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Arecibo Upgrade Notes
INSTRUMENTAL POLARIZATION:
POLARIZATION ISOLATION & ERRORS IN STOKES PARAMETERS
Jim Cordes & Joe Lazio
22 May 1996
Summary
The purpose of this memo is to:
(1) demonstrate how polarization isolation is related to errors in Stokes parameters;
(2) estimate a tolerable level of errors in Stokes parameters due to amplitude and phase mismatch
in and before the hybrid;
(3) emphasize that Stokes parameters will have to be corrected after the fact no matter how good
the hardware.
(4) estimate that isolation of ё 25 dB is needed, corresponding to phase errors ё 5 ffi and
power ratio ё 0:8 dB; it probably is not worth making heroic efforts to obtain increased
isolation, because software correction can better address the totality of systematic errors
that contaminate raw polarization data.
Stokes Parameter Matrix
In our 1992 Memo (Instrumental Polarization: Cross Coupling and Hybrid Conversion Issues)
we showed that the measured (primes) and true Stokes parameters are related in the following
way:
0
B
B
B
@
I 0
Q 0
U 0
V 0
1
C
C
C
A
=
0
B
B
B
@
G \DeltaG
2 0 0
\DeltaG
2 G 0 0
0 0 fl cos OE fl \Gammafl sin OE fl
0 0 fl sin OE fl +fl cos OE fl
1
C
C
C
A
0
B
B
B
@
I
Q
U
V
1
C
C
C
A
; (1)
where
G = 1
2 (G x + G y )
G x;y = jg x;y j 2
\DeltaG = G x \Gamma G y

-- 2 --
fl = (G x G y ) 1=2
OE fl = OE x \Gamma OE y \Gamma ffi OE;
g x;y are the complex gains of the x; y signals going into the hybrid, OE x;y is the phase of the complex
gain g x;y and ffi OE is the phase error in the hybrid.
Polarization Error
Consider pure RHCP of unit amplitude incident on the telescope. The true and measured
Stokes parameters are (using the V = LHCP \Gamma RHCP convention)
0
B B B
@
I
Q
U
V
1
C C C
A
=
0
B B B
@
1
0
0
\Gamma1
1
C C C
A
;
0
B B B
@
I 0
Q 0
U 0
V 0
1
C C C
A
=
0
B B B
@
G
\DeltaG=2
\Gammafl sin OE fl
\Gammafl cos OE fl
1
C C C
A
: (2)
The polarization error, defined as the error in the polarized intensity (\DeltaI pol =
[\DeltaQ 2 + \DeltaU 2 + \DeltaV 2 ] 1=2 , where \DeltaQ = Q 0 \Gamma Q, etc.) divided by the total intensity, I ,
is
ffiъ j \DeltaI pol
I
=
h
(\DeltaG=2) 2 + (fl sin OE fl ) 2 + (1 \Gamma fl cos OE fl ) 2
i 1=2
: (3)
Defining a gain ratio R = G y =G x and setting G x = 1, the polarization error is
ffi ъ = 1
2
h
(1 +R) 2 \Gamma 8
p
R cos OE fl + 4
i 1=2
: (4)
For a pure phase error, the polarization error reduces to ffi ъ = 2j sin OE fl =2j.
Polarization Isolation
Define the degree of isolation in terms of the (pseudo) LHCP that is measured in response to
the incident RHCP:
i j I 0
L
I 0
= I 0 + V 0
2I 0
: (5)
Substituting the gain ratio R j G y =G x , the isolation becomes
i = 1 +R \Gamma 2
p
R cos OE fl
2(1 + R)
: (6)
For a 1 dB amplitude mismatch and no phase error, Eq. 6 gives i = 0:0033 or \Gamma24:8 dB of isolation.
However, the errors in Stokes parameters are not so small: Eq. 4 indicates that the polarized flux
is in error by ffi ъ ё 22%. Each Stokes parameter is in error by an amount ё ffiъ=
p
3. Scientifically
useful results require precisions in Stokes parameters better than 1%. It does not appear practial
to strive for this precision in hardware because other effects, such as cross coupling in the feed,
require correction in software using a variety of calibration measurements using celestial sources.

-- 3 --
Figures
Figure 1 shows the polarization error ffi ъ plotted against the gain ratio R for different values
of phase error, OE fl . Note that as R gets larger, the polarization error becomces increasingly
less sensitive to phase errors. Also, for realistically achievable gain and phase mismatches, the
polarization error does not get better than ­15 dB.
Figure 2 shows contours of constanti against phase error (OE fl and the amplitude ratio, R.
What Level of Instrumental Polarization is Tolerable?
Examples given above suggest that better than 25 dB of isolation is needed. However, even
\Gamma30:8 dB isolation (or R of 0.5 dB) still gives 6% errors in individual Stokes parameters. The only
way to achieve 1% errors is to correct Stokes parameters in software by inverting Eq. 1. More
accurately, one needs to invert a similar matrix (given in our 1992 memo) that includes cross
coupling in the horn + OMT as well as hyrbrid errors.
A workable situation may be defined in terms of our experience with the line feed antennas.
There we had a level of cross coupling that gave about 10% errors in Stokes parameters. Through
tracking of polarized sources across parallactic angle, the cross coupling parameters could be
determined and the Stokes parameters corrected to about 1%.
With the more complicated situation involving the hybrid combined with cross coupling, the
hardware should be designed to yield ё ! 10% errors in Stokes parameters. This corresponds to an
amplitude ratio of 0.8 dB or less (0:8 ё ! R = G y =G x ё ! 1:2) or a phase error jOE fl j ё ! 0:1 rad = 5: ffi 7,
corresponding to an isolation i ё ! \Gamma26 dB. If this can be achieved, then the remaining correction
can be made in software.
How bad can the instrumental polarization be and still allow inversion of the matrix relation?
The bigger the off­diagonal terms in the 4 \Theta 4 matrix, the more unstable the correction and the
more accurately that the matrix elements need to be determined. For example, inversion of the
2 \Theta 2 submatrix involving I ; Q in Eq. 1 involves the determinant D = G 2
\Gamma (\DeltaG=2) 2 in the
denominator of the matrix elements. The elements become unwieldy as \DeltaG=2G ! 1.