I tried to work through this lecture and first of all, I'm curious, what is the reason for this:
We would like to use Green’s theorem with $v(x) = G(x − x0)$
As far as I understand, Green's theorem would also work with other fundamental solutions, for example a wavefront traveling in a single direction. Where would things go wrong when another solution is tried here?
Only thing I can imagine is that later it would be harder to let $R$ go to infinity and apply Sommerfeld's radiation condition. Is this the reason?
Apart from that, someone did someone a simulation of diffraction patterns here and the math looks way easier, it's basically just taking the FT of the field across the aperture plane and then propagating it along the $z$-axis by multiplying with $e^{-kzi}$. Where did the all the difficulties, like applying Green's theorem, dealing with singularities and discussing convergence of the integral for $R\to\infty$, disappear in this approach? Is the Helmholtz equation enough to guarantee a unique solution in $\mathbf{R}^3$ when the field is known along some plane?