# Polarization Properties of A Broadband Multi-Moded Concentrator

Alan Kogut  NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
Dale J. Fixsen  NASA Goddard Space Flight Center, Greenbelt, Maryland 20771,
USA  Robert S. Hill  NASA Goddard Space Flight Center, Greenbelt, Maryland
20771, USA

###### Abstract

We present the design and performance of a non-imaging concentrator for use in
broad-band polarimetry at millimeter through submillimeter wavelengths. A
rectangular geometry preserves the input polarization state as the
concentrator couples $f/2$ incident optics to a $2\pi$ sr detector.
Measurements of the co-polar and cross-polar beams in both the few-mode and
highly over-moded limits agree with a simple model based on mode truncation.
The measured co-polar beam pattern is nearly independent of frequency in both
linear polarizations. The cross-polar beam pattern is dominated by a uniform
term corresponding to polarization efficiency 94%. After correcting for
efficiency, the remaining cross-polar response is -18 dB.

###### pacs:

(030.4070) Modes (040.2235) Far infrared or terahertz; (080.4298) Nonimaging
optics; (220.1770) Concentrator; (260.5430) Polarization; (350.1260)
Astronomical optics.

##  I Introduction

The advent of bolometric detectors with background-limited sensitivity has
important implications for astronomical instrumentation at millimeter and
submillimeter wavelengths. Once the detector phonon noise falls below the
background from photon statistical fluctuations, further sensitivity gains can
only be realized by collecting additional photons. A common implementation
increases the effective detecting area through an array of individual sensors,
each coupled to the sky through an optical structure (feed horn, lenslet,
phased antenna array) that restricts 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.

An alternative design uses a much smaller number of detectors capable of
sensing multiple modes of the incident field. A multi-moded system uses non-
imaging optics (a “light bucket” or similar feed structure) to fix the beam
size independent of wavelength. Since the Lagrange invariant is conserved, in
order for the beam size to remain constant the number of modes must scale with
wavelength, $N_{\rm mode}=A\Omega/\lambda^{2}$, where $A$ is the detector
area, $\Omega$ is the solid angle, and $\lambda$ is the observing wavelength.
The different electromagnetic modes form an orthogonal basis set and add
incoherently at the detector. The number of detectors required to reach a
given sensitivity thus scales with etendu as $(A\Omega)^{-1}$, inversely
proportional to the number of modes.

Multi-moded systems can be particularly useful for observations of the linear
polarization of the cosmic microwave background (CMB), where the signals of
interest lie well below the photon background noise. Polarimetry adds an
additional level of complexity to the optical system. Multi-moded systems
typically require a concentrator or similar feed structure to couple light
from the sky to the detector while conserving etendu. Polarization sensitivity
can be achieved using a detector sensitive to both polarizations, with a
polarization diplexer (e.g. a wire grid analyzer) upstream of the detector.
Since the polarization separation occurs before light enters the concentrator,
the polarization properties of the concentrator do not matter. Alternatively,
the detector itself may be sensitive to a single linear polarization. This
avoids the need for a separate polarization diplexer, but requires the
concentrator to preserve the incident polarization state without cross-polar
mixing (see, e.g., the discussion in kusaka/etal:2014 ). Instruments in this
latter class include the planned PIXIE (kogut/etal:2011, ), MUSE
(kusaka/etal:2012, ) and LSPE (aiola/etal:2012, ) missions.

The design of non-imaging concentrators has been well studied for unpolarized
observations (winston:1970, ; winston/welford:1979, ; minano:1985, ;
ogallagher/etal:1987, ; ning/etal:1987, ; garcia-botella/etal:2009, ). Three-
dimensional concentrators formed as a surface of revolution can approach the
theoretical limit on concentration, but the resulting azimuthal symmetry
induces an undesirable cross-polar response. In this paper, we describe the
design and performance of a non-imaging concentrator using a rectangular
geometry suitable for CMB polarimetry.

Figure 1:  Schematic of the off-axis non-imaging rectangular concentrator. The
square entrance and exit apertures preserve incident polarization. Each of the
4 walls is an elliptical section to transform the $f/2$ incident beam to the
$2\pi$ sr solid angle at the detector while preserving etendu.

##  II Multi-moded rectangular concentrator

Figure 1 illustrates the concentrator, derived from the requirements of the
PIXIE instrument. We define a right-handed coordinate system $[x,y,z]$ with
the detector centered in the $xy$ plane and the concentrator extending in the
$+z$ direction. Light from the $f/2$ beam-forming optics is incident on the
concentrator at a 15∘ angle from the $\hat{z}$ axis. Within these constraints,
we use a rectangular geometry for the concentrator to minimize cross-polar
mixing of the incident light. Each of the 4 walls of the concentrator is an
elliptical section, optimized from an iterative ray-tracing algorithm to
maximize the total throughput in the geometric optics limit. We define the
ellipse in each plane as a quadratic equation

$\displaystyle M_{1}(x-c_{1})^{2}+M_{2}(z-c_{2})^{2}+2M_{3}(x-c_{1})(z-c_{2})$
$\displaystyle=$ $\displaystyle 1~{}~{}({\rm top~{}face})$ $\displaystyle
M_{1}(x-c_{1})^{2}+M_{2}(z-c_{2})^{2}+2M_{3}(x-c_{1})(z-c_{2})$
$\displaystyle=$ $\displaystyle 1~{}~{}({\rm bottom~{}face})$ $\displaystyle
M_{1}(y-c_{1})^{2}+M_{2}(z-c_{2})^{2}+2M_{3}(y-c_{1})(z-c_{2})$
$\displaystyle=$ $\displaystyle 1~{}~{}({\rm side~{}faces})$ (1)

for the top, bottom, and side faces, respectively. Table 1 lists the
coefficient values $M$ and $c$. The two side panels ($+\hat{y}$ and
$-\hat{y}$) are identical, while the top and bottom panels ($+\hat{x}$ and
$-\hat{x}$) have different elliptical sections to accommodate the off-axis
design. The projected entrance aperture thus appears approximately square.

Table 1: Concentrator Elliptical Figure Face | $c_{1}$ | $c_{2}$ | $M_{1}$ | $M_{2}$ | $M_{3}$
---|---|---|---|---|---
Top | -2.6043E+01 | 4.9987E+02 | 1.3455E-03 | 1.0025E-04 | 1.7423E-05
Bottom | 3.5040E+02 | 6.0139E+02 | 5.3181E-04 | 1.9821E-04 | -2.6392E-04
Left | -1.7033E+02 | 5.6460E+02 | 8.7479E-04 | 1.3043E-04 | -2.1488E-04
Right | 1.7033E+02 | 5.6460E+02 | 8.7479E-04 | 1.3043E-04 | 2.1488E-04
Note: Units are in mm. Values describe the larger scaled concentrator ($\S$3).

The exit aperture of the concentrator illuminates a pair of detectors, each
sensitive to a single linear polarization. Each detector covers the full
aperture; they are mounted one behind the other so that both are fully
illuminated. The two detectors are rotated by 90∘ such that one detector is
sensitive to polarization in the $\hat{x}$ direction and the other is
sensitive to polarization in the $\hat{y}$ direction. Each detector has etendu
$A\Omega=4$ cm2 sr. We model the co-polar and cross-polar response of the
concentrator using a time-reversed Monte Carlo approach. For each linear
polarization, we generate a set of 63 million outgoing rays, each originating
from a random position on the detector. The ray angular distribution is
uniform in azimuth, but takes into account the additional $cos^{2}(\theta)$
angular response of a thin resistive detector at the concentrator exit
aperturekusaka/etal:2014 . The distribution in polarization is uniform. A ray-
tracing algorithm follows each ray through multiple bounces within the
concentrator (assumed to be perfectly reflective) to determine the orientation
and polarization angle after leaving the concentrator.

The PIXIE concentrator does not view the sky directly, instead coupling to the
beam-forming optics through an $f/2$ coupling mirror. We extend the ray-
tracing algorithm to follow each ray out of the concentrator and and through
one reflection from the coupling mirror. The resulting set of rays
approximates the beam pattern of the concentrator/coupling mirror combination,
which can then be used as input through the remaining optical system. The co-
polar and cross-polar response of any downstream beam-forming optics can
readily be calculated, and and not treated here.

Figure 2:  Block diagram for beam pattern measurement.

Observations of the cosmic microwave background can span more than a decade in
frequency, from 30 GHz to 600 GHz. The corresponding number of modes within
the concentrator ranges from $N_{\rm mode}=4$ at the lowest frequency to
$N_{\rm mode}=1600$ at the highest frequency. The concentrator thus samples
the transition from a few-mode system to a highly over-moded system. Several
authors have discussed methods to estimate the beam pattern for non-imaging
concentrators in the few-mode limit (murphy/padman:1991, ; gleeson/etal:2002,
; thomas/withington:2013, ). Typically these methods require explicit mode
matching. Although mode matching successfully recovers the full phase and
amplitude information needed to generate far-field beam patterns, this method
is computationally expensive and is difficult to apply for off-axis geometries
where a full 3-dimensional treatment is required. To compare the modeled beam
patterns to measurement, we employ a simpler approximation. We use the beam
pattern from the ray-tracing algorithm to model the beam in the geometric
optics limit. We then mathematically expand the beam using an orthogonal set
of basis functions, and recompute the beam using a truncated subset of these
functions to account for mode loss at low frequencies. The rectangular
symmetry of the concentrator is well matched to the orthogonal basis functions
of a two-dimensional Fourier transform. We thus bin the co-polar or cross-
polar beam derived from the geometric optics model on a $6{{}^{\prime}\
}\times 6{{}^{\prime}\ }$ grid, and Fourier transform the binned beam map to
derive a set of complex Fourier coefficients. We sort the coefficients by
angular frequency and retain only the lowest $N_{\rm mode}$ values, setting
the remaining modes to zero. We then Fourier transform back to real space to
generate a model with the desired number of modes. Finally, we convolve the
mode-truncated pattern with the Airy pattern for the appropriate observing
frequency to approximate the effects of diffraction at the concentrator walls.

Figure 3:  Schematic showing the concentrator and coupling mirror mounted for
measurement in the GEMAC azimuth/elevation table, showing the relation between
the table and concentrator coordinate systems.

Figure 4:  Co-polar and cross-polar beam maps of the scaled concentrator /
coupling mirror system. The measurement of the scaled concentrator at 90 GHz
corresponds to 270 GHz for PIXIE, well within the geometric optics limit. The
transmitter is polarized in the $\hat{x}$ direction for the copolar map and
the $\hat{y}$ direction for the cross-polar map. The co-polar beam is well
described by a tophat with diameter 14∘. The cross-polar beam is dominated by
a similar tophat at -12 dB (corresponding to polarization efficiency 0.94),
with smaller-scale structure at the -18 dB level.

##  III Beam pattern measurements

We measured the co-polar and cross-polar beam patterns of the rectangular
concentrator in the Goddard Electromagnetic Anechoic Chamber (GEMAC). Figure 2
shows the experimental setup. A transmitter within the GEMAC uses a standard
rectangular gain feed to launch a single linear polarization toward a shaped
reflector. The reflector transforms the outgoing signal to a plane wave,
effectively placing the transmitter at infinity. We include a coupling mirror
in the GEMAC setup to convert the transmitted plane wave to the $f/2$ beam
generated by the PIXIE coupling mirror, and measure the copolar and cross-
polar beams from the concentrator/mirror system. The GEMAC coupling mirror
differs from the PIXIE mirror in that the PIXIE mirror uses an elliptical
surface to focus on the next mirror in the instrument while the GEMAC coupling
mirror uses a parabolic surface to focus on the transmitter at infinity. Since
the GEMAC setup excludes the remainder of the beam-forming optics, the
measured beam patterns represent only the concentrator/coupling mirror
combination and are not a measure of the final instrument beams on the sky.

The concentrator/mirror system is mounted on a table and can be moved in
azimuth and elevation. Figure 3 show the relative orientation of the table and
concentrator coordinate systems. Viewed from the detector in the concentrator
$xy$ plane, moving the table towards positive azimuth moves a source at
infinity toward positive $\hat{x}$ (as viewed through the coupling mirror).
Similarly, moving the table towards positive elevation moves a source at
infinity toward positive $\hat{y}$. To suppress reflected signals, all
surfaces except the coupling mirror and concentrator aperture were covered
with a single layer of Eccosorb AN72 microwave absorber. The absorber at the
edges of the coupling mirror acts as a beam stop, truncating the corners to
produce the desired tophat geometry.

We modulate the transmitted power by electronically switching the transmitted
power with a square wave at 20 Hz, and use a Thomas Keating THz absolute power
meter tk_meter  at the exit aperture of the concentrator to detect and
synchronously demodulate the signal. The power meter is insensitive to
polarization; a free-standing wire grid polarizer mounted between the power
meter and the concentrator exit aperture provides polarization sensitivity.
The polarizer is flush to the concentrator exit aperture while the power meter
is parallel to the exit aperture but 3 mm further back within a reflective
integrating cavity. The wire grid polarizer consists of copper-clad tungsten
wires 40 microns in diameter (36 $\mu$m tungsten and 1.3–2.5 $\mu$m copper) on
a 118 $\mu$m pitch. With the grid wires oriented parallel to the elevation
direction, the $\hat{y}$ polarization is reflected and the power meter is
sensitive to $\hat{x}$ polarization. The grid can be rotated by 90∘ so that
the wires are parallel to the azimuth direction. In this orientation, the
$\hat{x}$ polarization is reflected and the power meter is sensitive to
$\hat{y}$ polarization. The measured polarization isolation is better than -27
dB at frequencies 10 GHz to 300 GHz.

Table 2: Measured Map Power Frequency | Max Co-Pol | Max X-Pol | Noise Co-Pol | Noise X-Pol
---|---|---|---|---
(GHz) | ($\mu$W) | ($\mu$W) | ($\mu$W) | ($\mu$W)
11 | 1560 | 91 | 1.4 | 0.9
29 | 700 | 73 | 0.7 | 0.5
90 | 210 | 24 | 0.4 | 0.5

With the transmitter fixed to broadcast $\hat{x}$ polarization, we orient the
polarizer grid parallel to the elevation direction to measure the XX co-polar
response. We fix the elevation angle at $-10{{}^{\circ}}$ and continuously
sweep in azimuth from $-12{{}^{\circ}}$ to $+12{{}^{\circ}}$, recording data
from $-10{{}^{\circ}}$ to $+10{{}^{\circ}}$ to avoid possible startup
transients in the azimuth drive. We then increment the elevation by 1∘ and
repeat to map the concentrator/mirror response over a $20{{}^{\circ}}\times
20{{}^{\circ}}$ field. We then rotate the transmitter to the $\hat{y}$
polarization while leaving the polarizer grid parallel to the elevation
direction to measure the YX cross-polar response. Finally, we rotate the wire
grid polarizer by 90∘ and repeat to measure the YY co-polar and XY cross-polar
response. Since the off-axis concentrator is slightly different in the
$\hat{x}$ and $\hat{y}$ directions, we expect slightly different responses for
the 4 beam patterns.

We wish to measure the beam patterns at frequencies representative of both the
few-mode and highly-overmoded limits. The limited broadcast power available at
frequencies above 100 GHz makes direct beam mapping at these frequencies
impractical. We thus increase the dimensions of the nominal PIXIE concentrator
by a factor of three, and measure the larger (scaled) concentrator at lower
frequencies where higher broadcast power can be achieved. We measured all four
beam patterns (XX, YX, YY, and XY) at each of frequencies 11 GHz, 29 GHz, and
90 GHz, corresponding to CMB frequencies 33, 87, and 270 GHz. Table 2
summarizes the measurements. The maximum co-polar power measured at each
frequency ranged from 1500 $\mu$W at 11 GHz to 210 $\mu$W at 90 GHz with rms
noise 1.4 $\mu$W to 0.4 $\mu$W per 0.1∘ pixel. The noise floor is thus at
approximately -30 dB at each frequency.

---
Figure 5: Measured beam patterns for the scaled concentrator/coupling mirror
system compared to model predictions in the $\hat{x}$ direction. The solid
black line shows the co-polar (XX) beam while the dotted line shows the cross-
polar (YX) beam. Data are taken with $\hat{y}$ fixed at 0∘. Grey lines show
the model predictions after accounting for diffraction and mode loss.
Measurements at 11, 29, and 90 GHz correspond to 5, 34, and 324 modes
respectively, spanning the transition from a few-mode system to a highly over-
moded system.
---
Figure 6: Measured and model beam patterns for the copolar (XX) and cross-
polar (YX) beams in the $\hat{y}$ direction. Data are taken with $\hat{x}$
fixed at 0∘. Model and measured data are as in Fig 5.

At each frequency, we define the co-polar and cross-polar beam patterns as

$\displaystyle G_{c}(x,y)$ $\displaystyle=$
$\displaystyle\frac{P_{c}(x,y)\Omega}{\int P_{c}(x,y)\,d\Omega}$ (2)
$\displaystyle G_{x}(x,y)$ $\displaystyle=$
$\displaystyle\frac{P_{x}(x,y)\Omega}{\int P_{c}(x,y)\,d\Omega}$ (3)

where $P_{c}(x,y)$ is the co-polar power measured at each point $(x,y)$,
$P_{x}(x,y)$ is the measured cross-polar power, and

$\Omega=\frac{\left(\int P_{c}(x,y)\,d\Omega\right)^{2}}{\int
P^{2}_{c}(x,y)\,d\Omega}$ (4)

is the co-polar beam solid angle. With this choice of normalization the beam
solid angle is simply related to the beam pattern,

$\Omega=\int G_{c}(x,y)\,d\Omega\,,$ (5)

but with a peak amplitude greater than unity, $G_{\rm max}>1$ kraus:1986 .

Table 3: Beam Width Frequency | Co-Pol | Solid | FWHM | Equivalent
---|---|---|---|---
(GHz) | Map | Angle | (deg) | Tophat
|  | (sr) |  | (deg)
11 | XX | 0.0502 | 12.5 | 14.5
11 | YY | 0.0515 | 13.4 | 14.7
29 | XX | 0.0483 | 12.6 | 14.2
29 | YY | 0.0477 | 12.4 | 14.1
90 | XX | 0.0470 | 12.7 | 14.0
90 | YY | 0.0468 | 12.7 | 14.0

Figure 4 shows the co-polar (XX) and cross-polar (XY) beam pattern measured at
90 GHz in the GEMAC, corresponding to 270 GHz for PIXIE. The co-polar beam is
described by a tophat with diameter 14∘. To lowest order, cross-polar beam
resembles the co-polar beam but with amplitude reduced by 12 dB. Additional
higher-order structure in the cross-polar beam is also visible, reflecting the
fourfold symmetry of the concentrator walls.

Figure 5 compares the measured beam maps to the model. Each panel shows a one-
dimensional slice through the beams along the $\hat{x}$ (azimuth) axis at
$\hat{y}=0$. We show the co-polar XX beam pattern (transmitter broadcasting in
$\hat{x}$ and concentrator receiving in $\hat{x}$) as well as the cross-polar
YX pattern (transmitter broadcasting in $\hat{y}$ and concentrator receiving
in $\hat{x}$). The co-polar beams agree well with the model and show the
expected broadening from mode loss at low frequencies. The cross-polar
response is reduced by $\sim$12 dB from the co-polar response and is dominated
by the same tophat structure as the co-polar beam. We observe similar results
for slices in elevation ($\hat{x}=0$) (Figure 6). Note that the right-most
panels in Figures 5 and 6 correspond to horizontal or vertical slices through
the 90 GHz beam shown in Figure 4. The measured beam patterns show the
qualtitative features predicted by the mode-truncated model. Differences
between the measured and modeled beam patterns are dominated by an amplitude
scale factor which increases at lower frequency. Reflections from the mounting
plate beneath the concentrator/mirror system (Figure 3) could account for such
an effect.

---
Figure 7: On-axis response vs polarization angle measured at 11 GHz.

Diffraction and mode truncation combine to apodize the beam edges. The full
width at half maximum (FWHM) of the measured beam is 12.7∘. Power outside the
half-power point broadens the beam so that the FWHM does not fully describe
the co-polar beam. Table 3 shows the FWHM for each co-polar beam, along with
the diameter of an ideal circular tophat with the same solid angle as the
measured beam. Although the FWHM is nearly independent of frequency, the
equivalent tophat diameter increases slightly at the lowest frequency, showing
the effect of mode loss and diffraction.

###  III.1 Polarization Efficiency and Cross-Polar Residuals

A cross-polar response proportional to the co-polar beam is equivalent to a
loss in polarization efficiency, but does not generate systematic error when
mapping the polarized sky. Figure 7 illustrates this point. With the
concentrator held stationary, we record the received power at 11 GHz while the
transmitter rotates in polarization angle $\theta$. The measured power shows
the expected $\sin(2\theta)$ modulation with a polarization efficiency 94%
corresponding to an on-axis cross-polar response of -12 dB.

Table 4: Polarization Properties Frequency | Co-Pol | Cross-Pol | Polarization | Cross-Pol
---|---|---|---|---
(GHz) | Map | Map | Efficiency | (dB)
11 | XX | YX | 0.94 | -19
11 | YY | XY | 0.95 | -20
29 | XX | YX | 0.93 | -17
29 | YY | XY | 0.93 | -15
90 | XX | YX | 0.97 | -20
90 | YY | XY | 0.94 | -17

We estimate the non-uniform cross-polar response by fitting each cross-polar
beam map to the corresponding co-polar map,

$\alpha=\frac{\int G_{x}G_{c}\,d\Omega}{\int G_{c}^{2}\,d\Omega}$ (6)

and computing the rms amplitude of the residual

$R_{x}(x,y)=G_{x}(x,y)-\alpha G_{c}(x,y)~{}.$ (7)

Figure 8 compares the residual cross-polar response to the co-polar beam
measured at 90 GHz. The beams are shown on a linear scale to highlight the
cross-polar structure. The residual cross-polar response shows the effects of
the square aperture and walls. The asymmetry in the $\hat{x}$ direction
results from the corresponding asymmetry in the concentrator due to the off-
axis design (Figure 1). Table 4 lists the polarization efficiency and rms
cross-polar residuals measured for both polarizations at all three
frequencies. Averaged over the full beam, the polarization efficiency is 94%
and the rms cross-polar residual is -18 dB.

##  IV Conclusion

We have designed and tested an off-axis rectangular non-imaging concentrator
for use with multi-moded polarization-sensitive optical systems. The
concentrator preserves linear polarization and conserves the optical etendu
while transforming the beam from $f/2$ at the entrance aperture to $2\pi$ sr
at the detector ($f/0.5$). The measured co-polar beam pattern is nearly
independent of frequency in both linear polarizations. The co-polar beam is
well described by a tophat of angular diameter 14∘ with minimal internal
structure on smaller angular scales. The cross-polar response is dominated by
a uniform tophat, leading to polarization efficiency 94%. After removing the
uniform term, the residual cross-polar response is typically -18 dB.

---
Figure 8: Co-polar beam and cross-polar residual measured at 90 GHz. Contours
show the XX co-polar beam on a linear scale at amplitudes 0.01, 0.05, 0.1,
0.5, and 1. The YX cross-polar residuals are shown in color, with red positive
and blue negative.

Several improvements are possible. The design process restricted the figure
for each wall of the concentrator to remain an ellipse, while varying the
three foci to optimize the total throughput (co-polar and cross-polar). A more
ambitious design could add additional free parameters to the figure for each
wall, changing the simple ellipse to (e.g.) a low-order polynomial. Similarly,
the concentrator could be optimized to maximize the co-polar throughput while
simultaneously minimizing the cross-polar response. We have not yet attempted
such improvements.

We measured the co-polar and cross-polar beam patterns for an optical system
consisting of the concentrator plus a coupling mirror. The results are thus
representative of the beam patterns expected at output of the PIXIE instrument
and do not represent the PIXIE beams on the sky. The full PIXIE optical system
includes additional beam-forming mirrors to reduce the tophat diameter to the
design value 2.6∘kogut/etal:2011 . The beam-forming optics are optimized to
further symmetrize the beam patterns.

###### Acknowledgements.

We thank S. Dixon, P. Mirel, and A. Walts for assistance with beam pattern
measurements and E. Wollack for fruitful discussion.

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