Thanks for the comments. The derivation I presented earlier was essentially correct though I admit that the intermediate steps were not "clean" enough because I did it with "bare hand" without making reference to textbooks or online reources. I have revised the notations and some of the details, and used vectors in the derivation instead of moving to spherical coordinates at once. An analytical solution of circular aperture is also included as an example. See the next message.
1) Although diffraction spike is small, shouldn't wavefronts from the whole aperture (which subtend a significant angle) be integrated for each point on the spike? Higher order terms in expanding the phase term seem cannot be totally ignored base on a small rho.
We have indeed integrated da' over the whole wavefront for each point on spikes. Note that (x,y) is fixed on the focal plane but (x',y') are the integration variables which run over the aperture.
No, the higher order terms would not add up to a significant value. The reason is that when the higher order terms in the phase << 1, Exp(i higher order term) ~ 1,
Integrate[ Exp[ i k (useful terms) + i k (higher order term)]]
= Integrate[ Exp[ i k (useful terms)] x Exp [i k (higher order term)] ]
Thus the integrand is only modulated by a function very close to 1.
2) The final result looks like the form of Fresnel diffraction, the Fourier transform is modulated by an R-dependent phase term. This seems to show that Franuhofer approximation doesn't apply here.
No. it is not. The phase Exp(i kR) is an arbitrary constant phase which is independent of the position on the focal plane, it is not a modulation. In fact R ~ f, the focal length of the objective.
But from practical point of view, can we simply ignore the phase term (including possibly higher order phase terms) in final result because difference in R is so small within the extent of recordable diffraction spike?
In handling problems of wave optics, it is always the variation of the phase factor that determines the result. The reason is that k r = 2 Pi r / lamda is extremely senitive to very small changes in r because lamda ~ 10^-7 m is very small, i.e. delta r / lamda could be very large even if delta r is small in the ordinary sense. Thus the phase varies very rapidly over the domain of integration, leading to many cancellation and that is why wave optics problems are quite complicated.
By the way, can we simply do FT in spherical coordinates (centre on each image point) on the converging wavefront to get the result without going through those approximations?
The phase has to vary linearly with x', y' in order to identify the result with the FT. In comparsion with the rapidly changing phase, other modulation in the integrand is not important at all so many approximations can be made. In fact wave optics problem always involve these kinds of approximation.
(Just throw out some thoughts to those interested to think. No need to try reply in much mathematical detail for the sake of these questions.
Revised the derivation as follows anyway. Perhaps there are "cleaner" treatment in Fourier optics textbooks, but I just do it in that way that I can think of