Difference between revisions of "User:Tohline/Appendix/CGH/ParallelAperturesConsolidate"
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This means that <math>~\Theta_{i} = - \Theta_{( j_\mathrm{max} - 1 - i )}</math>. | This means that <math>~\Theta_{i} = - \Theta_{( j_\mathrm{max} - 1 - i )}</math>. | ||
The key expression under the summation therefore becomes, | |||
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<math>~a_j | <math>~~a_j e^{i \phi_j} \cdot \biggl[ \cos \biggl( \frac{j\pi y_1}{\mathfrak{L}}- \Theta_0 \biggr) - i \sin \biggl( \frac{j\pi y_1}{\mathfrak{L}}- \Theta_0 \biggr) \biggr] \, ,</math> | ||
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where, | |||
<div align="center"> | |||
<math>~\Theta_0 \equiv \frac{(j_\mathrm{max} - 1)}{2} \cdot \pi y_1 \biggl[ \frac{2w}{(j_\mathrm{max}-1) \lambda L} \biggr] = \frac{\pi y_1 w}{\lambda L} \, .</math> | |||
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=See Also= | =See Also= |
Revision as of 03:35, 20 March 2020
CGH: Consolidate Expressions Regarding Parallel Apertures
One-dimensional Apertures
From our accompanying discussion of the Utility of FFT Techniques, we start with the most general expression for the amplitude at one point on an image screen, namely,
<math>~A(y_1)</math> |
<math>~=</math> |
<math>~\sum_j a_j e^{i(2\pi D_j/\lambda + \phi_j)} \, , </math> |
and, assuming that <math>~|Y_j/L| \ll 1</math> for all <math>~j</math>, deduce that,
<math>~A(y_1)</math> |
<math>~\approx</math> |
<math>~\sum_j a_j e^{i[ 2\pi L/\lambda + \phi_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> |
where,
<math>~L</math> |
<math>~\equiv</math> |
<math>~ Z \biggl[1 + \frac{y_1^2}{Z^2} \biggr]^{1 / 2} \, . </math> |
Note that <math>~L</math> is formally a function of <math>~y_1</math>, but in most of what follows it will be reasonable to assume, <math>~L \approx Z</math>. Notice, as well, that this last approximate expression for the (complex) amplitude at the image screen may be rewritten in the form that will be referred to as our,
Focal-Point Expression | ||
---|---|---|
<math>~A(y_1)</math> |
<math>~\approx</math> |
<math>~ e^{i 2\pi L/\lambda } \sum_j a_j e^{i \phi_j} \cdot e^{-i \Theta_j } \, , </math> |
where,
<math>~\Theta_j</math> |
<math>~\equiv</math> |
<math>~\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \, .</math> |
Case 1
In a related accompanying derivation titled, Analytic Result, we made the substitution,
<math>~a_j </math> |
<math>~\rightarrow</math> |
<math>~a_0(Y) dY = a_0(\Theta) \biggl[ \frac{w}{2\beta_1} \biggr] d\Theta \, ,</math> |
where,
<math>~\frac{1}{\beta_1}</math> |
<math>~\equiv</math> |
<math>~\frac{\lambda L}{\pi y_1w} \, ,</math> |
and changed the summation to an integration, obtaining,
<math>~A(y_1)</math> |
<math>~\approx</math> |
<math>~ e^{i 2\pi L/\lambda }\biggl[ \frac{w}{2\beta_1} \biggr] \int a_0(\Theta) e^{i\phi(\Theta)} \cdot e^{-i \Theta } d\Theta \, . </math> |
If we assume that both <math>~a_0</math> and <math>~\phi</math> are independent of position along the aperture, and that the aperture — and, hence the integration — extends from <math>~Y_2 = -w/2</math> to <math>~Y_1 = +w/2</math>, we have shown that this last expression can be evaluated analytically to give,
<math>~A(y_1)</math> |
<math>~\approx</math> |
<math>~ e^{i [2\pi L/\lambda + \phi] }\biggl[ \frac{a_0 w}{2\beta_1} \biggr] \int_{\Theta_2}^{\Theta_1} e^{-i \Theta } d\Theta </math> |
|
<math>~=</math> |
<math>~ e^{i [2\pi L/\lambda + \phi] } \cdot a_0 w ~\mathrm{sinc}(\beta_1) \, . </math> |
We need to explicitly demonstrate that an evaluation of our Focal-Point Expression with <math>~a_j = 1</math>, gives this last sinc-function expression, to within a multiplicative factor of, something like, <math>~j_\mathrm{max}</math>.
Case 2
In our accompanying discussion of the Fourier Series, we have shown that a square wave can be constructed from the expression,
<math>~f(x)</math> |
<math>~=</math> |
<math>~ \frac{c}{L} + \sum_{n=1}^{\infty} \biggl( \frac{2}{n\pi} \biggr) \sin \biggl( \frac{n\pi c}{L} \biggr) \cos \biggl(\frac{n\pi x}{L}\biggr) </math> |
|
<math>~=</math> |
<math>~ \frac{2c}{L}\biggl\{\frac{1}{2} + \sum_{n=1}^{\infty} \mathrm{sinc} \biggl( \frac{n\pi c}{L} \biggr) \cos \biggl(\frac{n\pi x}{L}\biggr) \biggr\} \, . </math> |
Can we make this look like our above, Focal-Point Expression?
Let's start by setting
<math>~Y_j</math> |
<math>~=</math> |
<math>~\frac{j\cdot w}{(j_\mathrm{max}-1)} - \frac{w}{2} \, ,</math> |
for <math>~0 \le j \le (j_\mathrm{max}-1)</math>, in which case,
<math>~\Theta_j</math> |
<math>~\equiv</math> |
<math>~ \frac{2\pi y_1}{\lambda L} \biggl[ \frac{j\cdot w}{(j_\mathrm{max}-1)} - \frac{w}{2} \biggr] = \frac{2\pi y_1}{\lambda L} \biggl[ \frac{j\cdot w}{(j_\mathrm{max}-1)} \biggr] - \frac{2\pi y_1}{\lambda L} \biggl[ \frac{w}{2} \biggr] </math> |
<math>~=</math> |
<math>~ j \biggl[ \frac{2\pi y_1 w}{(j_\mathrm{max}-1) \lambda L} \biggr] - \frac{\pi y_1 w }{\lambda L} = j \cdot \Delta\Theta - \frac{(j_\mathrm{max} -1)}{2} \Delta\Theta \, ,</math> |
where,
<math>~\Delta\Theta \equiv \frac{\pi y_1}{\mathfrak{L}} \, ,</math> and <math>~\mathfrak{L} \equiv \biggl[ \frac{(j_\mathrm{max}-1) \lambda L}{2w} \biggr] \, .</math>
This means that <math>~\Theta_{i} = - \Theta_{( j_\mathrm{max} - 1 - i )}</math>.
The key expression under the summation therefore becomes,
<math>~a_j e^{i \phi_j} \cdot e^{-i \Theta_j } </math> |
<math>~=</math> |
<math>~~a_j e^{i \phi_j} \cdot \biggl[ \cos \biggl( \frac{j\pi y_1}{\mathfrak{L}}- \Theta_0 \biggr) - i \sin \biggl( \frac{j\pi y_1}{\mathfrak{L}}- \Theta_0 \biggr) \biggr] \, ,</math> |
where,
<math>~\Theta_0 \equiv \frac{(j_\mathrm{max} - 1)}{2} \cdot \pi y_1 \biggl[ \frac{2w}{(j_\mathrm{max}-1) \lambda L} \biggr] = \frac{\pi y_1 w}{\lambda L} \, .</math>
See Also
- Updated Table of Contents
- Tohline, J. E., (2008) Computing in Science & Engineering, vol. 10, no. 4, pp. 84-85 — 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 |