Difference between revisions of "User:Tohline/Appendix/CGH/Overview"
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As has been detailed in an [[User:Tohline/Appendix/CGH/ParallelApertures#OneDimensional_Apertureaccompanying discussion]], we  As has been detailed in an [[User:Tohline/Appendix/CGH/ParallelApertures#OneDimensional_Apertureaccompanying discussion]], we examine, first, the amplitude (and phase) of light which, after passing through an aperture of width <math>~w = (Y_1  Y_2) = 2c</math>, is incident on an image screen that is parallel to, and located a distance <math>~Z</math> from, the aperture. The amplitude at any location, <math>~y_1</math>, along the image screen is given by the expression,  
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<math>~A(  <math>~A(y_1)</math>  
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<math>~\sum_j  <math>~\sum_j  
\beta_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>  </math>  
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where, <math>~  where, <math>~\beta_j = \beta(Y_j)</math> is the specified brightness at each "j" point along the illuminated aperture,  
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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  
The green solid curves in [[User:Tohline/Appendix/CGH/ParallelApertures#Figure2Figures 2 & 3 of our accompanying, more detailed discussion]] display — for a fixed set of values of the parameterpair, <math>~(\lambda, w)</math> — how the amplitude of the incident light, <math>~(A\cdot A^*)^{1 / 2}</math>, varies across an image screen that is placed at two different distances from the aperture.  
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 [[#Simplificationsdiscussed below]], our expression for the amplitude simplifies nicely in situations where the quadratic term can be ignored. (The red dotted curves in [[User:Tohline/Appendix/CGH/ParallelApertures#Figure2Figures 2 & 3 of our accompanying discussion]] illustrate how the imagescreen diffraction pattern is altered as a result of this simplification.) Via a related simplification, we also will find that the various natural lengths of this problem can be related to one another via the expression,  
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CAUTION: Be sure that the "L" in these various expressions is, indeed, always the same quantity.  CAUTION: Be sure that the "L" in these various expressions is, indeed, always the same quantity.  
===Simplifications===  
If we assume that everywhere along the aperture the phase <math>~\phi_j = 0</math>; and if we assume that, for all <math>~j</math>,  
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<math>~\biggl \frac{Y_j}{L}\biggr \ll 1</math>  
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we can drop the quadratic term in favor of the linear one in the above expression for <math>~D_j</math> and [[User:Tohline/Appendix/CGH/ParallelApertures#Simplifydeduce that]],  
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<math>~A(y_1)</math>  
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<math>~\approx</math>  
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<math>~e^{i2\pi L/\lambda} \sum_j  
\beta_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]  
\, .  
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=See Also=  =See Also= 
Revision as of 22:19, 28 December 2017
CGH: Philosophical Overview
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Slit Diffraction
Single Aperture

As has been detailed in an accompanying discussion, we examine, first, the amplitude (and phase) of light which, after passing through an aperture of width <math>~w = (Y_1  Y_2) = 2c</math>, is incident on an image screen that is parallel to, and located a distance <math>~Z</math> from, the aperture. The amplitude at any location, <math>~y_1</math>, along the image screen is given by the expression,
<math>~A(y_1)</math> 
<math>~=</math> 
<math>~\sum_j \beta_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>~\beta_j = \beta(Y_j)</math> is the specified brightness at each "j" point along the illuminated aperture,
<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> 
The green solid curves in Figures 2 & 3 of our accompanying, more detailed discussion display — for a fixed set of values of the parameterpair, <math>~(\lambda, w)</math> — how the amplitude of the incident light, <math>~(A\cdot A^*)^{1 / 2}</math>, varies across an image screen that is placed at two different distances from the aperture.
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. (The red dotted curves in Figures 2 & 3 of our accompanying discussion illustrate how the imagescreen diffraction pattern is altered as a result of this simplification.) Via a related simplification, we also will find that the various natural lengths of this problem can be related to one another 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.
Simplifications
If we assume that everywhere along the aperture the phase <math>~\phi_j = 0</math>; and if we assume that, for all <math>~j</math>,
<math>~\biggl \frac{Y_j}{L}\biggr \ll 1</math>
we can drop the quadratic term in favor of the linear one in the above expression for <math>~D_j</math> and deduce that,
<math>~A(y_1)</math> 
<math>~\approx</math> 
<math>~e^{i2\pi L/\lambda} \sum_j \beta_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> 
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 