User:Tohline/Appendix/CGH/ParallelAperturesConsolidate
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,
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<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,
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<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,
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<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,
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<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,
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<math>~\Theta_j</math> |
<math>~\equiv</math> |
<math>~\biggl(\frac{2\pi y_1 Y_j}{\lambda L} \biggr) \, .</math> |
In a related accompanying derivation titled, Analytic Result, we made the substitution,
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<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,
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<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,
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<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> |
Case #1: 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,
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<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> |
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|
<math>~=</math> |
<math>~ e^{i [2\pi L/\lambda + \phi] } \cdot a_0 w ~\mathrm{sinc}(\beta_1) \, . </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)
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© 2014 - 2021 by Joel E. Tohline |