The Fresnel diffraction integral is:
[itex] A(x_0 , y_0 ) = \frac{i e^{ikz}}{λz} \int \int dx dy A( x , y ) e^{\frac{ik}{2z} [(x  x_0)^2 + (y  y_0)^2]} [/itex]
When we want to obtain the Fraunhofer diffraction integral from here, we need to somehow convert it to:
[itex] A(x_0 , y_0 ) = \frac{i e^{ikz}}{λz} \int \int dx dy A( x , y ) e^{\frac{+ik}{z} [x x_0 + y y_0]} [/itex]
So I thought we should do it as follows:
[itex] \frac{ik}{2z} [(x  x_0)^2 + (y  y_0)^2] = \frac{ik}{2z} [x^2 + x_0^2 + y^2 + y_0^2  2x x_0  2y y_0 ] [/itex]
And then it seems that we should neglect: [itex] x^2 + x_0^2 + y^2 + y_0^2 [/itex] since they're all much smaller than z.
Then we get the correct solution.
But I don't see why we could do that, and leave out the [itex]  2x x_0  2y y_0 [/itex]. After all they are of the same order... Please help!
