Editing Appendix/CGH/KAH2001 (section)

==SWMED==

In a paper titled, ''Dynamic holographic 3-D image projection'', [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract M. L. Huebschman, B. Munjuluri &amp; H. R. Garner (2003)] &#8212; Optics Express, vol. 11, no. 5, 437 - 445, hereafter, SWMED03 &#8212; describe the experimental 3-D projection system that was developed at [[Appendix/CGH/ZebraImaging#UT_Southwestern_Medical_Center_at_Dallas|UT's Southwestern Medical Center at Dallas]].  This is the research group that Richard Muffoletto and I visited circa 2004.

In &sect;4 of [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03] we find this general description:

The following integral expression is "<font color="orange">&hellip; the mathematical transform containing the wave physics of monochromatic light emanating from each object point, passing through the optical system and being superimposed at each point in the holographic plane.  It represents the integration over the object of spherical wave solutions of the [https://en.wikipedia.org/wiki/Helmholtz_equation Helmholtz form of the wave equation] &hellip; with additional phase shifts due to a spherical converging lens in the light pathway.</font>"

<table border="1" align="center" cellpadding="10" width="70%"><tr><td align="center">
Given the intensity, <math>~U(x', y', z')</math>, on various points across the objects in a 3-D scene, this integral expression generates the resulting intensity, <math>~U_s(x, y, 0)</math>, across the ''hologram plane'', i.e., immediately in front of the aperture.
</td></tr></table>

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

<tr>
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<math>~U_s(x, y, 0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\int_{\mathrm{V}'} U(x', y', z')~ \biggl[\frac{e^{-ikr} }{r}\biggr]~\times  \exp\biggl[ \frac{ik(x^2 + y^2)}{2f} \biggr]~dV^'
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
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  <td align="left">
<math>~
\int_{\mathrm{V}'} dV^' ~ \frac{U(x', y', z')}{\sqrt{ {z'}^2 + (x - x')^2 + (y - y')^2 }}~
\exp\biggl[-ik \biggl( \sqrt{ {z'}^2 + (x - x')^2 + (y - y')^2 } - \frac{x^2 + y^2}{2f} \biggr) \biggr] \, .
</math>
  </td>
</tr>
</table>

[The second of these two expressions has the form that appears as equation (1) in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03]; the first has been rewritten here in a form that can more easily be compared with related expressions that are found in the above subsections of this chapter.]  As is explained in [https://ui.adsabs.harvard.edu/abs/2003OExpr..11..437H/abstract SWMED03], "<font color="orange">&hellip; <math>~U_s(x, y, 0)</math> represents the intensity amplitude at a point in the hologram plane</font> &#8212; i.e., immediately in front of the aperture &#8212; <font color="orange"> and <math>~U(x', y', z')</math> is the intensity amplitude at a point on the objects in the 3-D scene of volume V' &hellip; The z'-axis is normal to the center of the hologram plane and extends through the center of the reconstructed 3-D scene volume.  The wave number of the light is given by k and f is the focal length of the converging lens.</font>"

"<font color="orange">The second term in the exponential represents the phase change due to the differing lengths of material the light traverses in the converging lens in our system &hellip;  Recall, a function of the converging lens is to produce the 3-D image reconstruction over a relatively near field.  Operating in the near field spatially disperses the depth information which is lost for reconstruction images that converge at infinity</font> &#8212; see [https://ui.adsabs.harvard.edu/abs/2001OptEn..40..926K/abstract T. Kreis, P. Aswendt, &amp; R. H&ouml;fling (2001)], which is [[#Optical_Field_in_the_Image_Plane|discussed above]]. <font color="orange">The phase information due to depth is conveyed via the first term in the exponential which is the radial distance from an object point to a hologram point.  We do not approximate this distance in our calculation, but we have incorporated the [https://en.wikipedia.org/wiki/Fresnel_diffraction#The_Fresnel_approximation Fresnel approximation] in the amplitude term in front of the exponential &hellip; Experimentally, we observed little change in image quality on reconstruction with this simplification.</font>"