Difference between revisions of "Polarization profile"

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= Measuring Beam Polarization Profile with p-Carbon Polarimeters =
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https://wiki.bnl.gov/rhicspin/upload/2/29/Polar.gif
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= Two-dimensional case =
 
= Two-dimensional case =
  

Revision as of 19:52, 28 November 2012

Measuring Beam Polarization Profile with p-Carbon Polarimeters

Polar.gif

Two-dimensional case

Let's assume the polarization and intensity profiles have a gaussian shape:

Since we are interested only in the width of the polarization profile with respect to the intensity one we can use the following relations:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sigma^2_{x,I} &\equiv \sigma^2_{y,I} \equiv 1 \\ \sigma^2_{x,P} &\equiv 1/R_x \\ \sigma^2_{y,P} &\equiv 1/R_y\end{align}}

Integrating from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\infty} to over both dimensions we get for the polarization weighted with intensity of either one or both beams:

As we normaly measure the average polarization given by it is trivial to get the equations for re-weighting factors and :

where

It is interesting to study the difference between the scale factors and . To make things easier we assume the same value for all 's which is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim 0.2} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{k_{DSA}}{k_{SSA,B} k_{SSA,Y}} = 1 + \frac{R^2}{4(1+R)}\end{align}}

where the last term gives a correction on the order of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lesssim 1\%} . Therefore, with good precision we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \langle P_B\cdot P_Y\rangle_{DSA} &\approx \langle P\rangle_{SSA,B} \langle P\rangle_{SSA,Y} \end{align}}


Time dependent P_SSA