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==One-dimensional Apertures==
==One-dimensional Apertures==


From our [[User:Tohline/Appendix/CGH/ParallelApertures#CGH:__Apertures_that_are_Parallel_to_the_Image_Screen|accompanying discussion]],  
From our accompanying discussion of the [[User:Tohline/Appendix/CGH/ParallelApertures#Utility_of_FFT_Techniques|''Utility of FFT Techniques'']], we start with the most general expression for the amplitude at one point on an image screen, namely,
<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>
</table>
and, assuming that <math>~|Y_j/L| \ll 1</math> for all <math>~j</math>, deduce that,
<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>~\sum_j
a_j e^{i[ 2\pi L/\lambda + \phi_j]}\biggl[ \cos\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) - i  \sin\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \biggr]
\, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<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 \biggl[1 + \frac{y_1^2}{Z^2}  \biggr]^{1 / 2} \, .
</math>
  </td>
</tr>
</table>


=See Also=
=See Also=

Revision as of 16:31, 17 March 2020

CGH: Consolidate Expressions Regarding Parallel Apertures

One-dimensional Apertures

From our accompanying discussion of the Utility of FFT Techniques, we start with the most general expression for the amplitude at one point on an image screen, namely,

<math>~A(y_1)</math>

<math>~=</math>

<math>~\sum_j a_j e^{i(2\pi D_j/\lambda + \phi_j)} \, , </math>

and, assuming that <math>~|Y_j/L| \ll 1</math> for all <math>~j</math>, deduce that,

<math>~A(y_1)</math>

<math>~\approx</math>

<math>~\sum_j a_j e^{i[ 2\pi L/\lambda + \phi_j]}\biggl[ \cos\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) - i \sin\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \biggr] \, , </math>

where,

<math>~L</math>

<math>~\equiv</math>

<math>~ Z \biggl[1 + \frac{y_1^2}{Z^2} \biggr]^{1 / 2} \, . </math>

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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