Difference between revisions of "Polarization profile"

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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  
+
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
 
</math>
 
</math>

Revision as of 10:03, 11 September 2012

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:

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 and :

where

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} k_{SSA} &= \frac{\sqrt{1 + R_x} \sqrt{1 + R_y}}{ \sqrt{1 + \frac{R_x}{2}} \sqrt{1 + \frac{R_y}{2}} }\\ k_{DSA} &= \frac{ \sqrt{1 + R_{x,B}} \sqrt{1 + R_{y,B}} \sqrt{1 + R_{x,Y}} \sqrt{1 + R_{y,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}}

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 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

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 }