Polarized Beam Patterns from a Multi-Moded Feed For Observations of the Cosmic Microwave Background

We measure linearly polarized beam patterns for a multi-moded concentrator and compare the results to a simple model based on geometric optics. We convolve the measured co-polar and cross-polar beams with simulated maps of CMB polarization to estimate the amplitude of the systematic error resulting from the cross-polar beam response. The un-corrected error signal has typical amplitude of 3 nK, corresponding to inflationary B-mode amplitude r ~ 10^{-3}. Convolving the measured cross-polar beam pattern with maps of the CMB E-mode polarization provides a template for correcting the cross-polar response, reducing it to negligible levels.


Introduction
Linear polarization of the cosmic microwave background provides a unique window into the physics of the early universe. Gravitational waves created during an inflationary epoch interact with the CMB at later times to impart a distinctive signature in linear polarization. For the simplest (single-field) inflation models, the amplitude of this B-mode signal depends on the energy scale of inflation as E = 1.06 × 10 16 r 0.01 1/4 GeV (1) where r is the power ratio of gravitational waves to density fluctuations. 1  The inflationary signal is faint compared to the fundamental limit imposed by photon noise statistics for a single-moded detector in a one-second integration. Even ideal (noiseless) detectors suffer from this limit; the only solution is to collect more photons. The light-gathering ability of a detector can be specified by its etendu AΩ, where A is the detector area and Ω is the solid angle. Increasing the etendu adds modes, thereby increasing both the signal and the photon noise.
The modes are independent, so that the noise increases as the square root of the etendu NEP ∝ (AΩ) 1/2 . But since the signal increases linearly with etendu, the signal-to-noise ratio improves as (AΩ) 1/2 , increasing the overall system sensitivity.
A common implementation for CMB polarization increases the instrument etendu using a focal plane tiled with thousands of detectors (for a recent review, see [3]). These systems typically couple the detector to the sky using optical structures (feed horn, lenslet, phased antenna array) which restrict the system response to a single electromagnetic mode at the sensor. Single-moded systems achieve diffraction-limited angular resolution with well-defined (Gaussian) beam profiles, but the large number of sensors required drives system-level complexity and cost, both directly and through the accompanying need for cryogenic multiplexing to reduce the wire count to cold stages of the instrument.
Multi-moded sensors provide an alternative design solution. The beam width of a diffractionlimited system observing a single polarization state in a single mode scales with wavelength, AΩ = λ 2 . In a multi-moded system, the etendu is fixed so that the number of modes scales as N mode = AΩ/λ 2 . The different electromagnetic modes form an orthogonal basis set and add incoherently at the detector. Assuming nearly equal occupancy for most modes, the detector sensitivity then scales as N 1/2 mode . For detector area A λ 2 , the number of modes is large, allowing a corresponding reduction in detector count compared to a single-moded system of comparable sen-sitivity.
The Primordial Inflation Explorer (PIXIE) 4, 5 is a mission concept to measure CMB polarization at levels r < 10 −3 . PIXIE combines four multi-moded detectors with a polarizing Fourier transform spectrometer (FTS) to achieve background-limited sensitivity across a broad frequency range. Figure 1 shows the optical design. A pair of primary mirrors 55 cm in diameter produce two co-aligned beams on the sky. Folding flats and secondary mirrors transfer the beams to the FTS, which mixes the beams and introduces an optical phase delay. The recombined beams are then routed to a pair of non-imaging concentrators, each of which contains two multi-moded bolometric detectors to sense the fringe pattern as a function of optical phase delay. The instrument etendu of 400 mm 2 sr corresponds to flat-topped beam with full width at half maximum 2.6 • on the sky, and is conserved throughout the optical path. An important aspect of the proposed instrument design is the use of non-imaging concentrators to couple light from the FTS to the detectors. Each detector is sensitive to a single linear polarization; any cross-polar response to the orthogonal polarization produces an effective rotation of the measured polarization with respect to the true polarization on the sky. The cross-polar response of single-moded structures is well understood. A corrugated circular feedhorn, for example, will typically exhibit cross-polarization at levels -25 dB with a distinctive quadrupolar (cloverleaf) angular pattern. 6 The cross-polar response of multi-moded structures is more complicated. We describe measurements and modeling of the co-polar and cross-polar beam patterns for a rectangular multimoded concentrator designed for the PIXIE polarimeter, and assess the impact of the measured cross-polar response for measurements of the inflationary polarization signal.

Polarized Beam Pattern Measurements
We measured the co-polar and cross-polar beam patterns of a scaled version of the PIXIE concentrator. Measurements took place within the Goddard Electromagnetic Anechoic Chamber (GEMAC) facility and are similar to previous measurements described in [7]. Briefly, we transmit power in a single linear polarization and use an unpolarized Thomas Keating THz absolute power meter to record the power at the concentrator as a function of angle from the point-source transmitter. A wire grating between the power meter and the exit aperture of the concentrator defines the polarization state at the power meter. The grating can be rotated to sample either theû orv polarization. An absorbing iris at entrance aperture of the concentrator truncates the corners of the aperture and serves as a beam stop to circularize the beam patterns.
Although PIXIE's FTS has frequency response to THz frequencies, the fixed noise of the power meter and the limited broadcast power available at frequencies above 100 GHz make direct mea-surements at higher frequencies impractical. Instead, we increase all dimensions of the PIXIE concentrator by a factor of three, and measure the beam patterns of the scaled concentrator at frequencies 10.8 GHz, 35 GHz, and 91 GHz corresponding to sky measurements of 32 GHz, 105 GHz, and 273 GHz for the un-scaled concentrator. The frequencies are selected to measure the beam pattern in the few-mode limit, the multi-moded geometric optics limit, and an intermediate case ( Table 1). Note that the frequency spectrum of the CMB polarization follows the derivative dB/dT of a blackbody spectrum, and peaks at frequencies near 270 GHz well into multi-moded operation. GEMAC data at 91 GHz sample the beam pattern near this peak.
The measured beam patterns include the response of the concentrator as well as the detector.
The PIXIE detectors use silicon wires degeneratively doped with phosphorus to absorb light in   The co-polar beam patterns measure the response of the concentrator to a linearly polarized plane wave, when the polarization of the incident beam matches the polarization accepted by the detector (e.g.û incident polarization measured by theû detector). We also measure the cross-polar response when the detector is orthogonal to the incident polarization (û incident polarization measured by thev detector). Figure 6 shows the cross-polar beam pattern measured at 35 GHz. As with The cross-polar beam pattern has a similar flat-topped shape as the co-polar beam, but at a reduced amplitude. The dominant effect of the cross-polar response is a modest reduction in polarization efficiency ( Table 2). The measured beam patterns correspond to polarization efficiency of ∼0.95, nearly independent of frequency. The polarization efficiency reduces the instrument response to a linearly polarized sky signal, and may be absorbed into the instrument calibration.  It thus affects instrument signal-to-noise estimates, but does not induce systematic error in the measured sky polarization.

Comparison with Model
We model the co-polar and cross-polar beam patterns using a ray-trace code in the geometric optics limit. As discussed in [7], we account for operation in the few-mode limit by first binning the modeled beam pattern on a rectangular grid, Fourier transforming the binned beam map, sorting the complex Fourier coefficients by angular frequency, and Fourier transforming back to real space using only the lowest N mode values. This removes the high spatial frequency information from the modeled beam pattern, approximating the effect of few-mode operation without computationally  We generate random realizations of CMB polarization using a standard ΛCDM cosmology, 9 evaluated to harmonic moment = 1500 (angular scale 7 ). We convolve the simulated Stokes Q and U maps with the scaled beam to generate the instrument response at fixed spin angle γ, then rotate the beam with respect to the sky and repeat for 32 angular steps uniformly covering spin angle [0, 2π]. Although the PIXIE fore-optics and FTS interfere the signals from two co-pointed beams, we ignore any additional cancellation of beam effects from the fore-optics and simply scale the beam patterns measured from the concentrator. The resulting signal is thus an upper limit to the systematic error expected from the full PIXIE optical system. beam (corresponding to 105 GHz sky signal). The "tilt" in the beam from the off-axis illumination modulates the response at the spin period. Since a true polarization signal is modulated twice per spin, the tilt does not contribute to the reconstructed polarization. We fit the simulated data to Compared to the inflationary B-modes, the m = 2 error from the measured cross-polar response corresponds to mean r = 1.2 × 10 −3 with standard deviation r = 0.6 × 10 −3 . Note that this represents an upper limit to the potential systematic error before correction. CMB polarization is dominated by the E-mode component. The measured cross-polar beam pattern may be convolved with maps of the E-mode polarization to correct the cross-polar response, further reducing its effect on B-mode searches.

Conclusion
We measure the co-polar and cross-polar beam patterns from a multi-moded concentrator designed for the PIXIE polarimeter. The beams provide a flat-topped pattern on the sky and are fully symmetrized between orthogonal linear polarizations. The cross-polar beam pattern has similar angular dependence as the co-polar beam; the dominant effect of the cross-polar response is a reduction in polarization efficiency to a measured efficiency of 95%. Structure in the cross-polar response on angular scales smaller than the 2.6 • co-polar beam creates a systematic error when reconstructing the polarization signal from the sky. Monte Carlo simulations show the un-corrected error signal to have mean amplitude 3.3 ± 1.7 nK, corresponding to an error δr = [1.2 ± 0.6] × 10 −3 for the inflationary B-mode signal. Convolving the measured cross-polar beam pattern with maps of the CMB E-mode polarization provides a template for correcting the cross-polar response, reducing it to negligible levels.