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>

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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