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Consider the setup below: enter image description here

In all cases the relationship between $u_o(x_o)$ and $u_f(x_f)$ is given by a Fourier transform. My question is, when is the same true for the relationship between $u_f(x_f)$ and $u_i(x_i)$?

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Strictly speaking, it's not true that in all cases the relationship is an exact Fourier Transform. Since we're dealing with electric-fields, unless the object is exactly one focal length away from the lens all other cases will have a quadratic factor that needs to be dealt with, unless you're interested in the purely incoherent case.

Also, I'm not sure if I understand your question regarding relating uf and uo.

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  • $\begingroup$ Hi thanks for your answer. Yes I agree you will get a quadratic factor (but this is a function of $x_0$ only and can be pulled out of the integral. My point about $u_f(x_f)$ and $u_0(x_0)$ was a typo, its meant to say $u_f(x_f)$ and $u_i(x_i)$ (I will change it) i.e. when can we relate these by a Fourier transform? I think that answer is always since the quadratic contributions cancel out but I am not sure. $\endgroup$ – Quantum spaghettification Feb 14 '16 at 20:11
  • $\begingroup$ Ui and Uf will be related via a Fourier Transform when the propagation distance is sufficiently large such that you're in the Fraunhofer regime. $\endgroup$ – JQK Feb 14 '16 at 20:32

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