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Arecibo Technical & Operations Memo ATOMS 2000-05
THE MUELLER MATRIX PARAMETERS FOR ARECIBO'S RECEIVER
SYSTEMS
December 21, 2000
Carl Heiles, Phil Perillat, Duncan Lorimer, Michael Nolan, Ramesh Bhat, Tapasi Ghosh, Murray
Lewis, Karen O'Neil, Chris Salter, Snezana Stanimiroc
ABSTRACT
During September 2000 we made extensive observations to obtain complete
characterizations of the polarization properties of all of Arecibo's receiver systems.
This memo describes a new method of deriving the Mueller matrices and describes
the matrix elements in terms of fundamental system parameters. These include:
G,
the fractional power gain error resulting from using incorrect cal values; , a measure
of the non-orthogonality of the feed outputs; , the tangent of which is the voltage
ellipticity of the feed's response; and two phase angles associated with the above
quantities. These parameters describe the system in physical terms and thus provide
a means for investigating the eects of design changes. Moreover, they allow one to
specify the transfer function for the voltages (known as the Jones matrix); in an array
of telescopes, this should allow one to make antenna-based instead of baseline-based
polarization corrections.
We also present the basic results for each of Arecibo's systems during the September
2000 measurements. Particular highlights include: (1) The very good performance of
LBW over a large frequency range; (2) The complete change of polarization sense of
LBN over the frequency range of its common usage; and (3) an irreconcilable dierence
between our results and radar results for SBN.
1. INTRODUCTION
In ATOMS 99-01 we considered and calibrated the polarization properties of the LBW
system, whose native polarization is dual linear. Here we treat all Arecibo systems using a revised
technique and parameter set. Some have native linear, some circular, and several|the turnstile
systems such as LBN|can be elliptical. The LBN system uses a turnstile for its orthomode
transducer, which is a relatively narrow-band device; it is tuned to give good dual circular near
1420 MHz, but as one departs from this frequency the polarization becomes dual elliptical.
The variety of native polarizations, especially the elliptical polarization of the LBN system,

{ 2 {
makes it desirable to generalize the techniques originally applied to the LBW system so that a
uniformly-dened and consistent set of Mueller matrices can be applied to all systems.
We begin by reviewing the basic theoretical concepts and developing the structure of our
treatment. x2 introduces the Stokes parameters, Mueller matrix, and Jones vector and matrix. x3
denes the Mueller matrices for the dierent eects we describe, including the transfer of radiation
from the source to the sky, the imperfections in the feed, the feed, and the amplier chain. x4
discusses the combined eects of the receiver components and calculates that matrix product. x5
describes the technique for evaluating the matrix elements from observations. Finally, x7 provides
the results from the September 2000 calibration observations, together with denitions of the
parameters.
2. STOKES, MUELLER, AND JONES
2.1. The Stokes and Jones vectors
Tinbergen (1996) provides an excellent discussion of polarization measurement. The
fundamental quantities are the four Stokes parameters (I; Q; U; V ), which we write as the
4-element Stokes vector
S =
2
6 6 6 6 4
I
Q
U
V
3
7 7 7 7 5
: (1)
The Stokes parameters are time averages of electric eld products; we use the terms \voltage"
and \electric eld" interchangeably because the radio telescope's feed converts one to the other.
The Jones vector represents the elds as orthogonal linear polarizations (EX ; E Y ), and considers
them as complex to account for their relative phase:
J =
"
EX
E Y
#
: (2)
Instructive special cases included pure linear and pure circular polarization. Orthogonal
linear polarizations are, obviously, (J X ; JY ) = ([1; 0]; [0; 1]) (In the text, we write these vectors
as rows instead of columns for typographical purposes). Orthogonal circular polarizations are
(J L ; JR ) =

[1;i] p
2
; [i;1] p
2

. Orthogonal polarizations satisfy, for example, JX 
J Y = 0. Here i =
p
1,
the bar over a symbol indicates the complex conjugate, and all products are time averages.
It is straightforward to relate the Stokes parameters to the Jones vector:

{ 3 {
I = EX 
EX +E Y 
E Y (3a)
Q = EX 
EX E Y 
E Y (3b)
U = EX 
E Y + 
EXE Y (3c)
iV = EX 
E Y 
EXE Y : (3d)
2.2. The Mueller and Jones matrices
When the elds pass through some device, such as a feed or an amplier, they suer amplitude
and phase changes. These modify the Stokes parameters. The Mueller matrix is the transfer
function between the input and output of the device:
S out = M
S in : (4)
The Mueller matrix is, in general, a 4  4 matrix in which all elements may be nonzero (but they
are not all independent). In the usual way, we write
M =
2
6 6 6 6
4
m II m IQ m IU m IV
m QI mQQ mQU mQV
mUI mUQ mUU mUV
m V I m V Q m V U m V V
3
7 7 7 7
5
: (5)
The matrix elements are just the partial derivatives, for example
m V Q = @V out
@Q in
   
I in ;U in
: (6)
Every Mueller matrix has its Jones matrix counterpart; the Jones matrix is the transfer
function for the voltages. We defer further discussion of Jones matrices until our treatment of
three specic cases of interest for radio astronomical systems.
3. THE RADIOASTRONOMICAL RECEIVER COMPONENTS AND THEIR
MATRICES
In this section, we consider the Mueller matrices of devices that are encountered by the
incoming radiation on its way from the sky to the correlator output. We consider three devices.

{ 4 {
The rst device encountered by the incoming radiation is the telescope (M sky ), whose azimuth
arm rotates with respect to the sky. The second device is the feed, which we split into two parts.
The rst part (MF ) has the ability to change the incoming linear to any degree of elliptical
polarization; by design, feeds are intended to produce either pure linear or pure circular, but in
practice the polarization is mixed, i.e. elliptical. The second part (M IFr ) describes imperfections
in such a feed, specically the production of nonorthogonal polarizations. The third device is the
amplier chain (MA ).
We describe Jones matrix of the incoming radiation in linear polarization, as in equation 2.
However, the feed matrix MF can radically change the polarization state; for example, as we shall
see a dual-circular feed changes the order of the Stokes parameters in equation 1. Thus the signal
voltages, after going through MF , are not intuitively described as (EX ; E Y ), because the (X; Y )
connote linear polarization. Therefore, for the voltages after the output of MF , we will use the
symbols (EA ; EB ), or simply (A; B), to emphasize the fact that the state of polarization can be
arbitrary.
We assume that the remaining device matrices M IFr and MA produce closely-matched
replicas of the input Stokes parameters because, by design (hopefully!), the imperfections in the
feed are small. We will retain only rst-order products of these imperfections.
3.1. Mueller matrix relating the radio source to the receiver input
Astronomical sources have linear polarization but very little circular polarization; we assume
zero. Moreover, we express the source polarization as a fraction of Stokes I. Thus,
S src =
2
6 6 6 6 4
1
Q src
U src
0
3
7 7 7 7 5
: (7)
A linearly polarized astronomical source has Stokes (Q src ; U src ) dened with respect to
the north celestial pole (NCP). The source polarization is conventionally specied in terms of
fractional polarization and position angle with respect to the NCP. We have
Q src = P src cos 2PA src (8a)
U src = P src sin 2PA src (8b)
P src = (Q 2
src + U 2
src ) 1=2 (8c)
PA src = 0:5 tan 1

U src
Q src

(8d)

{ 5 {
As we track a source, the parallactic angle PA az of the azimuth arm rotates on the sky. PA az
is dened to be zero at azimuth 0 and increase towards the east; for a source near transit, as at
Arecibo, PA az  az, where az is the azimuth angle of the source. The Stokes parameters seen by
the telescope are (Q sky ; U sky ), and are related to the source parameters by
MSKY =
2
6 6 6 6 4
1 0 0 0
0 cos 2PA az sin 2PA az 0
0 sin 2PA az cos 2PA az 0
0 0 0 1
3
7 7 7 7 5
: (9)
The central 2  2 submatrix is, of course, nothing but a rotation matrix.
3.2. Mueller matrix for a perfect feed providing arbitrary elliptical polarization
The feed modies the incoming voltages with its Jones matrix. Suppose that the feed
mixes these with arbitrary phase and amplitude, keeping the total power constant and retaining
orthogonality; this is a perfect feed that responds to elliptical polarization. Following Stinebring
(1982) and Conway and Kronberg (1969), we write the transfer equation for the feed as
"
E A;out
EB;out
#
=
"
cos  e i sin 
e i sin  cos 
# "
EX;in
E Y;in
#
: (10)
The feed can completely alter the polarization state, so the output Jones voltages are more
intuitively described by subscripts (A; B) instead of (X; Y ), which connote linear polarization.
Here  is the amount of coupling into the orthogonal polarization and  is the phase angle of that
coupling. For example, for a native linear feed  = 0 and  = 0; for a native circular feed  = 45 ô
and  = 90 ô . Using this with equation 3, we nd
MF =
2
6 6 6 6 4
1 0 0 0
0 cos 2 sin 2 cos  sin 2 sin 
0 sin 2 cos  cos 2  sin 2  cos 2 sin 2  sin 2
0 sin 2 sin  sin 2  sin 2 cos 2  + sin 2  cos 2
3
7 7 7 7 5
: (11)
Notice that in the right-bottom 3  3 submatrix, the o-diagonal transposed elements are of
opposite sign for two of the three pairs and the same sign for one. This is not an algebraic error!

{ 6 {
Some instructive special cases include:
(1) A dual linear feed:  = 0,  = 0, and M F is diagonal.
(2). A dual linear feed rotated 45 ô with respect to (X; Y ):  = 45 ô ,  = 0, and
MF =
2
6 6 6 6 4
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
3
7 7 7 7 5
: (12)
As expected, this interchanges Stokes Q and U , together with a sign change as bets rotation.
(3). A dual linear feed rotated 90 ô with respect to (X; Y ):  = 90 ô ,  = 0, and
MF =
2
6 6 6 6
4
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
3
7 7 7 7
5
: (13)
As expected, this reverses the signs of Stokes Q and U .
(4). A dual circular feed:  = 45 ô ,  = 90 ô , and
MF =
2
6 6 6 6 4
1 0 0 0
0 0 0 1
0 0 1 0
0 1 0 0
3
7 7 7 7 5
: (14)
The combination ( = 45 ô ;  = 90 ô ) permutes the order of the Stokes parameters in the output
vector, making it (I; V; U; Q). (EA 
EA EB 
EB ) provides Stokes V , instead of the Q written
in equation 3; in other words, it makes the feed native dual circular. With respect to linear
polarization,  = 45 ô has the same eect as in case (2), namely to interchange (Q; U) and change
the sign of Q, because it is equivalent to a feed rotation of 45 ô . More generally, if  = 90 ô then
orthogonal linear inputs produce orthogonal elliptical outputs with the ellipticity voltage ratio
equal to tan  (see Tinbergen 1996, Figure 2.1).
3.3. An important restriction: we set  = 90 ô
Generally speaking, we prefer either pure linear or pure circular feeds, i.e. we prefer either
the combination (; ) = (0 ô ; 0 ô ) or (; ) = (45 ô ; 90 ô ) in MF . If a feed is designed to produce

{ 7 {
circular polarization with (; ) = (45 ô ; 90 ô ) and instead produces elliptical polarization whose
major axis is aligned with X, then  = 90 ô but  6= 45 ô . If the ellipse is not aligned with X, then
 6= 90 ô , but this is equivalent to having  = 90 ô and physically rotating the feed. Thus, without
loss of generality, we can take  = 90 ô . This leads to great simplication in MF , whose restricted
form becomes
MFr =
2
6 6 6 6 4
1 0 0 0
0 cos 2 0 sin 2
0 0 1 0
0 sin 2 0 cos 2
3
7 7 7 7 5
: (15)
3.4. An imperfect feed
Again we follow Stinebring (1982) and Conway and Kronberg (1969), and represent the
imperfections of a feed by the Jones matrix
"
EX;out
E Y;out
#
=
"
1  1 e i 1
 2 e i 2 1
# "
EX;in
E Y;in
#
: (16)
Here the 's represent undesirable cross coupling between the two polarizations; for example, this
might be caused by the two linear probes not being exactly 90 ô apart. The 's are the phase
angles of these coupled voltages. This equation assumes that the feed is \good", meaning that we
need retain only rst-order terms in  (which makes the diagonal elements unity); however, for the
moment we allow the phases to be arbitrary.
After a little algebra, we nd the matrix for the Imperfect Feed to be
M IF =
2
6 6 6 6 4
1 0  cos  sin
0 1
 cos
 sin
 cos
 cos 1 0
 sin
 sin 0 1
3
7 7 7 7 5
; (17)
where  cos =  1 cos  1 +  2 cos  2 ;  sin =  1 sin  1 +  2 sin  2 ;
 cos =  1 cos  1  2 cos  2 ;

 sin =  1 sin  1  2 sin  2 . The imperfections in a good feed are completely specied by four
independent parameters.
The central 4-element submatrix is a rotation matrix that represents an error in position angle
of linear polarization (its diagonal elements are unity because of our rst-order expansion in );

{ 8 {
dierent amounts of in-phase mutual voltage coupling between the two probes causes an apparent
rotation. The o-diagonal elements
 cos are impossible to measure without calibration sources
whose position angles are accurately known. Moreover, a small rotation can also occur because of
mechanical inaccuracy in mounting the feed. In practice, these problems make it impossible to
separate this factor, so one might as well assume it is equal to zero.
Assuming
 cos = 0 is consistent with a more stringent assumption, namely that    1 =  2
and    1 =  2 so that, also, we have
 sin = 0. Physically, this is equivalent to assuming that
the coupling in a linear feed arises from the two probes being not quite orthogonal and that the
coupling in each has a the same relative phase. In other words, it makes the correlated output
non-orthogonal.
This assumption might seem to be too restrictive because, by requiring  1 =  2 , it eliminates
the possibility of imperfections inducing a change in the ellipticity of the polarization. However, it
leads to no loss in generality of our treatment, because the out-of-phase coupling case is included in
the feed matrix MF . Our restricted case of an imperfect feed is described by just two parameters:
M IFr =
2
6 6 6 6
4
1 0 2 cos  2 sin 
0 1 0 0
2 cos  0 1 0
2 sin  0 0 1
3
7 7 7 7
5
: (18)
3.5. The ampliers
The two polarization channels go through dierent amplier chains. Suppose these have
voltage gain (g A ; g B ), power gain (GA ; GB ) = (g 2
A ; g 2
B ), and phase delays ( A ; B ). The Jones
matrix is
"
E A;out
EB;out
#
=
"
g A e i A 0
0 g B e i B
# "
E A;in
EB;in
#
(19)
In practice, the amplier gains and phases are calibrated with a correlated noise source (the
\cal"). Thus, our amplier gains (GA ; GB ) have nothing to do with the actual amplier gains.
Rather, they represent the gains as calibrated by specied cal intensities, one for each channel. If
the sum of the specied cal intensities is perfectly correct, then the absolute intensity calibration
of the instrument is correct (Stokes I is correctly measured in absolute units). In our treatment,
GA +GB = 2 by necessity because we deal with fractional polarizations.
If the dierence is correct, then the dierence between the two polarization channels is zero
for an unpolarized source. This happy circumstance does generally not obtain. However, the

{ 9 {
relative cal intensities are known fairly well, which allows us to assume
G  GA GB  1 and
to carry g A g B to rst order only, meaning we take g A g B = 1. With this rst-order approximation,
we have
MA =
2
6 6 6 6 4
1
G
2 0 0

G
2 1 0 0
0 0 cos sin
0 0 sin cos
3
7 7 7 7 5
: (20)
The incorrect relative cal amplitudes produce coupling of Stokes I into Q through the nonzero
m QI ; the dierence between the relative cal and sky phases produces a transfer of power between
the two correlated outputs, as we now discuss.
The dierence between the amplier phases is also referred to the cal. Thus a phase dierence
= A B represents the phase dierence that exists between a polarized astronomical source
and the cal and has nothing to do with the amplier chains. The behavior of this phase dierence
depends on the native polarization of the feed.
For a perfect native dual linear feed, the phase of an astronomical source is independent of
PA az because the linearly polarized dipoles have no relative phase dierence ( = 0). Thus, is
independent of PA az . The desirable case = 0 means that (EA 
EB + 
EBEA ) contains pure Stokes
U and (EA 
EB 
EBEA ) pure Stokes V .
For a perfect native dual circular feed, the phase of an astronomical source (and therefore
) rotates as 2PA az (see discussion following equation 24). At PA az = 0, the condition = 0
produces the correctly dened Stokes parameters in equation 14, (EA 
EB + 
EBEA ) = U sky and
(EA 
EB 
EBEA ) = Q sky . If 6= 0, then the correctly dened Stokes parameters occur at
PA az = 
2 .
3.6. The correlator outputs
Our measured quantities are four time averaged voltage products from the digital correlator:
from autocorrelation, (EAEA ; EBEB ); from crosscorrelation, (EAEB ; EBEA ). In these products
we consider the second quantity to be delayed relative to the rst. Each correlation function has
N channels of delay. We Fourier transform (FT) these quantities to obtain spectra.
The autocorrelation functions are symmetric and thus their FT's are real, with no imaginary
components; we denote their FT's by (AA; BB). We combine the two measured crosscorrelation
functions into a single one with 2N channels; it has both negative and positive delays and is
generally not symmetric, so its FT is complex. We denote the real and imaginary parts of its FT
by (AB;BA). The Stokes parameters that we calculate from these products are as in equation 3,

{ 10 {
APB  AA+BB = EA 
EA +EB 
EB (21a)
AMB  AA BB = EA 
EA EB 
EB (21b)
AB = EA 
EB + 
EAEB (21c)
iBA = EA 
EB 
EAEB : (21d)
If we have an ideal native dual linear feed and receiver, then (APB;AMB;AB;BA) =
(I; Q; U; V ) sky ; for native circular, (APB;AMB;AB;BA) = (I; V; U; Q) sky .
4. THE SINGLE MATRIX FOR THE RADIOASTRONOMICAL RECEIVER
4.1. The general case with  = 90 ô
The observing system consists of several distinct elements, each with its own Mueller matrix.
The matrix for the whole system is the product of all of them. Matrices are not commutative, so
we must be careful with the order of multiplication.
We express the Jones vector of the incoming radiation in linear polarization. The radiation
rst encounters the feed, producing Stokes parameters as specied by MFr in equation 15. Next it
suers the restricted set of imperfections associated with M IFr (equation 18). Finally it proceeds
through the amplier chains, undergoing MA (equation 20). The product of these matrices, in this
order (MTOT = MA
M IFr
MFr ), produces the vector that the correlator sees, which we denote
by COR. In calculating COR, we ignore second order terms in the imperfection amplitudes
(;
G) but, of course, retain all orders in their their phases (; ) and also in the feed parameter
. This gives
MTOT =
2
6 6 6 6 4
1 ( 2 sin  sin 2 +
G
2 cos 2) 2 cos  (2 sin  cos 2 +
G
2 sin 2)

G
2 cos 2 0 sin 2
2 cos( + ) sin 2 sin cos cos 2 sin
2 sin( + ) sin 2 cos sin cos 2 cos
3
7 7 7 7 5
:(22)
The terms in the top row make I 6= 1 for a polarized source. If one derives fractional polarization,
for example Q
I , then it will be in error by amounts comparable to [(;
G)  (Q; U; V )]. For the
weakly polarized sources we use as calibrators, these products are second order and therefore are
of no concern.
However, for a strongly polarized source such as a pulsar, these terms are rst order. This
can be particularly serious for timing, because polarization variations across the pulse will produce
errors in the pulse shape. These eects can be eliminated by correcting for the Mueller matrix.

{ 11 {
4.2. Some instructive special cases
Below we discuss the observational determination of these parameters. For linear and circular
feeds there are degeneracies in the least square tting. To understand these points we will need to
consider two instructive special cases:
(1) A dual-linear feed with a slight elliptical component, meaning that [ = (0 ô ; 90 ô ) + ô]
with ô  1. We ignore second order products involving ô:
MTOT;lin =
2
6 6 6 6
4
1 
G
2 2 cos  2 sin 

G
2
1 0 2ô
2 cos( + ) 2ô sin cos  sin
2 sin( + ) 2ô cos sin  cos
3
7 7 7 7
5
: (23)
For terms with two signs, the top sign is for the 0 ô case and the bottom for the 90 ô case.
(2) As above, but for a dual-circular feed with [ = (45 ô ; 135 ô ) + ô]:
MTOT;circ =
2
6 6 6 6 4
1 2 sin  2 cos  
G
2

G
2 2ô 0 1
2 cos( + )  sin cos 2ô sin
2 sin( + )  cos sin 2ô cos
3
7 7 7 7 5
: (24)
Recall that in the circular case, the order of the Stokes parameters in the output vector is
permuted: [I; V; U; Q]. As with the imperfect linear, the imperfections in M IFr produce coupling
between Stokes I and (V; U; Q), represented by the nonzero elements in the left column. The
order of the terms in the column is independent of the feed polarization, but the Stokes parameters
are not, so the imperfections produce dierent eects for the two types of feed.
5. EVALUATING THE PARAMETERS IN THE MATRIX
We evaluate the parameters in equation 22 using observations of a polarized source tracked
over a wide range of position angle PA az . The source is described by S src and the Mueller matrix
for the radiation entering the feed by M sky , both described in section 3.1. The full Mueller matrix
is MTOT
M sky . The product of this matrix with S src results in a set of of four equations, one for
each element of the observed COR vector. Recalling that we dene the source Stokes parameters
as fractional (so that I src = 1) and that we assume V src = 0, they are expressed by the equation

{ 12 {
2
6 6 6 6
4
APB
AMB
AB
BA
3
7 7 7 7
5
= MTOT
MSKY

2
6 6 6 6
4
1
Q src
U src
0
3
7 7 7 7
5
: (25)
However, the equation for the rst element APB of COR is useless because we are using fractional
polarizations. This means that in equation 25 we divide each element of COR by APB. This
eliminates the position angle variation of APB by forcing it to be unity; we do this because it is
impossible to account perfectly for small gain variations in measurements of Stokes I at dierent
PA az . This also produces errors in the other elements of COR. However, these errors are second
order because they are products of
G and/or  with quantities (such as Q src ) that are already
rst order. Our whole treatment neglects second order products, so we can neglect these errors.
Discarding the rst equation in APB, we are left with three equations of the form
AMB = AAMB +BAMB cos 2PA az + CAMB sin 2PA az ; etc; (26)
where the coeôcients (A; B; C) are complicated function of the 5 parameters (; ; ;
G; ) and,
also, the two source Stokes parameters which are not known ab initio, so we have 7 unknown
parameters. We have 9 measured quantities, three (A; B; C) for each correlator output; these are
derived from least squares ts of (AMB;AB;BA) to PA az . We use nonlinear least squares tting
to solve for the 7 unknown parameters. In practice, we use numerical techniques to obtain the
relevant derivatives. An alternative tting technique that is useful when combining the results
from N src dierent sources into a grand average is to express the coeôcients AAMB , etc., in terms
of the (5 + 2N src ) unknown coeôcients, and to lump all observations of all sources together in one
grand nonlinear least squares t.
Nonlinear least square tting is often plagued by multiple minima, and the present case is
no exception when the polarization is nearly pure linear or circular. To discuss these cases we
temporarily assume (;
G) = 0, which makes the mathematics more transparent.
5.1. The nearly linear case
Equation 25 becomes
2
6 4
Q out
U out
V out
3
7 5 
2
6 4
AMB
AB
BA
3
7 5 =
2
6 4
1 0
2ô sin cos
2ô cos sin
3
7 5
"
cos 2PA az sin 2PA az
sin 2PA az cos 2PA az
# "
Q src
U src
#
(27a)

{ 13 {
and, when reduced further, becomes

AMB
AB
BA

=

 cos 2PAsrc  sin 2PAsrc
cos sin 2PAsrc  2ô sin cos 2PAsrc cos cos 2PAsrc  2ô sin sin 2PAsrc
sin sin 2PAsrc  2ô cos cos 2PAsrc sin cos 2PAsrc  2ô cos sin 2PAsrc
 h
Psrc cos 2PAaz
Psrc sin 2PAaz
i
:(27b)
For ô = 0 we have
2
6 4
AMB
AB
BA
3
7 5 =
2
6 4
 cos 2PA src  sin 2PA src
cos sin 2PA src cos cos 2PA src
sin sin 2PA src sin cos 2PA src
3
7 5
"
P src cos 2PA az
P src sin 2PA az
#
: (27c)
This shows that, for ô ! 0, one cannot distinguish between the two cases (; ) = (0 ô ; 0 ) and
(; ) = (90 ô ; 0 + 180 ô ), because the two bottom rows change sign for 0 + 180 ô . For these two
cases, the signs of (Q; U) change, which is equivalent to rotating the feed by 90 ô . The physical
interpretation is straightforward:  = 90 ô converts EX to E Y in equation 10, thus changing the
sign of Q; changing by 180 ô changes the sign of U ; the combination is equivalent to rotating the
feed by 90 ô .
In practice one must deal with this problem. For a conventional linear feed, loosely described
as two E-eld proves in a circular waveguide, the combination (; ) = (90 ô ; 180 ô ) is physically
unreasonable. However, for one of Arecibo's turnstiles operating in linear polarization, either
possibility can occur and one must make the appropriate choice based on frequencies well away
from where the feed is pure linear; for a turnstile we expect d
df  const:
5.2. The nearly circular case
Equation 25 is
2
6 4
V out
U out
Q out
3
7 5 
2
6 4
AMB
AB
BA
3
7 5 =
2
6 4
2ô 0
 sin cos
 cos sin
3
7 5
"
cos 2PA az sin 2PA az
sin 2PA az cos 2PA az
# "
Q src
U src
#
(28a)
and becomes
2
6 4
AMB
AB
BA
3
7 5 =
2
6 4
2ô cos 2PA src 2ô sin 2PA src
sin(2P A src  ) cos(2P A src  )
 cos(2P A src  )  sin(2P A src  )
3
7 5
"
P src cos 2PA az
P src sin 2PA az
#
: (28b)

{ 14 {
This shows that the angles 2PA src and are inextricably connected. We can determine only their
sum (for  = 45 ô ) or their dierence. It also shows that the two solutions  = (45 ô ; 45 ô ) are
degenerate as ô ! 0, because the bottom row changes sign for these two cases (thus ipping the
derived sign of Q src and thereby rotating the derived PA src by 90 ô ). The physical interpretation
of this degeneracy is straightforward: for a pure circular feed, the phase of a linearly polarized
source rotates with 2PA az and its absolute value depends both on the system phase and the
source position angle PA src .
In practice, one can only deal with this problem by having additional information, namely
knowing either or PA src . For the case of Arecibo's turnstile junctions, which are narrow band
devices, one can determine at frequencies well away from that where the feed is pure circular
and interpolate. Then, during the t, this value of should be xed. For a wideband circular
feed, there is no substitute for an independent calibration of the linearly polarized position angle,
either with a test radiator or with a source of known polarization.
5.3. Commentary
(1)  is the quadrature sum of the PA az -independent portions of the two correlated outputs
(AB;BA). This power is distributed between those outputs according to ( + ). In the near
linear case, can change by 180 ô by changing the choice for , and this also produces a 180 ô
change in .
(2) Consider a high-quality standard linearly polarized feed that has a correlated cal
connected by equal-length cables. Such a feed has   0 ô and the equal-length cables mean that
 0 ô . However, the solution yields  180 ô if the sign of AMB is incorrect, which can easily
happen if one interchanges cables carrying the two polarizations; this is equivalent to reversing the
handedness of PA az and PA src . For LBW at Arecibo, the sign of AMB is in fact incorrect, which
makes the derived  180 ô instead of the proper value  0 ô .
(3) For Arecibo's systems without correlated cals, is meaningless. has contributions at
r.f. (from the dierence in cable and electrical lengths to the rst mixer) and i.f. (the optical ber
to the control room and other i.f. cables). The latter dominates because d
df  0:1 rad MHz 1 ,
roughly constant among dierent systems. Moreover, each correlator board has its own ; thus,
without a correlated cal, one cannot compare phases in dierent correlator boards. This means
that, for dual circular feeds, one cannot derive Faraday rotation between dierent boards. One
cannot even derive Faraday rotation using the spectrum of a single board, because PA src depends
on , and the slope d
df from the system totally dominates that from Faraday rotation for nearly
all astronomical sources.
With the current phase calibration procedure, if there is a correlated cal then we measure cal

{ 15 {
and t it to a constant plus a slope d
df ; we subtract this t from the source phase and produce
corrected versions of (AB;BA). Thus, the only component left in the correlated products is the
dierence between source and cal phase, which is the same as in the equation 20.
If there is not a correlated cal, then we measure src and t it to a constant plus a slope d
df ;
we subtract the slope but not the constant from the source phase and produce corrected versions
of (AB;BA). While most of this slope is in the system, this procedure also subtracts away any
intrinsic slope caused by Faraday rotation.
6. APPLYING THE CORRECTION
One of the major reasons to determine the Mueller matrix elements is to apply it to
observations and obtain true Stokes parameters. There are two steps to this process.
6.1. Applying MTOT and M sky
The completely general form of equation 25 uses observed voltage products instead of
fractional ones and does not force V = 0. Thus, to derive the source Stokes parameters from the
data, we use
2
6 6 6 6 4
I src
Q src
U src
V src
3
7 7 7 7 5
= (MTOT
MSKY ) 1

2
6 6 6 6 4
APB
AMB
AB
BA
3
7 7 7 7 5
: (29)
6.2. Deriving true astronomical position angles
The position angle of the source polarization PA src is dened relative to the local
idiosyncrasies. For a dual linear feed these include the angle at which feed probes happen to be
mounted and, also, which feed probe happens to be dened as X (equivalent of A). For a dual
circular feed this includes the phase angle at which the correlated cal happens to be injected and
the angle at which the turnstile happens to be oriented.
Astronomers wish to express position angles in the conventional way, viz. with PA src
measured relative to the North Celestial Pole. There is also the possibility of its handedness, but
this is taken care of automatically in the tting process if the PA az is correctly dened. To satisfy
the astronomers' desire, we must apply a rotation matrix M astron .
For example, for LBW the feeds probes are oriented 45 ô with respect to the azimuth arm

{ 16 {
and the derived values of PA src need to be rotated by 45 ô ; we denote this angle by  astron . To
accomplish this, one applies M astron , which looks like MSKY in equation 9 with PA az replaced by
 astron ; for LBW,  astron = 45 ô . There is a sign ambiguity that is best determined by empirical
comparison with known astronomical position angles. At low frequencies, one must include the
eects of terrestrial ionospheric Faraday rotation, which is time variable.
7. RESULTS
Here we present the results, beginning with a table for all receivers and then a brief
commentary on each system. First, however, we reiterate the denitions of the parameters and
make a general remark about the relative cal values.

G is the error in relative intensity calibration of the two polarization channels. It results
from an error in the relative cal values (T calA ; T cal B ). Our expansion currently takes terms in
G
to rst order only. Therefore, in cases like LBN at 1415 MHz (which has
G  0:2), the other
terms will be aected.
The relative cal values should be modied to make
G = 0, keeping their sum the same. To
accomplish this, make T calA;modified = T calA

1
G
2

and T calB;modified = T calB

1 +
G
2

.
is the phase dierence between the cal and the incoming radiation from the sky; see the
discussion following equation 20. It redistributes power between (U; V ) for a dual linear feed and
between (Q; U) for a dual circular feed (equations 23 and 24).
 is a measure of the voltage ratio of the polarization ellipse. Generally, the electric
vector traces an ellipse with time; tan  is the ratio of major and minor axes of the voltage
ellipse. Thus, tan 2  is the ratio of the powers. If a source having fractional linear polarization
P src =
p
Q 2
src + U 2
src is observed with a native circular feed that has  6= 45 ô , then the measured
Stokes V will change with 2PA az and have peak-to-peak amplitude 4.
 is a measure of imperfection of the feed in producing nonorthogonal polarizations (false
correlations) in the two correlated outputs. Our expansion takes  to rst order only. The only
astronomical eect of nonzero  is to contaminate the polarized Stokes parameters (Q; U; V ) by
coupling Stokes I into them at level  2; the exact coupling depends on the other parameters.
For weakly polarized sources, this produces false polarization; for strongly polarized sources such
as pulsars, it also produces incorrect Stokes I.
 is the phase angle at which the voltage coupling  occurs. It works with  to couple I with
(Q; U; V ).
 astron is the angle by which the derived position angles must be rotated to conform with
the conventional astronomical denition.
Relative cal values: We found surprisingly large values of
G for some systems, particularly

{ 17 {
the workhorses LBN and LBW, which also exhibit signicant frequency dependence. We used these
observations to determine more accurate relative cal values. And we have requested engineering
to make new measurements of all the system cals. When new values become agreed upon, they
should be inserted into Phil's le from which our software obtains the values. Until that happens,
however, the values of
G obtained with the older, incorrect relative cal values should be used.
These are the values provided in the table below.
Table 1: ADOPTED MUELLER MATRIX PARAMETERS, SEPT 2000
RCVR FREQ
G     astron
430CH a 430 0.010 0 ô 47:6 ô 0.036 49:3 ô {
430G 430 {0.023 150 ô 0:5 ô 0.006 78:6 ô ?
610 610 ? 95 ô 1:3 ô 0.021 100 ô ?
LBW 1415 0.10 175:4 ô 0:25 ô 0.0015 148 ô 45 ô
LBN a;b 1415 0.034 43:6 ô 47:7 ô 0.003 21 ô {
SBN 2380 0.008 38 ô 39:9 ô 0.005 88 ô 15 ô
CB 5000 0.020 ? 1:2 ô 0.005 132 ô 78 ô
a : The system does not have a correlated cal, so has little relevance.
b : LBN has a turnstile junction and its properties vary rapidly with frequency.
7.1. Commentary on Individual Receivers
430CH is a turnstile junction without a correlated cal, so the parameters has no meaning.
We adjusted the relative cal values to obtain the low value of
G given here. Those values are
(lef t; right) = (tcalxx; tcalyy) = (27:4; 39:6) K.
430G was, in September 2000, a dual-linear feed. The week after we measured it it was
converted to a turnstile adjusted for circular polarization. Therefore, the numbers given here have
no relevance for current and future measurements.
610 provided contradictory results for
G on dierent days, as if the two i.f. cables had been
interchanged. According to engineering, this did not happen. Accordingly, we could not determine
any sensible value for
G.
LBW covers a huge frequency range, and our measurements cover 1150 ! 1666 MHz. We
discuss this receiver in more detail below.
LBN is a turnstile junction used over a large frequency range, and accordingly needs the
more detailed discussion presented below.
SBN is the radar turnstile feed, adjusted for circular polarization; it needs a longer discussion,
provided below.

{ 18 {
CB is a dual linear feed covering a huge frequency range, and our measurements cover the
relatively miniscule range 4500 ! 5400 MHz. We found a very rapid frequency dependence of
; it corresponded to a path length dierence of 0.5 meter or, perhaps, 2 meters. We notied
engineering that we suspected a problem in the network connecting the cal to the feed. They
indeed found a problem: a 2 db attenuator in one of the correlated cal cables. Accordingly,
the value of is indeterminant. By comparing polarization results for the sources B0017+154,
B1615+212, B2209+080 we estimated  astron = 78 ô . This value needs to be checked by repeated
observations and, also, determining the angle of the feed probes with respect to the azimuth arm;
 astron should equal this angle, or its complement.
7.2. LBN
LBN is a turnstile system without a correlated cal. Near 1415 MHz the polarization is dual
circular; the absence of a correlated cal means that the parameter has no meaning. The value
43:6 ô in our table is the value that happened to pop out of the t for the 1415 MHz measurement
and has no special meaning.
This system is commonly used over a large frequency range. A turnstile is a narrow-band
device, and over this range the polarization changes from linear to circular and back again. The
solid lines in Figure 1 exhibit the frequency dependence of
G, , , and  for the 25 MHz band
centered at four frequencies. These particular data were derived from the source B0017+154; we
obtained data for two additional sources, and the results are very close. The dashed lines are our
adopted analytic expression, which are dened by

G = 0:034 1:78  10 4 f 15 + 3:27  10 6 f 2
15 (30a)
 = 47:74 0:363f 15 (30b)
 = 0:0028 + 1:38  10 5 f 15 + 1:31  10 6 f 2
15 (30c)
 = 21 + 1:02f 15 (30d)
where f 15 = f 1415MHz and angles are in degrees.
The linear dependence of  on f 15 is just what's expected for a turnstile junction. However,
the variation of  is remarkably complicated, varying rapidly with frequency as one goes away
from the 1400 MHz center frequency, and we do not understand the reason. The scatter in  for
the 1375 MHz spectrum simply re ects the uncertainty in the angle, which is large because  is
small. Of course,
G simply re ects inaccurate relative cal values and not the properties of the
turnstile itself.

{ 19 {
Fig. 1.| Mueller matrix parameters versus frequency for LBN, together with our adopted analytic
approximations from equation 30.
7.3. LBW
LBW is a very wide band feed with native linear polarization. It has some problems with
resonances, which probably contribute to system temperature over the whole band (Heiles 1999).

{ 20 {
Otherwise, the feed properties are well-behaved and we take them to be frequency independent.
We approximate the current
G by

G = 0:100 + 0:015 cos(2f 20 =300:) (31)
where f 20 = f 1420MHz.
Figure 2 exhibits observational results for two sources, including the seven dierent center
frequencies and the 25-MHz spectrum covered by each correlator spectrum. The change of
position angle from Faraday rotation is obvious over the broad band covered by the data, and even
within a single spectrum. Regarding fractional polarization, the decrease toward low frequency
for B0017+154 is not surprising; this is caused by Faraday depolarization. However, that of
B1634+269 reaches a minimum near 1400 MHz, and this behavior, with a rise towards lower
frequencies, is not expected. We believe that our measurements are correct and that this observed
behavior is real.
7.4. SBN
SBN is the radar turnstile feed, adjusted for circular polarization, with a correlated cal; this
combination provides measurements of linear polarization that are free of gain errors, making it
very precise. We determined the angle  astron from the sources B2249+185 and B0134+329, taking
their position angles to be PA src = (47 ô  4 ô ; 58 ô  13 ô ) (Tabara and Inoue 1980); these produced
 astron = ( 17 ô ; 2 ô ). These are signicantly dierent! We adopt  astron = 15 ô because of the
larger error for B0134+329, and clearly this value is not very accurate. The error in this adopted
 astron produces a systematic error that applies to all sources and  astron should be bootstrapped
to a more accurate value by comparing future observations with known standards.
A mystery about SBN arises when comparing our value of  to the radar re ection from the
front surface of Venus. Pure circular polarization should convert 100% of the re ected power to
the orthogonal circular polarization, while pure linear should convert 0% to the orthogonal linear
polarization. Observations show that the former circumstance is very nearly attained, with only
0:1% of the power returning in the incident polarization. This corresponds to  = 44:99 ô . In
contrast, we derive  = 39:9 ô , for which the voltage ellipse has a major/minor axis ratio of 0.84
and the power ellipse a ratio of 0.70. These results are irreconcilable.
Our derivation of  involves a rather involved process that includes a nonlinear least squares
t. However, one can easily discern that  departs signicantly from 45 ô by simple inspection of
the data. Figure 3 exhibits the quantities (AMB;AB;BA) versus feed parallactic angle PA az for
the source B1749+096, which has linear polarization 5:7%. AMB is the dierence between the
two feed outputs and, if the feed is a pure dual-circular feed, should not respond to the linear
polarization in the source and, therefore, should have no dependence on PA az . However, the

{ 21 {
Fig. 2.| Polarizations of B1634+269 and B0017+154. Position angles have not been rotated by
 astron .
dependence is clear.

{ 22 {
Fig. 3.| Fractional correlator outputs versus the parallactic angle of the azimuth arm P az .
This work was supported in part by NSF grant 95-30590 to CH.
REFERENCES
Conway, R.G. and Kronberg, P.P. 1969, MNRAS, 142, 11
Stinebring, D. 1982, thesis, Cornell University.
Tabara, Tabara, H. and Inoue, M. 1980, A&A Suppl, 39, 379.
Tinbergen, J. 1996, Astronomical Polarimetry, Cambridge Univ Press.
This preprint was prepared with the AAS L A T E X macros v4.0.