User:Tohline/Appendix/CGH/ParallelApertures2D
CGH: 2D Rectangular Appertures that are Parallel to the Image Screen
This chapter is intended primarily to replicate §I.B from the online class notes — see also an updated Table of Contents — that I developed in conjunction with a course that I taught in 1999 on the topic of Computer Generated Holography (CGH) for a subset of LSU physics majors who were interested in computational science. This discussion parallels the somewhat more detailed one presented in §I.A on the one-dimensional aperture oriented parallel to the image screen.
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Utility of FFT Techniques
Consider the amplitude (and phase) of light that is incident at a location (x1, y1) on an image screen that is located a distance Z from a rectangular aperture of width w and height h. By analogy with our accompanying discussion in the context of 1D apertures, the complex number, A, representing the light amplitude and phase at (x1, y1) will be,
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<math>~A(x_1, y_1)</math> |
<math>~=</math> |
<math>~ \sum_j \sum_k a_{jk} e^{i(2\pi D_{jk} /\lambda + \phi_{jk})} \, , </math> |
where, here, the summations are taken over all "j,k" elements of light across the entire 2D aperture, and now the distance Djk is given by the expression,
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<math>~D^2_{jk}</math> |
<math>~\equiv</math> |
<math>~ (X_j - x_1)^2 + (Y_k - y_1)^2 + Z^2 </math> |
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<math>~=</math> |
<math>~ Z^2 + y_1^2 - 2y_1 Y_k + Y_k^2 + x_1^2 - 2x_1 X_j + X_j^2 </math> |
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<math>~=</math> |
<math>~ L^2 \biggl[1 - \frac{2(x_1 X_j + y_1 Y_k ) }{L^2} + \frac{X_j^2 + Y_k^2}{L^2} \biggr] \, , </math> |
and,
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<math>~L</math> |
<math>~\equiv</math> |
<math>~ [Z^2 + y_1^2 + x_1^2]^{1 / 2} \, . </math> |
If <math>~|X_j/L| \ll 1</math> and <math>~|Y_k/L| \ll 1</math> we can drop the quadratic terms in favor of the linear ones in the expression for Djk and deduce that,
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<math>~D_{jk}</math> |
<math>~\approx</math> |
<math>~ [Z^2 + y_1^2 + x_1^2]^{1 / 2} \, . </math> |
See Also
- Updated Table of Contents
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© 2014 - 2021 by Joel E. Tohline |