Difference between revisions of "User:Tohline/Appendix/CGH/ParallelAperturesConsolidate"

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<tr>
   <td align="right">
   <td align="right">
<math>~a_j e^{i \phi_j} </math>
<math>~a_j </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
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   </td>
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   <td align="left">
<math>~a_0(Y) dY = a_0(\Theta) \biggl[ \frac{\lambda L}{2\pi y_1} \biggr] d\Theta \, ,</math>
<math>~a_0(Y) dY = a_0(\Theta) \biggl[ \frac{w}{2\beta_1} \biggr] d\Theta \, ,</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
where,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{1}{\beta_1}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\lambda L}{\pi y_1w}  \, ,</math>
  </td>
</tr>
</table>
and changed the summation to an integration, obtaining,
and changed the summation to an integration, obtaining,


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   </td>
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   <td align="left">
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<math>~ e^{i 2\pi L/\lambda }\int a_0(Y) e^{-i \Theta } dY = e^{i 2\pi L/\lambda }\biggl[ \frac{\lambda L}{2\pi y_1} \biggr]  \int a_0(\Theta) e^{-i \Theta } d\Theta \, .
<math>~  
e^{i 2\pi L/\lambda }\biggl[ \frac{w}{2\beta_1} \biggr]  \int a_0(\Theta) e^{i\phi(\Theta)} \cdot  e^{-i \Theta } d\Theta
\, .
</math>
  </td>
</tr>
</table>
 
'''Case #1:''' &nbsp; If we assume that both <math>~a_0</math> and <math>~\phi</math> are independent of position along the aperture, and that the aperture &#8212; and, hence the integration &#8212; extends from <math>~Y_2 = -w/2</math> to <math>~Y_1 = +w/2</math>, we have shown that this last expression can be evaluated analytically to give,
 
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~A(y_1)</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
e^{i [2\pi L/\lambda + \phi] }\biggl[ \frac{a_0 w}{2\beta_1} \biggr]  \int_{\Theta_2}^{\Theta_1} e^{-i \Theta } d\Theta  
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{i [2\pi L/\lambda + \phi] } \cdot a_0 w ~\mathrm{sinc}(\beta_1) \, .  
</math>
</math>
   </td>
   </td>

Revision as of 20:50, 17 March 2020

CGH: Consolidate Expressions Regarding Parallel Apertures

One-dimensional Apertures

From our accompanying discussion of the Utility of FFT Techniques, we start with the most general expression for the amplitude at one point on an image screen, namely,

<math>~A(y_1)</math>

<math>~=</math>

<math>~\sum_j a_j e^{i(2\pi D_j/\lambda + \phi_j)} \, , </math>

and, assuming that <math>~|Y_j/L| \ll 1</math> for all <math>~j</math>, deduce that,

<math>~A(y_1)</math>

<math>~\approx</math>

<math>~\sum_j a_j e^{i[ 2\pi L/\lambda + \phi_j]}\biggl[ \cos\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) - i \sin\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \biggr] \, , </math>

where,

<math>~L</math>

<math>~\equiv</math>

<math>~ Z \biggl[1 + \frac{y_1^2}{Z^2} \biggr]^{1 / 2} \, . </math>

Note that <math>~L</math> is formally a function of <math>~y_1</math>, but in most of what follows it will be reasonable to assume, <math>~L \approx Z</math>. Notice, as well, that this last approximate expression for the (complex) amplitude at the image screen may be rewritten in the form,

<math>~A(y_1)</math>

<math>~\approx</math>

<math>~ e^{i 2\pi L/\lambda } \sum_j a_j e^{i \phi_j} \cdot e^{-i \Theta_j } \, , </math>

where,

<math>~\Theta_j</math>

<math>~\equiv</math>

<math>~\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \, .</math>

In a related accompanying derivation titled, Analytic Result, we made the substitution,

<math>~a_j </math>

<math>~\rightarrow</math>

<math>~a_0(Y) dY = a_0(\Theta) \biggl[ \frac{w}{2\beta_1} \biggr] d\Theta \, ,</math>

where,

<math>~\frac{1}{\beta_1}</math>

<math>~\equiv</math>

<math>~\frac{\lambda L}{\pi y_1w} \, ,</math>

and changed the summation to an integration, obtaining,

<math>~A(y_1)</math>

<math>~\approx</math>

<math>~ e^{i 2\pi L/\lambda }\biggl[ \frac{w}{2\beta_1} \biggr] \int a_0(\Theta) e^{i\phi(\Theta)} \cdot e^{-i \Theta } d\Theta \, . </math>

Case #1:   If we assume that both <math>~a_0</math> and <math>~\phi</math> are independent of position along the aperture, and that the aperture — and, hence the integration — extends from <math>~Y_2 = -w/2</math> to <math>~Y_1 = +w/2</math>, we have shown that this last expression can be evaluated analytically to give,

<math>~A(y_1)</math>

<math>~\approx</math>

<math>~ e^{i [2\pi L/\lambda + \phi] }\biggl[ \frac{a_0 w}{2\beta_1} \biggr] \int_{\Theta_2}^{\Theta_1} e^{-i \Theta } d\Theta </math>

 

<math>~=</math>

<math>~ e^{i [2\pi L/\lambda + \phi] } \cdot a_0 w ~\mathrm{sinc}(\beta_1) \, . </math>

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation