Editing Appendix/CGH/ParallelAperturesConsolidate (section)

===Case 1===
In a related accompanying derivation titled, [[Appendix/CGH/ParallelApertures#Analytic_Result|''Analytic Result'']], we made the substitution,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_j </math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~a_0(Y) dY = a_0(\Theta) \biggl[ \frac{w}{2\beta_1} \biggr] d\Theta \, ,</math>
  </td>
</tr>
</table>
where,
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  <td align="right">
<math>~\frac{1}{\beta_1}</math>
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<math>~\equiv</math>
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<math>~\frac{\lambda L}{\pi y_1w}  \, ,</math>
  </td>
</tr>
</table>

and changed the summation to an integration, obtaining,

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  <td align="right">
<math>~A(y_1)</math>
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<math>~\approx</math>
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  <td align="left">
<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>

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,

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<tr>
  <td align="right">
<math>~A(y_1)</math>
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  <td align="center">
<math>~\approx</math>
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  <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>

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  <td align="right">
&nbsp;
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<math>~=</math>
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<math>~ 
e^{i [2\pi L/\lambda + \phi] } \cdot a_0 w ~\mathrm{sinc}(\beta_1) \, . 
</math>
  </td>
</tr>
</table>

We need to explicitly demonstrate that an evaluation of our [[#FocalPoint|Focal-Point Expression]] with <math>~a_j = 1</math>, gives this last sinc-function expression, to within a multiplicative factor of, something like, <math>~j_\mathrm{max}</math>.