The polarized cosmic microwave background (CMB) contains valuable information about the composition, evolution, and geometry of our universe [1][2][3] and may hold direct evidence for the theory of inflation. [4][5][6][7] Observations from two all-sky space experiments, the Wilkinson Microwave Anisotropy Probe (WMAP) 8,9 and the Planck satellite, 10,11 have enlightened us with enormous details of the polarized microwave sky in the past two decades. Ground-based experiments, compared to space-borne missions, benefit from more frequent maintenance and upgrades, availability of higher angular resolution due to larger mirrors, and lower costs. These factors have enabled ground-based experiments to make substantial contributions to the field, particularly at intermediate and small angular scales. [12][13][14][15][16][17] However, ground-based observations are challenged by low-frequency 1/f variations associated with atmospheric, instrumental, and calibration drifts. These effects potentially complicate the recovery of large angular scale information from the sky. Polarization modulation is a proven technique to improve long-time instrumental stability. This technique usually involves manipulating the polarization of the sky signal while conserving the intensity and minimizing the mixing of the polarized and unpolarized components. 18 For millimeter and submillimeter experiments, the popular choices for the polarization modulators include the half-wave plate (HWP) and the variable-delay polarization modulator (VPM). 19 The HWP has been adopted in the design of various experiments such as the Atacama B-mode Search experiment (ABS), 20 the Balloon-borne Large Aperture Submillimeter Telescope for Polarimetry (BLASTPol), 21 the E and B Experiment (EBEX), 22 the Large-Scale Polarization Explorer (LSPE), 23 the LiteBIRD satellite, 24 MAXIPOL, 25 the Polarbear experiment (the predecessor of the Simons Array), 26 the Polarimeter fuer bolometer Kameras (PolKa) on Atacama Pathfinder EXperiment (APEX), 27 the Simons Observatory, 28 Spider, 29 etc. The VPM has been adopted in experiments such as Hertz, 30 the Primordial Inflation Polarization Explorer (PIPER), 31 and the Cosmology Large Angular Scale Surveyor (CLASS). 32 The CLASS telescope array (Figure 1) consists of single-frequency-band telescopes centered at 40 GHz and 90 GHz, as well as a dual-band 150/220 GHz telescope. 32,33 Located on Cerro Toco in the Atacama Desert of northern Chile (longitude 67 • W, latitude 23 • S), the 40 (90, 150/220) GHz band telescope has been operating since 2016 (2018, 2019). A second 90 GHz band telescope is to be deployed soon. The CLASS telescopes all adopt a similar diffraction-limited catadioptric optical design, 34 with a front-end polarization modulator as the first optical element. Thus far, the front-end modulator has been a VPM for all the telescopes. The VPM has significantly improved the stability of the observation, 35,36 enabling stable measurements even for the largest angular scales (ℓ < 20). 14,37,38 As the only ground-based experiment operating with VPMs, the CLASS telescopes provide a unique platform for comparing the performance of different types of polarization modulators, a possibility that has been investigated and tested. 39 Building on the initial concept introduced in Eimer et al. ( 2022) 39 (hereafter E22), in this paper, we present the design and in-lab characterization of a new front-end polarization modulator, a reflective half-wave plate (RHWP), for the CLASS 90 GHz band telescope that is in operation. The RHWP has been integrated into the CLASS 90 GHz telescope and started taking data in June 2024.
The structure of the paper is as follows: Section 2 introduces the operational formalism and the design of the RHWP system. Section 3 presents the one-dimensional profilometer we designed to characterize the RHWP properties. In Section 4 we characterize the resonance frequency of the wire array. Section 5 details the alignment of the wire array and the mirror. Results from mechanical tests are presented in Section 6. We summarize in Section 7.
A half-wave plate (HWP) modulator is designed to introduce a half-wavelength (π radians in phase) difference between two orthogonal linear polarized states in a coordinate basis fixed to the wave plate at a particular frequency. The axis along which the linear polarization has the largest phase velocity is called the fast axis, and the orthogonal counterpart is called the slow axis. The effect of an HWP on an incoming linearly polarized signal with polarization angle +α with respect to the fast axis is output by the HWP at angle -α. In effect, the polarization angle has been rotated by 2α. A more complete formalism is described in Section 2.1.
Taking advantage of this effect, the HWP has become a standard tool for polarization modulation. Among many possibilities, 18 common examples include single crystal birefringent plates, 40 multi-layer broad-band wave plate assemblies, 41,42 and meta-material embedded dielectric sheets. 43 Each method of producing a wave plate offers unique features, making them attractive for polarization modulation.
In this work, we have designed an HWP operating in reflection constructed of an array of wires in front of a fixed-distance mirror. This RHWP is notable for its straightforward construction, being suitable for 60 cm (or larger) front-end modulation applications, utilizing an open geometry realized with low emissivity materials, and no required cooling. The reflective design is compatible with commercial rotation drives and encoder readout systems. In this section we describe the operational formalism of the wave plate (Section 2.1) and the mechanical design (Section 2.2) of the RHWP system.
The Mueller matrix of a stationary (laboratory frame) ideal phase retarder with its normal along the z-axis and fast axis along the x-axis can be expressed as:
where ϕ is the phase delay between the fast and slow axis and ϕ = π for a half-wave plate (HWP). For an HWP with the fast axis rotated about the z-axis of the laboratory frame, its Mueller matrix in the laboratory frame is written as:
where
is the rotation matrix and γ is the angle of rotation. Therefore, for an ideal half-wave plate rotating with angular frequency ω (i.e., γ = ωt), the Stokes parameters ( ⃗ S = [I, Q, U, V ] T ) of the outgoing light are related to the incoming ones as:
which implies that ideally, the Stokes parameter I in is unchanged, V in flips its sign (without modulation), and Q in and U in (linear polarization) are modulated at 4× the rotation frequency of the HWP (Figure 2, panel (b)) as:
ν Sky RHWP Receiver (a) (b) 0°3 0°6 0°9 0°0 °60°1 20°1 80°F rom Sky To Receiver Wire Array Mirror θ z Path difference and the data (d(t)) collected by a horizontal/vertical (perpendicular/parallel to the x-axis) detector are:
where
are the Mueller matrices of horizontal and vertical polarizers.
For the CLASS project, one of the main purposes of using the HWP is to compare to the previous polarization modulator, the VPM, 19,44 to better understand the systematics. 39 Therefore, we prefer to retain the telescope's optical design, only replacing the VPM with the new modulator-meaning it needs to be reflective. Due to the success of the VPM in improving long-term stability, 35,36 we continue to use a wire-array-and-mirror design for the HWP. Figure 2 shows the model of the CLASS reflective half-wave plate (RHWP), which consists of a copper-plated tungsten wire array and an aluminum mirror. When linear polarization parallel to the wire direction reaches the RHWP, it is reflected at the wire array plane. In contrast, linear polarization perpendicular The wire array, attached to the outer rim of the support plate, is schematically illustrated with a sheet of thin strips over the mirror which is captured between the back of the support plate and the wire array. The actual wires are cylindrical and would not be visible at this scale. The RHWP and the encoder are coupled to the motor through a shaft. The motor is bolted on the mounting plate which connects to the telescope frame. Panel (b): A detailed view of the holding mechanism for the mirror. Three High Precision Screws (HPSs), mounted on the support plate, push against the v-grooves of the v-blocks, supporting the mirror's weight. They also adjust the mirror tilt and the wire-mirror separation (Section 5). The extension springs hold the v-blocks and the HPS mounting blocks together. The flattening ring is part of the mirror and sets the distance between the wire and the mirror.
to the wire direction passes through the wire array and is reflected at the mirror surface. This path difference between the two orthogonal linear polarization states results in a phase delay. For light with wavelength λ ≫ 2r where r is the wire radius, the geometric approximation of the phase delay can be expressed as:
where θ is the incidence angle, and z is the distance between the wire array and the mirror (or, the wire-mirror separation as in Section 5.2). For the CLASS 90 GHz band telescope, the incidence angle is θ = 22.2 • , 39,44 the band central frequency is 92 GHz, 45 and the phase delay for the band center is π when z = 0.88 mm. The nominal rotation frequency of the RHWP was chosen to be 2.5 Hz, 39 so the signal is modulated at 10 Hz.
Figure 3 provides a cross-sectional view of the final version of the RHWP system. The wire array and the mirror form the polarization modulator, both held by a support plate. A single shaft couples the support plate and the encoder to the motor, which is mounted on an aluminum mounting plate connected to the telescope frame.
In the following subsections, we briefly introduce the design of the main components of the RHWP system and present any major design upgrades since E22. 39
The specifications and manufacturing technique for the wire array are based on the fabrication of the VPM grids for CLASS. 44 The manufacturing technique is adapted from earlier efforts. [46][47][48] The wire array consists of ∼50 µm tungsten wire coated with 2 µm thick copper using a titanium adhesion layer. The nominal spacing is ∼175 µm, chosen to roughly optimize the transmission for the radiation polarized perpendicular to the wires and the reflection for radiation polarization parallel to the wires. 19 The grid was manufactured using a similar technique to the CLASS VPMs, employing a cylindrical mandrel with recessed bars. Grooves for the wires were cut into the bars using a modified CNC machine. The wire is then wrapped over the mandrel; the separation is set by the grooves. Once the wrapping is complete, the wires are epoxied to the grooved bars using Stycast 2850FT (with Catalyst 23 LV), cut, and unspooled from the mandrel. Further details of the wire fabrication specifically for the HWP can be found in E22. 39 After unspooling from the mandrel, the grooved bars were transferred to a stretching frame (Section 4.1) designed to tension the wires according to a desired profile. Finally, the wires were epoxied (Stycast 2850FT with Catalyst 23 LV) to the top surface of the support plate rim and cut from the grooved bars.
The mirror was designed to have three critical features:
1. A flat circular surface with a diameter of ∼ 59 cm to reflect the incoming light. 2. A flattening ring to ensure a uniform distance between the wire array and the flat surface. The flattening ring was designed to have a height of 0.88 mm and a width of 0.125 inch. 3. Dowel holes to locate the v-blocks (Figure 3, panel (b), more description can be found in Section 2.
There are three groups of four non-penetrating slip-fit dowel holes on the back of the mirror.
The mirror also needs to have a relatively low density but be stiff enough to bear the pressure from the wires when pushing the flattening ring against the wire array. We used Aluminum 6061 T6 as the material for the mirror, with a thickness of 0.4 inch. The mass of the mirror is ∼7 kg.
The support plate is also made of aluminum and has three main functions:
1. Provide a surface for the wires to be epoxied on and be stiff enough to not deform much under the tension from the wires. 2. Hold the mirror.
The final version of the support plate design has been simplified from that introduced in E22 39 to reduce lead time and cost. It includes a cylindrical wall on whose top surface the wires are epoxied, and a groove to increase the contact area of the epoxy. The back of the support plate has six windows, three of which are used to mount the high-precision screws (HPSs), and the remaining three windows provide access to monitor the back of the mirror. Three HPSs are used to hold and adjust the mirror position, each mounted on an aluminum block bolted to the support plate. They are evenly distributed on a ring with a 9-inch radius, centering on the support plate. We chose the Newport AJS127-0.5H for its sensitivity (0.56 µm/ • ), the axial load capacity (90 N), and the locking feature. The top of the high-precision screw pushes against the v-grooves of the v-blocks made of AISI 304 stainless steel. The v-blocks have the same dowel hole pattern as those on the back of the mirror and are epoxied (Stycast 2850FT with Catalyst 23 LV) on the back of the mirror, located with dowel pins. The v-blocks and the HPS mounting blocks are held together by extension springs. We used the Lee Spring LEM063B 02 M, for its appropriate size and tension, providing strong enough tension to hold the whole mirror without exceeding the load capacity of the HPSs. The tension provided by the six extension springs at the ideal position is ∼9 kg. The combination of the HPSs, v-blocks, and springs hold the mirror tightly to the support plate and allows for alignment adjustments. A circular indentation on the bottom of the support plate is used for mounting purposes (coupling to the shaft, Section 2.2.4). The mass of the support plate is ∼18 kg.
As in E22, 39 finite element analysis (FEA) simulations for the final RHWP design were conducted to understand the deformation of the support plate and eventually the mirror at different temperatures under the tension from the wire array (Section 4). The deformation scales for the support plate (≲ 80 µm) and the mirror (≲ 30 µm across the whole surface) were similar to the results in E22, 39 meaning that the mirror final surface is likely to be dominated by machining tolerances (expected to be < ±20 µm).
The coupling between the support plate, the encoder, and the motor has been simplified compared to the design in E22. 39 A single shaft couples the support plate and the encoder to the motor. The shaft is made of AISI 304 stainless steel to match the requirement for the coefficient of thermal expansion (CTE) of the encoder bore. The top cylinder of the shaft was used for the concentric alignment between the shaft and the support plate. As a ∼ 40 • C (-20 • C to 20 • C) thermal cycle is expected to happen for these parts, a 4 mil gap in diameter was left to accommodate the CTE mismatch between the support plate and the shaft. A confocal distance sensor (Section 3) was used for the center alignment of the support plate and the shaft, and an off-center of < 5 µm (radially) can be achieved. The mass of the shaft is ∼11 kg.
The servo system consists of a 16-pole Kollmorgen DH062M-12-1310 direct drive AC brushless servo motor, a Heidenhain RCN 2581 angle encoder, and an ABB Microflex e100 servo drive. The servo drive has an electromagnetic interference filter with ∼250 kHz cutoff on its motor power output to reduce the risk of interference from pulse width modulation motor drive currents. The encoder has an accuracy of 2 ′′ and features two encoder readout signals. A serial Endat signal with 28-bit resolution and one-volt peak-to-peak sine/cosine analog signals with 16384 cycles per revolution. The Endat readout is used for recording the position, while the analog signal is sent to the servo drive for feedback in the servo loop. The servo drive's position and velocity loops run at 4 kHz and upsample the encoder signal by a factor of 16 to 262144 counts per revolution, yielding the capability of positioning the RHWP to a resolution of 5 ′′ .
The controlling computer is a PICMG 1.3 single-board computer running the VxWorks real-time operating system mounted in a 13-slot mixed PCI/PCI Express passive backplane. This computer runs its own servo loop at 20 Hz and features two control modes: position and velocity. In the normal velocity mode, the computer simply sends constant velocity jog commands to the servo drive over a TCP/IP connection along with a watchdog signal that is used by the servo drive to cease motion if the connection is lost. It also reads the current position and velocity from the drive for use in the control loop along with various quantities such as bus voltage, motor current, and following error, which are then recorded for diagnostic purposes. Testing has shown that the RHWP maintains a constant rotation frequency of 2.5 Hz to an accuracy of within 0.005% (Section 6).
The controlling computer also records the position of the RHWP. This is done in an interrupt handler, which receives interrupts along with associated 32-bit consecutive serial numbers from a sync box running at ∼201 Hz. The interrupt handler reads the Endat serial position from the encoder using an Addi-Data APCIe-1711 PCI Express card. It also reads a time stamp from a Symmetricom bc637PCI-V2 GPS clock card, which has a resolution of 100 nanoseconds. Both are recorded to disk along with the serial number for each interrupt. The same sync box sends its interrupts and serial numbers to the multi-channel electronics, which record the detector data. The serial numbers are then used to synchronize the detector data with the position and time.
Figure 4 shows the mounting structure of the RHWP system. The mounting plate of the RHWP system is connected to the telescope optics cage through three 3/4-inch diameter threaded rods. The tilt and position of the RHWP assembly can be finely tuned by adjusting the length of the threaded rods. Each threaded rod is attached on a spherical bearing, allowing a maximum angle of 9 • between the rod and the normal of the attachment surface. The other end of the rod is attached to a mounting structure made of an aluminum Tslotted profile, which interfaces with the telescope beams. The pivot in the mounting structure allows additional elevation angle adjustment. The use of a T-slotted frame provides flexibility to adjust the position of the RHWP horizontally and vertically along the direction of the telescope beams. The motor and the RHWP are enclosed by the motor shroud and the RHWP shroud respectively, to separate the moving parts from the external environments.
We employed a one-dimensional profilometer to characterize the RHWP properties (Figure 5). We adapted the XY-gantry design from Harrington et al. (2018) 44 to suit our specific requirements. The sensor mounting block on the profilometer is supported by two 12 mm shafts and one ball screw with a 5 mm pitch, each exceeding 700 mm in effective length. A linear stage facilitates 25 mm of vertical sensor travel. Positioning of the sensor mounting block is achieved using a NEMA 23HS22-2804S stepper motor, controlled by an Arduino UNO R4 MINIMA board and DM254 micro-step driver. The position repeatability was achieved within 0.04 mm. The travel range of the sensor mounting block was limited by two optical endstops (LERDGE Optical Endstop-4001). The profilometer was bolted on the mounting plate using two T-slotted profiles to roughly set the distance between the sensor and the RHWP (Figure 5).
The profilometer was used to analyze the wire resonance frequency, mirror tilt relative to the rotation axis, and wire-mirror separation. Depending on the task, we employed either a microscope with a camera or a confocal distance sensor to scan the top surface of the RHWP. For microscopic inspections, we utilized a setup similar to Harrington et al. (2018): 44 a 10× objective and a 20× wide-field eyepiece provided 200× magnification, coupled with a 2-megapixel USB camera via a macro lens. This configuration provides a field-of-view of approximately 250 µm in diameter and a depth resolution of ∼10 µm, sufficient for measuring the distance between the wire and the mirror (0.88 mm). Alternatively, we employed the IFS2406-10 confocal distance sensor with an IFC2421 controller from MICRO-EPSILON. The confocal sensor offers a 10 mm measuring range and a dynamic resolution of 0.2 µm, suitable for simultaneous measurement of the wire array and mirror. With a 15 µm light spot diameter and a maximum ±13.5 • measuring angle, the sensor is capable of resolving the profile of individual wires (diameter ∼50 µm).
To scan the entire top surface of the RHWP, we integrated the one-dimensional linear motion of the profilometer with the RHWP rotation. The sensor trajectory relative to the RHWP system was determined using a portable coordinate measuring machine (CMM)
The wire resonance frequency must satisfy two constraints: it should exceed the signal band (∼10 Hz) to minimize vibration during RHWP operation, and it should remain low enough to prevent excessive deformation of the support plate due to wire tension. The relation between wire resonance frequency (f res ) and tension (T ) is:
where L is wire length, and ρ is the mass per unit length. Our final choice of the resonance frequency/wire tension profile is one step away from E22 39 In practice, we used a custom-designed stretching frame to adjust the wire array tension and a profilometer with a confocal distance sensor to measure the resonance frequency.
A custom-designed stretching frame was used to tension the wires to the desired profile. Figure 6 shows the model and a picture of the wire array being tensioned by the stretching frame.
The two pulling bars support the wire tension and are coupled to the rails through linear bearings. Two curved bars were used to achieve the desired tension profile (E22 39 and Figure 9). Each curved bar has a flat surface in contact with the pulling bar and a curved surface on the other side. The mandrel bars holding the wire array are attached to the curved bars and conform to the shape of the curved surfaces. The curved surface profile was chosen such that the wire tension profile matches the target as closely as possible when the curved bars are parallel to each other and support a total wire array tension of ∼ 3120 N. Following the coordinate system defined in Figure 6, the profiles of the curved surfaces are y ± = ∓0.0436x 2 + const where x is in inches, y is in mil, and the subscript ± indicates the curved bar in the y > 0 or y < 0 region. The y-difference between the middle and edge for both curved bars is ∼6.8 mil. The tension profile can also be finely adjusted by loosening the bolts that hold the pulling bar assemblies on each side. One of the pull bars is attached to two fixed load cells. The position of the other pull bar is adjustable via two 3/4-inch-diameter bolts.
FEA simulations were used to optimize the pull bar design by minimizing the static deformation of the mandrel bars when the wire array achieves the nominal tension of ∼ 3120 N, because if the mandrel bars deform significantly, the wire array tension profile deviates from the target. Simulation results (Figure 7) show that for the final design, the mandrel bar deformation along the wire direction is less than 0.16 mil at both sides, which is only 2% of the 6.8 mil y-difference target and therefore negligible.
After the wires were stretched to the desired profile by the stretching frame, the support plate was brought into contact with the wire array by three height adjustment bolts beneath it. The wire array was then permanently secured to the rim of the support plate with epoxy (Figure 6).
The method for measuring the wire resonance frequency is straightforward:
1. A speaker was used to excite the wires, with a function generator (Stanford Research DS340) controlling the speaker to perform phase-continuous linear frequency sweeps across a specific frequency range (single direction from low to high). 2. The confocal distance sensor was focused on a single wire, and the confocal distance data were recorded with and without excitation for a certain period.
0 5 10 15 20 Time [s] Distance [arbitrary] 10 1 10 2 10 3 Frequency [Hz] 10 -15 10 -10 10 -5 1 PG [arbitrary] Background Excited f res /2f res -300 -200 -100 0 100 200 300 Position [mm] 100 150 200 250 f res [Hz] optically active targeted measured -300 -200 -100 0 100 200 300 Position [mm] 0.6 0.8 1.0 Tension [N] optically active
Figure 9: Left: the measured resonance frequency profile (blue), which is mostly consistent with the target (black solid with gray region). The optically active (diameter <30 cm) region is marked with gray dashed lines. Right: the similar information for the tension.
3. The periodogram of the distance data was analyzed by comparing the data with and without excitation to identify any significant spikes. The frequency of the lowest significant spike was identified as the resonance frequency of the wire.
During data acquisition, the frequency sweep range changed from position to position. It was chosen such that the lowest frequency was always lower than half of the identified resonance frequency, ensuring that higher harmonics of the fundamental mode were not misidentified as the resonance frequency. The period of the frequency sweep was ∼2-3 seconds depending on the frequency range. A Hann window function was applied when computing the periodogram.
Due to time constraints, tension profile adjustments before epoxying the wires onto the support plate were based on a few (∼6 per iteration) measurements evenly distributed across the wire array. During these measurements, the speaker was placed on the pulling bar closer to the load cells. More detailed measurements were conducted after the RHWP was fully assembled. The left panel of Figure 8 shows the recorded distance of one wire without excitation (background) and with excitation. The wire vibration amplitude increased significantly as the speaker frequency swept across the resonance frequency. The periodogram of the excited case (Figure 8, right panel) shows spikes around 136.5 Hz and 273 Hz; the former is identified as the resonance frequency for that wire.
Figure 9 compares the measured resonance frequency profile (also converted to tension) to the targeted profile, and they are generally consistent with each other. The measured resonance frequency profile is also consistent with the two load cell readings. The load cells were not fully relied upon for adjusting the tension profile due to relatively large calibration uncertainty (∼10%) and coarse spatial resolution.
In this section, we present the in-lab alignment results, including the mirror tilt relative to the rotation axis, and the wire-mirror separation. The in-lab alignment was performed to validate the alignment methods, summarize the procedure, and prepare data pipelines. For safety, the mirror was sitting on the bottom of the support plate during transportation, and the alignment work was redone at the site.
Our current requirement on the mirror tilt relative to the rotation axis (or, the mirror wobbling for short) is based on its impact on the beam: the observed beam map (B obs (θ x , θ y )) is equivalent to the instrumental beam (B inst (θ x , θ y )) convolved with the trajectory of the mirror wobbling (K(θ x , θ y )) as:
where θ wobb is the angle between the mirror's normal direction and the rotation axis. We denote the beam smearing effect as ∆ FHWM , which is the difference between the full-width half maximum (FWHM) of B obs and B inst . Assuming a perfectly flat mirror, the convolution kernel can be expressed as:
where δ (2) is the two-dimension Dirac delta function.
The FWHM of the CLASS 90 GHz telescope is 36 ′ , 49 with a nominal pointing tolerance of 2 ′ . Practically, we can easily achieve θ wobb < 15 ′′ , which corresponds to a negligible ∆ FWHM = 0.6 ′′ .
Other systematic effects, including the T-to-P leakage, require more careful modeling and will be considered in future publications.
We used the confocal distance sensor to map the mirror-sensor distance and estimate θ wobb from it. The measurements were performed as follows:
1. Position the sensor at a desired location with the stepper. 2. Rotate the mirror at a constant speed of 60 • /s while keeping the sensor location fixed. 3. Record the sensor distance and RHWP angle encoder data for one minute (10 periods). 4. Repeat steps 1-3 until data from all sensor locations are collected.
For each sensor location, we will refer to the resulting mean-subtracted data as the "sensor distance residual". The profilometer is not perfectly rigid, and we noticed that the vertical position of the sensor at different locations changes significantly and found it hard to trace. Therefore, we analyzed the sensor distance residual data at each location independently. We obtained the relation between the sensor distance residual and the encoder position by interpolating the sensor distance residual to the encoder timestamps, due to the different sampling rates of the sensor (10 kHz) and the encoder (500 Hz). The RHWP rotated at 60 • /s, and the encoder sampling rate of 500 Hz provides an angular resolution of 7.2 ′ , which is sufficient for this task. Finally, using CMM measurements, we converted the encoder data and the sensor location into mirror x and y positions at each timestamp and mapped the sensor distance residual data. The normal of the best-fit plane of the sensor distance residual data at each radius provides the tilt orientation at that radius, and the angle between it and the z-axis was interpreted as θ wobb for each sensor location. During mirror tilt alignment, we iterate between taking measurements (the tilt orientation and θ wobb ) and adjusting the HPSs until satisfied.
The left panel of Figure 10 visualizes the relationship between the sensor distance residual and orientation for the final in-lab measurements. Some data were excluded from this analysis because when the wires are perpendicular to the direction of the connection between the sensor and the mirror center, the presence of the wire has a non-negligible influence on the sensor reading. We found that the sensor distance residual is dominated by the component with a 180 • period instead of a 360 • one, indicating that the mirror-sensor distance change is dominated by the mirror flatness imperfections rather than tilt. In the right panel of Figure 10, we present the ratio of the sensor distance residual and the rotation radius at each sensor location. Note that this reflects the inclination at different orientations and differs from θ wobb , which characterizes the overall tilt at each radius. The quadrupole pattern is consistent with the curves in the left panel, and the inclination scales at different radii are similar.
Table 1 lists the measurement properties. The uncertainty on the wobble, ∆θ wobb was propagated from the uncertainty in the sensor reading (∆d). We estimated ∆d at each sensor location through iterative fitting. In each iteration, we first fit a model η = A sin(2πξ/T + ξ 0 ) where η is the residual from the previous iteration (starting from the sensor distance residual) and ξ is the orientation. The period T is fixed to be 360 • /i where i is the fitting iteration index. The amplitude A and phase ξ 0 are the fitting parameters. We then subtracted the best-fit model from the current residual and calculated the standard deviation of the new residual. When the fractional difference in the standard deviation between iterations was less than 1%, we stopped iterating and assigned the last standard deviation as ∆d. We found that ∆d = 1 µm for all sensor locations. This is larger than the nominal dynamic uncertainty (0.2 µm) due to the RHWP system's vibration during rotation.
The wire-mirror separation (WMS, denoted with z as in Figure 2) directly affects the modulation efficiency.
-200 0 200 xpos [mm] -200 0 200 ypos [mm] HPS 0 HPS 1 HPS 2 0.00 0.02 0.04 0.06 WMS Residual [mm] Counts Median : 0.048 mm (optically active) -0.06 -0.03 0.03 0.06 [mm] -0.014 mm, which results in a modulation efficiency ϵ = 96.2 -0.4 +0.5 % with the estimated bandpass. 45 If the phase delay changes as ϕ = π + δϕ, the outgoing Stokes parameters are altered as:
where A = cos 4ωt cos 2 δϕ 2 , B = sin 4ωt cos 2 δϕ 2 .
The monochromatic modulation efficiency is given by: 50, 51
In reality, a nonzero δϕ ν (the phase-delay deviation for a light at frequency ν) can result from WMS misalignment and finite detector bandwidth as:
where ν 0 = 92 GHz is the bandcenter for the CLASS 90 GHz telescope, 45 and z 0 = 0.88 mm. Finally, the modulation efficiency across the band is:
where f (ν) is the bandpass.
We used the microscope to map the WMS by measuring the linear stage travel between focusing on the wires and the mirror. We adjusted the WMS using the HPSs based on the measured results until satisfied. The final characterization includes 48 measurements in total. The left panel of Figure 11 maps the WMS residual, and the right panel shows the histogram of the measurements in the optically active region (< 30 cm). 39 The median and 16/84th percentile of the WMS residual are 0.048 +0.013 -0.014 mm, resulting in an estimated modulation efficiency ϵ = 96.2 -0.4 +0.5 % with an estimated bandpass 45 (top-hat profile, 34 GHz width, centered at 92 GHz).
In-lab mechanical tests were performed to assess the rotation stability of the RHWP. The mirror surface was positioned at 26 • from vertical, simulating the nominal orientation for CMB scans at a 45 • elevation and a 0 • boresight angle. The motor was set to rotate at 2.5 Hz (900 • /s). Figure 12 shows results from a 2-minute operation. The RHWP angular velocity remained stable, with a fractional standard deviation of less than 0.005%. The power spectral density (PSD) of the angular velocity over the 2-minute rotation period showed no significant features near the signal band (∼10 Hz).
0 50 100 150 200 250 Frequency [Hz] 0.00 0.01 0.02 PSD [arbitrary] 0 10 20 10 -9 10 -5
Polarization modulation is an effective method for achieving long-time instrumental stability, critical for measuring the large angular scales of the CMB polarization. This work presents the design and in-lab performance test results of the first RHWP for the CLASS telescopes.
The design of the support plate and the coupling of RHWP and the drive system has been simplified compared to E22. 39 The mounting structure allows for adjustments in the orientation and the position of the RHWP system. A one-dimensional profilometer was used to characterize the RHWP properties, and a custom-designed stretching frame was employed to tension the wires to the desired profile.
For the in-lab test results, the wire frequency measurements closely match the target. The mirror tilt relative to the rotation axis was controlled to be within 15 ′′ at multiple radii, which corresponds to a negligible 0.6 ′′ beam smearing amplitude. The median and 16/84th percentile of the wire-mirror separation residual was maintained at 0.048 +0.013 -0.014 mm in the optically active region, which results in a modulation efficiency ϵ = 96.2 -0.4 +0.5 % with an estimated bandpass of 34 GHz. 45 Finally, the RHWP sustained a constant rotation frequency of 2.5 Hz to an accuracy of within 0.005%.
The RHWP has been successfully integrated into the existing 90 GHz telescope and started collecting data in June 2024. In addition to cosmological analyses, the data with the RHWP in operation will also be used to address sources of systematic errors, thereby enhancing the telescope's design and optimizing its performance to better achieve the scientific objectives.