometer's responsivity R det is where η is the optical efficiency, V bias is the RMS of AC bias voltage, L is the loop gain, and τ is the time constant of the TES bolometer. 21,22Here the optical efficiency depends on the telescope optics and the detector bandpass.The bias voltage is affected by the temperature of the focal plane and the optical loading from the atmosphere.The loop gain depends on the operation resistance of the TES bolometer.The time constant, depends on the loop gain, and its value for each bolometer should be determined through measurement.The calibration of R det is typically challenging and imprecise.Instead, we opt to calibrate the overall gain using optical reference signals.

Introduction of PB-2a Stimulator
The PB-2a stimulator was designed to be placed on the telescope as illustrated in Fig. 1.The stimulator is equipped with a ceramic heater inside.Wide frequency bandwidth optical signals are available by modulating a blackbody radiation source of ∼1000 K with a chopper of ∼300 K at arbitrary modulation speeds.The role of the stimulator is to evaluate the factor, which converts the input optical power to the temperature (in Kelvin unit) of the sky and to measure the time constant of the TES bolometers.The gain measured with the stimulator is calculated as G :¼ R det ðP stm ∕T stm Þ ¼ I stm ∕T stm , where I stm is the signal amplitude of the output current with the stimulator, and T stm is the effective temperature of the stimulator.T stm is defined with a situation in which a blackbody of a certain temperature fills the entire field-of-view of the TES bolometers.The signal from the stimulator is assumed to be equal to that from the blackbody.During CMB observation periods, gain calibrations are performed before and after each observation scan (typically over several hours) to evaluate the gain variation.The gain is assumed to change slowly and linearly during the observation scan, and the change could be calibrated by interpolation of calibrations.
The effective temperature T stm is corrected every few days using planets, 23 as the temperatures of those stellar objects are expected to be more stable.It is obtained by comparing the outputs from the stimulator and the planet.The detailed method is explained in Sec.6.1.
To accomplish the gain calibration by converting T stm to the sky optical temperature, we use the best fit of the WMAP-9 and the Planck power spectrum of CMB temperature anisotropies C TT l;ref 24,25 such that the observed anisotropies C TT l;obs match with reference C TT l;ref , and this paper is not primarily focused on the detailed method used to obtain the absolute gain for the subsequent physical analysis.

Gain Measurement with Stimulator
The electrical output for the stimulator signal is written as I obs ¼ G • T stm þ I noise .Here I noise represents the noise on the bolometer.The error in the gain estimation originates from an uncertainty of the stimulator signal and the bolometer noise in the electrical output.The uncertainty of the stimulator signal is proportional to the fluctuation of the stimulator heater temperature δT stm .The noise of the electrical output can be written as , where NET bolo represents the noise equivalent temperature of the bolometer, and t obs is the time spent on calibration.Therefore, the gain error is expressed as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 1 1 4 ; 6 4 0 Here δT stm ∕T stm denotes the stimulator signal fluctuation.When δT stm ∕T stm can be regarded sufficiently small, the following equation holds:

Time Constant Measurement with Stimulator
Another role of the stimulator is to evaluate the time constant due to the delay of the TES including electrothermal feedback.The time constant is measured as the decrease in amplitude observed at higher modulation frequencies.The observed signal with time constant τ (s) agrees well with the one-pole response function.It is written as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 1 1 4 ; 3 7 3 where f (Hz) denotes the signal-modulation frequency of the stimulator.

Calibration Requirements for POLARBEAR-2a
In this section, we estimate the required value of the instrument for measuring the gain variation at each observation.First, we explain the effect of the gain uncertainty on the CMB polarization observation.Let I be the electrical output from the bolometer.The output of the j'th bolometer is described using the gain G j as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 7 ; 1 1 4 ; 2 2 9 where T, Q, and U are the Stokes parameters of the optical source to be observed; T represents the nonpolarized component, which is also called intensity or temperature; Q and U represent linearly polarized components; and ϕ j is the polarization angle of the antenna.
The sky parameters T, Q, and U are reconstructed from the measured I j .In this regard, the detector gain G j must be known.However, in general, the measured gain G 0 j has an error.Let us define a fractional gain error δg j as δg j :¼ ðG 0 j − G j Þ∕G 0 j .The differentiation of the pair of orthogonal detectors provides linear polarization signals.In the case of two orthogonal detectors [their gains are G j ðj ¼ 1;2Þ], the difference between the two calibrated outputs is described as follows: where Ideally, the gains are properly calibrated: δg 1 ¼ δg 2 ¼ 0 holds.In such a case, the first term in the last line of Eq. ( 8) vanishes, and thus no systematic errors are present.However, when δg 1 ≠ δg 2 , a polarized component is induced from T even without Q and U.This is referred to as intensity-to-polarization leakage.
To mitigate the uncertainty of gains in each observation by measuring, we list the signal requirements including intensity and stability of the stimulator, as items (A) to (F) in Table 1.All these items must be satisfied to accomplish the calibration.For detailed estimates regarding each item, refer to Appendix B (Secs.10.1-10.6),In this case when a continuously rotating halfwave plate (HWP) is used, the list of the requirements is modified.It is discussed in Appendix C. Subsequently, we assume the following parameters unless otherwise specifically remarked.The noise level of a single bolometer: NET bolo ¼ 360 μK ffiffi s p , 11,17 measurement time: t obs ¼ 120 s, maximum chopper frequency: f max ¼ 44 Hz (refer to Sec. 5), and time constant of TES: τ ¼ 1 ms as the expected range is 1 to 5 ms. 26he requirement values for the stimulator are summarized in Table 1.In particular, the intensity of the stimulator must be > 5 mK [from item (C)].The stimulator's signal fluctuation must be smaller than 0.66% [from item (A)] in one observation scan and 0.03% [from item (D)] in one observation season (a year).The requirement on the season stability assumes that the heater temperature will be corrected using monitoring with thermometers.

Thermal Source
Figure 2 shows the schematic of the PB-2a stimulator.We selected a ceramic (alumina) heater (Sakaguchi MS-1000; maximum operation temperature, 1000°C, maximum voltage, 100 V AC, 25 × 25 mm 2 area) as a radiator because it can tolerate sufficiently high temperatures and is sufficiently large to cover the hole on the secondary mirror (diameter, 9 mm).The effective stimulator temperature T stm was approximated as ð1∕2ÞðT heater − T chop ÞðA pipe ∕A beam Þ, where T heater and T chop are a temperature of the heater surface and the chopper wheel, respectively.A pipe and A beam correspond to the area of the light pipe and the beam spot size on the secondary mirror, respectively.
Table 1 Summary of the requirements for the stimulator.Required value and fluctuation on the value are illustrated for each requirement.Here the T stm is assumed to be corrected with heater thermometer.The temperature unit is assumed to be calibrated with the temperature of the blackbody spectrum of the CMB.Substituting them with the designed values, the temperature T stm was approximated as 46 mK RJ .Here K RJ is thermodynamical Rayleigh-Jeans Kelvin temperature.
The heater was enclosed with inner and outer covers to improve thermal stability and prevent the heating of other components.The nominal operating voltage of the heater was set to 40 V.The power consumption of the heater was ∼31 W, and the temperature was ∼960 K at the nominal voltage.At the maximum tolerable voltage, the radiation intensity from the heater was high; however, the lifetime of the source heater was short.The nominal voltage was adjusted to consider this trade-off.

Optical Design
The basic idea of the optical system is to maximize the optical efficiency of the heater within the limited space behind the secondary mirror.Several types of optical systems have been considered.Crucial aspects of these are that (a) the chopper wheel is positioned perpendicular to the optical axis, or with a slanted angle, and (b) absorbers are applied on the surface of the detector side of the chopper.
Consequently, we selected a perpendicular chopper without an absorber.In this configuration, the mechanism is the simplest; thus, the size and cost can be minimized.Conversely, the back-traced rays from the detector are reflected at the chopper and return to the receiver side; hence, the effective temperature is uncertain because predicting where these rays terminate is challenging.Therefore, the stability of the reflection must be verified.
The selected design is illustrated in Fig. 3.A reflective straight pipe with a minimum wall thickness (inner diameter, 8 mm) was used as the light pipe, and an optical horn (Winston cone   horn) was placed on the heater and both sides of the gap.The diameter of the chopper wheel and horn aperture (diameter, 16 mm) were optimized to use the space in the box efficiently.The horn facing the heater was optimized to be a 10-mm aperture that maximally uses the area of the source heater.The horns were milled from aluminum blocks using an NC machine.

Modulation System
We adopted a rotation wheel chopper with a diameter of 160 mm, which was made of a 1-mmthick aluminum plate.A stepping motor (Oriental motor PKP225) was employed to drive a chopper wheel.The chopping frequency (motor frequency times the number of blades, six) covered more than the sampling Nyquist frequency (152-Hz sampling).The IR optical encoder (OMRON EE-SX1140; 4 μs both rising and falling times) was attached to read the chopper position.The encoder detected the timing when the chopper was in an open or closed position.The chopping information was combined with the clock at the microprocessor and sent to a housekeeping DAQ system, called "slowdaq."

Control System
A diagram of the control system is shown in Fig. 4. A single-board computer (Raspberry Pi) was used as the main controller of the system.The control system was used for the following five functions: (1) to control the AC voltage supply (Kikusui PCR500M) to the heater.The voltage, current, and power from the AC supply were constantly monitored at 10-s intervals.(2) To transmit control signals to the stepping motor controller (Oriental Motor EMP401) for the chopper.
(3) To communicate with the ADC module (LabJack U6 PRO), monitoring the temperature of the heater and inside the stimulator box using K-type thermocouples.(4) To receive chopper information from the microprocessor (Arduino Uno R3).The processor board calculates the chopper timing according to the IRIG-B timecode signal 27 and 16 MHz Arduino internal clock.
(5) To distribute stimulator slowdaq information to the site Ethernet LAN.

Mechanical Design
The stimulator was assembled into a cubic casing of ∼20 cm. Figure 5 shows the images under assembly.The casing had separate front and rear spaces.The front room contained only the stepping motor and cooling fans, and the rear room included the heater and others for temperature stability.The heater, inner cover, and support parts comprised the "heater unit," which was accessible from the back of the stimulator.This design was adopted to easily replace the heater with a spare in case of failure of one heater.The stimulator was attached to a telescope, as shown in Fig. 6. 5 Operation Method

Hardware Operation
The sequence of stimulator calibration was controlled by the SA main control system. 28The AC voltage supply for the heater was maintained at the nominal value.A discussion of the temperature stability is presented in Sec.6.2.Regular calibration runs with the stimulator were performed at the beginning and end of every 5 h of observation scans.In each run, the chopper was driven at seven different frequencies from low to high (5, 9, 13, 19, 29, 37, and 44 Hz), which were chosen referring to PB-1, not to collide with any other known modulation and noise frequencies (and their harmonics).The measurement time for each signal modulation frequency was set at 120 s.In some calibration runs, such as planet observations, the stimulator was used with a constant chopper frequency during the planet scan.

Data Analysis Method
The stimulator data were processed offline for each calibration runs.The time-ordered data were filtered using a high-pass filter and baseline subtraction at the beginning of the data analysis.Subsequently, the time-ordered data were chunked and stacked with the encoder timing information for each chopper frequency.The amplitude of the modulated signal from the stimulator was estimated by fitting the stacked data, as shown in Fig. 7.We used a series of sinusoidal functions: , where θ represents the phase of the chopper, and ϕ n is the phase offset.Finally, the fitted A 1 value was inferred as the chopping amplitude.

Results from Validation Tests
The results of this study were based on data acquired during the early PB-2a commissioning period since the first light in January 2019. 16After installing the stimulator in the telescope, we evaluated the basic performance of the stimulator.In this section, we report the measurement of the signal intensity and the temperature stability of the stimulator, time constant, and gain calibration precision.The evaluated values are listed in Table 2.

Stimulator Signal Intensity
The intensity or effective temperature of the stimulator (T stm ) was obtained by observing the planets with known sky temperatures.During the scans of the planets, the modulation signal from the stimulator was introduced simultaneously.It is obtained by comparing the outputs from the stimulator (I stm ) and the planet through Fig. 7 Example of the measured stimulator signal waveform at 5 Hz.The time-ordered data are chunked and stacked using the encoder timing information.The signal waveform is fitted with a series of sinusoidal functions.The cross-sectional area of the light pipe beam 29 is also plotted as a model.The light pipe cross section and signal waveforms are in good agreement.
Table 2 Summary of the measured characteristics of the stimulator.The signal effective temperate and source temperature fluctuation for one observation satisfy the required value listed in Table 1.The required value for the source temperature fluctuation is below the observed fluctuation without correction; in practice, the source temperature is corrected by monitoring.The source temperature fluctuation for the entire season is greater than the required value under no monitoring temperature conditions.However, in observations, heater temperature will be monitored and calibrated to satisfy fluctuation requirement.
where T p , I p , and Ω p are the brightness temperature, the measured amplitude, and solid angle of the planet, respectively.As a reference, we observed Jupiter, whose brightness temperature is T p ¼ 174.1 K RJ at 143 GHz, and T p ¼ 172.6 K RJ at 100 GHz. 30The telescope beam size (Ω beam ) is calculated with 5.2 and 3.5 arc min (FWHM) for 90 and 150 GHz; 16 hence the planets are regarded as a point source.
Across the focal plane, the calculated effective temperature of the stimulator was 46 AE 8 mK RJ at 90 GHz and 82 AE 16 mK RJ at 150 GHz, with 1σ spread, as shown in Fig. 8. Thus the measured effective temperature meets the requirements (>5 mK) over the entire focal plane.(The signal is more significant than the PB-1 stimulator signal 18 to 43 mK RJ 31 at the 150 GHz band.)However, unknown systematic biases remain in the measured temperature.Large sources may come from the amplitude estimation and spectra of planet sources.For the amplitude estimation, 20% error remains in waveform fitting with conservative evaluation.For the spectra of planet sources, specifically the brightness temperature spectrum of Jupiter varied by 10 K at the maximum within the frequency ranges of 90 and 150 GHz bands, 30 and we assumed the same relative amount error in our sensor responsivity spectrum.In total ∼23% of systematic error was estimated; however, the measured value still fulfilled the requirement, considering the worst assumption.
The stimulator is ideally an unpolarized light source, and its residual polarization should be sufficiently suppressed.After the stimulator was installed in the telescope, a HWP was placed between the stimulator and the receiver temporarily to assess the residual polarization of the stimulator's signal.Consequently, no obvious residual polarization signal of the stimulator above the current noise level was observed.The residual polarization was ≲3% with the upper limit based on the noise level.This aspect should be evaluated in the future after improving the noise level.

Stimulator Signal Fluctuation
Subsequently, we evaluated the response of the bolometer output to changes in the heater temperature near the nominal temperature (750 to 960 K).The change in bolometer electrical output was 0.15 ± 0.01%/K to the change in the heater temperature.Because the shift in the heater temperature typically varies within 3 K during a single scan, the effect on the variation in responsivity of the bolometer output was < 0.45%.This value satisfies the requirement of the temperature fluctuation (< 0.66%).
Figure 9 shows the observed heater temperature.The RMS of the heater temperature for the total season was 1.7 K, and when the seasonal fluctuation was removed, the RMS was 1.0 K.The average day-night modulation was 1.4 K (max.-min.).These values are greater than the required values for the entire season without correction.In the actual observations, the heater temperatures will be calibrated with the monitored values to satisfy the temperature fluctuation requirement.Requirement of 0.03% corresponds to 0.3 K for 960 K heater, and the precision of thermometry is better than this value.
The effect of reflection on the chopper (raised in Sec.4.2) was verified using the data of the different heater voltages, particularly at 0 V.At this voltage, the detector observed a modulation signal between the heater at the ambient temperature and the chopper reflection, and approximately one-tenth of the signal was detected for the nominal voltage.The relative signal was not strongly dependent on the location of the detector, and the long-term fluctuation was confirmed in the discussion above.The magnitude of the 0-V signal can be understood as follows.The reflected light terminates at the environment apparatus, inside the receiver and the sky (the primary mirror).

Validation of Chopper and Encoder
The accuracy of the modulation frequency was typically 0.2%.The intrinsic stability of chopping was of the order of 10 μs (<0.1% relative value compared to highest chopping frequency).The overall spread was smaller than 1 ms in RMS when recorded at 5 Hz, the lowest chopper frequency (corresponding to 0.5% relative frequencies), which is dominated by the manufacturing imperfection of the chopper wheel.The intrinsic sensor jitter was estimated as <1 μs.The chopper frequency and readout timing requirements were confirmed to be satisfactory.

Demonstration of Calibration 7.1 Time Constant
The time constant of the bolometer was measured by changing the chopper frequencies and fitting for the value of the stimulator amplitude at each frequency.The one-pole model Eq. ( 6) used for fitting is shown as the response at frequency AðfÞ ¼ A 0 ½1 þ ð2πτfÞ 2 −1∕2 , where f is the modulation frequency, A 0 is the signal amplitude, and τ is the time constant of the bolometer.An example of the fitting of each bolometer response for each frequency is shown in Fig. 10.
In well-fabricated and properly tuned TESs, the resolution of the time constant δτ achieved ∼0.01 ms for τ ∼ 1 ms (close to the design) case. 32The estimated A 0 was used for the relative gain calculation.

Gain Calibration Precision
Figure 11 presents a scatter plot of the precision of the gain and inverse of the signal-to-noise ratio.Noise is defined as the white noise value converted by the effective temperature of the stimulator's signal.The gain calibration precisions in the commissioning tests were 0.3% (90 GHz) and 0.2% (150 GHz) per observation.The gain calibration accuracy depends on the ratio of the gain to the statistical error obtained by fitting the response to the stimulator signal with the one-pole model, as in Sec.7.1.The gain calibration precision improved as the noise level of the detector decreased, as indicated by Eq. ( 3).Based on the signal fluctuation evaluation test in Sec.6.2, the signal fluctuation was <0.45%.If the noise level is improved to the design value (NET bolo ¼ 360 μK ffiffi s p ), and signal fluctuation is 0.4%, the gain calibration accuracy of 0.4% (90 GHz) and 0.4% (150 GHz) are expected.At sufficiently low noise levels, the fluctuation of the heater source temperature limits the gain calibration precision.If δT stm ∕T stm ¼ 0.1%, the gain calibration precision is expected to be ∼0.1% (90 GHz) and 0.2% (150 GHz).

Conclusion
Calibration of the detector gain is essential for CMB observations.We developed an advanced gain calibration system called a stimulator to perform periodic relative gain calibration of the TES for PB-2a.The system employed a thermal ceramic heater (∼960 K) as a blackbody radiator to cover the entire frequency bands of the observation.In addition, a frequency-independent reflective horn allows simultaneous calibration at multiple frequencies.First, we established the requirements for the instrumental system based on the accuracy of the gain measurements in our experiment.This approach for estimating the requirements can be adapted for other CMB experiments.Early commissioning tests at an observation site in Chile confirmed that our system satisfied these requirements.
The stimulator contributes to the calibration of the detector gain for PB-2a observations.Furthermore, this calibration system applies to next-generation experiments with a larger number of bolometers and different observation frequencies, such as the Simons Observatory. 33,34g. 11 Scatter plot of the ratio of the gain and the gain error (y axis) and inverse of signal-to-noise ratio (x axis) for each bolometer.The dashed vertical line shows the expected SNR.Curves indicate the relationships between SNR and gain precision, and the different colors denote the different assumed stimulator temperature variations.The trends of measured gain precision agree with model used for the estimation.

Appendix A: PB-1 Stimulator
The stimulator that we referred to for the design in PB-2a was the one utilized in the PB-1 experiment.A schematic of the PB-1 stimulator is illustrated in Fig. 12.The basic components are similar to those of PB-2a.
For long-term observations, the heater source was replaced with a silicon nitride source (SUPCO Hot Surface Ignitor, SSN2000; maximum voltage, 120 V), which has a better environmental tolerance.The radiation from the heater was collected by a corrugated conical horn optimized for 150 GHz band.An optical chopper was located next to the horn.The chopper has two blades and comprises an aluminum plate with a foam-type blackbody (Eccosorb ® AN series) applied on the receiver-side face.The chopper was driven by a low-torque stepping motor (Oriental Motor CMK235).
10 Appendix B: Derivation of Values of Requirements 10.1 Intensity-to-Polarization Leakage from Gain Mismatch As mentioned in Sec. 3, gain miscalibrations result in systematic errors in CMB polarization measurements.The errors caused by the difference in gain calibration of two orthogonal detectors Δδ :¼ δg 1 − δg 2 in the power spectra of E-mode and B-mode polarizations are both hðΔδÞ 2 if 1 ⋆C TT l , where f 1 ð0 < f 1 < 1Þ is a parameter that depends on the scanning strategy of observation. 35For example, when Δδ after averaging over detectors and observations is 1% and f 1 ¼ 1, 10 −4 of the power spectrum C TT l originating from CMB temperature anisotropies leaks into both E-mode and B-mode polarizations.To mitigate the leakage from nonpolarized to polarized signal under 10% of the statistical uncertainty in the B-mode power spectrum, the difference in gain calibration Δδ∕2 must be calibrated to be < 0.019% 36 after averaging over detectors and observations.
Assuming no systematic error from the instruments, the uncertainty of the gain for each pixel δg bolo should be smaller than the fractional gain mismatch for a single observation.Here we are assuming 600-times constant elevation scan observations in a year, and the number of pixels N pixel is 3600.From demand δg bolo < 39.5% and Eq. ( 4), the signal intensity of stimulator T stm must be larger than 0.1 mK.In addition, from δg array < 39.5%∕ ffiffiffiffiffiffiffiffiffiffiffi N pixel p and Eq. ( 5), the signal fluctuation δT stm ∕T stm must be < 0.66%.

IR source
Corrugated horn Chopper Fig. 12 Stimulator was used for PB-1.Radiation from IR source passes a corrugated horn, chopper, and a light pipe with a diameter of 1/4 in.A motor and an encoder system are equipped for the chopper.

Monitoring Gain Variation During Observation
To accurately monitor the gain variation during scan observations, the signal fluctuation of the stimulator must be smaller than the gain-change uncertainty, denoted as δg scan .Gain variation during observation can be caused by detector nonlinearity, where gain-change uncertainty can be expressed as δg scan ¼ kδT sky .A temperature coefficient k is known to be k ¼ −1%∕K, and the typical loading variation from sky δT sky is ∼1 K, from the PB-1 experiment data. 37The uncertainty of the measured gain per observation δg bolo must be < δg scan ¼ 1%.According to Eq. ( 5), signal fluctuation δT stm ∕T stm must be < 1%.In addition, from Eq. ( 4), T stm must be >3 mK.

Designed Time Constant Value
We assume a simple case to estimate the time constant uncertainty.The estimated error is defined as δτ ¼ ðdI obs ∕dτÞ −1 δI obs .From Eq. ( 6) and considering the maximum chopper frequency f max , the estimated error of the time constant is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 0 ; 1 1 4 ; 5 7 8 In accordance with Eq. ( 10), when the maximum signal modulation frequency f max is 44 Hz, time constant value τ ¼ 1 ms, and required calibration accuracy δτ < 0.1 ms, the required temperature intensity T stm is higher than 5 mK.

Gain Error by Stimulator < Absolute Gain Error
The calibration accuracy of the gain by the stimulator measurement must be better than the calibration accuracy of the absolute scale determined with CMB power spectrum C TT l .The sample variance of the observable field in the universe limits the ultimate accuracy of the absolute scale: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 1 ; 1 1 4 ; 4 4 2 where N mode and f sky denote the number of modes and the observable sky fraction (over 4π), respectively.The l min and l max in the summation indicate the minimum and maximum multipoles of the analysis window.Based on the relationship described by Eq. ( 11) and assuming f sky ¼ 65% and l max ¼ 2000, δg abs is ∼0.03%.From δg array < g abs ffiffiffiffiffiffiffiffiffi N obs p and Eq. ( 5), signal fluctuation δT stm ∕T stm must be < 0.76%.From demand δg bolo < g abs ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi N obs N pixel p and Eq. ( 5), the required temperature intensity was T stm > 0.1 mK.

Accuracy of the Time Constant Measurement for Chopper Frequency
Accuracy of the time constant depends on f max , as seen in Eq. (10).We request the PB-2a stimulator to be able to cover that of the PB-1 stimulator: 44 Hz.The accuracy also depends on the uncertainty of chopper frequency.The error propagation on τ by the frequency error can be calculated using Eq. ( 6) as δτ∕τ ¼ δf∕f.We require the chopper frequency uncertainty to be at the same level as the statistical uncertainty of the time-constant measurement (1% relative) in the entire chopper operation frequency.

Encoder Timing Readout
To properly use the stimulator modulation timing information, timing synchronization is required between the bolometer and stimulator DAQ systems.Therefore, the timing resolution is chosen to be substantially better than the chopping period at the maximum frequency, to avoid misidentifying one chopper's timing with another.The signal interval between opening and closing of the chopper is t chop ¼ 1∕2f max .Assuming f max ¼ 44 Hz and t chop ≈ 10 ms, the chopper timing resolution should be better than this number; for example, δt chop shall be < 0.1 ms.
11 Appendix C: Application of the Stimulator with Using HWP SA uses a HWP in front of the receiver to mitigate 1/f noise. 37,38In the configuration with the HWP, as two linearly polarized components can be reconstructed from a single detector, other stimulator requirements are needed.Specifically, a new condition must be considered instead of item Sec.10.1 in Appendix B. When linearly polarized light with a polarization angle of ϕ is incident to the rotating HWP at angle θ HWP , the polarization angle of the transmitted light is 2θ HWP − ϕ.The polarization angle of the transmitted light rotates at θ HWP ¼ ω HWP t when the HWP rotates at a speed of ω HWP .Therefore, the amplitude observed by the j'th bolometer (whose sensitive polarization angle is ϕ j ) can be described as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 2 ; 1 1 7 ; 5 0 5 For an ideal HWP, only the polarization components Q and U are modulated and not T.

Optical Leakage from Intensity to 4f Signal
This item is set to calibrate the optical leakage, that is, polarization converted from T component owing to imperfections of the optical elements on the sky side of the HWP, such as the primary mirror of the telescope.In PB-1, the optical leakage δλ was estimated δλ < 0.06%. 37In PB-2a, we assume that δλ is calibrated to be <0.01%.The uncertainty of optical leakage is estimated as δλ ≈ A ð4Þ δg∕ðδT sky ffiffiffiffiffiffiffiffiffi N obs p Þ.Here we assume that the variation in sky temperature is δT sky ∼ 1 K, the intensity of the 4th harmonic term of the HWP synchronous signal (4f signal) is A ð4Þ ∼ 0.1 K and the number of observation scan N obs ¼ 600 as used in Appendix B. Substituting values into the equation, the requirement for the detector gain δg is < 2.5%.From Eq. ( 4), the signal intensity of stimulator T stm must be larger than 1 mK.From Eq. ( 5), signal fluctuation δT stm ∕T stm must be < 2.5%.

Polarization Angle Error from the Time Constants Error
When there is uncertainty in the measurement time constant of the TES bolometer (δτ), it leads to uncertainty in the polarization angle (δθ ≈ 2ω HWP δτ).Uncertainty in the polarization angle induces an incorrect B-mode polarization pattern.To calibrate the polarization angle with accuracy δθ < 0.1 deg, 39 δτ must be calibrated to be δτ < δθ∕ð2ω HWP Þ ≈ 0.07 ms with the HWP rotating at 2 Hz.Subsequently, the signal intensity of the stimulator T stm should be > 7 mK, as derived from Eq. (10).Table 3 summarizes the requirements for the stimulator in applications using the HWP.As presented in Table 2, the stimulator also satisfies the required values for measurements using the HWP.

Fig. 1
Fig. 1 Cross-sectional image of the stimulator on the PB-2a telescope.The figure on the right side provides a zoomed-in view around the stimulator.The stimulator was placed behind the secondary mirror.Its light covers the detectors on the focal plane through a light pipe penetrating a 9-mm diameter hole in the secondary mirror.

Fig. 3
Fig.3Internal optical system of the stimulator.We applied the optical horns (Winston cone horn) at the heater and both sides of the gap for improving optical efficiency.Note that the figure is not to scale.

Fig. 2
Fig. 2 Cross-sectional image of the PB-2a stimulator.Images in two different planes are shown.Units are in millimeters.(a) A plane parallel to the optical axis.(b) A plane perpendicular to the optical axis and including the heater.

Fig. 6
Fig.6Image depicting mounting of the stimulator on the telescope.The stimulator (painted in pink) is mounted on the base plate (dark gray).The base plate is attached to the insulating blocks (blue) and L-shaped angle beam.The L-shaped beam is bolted on the telescope's boom as well as the secondary mirror.

Fig. 5
Fig. 5 Pictures of the stimulator in lab assembly.(a) Top panel and service window are disassembled.(b) A connector panel is removed; in addition, outer and inner heater covers are removed to show the heater structure.

Fig. 8
Fig.8Effective temperature of the stimulator for each bolometer.

Fig. 9
Fig. 9 (a) Observed heater temperature stacked in the 24-h range.The average is indicated using orange markers.The red dashed lines show the width of 0.66% power variation.(b) Temperature history after last heater modification.

Fig. 10
Fig. 10 Demonstration to measure the time constant on a typical TES at the observation site.The time constant is estimated by fitting with the one-pole model of Eq. (6).
Let δg array be the relative array error averaged over all detectors.Under the assumption of independent error in each channel, δg array ¼ δg bolo ∕ Connection diagram of stimulator controller.The controller box was installed inside the co-moving rack of the telescope.Red (double/single) lines show (AC/DC) power cables, black dashed lines show digital signal lines (Ethernet cable and USB cable are included), and green chain lines show (K-type) thermocouple leads.

Table 3
System requirement summary for stimulator using HWP.We assumed the noise level NET bolo ¼ 360 μK ffiffi ffi s p and measurement time t obs ¼ 120 s.Here we assumed the temperature unit as the temperature after calibrating with the temperature of the blackbody spectrum of the CMB.