MNRAS
446,
1252–1267 (2015)
doi:10.1093/mnras/stu2172
Astronomical receiver modelling using scattering matrices
O. G. King,
1
,
2
‹
Michael E. Jones,
2
C. Copley,
2
,
3
R. J. Davis,
4
J. P. Leahy,
4
J. Leech,
2
S. J. C. Muchovej,
1
T. J. Pearson
1
and Angela C. Taylor
2
1
California Institute of Technology, Pasadena CA 91125, USA
2
Sub-department of Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK
3
SKA South Africa, Park Road, The Park, Pinelands, Cape Town, 7405, South Africa
4
Jodrell Bank Centre for Astrophysics, School of Physics & Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
Accepted 2014 October 16. Received 2014 October 16; in original form 2014 September 5
ABSTRACT
Proper modelling of astronomical receivers is vital: it describes the systematic errors in the
raw data, guides the receiver design process, and assists data calibration. In this paper, we
describe a method of analytically modelling the full signal and noise behaviour of arbitrarily
complex radio receivers. We use electrical scattering matrices to describe the signal behaviour
of individual components in the receiver, and noise correlation matrices to describe their noise
behaviour. These are combined to produce the full receiver model. We apply this approach to a
specified receiver architecture: a hybrid of a continuous comparison radiometer and correlation
polarimeter designed for the C-Band All-Sky Survey. We produce analytic descriptions of the
receiver Mueller matrix and noise temperature, and discuss how imperfections in crucial
components affect the raw data. Many of the conclusions drawn are generally applicable to
correlation polarimeters and continuous comparison radiometers.
Key words:
instrumentation: polarimeters – methods: analytical – techniques: polarimetric –
techniques: radar astronomy.
1 INTRODUCTION
Astronomical receiver modelling has many purposes. Perhaps the
most important is understanding the data produced by the receiver:
we want to know how the raw data values produced by the receiver
relate to the astronomical signal of interest. We may, for instance,
wish to know how sensitive the receiver will be or what systematic
errors will be present in the data. If the receiver is simple it may be
straightforward to describe the raw data, but radio receiver archi-
tectures are often complex and difficult to model accurately. One
approach to the complexity is to calculate the receiver response
by numerical simulation. A more powerful approach is to model
the receiver behaviour analytically: this in principle allows us to
describe exactly how particular instrumental parameters affect the
data, with no confusion as to what a particular artefact in the output
is caused by.
Most analytic and semi-analytic approaches to characterizing
systematic effects in receivers have employed Jones matrices to de-
scribe receiver components and Mueller matrices to characterize the
effects of receiver imperfections on the observed signal, e.g. Heiles
et al. (
2001
), Carretti et al. (
2001
), Hu, Hedman & Zaldarriaga
E-mail:
ogk@astro.caltech.edu
(
2003
), O’Dea, Challinor & Johnson (
2007
). In this formulation the
propagation of radiation through a receiver can be described by a
2
×
2 Jones matrix
J
. The total instrument Jones matrix is found
by cascading (multiplying) the Jones matrices of the components
in the instrument. If the receiver is polarization sensitive, the effect
of the instrument on the true polarization vector can be found by
calculating the Mueller matrix that describes the instrument. While
powerful, a shortcoming of this approach is that it does not include
the effect of noise produced by components in the receiver – i.e. it
does not model the sensitivity of the instrument.
In this paper, we describe a method for modelling astronomical
receivers that produces a full signal and noise description of the in-
strument. Each component in a receiver is described by an electrical
scattering matrix and a noise correlation matrix. These describe
the signal response and noise properties of the component, respec-
tively. Scattering matrices provide a full description of the reflection
and transmission of electromagnetic waves incident on a compo-
nent; noise correlation matrices describe the noise produced by
the component (Zmuidzinas
2003
; Pozar
2005
). A component may
have multiple ports (input and outputs); at radio and millimeter
wavelengths these are easily understood as guided electromag-
netic waves, while in a quasi-TEM optical system they might be
thought of as different polarization states of the electromagnetic
wave. We build a network of components by connecting their ports
C
2014 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
Astronomical receiver modelling
1253
appropriately to describe the signal flow in the receiver. The re-
sponse of the full network (the whole receiver) can then be cal-
culated and its signal and noise response described by a single
scattering matrix and noise correlation matrix, respectively.
We will analyse a specific radio receiver architecture in this paper:
the C-Band All-Sky Survey (C-BASS) receiver (King et al.
2014
).
This receiver measures the full polarization state of the instrument.
It is a hybrid of two commonly used architectures – a continuous
comparison radiometer to measure the total intensity of the signal;
and a correlation polarimeter to measure the linear polarization
state of the signal. It can also measure the circular polarization
state, though the architecture is not optimized to do this. Many
of the results we obtain are applicable to correlation polarimeters
and continuous comparison radiometers in general, rather than the
specific C-BASS implementation described here.
In Section 2, we introduce the framework (Stokes parameters
and Mueller matrices) we will use throughout the paper to describe
the signal response of the receiver. In Section 3, we introduce the
method, describing scattering matrices, noise correlation matrices,
and how they can be used to derive a useful description of the
instrument behaviour. In Section 4, we describe the specific C-BASS
receiver architecture. In Section 5 we analyse the receiver, produce
exact descriptions of the instrument signal and noise behaviour, and
discuss how imperfections in crucial components affect the output
data. In Section 6, we test some of the predictions made in Section 5,
and fit some model parameters to the measured instrument response.
We conclude in Section 7.
2 STOKES PARAMETERS
The Stokes parameters are a convenient and powerful way of de-
scribing the state of polarization of an electromagnetic signal.
I
describes the total intensity of the signal,
Q
and
U
describe the
linear polarization state, and
V
describes the circular polarization
state. Stokes parameters are defined relative to a local coordinate
system;
Q
represents the degree of linear polarization parallel and
perpendicular to the local coordinate axes, while
U
represents the
linear polarization at 45
◦
to these axes. Table
1
lists the definitions of
the Stokes parameters in two commonly used bases of the electric
field vector: the orthogonal linear modes
E
x
(
t
)and
E
y
(
t
), and the
orthogonal circular modes
E
l
(
t
)and
E
r
(
t
).
Table 1.
The Stokes parameters written in terms of orthogonal linear
and orthogonal circular bases of the electric field vector.
β
=
4
Rk
B
is
a proportionality constant to place the Stokes parameters in units of
antenna temperature (Appendix A). The circular polarization bases
are related to the linear polarization bases by the equations
E
l
=
E
x
+
iE
y
/
√
2and
E
r
=
E
x
−
iE
y
/
√
2.
i
is the imaginary num-
ber, while
{
x
}
and
{
x
}
are the real and imaginary parts of
x
,
respectively.
Linear basis
Circular basis
β
I
|
E
x
(
t
)
|
2
+|
E
y
(
t
)
|
2
|
E
l
(
t
)
|
2
+|
E
r
(
t
)
|
2
2
{
E
r
(
t
)
E
∗
l
(
t
)
}
β
Q
|
E
x
(
t
)
|
2
−|
E
y
(
t
)
|
2
E
∗
l
(
t
)
E
r
(
t
)
+
E
l
(
t
)
E
∗
r
(
t
)
2
{
E
x
(
t
)
E
∗
y
(
t
)
}
−
2
{
E
r
(
t
)
E
∗
l
(
t
)
}
β
U
E
∗
x
(
t
)
E
y
(
t
)
+
E
x
(
t
)
E
∗
y
(
t
)
i
E
∗
l
(
t
)
E
r
(
t
)
−
E
l
(
t
)
E
∗
r
(
t
)
2
{
E
x
(
t
)
E
∗
y
(
t
)
}
β
V
|
E
l
(
t
)
|
2
−|
E
r
(
t
)
|
2
i
E
∗
x
(
t
)
E
y
(
t
)
−
E
x
(
t
)
E
∗
y
(
t
)
The coherency vector (Born & Wolf
1964
) describes the state
of an electromagnetic signal by including all possible correlations
between its orthogonal electric field modes:
e
=
⎡
⎢
⎢
⎢
⎢
⎣
E
x
(
t
)
E
∗
x
(
t
)
E
x
(
t
)
E
∗
y
(
t
)
E
y
(
t
)
E
∗
x
(
t
)
E
y
(
t
)
E
∗
y
(
t
)
⎤
⎥
⎥
⎥
⎥
⎦
=
E
(
t
)
⊗
E
∗
(
t
)
,
(1)
where
E
(
t
) is the complex vector of the orthogonal linear elec-
tric field modes
E
x
(
t
)and
E
y
(
t
) of the signal,
...
indicates time
averaging, and
⊗
indicates the Kronecker tensor product.
The Stokes vector [
I
,
Q
,
U
,
V
] is a representation of the coherency
vector in an abstract space. The Stokes vector
e
S
is obtained from
the coherency vector
e
by
e
S
=
⎡
⎢
⎢
⎢
⎢
⎣
I
Q
U
V
⎤
⎥
⎥
⎥
⎥
⎦
=
Te
,
(2)
where
T
=
⎡
⎢
⎢
⎢
⎢
⎣
1001
100
−
1
0110
0
−
ii
0
⎤
⎥
⎥
⎥
⎥
⎦
.
(3)
We see that
T
is a coordinate transformation of the coherency vector
to the abstract Stokes frame (Hamaker, Bregman & Sault
1996
).
The action of an optical element, or indeed the entire instrument,
on the Stokes vector of the astronomical signal can be represented
by a Mueller matrix. Suppose that the incident Stokes vector is given
by
e
S
, and the Stokes vector of the signal after it has passed through
an optical element is
e
S
m
. The Mueller matrix
M
that describes the
optical element is then defined as
e
S
m
=
M
e
S
.
(4)
The elements of the Mueller matrix are given by
M
=
⎡
⎢
⎢
⎢
⎢
⎣
M
II
M
IQ
M
IU
M
IV
M
QI
M
QQ
M
QU
M
QV
M
UI
M
UQ
M
UU
M
UV
M
VI
M
VQ
M
VU
M
VV
⎤
⎥
⎥
⎥
⎥
⎦
.
(5)
The diagonal elements of the Mueller matrix encode the sensitivity
to each Stokes parameter, while the off-diagonal elements encode
the leakage between Stokes parameters. If the instrument Mueller
matrix elements are constant and measurable, the Mueller matrix
can be inverted and applied to the data to return a leakage-free data
stream.
3 RECEIVER MODELLING USING
SCATTERING AND NOISE MATRICES
3.1 Scattering matrix modelling
In the scattering matrix formulation any arbitrary component or
network of components (excluding detectors) can be described by a
frequency-dependent scattering matrix
S
(
ν
). We omit the frequency
dependence from now on for brevity.
MNRAS
446,
1252–1267 (2015)
1254
O. G. King et al.
The scattering matrix relates the incident, reflected, and transmit-
ted waves that travel on transmission lines attached to the
N
ports
of a linear network. It provides a complete description of an
N
-port
network as seen at its
N
ports (Pozar
2005
). This formulation can
be extended to optical systems and used to describe instruments
that contain both optical components, such as lenses or mirrors, and
microwave circuit techniques, such as horns, transmission lines,
filters, etc. (Zmuidzinas
2003
).
3.1.1 Scattering and noise matrices
Consider an arbitrary
N
-port network. We denote the incident wave
at port
i
by
V
+
i
, the reflected wave by
V
−
i
, and the noise wave
produced by the network at that port by
c
i
. These quantities are
related by the scattering matrix
S
and noise wave vector
c
as
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
V
−
1
V
−
2
.
.
.
V
−
N
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
S
11
S
12
···
S
1
N
S
21
.
.
.
.
.
.
S
N
1
···
S
NN
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
V
+
1
V
+
2
.
.
.
V
+
N
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
c
1
c
2
.
.
.
c
N
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(6)
The scattering matrix
S
is unitary if the device is lossless and recip-
rocal networks have symmetric scattering matrices (Pozar
2005
).
The noise wave voltages
c
i
of an
N
-port network are com-
plex time-varying random variables characterized by a correlation
matrix
C
C
=
c
⊗
c
†
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
|
c
1
|
2
c
1
c
∗
2
···
c
1
c
∗
N
c
2
c
∗
1
.
.
.
.
.
.
c
N
c
∗
1
···
|
c
N
|
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
(7)
where
†
indicates the conjugate transpose operation, and
c
is a
vector with elements
c
i
. The diagonal terms of
C
give the noise
power deliverable at each port per unit bandwidth. The off-diagonal
terms are correlation products. The noise correlation matrix
C
for a
passive network is determined by its scattering matrix
S
(Wedge &
Rutledge
1991
)
C
=
kT
(
I
−
SS
†
)
,
(8)
where
k
is Boltzmann’s constant,
T
is the physical temperature of
the network, and
I
is the identity matrix. The noise correlation
matrix for an active network can be determined by measurement or
modelling.
3.1.2 Solving the network response
We can build a network of
N
-port devices, connected by nodes, and
assign scattering matrices and noise correlation matrices to each
device. Once all the components in a receiver have been described
by scattering and noise correlation matrices, we can then calculate
the scattering matrix and the noise wave vector that describe the
whole receiver as seen at its inputs and outputs using the
MATLAB
1
package
SNS
2
(King
2010
). It implements algorithms that solve for
1
http://www.mathworks.com
2
Download at
https://github.com/kingog/SNS
.
Figure 1.
An arbitrary receiver, where orthogonal linear polarization volt-
ages
E
x
(
t
)and
E
y
(
t
) are presented at ports 1 and 2, respectively, while ports
3to
N
are the output ports.
D
is the output at port
m
, and is connected to a
power detector. The receiver is described by scattering matrix
S
and noise
wave vector
c
.
the network response and can operate on both analytic and numeric
descriptions of scattering and noise correlation matrices. The
SUPER
-
MIX
software package (Ward et al.
1999
) can also solve for the noise
and signal response of a microwave network, but only numerically:
it cannot provide an analytic description of the outputs.
3.1.3 Interpreting the measured power
We now have a scattering matrix and a noise wave vector that de-
scribe an arbitrary receiver architecture, excluding power detectors.
The inputs to the scattering matrix are electric field vector elements
from the sky, and the noise wave vector describes the noise at each
output of the receiver. The outputs are connected to power detectors
– often square-law diodes in radio receivers. We now describe how
to rewrite the power detected at each output in terms of Mueller
matrices.
Consider the arbitrary receiver shown in Fig.
1
. Orthogonal linear
polarizations
E
x
(
t
)and
E
y
(
t
), representing either signals in transmis-
sion lines, orthogonal electric field modes in waveguide, or orthog-
onal electric field modes in free space, are connected to ports 1
and 2 of the receiver, respectively. Receiver output
D
at port
m
is connected to a power detector. The receiver is described by the
scattering matrix
S
and the noise wave vector
c
.Wewillshowhow
to rewrite the power per unit bandwidth detected at output
D
as
P
D
=
P
D,S
+
P
D,N
,
(9)
where
P
D
,
S
=
k
B
(
M
DI
I
+
M
DQ
Q
+
M
DU
U
+
M
DV
V
) is the contribu-
tion of the Stokes parameters that describe the sky signal voltages
E
x
(
t
),
E
y
(
t
) to the power detected at the diode and
P
D
,
N
is the
contribution of receiver noise.
The Mueller matrix elements and the sky signal Stokes parame-
ters in equation (9) are frequency dependent. The shape of the sky
signal spectrum is required to obtain the band-integrated power.
Stokes contribution:
the contribution
E
m
(
t
) to output port
m
from
the input sky signal is given by
E
m
(
t
)
=
S
m
1
E
x
(
t
)
+
S
m
2
E
y
(
t
)
.
(10)
The power contained in the signal
E
m
(
t
) is then measured. At radio
wavelengths this might be achieved through the use of a square-law
detector diode. The measured power
P
D
,
S
is given by
P
D,S
=
α
E
m
(
t
)
E
m
(
t
)
∗
=
α
|
E
x
(
t
)
|
2
|
S
m
1
|
2
+|
E
y
(
t
)
|
2
|
S
m
2
|
2
+
E
x
(
t
)
E
∗
y
(
t
)
S
m
1
S
∗
m
2
+
E
∗
x
(
t
)
E
y
(
t
)
S
∗
m
1
S
m
2
,
(11)
MNRAS
446,
1252–1267 (2015)
Astronomical receiver modelling
1255
where
α
=
α
D
/
(4
R
) scales the voltage squared units to power per
unit bandwidth (see the Nyquist theorem equation A3) and con-
tains a factor
α
D
dependent on the power detection method and
post-detector gain. We assume that the instrument scattering matrix
parameters are constant during the averaging time period. Now let
P
D,S
=
k
B
(
M
DI
I
+
M
DQ
Q
+
M
DU
U
+
M
DV
V
)
=
1
4
R
M
DI
|
E
x
(
t
)
|
2
+|
E
y
(
t
)
|
2
+
M
DQ
|
E
x
(
t
)
|
2
−|
E
y
(
t
)
|
2
+
M
DU
E
x
(
t
)
E
∗
y
(
t
)
+
E
∗
x
(
t
)
E
y
(
t
)
−
iM
DV
E
x
(
t
)
E
∗
y
(
t
)
−
E
∗
x
(
t
)
E
y
(
t
)
,
(12)
where we have used the definition of the Stokes parameters in a
linear basis given in Table
1
.
By comparing equations (11) and (12), we can obtain the contri-
bution of each Stokes parameter to the power measured at output D
in terms of the scattering matrix parameters:
M
DI
=
α
D
2
|
S
m
1
|
2
+|
S
m
2
|
2
M
DQ
=
α
D
2
|
S
m
1
|
2
−|
S
m
2
|
2
M
DU
=
α
D
2
S
m
1
S
∗
m
2
+
S
∗
m
1
S
m
2
M
DV
=
iα
D
2
S
m
1
S
∗
m
2
−
S
∗
m
1
S
m
2
.
(13)
Noise contribution:
we now describe how to derive the power con-
tributed to output
D
by the receiver noise using the noise wave
vector returned by the network solving algorithm. We also show
how to rewrite it referenced to the input of the receiver, i.e. as a
receiver noise temperature.
If the noise wave vector of the receiver is given by
c
, then the
noise power measured at the output
D
(port
m
of the scattering
matrix) in a 1 Hz bandwidth is given by
P
D,N
=
α
c
m
c
∗
m
,where
c
m
is the noise wave vector element corresponding to output
D
.
We decompose the noise power seen at output
D
into the power
contributed by each noisy component. Suppose that component
k
(of
M
total noisy components in the receiver) is specified by a scattering
matrix
S
k
and a noise wave vector
c
k
.
c
m
is given by
c
m
=
M
k
=
1
c
k
,
where
c
k
=
b
k
c
k
.
(14)
c
k
is the weighted contribution of the elements of the noise wave
vector
c
k
to the total noise wave signal seem at port
m
.
b
k
is a row
vector containing the weights; it is some function of the receiver
response and is calculated during the network solving step.
Noise waves from different devices are usually not correlated
3
:
c
k
i
(
c
p
j
)
∗
=
0for
k
=
p
.So,
P
D
,
N
is given by the sum of the indi-
vidual component contributions:
P
D,N
=
α
M
k
=
1
P
k
D,N
,
where
P
k
D,N
=
C
k
·
b
k
⊗
(
b
k
)
†
.
(15)
Here
C
k
is the noise correlation matrix for component
k
and
·
is the
matrix dot product.
3
However, common temperature fluctuations of the amplifiers can cause a
correlated noise component.
We have rewritten the sky contribution
P
D
,
S
to the detected power
in terms of the Stokes parameters. If we want to rewrite the noise
contribution in a way that we can directly compare to the Stokes con-
tributions, we can turn it into a receiver temperature by referencing
it to the receiver input. The receiver temperature
T
D
of output
D
is defined as the temperature of a thermal source seen equally at
each receiver input that, for a noiseless receiver, produces the same
power
P
D
,
N
at output
D
as the noise does:
P
D,N
=
α
|
S
m
1
|
2
+|
S
m
2
|
2
T
D
∴
T
D
=
M
k
=
1
P
k
D,N
|
S
m
1
|
2
+|
S
m
2
|
2
.
(16)
Noise variance:
the previous section described the power seen at
a particular detector due to noise produced by components in the
receiver. The variance of the power signal can be obtained from the
radiometer’s equation:
σ
D
=
P
D,S
+
P
D,N
|
S
m
1
|
2
+|
S
m
2
|
2
√
ντ
,
(17)
where
ν
is the signal bandwidth and
τ
is the integration time.
We have turned the detected power
P
D
,
S
+
P
D
,
N
into an antenna
temperature by referencing it to the input using the gain term
|
S
m
1
|
2
+|
S
m
2
|
2
.
4 C-BASS RECEIVER MODEL
We apply the scattering matrix modelling approach to the northern
C-BASS receiver (described in King et al.
2014
).
The northern C-BASS receiver is a combination of a continu-
ous comparison radiometer and a correlation polarimeter, shown in
Fig.
2
. The radiometer measures the powers of both orthogonal cir-
cular polarizations independently by correlating them against two
independent stabilized thermal loads, so fluctuations in the mea-
sured quantities as the telescope scans the sky track the true sky
brightness. This continuous-comparison radiometer architecture re-
duces the 1
/
f
gain fluctuations, at the expense of a
√
2 higher white
noise level due to the thermal noise of the load.
The linearly polarized Stokes parameters
Q
and
U
are measured
simultaneously by a complex cross-correlation between the orthog-
onal circular polarizations. This is exactly the same process that
is used in interferometric polarimeters, except that in this case the
orthogonally polarized signals are from the same antenna rather
than two different antennas. The 1
/
f
gain fluctuations due to the
low noise amplifiers are uncorrelated between amplifiers and so are
removed in the correlation operation.
The model of the receiver used in the systematic error analysis
is shown in Fig.
2
. Components are modelled as
N
-port devices
(labelled P1 to P36) connected by central nodes (labelled c1 to
c44). The inputs to the receiver are orthogonal linear electric field
vectors from the horn;
E
x
(
t
) is connected to input node I1 and
E
y
(
t
)is
connected to input node I2. Output nodes O1 to O12 are connected
to power detectors. In this model we assume that the orthomode
transducer (OMT), cold reference loads, calibration noise diode
and attenuator, and amplifiers produce noise. Every component will
produce noise, but these components will dominate the noise budget.
The model shown in Fig.
2
accurately represents the action of the
receiver, though it is not a facsimile of the actual receiver diagram.
Long gain chains, composed of multiple amplifiers, attenuators, fil-
ters, isolators, and slope compensators are represented by a single
amplifier. Not all components are modelled as producing noise,
MNRAS
446,
1252–1267 (2015)
1256
O. G. King et al.
Figure 2.
A model of the C-BASS receiver used in the analysis. It is a hybrid of a continuous comparison radiometer and a correlation polarimeter. Components
are modelled as
N
-port devices (numbered P1 to P36) connected by central nodes (numbered c1 to c44). The inputs to the receiver are orthogonal linear electric
field vectors from the horn;
E
x
(
t
) is connected to input node I1 and
E
y
(
t
) is connected to input node I2. Output nodes O1 to O12 are connected to power
detectors. The OMT, cold reference loads, calibration noise diode and attenuator, and amplifiers produce noise.
though all significant sources are modelled. A fully parametrized
component-for-component reproduction of the real receiver is not
suitable for analytic description; the elements of the Mueller matrix
become so complex as to be meaningless and would fail to illumi-
nate the important lessons that can be learnt using a simpler, but
representative, model.
4.1 Receiver data channels
In Section 3.1.3, we described how to rewrite the power detected at
each receiver output in terms of contributions from the sky (writ-
ten in terms of Stokes parameters) and contributions from noisy
components in the receiver. This allows us to create a description
of each receiver data channel in a framework that is powerful and
natural to radio astronomy. We will express the vector
r
of receiver
data channels (power per unit bandwidth) in the form
r
=
k
B
M
corr
M
OMT
e
S
+
N
OMT
+
N
corr
=
k
B
M
rec
e
S
+
N
rec
,
where
M
rec
=
M
corr
M
OMT
N
rec
=
M
corr
N
OMT
+
N
corr
.
(18)
Here, we have split the receiver into two parts at the point where
the noise diode calibration signal is injected. We refer to the pre-
calibration signal injection part of the receiver as the OMT section,
and the post-signal injection part as the correlator section.
M
OMT
and
M
corr
are the Mueller matrices for the OMT and correlator sec-
tions, respectively. The contributions of the noise sources to each
data stream are given by
N
OMT
and
N
corr
, which describe the OMT
section and correlator section (amplifiers, lossy components, cali-
bration noise sources), respectively. The action of the telescope op-
tics (reflectors and horn) can be included by prepending its Mueller
matrix to the Mueller matrix chain.
In this receiver outputs O1 to O12 are connected to detector
diodes whose outputs are described by equation (9). We subtract
the signals from adjacent detectors to give us six data streams la-
belled I1 (O2
−
O1), I2 (O12
−
O11), U1 (O3
−
O4), Q1 (O6
−
O5),
U2 (O7
−
O8), and Q2 (O9
−
O10). These can each be written in
terms of Stokes parameter contributions and noise contributions.
We then calculate the raw receiver data streams:
r
=
⎡
⎢
⎢
⎢
⎢
⎣
r
I
r
Q
r
U
r
V
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎣
I1
+
I2
(Q1
+
Q2)
/
2
(U1
+
U2)
/
2
I1
−
I2
⎤
⎥
⎥
⎥
⎥
⎦
.
(19)
4.2 Analysis procedure
In the analysis that follows we construct a receiver model using
SNS
and the
MATLAB
symbolic algebra toolbox in which all the com-
ponents have fully parametrized scattering matrices as described
in Section 4.3. We perform the computationally expensive steps of
calculating the analytic receiver scattering matrix and noise wave
vector, and deriving the Mueller matrix
M
rec
and noise vector
N
rec
,
only once. The elements of
M
rec
and
N
rec
are algebraic expressions
that contain a full description of the receiver. We explore the ef-
fect of imperfections in particular components on the instrument
performance by removing unwanted parameters – this is achieved
by substituting ‘perfect’ values for the error parameters (0 or 1,
depending) and simplifying the resultant expressions.
4.3 Component models
In general, there are three levels of non-ideality in the scattering
matrix that describes a component: those implicit in the design,
random variations from device to device, and measurement er-
rors. For the components in the C-BASS receiver, the non-ideal
behaviour implicit in the design was generally dominant: device to
device variations were substantially lower and measurement error
was negligible. We assume in this analysis that all errors are im-
plicit to the design, and hence we can describe nominally identical
components with the same matrices.
MNRAS
446,
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Astronomical receiver modelling
1257
A comment on notation: amplitude balance errors are denoted
by the
δ
symbol, and are zero for an ideal component. Phase errors
are denoted by
φ
(zero for an ideal component), and transmission
amplitudes are denoted by
α
(one for an ideal component). This
convention was adopted to make it easy to verify that the derived
Mueller matrices were sensible – a quick visual substitution for
ideal component values should result in the identity matrix.
4.3.1 Circularizing OMT
We model the circularizing OMT (P1) as a component that accepts
as inputs the orthogonal linear components of the electric field from
the sky and produces at its outputs orthogonal circularly polarized
signals. In the C-BASS receiver, this is achieved by first extracting
orthogonal linear TE
11
modes from a circular waveguide using four
rectangular probes (Grimes et al.
2007
;Kingetal.
2014
). The out-
of-phase signals from opposite pairs of probes are then combined
with a 180
◦
phase shift, using two 180
◦
hybrids, to obtain two
orthogonal linear polarizations. Finally, these linear polarizations
are then passed through a 90
◦
hybrid to produce two orthogonal
circular polarizations. In our case, the two 180
◦
hybrids and single
90
◦
hybrid were fabricated on a single planar substrate to form a
device known as a linear-to-circular converter. For simplicity, we
hereinafter refer to the combination of the four-probe linear OMT
and the linear-to-circular converter as the
circularizing OMT
.
The scattering matrix for an ideal circularizing OMT that accepts
E
x
(
t
) at port 1 and
E
y
(
t
) at port 4 and returns the circular polarization
signals at ports 2 and 3 is the same as that for an ideal 90
◦
hybrid:
Ideal
S
o
=
1
√
2
⎡
⎢
⎢
⎢
⎢
⎣
01
i
0
100
i
i
001
0
i
10
⎤
⎥
⎥
⎥
⎥
⎦
.
(20)
Any real OMT will not perfectly convert linear polarizations
into orthogonal circular polarizations. In practice, the non-ideal
behaviour can arise due to mismatches in probe dimensions, probe
angles or non-ideal performance of the 180
◦
and 90
◦
hybrids which
make up the linear-to-circular converter. We may model this non-
ideal behaviour by referring to a somewhat simplified, conceptual
representation of the circularizing OMT (Fig.
3
). Linear polarized
signals can be thought of as being presented to a linear OMT at
ports 1 and 4. The
x
output (port 2) of the OMT is perfectly aligned
with the
x
component of the sky signal, but the
y
output (port 3) is
rotated by
φ
⊥
from the nominal (perpendicular) orientation, leading
to leakage of
E
x
into the
y
output. Each probe has a gain
α
x
,
y
(ideally
1), and there is a phase shift
φ
y
in the
y
line relative to the
x
line. This
Figure 3.
Scattering matrix model of the circularizing OMT. Linear po-
larization signals are presented to a linear OMT. The
x
-axis of the OMT
is perfectly aligned with the
x
component of the sky signal, but the
y
-axis
is rotated by
φ
⊥
from nominal, leading to leakage of
E
x
into the
y
output.
Each probe has gain
α
x
,
y
(ideally 1), and there is a phase shift in the
y
line
φ
y
relative to the
x
line. The linear OMT is followed by a 90
◦
hybrid to
circularise the voltages.
simplified linear OMT is then followed by a standard, imperfect, 90
◦
hybrid (described by equation 25). The fully parametrized model of
the imperfect circularizing OMT is then given by
S
o
=
1
√
2
⎡
⎢
⎢
⎢
⎢
⎣
0
S
lx
S
rx
0
S
lx
00
S
ly
S
rx
00
S
ry
0
S
ly
S
ry
0
⎤
⎥
⎥
⎥
⎥
⎦
S
lx
=
√
α
x
√
1
+
δ
+
i
√
α
y
√
1
−
δ
sin
φ
⊥
e
−
i(
φ
90
+
φ
y
)
S
rx
=
i
√
α
x
√
1
−
δ
e
−
i
φ
90
+
√
α
y
√
1
+
δ
sin
φ
⊥
e
−
i
φ
y
S
ly
=
i
√
α
y
√
1
−
δ
cos
φ
⊥
e
−
i(
φ
90
+
φ
y
)
S
ry
=
√
α
y
√
1
+
δ
cos
φ
⊥
e
−
i
φ
y
.
(21)
As this is a passive component the noise correlation matrix can be
determined using equation (8). We emphasize that the parameters
introduced in equation (21) are simply describing the non-ideality
of the complete circularizing OMT, and will not necessarily corre-
spond to e.g. the physical probe angles within the four-probe OMT
itself.
4.3.2
180
◦
hybrid
The 180
◦
hybrid (P4, P5, P16, P19, P20, P21, P28, and P30 in Fig.
2
)
is a component that combines two incoming voltages, producing at
one output port the sum of the inputs and at the other the difference
of the inputs. It is sometimes called a ‘magic tee’ when implemented
in waveguide. The scattering matrix for an ideal 180
◦
hybrid, with
ports 1 and 4 being the inputs and ports 2 and 3 being the outputs,
is given by
Ideal :
S
180
=
1
√
2
⎡
⎢
⎢
⎢
⎢
⎣
0110
100
−
1
1001
0
−
11 0
⎤
⎥
⎥
⎥
⎥
⎦
.
(22)
A well-designed hybrid can closely approximate the ideal be-
haviour, but the two most significant types of imperfect behaviour
that will remain are amplitude and phase imbalances. Amplitude
imbalances occur if the power from one input port is not equally
split between the output ports. We use the parameter
δ
180
to des-
ignate this imbalance. Phase imbalances occur if there are phase
errors in the ‘sum’ and ‘difference’ outputs:
φ
indicates the phase
error in the summation operation and
φ
indicates the phase error in
the difference operation. The symmetric, unitary, scattering matrix
for such a hybrid is given by
Error :
S
180
=
1
√
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
···
√
1
+
δ
180
0
√
1
−
δ
180
e
−
i
φ
00
.
.
.
0
−
√
1
−
δ
180
e
−
i
φ
√
1
+
δ
180
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(23)
Note that this matrix is equal to the ideal matrix if
δ
180
=
φ
=
φ
=
0.
MNRAS
446,
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O. G. King et al.
The C-BASS receiver shown in Fig.
2
contains several stages
of 180
◦
hybrids. Those in the first stage, between the circularizing
OMT and the first amplification stage, are called the cold hybrids
as they are located in the receiver cryostat. Those in later stages are
called warm hybrids.
4.3.3 90
◦
hybrid
A90
◦
hybrid (P29 and P31) combines two input voltages with equal
amplitude but with a 90
◦
phase difference. They are commonly
used to introduce 90
◦
phase shifts to a voltage (in correlators, for
instance), and to convert linear polarization signals to a circular
basis in correlation polarimeters. The scattering matrix for an ideal
90
◦
hybrid is given by
Ideal :
S
90
=
1
√
2
⎡
⎢
⎢
⎢
⎢
⎣
01
i
0
100
i
i
001
0
i
10
⎤
⎥
⎥
⎥
⎥
⎦
.
(24)
As with the 180
◦
hybrid, both amplitude and phase imbalances
can occur in these hybrids. However, the C-BASS 90
◦
hybrids were
implemented as microstrip branch-line couplers. These are symmet-
ric structures that remove the need for having two different phase
errors for the outputs as we do with the 180
◦
hybrid. We describe
the amplitude imbalance using the parameter
δ
90
and the phase im-
balance using the parameter
φ
90
. The scattering matrix for the 90
◦
hybrid is then given by the symmetric, unitary, matrix:
Error :
S
90
=
1
√
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
···
√
1
+
δ
90
0
i
√
1
−
δ
90
e
−
i
φ
90
00
.
.
.
0
i
√
1
−
δ
90
e
−
i
φ
90
√
1
+
δ
90
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(25)
4.3.4 Directional coupler
Directional couplers (P2 and P3) are used to inject a calibration
signal from a broad-band noise source into the signal path. This
is used to calibrate the instrument by measuring the gain of the
receiver and the polarization vector rotation angle of the receiver.
The scattering matrix for an ideal directional coupler where the
through signal is connected to port 1, the coupled signal is connected
to port 3 and the output is port 2 is given by
S
cpl
=
⎡
⎢
⎢
⎣
01 0
10
√
D
0
√
D
0
⎤
⎥
⎥
⎦
.
(26)
Here,
D
is the power coupling factor. Technically, if the coupler
is lossless, the through-parameter should be
S
21
=
S
12
=
√
1
−
D
,
but if the coupling factor
D
is sufficiently low (say
D
=−
30 dB) it
can be assumed to be equal to 1.
4.3.5 Cold reference loads and terminations
The cold reference loads are modelled as perfectly matched
1-port terminations. Their scattering matrices and noise correlation
matrices are given by
S
load
=
0
C
load
=
kT
load
1
,
(27)
where
T
load
is the physical temperature of the load, and
1
is a
1
×
1 unit matrix. The cold reference loads P32 and P33 have
physical temperatures
T
A
and
T
B
, respectively. Ideally
T
A
=
T
B
,but
there may be some small remaining temperature difference. The
terminations P37 and P38 contribute a negligible amount of noise
so they are assigned a 0 K physical temperature.
4.3.6 Noise diode
The noise diode (P34) is modelled as a termination at physical
temperature
T
ND
. When the noise diode is turned off
T
ND
[off]
=
290 K. When the noise diode is turned on the temperature changes
to
T
ND
[on]
=
290(1
+
10
ENR
/
10
), where ENR is the excess noise
ratio of the noise diode in dB.
4.3.7 Attenuator
The noise diode is followed by an attenuator (P35) to control the
injected signal level. The scattering matrix and noise correlation
matrix for a perfect attenuator are given by
S
att
=
0
√
L
C
att
=
kT
amb
1
−
L
0
01
−
L
,
(28)
where
T
amb
is the temperature of the attenuator and
L
is the power
transmission.
4.3.8 Amplifier
An amplifier (P6, P7, P8, P9, P24, and P25) working in the linear
regime (i.e. uncompressed) increases the amplitude of the incoming
voltage wave by some factor. However, this comes at the expense
of adding noise to the signal. The ideal scattering matrix and noise
correlation matrix for such an amplifier where the input is port 1
and the output is port 2 is given by
S
amp
=
00
g
amp
0
C
amp
=
00
0
kT
amp
|
g
amp
|
2
.
(29)
Here,
g
amp
is the complex voltage gain of the amplifier. It has a power
gain factor of
|
g
amp
|
2
and introduces a phase shift of
∠
g
amp
to the
wave. The noise power produced at the output port of the amplifier
is given by
kT
amp
|
g
amp
|
2
,where
T
amp
is the noise temperature of the
amplifier.
Note that in reality amplifiers produce noise at their input port
as well and this noise is correlated with the output noise. If this
noise were to leak through to other signal chains through the pre-
amplifier components, it would introduce a spurious correlation to
the signal. The resulting offset in the polarization channels will be
constant with time and telescope pointing. However, if the first-
stage amplifiers are preceded by isolators that severely attenuate
signals travelling in the reverse direction while allowing signals in
MNRAS
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Astronomical receiver modelling
1259
the forward direction to pass nearly unattenuated, this small but
constant offset can be substantially reduced.
4.3.9 Power divider
The receiver in Fig.
2
contains both 2-way (P10, P11, P12, P13, and
P36) and 4-way (P26 and P27) power dividers. If port 1 is the input
power then they have ideal scattering matrices given by
S
2-way
=
1
√
2
⎡
⎢
⎣
011
100
100
⎤
⎥
⎦
(30)
S
4-way
=
1
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
01111
10000
10000
10000
10000
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(31)
4.3.10 Phase switch
Phase switches (P14, P15, P17, P18, P22, and P23) are used to
introduce a 0
◦
or 180
◦
phase shift to a signal. They remove the effects
of post-detection gain variations, as demonstrated in Section 5.2.
The scattering matrix for an ideal phase switch is given by
Ideal :
S
ps
=
0
±
1
±
10
.
(32)
The sign of the transmitted voltage is switched.
In a real phase switch, we can have two sources of error: transmis-
sion amplitude differences between the two phase switch states, and
a phase error (i.e. retard the phase by something other than 180
◦
).
A more realistic time-dependent phase switch model is given by
Error :
S
ps
(
t
)
=
0
α
0
,
1
(
t
)e
−
i
φ
0
,
1
(
t
)
α
0
,
1
(
t
)e
−
i
φ
0
,
1
(
t
)
0
.
(33)
We explicitly denote the time dependence as a reminder that this is a
rapidly varied quantity. Of course, all receiver parameters are time-
dependent to some extent. The two phase switch states are denoted
by subscripts ‘0’ for the zero-phase shift state and ‘1’ for the 180
◦
phase shift state. The amplitude of the transmission is given by
α
0
,
1
(
t
) (ideally
=
1 in both states) and the phase shift is denoted
by
φ
0, 1
(
t
) (ideally
φ
0
=
0,
φ
1
=
π
). A further consideration is that
two phase switches may have different responses in supposedly
equal states.
4.3.11 Post-detection gain
Each detector diode output voltage, which is proportional to the
input power to the detector diode, is transported, amplified, filtered
and digitized by a different chain of electronics. These may have
different gains. We model this by granting each detector diode a
different responsivity
α
in equations (11) and (13).
5 RECEIVER ANALYSIS
At this point in the analysis, we have used the procedure outlined in
Section 3 to calculate the receiver Mueller matrix and noise vector of
the model described in Section 4. The elements in the Mueller matrix
and noise vector are complicated analytic expressions containing
the variables described above. We will now simplify these analytic
expressions to explore various aspects of the receiver performance.
5.1 Ideal receiver behaviour
The receiver will behave in an ideal fashion when all the components
are perfect. There is no amplitude or phase difference between the
amplifiers: all voltage gains are equal to some gain factor
g
.The
receiver Mueller matrix and noise vector, after dividing by the factor
|
g
|
2
,aregivenby
M
rec
=
⎡
⎢
⎢
⎢
⎢
⎣
1000
0100
0010
0001
⎤
⎥
⎥
⎥
⎥
⎦
(34)
N
rec
=
⎡
⎢
⎢
⎢
⎢
⎣
−
(
T
B
+
T
A
)
+
DLT
ND
+
D
(1
−
L
)
T
amb
0
DLT
ND
+
D
(1
−
L
)
T
amb
T
B
−
T
A
⎤
⎥
⎥
⎥
⎥
⎦
.
(35)
As expected, the ideal receiver Mueller matrix is the identity
matrix. There is no leakage between Stokes parameters, and all data
channels have identical gains. An offset term
−
(
T
A
+
T
B
) appears
in the
r
I
channel, indicating that what we measure is the difference
between the sky total intensity and the reference load temperature.
An unwanted offset of
T
B
−
T
A
appears in the
r
V
channel; this
should, however, be zero if the reference loads are held at the same
physical temperature.
An offset term
D
[
LT
ND
+
(1
−
L
)
T
amb
] appears in both the
r
I
and
r
U
channels. This is due to the calibration signal injection system,
which is used to measure the instrument response by injecting a
signal with known properties. This is discussed in more detail in
the correlation receiver section, Section 5.4.
5.2 Role of phase switching
Phase switching performs two roles in this receiver: it reduces the
leakage of total intensity into the polarization channels and it mod-
ulates the slowly varying sky signal at a high frequency. This mod-
ulation allows some undesired low-frequency signals that would
otherwise contaminate the sky signal, such as low-frequency mains
pickup, to be reduced by high-pass filtering prior to demodulation
(or by low-pass filtering after demodulation). We will explore the
role of phase switching in reducing leakage of total intensity to the
polarization channels by considering a series of phase switching
scenarios.
There are three pairs of phase switches in the receiver in Fig.
2
.
The first pair, P14 and P15, switch the I1 output. The second pair,
P17 and P18, switch the I2 output. The third pair, P22 and P23
switch the Q1, Q2, U1, and U2 outputs. The pairs are switched
independently with orthogonal Walsh functions. We consider the
general case where all phase switch states have both amplitude and
phase errors, and the errors are different in the two phase switches.
Since only the difference between the phase switches is important
(any shared amplitude or phase error is mathematically degenerate
with differing amplifier gains), we can make the 0 state for one of the
phase switches an ideal scattering matrix. The scattering matrices
MNRAS
446,
1252–1267 (2015)
1260
O. G. King et al.
Table 2.
The scattering matrices for phase switches
1 and 2 in both phase switch states.
S
I
is the 2
×
2
ideal transmission matrix. For ideal phase switches
1
=
2
=
3
=
φ
1
=
φ
2
=
φ
3
=
0.
State
i
S
PS1
,i
S
PS2
,i
0
S
I
√
1
−
1
e
−
i
φ
1
S
I
1
√
1
−
2
e
−
i(
π
+
φ
2
)
S
I
√
1
−
3
e
−
i(
π
+
φ
3
)
S
I
for both phase switches in a pair in both phase switch states are
showninTable
2
.
We assign the power detectors on outputs O1 to O12 responsivity
coefficients
α
1
to
α
12
, respectively. These represent the multipli-
cation of differing detector diode sensitivities and differing post-
detection gains. We make all the other components in the receiver
perfect and make all gains equal to 1.
The action of phase switching can be revealed by looking at the
first column of the instrument Mueller matrix. This encodes the
contribution of the total intensity (Stokes
I
) to each of the four raw
receiver data channels. For an ideal receiver only the first element
is non-zero (see equation 34).
5.2.1 No phase switching
If the receiver had no phase switches – i.e. assign the ideal transmis-
sion matrix to P14, P15, P17, P18, P22 and P23 – the first column
of the receiver Mueller matrix is
⎡
⎢
⎢
⎢
⎢
⎣
M
II
M
QI
M
UI
M
VI
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎣
1
2
(
α
2
+
α
12
)
1
4
(
α
6
−
α
5
+
α
9
−
α
10
)
1
4
(
α
3
−
α
4
+
α
7
−
α
8
)
1
2
(
α
2
−
α
12
)
⎤
⎥
⎥
⎥
⎥
⎦
.
(36)
Because we have set the amplitudes and phases of the gain chains
to be equal, the load signal
T
A
appears exclusively at O1 and has
the coefficient
α
1
in the noise vector
N
rec
. Similarly,
T
B
appears
exclusively at output O11 and has the coefficient
α
11
.
Without phase switching, the total intensity leaks into the po-
larization signal channels
if
there is an amplitude difference in the
post-warm hybrid power detection hardware. This is very likely to
be the case, so analogue correlation polarimeters must be phase
switched.
5.2.2 Asymmetric phase switching
We now take the simplest approach to phase switching and ‘jam’
one phase switch in each pair in a constant state, switching only
the other. We call this asymmetric phase switching. We use the
scattering matrices listed in Table
2
for the phase switches and
calculate the first column of the receiver Mueller matrix:
M
II
=
(
1
−
3
)(
α
1
−
α
2
+
α
11
−
α
12
)
16
+
1
8
(
α
1
+
α
2
+
α
11
+
α
12
)(
1
−
1
cos
φ
1
+
1
−
3
cos
φ
3
)
M
QI
=
(
1
−
3
)(
α
5
−
α
6
+
α
10
−
α
9
)
16
M
UI
=
(
1
−
3
)(
α
4
−
α
3
+
α
8
−
α
7
)
16
M
VI
=
(
1
−
3
)(
α
1
−
α
2
−
α
11
+
α
12
)
16
+
1
8
(
α
1
+
α
2
−
α
11
−
α
12
)(
1
−
1
cos
φ
1
+
1
−
3
cos
φ
3
)
.
(37)
Introducing asymmetric phase switching has reduced the total
intensity to polarization leakage – the leakage of Stokes
I
into
Q
has
been multiplied by a factor of (
1
−
3
)
/
4
1 – but not removed
it entirely.
5.2.3 Symmetric phase switching
The most general form of phase switching is when we switch both
phase switches, spending an equal amount of time in each phase
switch state for each data sample.
4
We average the states in which
both are ‘high’ or both are ‘low’ to form a composite 0 state, and
do the same for the states in which they are switched in an opposite
sense to form a composite 1 state. The first column of the receiver
Mueller matrix is now
M
II
=
α
1
+
α
2
+
α
11
+
α
12
16
1
−
1
cos
φ
1
+
1
−
3
cos
φ
3
+
1
−
2
1
−
1
cos(
φ
1
−
φ
2
)
+
1
−
3
cos(
φ
2
−
φ
3
)
M
QI
=
0
M
UI
=
0
M
VI
=
α
1
+
α
2
−
α
11
−
α
12
16
1
−
1
cos
φ
1
+
1
−
3
cos
φ
3
+
1
−
2
1
−
1
cos(
φ
1
−
φ
2
)
+
1
−
3
cos(
φ
2
−
φ
3
)
.
(38)
Symmetric phase switching stops the leakage of total intensity
into the polarization channels. Imperfections in the phase switches
and differences in the power detection chains now merely manifest
themselves as reductions in the gain of the receiver.
In summary, a major role of phase switching in a correlation
polarimeter is to compensate for differences in the power detection
chains. If uncorrected, these differences would result in a leakage
of total intensity into the polarization channels. While asymmetric
phase switching reduces this leakage, it does not remove it entirely
thanks to imperfections in the phase switches themselves – sym-
metric phase switching is needed to do this. This result remains true
if we reintroduce imperfections to all the post-OMT components.
Stokes leakage due to imperfections in the OMT cannot be reduced
by this type of phase switching as the Mueller matrices are cascaded
– see Section 5.3.
5.3 Circularizing OMT errors
The circularizing OMT is perhaps the most critical component in a
polarimeter. Cleanly extracting orthogonal modes from a waveguide
or free-space wave without some difference between the treatment
of the modes, or leakage between them, is extremely difficult. Leak-
age between Stokes parameters caused by the OMT is impractical
4
This is sometimes called ‘double demodulation’.
MNRAS
446,
1252–1267 (2015)