Editing Appendix/CGH/ParallelApertures2D (section)

==Analytic Results==
Now, as in [[Appendix/CGH/ParallelApertures#Parallels_With_Example_.232|&sect;I.A]], we would like to consider the case where the rectangular aperture is divided into an infinite number of divisions in both the X and Y dimensions and convert the summations in this last expression into integrals whose limits in both directions are given by the edges of the aperture.  If we specifically consider the case where the aperture is assumed to be uniformly bright (''i.e.,'' a<sub>jk</sub>  = a<sub>0</sub> dX DY, and a<sub>0</sub> is the brightness per unit area), and the phase <math>~\phi_{jk} = 0</math> at all locations on the aperture, we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A(x_1, y_1)</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~ A_0 a_0 \iint \exp\biggl\{ -i \biggl[ \frac{2\pi(x_1 X_j + y_1Y_k}{\lambda L} \biggr] \biggr\} dX dY
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ A_0 a_0 
\int \exp\biggl\{ -i \biggl[ \frac{2\pi x_1 X_j }{\lambda L} \biggr] \biggr\} dX
\cdot \int \exp\biggl\{ -i \biggl[ \frac{2\pi y_1Y_k}{\lambda L} \biggr] \biggr\} dY
\, .
</math>
  </td>
</tr>
</table>

Both of these integrals can be completed in the same fashion as described in [[Appendix/CGH/ParallelApertures#Parallels_With_Example_.232|&sect;I.A]] for the 1D slit, giving,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A(x_1, y_1)</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~ A_0 a_0 (wh) e^{-i\Theta_1} ~ e^{-i\vartheta_1} \mathrm{sinc}(\alpha_1) \mathrm{sinc}(\beta_1)
\, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\alpha_1</math>
  </td>
  <td align="center">
<math>~\equiv </math>
  </td>
  <td align="left">
<math>~\frac{\pi x_1 (X_1 - X_2)}{\lambda L}= \frac{\pi x_1 h}{\lambda L} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Theta_1</math>
  </td>
  <td align="center">
<math>~\equiv </math>
  </td>
  <td align="left">
<math>~\frac{\pi x_1 (X_1 + X_2)}{\lambda L} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta_1</math>
  </td>
  <td align="center">
<math>~\equiv </math>
  </td>
  <td align="left">
<math>~\frac{\pi y_1 (Y_1 - Y_2)}{\lambda L}= \frac{\pi y_1 w}{\lambda L} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\vartheta_1</math>
  </td>
  <td align="center">
<math>~\equiv </math>
  </td>
  <td align="left">
<math>~\frac{\pi y_1 (Y_1 + Y_2)}{\lambda L} \, .</math>
  </td>
</tr>
</table>

It is worth noting that this derivation closely parallels the one presented in &sect;8.5.1 (p. 393) of Born &amp; Wolf (1980) (see the [[#See_Also|reference below]]).  Specifically, our double-integral expression is identical to the "Fraunhofer diffraction integral" written down by Born &amp; Wolf at the beginning of their &sect;8.5.1; and if we follow Born &amp; Wolf's lead and position the origin of our coordinate system at the center of the rectangle, then <math>~\Theta_1 = \vartheta_1 = 0</math> and the intensity <math>~I(P)</math> at point <math>~P</math> that is given by their equation (1) precisely matches the expression for the square of the amplitude, <math>~A^*A(x_1, y_1)</math>, that is obtained from our last expression.