Editing Appendix/CGH/ParallelAperturesHolograms (section)

==Light from the Object==

The discussions in &sect;&sect;[[Appendix/CGH/ParallelApertures|I.A]] and [[Appendix/CGH/ParallelApertures2D|I.B]] illustrate how light that originates from an extended object (aperture) illuminates various points along the image screen.  For example, from the perspective of the image screen, the front vertical edge of the red cube shown above in Figure I.2 looks like a vertical, one-dimensional "slit" aperture of width "w", as illustrated in [[Appendix/CGH/ParallelApertures#General_Concept|Figure I.1 from our &sect;I.A]].  Hence, the complex amplitude of light that is striking various points "y" along any vertical column of the image screen (hologram) ''due solely to the light coming from this edge of the cube'' is given by the expression derived in &sect;[[Appendix/CGH/ParallelApertures|I.A]], namely,

<table border="1" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>A(Y)</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\biggl[A_0 a_0 w \biggr] \mathrm{sinc}\beta \, ,</math>
  </td>
</tr>
</table>

<table border="1" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A(y)</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>~
[A_0 a_0 w] \mathrm{sinc}(\beta) \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\beta</math>
  </td>
  <td align="center">
<math>~
\equiv
</math>
  </td>
  <td align="left">
<math>~
\frac{\pi y w}{\lambda L} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~L</math>
  </td>
  <td align="center">
<math>~
\equiv
</math>
  </td>
  <td align="left">
<math>~
[Z^2 + y^2]^{1 / 2} \, ,
</math>
  </td>
</tr>
</table>
Z is the shortest distance measured between the front vertical edge of the cube and the selected column of the image screen, and for purposes of illustration, we have set to zero the offset phase angle, <math>~\vartheta_1</math>, that appears in our previously derived analytic expression for the amplitude.

However, as depicted in Figure I.2, the image screen is not being illuminated by just one vertical edge of the cube but, instead, by one entire two-dimensional face of the cube (the side of the cube that is facing to the right and is therefore not visible in the figure).  From the perspective of the image screen (hologram), the surface of the cube looks like a two-dimensional aperture of width "w" and height "w."  Therefore, the more complete expression for the complex amplitude of light striking the image screen is,

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

<tr>
  <td align="right">
<math>~A(x, y)</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
[A_0~a_0~w^2] \mathrm{sinc}(\alpha)~\mathrm{sinc}(\beta) \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\alpha</math>
  </td>
  <td align="center">
<math>~
\equiv
</math>
  </td>
  <td align="left">
<math>~
\frac{\pi x w}{\lambda L} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta</math>
  </td>
  <td align="center">
<math>~
\equiv
</math>
  </td>
  <td align="left">
<math>~
\frac{\pi y w}{\lambda L} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~L</math>
  </td>
  <td align="center">
<math>~
\equiv
</math>
  </td>
  <td align="left">
<math>~
[Z^2 + y^2 + x^2]^{1 / 2} \, ,
</math>
  </td>
</tr>
</table>
as derived in our [[Appendix/CGH/ParallelApertures2D#Analytic_Results|accompanying discussion of two-dimensional apertures that are parallel to the image screen]].  (Again, as in our separate discussion, for the sake of illustration we have set the two offset phase angles, <math>~\Theta_1 = \vartheta_1 = 0</math>.)