THE Q/U IMAGING EXPERIMENT INSTRUMENT

The telescope design requirements include: a wide field of view, excellent polarization characteristics, minimal beam distortion, minimal instrumental polarization, minimal spillover, and low sidelobes that could otherwise generate spurious polarization. The latter requirements have often been met by CMB experiments by using either classical, dual offset Cassegrain antennas (e.g., Barkats et al. 2005), Gregorian antennas (e.g., Meinhold et al. 1993), or shaped reflectors (e.g., Page et al. 2003b). QUIET is the first CMB polarization experiment to take advantage of the wide field of view enabled by a classical Dragonian antenna (Imbriale et al. 2011). An additional advantage of the classical Dragonian antenna is that it satisfies the Mizuguchi condition (Mizugutch et al. 1976), which, when combined with the very low cross-polar characteristics of the conical corrugated feed horns, yields very low antenna contribution to the instrumental polarization. As pointed out by Chang & Prata (2004), a classical Dragonian antenna affords two natural geometries, a front-fed design and a side-fed (or crossed) design. QUIET uses the side-fed design because it allows for the use of a larger cryostat, and hence focal plane array, without obstructing the beam.

3.1.1. Telescope Design

The design of the reflectors follows the procedure outlined by Chang & Prata (2004) and is augmented with a physical optics program (Imbriale & Hodges 1991) to predict beam patterns. This procedure relies on the specification of the first five design parameters given in the top half of Table 3 and shown in Figure 3. Once these parameters are specified, a number of other useful parameters can be calculated including the MR focal length and the SR eccentricity, and these are listed in the lower half of Table 3. The actual MR circular diameter was decreased slightly to 1400 mm, as noted by the actual value in Table 3, in order to avoid beam blockage from the SR. Similarly, the actual SR circular diameter was increased slightly, also to 1400 mm, and this resulted in an increased value of the actual SR edge angle given by 20° in Table 3. The oversized SR reduces feed spillover for the horns on the edge of the array. The design values (not the actual values) shown in the top half of Table 3 were used to establish the calculated values shown in the lower half of Table 3.

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Table 3. Telescope Design Parameters

Description, Parameter	Design/Actual Value
MR circular aperture diameter, D	1470/1400 mm
SR edge ∠, θe	17°/20°
MR-SR separation, ℓ	1270 mm
MR offset ∠, θ0	−53°
∠ between MR and horn axes, θp	−90°
	Calculated Value
MR focal length, F	4904.1 mm
SR eccentricity, e	2.244
∠ between SR and MR axis, β	−6337
SR interfocal distance, 2c	6516.1 mm
MR offset distance, d0	4890.2 mm

Notes. The design values in the top half of the table were used to establish the calculated values in the lower half of the table. For the first two parameters, the actual values listed supersede the design values for the purpose of fabrication. Negative angles are measured clockwise with respect to the vertical axis shown in Figure 3.

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3.1.2. Telescope Fabrication and Alignment

The requirements for the feed horns include high beam symmetry, efficiency, gain, and bandwidth, as well as low sidelobes and cross-polarization. These requirements are satisfied by conical, corrugated feed horns (Kay 1962; Clarricoats & Olver 1984). Standard production techniques for corrugated feed horns (e.g., computer-numerically-controlled lathe machining and electroforming) are prohibitively costly for the large number of feeds for the W-band array. A lower-cost option is described in the next subsection.

3.2.1. Platelet Array Design

A 91-element W-band and a 19-element Q-band platelet array of hexagonally-packed, conical, corrugated feed horns were designed for QUIET (Gundersen & Wollack 2009; Imbriale et al. 2011). Each array is machined from aluminum 6061-T6 and consists of a number of thin platelets each with a single corrugation, a number of thick plates each with multiple corrugations, and a base plate. The assembly of platelets and plates is then diffusion bonded together. Table 4 provides the parameters of each array.

Table 4. Platelet Array Design Parameters

Frequency	No. of	L × W × H	Mass	Aperture	Throat	No. of	Horn	Semi-flare
(band/GHz)	Feeds	(mm × mm × mm)	(kg)	Diameter	Diameter	Grooves	Separation	Angle
				(mm)	(mm)		(mm)	(degrees)
Q/39–47	19	281.7 × 427.3 × 370.1	43.7	71.78	6.69	104	76.20	76
W/89–100	91	129.1 × 427.8 × 370.5	20.6	31.62	2.97	103	35.56	76

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Due to the side-fed geometry of the telescope, the feed horns must have relatively high gain (≃ 27 to 28 dB) in order to provide a low edge taper of ⩽ − 30 dB for both the Q- and W-band systems. This dictates the aperture size of the feed horns and hence the horn-to-horn spacing. For the W-band horns, this spacing is commensurate with the size of the modules. Most of the dimensions of the Q-band horns are scaled by the ratio of the frequencies (∼ 90/40 = 2.25), which results in a Q-band horn spacing that is larger than the Q-band modules. These horn spacings give rise to angular separations of 175 (082) between adjacent beams in the Q (W) systems and result in fields of view of 70 and 82 for the Q and W systems, respectively.

The number of corrugations is fixed at three per wavelength for each horn and a semi-flare angle of 76 is chosen using a design procedure that ensures both acceptable cross-polar levels and return loss (Hoppe 1987, 1988). This optimization procedure also adjusts the depth of the first six corrugations of each horn in order to reduce the predicted reflection coefficient to better than −32 dB over the full anticipated band of operation.

3.2.2. Platelet Array Testing

A vector network analyzer (VNA) was used to measure the return loss of each horn in each array. Each measurement consisted of attaching one horn in a platelet array to one port of the VNA using a commercially available circular-to-rectangular transition. A sheet of microwave absorber was placed at 45° in front of the horns at a distance of ≃ 1 m. The return losses for five of the 19 Q-band horns are shown in Figure 4 and are similar for the W-band feed horns. Maximum reflection strengths (negative return loss) are listed in Table 5. For comparison, individual electroformed horns that are identical in design to the Q- and W-band horns were fabricated. The array values in Table 5 are comparable to but not quite as good as the electroformed horns or the theoretical predictions, both of which were < − 30 dB across the band.

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Table 5. Measured Platelet Array Performance

Frequency	FWHM	Gain	Crosspol	Reflection	Insertion
(band/GHz)	(deg)	(dB)	E/H	Strength	Loss
			(dB)	(dB)	(dB)
Q/39–47	8.3–6.9	27.2–28.5	< − 34/ − 29	< − 25	< − 0.1
W/89–100	8.3–7.4	27.1–28.0	< − 31/ − 29	< − 24	< − 0.1

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Beam patterns were measured for all 91 horns in the W-band array and 13 out of 19 horns in the Q-band array. A synthesizer combined with × 3 and × 6 multipliers generated the source signals at 40 and 90 GHz, respectively. A standard gain horn was used as a source antenna. The platelet arrays were mounted on an azimuth-elevation mount so that the source was in the far-field of the platelet array horns. The source signals were modulated at 1 kHz and a lock-in amplifier connected to a detector diode on the platelet array detected the signal. A co-aligned alignment laser ensured that the source horn and platelet array horn were parallel and axially aligned to each other. A digital protractor with an accuracy of ±005 ensured that the source and receiver horn's polarization axes were coincident for the copolar patterns or perpendicular to each other for the cross-polar patterns. Several measurements were made on each horn including E- and H-plane copolar patterns as well as their corresponding cross-polar patterns. The patterns were taken by keeping the source horn static and rotating the platelet array horn in azimuth about a vertical axis that intersected the horns phase center. A detailed description of this procedure is given in Clarricoats & Olver (1984).

The beam patterns of typical Q-band and W-band horns are shown in Figure 5. This figure shows both E- and H-plane copolar patterns as well as cross-polar patterns for the platelet feeds and for an electroformed feed with identical design parameters. The figure also shows the theoretical model responses. In all cases the E- and H-plane copolar patterns are consistent with both the model and the electroformed feed measurements out to the −30 dB level. Upper limits of −34 (−31) dB are set on the E-plane cross-polar levels for Q-band (W-band). The H-plane cross-polar patterns are not in as good agreement with the model, which predicts both E- and H-plane cross-polar levels at the < − 40 dB level. The largest discrepancies are similar in shape to the Q-band H-plane cross-polar measurements shown in Figure 5 and have a non-null cross-polar boresight response. This type of response is typical of angular misalignment between the source and receiver probes, and the level of the response is consistent with the precision of the digital level. The W-band H-plane cross-polar response does have a null on boresight and is likely the true cross-polar response. The fact that the cross-polar responses of the individual feed horns in the platelet array are consistently higher than the corresponding electroformed horns' responses suggests that either the machining or the diffusion bonding process leads to somewhat compromised performance. However, none of the measured feeds has cross-polar levels > − 29 dB. Table 5 summarizes the results of the beam pattern measurements.

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Upper limits on the insertion loss were obtained during the return loss measurements of both the W-band and Q-band platelet arrays by placing a flat aluminum plate in front of the horn and generating an effective short. In both cases the measured reflection strength allows a lower bound to be set on the feeds' room temperature transmission efficiency of >99%. Assuming solely ohmic losses, this transmission efficiency is expected to increase to >99.5% upon cooling to 25 K as the electrical resistivity of the horns decreases with temperature (Clark et al. 1970).

An absorbing, comoving ground screen is employed to shield the instrument from varying ground and Sun pick-up and provide a stable, essentially unpolarized emission source that does not vary during a telescope scan. The ground screen structure (Figure 1(b)) consists of two parts: the lower ground screen is an aluminum box that encloses both reflectors and the front half of the cryostat; the upper ground screen (UGS) is a cylindrical tube that attaches to the lower ground screen directly above the MR. The external surface of the ground screen is coated in white paint in order to reduce diurnal temperature variations and to minimize radiative loading. Following the approach used by the BICEP experiment (Takahashi et al. 2010), the interior of the ground screen is coated with a broadband absorber³² that absorbs radiation and re-emits it at a constant temperature, allowing the ground screen to function as an approximately constant Rayleigh–Jeans source in both Q- and W-bands. The UGS was installed in 2010 January (a third of the way through the W-band season), and it was particularly useful in eliminating spillover past the SR, as shown in Sections 3.5.2 and 3.5.3.

The main beam profiles are primarily determined from observations of Jupiter. Additional observations of Taurus A (hereafter Tau A) are performed to check the main lobe response, to measure the polarized responsivity, to determine the polarization angles, and to characterize instrumental polarization. Tau A and Jupiter are used for main beam characterization since they are, respectively, the brightest polarized and unpolarized, compact sources in the sky. Figure 6 shows beam patterns of Jupiter for a differential-temperature assembly in each of the Q- and W-band arrays. These measurements are consistent with the lower signal-to-noise main beam profiles measured using Tau A once the slightly different instrumental bandpasses, source spectra, and positions in the focal plane are taken into account. The main beam is used to compute the main beam solid angle Ω_B, the main beam forward gain, G_m = 4π/Ω_B, and the telescope sensitivity,

$$\begin{equation} \Gamma =\frac{10^{-20}c^2}{2k_{\mathrm{B}}\nu _{\mathrm{e}}^2\Omega _{\mathrm{B}}}\; \mu \mathrm{K\; Jy}^{-1} \end{equation} \tag{ 1 }$$

in terms of the effective frequency,

$$\begin{equation} \nu _{\mathrm{e}}=\frac{\int \nu f(\nu)\sigma (\nu)d\nu }{\int f(\nu)\sigma (\nu)d\nu } \end{equation} \tag{ 2 }$$

for a given instrumental bandpass f(ν) and source spectrum σ(ν). Equations (1) and (2) explicitly assume G_m∝ν². The source spectra of Tau A and Jupiter are based on the Wilkinson Microwave Anisotropy Probe (WMAP) measurements (Weiland et al. 2011). A Tau A source spectrum with σ∝ν^−0.302 is assumed for the calculation of the effective frequency for the Tau A measurements. An empirical fit to WMAP's measurements of Jupiter's brightness temperatures yields a source spectrum of the form

$$\begin{eqnarray} \sigma (\nu _{\mathrm{GHz}})&=&\frac{2k_{\mathrm{B}}\nu _{\mathrm{GHz}}^2}{c^2}\left(96.98+2.175\nu _{\mathrm{GHz}} -2.219\times 10^{-2}\nu _{\mathrm{GHz}}^2\right. \nonumber\\ && \left. +8.217\times 10^{-5}\nu _{\mathrm{GHz}}^3 \right). \end{eqnarray} \tag{ 3 }$$

Similarly a source spectrum of the form

$$\begin{equation} \sigma (\nu)\propto \nu ^4e^x(e^x-1)^{-2} \end{equation} \tag{ 4 }$$

is used to compute the effective frequency for unresolved CMB fluctuations, where x = hν/k_BT_CMB.

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Table 6 provides a summary of the mean values of these quantities for the Q-band and W-band polarization and differential-temperature modules for a source spectrum of the form given in Equation (4). The Q-band total power values are for the lone Q-band differential-temperature assembly, while the Q-band polarization values are for the central pixel, which is typical for the array. Both the W-band total power and polarization values shown in Table 6 are averaged over the respective differential-temperature and polarization array elements using an inverse-variance weighting.

The shape of the main beam and its uncertainties are used to compute the instrumental window function and its associated uncertainties (Monsalve 2010). Initially, an arbitrarily oriented, two-dimensional, elliptical Gaussian beam is fit to the data shown in Figure 6. If σ_a and σ_b represent the beam widths of the semi-major and semi-minor axes of the elliptical Gaussian (with σ_a ⩾ σ_b), then the elongation is defined by = (σ_a − σ_b)/(σ_a + σ_b). Typical elongations were found to be <0.02 and averaged about 0.01. This low elongation, and the fact that the CMB scans use a combination of natural sky rotation and deck angle rotation, imply that the beams are well described by an axially-symmetric beam. The symmetrized beam is expressed as a one-dimensional Hermite expansion (Monsalve 2010), and this expansion is used to compute the beam transfer function and its covariance matrix (Page et al. 2003a).

Two different methods are used to measure sidelobes. These included pre-deployment antenna range measurements and in situ measurements of a bright, near-field source. In addition, unintentional measurements of the Sun in the sidelobes also enabled their characterization. These three measurements and their results are discussed in more detail here.

3.5.1. Antenna Range Measurements of Sidelobes

The telescope was installed on the Jet Propulsion Laboratory's Mesa Antenna Measurement Facility for measurements of both the main lobe and far sidelobes at both 40 and 90 GHz. The telescope was mounted on an elevation-over-azimuth positioner with 4'' pointing accuracy. Individual electroformed versions of the Q- and W-band horns, described in Section 3.2.2, were used for the range measurements. The range measurements were conducted before the ground screens were fabricated, so the sidelobe results are only appropriate for the telescope in its bare configuration. The measurements made use of the facility's Scientific Atlanta model 1797 heterodyne receiver system, which enabled repeatable measurements down to −90 dB of the peak power level. A combination of a source synthesizer, multiplier and amplifier was used to generate ≃ 100–200 mW of power at each frequency. The sources were separately connected to corrugated feeds at the focus of a small Cassegrain antenna at a distance of 914 m from the telescope. Due to limitations of the mount, only a simple principal plane cut within ±90° of the telescope boresight (in the plane shown in Figure 3) was performed for a number of arrangements of the source/receiver antennas. These arrangements included moving the receiver horn to a few positions in the focal plane and rotating the source and receiver horns for both E- and H-plane cuts.

The results for one feed horn position for each of the Q- and W-band arrays are shown in Figure 7. In each case the feed horn position that was tested corresponds to the top row of the respective platelet array, furthest from the MR and directly above the central feed horn. Cross-polar measurements were not made on the antenna range since they are made during routine calibrations. The main lobe beam sizes compare well with initial theoretical predictions (Imbriale et al. 2011); however, the near-in (i.e., within ±5° of the main lobe) sidelobe levels do not. It is also shown in that paper that the predicted envelope of near-in sidelobes matches well with the observations once the measured reflector surface imperfections are included. The surface imperfections caused the near-in sidelobe levels to increase by as much as 15 dB in some regions.

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The two dominant far sidelobes are the SR spillover lobe and the "triple reflection" lobe, both predicted in Imbriale et al. (2011). The SR spillover lobe is broad, arises from direct coupling into the feed horn, and is located ∼70° from boresight. The triple reflection lobe is due to an additional reflection off the SR (as indicated in Figure 9) and it is located ≃ 50° from boresight in the opposite direction from the SR spillover lobe. The amplitudes of each lobe for the W-band case are −60 to −62 dB, while they are −58 to −59 dB for the Q-band measurement. These amplitudes are both 5–7 dB above the uncorrected predictions of Imbriale et al. (2011).

3.5.2. Source Measurements of Sidelobes

The performance of the UGS was assessed using the W-band array in 2010 January. For these measurements, a polarized, modulated 92 GHz oscillator was placed in the near field of the telescope at a distance of approximately 15 m. The telescope was scanned over its entire azimuth and elevation range at four different deck angles (0°, 90°, −90°, −180°). The top and middle panels in Figure 8 show measurements before and after the installation of the UGS, respectively. The main sidelobe feature at the bottom of the top map corresponds to the line of sight over the SR. This feature is clearly removed by the UGS. The remaining sidelobes were caused by holes in the floor of the lower ground screen below the SR. A third measurement taken after placing an absorber over these holes (bottom panel in Figure 8) verifies this and displays the sidelobe performance in the final ground screen configuration. The UGS was not in place during any of the Q-band observing season nor during the first third of the W-band observing season.

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3.5.3. Sun Measurements of Sidelobes

Before the installation of the UGS, the Sun was occasionally detected in the sidelobes. This is particularly apparent once the data are binned into maps in "telescope boresight-centered" coordinates (Chinone 2011). The Cartesian basis of this coordinate system has $\hat{i}$ oriented along the feed horn boresight, $\hat{k}$ oriented along the telescope boresight, and $\hat{j}=\hat{k}\times \hat{i}$. If $\hat{s}$ is directed toward the Sun, the corresponding spherical coordinates of the Sun are defined to be $\theta =\cos ^{-1}(\hat{s}\cdot \hat{k})$, and $\phi =\tan ^{-1}(\hat{s}\cdot \hat{j}/\hat{s}\cdot \hat{i})$. In these coordinates, the telescope would be pointed directly at the Sun at θ = 0, but it never is (intentionally) pointed closer than ∼30°. These Sun-centered sidelobe maps are shown in Figure 9 for a top-row module (as the SR spillover is dependent on focal plane position, and the top row has the strongest coupling). Figure 9(a) shows the optical path of these sidelobes before the installation of the UGS for a Q-band horn, and Figures 9(c) and (d) show measurements of the far sidelobes for the W-band system before and after installation of the UGS, respectively. Figure 9(d) confirms that both far sidelobes are eliminated by the UGS. The ϕ = 0° − 180° horizontal line in Figure 9 corresponds to the principal plane measurement shown in Figure 7, and both show the SR spillover lobe and triple reflection lobe before the installation of the UGS. The amplitudes of the two far sidelobes measured with the Sun are consistent with the ∼ − 60 dB levels obtained with the range measurements shown in Figure 7. Data with the Moon or Sun in the sidelobes were excised in the Q-band analysis (QUIET Collaboration et al. 2011) as well as during the first third of the W-band season (QUIET Collaboration et al. 2012). The addition of the UGS for the W-band data, in combination with azimuth filtering and data rejection used for the Q-band data, makes the spurious polarization signal due to sidelobes a negligible effect on the B-mode measurements.

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The leakage beams quantify both the Q and U detector diodes'³³ responses to an unpolarized source, as well as the leakage that can convert a sky Q into a measured U or a sky U into a measured Q. In order to assess these various forms of leakage, daily observations of Jupiter and/or Tau A were performed. These produce beam maps that are subsequently decomposed into their respective beam Mueller fields following O'Dea et al. (2007). The beam Mueller fields are related to the copolar and cross-polar components of the dual, orthogonal polarizations supported by the feed system. For a linearly polarized source with Stokes parameters I_src, Q_src, U_src (assuming V_src = 0), degree of linear polarization $p=(Q_{\mathrm{src}}^2+U_{\mathrm{src}}^2)^{1/2}/I_{\mathrm{src}}$, and position angle γ_PA = (1/2)tan ⁻¹(− U_src/Q_src), the output voltage d_Q of a Q diode as a function of instrumental flux density gain g_Q and instrumental position angle ψ is given by

$$\begin{eqnarray} d_{\mathrm{Q}} &=& g_{\mathrm{Q}}\,e^{-\tau }\,I_{\mathrm{src}}\,\lbrace m_{\mathrm{QI}}+p\,m_{\mathrm{QQ}}\,\cos (2[\gamma _{\mathrm{PA}}-\psi ]) \nonumber\\ && +p\, m_{\mathrm{QU}}\,\sin (2[\gamma _{\mathrm{PA}}-\psi ])\rbrace , \end{eqnarray} \tag{ 5 }$$

where m_QI and m_QU are the Mueller fields representing the I-to-Q and U-to-Q leakage beams, m_QQ is the extracted Q polarization beam and τ is the opacity with typical values given in Figure 2. Similarly, the output voltage of a U diode is given by

$$\begin{eqnarray} d_{\mathrm{U}} &=& g_{\mathrm{U}}\, e^{-\tau }\, I_{\mathrm{src}}\, \lbrace m_{\mathrm{UI}}+p\, m_{\mathrm{UU}}\,\sin (2[\gamma _{\mathrm{PA}}-\psi ]) \nonumber\\ && +p\, m_{\mathrm{UQ}}\,\cos (2[\gamma _{\mathrm{PA}}-\psi ])\rbrace , \end{eqnarray} \tag{ 6 }$$

where m_UI and m_UQ are the corresponding leakage beams, and m_UU is the U polarization beam. In each of these expressions, the factor g is the product of the receiver responsivity (described in Section 8.4) and the telescope sensitivity Γ given by Equation (1). The instrumental position angle is given by ψ = η + ϕ_d where η is the parallactic angle of the beam center and ϕ_d is the deck angle.³⁴ For a number of sources the parallactic angle coverage is not very large, so beam maps at various deck angles are necessary in order to vary the outputs of the Q and U detector diodes. This is particularly true for Tau A, which (due to the mount's lower elevation limit) is only tracked over a limited azimuth range, and this translates into a limited range of parallactic angles. Figure 10 shows the results of this extraction of the leakage and polarized beams for a Q and U diode pair behind the central W-band horn. A similar figure is shown in Monsalve (2010) for the Q-band system.

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The m_QI and m_UI Mueller fields are of particular importance since they characterize the instrumental polarization. Instrumental polarization can be generated by any of the elements in the optical path including the reflectors, the curved cryostat window, the IR blocker, the feed horns, the septum polarizers and the modules themselves. In Appendices A and B, specific expressions are derived for these leakage terms for the modules and the septum polarizers. These two elements are the primary cause of the monopole leakage contribution to the m_QI and m_UI Mueller fields. The median W-band monopole leakage is 0.25% and is lower than the median Q-band monopole leakage. These Q- and W-band leakages measured with Jupiter and Tau A are consistent with those obtained from sky-dip measurements that are described in Section 8.4. As reported in QUIET Collaboration et al. (2011), the Q-band monopole leakage is the largest systematic error in the B-mode measurement at ℓ ∼ 100 where it begins to dominate the constraint on r at levels of r < 0.1. A naïve estimate of the impact of this leakage would cause it to dominate at a much higher level; however, a combination of sky rotation and frequent boresight rotation suppresses this systematic by some two orders of magnitude. The origins of the Q-band monopole leakage are described in more detail in Section 5.1.

The monopole leakage refers to the s₀₀ term in the two-dimensional Hermite expansion of these leakage beams given by b_leak(x, y) (Monsalve 2010). Here and in Figure 10 the coordinates (x = sin θsin ϕ, y = sin θcos ϕ) are telescope boresight-centered coordinates defined in Section 3.5.3. The leakage beams can be expressed as

$$\begin{equation} b_{{\rm leak}}(x,y)=\sum _{j=0}^2\sum _{i=0}^2 s_{ij} \, f_{ij}(x,y), \end{equation} \tag{ 7 }$$

where s_ij are the fit coefficients and the normalized basis functions f_ij(x, y) are

$$\begin{equation} f_{ij}(x,y)=\left(\frac{1}{\sqrt{2^{i+j}i!j!\pi \sigma ^2}}\right){\mathrm{e}}^{-\frac{1}{2\sigma ^2}\left[x^2+y^2 \right]}H_{i}\left(\frac{x}{\sigma }\right)H_{j}\left(\frac{y}{\sigma }\right), \end{equation} \tag{ 8 }$$

where σ is the Gaussian width of the symmetrized beam described in Section 3.4 and the H_i and H_j are Hermite polynomials.

Higher-order leakage terms, including dipole (s₀₁ or s₁₀) and quadrupole leakages (s₁₁ or (s₂₀ − s₀₂)/2), can also arise due to the off-axis nature of the telescope and the imperfectly matched E- and H-plane feed horn patterns. The full array drift scans of Jupiter are particularly useful in measuring these quantities for every diode in the W-band array. Histograms of the peak amplitudes complete to i = j = 2 are shown in Figure 11 for the W-band array (similar results are provided for the central pixel of the Q-band array in Monsalve 2010). Additional terms in the expansion are also included, but they are consistently less than 0.1%. Leakages above 1% are quite rare and typical values are in the 0.2%–0.4% range. The W-band dipole and quadrupole leakages are typically slightly higher than those in Q-band. The systematic effects that these leakage beams generate for power spectrum estimation are provided for the Q-band results (QUIET Collaboration et al. 2011) and in the W-band analysis (QUIET Collaboration et al. 2012).

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The m_UQ and m_QU Mueller fields measure the leakage of the incident Q Stokes parameter into the measured U Stokes parameter or the incident U Stokes parameter into the measured Q Stokes parameter. Curved reflector surfaces, imperfections in the septum polarizer, and imperfections in the phase switch are potential sources of this leakage. These primarily give rise to monopole leakage and effectively rotate the instrumental position angle. In the case that the ratios m_QU/m_QQ and m_UQ/m_UU are constant over the extent of the beam, the m_UQ and m_QU Mueller fields can be absorbed into the expressions for the two diode outputs with the definition of detector angles ψ_Q and ψ_U. The detector angles are defined by replacing the last two terms in each of Equations (5) and (6) with a single term as follows:

$$\begin{eqnarray} && p\, m_{\mathrm{QQ}}\,\cos (2[\gamma _{\mathrm{PA}}-\psi -\psi _{\mathrm{Q}}]) \equiv p\, m_{\mathrm{QQ}}\,\cos (2[\gamma _{\mathrm{PA}}-\psi ]) \nonumber\\ && \quad +p\, m_{\mathrm{QU}}\,\sin (2[\gamma _{\mathrm{PA}}-\psi ]) \end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray} && p\, m_{\mathrm{UU}}\,\sin (2[\gamma _{\mathrm{PA}}-\psi -\psi _{\mathrm{U}}])\equiv p\, m_{\mathrm{UU}}\,\sin (2[\gamma _{\mathrm{PA}}-\psi ]) \nonumber\\ && \quad +p\, m_{\mathrm{UQ}}\,\cos (2[\gamma _{\mathrm{PA}}-\psi ]), \end{eqnarray} \tag{ 10 }$$

respectively. A Hermite decomposition of the m_QU and m_UQ Mueller fields shown in Figure 10 shows that they are simply related by a multiplicative factor to the m_QQ and m_UU fields. Thus they can be represented in terms of single-valued detector angles, ψ_Q and ψ_U and are not a source of systematic error. In order to achieve the maximum benefit of simultaneous Q/U detection, it is an important feature that the detector angles are separated by nearly integer multiples of 45° for each of the four diodes in a given module. This is the case for both the Q-band and W-band instruments (and the W-band instrument detector angles are shown in Section 8.5).

The Q-band and W-band receiver arrays each has a dedicated cryostat (Figure 12). In each cryostat, cryogenic temperatures are achieved with two Gifford-McMahon dual-stage refrigerators. The first stage of the refrigerators provide cooling power to a radiation shield, maintained at ∼50 K (∼80 K) for the Q-band (W-band) cryostat. The difference in shield temperature between the W-band and Q-band instruments was not anticipated from the cryostat design, but ultimately did not greatly impact the module temperatures. Infrared radiation is reduced with 10 cm thick, 3 lb density polystyrene foam (Table 7) attached to the top of the radiation shield. The first stages of the refrigerators also provide a thermal break for the electrical cables. The second stages of the refrigerators provide cooling power for the feed horn array and the modules. The two stages are thermally isolated by G-10 rings.

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The cryogenic performance of the Q-band array is consistent with the design goals of (1) 20 K module temperatures and (2) that the module temperatures remain constant during a CES (CESes are described in Section 2) to within ±0.1 K. A temperature sensor located on an edge module in the Q-band cryostat had a mean temperature of 20.0 K with a standard deviation of 0.3 K throughout the season and a deviation of 0.02 K within a CES.

For the W-band array, additional heat loads from the active components and conduction through cabling from a factor of five more modules contribute to slightly higher module temperatures compared with the Q-band array. Taking this into consideration, the W-band modules were still warmer than expected by ∼3 K, likely as a result of both higher shield temperatures and a minor vacuum leak. A temperature sensor placed directly on the central polarimeter of the W-band array had a mean temperature of 27.4 K with a standard deviation of 1.0 K throughout the season, and a mean variation within a CES of 0.12 K. For each receiver array, both the variation of the module temperatures within a CES and throughout the season had a negligible impact on the responsivity (QUIET Collaboration et al. 2011, 2012).

The vacuum windows for the Q-band and W-band cryostats are each ∼56 cm in diameter, the largest vacuum window to date for any CMB experiment. The vacuum windows must be strong enough to withstand atmospheric pressure while maximizing transmission of signal and minimizing instrumental polarization.

Ultra-high molecular weight polyethylene (UHMW-PE) was chosen as the window material after stress-testing a variety of window materials and thicknesses. The index of refraction was expected to be 1.52 (Lamb 1996). To make a well-matched anti-reflection coating for the UHMW-PE in the QUIET frequency bands, the window was coated with expanded teflon, which has an index of refraction of 1.2 (Benford et al. 2003). The teflon was adhered to the UHMW-PE window by placing an intermediate layer of low-density polyethylene (LD-PE) between the teflon and the UHMW-PE. The plastics were heated above the melting point of LD-PE while applying pressure with a clamping apparatus in a vacuum chamber to avoid trapping air bubbles between the material layers (the window material properties are included in Table 7). The band-averaged transmission was expected to improve from 90% to 99% for the Q-band array and from 91% to 98% for the W-band array by adding this anti-reflection coating to the windows.

An anti-reflection coated sample for the W-band window was measured using a VNA. The envelope of the transmission and reflection response were fitted to obtain values for the optical properties and material thicknesses (shown in Table 7, they matched literature values). The expected contributions to the system noise from loss were computed using published loss tangent values (Lamb 1996): ∼3 K (∼4 K) for the Q-band (W-band) windows. These values were confirmed within ∼1 K by placing a second window over the main receiver window and measuring the change in instrument noise.

The curvature of the window under vacuum pressure could introduce cross-polarization. A physical optics analysis of the W-band window was performed with the General Reflector Antenna Analysis (GRASP)³⁵ package to investigate the effect of the curved surface on the transmission properties of the window. For these simulations we use a window curvature determined from measurements of the deflection of the window under vacuum, ∼7.5 cm. With a curved window, the central feed horn has negligible instrumental polarization. The edge pixel has 0.16% additional cross-polarization, where this is defined as leakage from one linear polarization state into the other linear polarization state. This −28 dB cross-polarization is of the same order as expected cross-polarization from the horns alone and would contribute indirectly to the cross-polarization coefficients m_QU and m_UQ given in Section 3.6.

Each QUIET polarimeter assembly consists of (1) a septum polarizer, (2) a waveguide spreader, and (3) a module containing the integrated package of HEMT-based MMIC devices (Figure 14). The septum polarizer consists of a short circular-to-square transition into a square waveguide containing a septum (a thin aluminum piece with a stepped profile) in the center, which adds a phase lag to one of the propagating modes (Bornemann & Labay 1995). Given an incident electric field with linear orthogonal components E_x and E_y, where the x and y axis orientations are defined by the septum, the septum polarizer assembly sends a left-circularly polarized component $L=(E_{\rm x} + i E_{\rm y})/\sqrt{2}$ to one output port, and a right-circularly polarized component $R=(E_{\rm x} - i E_{\rm y})/\sqrt{2}$ to the other output port. Thus the septum's spatial orientation is used to define the instrumental position angle. The output ports of the septum polarizer are attached to a waveguide spreader, which transitions from the narrow waveguide spacing of the septum-polarizer component to the wider waveguide separation of the module waveguide inputs. A more thorough mathematical description of the septum polarizer is given in Appendix B.1.

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The scattering matrices, gains, and the temperature-to-polarization (monopole) leakage terms of both the Q-band and W-band septum polarizers are derived from VNA measurements. The scattering matrix and derived quantities for these terms are presented in Appendix B.1. Spectrum analyzer measurements of the Q-band modules in the laboratory show a degradation in the return loss near the low frequency end of the module's bandpass. When this return loss power is reflected off the septum polarizer and back into the module, it is amplified in the LNAs in the module legs and sent back out of the module to reflect again. This sets up an oscillation that renders the module incapable of measuring input signals. Therefore, a bandpass mismatch between the septum polarizer and module was deliberately introduced to send this return loss to the sky and prevent oscillations in the module output. The bandpass mismatch leads to an enhancement in the differential loss between the E_x and E_y transmissions at 47 GHz, causing a temperature-to-Stokes Q leakage of ∼1%, averaged over the module's bandpass. This estimate is consistent with leakage values derived from Tau A measurements (Section 3.6). W-band VNA measurements show no return loss degradation (measurements indicate −30 dB return loss, compared with −19 dB for the Q-band septum polarizers), and therefore no bandpass adjustments are needed. The VNA measurements predict a smaller leakage of ∼0.3%, so that it is subdominant to leakage due to optics. These measurements are consistent with monopole leakage values obtained from on-sky calibrators (see Section 3.6 and Figure 11). Note that since the optics leakage has a random direction relative to the polarimeter assembly leakage, the combined leakage averages to a smaller value and is randomly distributed both in sign and amplitude among modules. The isolation between the left- and right-circularly polarized ports was measured to be −22 dB for the Q-band septum polarizers, and −28 dB for the W-band septum polarizers.

The QUIET modules are used in the polarimeter and differential-temperature assemblies, functioning as pseudo-correlation receivers so that the output is a product (rather than sum or difference) of gain terms. The modules employ a high speed switching technique to reduce 1/f noise, and are an improvement on classical Dicke-switched radiometers (Dicke 1946) in that we switch between two sky signals, yielding an improvement of $\sqrt{2}$ in sensitivity (Mennella et al. 2003).

In a polarimeter assembly, the module receives as inputs the left (L) and right (R) circularly polarized components of the incident radiation, and measures the Stokes parameters Q, U and I, defined as

$$\begin{eqnarray} I &=& |L|^2 + |R|^2, \nonumber \\ Q &=& 2 \, {\rm Re}(L^{*}R), \nonumber \\ U &=& -2 \, {\rm Im}(L^{*}R), \nonumber \\ V &=& |L|^2 - |R|^2, \end{eqnarray} \tag{ 11 }$$

where the * denotes complex conjugation and we expect V to be zero but do not measure it.

Figure 15(a) shows a schematic of the QUIET module, in which L and R traverse separate amplification "legs" (called legs A and B). A phase switch in each leg allows the phase to be switched between 0°(+1) and 180°(−1).³⁶ The outputs of the two amplification legs are combined in a 180° hybrid coupler, which, for voltage inputs a and b, produces $(a+b)/\sqrt{2}$ and $(a-b)/\sqrt{2}$ at its outputs. The hybrid coupler outputs are split, with half of each output power going to detector diodes D₁ and D₄, respectively. The other halves of the output powers are sent to a 90° coupler, which, for voltage inputs $\bar{a}$ and $\bar{b}$, produces $(\bar{a}+i\bar{b})/\sqrt{2}$ and $(\bar{a}-i\bar{b})/\sqrt{2}$ at its outputs. The outputs of this 90° coupler are each detected in diodes D₂ and D₃, respectively. The detector diodes are operated in the square-law regime, and so their output voltages are proportional to the squared input magnitudes of the electric fields.

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Table 8 shows the idealized detector diode outputs for the two states of leg B with the leg A state held fixed. The diode outputs are averaged and demodulated by additional warm electronics (see Section 6). Given a diode output of I ± Q(U), the averaging and demodulation operations return I and Q(U), respectively.³⁷ The Stokes parameters can be self-consistently expressed in units of temperature as follows (Staggs et al. 2002). Let T_x (T_y) be the brightness temperature of a source that emits the observed value of $\langle E^2_x \rangle$ ($\langle E^2_y \rangle$). The Stokes parameters in temperature units become

$$\begin{eqnarray} &&I_T = \frac{1}{2} \cdot (T_x + T_y), \nonumber \\ &&Q_T = \frac{1}{2} \cdot (T_x - T_y). \end{eqnarray} \tag{ 12 }$$

For completeness, the voltage V_Q1 appearing at the Q₁ diode would measure

$$\begin{equation} V_{Q1} = g \cdot \left(\frac{1}{2}(T_x + T_y) \pm \frac{1}{2}(T_x - T_y) \right), \end{equation} \tag{ 13 }$$

where ± indicates the states of leg B, and g is the responsivity constant extracted using calibration tools and procedures described in Sections 7 and 8.

Table 8. Idealized Detector Diode Outputs for a Polarimeter Assembly

Diode	Raw Output	Average	Demodulated
D1	$\propto \frac{1}{4}(I \pm Q)$	∝ $\frac{1}{4}I$	∝ $\frac{1}{2}Q$
D2	$\propto \frac{1}{4}(I \mp U)$	∝ $\frac{1}{4}I$	∝ $-\frac{1}{2}U$
D3	$\propto \frac{1}{4}(I \pm U)$	∝ $\frac{1}{4}I$	∝ $\frac{1}{2}U$
D4	$\propto \frac{1}{4}(I \mp Q)$	∝ $\frac{1}{4}I$	∝ $-\frac{1}{2}Q$

Note. Results are shown for the two states of leg B, with the leg A state held fixed.

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In practice, the phase of leg B is switched at 4 kHz, reducing the 1/f knee frequency from the LNAs once the signal is demodulated in the Q and U outputs (this is discussed in greater detail in Section 8.7). The choice of circularly-polarized inputs thus allows for the simultaneous measurement of both Stokes Q and U from differencing the same detector diode, giving an advantage in reduced systematic errors and typically lower knee frequencies over current incoherent detectors. However the phase switches do not reverse the sign of I; therefore the I output suffers from significant 1/f noise and so is not used to measure the temperature anisotropy.

The amplifier gains and transmission coefficients are represented by the proportionality symbols in Table 8. In practice, the transmission through leg B is not exactly identical between the two leg B states, leading to additional free parameters needed to characterize the module. If the leg B transmission differences are not accounted for, they lead to instrumental (i.e., false) polarization. This is resolved by modulating the phase of leg A at 50 Hz during data taking, and performing a double demodulation procedure on the offline data. Imperfections in the optics and the septum polarizer introduce additional offsets and terms proportional to I. These effects are discussed in Appendix B.

In practice, the signal pseudo-correlation is implemented in a single small package as shown in Figure 15(b) (Kangaslahti et al. 2006; Cleary 2010). The LNAs, phase switches and hybrid couplers are all produced using Indium–Phosphide (InP) fabrication processes. Three LNAs, each with gain ∼25 dB, are used in each of the two legs. When the input amplifiers are packaged in individual amplifier blocks and cryogenically cooled to ∼20 K, they exhibit noise temperatures of about 18 K (50–80 K) for the Q-band (W-band). The phase switches operate by sending the signal down one of two paths within the phase switch circuit, one of which has an added length of λ/2 (i.e., 180° shift). Two InP PIN (p-doped, intrinsic-semiconductor, n-doped) diodes control which path the signal takes. The signals go through band-defining passive filters made from alumina substrates, and are then detected by commercially-available Schottky detector diodes downstream of the hybrid couplers. The amplifiers and phase switches are specific to each band, and hence unique to each array. The detector diodes are capable of functioning at both 40 GHz and 90 GHz, and so are identical between the two arrays.

The module components are packaged into clamshell-style brass housings, precision-machined for accurate component placement and signal routing. To provide bias for active components and readout of diodes, the housing has feedthrough pins connecting to the module components via microstrip lines on alumina substrates and wire bonds. Miniature absorbers and an epoxy gasket between the two halves of the clamshell are used to suppress cross talk between the RF and DC components. All Q-band modules and roughly 40% of W-band modules were assembled by hand. For the remaining W-band modules, the components and substrates were automatically placed in the housings by a commercial contractor using a pick-and-place machine; the wire bonding, absorber and epoxy gasket were then finished by hand.

The differential-temperature assemblies are grouped into pairs of assemblies, with waveguide components that mix two neighboring horn signals into two neighboring modules. Figures 16(a) and (b) show the schematic and implementation of these assemblies. An orthomode transducer (OMT) located after feed horn A outputs the linear polarizations E_Ax and E_Ay. One of these polarizations, E_Ay, enters a waveguide 180° coupler (a "magic-tee") and is combined with E_Bx from the adjacent feed horn. The magic-tee outputs are coupled to a module's inputs. The OMTs were reused from CAPMAP (Barkats et al. 2005) while the waveguide routing and magic-tees were made by Custom Microwave, Inc. Note that the differential-temperature assembly design resembles that of WMAP (Jarosik et al. 2003b), with the significant differences being in the feed horn separation and the implementation of the LNAs, both due to advances in MMIC HEMT LNAs and planar circuitry that enabled QUIET's cryogenically cooled integrated compact array design.

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For an ideal differential-temperature assembly, the demodulated Q diodes (D₁ and D₄) measure $E^{2}_{\rm Ax}-E^{2}_{\rm By}$, while their counterparts in the adjacent differential-temperature assembly measure $E^{2}_{\rm Ay}-E^{2}_{\rm Bx}$. The difference of demodulated Q diode outputs from adjacent differential-temperature assemblies measure the beam-differenced total power $(E^{2}_{{\rm A}x} + E^{2}_{{\rm A}y}) - (E^{2}_{{\rm B}x} + E^{2}_{{\rm B}y}) = I_{A} - I_{B}$ (see Table 9). The demodulated U diodes (D₂ and D₃) would measure zero for an ideal assembly. However, frequency-dependent unequal path lengths (ϕ) in the two legs of a module mix some of the temperature difference signal from the Q diodes (mixing ∝cos (ϕ)) to the U diodes (mixing ∝sin (ϕ)). We measure ∼15–30% of the signal on the U-diodes (ϕ ∼ 10°–20°).

Table 9. Idealized Detector Diode Outputs for a Differential-temperature Assembly

	Mod 1	Mod 2
D1	$\propto E^{2}_{{\rm A}y} (E^{2}_{{\rm B}x})$	$\propto E^{2}_{{\rm B}y} (E^{2}_{{\rm A}x})$
D4	$\propto E^{2}_{{\rm B}x} (E^{2}_{{\rm A}y})$	$\propto E^{2}_{{\rm A}x} (E^{2}_{{\rm B}y})$
demod(D1, Mod1) −
demod(D1, Mod2)	$(E^{2}_{{\rm A}x} + E^{2}_{{\rm A}y}) - (E^{2}_{{\rm B}x} + E^{2}_{{\rm B}y})$

Notes. Outputs of D₁ and D₄ corresponding to a leg B state of +1(− 1), with leg A fixed at +1. Also shown is the difference of the demodulated D₁ signals from two modules. The outputs of D₂ and D₃ are zero for an ideal assembly (see the text).

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Finally, we note that the sum of demodulated Q diode outputs from adjacent modules is Q_A + Q_B, where Q is the Stokes Q parameter seen by the respective horns. Thus one can in principle extract polarization information from the differential-temperature assemblies. However, as these assemblies form a small fraction of the array, the sensitivity gain is marginal and so this was not explored further in the analyses.

The electrical connection to, and protection of, the modules is provided by Module Assembly Boards (MABs). Each MAB is a printed circuit board with pin sockets for seven modules. Voltage clamps and RC low-pass filters protect the sensitive components inside the module from damage. The Q-band (W-band) modules require 28 (23) pins for grounding, biasing active components, and measuring the detector diode signals. All of these electrical connections are routed to the outside the cryostat. After the MAB protection circuitry, these signals travel on high density flexible printed circuits (FPC), which bring them out of the cryostat through Stycast-epoxy–filled hermetic seals. An additional layer of electronic protection circuitry is provided by the array interface boards, which also adapt the FPC signals to board-edge connectors and route to the Bias and Readout systems.

All biasing is accomplished by custom circuit boards. The amplifier bias boards provide voltage and current to power the amplifiers in the modules. Each of these bias signals is controlled by a 10-bit digital-to-analog converter (DAC), which allows the biases to be tuned for optimal performance of each amplifier. Phase switch boards provide control currents to the PIN diodes in the phase switches. The control current is switched by the board at 4 kHz for one phase switch and 50 Hz for the other phase switch, generating the modulation described in Section 5.2. The data taken during the switch transition time are discarded in the Readout system. A housekeeping board monitors the bias signals at ∼1 Hz for each item being monitored. The housekeeping board multiplexes these items, switching only during the phase switch transitions when data will be discarded.

The Q-band amplifier bias boards are designed to operate at 25°C so the enclosure is thermally regulated at that temperature. The W-band amplifier bias boards use a different design that is much less temperature sensitive. Therefore, the enclosure regulation was less strict and the temperature was allowed to vary between 35°C and 40°C, depending on the time of year, to reduce the power needed for regulation. For both the Q-band and W-band observing seasons, the enclosure temperature remained within the regulation setpoint for ∼90% of the time. For the Q-band system, the excursions primarily affect the drain-current bias supplied by the amplifier bias boards, which changes the detector responsivity by ∼2%/°C. This effect is taken into account with an enclosure-temperature dependent responsivity model (QUIET Collaboration et al. 2011).

The Readout system first amplifies each module's detector diode output by ∼130 in order to match the voltage range of the digitizers. The noise of this warm preamplifier circuit does not contribute significantly to the total noise. This is determined in situ at the site by selectively turning off the LNAs in the module and seeing that the total noise decreases by roughly two orders of magnitude. For the W-band array, the preamplifier noise contributes less than 2% to the total noise in the quadrature sum. The amplifier chain also low-pass filters the signal at ∼160 kHz to prevent aliasing in digitization. Each detector diode output is digitized by a separate 18-bit Analog Devices AD7674 (Analog–Digital Converter) ADC with 4V dynamic range at a rate of 800 kHz. Each ADC Board has a field-programmable gate array (FPGA) that accumulates the samples from the 32 ADCs on that Board. The FPGA on one ADC Board, designated the "Master ADC Board," generates the 4 kHz and 50 Hz signals used by the Bias system to modulate the phase switch control currents. This signal is also distributed to all ADC Boards, and the FPGA on each ADC Board uses it to demodulate the detector diode data synchronously with the phase switch modulation.

Figure 17 summarizes the organization of data performed by the FPGA. The FPGA organizes the 800 kHz detector diode data into continuous 10 ms blocks (i.e., 100 Hz time streams), itself organized into continuous 125 μs blocks. These 10 ms blocks contain an equal sampling of both 4 kHz clock states. In the "TP" stream, the 800 kHz data within a 10 ms block are averaged, regardless of the 4 kHz clock state. This stream is sensitive to Stokes I and is used for calibration and monitoring. In the "demodulated" stream, data within a 125 μs block have the same 4 kHz phase state, and are averaged. Averaged data from sequential 125 μs blocks are differenced, thus forming the polarization-sensitive data stream. Offline, two adjacent 10 ms blocks in the demodulated stream are differenced to form the "double-demodulated" (50 Hz) stream. The W-band ADC firmware was upgraded to include an additional specially demodulated 100 Hz data stream, called the "quadrature stream." Unlike the usual demodulated stream, data within a 125 μs block populate equally both 4 kHz phase states, and are averaged. When these averaged data are differenced, the result has the same noise as demodulated data but has no signal. The quadrature stream is used to monitor potential contamination and to understand the detector noise properties.

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As noted earlier, the data are masked at the phase switch transition. Masking 14% of the samples around the transition is found to be adequate to remove contamination in the data stream.

The ADC Boards have a small nonlinearity in their response. At intervals of 1024 counts, the ADC output has a jump discontinuity between 1 and 40 counts, affecting ∼14% of the data. This jump is shown schematically in Figure 18. When the 800 kHz data stream value falls at a discontinuity, the jump in the output signal will trickle into the 100 Hz stream. This nonlinearity is corrected in the 100 Hz stream. The correction is statistical in nature, based on the width of the 800 kHz noise and its proximity to the discontinuity (Bischoff 2010). This nonlinearity, if uncorrected, causes a variation of responsivity during a CES and a systematic effect similar to the leakage of temperature to polarization. For the Q-band, the correction reduces the ADC nonlinearity to contribute at most 3% to the leakage bias systematic error, and at most 50% to the CES responsivity systematic error. For the W-band, the residual ADC nonlinearity adds 40% in quadrature to the leakage bias systematic error. The effect on the CES responsivity is <1%, negligible compared with other errors in the gain model. These affect r at a level below 0.01 for the W-band.

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The Readout system ensures that the housekeeping data and 100 Hz data from the detectors are synchronized to each other and to the mount motion encoder readout. Synchronization is achieved by distributing the same GPS-derived IRIG-B³⁸ time code to both the receiver and mount electronics. In the Readout system, the time code is decoded by a Symmetricom TTM635VME-OCXO timing board. Clock signals of 1 Hz and 10 MHz, locked to the IRIG-B time code, synchronize the readout of all ADC Boards. The timing board provides the GPS-derived time to the Data Management system so that each datum is assigned a time stamp.

The Data Management system sends commands to the Bias system to prepare for observation, acquires the data from the Readout system, writes them to disk, and creates summary plots of the detector diode signals and housekeeping data for display in real time. The complete data are written to disk and DVDs in the control room at the observation site at a rate of ∼8 GB day⁻¹ for the Q-band array. W-band array data are written to blu-ray optical discs at a rate of ∼35 GB day⁻¹. A subset of ∼10% of the data were transferred by internet every day to North America for more rapid analysis and monitoring. The DVDs or blu-ray discs were mailed weekly to North America.

The polarized response of the receiver in the laboratory is measured with the "optimizer," a reflective plate and cryogenic load that rotate around the boresight of the cryostat (Figure 19). The optimizer was used to verify that the responsivities derived from unpolarized measurements with cryogenic loads were not substantially different from the polarized responsivities, and hence that the projections of instrument sensitivity (which were made from unpolarized measurements) are valid for the Q-band array. For the W-band array, the optimizer was used to select functioning modules for the final array configuration.

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The plate is oriented at angle β from the plane of the feed horns and reflects radiation from the cryogenic load into the window of the cryostat. The resulting amplitude of the Stokes Q vector can be computed in temperature units via (Barkats et al. 2005):

$$\begin{eqnarray} Q & = & \frac{1}{2} \cdot \frac{ 4 \pi \delta }{\lambda }(\cos \beta - \sec \beta)(T_{\rm plate} - T_{\rm load})\sin (2\alpha t), \nonumber \\ \delta & = & \sqrt{\frac{\rho }{\mu _0 \pi \nu }}, \end{eqnarray} \tag{ 14 }$$

where δ is the skin depth, ρ is the bulk resistivity of the metal plate, μ₀ is the permeability of free space, ν and λ correspond to the center frequency and wavelength of the detector bandpass, t is time, and T_plate and T_load are the temperatures of the plate and cryogenic load. This apparatus rotates at an angular speed α around the boresight of the cryostat so that the resulting polarized signal will rotate between the Stokes Q and U at an angular speed of 2α. Polarization signals that do not rotate with the system (such as thermal emission from objects in the laboratory) will be detected at a rate of α, and so can be removed.

The predicted polarized emission from Equation (14) and the measured voltages on the detector diodes are used to calculate the polarized responsivities for polarimeters whose beams primarily sample the reflected cryogenic load. Various plate materials (aluminum, stainless steel, and galvanized steel) and two thermal loads (liquid nitrogen and liquid argon) are used to obtain multiple estimates of the polarized responsivity. The loads are too small to fill the entire array beam, so only the measurements from the central polarimeter (Q-band) or inner two rings (W-band) are used.

A "sparse wire grid" (Tajima et al. 2012), a plane of parallel wires held in a large circular frame with the same diameter as the cryostat window, was used to impose and modulate a polarization signal onto the array. For the polarization parallel to the grid wires, a fraction of the rays that would ordinarily pass through the telescope to the cold sky are instead scattered to large angles, mostly terminating on the warm ground shield. The grid is placed as close to the cryostat aperture as possible to minimize interference with the telescope optics and to ensure that it covers the field of view of each detector (Figure 20).

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With this geometry, the polarized signal directed parallel to the wires is empirically found to be ∼2 K. The circular frame rotates about the cryostat boresight axis via a small motor, allowing for modulation of the injected polarized signal at a constant frequency. The wire grid was used for calibration measurements in the laboratory and three times during the observing season: at the end of the Q-band observing season and at the beginning and end of the W-band season. The grid was not mounted on the cryostat during sky observations.

An example of the data taken with the rotating grid is shown in Figure 21. In the ideal case in which the intensity of the reflected radiation is isotropically uniform over the array, the polarized signal D(θ) from each detector diode would exhibit a sinusoidal dependence at twice the frequency of θ, the angle about the cryostat boresight axis between the wires and a fixed point on the cryostat. The measured polarization signal has an additional dependence on θ due to rays terminating at different temperatures in the non-uniform ground screen. This variation appears in both the polarized data stream D and the total power data stream I as a function of θ, and so this variation can be measured in the I data stream and accounted for in the D data stream. Each detector diode data stream is fitted to the form

$$\begin{eqnarray} D(\theta) &=& D_0 + (D_2 + \eta [I (\theta)-I_0] ) \cos [2(\theta - \gamma)], \end{eqnarray} \tag{ 15 }$$

where D(θ) and I(θ) are the double-demodulated polarization and total power signals, respectively. Here, I₀ is the average of I(θ) over all angles θ, and D₀ is an offset term discussed in Appendix B.1. The fit extracts γ, the angle θ that maximizes D(θ), D₂, the polarization amplitude (in mV), and η, a dimensionless constant relating the total power to polarization responsivity. Since the fixed point on the cryostat used to define θ can be arbitrarily chosen, only the relative γs amongst the detector diodes are relevant; they are just the relative detector angles. The values of D₂ indicate the spread of polarized responsivities. For the W-band, their relative ratios agree with ones derived from Tau A observations at the level of 20%. The precision of this agreement is limited by the statistical errors of Tau A observations for the off-center detectors.

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For the Q-band array, the amplifiers were biased manually for each module using a room temperature blackbody load in front of the cryostat. The phase switches were turned on separately, so that the signal only propagated through the module leg with the phase switch on. The bias values for the phase switches were chosen to equalize the signal measured on the two separate legs of the module, to keep the contribution to the module noise from the first-stage amplifier low, and for an adequate signal level, which should not compress the amplifiers. These bias settings were chosen once at the beginning of the season, and kept fixed during the observing season.

For the W-band array, biasing the modules by hand was not feasible due to the large number of modules compared with the Q-band array, and so an automatic method was developed. A sinusoidal polarized signal was injected during module biasing by continually rotating the sparse wire grid. Amplifier bias settings were found by maximizing the amplitude of the sinusoid relative to the time-stream noise. The bias settings were sampled via a computer-based downhill simplex algorithm and optimum values were found for all modules within a few hours. As with the Q-band array, the bias settings were kept fixed during the W-band observing season. Because the settings were chosen to enhance signal-to-noise ratio, balance between the legs was not explicitly prioritized (the consequences of this are discussed in the next section).

One source of leakage from total power into polarization from the module stems from differential power transmission between the two phase switch states within a given leg (Appendix B). We found that double demodulating (described in Section 5.2) typically reduced the root-mean-square of leakage from 0.8% to 0.4% for the W-band modules (Figure 22). The improvement was smaller for the Q-band array, <0.1%, likely because it was dominated by other sources of leakage (Sections 3.6 and 5.1) and because the phase switches had been balanced during bias optimization.

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The module is not the only source of leakage between temperature and polarization. Instrumental polarization from the mirrors, window, horns, etc. can be calibrated from large and small sky dips (elevation nods of ±20° and ±3° amplitude) with 0.3% precision for each sky dip as the signal from the changing atmospheric temperature leaks into the polarized data stream. The median monopole leakage was 0.2% for the W-band array, which is consistent with leakage measurements from Jupiter (Section 3.6). The median monopole leakage was 1.0% and 0.2% for the Q-diodes and U-diodes for the Q-band array, respectively, which are also consistent with measurements from other calibrators. The discrepancy in the monopole leakage between the two diodes for the Q-band array was anticipated from the measurements of the septum polarizers (Section 5.1).

Typical bandpasses for the Q-band and W-band arrays are shown with the spectrum of the atmosphere in Figure 2. Central frequencies and bandwidths are computed from discrete frequency steps as

$$\begin{eqnarray} \mathrm{Central frequency} & \equiv & \sum \limits _{i} I_{i}\nu _{i} \over \sum \limits _{i} I_{i}, \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} \mathrm{Bandwidth} & \equiv & [\sum \limits _{i} I_{i}]^{2} \Delta \nu \over \sum \limits _{i} I_{i}^{2}, \end{eqnarray} \tag{ 17 }$$

where I_i is the measured intensity from a detector diode for each frequency, ν_i, and Δν is the frequency step of the signal generator (100 MHz).

For the Q-band array, bandpasses were measured for each diode in the laboratory during the course of array testing and in end-of-season calibration measurements at the site. The laboratory measurement was performed by injecting a polarized carrier-wave signal from a signal generator with a standard-gain horn over a 35–50 GHz range. The signal was injected into the receiver array through the cryostat window without additional imaging optics, with the horn approximately 3 m away from the window. Sweeps were performed at least eight times. The average bandwidth and central frequency of the polarization modules are given in Table 11. The statistical errors on this measurement are obtained by finding the standard deviation between the eight measurements for a given module, and then averaging that standard deviation for all modules.

Bandpasses were also measured at the site for the Q-band array by reflecting the swept signal from a small (∼1 cm²) plate into the primary mirror. While measurements performed in the laboratory and at the site are consistent with each other, the variation in bandpass shape between the two days of data taking at the site showed that the systematic errors were larger in the experimental setup at the site, so laboratory measurements were used where available. Although the amplifier bias settings were different between the laboratory and the site measurements, a review of laboratory measurements revealed that changing the amplifier bias over the range of interest had no significant effect on the bandpasses. The systematic error in Table 11 is the average of the difference between the site and lab bandpasses.

For the W-band array, bandpasses were measured at the site at the end of the observing season and the central frequency and bandwidth are also given in Table 11. A standard-gain horn was mounted beside the secondary mirror, so it could illuminate the cryostat window from ∼1.5 m away. The signal generator was swept over 72–120 GHz, while the phase switches were held constant (no switching). In this configuration the signal can be sent down each module leg separately. The responses at each frequency bin for each module leg were combined to emulate the power combinations occurring in the module:

$$\begin{eqnarray} &&P_{\rm pol} = P_{\rm A}P_{\rm B} \cos (2(\phi - \gamma)), \end{eqnarray} \tag{ 18 }$$

where P_A and P_B are the measured bandpasses for the signal traveling through module legs A and B, respectively, ϕ is the detector angle (for example, a Q diode might have ϕ = 90° and a U diode might have ϕ = 45°), and γ is the angle of the polarized input from the signal generator. The measured signal is only dependent on the difference between the two angles. Systematic errors have two main sources: the accuracy with which the spike that was used to indicate the beginning of a sweep can be detected, and from reconstructing the bandpass for both module legs biased from data in which only one leg is biased. The first was computed by noting that the timing was accurate to 1.5 ms, which corresponded to 0.7 GHz during the sweep measurement. The second was computed by comparing measurements performed with both legs biased and the reconstruction from single-leg bandpasses from the total power stream. Because the total power stream does not have a dependence on detector angle ϕ, the two should be identical and the difference represents the systematic error in the measurement. The systematic error was found to be 0.3 GHz for the central frequency and 0.9 GHz for the bandwidth.

The responsivities were characterized for the differential-temperature modules and the polarization modules separately with different calibration sources. Responsivities of the differential-temperature modules are computed from calibration observations of Jupiter, RCW38, and Venus, one of which was observed ∼once per week for the Q-band receiver, and once a day for the W-band receiver. The responsivities could depend on the temperature of the bias boards, as described in Section 6.2, and so a temperature dependent responsivity model was developed for the Q-band receiver (QUIET Collaboration et al. 2011) and was found to be negligible for the W-band system (QUIET Collaboration et al. 2012).

For the Q-band array, the absolute polarimeter responsivity for the central horn was determined from Tau A measurements performed every two days. Relative responsivity values among the polarization modules were measured from observations of the Moon (performed once per week). Sky-dip measurements (elevation nods of ∼6° for "normal" sky dips, and ∼40° for "large" sky dips) are also used to obtain the relative total power responsivities of both the differential-temperature and polarized modules before each CES for the Q-band array ("flat fielding"). These frequent (once every ∼1.5 hr) responsivity measurements provide relative responsivity tracking for the differential-temperature and polarized modules on short time-scales. The relative responsivities were checked with an end-of-season wire grid measurement and measurements of Tau A with off-center modules. For the W-band array, the Moon is too bright for relative responsivity calibration, so measurements from the wire grid and Tau A from off-center modules were used.

One potential concern when using amplifiers is signal compression: an input-dependent responsivity that is greatly reduced at high input powers. Compression is typically manifested as different responsivity values for different load temperatures. For an ideal radiometer, the combination of sky temperature and typical measurement signals will not compress the amplifiers. However, calibration sources are typically much warmer than the CMB and so compression can have important consequences when deriving responsivities from astronomical or other sources (for example, the Moon is ∼223 K; Ulich et al. 1973). For the Q-band array, responsivity measurements in the laboratory and at the site with different calibration sources were all consistent with each other, confirming that the modules were not operating in a compressed regime. Laboratory responsivity studies of the W-band modules using liquid nitrogen as a cold load showed some evidence for compression. In the field, the W-band modules exhibited compression during observations of the Moon. The emission from the ∼05 Moon varies across its face (Ulich et al. 1973); polarized responsivities for a single detector could vary between the brightest and darkest portions of the Moon's face by 20% (worst case 50%). Ultimately this meant we could not use the Moon to calibrate the responsivities for the W-band receiver as we could with the Q-band receiver or we would have misestimated our responsivity (and hence our noise) by at least 20%.

Compression affects the polarized signal and the total power signal differently (Appendix A). Since the sky dips measure total power responsivity only, this complicates the use of sky dips to track relative polarized responsivity for the W-band array. As a result, daily Tau A measurements of the central module were used to measure fast variations. Relative responsivities between the central module and the other modules are obtained from additional Tau A measurements and an end-of-season polarization grid measurement and these were used to extrapolate absolute responsivities to all modules. The resulting instrumental systematics from using a single module to track fast variations is discussed in QUIET Collaboration et al. (2012).

Additional laboratory studies performed after deployment explain why the W-band modules were operated in a compressed regime: passive hybrids in the W-band modules had as much as twice the expected loss. To compensate for this loss, the amplifiers were biased for higher gain to overcome the post-detector noise. As a result, the gain was large enough that it contributed a significant fraction of the power required to compress the amplifiers. Modules with new passive components having lower loss have been produced. These modules exhibit little compression and have noise temperatures closer to the ∼50 K intrinsic W-band amplifier noise (Reeves 2012). This indicates we can gain a factor of two in sensitivity for W-band modules with the new, less lossy components.

Absolute polarized detector angles were measured for the central module of each array through observations of Tau A, whose position angle is known to 02 precision from IRAM measurements (Aumont et al. 2010). For the Q-band array, the absolute angle shifted by as much as 2° due to jumps in the deck position during the first half of the Q-band season due to slippage in the deck encoder. The systematic uncertainties related to the encoder jumps are discussed in QUIET Collaboration et al. (2011). The Q-band angle calibration relied on weekly Moon observations and an end-of-season sparse wire grid measurement to find the relative angles of the diodes. The relative angles between one of the diodes of the central module and every other diode from all ∼35 Moon measurements deviated less than 02 from nominal, indicating that the relative angles remained nearly constant during the season. Relative detector angles are not affected by the encoder jumps.

The W-band array had a smaller, more efficient Tau A scan trajectory and was able to make measurements with all modules over the course of the season to obtain absolute angle calibration. The variance of detector angles for the central module from repeated measurements of Tau A is 03. The relative angles among the diodes were confirmed with end-of-season wire grid measurements for both arrays to within 09.

Relative angles for all diodes in the W-band array are shown in Figure 23. Systematic errors in the absolute angle are the largest source of systematic errors for the W-band array, which would limit the measurement of r to 0.01 at ℓ ∼ 100 (QUIET Collaboration et al. 2012).

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The telescope pointing model is derived by fitting a physical model of the three-axis mount and telescope to astronomical observations (Næss 2012). The orientations of individual feed horns are determined by observations of the Moon and Jupiter. Then, holding the focal plane layout fixed, the parameters of the dynamical mount model are determined from observations of Jupiter, Venus, RCW38 (W-band only), and the Galactic plane.³⁹ Optical observations are taken regularly with a co-aligned star camera and used to monitor the time evolution of the pointing model. Except for the mechanical problem with the deck-angle encoder during the first two months of Q-band observations (QUIET Collaboration et al. 2011), no significant trends are found.

The residual scatter after all pointing corrections is 35 rms in the Q-band observations (QUIET Collaboration et al. 2011) and 51 FWHM (22 rms) in the W-band observations (QUIET Collaboration et al. 2012). The larger residuals compared with the beam size in the W-band pointing could not be modeled from our pointing variables, but we explicitly account for them in the window function for the high-resolution W-band data, as described in QUIET Collaboration et al. (2012), where we also show that this is a negligible contribution to the constraint on r. In order to validate the pointing model, a high-resolution W-band map of PNM J538-4405 (Gold et al. 2011; a particularly bright point source in the QUIET observing field CMB-2) was produced and both its apparent position and angular size was found to be consistent with the assumed beam profile and estimated uncertainty.

Noise measurements at the site were obtained from a noise spectrum fit to the Fourier-transform of the double-demodulated time stream for each CES. The measured noise floor should be proportional to the combination of module noise temperature, atmospheric temperature, contributions from optical elements, and CMB temperature. A power law with a flat noise floor was assumed for the functional form of the noise spectrum,

$$\begin{eqnarray} &&N(\nu) = \sigma _{0} \left[ 1+ \left(\frac{\nu }{\nu _{\mathrm{knee}}} \right)^{\alpha } \right], \end{eqnarray} \tag{ 19 }$$

where N(ν) and σ₀ have units ${\rm V}/\sqrt{\mathrm{Hz}}$, ν is frequency, σ₀ is the white noise level, α is the slope of the low frequency end of the spectrum, and ν_knee is the knee frequency. A typical noise power spectrum for a W-band module is given in Figure 24, which also shows the effects on the noise of demodulating and double demodulating the time streams. After double demodulation, the median knee frequency is 5.5 mHz (10 mHz) for the Q-band (W-band) array; thus the noise is white at the scan frequencies of the telescope, 45–100 mHz.

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The white noise is correlated among detector diodes within a given module. The correlation between Q and U diodes is expected (Bischoff 2010); the theoretical expectation and typical measured correlations are given in Table 12. The measured correlation coefficients are larger than theoretically anticipated; the source is unknown but could come from unequal transmission in the coupling hybrid in the module, or from leakage of the atmosphere causing residual 1/f noise. However, the noise correlation among diodes is easily treated in the data analysis (QUIET Collaboration et al. 2011), and more importantly does not impact the measured polarized signal, which is a difference between diode signals: (Q₁ − Q₂) and (U₁ − U₂).

The system noise is given by

$$\begin{eqnarray} T_{\rm system} &=& T^{\prime }_{{\rm CMB}} + T_{\mathrm{atm}} +\frac{T_{\mathrm{R}}}{G_{\mathrm{atm}}} + \frac{T_{\mathrm{W}}}{G_{\mathrm{atm}}G_{\mathrm{R}}} \nonumber \\ & &+ \frac{T_{\mathrm{IR}}}{G_{\mathrm{atm}}G_{\mathrm{R}}G_{\mathrm{W}}} + \frac{T_{\mathrm{H}}}{G_{\mathrm{atm}}G_{\mathrm{R}}G_{\mathrm{W}}G_{\mathrm{IR}}} \nonumber\\ & & +\frac{T_{\mathrm{SP}}}{G_{\mathrm{atm}}G_{\mathrm{R}}G_{\mathrm{W}}G_{\mathrm{IR}}G_{\mathrm{H}}} \nonumber\\ & & + \frac{T_{\mathrm{module}}}{G_{\mathrm{atm}}G_{\mathrm{R}}G_{\mathrm{W}}G_{\mathrm{IR}}G_{\mathrm{H}}G_{\mathrm{SP}}}, \end{eqnarray} \tag{ 20 }$$

where T_atm is the effective atmospheric temperature, G_atm = e^−τ is the transmission through the atmosphere where τ is atmospheric opacity, $T^{\prime }_{\mathrm{CMB}}$ is the brightness temperature of the CMB, T_module is the noise temperature of a QUIET module, {T_R, G_R}, {T_W, G_W}, {T_IR, G_IR}, {T_H, G_H}, and {T_SP, G_SP} are the effective noise temperatures and gains for both reflectors (including ohmic and spillover contributions), window, IR blocker, horns, and septum polarizers, respectively (Table 13).

The system noise can be found from the total power time streams taken during sky dips. During a sky dip, the sky temperature seen by the receiver changes with telescope elevation. Using an atmospheric model, the change in signal with this model-dependent change in sky temperature allows us to estimate the system noise. The contribution to instrument noise due to the module alone can be estimated by subtracting assumed or measured values for all other known instrument noise sources (Table 13). All components other than the modules are lossy; thus their noise temperatures are given by (1/G − 1) × T_phys, where G is the gain of the component and T_phys is its physical temperature. The full receiver noise was found to be 38 K for the Q-band system and 109 K for the W-band system, yielding extrapolated module temperatures of 15 K and 77 K for a Q-band and W-band module, respectively. Measurements of the Q-band amplifiers give noise values of ∼18 K; the most likely source of the discrepancy (that the module noise temperature is higher than a single LNA noise temperature) is that the loss in the septum polarizer was overestimated. Similar measurements for the W-band module give amplifier noise values of 50 K. The discrepancy between the W-band module and amplifier temperatures stems from operating them uncompressed in the laboratory (this is explained in greater detail in Section 8.4).

The sensitivity for the polarization response, S_pol (μKs^1/2), is calculated as the ratio of the white noise level to the responsivity. For the Q-band array, after data selection (QUIET Collaboration et al. 2011), the sensitivity is 69 μK s^1/2 corresponding to an average module sensitivity of 275 μK s^1/2. For the W-band array, the array sensitivity is 87 μK s^1/2 (QUIET Collaboration et al. 2012), corresponding to an average module sensitivity of 756 μK s^1/2. A histogram of module sensitivity is shown in Figure 25. Both values are given in thermodynamic units, so that the power detected by the receiver has been corrected from a Rayleigh–Jeans approximation to correspond to fluctuations in the blackbody temperature of the CMB. Functionally this is performed by dividing by C_RJ, which is 0.95 (0.79) for the Q-band (W-band) central frequencies. These values can be compared to the expected sensitivity per module, S_pol, from the radiometer equation (Kraus 1986):

$$\begin{equation} S_{\mathrm{pol}} = \frac{1}{C_{\mathrm{RJ}}} \times \frac{T_{\mathrm{instrument}}}{\sqrt{2 \Delta \nu } G_{\mathrm{total}}(1-f_{\mathrm{mask}})}. \end{equation} \tag{ 21 }$$

Using the measured values for T_system and the atmospheric gain, G_atm (Table 13), the bandwidths Δν (Section 8.3), the Rayleigh–Jeans correction C_RJ for the CMB, and the fraction of the data masked during the phase switch transitions, f_mask (14%; Section 6), sensitivity values of 310 μK s^1/2 for the Q band, and 913 μK s^1/2 for the W band were found. Errors in bandpasses and the atmospheric temperature contribute directly to the difference between the two methods of computing the sensitivity S_pol. A potential explanation for the greater discrepancy between these methods for the W-band array (∼30%) compared with the Q-band (∼11%) array is that T_rec is measured from the total power stream during sky dips, which could be compressed as much as 30% (Appendix A) in the W-band data stream. This compression inflates the noise temperature by the same compression factor, although does not impact the polarization stream.

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As shown above, it can be assumed that the module does not generate any instrumental polarization on its own since the double demodulation procedure nulls out this effect. However, the interaction between the module and septum polarizer can cause irreducible instrumental polarization and offsets; this section derives these couplings. Since the module does not generate instrumental polarization on its own, the module measures:

$$\begin{equation} Q_m = 2\Re (L_m^*R_m) \end{equation} \tag{ B3 }$$

and

$$\begin{equation} U_m = -2\Im (L_m^*R_m), \end{equation} \tag{ B4 }$$

where L_m and R_m are the signals transmitted into the module inputs. Without loss of generality, all the constant factors are absorbed into the responsivity and set to unity. The signals transmitted into the module inputs need not be the same as the L and R components at the septum polarizer input; this difference is a cause of instrumental polarization.

The effect of the septum polarizer is described by a 4 × 4 complex scattering matrix S:

$$\begin{equation} \left(\begin{array}{@{}c@{}}E_x^{\prime } \\ L^{\prime } \\ R^{\prime } \\ E_y^{\prime }\end{array} \right) = S \cdot \left(\begin{array}{@{}c@{}}E_x \\ L_r \\ R_r \\ E_y\end{array}\right), \end{equation} \tag{ B5 }$$

$$\begin{equation} S = \left(\begin{array}{@{}cccc@{}}r_1 & \frac{e^{i\gamma }}{\sqrt{2}}\tau _{21} & \frac{e^{i\gamma }}{\sqrt{2}}\tau _{31} & r_{41} \\ \frac{e^{i\gamma }}{\sqrt{2}}\tau _{21} & r_2 & c & i\frac{e^{i\gamma }}{\sqrt{2}}\tau _{24} \\ \frac{e^{i\gamma }}{\sqrt{2}}\tau _{31} & c & r_3 & -i\frac{e^{i\gamma }}{\sqrt{2}}\tau _{34} \\ r_{41} & i\frac{e^{i\gamma }}{\sqrt{2}}\tau _{24} & -i\frac{e^{i\gamma }}{\sqrt{2}}\tau _{34} & r_4 \end{array} \right), \end{equation} \tag{ B6 }$$

where E_x and E_y are electric field components at the septum polarizer input port; L' and R' are the fields at the two septum polarizer output ports; $E_x^{\prime }$ and $E_y^{\prime }$ are the electric fields emitted from the septum polarizer back toward the feed horn; L_r and R_r are signals reflected (or emitted) from the module inputs traveling back toward the septum polarizer output ports; e^iγ is the propagation phase shift; τ_ij and r are transmission and reflection coefficients, respectively; c is a measure of the isolation between the output ports. For an ideal septum polarizer, τ = 1 and r = c = 0. Symmetry across the septum implies τ₂₁ = τ₃₁, τ₂₄ = τ₃₄, r₂ = r₃, and r₄₁ = 0, although manufacturing errors can cause these conditions to be violated. As described in Sections 5.1 and 8, there are small departures from ideal operation. In this section these departures are computed up to second order. Note that the scattering matrix is frequency-dependent. The analysis given here is strictly for a single frequency. In practice, the result should be averaged with the effective bandpass. The median value of the Q(W)-band, band-averaged return loss (= −20log |r|) for the septum polarizers is 19(30) dB, while the median value of the Q(W)-band, band-averaged isolation (= −20log |c|) is 22(28) dB. Another quantity of interest is median value of the band averaged (linear) axial ratio of the septum polarizers. This is measured to be 1.12(1.07) for Q(W) band and implies a cross-polar discrimination of 24.9(29.4) dB.

A perturbative expansion is used to derive L_m and R_m which are the fields transmitted into the module inputs due to a sky source consisting of fields E_x and E_y. Here a noiseless module is assumed. The case of a noise signal from the module is described later. To lowest order, the S matrix applied to the column vector (E_x, 0, 0, E_y) yields (0, L', R', 0), where

$$\begin{eqnarray} L^{\prime } & = & \frac{e^{i\gamma }}{\sqrt{2}}\left(\tau _{21}\frac{L+R}{\sqrt{2}} + \tau _{24}\frac{L-R}{\sqrt{2}}\right)\nonumber \\ & = & \frac{e^{i\gamma }}{2}\left[ (\tau _{21}+\tau _{24})L + (\tau _{21}-\tau _{24})R\right]. \end{eqnarray} \tag{ B7 }$$

Similarly,

$$\begin{equation} R^{\prime } = \frac{e^{i\gamma }}{2}\left[ (\tau _{31} + \tau _{34})R + (\tau _{31} - \tau _{34})L\right]. \end{equation} \tag{ B8 }$$

where $E_x = (L+R)/\sqrt{2}$ and $E_y = (L-R)/(i\sqrt{2})$. However, L_m and R_m differ from L' and R' due to reflection at the module input. Let r_L (r_R) be the reflection coefficient at the module's L (R) input. Then the S matrix applied to (E_x, r_LL', r_RR', E_y) yields (−, L_m, R_m, −) where

$$\begin{equation} L_m = (1 + r_2r_L)L^{\prime } + cr_RR^{\prime }. \end{equation} \tag{ B9 }$$

$$\begin{equation} R_m = (1 + r_3r_R)R^{\prime } + cr_LL^{\prime }. \end{equation} \tag{ B10 }$$

and for simplicity, the expressions for the first and fourth component are omitted. The module output is

$$\begin{equation} L_m^{*}R_m = L^{\prime *}R^{\prime }(1 + r_3r_R + r_2^{*}r_L^{*}) + L^{\prime *}L^{\prime }cr_L + R^{\prime *}R^{\prime }c^{*}r_R^{*} \quad \end{equation} \tag{ B11 }$$

where r_i, c, r_R and r_L are assumed to be small, and terms above second order are dropped.

In the following, the right-hand side of Equation (B11) is simplified into the underlying physics parameters Q and U in order to identify the sources of instrumental polarization. The terms L'*L' and R'*R' need only be calculated to leading order since they appear in Equation (B11) multiplied by the second order terms cr_L and $c^{*}r_R^{*}$. To leading order, L'*L' = L*L and R'*R' = R*R since τ_ij ≈ 1.

The first term in Equation (B11) is expanded by substituting Equations (B7) and (B8) and using L*R = (Q − iU)/2, LL* = (I + V)/2, and RR* = (I − V)/2 to obtain

$$\begin{eqnarray} L^{\prime *}R^{\prime } &= & \frac{1}{4}((\tau _{21}^{*}\tau _{31} + \tau _{24}^{*}\tau _{34})Q - i(\tau _{21}^{*}\tau _{34} + \tau _{24}^{*}\tau _{31})U \nonumber \\ && + (\tau _{21}^{*}\tau _{31}-\tau _{24}^{*}\tau _{34})I + (\tau _{24}^{*}\tau _{31} -\tau _{21}^{*}\tau _{34})V). \qquad \end{eqnarray} \tag{ B12 }$$

The first two terms are the expected response to Q and U. The presence of τ_ij in these terms parameterizes the imperfections in the septum polarizer transmissions. These terms reduce the gain to Q and U, and in general cause mixing between Q and U. In practice, the gain is absorbed into the calibration of the total system responsivity,⁴⁰ and the Q/U leakage is absorbed into the detector angle as defined in Equations (9) and (10). Therefore, these two terms do not cause instrumental polarization, and these imperfections can be neglected in the following discussion. By the same argument, the terms r₃r_R and $r_2^{*}r_L^{*}$ in Equation (B11) can be ignored since their only effect is to change the gain and detector angle.

The third and fourth terms represents I → Q/U and V → Q/U leakage, respectively. Since V ≪ I for reasonable sources and the coefficients have the same order, these circular polarization leakages are neglected. Combining these simplifications, the right-hand-side of Equation (B11) becomes:

$$\begin{eqnarray} L_m^{*}R_m &= & \frac{1}{4}\left[ 2\widetilde{Q} -2\widetilde{U} + (\tau _{21}^{*}\tau _{31}-\tau _{24}^{*}\tau _{34})I \right] \nonumber \\ && + \frac{1}{2}(cr_L + c^{*}r_R^{*})I, \end{eqnarray} \tag{ B13 }$$

where $2\widetilde{Q}=(\tau _{21}^{*}\tau _{31} + \tau _{24}^{*}\tau _{34})Q$ and $2\widetilde{U}=i(\tau _{21}^{*}\tau _{34} + \tau _{24}^{*}\tau _{31})U$. Using Equation (B3) and ignoring U → Q leakage, the module output is

$$\begin{equation} Q_m = \Re (\widetilde{Q}) + \frac{1}{2}\Re (\tau _{21}^{*}\tau _{31}-\tau _{24}^{*}\tau _{34}) I + \Re (cr_L + c^{*}r_R^{*})I, \end{equation} \tag{ B14 }$$

where the first term is the expected response, the second term is I → Q leakage due to differential loss, and the third term is leakage caused by reflections at the module inputs coupling with the septum polarizer cross talk. Similarly, using Equation (B4) and ignoring the Q → U leakage:

$$\begin{equation} U_m = \Im (\widetilde{U}) - \frac{1}{2}\Im (\tau _{21}^{*}\tau _{31}-\tau _{24}^{*}\tau _{34}) I - \Im (cr_L + c^{*}r_R^{*})I. \end{equation} \tag{ B15 }$$

In summary, the two equations above describe the measurements of a sky signal in the absence of noise from the module.

Now consider the case of noise emitted from the module inputs, reflecting from the septum polarizer and returning into the module. Module noise stems primarily from the HEMT-based first-stage LNAs. Since the sky signal and module noise are relatively incoherent, they decouple and the sky signal can be neglected in the following. Let the module noise fields be given by the column vector (0, L_r, R_r, 0). Applying the S matrix, the vector (−, L_m, R_m, −) is obtained where

$$\begin{equation} L_m = L^{\prime } = r_2L_r + cR_r \end{equation} \tag{ B16 }$$

$$\begin{equation} R_m = cL_r + r_3R_r. \end{equation} \tag{ B17 }$$

The output is

$$\begin{equation} L_m^{*}R_m = r_2^{*}L_r^{*}cL_r + c^{*}R_r^{*} r_3R_r \end{equation} \tag{ B18 }$$

because the L_rR_r terms average to zero due to the fact that the two amplifier noises are uncorrelated. Thus each output acquires an offset

$$\begin{equation} Q_m = 2L_r^{*}L_r\Re (r_2^{*}c) + 2 R_r^{*}R_r\Re (c^{*}r_3). \end{equation} \tag{ B19 }$$

$$\begin{equation} U_m = -2L_r^{*}L_r\Im (r_2^{*}c) - 2 R_r^{*}R_r\Im (c^{*}r_3) \end{equation} \tag{ B20 }$$

The offset is independent of the input I; however, it is modulated by gain fluctuations so the offset also contributes to 1/f noise at a level suppressed by the product of the cross-talk and reflection coefficients in the OMT and so would not dominate.