Actually it was just stipulated above that a ≤≤ b, so that chi IS small for r ≤ a. But of course not for r ≥≥ a, could never be if you're integrating to ∞. I think the book this was copied from might have said "if .... chi at r=a IS negligible....." (en)
Actually it can be solved analytically! This is the case for the on-axis amplitude which is easier. (en)
Going to polar coordinates, from what?? This is an example of where someone switched to a different textbook without reading over the whole thing. For the paraxial regime a simpler construction for above would assume collimated light so g=∞; in fact this is what the result written below appears to have assumed: a simpler model. (en)
