Editing Appendix/Ramblings/FourierSeries (section)

==One-Dimensional Aperture==
===General Concept===

Here we draw from our [[Appendix/CGH/ParallelApertures#Utility_of_FFT_Techniques|accompanying discussion of single-slit apertures]] in the context of computer-generated holography and, for the time being, retain our option to specify a phase angle, <math>~\phi_j</math>, that varies with position across the aperture.  Recognizing from that discussion that <math>~D_j \approx L_1[1 - y_1 Y_j/L_1^2]</math>, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A(y_1)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sum_j
a_j e^{i(2\pi D_j/\lambda + \phi_j)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\sum_j
a_j \mathrm{exp}\biggl\{i\biggl[
\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_j}{L_1 \lambda} + \phi_j
\biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sum_j
a_j \mathrm{exp}\biggl\{i\biggl[
\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} - \frac{2\pi j y_1 \Delta Y}{L_1 \lambda} + \phi_j
\biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sum_j
a_j \mathrm{exp}\biggl\{i\biggl[
\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} + \phi_j
\biggr] \biggr\} \biggl[ \cos\biggl(\frac{2\pi y_1 j \Delta Y}{\lambda L_1} \biggr) - i  \sin\biggl(\frac{2\pi y_1 j \Delta Y}{\lambda L_1} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathrm{exp}\biggl\{i\biggl[
\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} 
\biggr] \biggr\}
\sum_j a_j e^{i\phi_j} 
\biggl[ \cos\biggl(\frac{2\pi y_1 j \Delta Y}{\lambda L_1} \biggr) - i  \sin\biggl(\frac{2\pi y_1 j \Delta Y}{\lambda L_1} \biggr) \biggr] \, ,
</math>
  </td>
</tr>

</table>
</div>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~L_1</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[ Z^2 + y_1^2 ]^{1 / 2} \, ,</math>
  </td>
</tr>
</table>
and we have inserted the following expression in order to identify discrete locations along the aperture,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~Y_j</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~Y_0 + j\Delta Y \, .</math>
  </td>
</tr>
</table>

This is precisely the same as the [[#Standard|above-defined]],
<div align="center" id="StandardExpression">
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center">''Standard'' Fourier Series Expression</th>
</tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a_0}{2} + \sum_{n=1}^{\infty} \biggl[ a_n\cos \biggl(\frac{n\pi x}{L}\biggr) + b_n\sin  \biggl(\frac{n\pi x}{L}\biggr) \biggr] 
</math>
  </td>
</tr>
</table>
</td></tr></table>
</div>
if we accept the following associations:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~n</math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~j</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{x}{L}</math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~\frac{2 y_1 \Delta Y}{\lambda L_1}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{a_{n=0}}{2}</math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~\mathrm{exp}\biggl\{i\biggl[
\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} + \phi_0
\biggr] \biggr\} ~a_{j=0} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a_n</math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~a_j ~ \mathrm{exp}\biggl\{i\biggl[
\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} + \phi_j
\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_n</math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~-i a_j~\mathrm{exp}\biggl\{i\biggl[
\frac{2\pi L_1}{\lambda} - \frac{2\pi y_1 Y_0}{L_1 \lambda} + \phi_j
\biggr] \biggr\} 
</math>
  </td>
</tr>
</table>