Editing Appendix/CGH/ParallelApertures2D (section)

==Utility of FFT Techniques==
Consider the amplitude (and phase) of light that is incident at a location (x<sub>1</sub>, y<sub>1</sub>) on an image screen that is located a distance Z from a rectangular aperture of width ''w'' and height ''h''.  By analogy with [[Appendix/CGH/ParallelApertures#Utility_of_FFT_Techniques|our accompanying discussion in the context of 1D apertures]], the complex number, A, representing the light amplitude and phase at (x<sub>1</sub>, y<sub>1</sub>) will be,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A(x_1, y_1)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \sum_j \sum_k
a_{jk} e^{i(2\pi D_{jk} /\lambda + \phi_{jk})} 
\, ,
</math>
  </td>
</tr>
</table>
where, here, the summations are taken over all "j,k" elements of light across the entire 2D aperture, and now the distance D<sub>jk</sub> is given by the expression,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~D^2_{jk}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
(X_j - x_1)^2 + (Y_k - y_1)^2 + Z^2  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
Z^2 + y_1^2 - 2y_1 Y_k + Y_k^2 + x_1^2 - 2x_1 X_j + X_j^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
L^2 \biggl[1 - \frac{2(x_1 X_j + y_1 Y_k ) }{L^2}  + \frac{X_j^2 + Y_k^2}{L^2} \biggr] \, ,
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~L</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
[Z^2 + y_1^2  + x_1^2]^{1 / 2} \, .
</math>
  </td>
</tr>
</table>

If <math>~|X_j/L| \ll 1</math> and <math>~|Y_k/L| \ll 1</math> we can drop the quadratic terms in favor of the linear ones in the expression for  D<sub>jk</sub> and deduce that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~D_{jk}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
L \biggl[1 - \frac{2(x_1 X_j + y_1 Y_k ) }{L^2}  \biggr]^{1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
L \biggl[1 - \frac{(x_1 X_j + y_1 Y_k ) }{L^2}  \biggr] \, .
</math>
  </td>
</tr>
</table>

Hence, the double-summation expression for the amplitude at screen location (x<sub>1</sub>, y<sub>1</sub>) becomes,
<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 \sum_j \sum_k
a_{jk} e^{i\phi_{jk} } \cdot \exp\biggl\{ -i \biggl[ \frac{2\pi(x_1 X_j + y_1Y_k}{\lambda L} \biggr] \biggr\}
\, ,
</math>
  </td>
</tr>
</table>
where, 
<div align="center">
<math>~A_0 \equiv e^{i(2\pi L/\lambda)}</math>.
</div>
When written in this form, it should be apparent why discrete Fourier transform techniques &#8212; specifically, 2D-FFT techniques &#8212; are useful tools for evaluation of the complex amplitude, A(x<sub>1</sub>, y<sub>1</sub>).

<font color="red"><b>NOTE:</b></font>  If x<sub>1</sub> and/or y<sub>1</sub> are ever comparable in size to Z &#8212; which may be the case for large apertures or for apertures tilted by nearly 90&deg; to the image screen &#8212; then the variation of L with image screen position cannot be ignored and, accordingly, the coefficient A<sub>0</sub> cannot be moved outside of the double summation.