Editing Appendix/CGH/ParallelApertures (section)

===More Attention to Detail===

====Theory====

Guided by our [[Appendix/Ramblings/FourierSeries#One-Dimensional_Aperture|accompanying discussion of the relationship between a one-dimensional aperture and the Fourier Series]], let's begin again with the [[#Utility_of_FFT_Techniques|above general summation expression]],
<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, <math>~0 \le j \le j_\mathrm{max}\, ,</math>
<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} \approx Z \biggl(1 + \frac{y_1^2}{2Z^2}  \biggr) \, ,</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 = -\frac{w}{2} + j\biggl(\frac{w}{j_\mathrm{max}}\biggr) \, .</math>
  </td>
</tr>
</table>

Now, let's adopt the following conventions:
<ol>
  <li>Specify the values of the parameters, <math>~\lambda, w, c/\mathfrak{L}_0,</math> and <math>~Z_0</math>, where,

<table border="0" align="center">
<tr><td align="center"><math>~\mathfrak{L} \equiv \lambda Z j_\mathrm{max}/(2w) </math> &nbsp;  &nbsp;  &hellip; hence &hellip;  &nbsp;   &nbsp; <math>~\mathfrak{L}_0 = \lambda Z_0 j_\mathrm{max}/(2w) \, .</math></td></tr>
</table>

  </li>
  <li>In specifying the properties of the light as it leaves the aperture, ensure that ''neither'' <math>~a_j</math> nor <math>~\phi_j</math> depends on the distance between the aperture and the image plane, <math>~Z</math>.</li>
  <li>It is best to evaluate the amplitude on the image screen over the range, <math>~-\mathfrak{L}_0 \le y_i \le \mathfrak{L}_0 \, .</math></li>
  <li>Expect the amplitude to be <math>~\sim 2c/\mathfrak{L}_0 \, .</math></li>
</ol>

With this in mind, the expression for the image-screen amplitude can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A(y_1)</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\mathrm{exp}\biggl\{i\biggl[
\frac{2\pi Z}{\lambda}\biggl(1 + \cancelto{0}{\frac{y_1^2}{2Z^2}} \biggr) + \frac{\pi y_1 w}{Z \lambda} \biggl( \cancelto{1}{\frac{Z}{L_1} } \biggr)
\biggr] \biggr\}
\sum_j a_j e^{i\phi_j} 
\biggl\{ \cos\biggl[ \frac{(\pi j y_1 ) 2w}{\lambda Z j_\mathrm{max}} \biggl( \cancelto{1}{\frac{Z}{L_1} } \biggr) \biggr] - i  \sin\biggl[ \frac{(\pi j y_1 ) 2w}{\lambda Z j_\mathrm{max}} \biggl( \cancelto{1}{\frac{Z}{L_1} } \biggr) \biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\mathrm{exp}\biggl\{i\biggl[
\frac{2\pi Z}{\lambda}+ \frac{\pi y_1 w}{Z \lambda} 
\biggr] \biggr\}
\sum_j a_j e^{i\phi_j} 
\biggl\{ \cos\biggl( \frac{\pi j y_1  }{\mathfrak{L}}  \biggr) - i  \sin\biggl( \frac{\pi j y_1 }{\mathfrak{L}} \biggr) \biggr\} \, .
</math>
  </td>
</tr>

</table>
</div>

<table border="1" cellpadding="8" align="center" width="85%"><tr><td align="left">
<div align="center"><font color="red">First (Misguided) Attempt</font></div>
Next, let's set <math>~\phi_j = - 2\pi Z_0/\lambda</math>, which is independent of <math>~j</math> and therefore can be shifted outside of the summation.  We have, then,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A(y_1)</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\mathrm{exp}\biggl\{i\biggl[
\frac{2\pi (Z - Z_0)}{\lambda}+ \frac{\pi y_1 w}{Z \lambda} 
\biggr] \biggr\}
\sum_j a_j  
\biggl\{ \cos\biggl( \frac{\pi j y_1  }{\mathfrak{L}}  \biggr) - i  \sin\biggl( \frac{\pi j y_1 }{\mathfrak{L}} \biggr) \biggr\} \, .
</math>
  </td>
</tr>

</table>

</td></tr></table>

Let's set,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\phi_j</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2\pi Z_0}{\lambda} - \frac{\pi y_1 w}{Z_0 \lambda} \, ,</math>
  </td>
</tr>
</table>

which is independent of <math>~j</math> and therefore can be shifted outside of the summation.  We have, then,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A(y_1)</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\mathrm{exp}\biggl\{i\biggl[
\frac{2\pi (Z - Z_0)}{\lambda}+ \frac{\pi y_1 w}{Z \lambda} 
\biggr] \biggr\}
\sum_j a_j  
\biggl\{ \cos\biggl( \frac{\pi j y_1  }{\mathfrak{L}}  \biggr) - i  \sin\biggl( \frac{\pi j y_1 }{\mathfrak{L}} \biggr) \biggr\} \, .
</math>
  </td>
</tr>

</table>

Using this general expression, we should specifically expect the amplitude across the image plane at Z<sub>0</sub> to be a step function if we insert the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_j</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{j \pi} \sin\biggl( \frac{j\pi c}{\mathfrak{L}_0} \biggr)  = \biggl(\frac{2c}{\mathfrak{L}_0} \biggr) \mathrm{sinc}\biggl( \frac{j\pi c}{\mathfrak{L}_0} \biggr) \, .</math>
  </td>
</tr>
</table>

Hopefully, the amplitude of this image will die off quickly as we insert values of <math>~Z \ne Z_0 \, .</math>

====Example #1====

Let's try an example using the parameters listed at the top of Figure 4.

<div align="center" id="Figure4">
<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">Figure 4: &nbsp; <math>~w = 1~\mathrm{mm}; Z_0 = 10~\mathrm{m}; \lambda = 500~\mathrm{nm}; j_\mathrm{max} = 51</math></th>
</tr>
<tr>
  <td align="center">
[[File:SingleSlit02B.png|700px|Second Plot of Single-Slit Diffraction Pattern]]
  </td>
</tr>
</table>
</div>