Polarization profile

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Revision as of 14:27, 9 August 2012 by Dmitri Smirnov (talk | contribs)
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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:

Integrating from 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 :

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\rangle_{SSA} &= k_{SSA} \times \langle P \rangle \\ \langle P_B\cdot P_Y\rangle_{DSA} &= k_{DSA} \times \langle P_B \rangle \cdot \langle P_Y \rangle\end{align}}

where

It is interesting to study the difference between the scale factors and . To make things easier we assume the same value for all 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 R} '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 . 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}}