User:Tohline/Appendix/CGH/ParallelAperturesConsolidate
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CGH: Consolidate Expressions Regarding Parallel Apertures
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Onedimensional 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,



and, assuming that for all , deduce that,



where,



Note that is formally a function of , but in most of what follows it will be reasonable to assume, . Notice, as well, that this last approximate expression for the (complex) amplitude at the image screen may be rewritten in the form that will be referred to as our,
FocalPoint Expression  




where,



Case 1
In a related accompanying derivation titled, Analytic Result, we made the substitution,



where,



and changed the summation to an integration, obtaining,



If we assume that both and are independent of position along the aperture, and that the aperture — and, hence the integration — extends from to , we have shown that this last expression can be evaluated analytically to give,






We need to explicitly demonstrate that an evaluation of our FocalPoint Expression with , gives this last sincfunction expression, to within a multiplicative factor of, something like, .
Case 2
In our accompanying discussion of the Fourier Series, we have shown that a square wave can be constructed from the expression,






Can we make this look like our above, FocalPoint Expression?
Let's start by setting



for , in which case,








where,
and
This means that .
The key expression under the summation therefore becomes,



where,
Now, what is the argument of the sinc function? By default, it needs to be something along the lines of,



Then, as varies from to , the argument goes from to . In an effort to make the function exhibit reflection symmetry as we move from one side of the aperture to the next, let's subtract half of this upper limit; that is, let's modify the argument of the sinc function to read,



This means that in our above, FocalPoint Expression we want to set,



This therefore gives the following,
FocalPoint Expression for a Square Wave  




This exhibits a very desirable feature: Both the sinc function and the sine function — and, hence, also their product — have reflection symmetry about the summation index, . As a result, if the overall phase factor, , behaves in an appropriately simple way — for example, if it is zero everywhere — then under the summation the sine term will sum to zero and leave only the desired — and real — product, . Try this out in Excel to see if it works!
This could use a little more manipulation. Let's define the alternate summation index,



in which case we can write,












Finally, recalling that,



let's set …






As a result, we have,



Therefore, a clean square wave will appear only if .
See Also
 Updated Table of Contents
 Tohline, J. E., (2008) Computing in Science & Engineering, vol. 10, no. 4, pp. 8485 — 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)
© 2014  2021 by Joel E. Tohline 