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/6/6c/Profile.gif<br>
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https://wiki.bnl.gov/rhicspin/upload/4/41/Intens.gif<br>
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https://wiki.bnl.gov/rhicspin/upload/2/29/Polar.gif
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= Two-dimensional case =
 
= Two-dimensional case =
  
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<math>
 
<math>
P_{SSA} = (1 + \frac12 R(t)) 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  
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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
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=P_0 (1 + \frac12 R_0) + (P' + \frac12 (R_0 P' + R' P_0) ) t
 
</math>
 
</math>

Latest revision as of 20:05, 28 November 2012

Measuring Beam Polarization Profile with p-Carbon Polarimeters

Profile.gif
Intens.gif
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:

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:

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{\iint P(x,y) I(x,y) dx dy}{\iint I(x,y) dx dy} &= \frac{P_{0}}{\sqrt{1 + R_x} \sqrt{1 + R_y}} \\ \frac{\iint P(x,y) I_B(x,y) I_Y(x,y) dx dy}{\iint I_B(x,y) I_Y(x,y) dx dy} &= \frac{P_{0}}{ \sqrt{1 + \frac{R_x}{2}} \sqrt{1 + \frac{R_y}{2}} }\\ \frac{\iint P_B(x,y) P_Y(x,y) I_B(x,y) I_Y(x,y) dx dy}{\iint I_B(x,y) I_Y(x,y) dx dy} &= \frac{P_{0,B} P_{0,Y}}{\sqrt{1 + \frac{R_{x,B}}{2} + \frac{R_{x,Y}}{2} } \sqrt{1 + \frac{R_{y,B}}{2} + \frac{R_{y,Y}}{2} }}\end{align}}

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

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