User:Tohline/Appendix/CGH/ParallelApertures2D
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CGH: 2D Rectangular Apertures 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 onedimensional 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 (x_{1}, y_{1}) 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 (x_{1}, y_{1}) will be,



where, here, the summations are taken over all "j,k" elements of light across the entire 2D aperture, and now the distance D_{jk} is given by the expression,









and,



If and we can drop the quadratic terms in favor of the linear ones in the expression for D_{jk} and deduce that,






Hence, the doublesummation expression for the amplitude at screen location (x_{1}, y_{1}) becomes,



where,
.
When written in this form, it should be apparent why discrete Fourier transform techniques — specifically, 2DFFT techniques — are useful tools for evaluation of the complex amplitude, A(x_{1}, y_{1}).
NOTE: If x_{1} and/or y_{1} are ever comparable in size to Z — which may be the case for large apertures or for apertures tilted by nearly 90° to the image screen — then the variation of L with image screen position cannot be ignored and, accordingly, the coefficient A_{0} cannot be moved outside of the double summation.
Analytic Results
Now, as in §I.A, we would like to consider the case where the rectangular aperture is divided into an infinite number of divisions in both the X and Y dimensions and convert the summations in this last expression into integrals whose limits in both directions are given by the edges of the aperture. If we specifically consider the case where the aperture is assumed to be uniformly bright (i.e., a_{jk} = a_{0} dX DY, and a_{0} is the brightness per unit area), and the phase at all locations on the aperture, we can write,






Both of these integrals can be completed in the same fashion as described in §I.A for the 1D slit, giving,



where,












It is worth noting that this derivation closely parallels the one presented in §8.5.1 (p. 393) of Born & Wolf (1980) (see the reference below). Specifically, our doubleintegral expression is identical to the "Fraunhofer diffraction integral" written down by Born & Wolf at the beginning of their §8.5.1; and if we follow Born & Wolf's lead and position the origin of our coordinate system at the center of the rectangle, then and the intensity at point that is given by their equation (1) precisely matches the expression for the square of the amplitude, , that is obtained from our last expression.
See Also
 Updated Table of Contents
 Born, M. and Wolf, E. (1980) Principles of Optics, 3^{rd} Edition. New York: Pergamon Press. [See especially their §§8.5 and 8.10.] A link to the 6^{th} edition of this book can be found here.
© 2014  2020 by Joel E. Tohline 