Editing Appendix/CGH/ParallelApertures (section)

===Location of the Dark Fringe(s)===

<span id="DarkFringe">As has been described</span> in [[#See_Also|multiple online references]], the locations across the image screen where complete ''destructive'' interference occurs can be determined straightforwardly using geometric relationships.  For example, in terms of the angle, <math>~\theta</math>, defined such that,
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<math>~\tan\theta = \frac{y_1}{Z} \, ,</math>
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the distance from the central brightness peak to the first location where the brightness/amplitude goes to zero &#8212; ''i.e.'', the location of the first ''dark fringe'' &#8212; is given by the relation,
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<math>~\sin\theta = \frac{\lambda}{w} \, .</math>
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(For each successive fringe, labeled by the positive integer, <math>~m</math>, the relation is, <math>~\sin\theta_m = m\lambda/w.</math>) Hence, acknowledging that usually <math>~\lambda/w \ll 1</math>, we find that,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~y_1\biggr|_{1^\mathrm{st} \mathrm{fringe}}</math>
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  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~Z \tan\theta = Z \sin\theta [ 1 - \sin^2\theta ]^{-1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\lambda Z}{w} \biggl[ 1 - \biggl(\frac{\lambda }{w}\biggr)^2 \biggr]^{-1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
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  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{\lambda Z}{w} \, .</math>
  </td>
</tr>
</table>
</div>

In the context of Figure 3, this means that, <math>~y_1|_{1^\mathrm{st} \mathrm{fringe}} = 5</math> millimeters, while, in Figure 2, <math>~y_1|_{1^\mathrm{st} \mathrm{fringe}} = 0.5</math> millimeters &#8212; in both cases, this is in agreement with the plotted ''linearized'' amplitude curves.