Difference between revisions of "User:Tohline/Appendix/CGH/Overview"
Line 23:  Line 23:  
<tr>  <tr>  
<td align="right">  <td align="right">  
<math>~A(y_1)</math>  <math>~A(n\Delta y_1)</math>  
</td>  </td>  
<td align="center">  <td align="center">  
Line 75:  Line 75:  
</table>  </table>  
</div>  </div>  
Notice that, in the definition of <math>~D_j</math>, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, <math>~Y_j/L</math>, as well as a term that depends linearly on this ratio. As discussed below, our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored. Via a related simplification, we will find that the various natural lengths of this problem are can be related via the expression,  Notice that, in the definition of <math>~D_j</math>, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, <math>~Y_j/L</math>, as well as a term that depends linearly on this ratio. As discussed below, our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored. Via a related simplification, we also will find that the various natural lengths of this problem are can be related via the expression,  
<div align="center">  <div align="center">  
<table border="0" cellpadding="5" align="center">  <table border="0" cellpadding="5" align="center">  
Line 81:  Line 81:  
<tr>  <tr>  
<td align="right">  <td align="right">  
<math>~y_1\biggr_{1^\mathrm{st} \mathrm{fringe}}</math>  <math>~y_1\biggr_{1^\mathrm{st} \mathrm{fringe}} = \frac{L \Delta y_1}{c}</math>  
</td>  </td>  
<td align="center">  <td align="center">  
Line 87:  Line 87:  
</td>  </td>  
<td align="left">  <td align="left">  
<math>~\frac{\lambda Z}{  <math>~\frac{\lambda Z}{2c} \, .</math>  
</td>  </td>  
</tr>  </tr>  
</table>  </table>  
</div>  </div>  
CAUTION: Be sure that the "L" in these various expressions is, indeed, always the same quantity.  
=See Also=  =See Also= 
Revision as of 04:55, 28 December 2017
CGH: Philosophical Overview
 Tiled Menu  Tables of Content  Banner Video  Tohline Home Page  
Slit Diffraction
Single Aperture

As has been detailed in an accompanying discussion, we consider, first, the amplitude (and phase) of light that is incident at a location <math>~y_1</math> on an image screen that is located a distance <math>~Z</math> from a slit of width <math>~w = (Y_1  Y_2) = 2c</math>. The amplitude is given by the expression,
<math>~A(n\Delta y_1)</math> 
<math>~=</math> 
<math>~\sum_j a_j \biggl[ \cos\biggl(\frac{2\pi D_j}{\lambda} + \phi_j \biggr) + i \sin\biggl(\frac{2\pi D_j}{\lambda} + \phi_j \biggr) \biggr] \, , </math> 
where,
<math>~D_j</math> 
<math>~=</math> 
<math>~ L \biggl[1  \frac{2y_1 Y_j}{L^2} + \frac{Y_j^2}{L^2} \biggr]^{1 / 2} \, , </math> 
and,
<math>~L</math> 
<math>~\equiv</math> 
<math>~ [Z^2 + y_1^2 ]^{1 / 2} \, . </math> 
Notice that, in the definition of <math>~D_j</math>, the expression inside the square brackets involves a term that depends quadratically on the dimensionless length scale, <math>~Y_j/L</math>, as well as a term that depends linearly on this ratio. As discussed below, our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored. Via a related simplification, we also will find that the various natural lengths of this problem are can be related via the expression,
<math>~y_1\biggr_{1^\mathrm{st} \mathrm{fringe}} = \frac{L \Delta y_1}{c}</math> 
<math>~\approx</math> 
<math>~\frac{\lambda Z}{2c} \, .</math> 
CAUTION: Be sure that the "L" in these various expressions is, indeed, always the same quantity.
See Also
 Tohline, J. E., (2008) Computing in Science & Engineering, vol. 10, no. 4, pp. 8485 — Where is My Digital Holographic Display? [ PDF ]
 Diffraction (Wikipedia)
 Various Google hits:
 Single Slit Diffraction (University of Tennessee, Knoxville)
 Diffraction from a Single Slit; Young's Experiment with Finite Slits (University of New South Wales, Sydney, Australia)
 Single Slit Diffraction Pattern of Light (University of British Columbia, Canada)
 Fraunhofer Single Slit (Georgia State University)
© 2014  2021 by Joel E. Tohline 