The MEASUREMENT EQUATION of a generic radio telescope AIPS++ Implementation Note nr 185

28 Pages·2011·0.24 MB·English

The MEASUREMENT EQUATION
of a generic radio telescope
AIPS++ Implementation Note nr 185
J.E.Noordam
([email protected])
15 February 1996, version 2.0
File: /aips++/nfra/185.latex SymbolsFile: /aips++/nfra/megi-symbols.tex
Abstract: This note is a step towards an ‘o(cid:14)cial’ AIPS++ description of the Mea-
surementEquation, basedonanagreedset ofnamesandconventions. Thelatter have
been de(cid:12)ned in a separate TeX (cid:12)le, and can (should) be used in subsequent AIPS++
documents to ensure consistency.
Contents
1 INTRODUCTION 3
2 THE M.E. FOR A SINGLE POINT SOURCE 4
2.1 The feed-based instrumental Jones matrices . . . . . . . . . . . . . . . . . . . . . 4
2.2 The Jones matrix of a Tied Array feed . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Jones matrices for multiple beams . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 THE FULL MEASUREMENT EQUATION 6
3.1 Summing and averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 interferometer-based e(cid:11)ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 POLARISATION COORDINATES 8
5 GENERIC FORM OF JONES MATRICES 10
5.1 Ionospheric Faraday rotation (F((cid:26)~;~r)) . . . . . . . . . . . . . . . . . . . . . . . . 10
i i
5.2 Atmospheric gain (T((cid:26)~;~r)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
i i
1
5.3 Fourier Transform kernel (K(~r:(cid:26)~)) . . . . . . . . . . . . . . . . . . . . . . . . . . 11
i i
5.4 Projection matrix (P) if (cid:13) = (cid:13) . . . . . . . . . . . . . . . . . . . . . . . . . . 11
i xa yb
5.5 Projection matrix (P) if (cid:13) = (cid:13) . . . . . . . . . . . . . . . . . . . . . . . . . . 12
i xa yb
6
5.6 Voltage primary beam (E((cid:26)~)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
i
5.7 Position-independent receptor cross-leakage (D ) . . . . . . . . . . . . . . . . . . 14
i
5.8 Commutation (Y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
i
5.9 Hybrid (H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
i
5.10 Electronic gain (G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
i
5.11 Do we need a con(cid:12)guration matrix (C)? . . . . . . . . . . . . . . . . . . . . . . . 16
i
6 THE ORDER OF JONES MATRICES 18
6.1 Overview of commutation properties . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.2 Overview of Jones matrix forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.3 Allowable changes of order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.4 VisJones and SkyJones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.4.1 Tied Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
A APPENDIX: CONVENTIONS 23
A.1 Some de(cid:12)nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A.2 Labels, sub- and super-scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A.3 Coordinate frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A.4 Matrices and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A.5 Miscellaneous parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2
1 INTRODUCTION
The matrix-based Measurement Equation (ME) of a Generic Radio Telescope was developed
by Hamaker, Bregman andSault [2] [3], based on earlier work by Bregman [1]. After discussion
by Noordam [5] and Cornwell [6] [7] [8] [9] [10] [11], the M.E. has been adopted as the generic
foundation of the uv-data calibration and imaging part of AIPS++. In the not too distant
future, an ‘o(cid:14)cial’ AIPS++description of the ME will be needed, with agreed conventions and
nomenclature (see Appendix A). This note is a step towards that goal.
The heart of the M.E. is formed by the 2 2 feed-based ‘Jones’ matrices, which describe the
(cid:2)
e(cid:11)ects of various parts of the observing instrument on the signal. The main section of this
document is devoted to describing the basic form of the Jones matrices in linear and circular
polarisation coordinates. Another section discusses the conditions underwhich their order may
be modi(cid:12)ed (matrices do not always commute).
It is expected that the details of the M.E. (and of this note) will be re(cid:12)ned during the (cid:12)rst few
iterations of design and implementation of AIPS++. But the structure of the M.E. formalism
as presented here appears to be rich enough to accomodate all existing and planned radio
telescopes. Thisincludes‘exotic’oneslikecylindricalmirrors,phasedarrays, andinterferometer
arrays with very dissimilar antennas. Further re(cid:12)nements should only require the addition of
new Jones matrices, or devising new expressions for existing matrix elements.
In order to test this bold assertion, the various institutes might endeavour to model their own
telescopes in terms of the precise and common language of the M.E., using this note as a
reference. The following ‘rules’ are probably good ones:
In modelling an instrument, stay as close to the actual physical situation as possible.
(cid:15)
Violations of this principle, for whatever reasons, will lead to problems sooner or later.
ItiscounterproductivetotryandsimplifytheM.E.tomakeit‘lookmoretractable’. This
(cid:15)
practice introduces hidden assumptions, which tend to be forgotten by the programmer,
and unknown to the user.
Use the suggested nomenclature and conventions.
(cid:15)
It is also good to realise that there are two basic forms of ME, which should not be confused:
In the physical form, each instrumental e(cid:11)ect is modelled separately by its own matrix. This is
useful for simulation purposes. In the mathematical form, e(cid:11)ects are ‘lumped together’ if they
cannot be solved for separately. Example: the various contributions to the receiver gain, and
tropospheric gain.
Acknowledgements: Theauthorhasgreatlybene(cid:12)tedfromdetaileddiscussionswithJayaram
Chengalur, Jaap Bregman, Johan Hamaker, Tim Cornwell, Wim Brouw and Mark Wieringa.
3
2 THE M.E. FOR A SINGLE POINT SOURCE
For the moment, it will be assumed that there is a single point source at an arbitrary position
(direction) (cid:26)~ = (cid:26)~(l;m) w.r.t. the fringe-tracking centre, and that observing bandwidth and
integration time are negligible. Multiple and extended sources, and the e(cid:11)ects of non-zero
bandwidth and integration time will be treated for the Full Measurement Equation in section
3.
For a given interferometer, the measured visibilities can be written as a 4-element ‘coherency
vector’ V~ , which is related to the so-called ‘Stokes vector’ I~(l;m) of the observed source by a
ij
matrix equation,
v I
ipjp
v Q
V~ = 0 ipjq 1 = (J J ) S 0 1 (1)
ij viqjp i(cid:10) j(cid:3) U
B C B C
BB viqjq CC BB V CCl;m
@ A @ A
The subscripts i and j are the labels of the two feeds that make up the interferometer. The
subscripts p and q are the labels of the two output IF-channels from each feed.1
The ‘Stokes matrix’ S is a constant 4 4 coordinate transformation matrix. It is discussed in
(cid:2)
detail in section 4 below. The real heart of the M.E. is the ‘direct matrix product’ J J of
i j(cid:3)
(cid:10)
two 2 2 feed-based Jones matrices.
(cid:2)
The ‘Stokes-to-Stokes’ transmission of a Stokes vector through an ‘optical’ element may be
described by multiplication with a 4 4 Mueller matrix [2] [3]. Using equation 1:
ij
(cid:2) M
I~out(l;m) = S 1 V~ = S 1 (J J ) S I~in(l;m) = (l;m) I~in(l;m) (2)
(cid:0) ij (cid:0) i j(cid:3) ij
(cid:10) M
Mueller matrices are useful in simulation, when studying the e(cid:11)ect of instrumental e(cid:11)ects on a
test source I~(l;m). They can be easily generalised to the full M.E. (see section 3).
2.1 The feed-based instrumental Jones matrices
Itwillbeassumed(forthemoment)thatallinstrumentale(cid:11)ects canbefactoredinto feed-based
contributions, i.e. any interferometer-based e(cid:11)ects are assumed to be negligible (see section 3).
The 4 4 interferometer response matrix J then consists of a ‘direct matrix product’2 J J
ij i j(cid:3)
(cid:2) (cid:10)
of two 2 2 feed-based response matrices, called ‘Jones matrices’. The reader will note that
(cid:2)
this factoring is the polarimetric generalisation of the familiar ‘Selfcal assumption’, in which
the (scalar) gains are assumed to be feed-based rather than interferometer-based.
The 2 2 Jones matrix J for feed i can be decomposed into a product of several 2 2 Jones
i
(cid:2) (cid:2)
matrices, each of which models a speci(cid:12)c feed-based instrumental e(cid:11)ect in the signal path:
1The generic IF-channel labels p and q are known as X and Y for WSRT and ATCA, and R and L for the
VLA. They should not be confused with the two receptors a and b, since the signal in an IF-channel may be a
linear combination of the receptor signals.
2Also called the outer matrix product,or tensor product, or Kroneckerproduct. See [2].
4
J = G [H] [Y] B K T F = G [H] [Y] (D E P) K T F (3)
i i i i i i i i i i i i i i i i i
in which
F((cid:26)~;~r) ionospheric Faraday rotation
i i
T((cid:26)~;~r) atmospheric complex gain
i i
K((cid:26)~:~r) factored Fourier Transform kernel
i i
P projected receptor orientation(s) w.r.t. the sky
i
E((cid:26)~) voltage primary beam
i
D position-independent receptor cross-leakage
i
[Y] commutation of IF-channels
i
[H] hybrid (conversion to circular polarisation coordinates)
i
G electronic complex gain (feed-based contributions only)
i
Matrices between brackets ([ ]) are not present in all systems. B is the ‘Total Voltage Pattern’
i
of an arbitrary feed, which is usually split up into three sub-matrices: D E P. Jones matrices
i i i
that model ‘image-plane’ e(cid:11)ects dependon the source position (direction) (cid:26)~. Some also depend
on the antenna position~r. Of course most of them depend on time and frequency as well. The
i
various Jones matrices are treated in some detail in section 5.
Since the Jones matrices do not always commute with each other, their order is important. In
principle, they should be placed in the ‘physical’ order, i.e. the order in which the signal is
a(cid:11)ected by them while traversing the instrument. In practice, this is not always possible or
desirable. Section 6 discusses the implications of choosing a di(cid:11)erent order.
2.2 The Jones matrix of a Tied Array feed
The output signals from the two IF-channels of a ‘tied array’ is the weighted sum of the IF-
channel signals from n individual feeds. A tied array is itself a feed (see de(cid:12)nition in appendix
A), modelled by its own Jones matrix. For a single point source, we get:
Jtied array = Q w J (4)
i i in in
n
X
and for an interferometer between two tied arrays i and j with n and m constituent feeds
respectively:
J = (J J ) = (Q Q ) w w (J J ) (5)
ij i j(cid:3) i j(cid:3) in jm in jm(cid:3)
(cid:10) (cid:10) (cid:10)
n m
X X
See also section 6.4. The matrix Q models electronic gain e(cid:11)ects on the added signal of the tied
i
array feed i. The Q can be solved by the usual Selfcal methods, in contrast to instrumental
i
errors in the constituent feeds before adding. The latter will often cause decorrellation, and
thus closure errors in an interferometer.
5
Since a tied array feed can be modelled by a Jones matrix, it can be combined with any other
type of feed to form an interferometer. Examples are the use of WSRT and VLA as tied
arrays in VLBI arrays. Note that this is made possible by factoring the Fourier Transform
kernel K (~u :(cid:26)~) into K(~r:(cid:26)~) and K(~r:(cid:26)~), and including the latter in the Jones matrices of the
ij ij i i j j
individual feeds (see equ 28).
Obviously, the primary beam of a tied array can be rather complicated, but it is fully modelled
by equ 4. Moreover, the contributing feeds in a tied array are allowed to be quite dissimilar. It
is noreven necessary fortheir receptors (dipoles) to bealigned with each other! Thus, equation
4canalso beusedtomodel‘di(cid:14)cult’telescopes like Ooty orMOST,oranelement ofthefuture
Square Km Array (SKAI). This puts the crown on the remarkable power of the Measurement
Equation.
2.3 Jones matrices for multiple beams
Using the de(cid:12)nition in appendix A, each beam in a multiple beam system should be treated
like a separate logical feed, modelled by its own Jones matrix. Any communality between them
can be modelled in the form of shared parameters in the expressions for the various matrix
elements.
3 THE FULL MEASUREMENT EQUATION
3.1 Summing and averaging
For k ‘real’ incoherent sources, observed with a ‘real’ telescope, equ 1 becomes:
1 1
V~ = dt df dldm J J S I~(l;m) (6)
ij i j(cid:3)
(cid:1)t(cid:1)f (cid:1)l(cid:1)m (cid:10)
Z Z k Z
X
Thevisibilityvector V~ isintegratedovertheextentofthesources( dldm), overtheintegration
ij
time ( dt) and over the channel bandwidth ( df). Integration over the aperture ( dudv) is
R
taken care of by the primary beam properties.
R R R
There are only four integration coordinates, whose units are determined by the (cid:13)ux density
units in which I~ is expressed: energy=sec=Hz=beam. These coordinates de(cid:12)ne a 4-dimensional
‘integration cell’. If the variation of V~(f;t;l;m) is linear over this cell, integration is not neces-
sary:
V~ = V~ (f ;t ;l ;m ) (7)
ij 0k 0 0 0 0
k
X
in which V~ is the value for source k at the centre of the cell, for (cid:1)f = 1 Hz and (cid:1)t = 1 sec.
0k
If the variation of V~(f;t;l;m) over the cell can be approximated by a polynomial of order 3,
(cid:20)
then it is su(cid:14)cient to calculate only the 2nd derivative(s) at the centre of the cell:
6
1 @2V~ @2V~ @2V~ @2V~
V~int = V~ + ( 0k ((cid:1)f)2 + 0k ((cid:1)t)2 + 0k ((cid:1)l)2 + 0k ((cid:1)m)2) (8)
0k 12 @f2 @t2 @l2 @m2
k
X
Hereitisassumedthatthe2ndderivativesarebeconstantoverthecell,i.e. thecross-derivatives
@V~0 are zero.
@p1@p2
3.2 interferometer-based e(cid:11)ects
Until now, we have assumed that all instrumental e(cid:11)ects could be factored into feed-based
contributions, i.e. we have ignored any interferometer-based e(cid:11)ects. This is justi(cid:12)ed for a well-
designed system, provided that the signal-to-noise ratio is large enough (thermal noise causes
interferometer-based errors, albeit with a an average of zero). However, if systematic errors do
occur, they can be modelled:
V~0 = X (A~ + M V~ ) (9)
ij ij ij ij ij
The4 4diagonalmatrixX,the‘Correlatormatrix’,representsinterferometer-basedcorrections
(cid:2)
that are applied to the uv-data in software by the on-line system. Examples are the Van Vleck
correction. In the newest correlators, it approaches a constant (x).
x 0 0 0
ipjp
0 x 0 0
X = 0 ipjq 1 x (10)
ij 0 0 xiqjp 0 (cid:25) U
B C
B 0 0 0 xiqjq C
B C
@ A
The 4 4 diagonal matrix M represents multiplicative interferometer-based e(cid:11)ects.
(cid:2)
m 0 0 0
ipjp
0 m 0 0
M = 0 ipjq 1 (11)
ij 0 0 miqjp 0 (cid:25) U
B C
B 0 0 0 miqjq C
B C
@ A
The4-element vector A~ representsadditive interferometer-based e(cid:11)ects. Examplesare receiver
ij
noise, and correlator o(cid:11)sets.
a
ipjp
a
A~ = 0 ipjq 1 ~0 (12)
ij aiqjp (cid:25)
B C
B aiqjq C
B C
@ A
In some cases, interferometer-based e(cid:11)ects can be calibrated, e.g. when they appear to be
constant in time. It will be interesting to see how many of them will disappear as a result of
better modelling with the Measurement Equation. In any case, it is desirable that the cause of
interferometer-based e(cid:11)ects is properly understood (simulation!).
7
4 POLARISATION COORDINATES
In the 2 2 signal domain, the electric (cid:12)eld vector E~ of the incident plane wave can be
(cid:2)
represented either in a linear polarisation coordinate frame (x;y) or a circular polarisation
coordinate frame (r;l). Jones matrices are linear operators in the chosen frame:
v e e
V~+ = ip = J+ x or V~ = J r (13)
i v i e i(cid:12) (cid:12)i e
iq ! y ! l !
For linear polarisation coordinates, equation 1 becomes:
e e
x (cid:3)x
e e
V~ij+ = (J+i (cid:10)J+j (cid:3)) (E~ (cid:10)E~(cid:3)) = (J+i (cid:10)J+j (cid:3)) 0 eyxe(cid:3)y(cid:3)x 1 = (J+i (cid:10)J+j (cid:3)) S+ I~(l;m) (14)
B C
B eye(cid:3)y C
B C
@ A
andthereisasimilarexpressionforcircularpolarisationcoordinates. Thus,asemphasisedin[2],
theStokesvectorI~(l;m)andthecoherencyvector V~ representthesamephysicalquantity,butin
ij
di(cid:11)erent abstract coordinate frames. A ‘Stokes matrix’ Sis a coordinate transformation matrix
in the 4 4 coherency domain: S+ transforms the representation from Stokes coordinates
(cid:2)
(I,Q,U,V) to linear polarisation coordinates (xx;xy;yx;yy). Similarly, S transforms to circular
(cid:12)
polarisation coordinates (rr;rl;lr;ll). Following the convention of [4], we write:3
1 1 0 0 1 0 0 1
1 0 0 1 i 1 0 1 i 0
S+ = 0 1 S = 0 1 (15)
(cid:12)
2 0 0 1 i 2 0 1 i 0
B (cid:0) C B (cid:0) C
B 1 1 0 0 C B 1 0 0 1 C
B (cid:0) C B (cid:0) C
@ A @ A
S-matrices are almost unitary, i.e. except for a normalising constant: (S) 1 = 2(S) T. S cannot
(cid:0) (cid:3)
be factored into feed-based parts. The two Stokes matrices are related by:
S(cid:12) = ( (cid:3)) S+ S+ = ( (cid:0)1 ( (cid:0)1)(cid:3)) S(cid:12) (16)
H(cid:10)H H (cid:10) H
with4
1 1 i 1 1 1
= 1 = (18)
(cid:0)
H p2 1 i ! H p2 i i !
(cid:0) (cid:0)
3Inonein(cid:13)uentialbook[12],thefactor0:5isomittedfromS(cid:12). Thisisclearlyincorrect,sinceasinglereceptor
can nevermeasure more thanone half of the total (cid:13)uxof an unpolarised source.
4Onemightarguethatamoreconsistentform of wouldbeanexpressioninterms of the (cid:25)=4ellipticities
H (cid:6)
thatare intrinsic to a circular receptor:
alternative 1 1 i 1 0
= Ell((cid:25)=4; (cid:25)=4) = = (17)
H (cid:0) p2 i 1 0 i H
(cid:18) (cid:19) (cid:18) (cid:19)
However, achoice for a di(cid:11)erent shouldnotbemade lightly, sinceit would a(cid:11)ectthe deeplyentrenchedform
H
of the Stokesmatrices.
8
Most Jones matrices will have the same form in both polarisation coordinate frames. But if
a Jones matrix is expressed in terms of parameters that are de(cid:12)ned in one of the two frames,
it will have two di(cid:11)erent but related forms. This is the case for Faraday rotation F , receptor
i
orientation P, and receptor cross-leakage D, in which the orientation w.r.t. the x;ccY frame
i i
plays a role. The two forms of a Jones matrix A can be converted into each other by the
coordinate transformation matrix and its inverse:
H
A = A+ 1 A+ = 1 A (19)
(cid:12) (cid:0) (cid:0) (cid:12)
H H H H
The conversion may be done by hand, using (the elements a;b;c;d may be complex):
a c (a+b) i(c d) (a b)+i(c+d)
(cid:0)1 = 0:5 (cid:0) (cid:0) (cid:0) (20)
H d b ! H (a b) i(c+d) (a+b)+i(c d) !
(cid:0) (cid:0) (cid:0)
a c (a+b+c+d) i(a b c+d)
(cid:0)1 = 0:5 (cid:0) (cid:0) (21)
H d b ! H i(a b+c d) (a+b c d) !
(cid:0) (cid:0) (cid:0) (cid:0) (cid:0)
Applying these general expressions to rotation Rot((cid:11)) and ellipticity Ell((cid:11); (cid:11)) matrices (see
(cid:0)
Appendix for their de(cid:12)nition), the conversions are:
Rot((cid:11)) 1 = Diag(expi(cid:11);exp i(cid:11))
(cid:0) (cid:0)
H H
Rot((cid:11);(cid:12)) 1 = see equation 34
(cid:0)
H H
Ell((cid:11); (cid:11)) 1 = Rot((cid:11)) (22)
(cid:0)
H (cid:0) H
1 Rot((cid:11)) = Ell((cid:11); (cid:11))
(cid:0)
H H (cid:0)
1 Ell((cid:11); (cid:11)) = Diag(expi(cid:11);exp i(cid:11)) (23)
(cid:0) (cid:0)
H (cid:0) H
Usually, all matrices in a ‘Jones chain’ will be de(cid:12)ned in the same coordinate frame. An
exception is the case where linear dipole receptors are used in conjunction with a ‘hybrid’ H
i
to create pseudo-circular receptors:
J = G H D+ E+ P+ K+ T+ F+ (using S=S+)
i (cid:12)i i i i i i i i
= G (H D+ 1) E+ P+ K+ T+ F+ (using S=S+)
(cid:12)i i i H(cid:0) H i i i i i
= G D E+ P+ K+ T+ F+ (using S=S+)
(cid:12)i (cid:12)i H i i i i i
= G D E P K T F (using S=S+)
(cid:12)i (cid:12)i (cid:12)i (cid:12)i (cid:12)i (cid:12)i (cid:12)i H
= G D E P K T F (using S=S ) (24)
(cid:12)i (cid:12)i (cid:12)i (cid:12)i (cid:12)i (cid:12)i (cid:12)i (cid:12)
in which H represents an electronic implementation of the coordinate transformation matrix
i
. All these expressions are equivalent in the sense that, in conjunction with the indicated
H
Stokes matrix, theyproducea coherency vector incircular polarisation coordinates. Thechoice
of which expression to use depends on whether one wishes to model the feed explicitly in terms
of its physical (dipole) properties, or whether one wishes to regard is as a ‘black box’ circular
feed with unknown internal structure.
9
5 GENERIC FORM OF JONES MATRICES
In this section, the ‘generic’ form of various 2 2 feed-based instrumental Jones matrices
(cid:2)
(operators) will be treated in some detail.
Itwillbenotedthatforeachmatrix, the4elementshavebeengivenan‘o(cid:14)cial’name(e.g. f ).
ixx
The (possibly naive) idea is that, if the structure of the Measurement Equation is more or less
complete, these‘standard’matrixelements couldbereferredto explicitlybytheiro(cid:14)cial names
in other AIPS++ documents (and code), for instance to replace them with speci(cid:12)c expressions
for particular telescopes or purposes.
The subscript convention is as follows: y is an element of matrix Y for feed i, which models
ibp
the ‘coupling factor’ for the signal going from receptor b to IF-channel p. Where possible,
the expressions have been reduced to matrices like the diagonal matrix (Diag), rotation matrix
(Rot) etc. These are de(cid:12)ned in the Appendix.
5.1 Ionospheric Faraday rotation (F((cid:26)~;~r))
i i
The matrix F+ represents (ionospheric) Faraday rotation of the electric vector over an angle (cid:31)
i i
w.r.t. the celestial x;y-frame. Since (cid:31) is de(cid:12)ned in one of the polarisation coordinate frames,
i
there will be two di(cid:11)erent forms for F (see also section 4). For linear polarisation coordinates:
i
f f
F+((cid:26)~;~r) = ixx iyx = Rot((cid:31)) (25)
i i f f i
ixy iyy !
Incircularpolarisationcoordinates,thematrixF isadiagonalmatrixwhichintroducesaphase
(cid:12)i
di(cid:11)erence, or rather a delay di(cid:11)erence. It expresses the fact that ionospheric Faraday rotation
is caused by a (strongly frequency-dependent) di(cid:11)erence in propagation velocity between right-
hand and left-hand circularly polarised signals when travelling through a charged medium like
the ionosphere. In terms of the Faraday rotation angle (cid:31) (see above), we get:
i
f f
F(cid:12)i ((cid:26)~;~ri) = fiirrrl fiillrl ! = H F+i H(cid:0)1 = Diag(expi(cid:31)i;exp(cid:0)i(cid:31)i) (26)
In principle, the Faraday rotation angle is a function of source direction and feed position:
(cid:31) = (cid:31)((cid:26)~;~r). However, Faraday rotation is a large-scale e(cid:11)ect, so it will usually have the same
i i i
value for all sources in the primary beam: (cid:31) = (cid:31)(~r). For arrays smaller than a few km, the
i i
rotation angle will usually also be the same for all feeds: (cid:31) = (cid:31)(t). These assumptions reduce
i
the number of independent parameters considerably.
5.2 Atmospheric gain (T((cid:26)~;~r))
i i
The matrix T+ represents complex atmospheric gain: refraction, extinction and perhaps non-
i
isoplanaticity. Since T+ does not depend on a polarisation coordinate frame, there is only one
i
form:
10