Dodgy step in the Far field approximation

Loro

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!

sam_bell

There might be an assumption that the aperture is small compared to the image space (x0,y0). Considering this is a far-field approximation, that tends to make sense.

Loro

Thanks,

It does, but then we couldn't neglect [itex] x_0^2 + y_0^2 [/itex]

mfb

Those terms do not depend on the integration variables, it is possible to pull them out of the integral. They give a prefactor, which might be irrelevant, or accounted for in some other way.

Loro

They're just a part of a phase! Got it. Thanks :)

Loro

Hold on, but wouldn't that mean that Fraunhofer approximation works best away from the optical axis - where we're allowed to say: [itex] x_0 , y_0 >> x , y [/itex] ? (I don't think that's the case)