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,

<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,

<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>

In a related accompanying derivation titled, Analytic Result, we made the substitution,

<math>~a_j e^{i \phi_j} </math>

<math>~\rightarrow</math>

<math>~a_0(Y) dY \, ,</math>

and changed the summation to an integration, obtaining,

<math>~A(y_1)</math>

<math>~\approx</math>

<math>~ e^{i 2\pi L/\lambda } \int a_0(Y) e^{-i \Theta } dY \, , </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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