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Let S, the source point, O, the origin, and P, the observation point, are all in a straight line. a is the distance from S to O, b from O to P, and $r_1$ from S to whichever part of the aperture we're interested in and $r_2$ from that to P.

Geometry

$r_1 + r_2 = \sqrt{a^2+x^2+y^2}+\sqrt{b^2+x^2+y^2}$

$=a+b+\frac{x^2+y^2}{2a}+\frac{x^2+y^2}{2b}+$higher order terms

writing

$1/R=1/a+1/b$

optical path = constant +$\frac{x^2+y^2}{2R}$

giving

$\psi_p = \int\int\frac{h(x,y)K(\theta)exp(ik\frac{x^2+y^2}{2R})}{r_1r_2}dxdy$

where K is the obliquity factor and h is the aperture function.

assuming K=1 and variations in $r_1$ and $r_2$ negligible,

$\psi_p \propto \int\int h(x,y) exp(\frac{x^2+y^2}{2R})dxdy$

This is the derivation given to me in my notes, which I'm sort of happy about.

However, I have two questions:

  1. When this is used for an edge, surely both of the assumptions break down and we cannot use Fresnel diffraction anymore? My notes goes in depth about how to use this for an edge diffraction (using Cornu spiral etc)

  2. When do we know whether Fresnel or Fraunhofer diffraction to use? I am given that if $\frac{x^2+y^2}{\lambda}<<R$ then I can use Fraunhofer, but in the edge case surely this is the case since x goes to infinity but the note uses the edge diffraction as an example of a Fresnel diffraction.

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  • $\begingroup$ can u provide a diagram labelling all the points you have defined. There is some ambiguity in the definitions? $\endgroup$ – Rishabh Jain Dec 7 '17 at 14:38
  • $\begingroup$ @RishabhJain done - yeah sorry I should have done it when I originally posted the question! $\endgroup$ – RelativisticDolphin Dec 7 '17 at 14:55

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