Editing Appendix/CGH/ParallelApertures (section)

===Parallels With Example #2===

This "one dimensional aperture" analysis should exhibit features that strongly resemble the features that appear in our accompanying discussion of the Fourier series associated with a "[[Appendix/Ramblings/FourierSeries#Example_.232|square wave]]".  In both cases &#8212; after performing both a Fourier transform ''and'' the inverse transform &#8212; the ultimate series expression that will represent the (square wave) amplitude across the aperture will take the form,

<div align="center" id="StandardExpression">

<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>
</div>
This function clearly repeats itself at spatial intervals of <math>~x \pm 2L</math>.  Hence, we must acknowledge that, even if the initial state is intended to represent a ''single'' aperture, the inverse transform will produce an infinite set of identical apertures that are spaced (center-to-center) at intervals of <math>~2L</math>.  We can presumably arrange to have successive apertures of width, <math>~2c</math>, widely spaced from one another by picking a value of <math>~|c/L | \ll 1</math>.  (Reference, also, frame ''a'' of [[#Figure5|Figure 5]], below, which depicts a uniformly illuminated (yellow), ''two''-dimensional aperture whose horizontal width, as labeled, is <math>~2c_0 ;</math> the aperture has been cut into a mat of width, <math>~2L_0</math>.)

In the "square wave" analysis in which the brightness across the aperture is specified by a continuous function, the amplitude, <math>~a_n</math>, of each Fourier mode, <math>~n</math>, is given by the expression,

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

<tr>
  <td align="right">
<math>~a_n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{L} \int_{-c}^{c} \cos\biggl( \frac{n\pi x}{L} \biggr) dx
= \biggl(\frac{2c}{L} \biggr) \mathrm{sinc}(\alpha_n)\, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~\alpha_n \equiv n\pi c/L</math>.
On the other hand, when a ''discrete'' Fourier transform is used, the analogous Fourier amplitude is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{R}e[A(n\Delta y_1)]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sum_{j=-j_\mathrm{max}}^{j_\mathrm{max}}
\cos\biggl[n\Delta y_1 \biggl(\frac{2\pi}{\lambda \mathcal{D}} \biggr)\frac{j c}{j_\mathrm{max}} \biggr] 
\, .
</math>
  </td>
</tr>
</table>
Comparing the two expressions, we recognize first that the integer, <math>~n</math>, has the same meaning in both; and, second, that <math>~x \leftrightarrow jc/j_\mathrm{max}</math>.  Therefore &#8212; after recognizing that <math>~\mathcal{D} \approx Z</math> &#8212; it must also be true that,
<div align="center">
<math>~\frac{1}{L} \leftrightarrow \frac{2\Delta y_1}{\lambda \mathcal{D}} \approx \frac{2\Delta y_1}{\lambda Z} \, .</math>
</div>

Next we notice, from the "square wave" analysis, that since the amplitude of the the diffraction pattern, <math>~a_n</math>, varies as <math>~\mathrm{sinc}(\alpha_n)</math>, the first dark fringe will arise when <math>~\alpha_n = \pi</math>, that is, when <math>~n = n_\pi \equiv L/c</math>.  But, [[#DarkFringe|as explained above]], from geometric arguments associated with the "one dimensional aperture" analysis, we expect the first dark fringe on the image screen to arise when,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{y_1}{Z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\lambda}{2c} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{2}{\lambda Z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{y_1 c} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\frac{\Delta y_1}{y_1 c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2\Delta y_1}{\lambda Z} ~~\rightarrow ~~ \frac{1}{L} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{L}{c} ~~~\Rightarrow~~ n = n_\pi \, ,</math>
  </td>
</tr>
</table>
</div>
where this last relation has been derived by recognizing that, quite generally by design, <math>~y_1 = n\Delta y_1</math>.  So, these two separate ways of identifying the location of the first dark fringe agree with one another.


COMMENT: &nbsp;  Throughout our discussion of computer-generated holography, we will find it necessary to construct a discretized image screen and, hence, will need to discuss the corresponding discretized modal amplitude ("sinc") function.  This discretized amplitude function can be treated as a multiple-slit source function &#8212; reference, for example, frame ''c'' of [[#Figure5|Figure 5]], below, which displays a 6 &times; 6 horizontal &times; vertical aperture layout &#8212; and, via an ''inverse'' Fourier transform, be used to regenerate the original square-wave (or analogous) function.  This square wave of width, <math>~2c</math>, will necessarily be accompanied by multiple duplicate images that are spaced a (center-to-center) distance, <math>~2L</math>, apart.  The result that we have just derived tells us that the relative spacing of these duplicate images will be large, relative to the width of the original square wave, if the discretization of the image screen is done in such a way as to ensure that <math>~n_\pi</math> is a large number.