# Difference between revisions of "Polarization profile"

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## Latest revision as of 20:05, 28 November 2012

# Measuring Beam Polarization Profile with p-Carbon Polarimeters

# 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 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 **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 \langle P \rangle}**
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 **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 k_{SSA}}**
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 .

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

# Time dependent P_SSA

**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 P_{SSA} = \left(1 + \frac12 R(t)\right) P(t) = (1 + \frac12 R_0 + \frac12 R' t)(P_0 + P't) \approx P_0 + P't + \frac12 R_0 P_0 + \frac12 (R_0 P' + R' P_0) t =P_0 (1 + \frac12 R_0) + (P' + \frac12 (R_0 P' + R' P_0) ) t }**