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%\Huge\scaletoheight{0.5in}{A HEURISTIC INTRODUCTION}

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%Large, LARGE, huge, Huge

\begin{document}

\Huge
  \begin{center}
\vspace{1.5in}
{\bf AN HEURISTIC INTRODUCTION TO RADIOASTRONOMICAL POLARIZATION}
\end{center}
\vspace{0.5in}

\huge
\begin{center}
{\bf CARL HEILES}
\end{center}

\LARGE
\begin{center}
{\bf Astronomy Department, UC Berkeley}
\end{center}

\clearpage
\Huge
\begin{center} {\bf OUTLINE} \end{center} \begin{itemize}
\item Polarization: the unique probe of magnetic fields in the Universe
\item Quantifying polarization with Stokes parameters
\item Radio beats optical: We measure all Stokes parameters simultaneously
\item The Mueller Matrix relates the CAL to the SKY
\item The receiver system introduces calibratable effects in both gain and
phase
\item Beam effects: squint, squash, more distant sidelobes---Arecibo and GBT
\item My website: a ``paradise'' of tutorials and documentation
\item Three things to remember
\end{itemize}

\clearpage

\Huge {\bf WHAT PRODUCES RADIO POLARIZATION?} \begin{itemize}
\item Cyclotron emission---electrons spiraling in a {\it magnetic field}
\item Synchrotron emission---relativistic electrons spiraling in a {\it magnetic field}
\item mm-wave and IR emission from {\it magnetic}ally aligned dust grains
\item The Goldreich/Kylafis effect: spectral-line scattering of 
  anisotropic radiation by atoms/molecules in a {\it magnetic field}
\item Zeeman splitting of spectral lines by atoms/molecules in a
  {\it magnetic field}.
\item Blackbody Emission from a solid planetary surface (remember the
  ``Brewster angle''?
\item Scattering of radiation (remember the polarization of blue sky?
\end{itemize}

Note the prevalence of the {\it magnetic field} in producing polarization! 
Why do we care about cosmic magnetism?

\clearpage

\begin{center}   
\includegraphics[scale=1.0]{mag_fundamentalsv1_p0.eps}
\end{center}
\clearpage

\begin{center}   
\includegraphics[scale=1.0]{mag_fundamentalsv1_p1.eps}
\end{center}
\clearpage

\begin{center}   
\includegraphics[scale=1.0]{mag_fundamentalsv1_p2.eps}
\end{center}
\clearpage

\begin{center}   
\includegraphics[scale=1.0]{mag_fundamentalsv1_p0a.eps}
\end{center}
\clearpage

\begin{center}   
\includegraphics[scale=1.0]{mag_fundamentalsv1_p0b.eps}
\end{center}
\clearpage

\begin{center}   
\includegraphics[scale=1.0]{mag_fundamentalsv1_p3.eps}
\end{center}
\clearpage

\begin{center}   
\includegraphics[scale=1.0]{mag_fundamentalsv1_p4.eps}
\end{center}
\clearpage

\begin{center}   
\includegraphics[scale=1.0]{mag_fundamentalsv1_p5.eps}
\end{center}
\clearpage

\begin{center}   
\includegraphics[scale=1.0]{mag_fundamentalsv1_p6.eps}
\end{center}
\clearpage

\noindent {\bf Spectral line CIRCULAR polarization: Zeeman Splitting of
  an  OH Megamaser in a distant ULIRG}
\begin{center}   
\includegraphics[width=10in, height=5.5in, keepaspectratio=false]{12032+1707_a2119plot.ps}
\end{center}

\clearpage

\noindent {\bf Spectral line LINEAR polarization of OH Megamaser
  emission in a less-distant ULIRG}
\begin{center}   
\includegraphics[bb= 60 290 545 600,clip, width=10in, height=5.5in,
  keepaspectratio=false]{carls_linpol_p8.ps} 
\end{center}
\clearpage

\noindent {\bf LINEAR polarization of mm-wave dust emission and CO 2-1
  in a protostellar cloud}
\begin{center}   
\includegraphics[bb= 310 480 570 750, clip,scale=1.6]{lai_girart_crutcher_fig1.ps}
\end{center}
\clearpage

\noindent {\bf LINEAR polarization of planetary thermal emission (theory)}
\begin{center}   
%\includegraphics[bb= 0 0 515 380,clip,scale=1.1]{moonpol_p3.ps}
\includegraphics[bb= 90 366 515 730,clip,scale=1.1]{moonpol_p3.ps}
\end{center}
\clearpage

\noindent {\bf LINEAR polarization of lunar thermal emission
  (observation by the 300 foot)}
\begin{center}   
%\includegraphics[bb= 0 -280 515 80,clip,scale=1.2]{moonpol_fig6a.ps}
\includegraphics[bb= 0 100 515 455,clip,scale=1.1]{moonpol_fig6a.ps}
\end{center}
\clearpage

\noindent {\bf Linear polarization of synchrotron emission from M51---a
  strong spiral density wave and synchrotron emission/polarization}
\begin{center}   
\includegraphics[scale=0.85]{m51_pol.eps}
\end{center}

%\clearpage

%\clearpage

\begin{center}   
\includegraphics[scale=.75]{ham_p1.ps} 
\end{center}

\clearpage

\Huge
Their equation (1): \hfill
\begin{figure}[h!]
\begin{center}   
\includegraphics[bb= 355 0 494 140,clip,scale=3.0]{ham_p1.ps} 
\end{center}
%\caption{hi}
\end{figure}

\vspace{-2.5in}
\noindent (The four {\bf STOKES PARAMETERS}). 
They look {\it awfully} complicated...

\clearpage

But it's not {\it that} complicated!

        Stokes parameters are linear combinations of power measured in
{\it orthogonal polarizations}. There are four:
\vspace{0.25in}
$$I = E_X^2 + E_Y^2 = E_{0^\circ}^2 + E_{90^\circ}^2 $$
$$Q = E_X^2 - E_Y^2 = E_{0^\circ}^2 - E_{90^\circ}^2 $$
$$U =  E_{45^\circ}^2 - E_{-45^\circ}^2 $$
$$V = E_{LCP}^2 - E_{RCP}^2 $$

\vspace{0.25in}
\noindent We like to write the {\it Stokes vector}
\vspace{0.25in}
\begin{eqnarray*} 
\label{IQUV}
{\bf S} = \left[ 
\begin{array}{c} 
    I \\ Q \\ U \\ V \\
\end{array} 
\; \right] \; .
\end{eqnarray*} 


\clearpage

\begin{center} 
{\bf STOKES PARAMETERS: BASICS}
\end{center}

$$I = E_X^2 + E_Y^2 = E_{0^\circ}^2 + E_{90^\circ}^2 $$
\vspace{-0.3in}
$$Q = E_X^2 - E_Y^2 = E_{0^\circ}^2 - E_{90^\circ}^2 $$
\vspace{-0.3in}
$$U =  E_{45^\circ}^2 - E_{-45^\circ}^2 $$
\vspace{-0.3in}
$$V = E_{LCP}^2 - E_{RCP}^2 $$

The first, Stokes $I$, is total intensity. It is the sum of {\it any two
  orthogonal} polarizations\footnote{\Huge Some ill-advised people (like at the
  \textcolor{red}{VLA}) define $I$ as
  the {\it average} instead of the {\it sum}. \textcolor{red}{BE CAREFUL!}}.

The second two, Stokes $Q$ and $U$,  completely specify linear polarization.

The last, Stokes $V$, completely specifies circular polarization.

%\enlargethispage{30pt}

\clearpage

\begin{center} 
  {\bf CONVENTIONAL LINEAR POL PARAMETERS}
\end{center}
\vspace{0.2in}
$$\hspace{-4.0in}{Q \over I} = p_{QU}  \cos (2 \chi)$$
\vspace{0.1in}
$$\hspace{-4.0in}{U \over I} = p_{QU}  \sin (2 \chi)$$
\begin{wrapfigure}{r}[2.5in]{5.8in}
\vspace{-3.5in}
\includegraphics[width=2.5in]{fig3.ps}
\end{wrapfigure}

\vspace{1.0in}
\noindent FRACTIONAL LINEAR POLARIZATION:
$$\hspace{1.0in} p_{QU} = \left[ \left({Q \over I}\right)^2 + \left({U \over I}\right) ^2
 \right]^{1/2} $$
\noindent POSITION ANGLE OF LINEAR POLARIZATION:
$$\chi = 0.5 \tan^{-1} {U \over Q}$$
%\enlargethispage{30pt}

\clearpage
\Huge
\begin{center} 
{\bf HEY!!! LINEAR POLARIZATION ``DIRECTION'' ??}
\end{center}

Look at the figure again:
\begin{center}
\includegraphics[height=2.5in]{fig3.ps}
\end{center}


\noindent THERE'S NO ARROWHEAD ON THAT ``VECTOR''!! That's because it's
the angle $2 \chi$, not $\chi$, that's important.

\noindent Moral of this story: \begin{itemize}

\item {\bf NEVER} say ``linear polarization {\bf DIRECTION}''.

\vspace{0.3in}
\item \textcolor{red}{{\bf INSTEAD}, always say ``linear polarization
  {\bf ORIENTATION}''}.

\end{itemize}

\clearpage

%\begin{center}
%%\includegraphics[scale=0.5,angle=-90]{starpol.ps}
%\includegraphics{starpol.ps}
%\end{center}
%
%\clear

\begin{center} {\bf OTHER CONVENTIONAL POLARIZATION PARAMETERS}
\end{center}

FRACTIONAL CIRCULAR POLARIZATION:

$$p_V = {V \over I}$$

TOTAL FRACTIONAL POLARIZATION:

$$p = \left[ \left({Q \over I}\right)^2 +
        \left({U \over I}\right) ^2 +
        \left({V \over I}\right) ^2  \right]^{1/2}
$$

If both $p_{QU}$ and $p_V$ are nonzero, then the polarization
is {\it elliptical}.

\clearpage
\begin{center}{\bf THE (NON) SENSE OF CIRCULAR POLARIZATION} 
\vspace{-0.15in}
How is Right-hand Circular Polarization defined? \end{center}
\begin{itemize}
\item \vspace{-0.15in} If you're a physicist: clockwise as seen by the {\it receiver}.

\item If you're an electrical engineer: the IEEE convention, clockwise
  as seen by the {\it transmitter}. {\it Hey!!! what does the receiver
  see???}

\item If you're a radio astronomer: the technical roots are in microwave
  engineering, so it's the IEEE convention.  Probably!! {\it You'd
  better check with your receiver engineers!} Or, to be {\it really}
  sure, {\it measure it yourself} by transmitting a helix from a known
  vantage point (and remember that $V$ changes sign when the signal
  reflects from a surface!).

\enlargethispage{0.5in}
\item If you're an optical astronomer: you read it off the label of the
  camera and you have no idea (your main goal is the grant money, so
  getting the science right is too much trouble). 
\end{itemize}

\clearpage

\begin{center}{\bf THE (NON) SENSE OF STOKES $V$}

OK\dots Now that we have RCP straight, how about Stokes $V$?
\end{center}
\begin{itemize}

\item If you're a physicist: \underline{$V$ = $RCP$ -- $LCP$}. 

\item If you're an electrical engineer: there's no IEEE
  convention. Radio astronomers' convention is, historically, from Kraus
  (e.g. his ``ANTENNAS'' or his ``RADIO ASTRONOMY''): \underline{$V$ =
  $LCP$ -- $RCP$}.  {\it Hey! With Kraus's definition of $V$, do
  physicists and engineers agree???}

\item If you're an official of the International Astronomical Union
  (IAU): The IAU uses the IEEE convention for RCP\dots, \\
and it defines \underline{$V$ = $RCP$ -- $LCP$},
  meaning that, for $V$,  the IAU {\it differs from both the physicist
  and the Kraus convention}!.
\end{itemize}

\Huge 
\begin{center} \textcolor{red}{\bf IS ALL THIS PERFECTLY CLEAR?} \end{center}

\clearpage

WE'RE NOT THE ONLY ONES WHO ARE CONFUSED! In his thesis, Tim Robishaw
traced historical use of $V$ by astronomers in his thesis. Lets take a
look:

(separate pdf file).
\Huge 

\clearpage

%==============================#################################
\begin{center}   
%\includegraphics[bb= 100 -200 550 310,clip,scale=1.0]{tim_p41.ps}
%\includegraphics[bb= 100 0 550 510,clip,scale=1.0]{tim_p41.ps}
\vspace{-1.0in}
\includegraphics[bb= 100 100 550 720,clip,scale=0.8]{tim_p41.ps}
\end{center}
\clearpage

\begin{center}   
\includegraphics[bb= 100 100 550 720,clip,scale=0.8]{tim_p42.ps}
\end{center}
\clearpage

\begin{center}   
\includegraphics[bb= 100 100 550 720,clip,scale=0.8]{tim_p43.ps}
\end{center}
\clearpage

\begin{center}   
\includegraphics[bb= 100 100 550 720,clip,scale=0.8]{tim_p44.ps}
\end{center}
\clearpage

%\begin{center}   
%\includegraphics[bb= 100 100 550 710,clip,width=10in, height=5in, keepaspectratio=false]{tim_p44.ps}
%\vspace{-3in}
%\includegraphics[bb= 100 -200 550 310,clip,scale=1.0]{tim_p44.ps}
%\includegraphics[scale=0.4]{tim_p44.ps}
%\includegraphics[scale=0.75]{tim_p44.ps}
%\includegraphics[bb= 355 0 494 140,clip,scale=3.0]{ham_p1.ps} \end{center}
%\end{center}
\clearpage


\begin{center} {\bf PULSAR ASTRONOMERS vs. THE WORLD} \end{center}

The WORLD considers positive velocies as moving away from us. This is
like the IEEE definition of circular polarization; we regard ourselves
as the transmitter.

Similarly, the WORLD considers positive MAGNETIC FIELDS as pointing AWAY
FROM us.

But...PULSAR ASTRONOMERS???

They define positive magnetic fields as producing positive Faraday
Rotation. A positive Faraday rotation results from a magnetic field that
points TOWARDS us. So...

PULSAR ASTRONOMERS consider positive magnetic fields as pointing TOWARDS
us...opposite to the WORLD view !!!

\clearpage

\begin{center} {\bf RADIOASTRONOMICAL FEEDS} \end{center}

Feeds are normally designed to approximate pure linear or
circular---known as {\it native linear} or {\it native circular}.

Generally speaking, native linear feeds are intrinsically accurate and
provide true linear. However, {\it native circular feeds are less accurate
and their exact polarization response is frequency dependent}.

%\vspace{1in}
\clearpage

	At the {\it GBT}: \begin{itemize}

	\item Feeds below 8 GHz are native linear.

	\item Feeds above 8 GHz are native circular. For the 8-10 GHz
receiver, the response changes from pure circular at 8 GHz to $14\%$
elliptical at 10 GHz.

\end{itemize}

	At {\it ARECIBO}: \begin{itemize}

	\item Feeds at 1-2 GHz and 4-6 GHz are native linear.
In fact, most feeds are native linear. 

	\item However, a couple are native circular. At Arecibo, circular is
achieved with waveguide turnstile junctions, which can be tuned to
produce very accurate polarization at the center frequency. However,
these are narrow band devices: the feeds become increasingly elliptical,
changing to linear and back again over frequency intervals $\sim 100$
MHz!
\end{itemize}

\clearpage

\begin{center} {\bf \textcolor{red}{REAL} RADIO ASTRONOMERS MEASURE ALL
STOKES PARAMETERS SIMULTANEOUSLY! } \end{center}

	Extracting two orthogonal polarizations provides {\it all} the
information; you can synthesize {\it all} other E fields from the two
measured ones!

	Example: Sample $(E_X,E_Y)$ and synthesize $E_{45}$ from
$(E_X,E_Y)$:

\begin{center}
\includegraphics{fig4.ps}
\end{center}

\enlargethispage{30pt}
\vspace{-0.6in}
To generate $E_{45}$, add $(E_X,E_Y)$ with no phase difference.

	To generate $E_{LCP}$, add $(E_X,E_Y)$ with a $90^\circ$ phase
difference.

\clearpage
\begin{center} {\bf CARRYING THROUGH THE ALGEBRA FOR THE TWO LINEARS
\dots} \end{center}

It's clear that
\vspace{-0.4in}
$$E_{45^\circ} = { E_{0^\circ} + E_{90^\circ} \over \sqrt 2} $$

$$E_{-45^\circ} = { E_{0^\circ} - E_{90^\circ} \over \sqrt 2}$$

Write the two linear Stokes parameters:

$$Q = E_X^2 - E_Y^2 = E_{0^\circ}^2 - E_{90^\circ}^2$$

$$U =  E_{45^\circ}^2 - E_{-45^\circ}^2 \textcolor{red}{ = 2E_XE_Y}$$

{\boldmath {\bf STOKES $U$ IS GIVEN BY THE CROSSCORRELATION} $E_XE_Y$}

To get $V$, throw a $90^\circ$ phase factor into the correlation. 

\clearpage

\begin{center} {\bf DOTTING THE I'S AND CROSSING THE T'S GIVES\dots}
\end{center}

	Carrying through the algebra and paying attention to complex
conjugates and extracting the real part of the expressions yields (for
sampling linear polarization $(X,Y)$:

%\vspace{-0.1in}
$$I = E_X \overline{ E_X} + E_Y \overline{ E_Y} \equiv {\bf XX} $$
\vspace{-0.8in}

$$Q = E_X \overline{ E_X} - E_Y \overline{ E_Y} \equiv {\bf YY} $$
\vspace{-0.8in}

$$U = E_X \overline{ E_Y} + \overline{ E_X} E_Y  \equiv {\bf XY} $$  
\vspace{-0.8in}

$$iV = E_X \overline{ E_Y} - \overline{ E_X} E_Y  \equiv {\bf YX} $$

\noindent The overbar indicates the complex conjugate. These products  
are time averages; we have omitted the indicative $\langle \rangle$
brackets to avoid clutter. 

\clearpage

{\bf IMPORTANT FACT for NATIVE LINEAR FEEDS:}: 

	Stokes $I$ and $Q$ are the sum and difference of {\it
self-products}. These {\it self-products} are large---equal to the full
system temperature---so \textcolor{red}{$Q$ is the difference between
two large numbers}, and is correspondingly inaccurate.

	Stokes $U$ and $V$ are sums and differences of {\it cross
products}. These {\it cross products} are small---in fact, in the
absence of polarization they should equal {\it ZERO}! So...

\noindent For small polarizations (the usual case!),
\textcolor{red}{{\it cross} products are {\it much less subject to
error} than self-products}.

%\vspace{1.5in} \begin{center} and\dots \end{center}
%\vspace{0.4in}

\begin{center}{\bf COROLLARY:}\end{center}

\noindent \textcolor{red}{To accurately measure small {\it linear} polarization, use a
dual {\it circular} feed (for which $Q$ and $U$ are cross products); to
accurately measure small {\it circular} polarization, use a dual {\it
linear} feed (for which $V$ is a cross product).}

\clearpage

\begin{center} {\bf THE MUELLER MATRIX} \end{center}

	The output terminals of a native linear feed provides voltages
that sample the E-fields $E_X$ and $E_Y$; this it provides directly the
Stokes parameters $I$ (from ${\bf XX} + {\bf YY}$) and $Q$ (from ${\bf
XX} - {\bf YY}$). 

	Rotating the feed by $45^\circ$ interchanges ${\bf
XX}$ and ${\bf YY}$, so it provides directly $I$ and $U$. 

	A native
circular feed adds a $90^\circ$ phase to $X$ (or $Y$) and provides
directly the Stokes parameters $I$ (from {\bf XX}$+ {\bf YY}$) and $V$
(from {\bf XX}$- {\bf YY}$). We express these interchanges of power
among Stokes parameters with the {\it Mueller matrix} $\bf M$.

\enlargethispage{30pt}
\vspace{-0.5in}
\begin{eqnarray}
\label{IQUVsrc} 
\left[
\begin{array}{c}   
   {\bf XX} \\ {\bf YY} \\ {\bf XY} \\ {\bf YX} \\
\end{array}
\; \right] = {\bf M \cdot} 
 \left[
\begin{array}{c}   
    I \\ Q \\ U \\ V \\ 
\end{array}
\right]
\end{eqnarray}

\clearpage

	Some examples of Mueller matrices:

        {\bf (1)} A dual linear feed: $\bf M$ is unitary.

        {\bf (2)}. A dual linear feed rotated $45^\circ$: $Q$ and $U$
interchange, together with a sign change as befits rotation:
\vspace{-0.0in}
\begin{eqnarray}
\label{feedmatrixf2}
{\bf M} = \left[
\begin{array}{cccc}
 1 & 0 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & -1 & 0 & 0 \\
 0 & 0 & 0 & 1 \\ 
\end{array}
\; \right] \; .   
\end{eqnarray}
 
        {\bf (3)}. A dual linear feed rotated $90^\circ$, which reverses
the signs of $Q$ and $U$:
\vspace{-0.2in}
\begin{eqnarray}
\label{feedmatrixf3}
{\bf M} = \left[
\begin{array}{cccc}
 1 & 0 & 0 & 0 \\
 0 & -1 & 0 & 0 \\
 0 & 0 & -1 & 0 \\
 0 & 0 & 0 & 1 \\
\end{array}
\; \right] \; .
\end{eqnarray}

\clearpage

	{\bf (4)} The above are special instructive cases. As an alt-az
telescope tracks a source, the feed rotates on the sky by the {\it
parallactic angle}  $PA_{az}$ \textcolor{red}{(What's {\it that}?)}. 
\vspace{-0.0in}
\begin{eqnarray}
\label{skymatrix}
{\bf M_{SKY}} = \left[
\begin{array}{cccc}
 1 &     0     &     0    & 0 \\
 0 &  \cos 2PA_{az} & \sin 2PA_{az} & 0 \\
 0 & -\sin 2PA_{az} & \cos 2PA_{az} & 0 \\
 0 &     0     &    0     & 1 \\
\end{array}
\; \right] \; .
\end{eqnarray}

\noindent The central $2 \times 2$ submatrix is, of course, nothing but
a rotation matrix. $\bf M_{SKY}$ doesn't change $I$ or $V$.

        {\bf (5)}. A dual circular feed, for which $V = {\bf XX}-
{\bf YY}$:
\vspace{-0.0in}
\begin{eqnarray}
\label{feedmatrixf4}
{\bf M} = \left[
\begin{array}{cccc} 
 1 & 0 & 0 & 0 \\   
 0 & 0 & 0 & 1 \\
 0 & 0 & 1 & 0 \\  
 0 & -1 & 0 & 0 \\
\end{array}
\; \right] \; .
\end{eqnarray}

\clearpage
%\vspace{-1in}

For a perfect system that has $\bf M$ unitary, as we track a linearly
polarized source across the sky the parallactic angle $PA$ changes. This
should produce:
\begin{itemize}

	\item For $\bf XX-YY$, $[\cos 2(PA_{AZ} + PA_{SRC})]$ centered at zero;

	\item For $\bf XY$, $[\sin 2(PA_{AZ} + PA_{SRC})]$ centered at zero;

	\item For $\bf YX$, zero (most sources have zero circular polarization).
\end{itemize}

\noindent The extent to which these idealizations are not realized defines the
Mueller matrix elements.

Let's look at some {\it real data}: first native linear, then native
circular. 


%\clearpage

%\begin{center} {\bf REAL DATA: NATIVE LINEAR POLARIZATION} \end{center}
%
    \begin{center}
\includegraphics[scale=.825]{Rcvr1_2_1420_bd0_3C286_m4_2d_3-AUG-2003.ps}
\end{center}

%\clearpage

%\begin{center} {\bf REAL DATA: NATIVE CIRCULAR POLARIZATION} \end{center}

\begin{center}
\includegraphics[scale=.825]{Rcvr8_10_10000_bd0_3C138_m4_2d_12-JAN-2003.ps}
\end{center}

\begin{center} {\bf THE SINGLE MATRIX FOR THE RADIOASTRONOMICAL RECEIVER}
\end{center}

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.

\LARGE
%\huge
\begin{eqnarray*}
{\bf M_{TOT}} = \left[
%{\bf M_{TOT}} = \biggl[
\begin{array}{cccc}
                    1        & (-2 \epsilon \sin\phi \sin2\alpha +
{\Delta G \over 2} \cos 2\alpha) &  
  2 \epsilon \cos\phi &  (2 \epsilon \sin\phi \cos2\alpha + {\Delta G
\over 2} \sin 2\alpha) \\
{\Delta G \over 2}           &   \cos 2\alpha  &  0    &  \sin 2\alpha 
\\
2\epsilon \cos(\phi + \psi)  & \sin 2\alpha \sin\psi & \cos\psi  & 
-\cos2\alpha \sin\psi  \\
2\epsilon \sin(\phi + \psi)  & -\sin 2\alpha \cos\psi &  \sin \psi  & 
\cos2\alpha \cos\psi
\end{array}
\; \right] \; .
\end{eqnarray*}

\Huge
\vspace{0.3in} NOTE: The Mueller matrix has 16 elements, but {\it ONLY 7
	INDEPENDENT PARAMETERS}. \underline {The matrix elements are not
	all independent}.

\clearpage

        {\boldmath $\Delta G$} is the error in relative intensity
calibration of the two polarization channels. It results from an error
in the relative cal values $(T_{calA}, T_{calB})$. 

        {\boldmath $\psi$} is the phase difference between the cal and
the incoming radiation from the sky (equivalent in spirit to $L_X -
L_Y$ on our block diagram..

        {\boldmath $\alpha$} is a measure of the voltage ratio of the  
polarization ellipse produced when the feed observes pure linear
polarization. 

        {\boldmath $\chi$} is the relative phase of the two voltages
specified by $\alpha$. 

        {\boldmath $\epsilon$} is a measure of imperfection of the feed
in producing nonorthogonal polarizations (false correlations) in the two
correlated outputs. 

        {\boldmath $\phi$} is the phase angle at which the voltage
coupling $\epsilon$ occurs. It works with $\epsilon$ to couple $I$ with
$(Q,U,V)$.

        {\boldmath $\theta_{astron}$} is the angle by which the derived
position angles must be rotated to conform with the conventional
astronomical definition.


\clearpage
\begin{center} {\bf THE MUELLER MATRIX REFERS \underline{ONLY} TO THE
    THICK CIRCUIT BELOW (RELATES CAL TO SKY)}
\end{center}
\vspace{-0.4in}
\begin{center}
\includegraphics[width=8.0in, height=6.0in, keepaspectratio=false]{fig1a.ps}
\end{center}
\vspace{-0.6in}
\begin{center} {\bf THE THIN CIRCUIT IS EASIER: NO COUPLING! EACH CHANNEL
    HAS A \underline{GAIN} AND \underline{PHASE}}
\end{center}

\clearpage

\Huge
\begin{center} {\bf THE CAL...OUR CONVENIENT INTENSITY AND PHASE REFERENCE}
\end{center}

We determine the Mueller matrix elements by comparing the cal deflection
with the deflection of an astronomical source of known polarization
(i.e., a ``polarization calibrator''; the best is 3C286). {\bf THIS
  CALIBRATES THE THICK CIRCUIT}

The signal and cal share common paths below its injection point (thin
circuit). The thin circuit affects both equally. We use the cal to
determine the properties of the thin circuit, minute-by-minute (or whatever). 

\clearpage

\Huge
\begin{center} {\bf THE THIN CIRCUIT HAS ACTIVE ELEMENTS AND IS THUS TIME-VARIABLE}\end{center}

\noindent Amplifier gains are {\it complex}: {\bf amplitude (GAIN)} and {\bf PHASE}.

\noindent The {\bf GAIN} calibration is just like {\it ordinary nonpolarized observations}.

\noindent The {\bf PHASE} calibration\dots It's the {\bf PHASE
DIFFERENCE} that matters. Cable lengths introduce phase delays. Cables
are never identical! Since 

\vspace{-.5in}
$$ \Delta \phi = {2 \pi (D_X - D_Y) \over \lambda} \ ,$$ 

\noindent \textcolor{red}{\bf THE PHASE DIFFERENCE DEPENDS ON FREQUENCY!}

\vspace{-.25in}
$$ {d \Delta \phi \over df} = { 2 \pi \ \Delta D  \over c} \approx 
	0.3 \ {\rm rad \over MHz} $$

\noindent (At Arecibo; it's almost as large for the GBT). This corresponds to 

\vspace{-.25in}
$$ \Delta D \approx 20 \ {\rm m \ \ \textcolor{red}{!!!!!}} $$
%{\rm \textcolor{red}!!!!}} 

\clearpage

\Huge
\begin{center} {\bf THIS HAS CONSEQUENCES!} \end{center}
 \begin{itemize}

	\item You must include the frequency dependence when you
	  calibrate your data. This is a bit tricky.

	\item You cannot do continuum polarization over significant
bandwidths without including $ {d \Delta \phi \over df} $. In particular,
continuum observing usually uses large bandwidth---and the phase can
easily wrap over multiple $2\pi$ intervals. \textcolor{red}{\bf DISASTER!
  EVEN POLARIZED SOURCES WILL APPEAR UNPOLARIZED!!}

\end{itemize}

\clearpage

\begin{center} {\bf POLARIZED BEAM EFFECTS: BEAM SQUINT}
\end{center}

\begin{center}
\includegraphics[scale=0.85]{figsquint.ps}
\end{center}

\clearpage

\begin{center} {\bf POLARIZED BEAM EFFECTS: BEAM SQUASH}
\end{center}

\begin{center}
\includegraphics[scale=0.85]{figsquash.ps}
\end{center}

\begin{center}   
\includegraphics[scale=.75]{mkfig1_i.ps} 
\end{center}

\begin{center}   
\includegraphics[scale=.75]{mkfig1_q.ps} 
\end{center}

\begin{center}   
\includegraphics[scale=.75]{mkfig1_u.ps} 
\end{center}

\begin{center}
\includegraphics[scale=.75]{mkfig1_v.ps}
\end{center}

\clearpage

\begin{center} {\bf POLARIZED BEAM EFFECTS: DISTANT SIDELOBES}
\end{center}

\begin{center}
\includegraphics[width=7in, height=6in, keepaspectratio=false]{windmill.ps}
\end{center}
\noindent(Stokes $V$ from the Hat Creek 85-footer. Image is $120^\circ
\times 120^\circ$)

\clearpage

Even the
GBT is not sidelobe-free. Here's an approximate image of the secondary
spillover in Stokes $I$---and there are also serious near-in lobes. All
are {\it highly polarized!!!} \ (Robishaw \& Heiles 2009, PASP, 121, 272)

\begin{center}   
\includegraphics[scale=.85]{f7.eps} 
\end{center}

\clearpage


\begin{center} {\bf THE EFFECT ON ASTRONOMICAL POLARIZATION MEASUREMENTS}
\end{center}

        Large-scale features have spatial structure of {\it Stokes I}.
Sidelobes in Stokes $Q$, $U$, and $V$ see this structure. The polarized
beam structure interacts with the Stokes $I$ derivatives to produce
\textcolor{red} {\bf FAKE RESULTS} in the {\it polarized} \ Stokes
parameters $(Q,U,V)$. 

        Correcting for these effects is a complicated business. First,
you have to measure them; they are weak, so this is difficult. (At the
GBT, Robishaw and Heiles (2009) used the Sun.) They may well be time
variable, particularly at Arecibo where the telescope geometry changes
as the telescope tracks. Finally, the polarized sidelobes rotate on the
sky as the parallactic angle changes---and distant sidelobes might see
the ground instead of the sky.

\Huge
\begin{center}\textcolor{red}{\bf IT'S REALLY HARD TO ACCURATELY 
MEASURE POLARIZATION OF EXTENDED EMISSION!!} \end{center}


\huge
\begin{center} {\bf MY WEBSITE: SOME DOCUMENTATION.} \end{center}

You might find my website useful; it
contains instructional handouts and practical IDL software.

\begin{verbatim} http://astro.berkeley.edu/~heiles/ \end{verbatim}

\noindent It has sections on (a partial list): \begin{itemize}
\item {\it Radio Astronomical Techniques and Calibration} [specific intensity,
  spectral lines, polarization, characterizing the telescope beam
  (including ``Spider scans''), 
  LSFS (Least-Squares Frequency Switching)]
\item {\it IDL Procedures and Instructional Handouts} [Introductory tutorial;
datatypes]
\item Downloading my set of IDL procedures
\item {\it Principles of Imaging and Projections} [Four tutorials, including
  use of color]
\item {\it Handouts on Numerical Analysis} [Least squares, Fourier, Wavelets]
\end{itemize}

%\enlargethispage{50pt} 

In addition, we are currently working on two coherent
practical writeups of ``how to do polarization calibration and data
analysis'' for the GBT and Arecibo. We have melded our
polarization software into {\tt gbtidl}, and THIS VERY WEEK are making
our Arecibo software more user-friendly (we hope!)

\clearpage

\Huge
\begin{center}{\textcolor{red}{\bf REMEMBER THIS \# 1: AVERAGING LINEAR
POLARIZATIONS!!!}} \end{center}

\huge
Suppose you average two polarization observations together:

Observation 1 has $p = 13.6\%$ and $\chi = 2 ^\circ$

Observation 2 has $p = 13.7\%$ and $\chi = 178 ^\circ$

NOTE THAT THE POSITION ANGLES AGREE TO WITHIN 4 DEGREES.

If you average $p$ and $\chi$, you get $p = 13.65\%$ and
\textcolor{red}{$\chi = 90^\circ$}.

\begin{center} ======== \textcolor{red}{\bf THIS IS INCORRECT!!!!!!!!!} ========\end{center}

        There is only one {\it proper} way to combine polarizations, and
that is to use the Stokes parameters. The reason is simple: because of
conservation of energy, powers add and subtract. 

\Huge
\noindent What you must {\bf always} do is \textcolor{red}{convert the fractional
polarizations and position angles to Stokes parameters, average the
Stokes parameters, and convert back}.

\clearpage

\Huge

\begin{center} {\textcolor{red}{\bf REMEMBER THIS \# 2: SHOULD YOU GENERATE CIRCULARS WITH A
POST-AMP HYBRID???}}
\end{center}

\huge
%\enlargethispage{30pt}
\enlargethispage{50pt}
Some astronomers believe that source fluxes (that's Stokes $I$)
are better measured with circular polarization. Receiver engineers
accommodate them with a post-amp hybrid:
\vspace{-0.4in}
\begin{center}
\includegraphics{fig2.ps}
\end{center}
\vspace{-0.5in}
\noindent Using this system properly requires a much more complicated
calibration procedure. Example: If $G_Y = 0$, the system still appears
to work!!

\begin{center} \Huge \textcolor{red}{\bf JUST SAY NO!!!!!!} \end{center}

%\Large
\huge

\clearpage

\Huge
\begin{center} {\textcolor{red}{\bf REMEMBER THIS \#3: CROSSCORRELATION VERSUS DIFFERENCING!!!}}
\end{center}

\begin{center}
\includegraphics[bb= 80 380 550 750,clip,scale=0.85]{plotpa.ps}
\end{center}

\noindent Look at the wild fluctuations for the 1420 MHz position angle
measurement (native linear---differencing). This {\it NEVER} happens
with crosscorrelation (9495 MHz). Crosscorrelation is also less noisy 
(At 1420 MHz 3C286 is $\sim
1.5 T_{sys}$; at 9493 MHz 3C286 is $\sim 0.5 T_{sys}$).

%	Crosscorrelation (9493 MHz) gives better results than
%%differencing (1420 MHz) for position angle measurements. You {\it NEVER}
%see wild fluctuations with crosscorrelation, while they are common with
%differencing. ( (POL for 9493 MHz is displaced downward by
%4 for clarity).

%	Crosscorrelation (9493 MHz) gives better results than
%differencing (1420 MHz) for position angle measurements. You {\it NEVER}
%see wild fluctuations with crosscorrelation, while they are common with
%differencing. (At 1420 MHz 3C286 is $\sim 1.5 T_{sys}$; at 9493 MHz
%3C286 is $\sim 0.5 T_{sys}$). (POL for 9493 MHz is displaced downward by
%4 for clarity).


\end{document}


\begin{wrapfigure}{r}[1.5in]{5.8in}
\vspace{-1.5in}
\includegraphics[width=4.5in]{gbt_vs_hc_rweil.ps}
\end{wrapfigure}

POLARIZATION IS UNIQUE! \\
MORE AND MORE!
============================================================
\clearpage

