Hello all,
I have started reading about diffractive optics. The papers and books I have read till now, all claim that
"It is extremely difficult (impossible?) to find closed-form diffraction solutions
using the Rayleigh–Sommerfeld expression for most apertures. The Fresnel
expression is more tractable, but solutions are still complicated even for simple cases such as a rectangular aperture illuminated by a plane wave."
I want to know why does this happen?
Look at the Sommerfeld problem, which is arguably the simplest conceivable diffraction problem: you consider a two-dimensional system, with a plane wave obliquely incident on a half line. There are 3 different regions in which the behaviour is qualitatively different
1) the reflection region, where the wave is roughly given by a superposition of two waves, one incident, one reflected.
2) the transmission region, in which one wave goes through
3) the shadow region.
The exact solution must account for these 3 different behaviours, and display a continuous transition between all of them. Sommerfeld's solution does exactly that, but it should not be a surprise that, in order to be able to do that, it is fairly complicated.