In this appendix we calculate the relationship between temperature fluctuations on the sky to optical power fluctuations seen by a detector. In the Rayleigh-Jeans limit, the intensity of a blackbody spectrum with temperatureTRJ is:
I⌫ = 2kTRJ
⌫2 c2
W
m2Hz sr , (A.14)
or, integrated over a finite bandwidth (to first order in ⌫):
I'2kTRJ⌫2 ⌫ c2
W
m2sr . (A.15)
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To calculate total intensity (instead of flux), we need to calculateA⌦pix, the solid angle subtended by a given detector multiplied by the effective light-collecting area. Since our detectors are single- moded, this is just equal to 2, up to factors of order unity. Here’s why: single moded optical systems preserveA⌦, which means that rather than calculating this quantity at the aperture (which is hard, because of the non-trivial illumination of the primary optic), we can calculate the quantity at the antenna. This is much easier, as the illumination pattern of the antenna is a top-hat. Differences can be accounted for by an overall efficiency factor⌘, which includes the total throughput.
⌦pix can be easily calculated for a top-hat of width w at a wavelength with the well-known formula:
⌦=
✓ 1.44w
◆2
. (A.16)
A⌦is then simply(1.44 )2.
Returning to our brightness fluctuation, we multiply byA⌦for a single detector and an overall optical efficiency factor⌘= 0.4and take the trivial derivative with respect toTRJ:
dI
dTRJ '2k⌫2⌘A⌦ ⌫
c2 = 4.14k⌘ ⌫'8⇥10 13
W KRJ
. (A.17)
To convert to TCMB, we multiply by a factor of 1.7, which is the ratio of the integral of a 3 K blackbody over this band to the Rayleigh-Jeans approximation. We finally arrive at the brightness temperature fluctuation relation:
dI
dTCMB '1⇥10 12
W KCMB
. (A.18)
Appendix B
Dielectric sheet calibrator model
Deriving precise polarization angle measurements using the dielectric sheet calibrator described in Section 3.4.1 requires an accurate model of the reflected and transmitted power through the beam splitter as a function of several model parameters. We describe that model below, following closely the model presented in (Takahashi 2010).
We will define the following local coordinate system in which to establish our model: +zincreases along the boresight toward the sky with z = 0 corresponding to the plane of the detectors. The positiveyaxis is aligned with✓= 0, as in Figure 3.19, withDKincreasing CCW. This puts detector tile 1 in the +x,+y quadrant of our coordinate system. We work in a coordinate frame in which the focal plane is fixed, so that polarization modulation results from motion of the dielectric sheet relative to the coordinate frame. We let represent the rotational position of the dielectric sheet with respect to the focal plane, equal to theDK angle of the telescope, with zero defined as usual.
As a practical matter, we use the telescope command coordinates because we are interested in the DK angle with respect to the ground, rather than the telescope topocentric horizontal coordinates (as in Equation C.3). Usually, these are very close to identical, but since this measurement is taken very near the coordinate singularity at zenith, the discrepancy between them can be large.
The tilt tof the dielectric sheet is measured from horizontal. This is nominally 39 degrees plus the measured tilt at the start of the scan, which is typically close to 6.4 degrees. The dielectric sheet is made from Mylar, which has a nominal index n = 1.83 (Lamb 1996). The Bicep2 data were taken with sheet thicknesses of 1 and 2 mil, measured in the lab with a micrometer. We assume a nominal band center = 2.07mm (145 GHz).
Calculating the normal vector of the dielectric sheet requires some of the same geometric ma- chinery used in Appendix C. We begin with the calibrator sheet’s normal vector co-aligned withˆz,
168 and then apply the following rotation matrices:
Rt= 2 66 64
1 0 0
0 cost sint 0 sint cost
3 77 75,R =
2 66 64
cos 0 sin
0 1 0
sin 0 cos 3 77 75,R =
2 66 64
cos sin 0
sin cos 0
0 0 1
3 77 75.
R is the rotation of the calibrator with respect to the focal plane due toDK rotation,R accounts for a potential lateral tilt1 of the dielectric sheet (typically small), and Rt accounts for the tilt angle of the dielectric sheet, typically close to 45 degrees. These rotation matrices obviously do not commute. The correct sequence of rotations can be applied2 to calculate the normal vector:
ˆ
n= ˆn0RtR R . (B.1)
In vector notation, this reduces to:
ˆ
n= (cos sin cost sintsin )ˆx+ ( costsin sin + sintcos )ˆy+ costcos ˆz. (B.2) The angle of incidence of each detector’s beam can be calculated by considering the ray of each detector as it exits the aperture. This is parameterized by rp and ✓p, the radial distance of the detector’s centroid from the boresight and its bearing, respectively. The bearing angle✓p increases CCW from the x axis of our coordinate system. These are equivalent to the beam centroids as measured from the boresight. The resulting ray is:
ˆb(rp,✓p) = sin(rp) cos(✓p)ˆx+ sin(rp) sin(✓p)ˆy+ cos(rp)ˆz. (B.3)
We calculate the angle of incidence with the dielectric sheet as:
✓i= arccos(ˆb·n).ˆ (B.4)
To simplify the notation, we introduce✓n, the angle of incidence within the dielectric:
✓n = arcsin(sin(✓i)/n). (B.5)
Similarly, we define:
=4⇡nd
cos(✓n). (B.6)
1In theBicep2observation logbook, the sign of the measured lateral tilt is recorded with a forward or backslash.
In our definition of , a backslash corresponds to positive , while a forward slash corresponds to negative .
2One might be tempted to switch the rotations oftand , but doing so would be incorrect: if we slew the telescope to a new elevation, we will still measure the same lateral tilt angle .
The last piece we need is the thing we are trying to measure: the axes of sensitivity of detectorsA andB. We parameterize the polarization angle, by which we mean the axes of maximum sensitivity, by . In the case that = 0, the polarization axes will be co-aligned with the vector that connects the boresight and✓p. The polarization angle is therefore referenced from ✓p. We represent the orientation of the linear polarization axes forAandBwith a “headed” vector, expressed in our local coordinate frame as:
A( ) = ( sin( ) cos(rˆ p) cos(✓p) cos( ) sin(✓p))ˆx (B.7) + cos(✓p) cos( ) sin( ) cos(r) sin(✓)ˆy (B.8)
+ sin( ) sin(r)ˆz (B.9)
B( ) = (cos( ) cos(rˆ p) cos(✓p) sin( ) sin(✓p))ˆx (B.10) + cos(✓p) sin( ) + cos( ) cos(r) sin(✓)ˆy (B.11)
cos( ) sin(r)ˆz. (B.12)
Note that in this step we have assumed perfect orthogonality betweenA andB, so that the polar- ization axes can be described by a single angle, rather than fitting the two angles separately. We make this assumption because of the large common-mode atmospheric contamination in Aand B. By performing a fit to the pair-difference signal rather than individually fitting A and B, we can reject the variable atmospheric signal and vastly improve the fit. We also note that while we have represented the polarization pair with the single angle , we can readily recover the polarization angles ofAandB:
A= (B.13)
B= +⇡/2. (B.14)
For the sake of clarity, we will also introduce an angle↵to visualize deviations of the polarization angle from nominal. We define this angle to be:
↵= ( +✓p) + 90. (B.15)
In this definition,↵will be close to 0 for allAdetectors, and close to±90for allB detectors.
We can now calculate the coupling fraction of polarizationsAandB to the reflected TE mode:
fA,TE= ˆA· ˆb⇥ˆn
|ˆb⇥ˆn| (B.16)
fB,TE= ˆB· ˆb⇥ˆn
|ˆb⇥ˆn|. (B.17)
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From orthogonality, the coupling fraction to the TM mode can be calculated from the coupling to the TE mode as:
fA,TM=q
1 fA,TE2 (B.18)
fB,TM=q
1 fB,TE2 . (B.19)
As computed in Takahashi 2010, we can find the reflection coefficient of the TE and TM modes by a superposition of the reflections that occur upon entering and exiting the dielectric material:
RTE,TM= r21+r22+ 2r1r2cos
1 +r21r22+ 2r1r2cos . (B.20) Here, r1 and r2 are the reflection coefficients from entering and exiting the dielectric material, calculated as:
r1,TE=cos✓i ncos✓n
cos✓i+ncos✓n
, r2,TE= r1,TE (B.21)
r1,TM= ncos✓i cos✓n
ncos✓i+ cos✓n
, r2,TM = r1,TM. (B.22)
The reflected modes seen by detectorsAandB can finally be calculated:
RA=RTEfA,TE2 +RTMfA,TM2 =RTM+ (RTE RTM)fA,TE2 (B.23) RB=RTEfB,TE2 +RTMfB,TM2 =RTM+ (RTE RTM)fB,TE2 . (B.24) By pair differencing, we reject the transmitted modes through the beam splitter, and we are left with the purely polarized intensity, which is:
Idi↵= I(RA RB). (B.25)
This model thus reduces to two free parameters: the angle of the polarization axes,↵, and an ampli- tude that corresponds to the effective brightness difference between the warm absorber surrounding the dielectric sheet and the sky, I.
Appendix C
Three-dimensional pointing model for Bicep2 beam mapping
In this appendix, we document a fully general pointing model used for analyzing beam maps taken at the South Pole. Bicep2employs a folding flat mirror to redirect the main beam to a point near the horizon, which sits 2.98 m above the intersection of theAZ, EL, andDK rotation axes. With a source distance of⇠200 m, the movement of the mirror introduces parallax effects that must be taken into account.
Our approach differs somewhat from the offline pointing model, used to reconstruct the boresight position on the sky using fits derived from star pointing. There, the analysis focuses on a variety of coordinate transformations, given various tilt, offset, and flexure terms. Rather than approaching the problem as a series of coordinate transformations, we instead take a “ray tracing” approach, treating the boresight as a vector in a fixed coordinate system. We represent each of the detectors’
chief rays with a vector, and then perform a series of operations on this “ray bundle” to recover the projected elevation and azimuth of the detector centroid on a sphere with a radius equal to the source distance. The origin of our coordinate system is the intersection of the elevation, azimuthal, and boresight axes, which we assume to be coincident. This is a good approximation for Bicep2, but not necessarily so for Keck. However, this can be accounted for by simple vector addition.
We use the same definitions as in Yoon 2007 for a few distinct coordinate systems that will be referenced:
[AZ, EL, DK]0, Raw encoder counts (C.1) [AZ, EL, DK]c, Command coordinates (C.2) [AZ, EL, DK], Topocentric ideal horizontal coordinates. (C.3)
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