Editing Appendix/CGH/KAH2001 (section)

==Optical Field in the Image Plane==

In a paper titled, ''Hologram reconstruction using a digital micromirror device'', [https://ui.adsabs.harvard.edu/abs/2001OptEn..40..926K/abstract T. Kreis, P. Aswendt, &amp; R. H&ouml;fling (2001)] &#8212; Optical Engineering, vol. 40, no. 6, 926 - 933, hereafter, KAH2001 &#8212; present some background theoretical development that was used to underpin work of the group at [[Appendix/CGH/ZebraImaging#UT_Southwestern_Medical_Center_at_Dallas|UT's Southwestern Medical Center at Dallas]] that Richard Muffoletto and I visited circa 2004.

This same integral expression &#8212; with a slightly different leading normalization factor &#8212; appears as equation (5) of [https://ui.adsabs.harvard.edu/abs/2001OptEn..40..926K/abstract KAH2001].  Referring to it as the ''Fresnel transform'' expression for the "optical field, <math>~B(x, y)</math>, in the image plane at a distance <math>~d</math> from the [aperture]," they write,

<table border="1" align="center" cellpadding="10" width="65%"><tr><td align="center">
Given the intensity immediately in front of the aperture, <math>~U(\xi, \eta)</math>, this integral expression generates the intensity, <math>~B(x, y)</math>, on the image plane whose distance from the aperture is, <math>~d</math>.
</td></tr></table>

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

<tr>
  <td align="right">
<math>~B(x,y)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{e^{i k d}}{i k d} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} U(\xi,\eta) \times \exp\biggl\{ \frac{i \pi}{d \lambda} \biggl[ (x - \xi)^2 + (y-\eta)^2 \biggr] \biggr\} d\xi d\eta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{e^{i k d}}{i k d} \biggr] I_\xi(x) \cdot I_\eta(y) \, ,
</math>
  </td>
</tr>
</table>
with,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~I_\xi(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int_{-\infty}^{\infty} V(\xi) \times \exp\biggl[ \frac{i \pi}{d \lambda} (x - \xi)^2 \biggr] d\xi \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~I_\eta(y)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int_{-\infty}^{\infty} W(\eta) \times \exp\biggl[ \frac{i \pi}{d \lambda} (y - \eta)^2 \biggr] d\eta \, ,
</math>
  </td>
</tr>
</table>
and where "&hellip; the optical field immediately in front of the [aperture]" is assumed to be of the form, <math>~U(\xi,\eta) = V(\xi)\cdot W(\eta)</math>.  Following [https://ui.adsabs.harvard.edu/abs/2001OptEn..40..926K/abstract KAH2001] &#8212; especially the discussion associated with their equations (7) - (10) &#8212; if we make the substitutions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu \equiv \frac{x}{d\lambda} \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\alpha \equiv \frac{\sqrt{2} \xi}{ \sqrt{d \lambda} }  - \sqrt{2d\lambda} ~\mu </math>&nbsp; &nbsp; &nbsp;
<math>~\Rightarrow ~~~ d\xi = \biggl(\frac{d \lambda}{2}\biggr)^{1 / 2} d\alpha \, ,</math>
  </td>
</tr>
</table>

<span id="Eq10KAH2001">the expression for <math>~I_\xi(x)</math> may be written as,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~I_\xi(x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int_{-\infty}^{\infty} V(\xi) \times \exp\biggl\{  \frac{i \pi}{d \lambda} \biggl[ d \lambda \mu - \frac{\sqrt{d\lambda}}{\sqrt{2}} \biggl( \alpha + \sqrt{2 d\lambda}~\mu \biggr) \biggr]^2 \biggr\} \biggl( \frac{d \lambda}{2}\biggr)^{1 / 2} d\alpha
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int_{-\infty}^{\infty} V(\xi) \times \exp\biggl\{  i \pi d \lambda \biggl[ \mu - \frac{1}{\sqrt{2d \lambda}} \biggl( \alpha + \sqrt{2 d\lambda}~\mu \biggr) \biggr]^2 \biggr\} \biggl( \frac{d \lambda}{2}\biggr)^{1 / 2} d\alpha
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl( \frac{d \lambda}{2}\biggr)^{1 / 2} 
\int_{-\infty}^{\infty} V(\xi) \times \exp\biggl[ \frac{i \pi \alpha^2}{2} \biggr] d\alpha \, .
</math>
  </td>
</tr>
</table>

The expression for <math>~I_\eta(y)</math> may be rewritten similarly.