2.1.

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 $\sim 1000\mathrm{K}$ with a chopper of $\sim 300\mathrm{K}$ 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.

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.

The gain measured with the stimulator is calculated as $G:=R_{\det }(P_{\mathrm{stm}}/T_{\mathrm{stm}})=I_{\mathrm{stm}}/T_{\mathrm{stm}}$, where $I_{\mathrm{stm}}$ is the signal amplitude of the output current with the stimulator, and $T_{\mathrm{stm}}$ is the effective temperature of the stimulator. $T_{\mathrm{stm}}$ 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 $T_{\mathrm{stm}}$ 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 $T_{\mathrm{stm}}$ to the sky optical temperature, we use the best fit of the WMAP-9 and the Planck power spectrum of CMB temperature anisotropies $C_{\ell ,\mathrm{ref}}^{\mathrm{TT}}$^24,25 such that the observed anisotropies $C_{\ell ,\mathrm{obs}}^{\mathrm{TT}}$ match with reference $C_{\ell ,\mathrm{ref}}^{\mathrm{TT}}$, and this paper is not primarily focused on the detailed method used to obtain the absolute gain for the subsequent physical analysis.

2.2.

Gain Measurement with Stimulator

The electrical output for the stimulator signal is written as $I_{\mathrm{obs}}=G·T_{\mathrm{stm}}+I_{\mathrm{noise}}$. Here $I_{\mathrm{noise}}$ 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 $\delta T_{\mathrm{stm}}$. The noise of the electrical output can be written as $\delta I_{\mathrm{obs}}=G·{\mathrm{NET}}_{\mathrm{bolo}}/\sqrt{t_{\mathrm{obs}}}$, where ${\mathrm{NET}}_{\mathrm{bolo}}$ represents the noise equivalent temperature of the bolometer, and $t_{\mathrm{obs}}$ is the time spent on calibration. Therefore, the gain error is expressed as follows:

Here $\delta T_{\mathrm{stm}}/T_{\mathrm{stm}}$ denotes the stimulator signal fluctuation. When $\delta T_{\mathrm{stm}}/T_{\mathrm{stm}}$ can be regarded sufficiently small, the following equation holds:

Let $\delta g_{\mathrm{array}}$ be the relative array error averaged over all detectors. Under the assumption of independent error in each channel, $\delta g_{\mathrm{array}}=\delta g_{\mathrm{bolo}}/\sqrt{N_{\mathrm{bolo}}}$ stands, where $N_{\mathrm{bolo}}$ is the number of bolometers. When the array noise ${\mathrm{NET}}_{\mathrm{array}}\approx {\mathrm{NET}}_{\mathrm{bolo}}/\sqrt{N_{\mathrm{bolo}}}$ is sufficiently small, $\delta g_{\mathrm{array}}$ can be written as follows:

2.3.

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 $\tau$ (s) agrees well with the one-pole response function. It is written as follows:

4.1.

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, $25\times 25{\mathrm{mm}}^2$ 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 $T_{\mathrm{stm}}$ was approximated as $(1/2)(T_{\mathrm{heater}}-T_{\mathrm{chop}})(A_{\mathrm{pipe}}/A_{\mathrm{beam}})$, where $T_{\mathrm{heater}}$ and $T_{\mathrm{chop}}$ are a temperature of the heater surface and the chopper wheel, respectively. $A_{\mathrm{pipe}}$ and $A_{\mathrm{beam}}$ correspond to the area of the light pipe and the beam spot size on the secondary mirror, respectively.

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.

Substituting them with the designed values, the temperature $T_{\mathrm{stm}}$ was approximated as $46{\mathrm{mK}}_{\mathrm{RJ}}$. Here ${\mathrm{K}}_{\mathrm{RJ}}$ 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 $\sim 31\mathrm{W}$, and the temperature was $\sim 960\mathrm{K}$ 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.

4.2.

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.

Fig. 3

Internal 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.

4.3.

Modulation System

We adopted a rotation wheel chopper with a diameter of 160 mm, which was made of a 1-mm-thick 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; $4\mu \mathrm{s}$ 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.”

4.4.

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 signal²⁷ and 16 MHz Arduino internal clock. (5) To distribute stimulator slowdaq information to the site Ethernet LAN.

Fig. 4

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.

4.5.

Mechanical Design

The stimulator was assembled into a cubic casing of $\sim 20\mathrm{cm}$. 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.

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. 6

Image 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.

5.1.

Hardware Operation

The sequence of stimulator calibration was controlled by the SA main control system.²⁸ The 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.

5.2.

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: $A(\theta )=\sum _{n=1}^7{A}_n\sin (n\theta +{\varphi }_n)\hspace{0.17em}[0\leqq \theta <2\pi ]$, where $\theta$ represents the phase of the chopper, and ${\varphi }_n$ is the phase offset. Finally, the fitted $A_1$ value was inferred as the chopping amplitude.

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 beam²⁹ is also plotted as a model. The light pipe cross section and signal waveforms are in good agreement.

6.1.

Stimulator Signal Intensity

The intensity or effective temperature of the stimulator ($T_{\mathrm{stm}}$) 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 ($I_{\mathrm{stm}}$) and the planet through

Across the focal plane, the calculated effective temperature of the stimulator was $46\pm 8{\mathrm{mK}}_{\mathrm{RJ}}$ at 90 GHz and $82\pm 16{\mathrm{mK}}_{\mathrm{RJ}}$ at 150 GHz, with $1\sigma$ spread, as shown in Fig. 8. Thus the measured effective temperature meets the requirements ($>5\mathrm{mK}$) over the entire focal plane. (The signal is more significant than the PB-1 stimulator signal 18 to $43{\mathrm{mK}}_{\mathrm{RJ}}$³¹ at the 150 GHz band.)

Fig. 8

Effective temperature of the stimulator for each bolometer.

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 $\sim 23\%$ 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 $\lesssim 3\%$ with the upper limit based on the noise level. This aspect should be evaluated in the future after improving the noise level.

6.2.

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.

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.

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).

6.3.

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 $10\mu \mathrm{s}$ ($<0.1\%$ 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 $<1\mu \mathrm{s}$. The chopper frequency and readout timing requirements were confirmed to be satisfactory.

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 $A(f)=A_0[1+(2\pi \tau f{)}^2{]}^{-1/2}$, where $f$ is the modulation frequency, $A_0$ is the signal amplitude, and $\tau$ is the time constant of the bolometer. An example of the fitting of each bolometer response for each frequency is shown in 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).

In well-fabricated and properly tuned TESs, the resolution of the time constant $\delta \tau$ achieved $\sim 0.01\mathrm{ms}$ for $\tau \sim 1\mathrm{ms}$ (close to the design) case.³² The estimated $A_0$ was used for the relative gain calculation.

7.2.

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 (${\mathrm{NET}}_{\mathrm{bolo}}=360\mu \mathrm{K}\sqrt{\mathrm{s}}$), 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 $\delta T_{\mathrm{stm}}/T_{\mathrm{stm}}=0.1\%$, the gain calibration precision is expected to be $\sim 0.1\%$ (90 GHz) and 0.2% (150 GHz).

Fig. 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.

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 $\mathrm{\Delta }\delta :=\delta g_1-\delta g_2$ in the power spectra of $E$-mode and $B$-mode polarizations are both $⟨(\mathrm{\Delta }\delta {)}^2⟩f_1\star C_{\ell }^{\mathrm{TT}}$, where $f_1(0<f_1<1)$ is a parameter that depends on the scanning strategy of observation.³⁵ For example, when $\mathrm{\Delta }\delta$ after averaging over detectors and observations is 1% and $f_1=1$, ${10}^{-4}$ of the power spectrum $C_{\ell }^{\mathrm{TT}}$ 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 $\mathrm{\Delta }\delta /2$ 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 $\delta g_{\mathrm{bolo}}$ 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 $N_{\mathrm{pixel}}$ is 3600. From demand $\delta g_{\mathrm{bolo}}<39.5\%$ and Eq. (4), the signal intensity of stimulator $T_{\mathrm{stm}}$ must be larger than $0.1\hspace{0.17em}\mathrm{mK}$. In addition, from $\delta g_{\mathrm{array}}<39.5\%/\sqrt{N_{\mathrm{pixel}}}$ and Eq. (5), the signal fluctuation $\delta T_{\mathrm{stm}}/T_{\mathrm{stm}}$ must be $<0.66\%$.

10.2.

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 $\delta g_{\mathrm{scan}}$. Gain variation during observation can be caused by detector nonlinearity, where gain-change uncertainty can be expressed as $\delta g_{\mathrm{scan}}=k\delta T_{\mathrm{sky}}$. A temperature coefficient $k$ is known to be $k=-1\%/\mathrm{K}$, and the typical loading variation from sky $\delta T_{\mathrm{sky}}$ is $\sim 1\mathrm{K}$, from the PB-1 experiment data.³⁷ The uncertainty of the measured gain per observation $\delta g_{\mathrm{bolo}}$ must be $<\delta g_{\mathrm{scan}}=1\%$. According to Eq. (5), signal fluctuation $\delta T_{\mathrm{stm}}/T_{\mathrm{stm}}$ must be $<1\%$. In addition, from Eq. (4), $\hspace{0.17em}{T}_{\mathrm{stm}}$ must be >$3\mathrm{mK}$.

10.3.

Designed Time Constant Value

We assume a simple case to estimate the time constant uncertainty. The estimated error is defined as $\delta \tau =(d{I}_{\mathrm{obs}}/d\tau {)}^{-1}\delta I_{\mathrm{obs}}$. From Eq. (6) and considering the maximum chopper frequency $f_{\max }$, the estimated error of the time constant is

In accordance with Eq. (10), when the maximum signal modulation frequency $f_{\max }$ is 44 Hz, time constant value $\tau =1\mathrm{ms}$, and required calibration accuracy $\delta \tau <0.1\hspace{0.17em}\mathrm{ms}$, the required temperature intensity $T_{\mathrm{stm}}$ is higher than $5\hspace{0.17em}\mathrm{mK}$.

10.4.

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 $C_{\ell }^{\mathrm{TT}}$. The sample variance of the observable field in the universe limits the ultimate accuracy of the absolute scale:

Based on the relationship described by Eq. (11) and assuming $f_{\mathrm{sky}}=65\%$ and ${\ell }_{\max }=2000$, $\delta g_{\mathrm{abs}}$ is $\sim 0.03\%$. From $\delta g_{\mathrm{array}}<g_{\mathrm{abs}}\sqrt{N_{\mathrm{obs}}}$ and Eq. (5), signal fluctuation $\delta T_{\mathrm{stm}}/T_{\mathrm{stm}}$ must be $<0.76\%$. From demand $\delta g_{\mathrm{bolo}}<g_{\mathrm{abs}}\sqrt{N_{\mathrm{obs}}{N}_{\mathrm{pixel}}}$ and Eq. (5), the required temperature intensity was $T_{\mathrm{stm}}>0.1\mathrm{mK}$.

10.5.

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 $\tau$ by the frequency error can be calculated using Eq. (6) as $\delta \tau /\tau =\delta f/f$. 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.

10.6.

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 $t_{\mathrm{chop}}=1/2f_{\max }$. Assuming $f_{\max }=44\mathrm{Hz}$ and $t_{\mathrm{chop}}\approx 10\mathrm{ms}$, the chopper timing resolution should be better than this number; for example, $\delta t_{\mathrm{chop}}$ shall be $<0.1\mathrm{ms}$.

11.1.

Optical Leakage from Intensity to $\mathbf{4}f$ 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 $\delta \lambda$ was estimated $\delta \lambda <0.06\%$.³⁷ In PB-2a, we assume that $\delta \lambda$ is calibrated to be <0.01%. The uncertainty of optical leakage is estimated as $\delta \lambda \approx A^{(4)}\delta g/(\delta T_{\mathrm{sky}}\sqrt{N_{\mathrm{obs}}})$. Here we assume that the variation in sky temperature is $\delta T_{\mathrm{sky}}\sim 1\hspace{0.17em}\mathrm{K}$, the intensity of the 4th harmonic term of the HWP synchronous signal ($4f$ signal) is $A^{(4)}\sim 0.1\hspace{0.17em}\mathrm{K}$ and the number of observation scan $N_{\mathrm{obs}}=600$ as used in Appendix B. Substituting values into the equation, the requirement for the detector gain $\delta g$ is $<2.5\%$. From Eq. (4), the signal intensity of stimulator $T_{\mathrm{stm}}$ must be larger than 1 mK. From Eq. (5), signal fluctuation $\delta T_{\mathrm{stm}}/T_{\mathrm{stm}}$ must be $<2.5\%$.

11.2.

Polarization Angle Error from the Time Constants Error

When there is uncertainty in the measurement time constant of the TES bolometer ($\delta \tau$), it leads to uncertainty in the polarization angle ($\delta \theta \approx 2{\omega }_{\mathrm{HWP}}\delta \tau$). Uncertainty in the polarization angle induces an incorrect $B$-mode polarization pattern. To calibrate the polarization angle with accuracy $\delta \theta <0.1\mathrm{deg}$,³⁹ $\delta \tau$ must be calibrated to be $\delta \tau <\delta \theta /(2{\omega }_{\mathrm{HWP}})\approx 0.07\mathrm{ms}$ with the HWP rotating at 2 Hz. Subsequently, the signal intensity of the stimulator $T_{\mathrm{stm}}$ should be $>7\mathrm{mK}$, 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.

Table 3

System requirement summary for stimulator using HWP. We assumed the noise level NETbolo=360  μKs and measurement time tobs=120  s. Here we assumed the temperature unit as the temperature after calibrating with the temperature of the blackbody spectrum of the CMB.