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Most sources I've looked at treat far field diffraction and near field diffraction as separate cases, with the former using the Fraunhofer approximation and the latter the Fresnel approximation. In particular, e.g. the wikipedia articles on the two cases specify incommensurable conditions for the two, namely

$F << 1$ for Fraunhofer approximation

$F >> 1$ and at the same time $F \theta^2 / 4 << 1$ for Fresnel approximation

where $F = a^2/L \lambda $ is the Fresnel number and $\theta$ is the maximal diffraction angle considered.

Loosely, my understanding of the two approximations is that:

Fraunhofer approximation: the observation point is far enough away from the aperture that all paths (lines) going from the aperture to a given observation point are assumed parallel. To get the diffraction pattern we just have to look at the phase difference due to different lengths of parallel lines

Fresnel approximation: the observation point is closer, so that paths from different parts of the aperture to a given observation point are no longer parallel. To get the diffraction pattern, we have to take this into account, and specifically we ignore any phase terms of 3rd or higher order in the Taylor series. Thus the Fresnel approximation is still limited to small-ish angle, and not too close to the aperture.

Based on that understanding, is it correct to say that, actually, the Fresnel approximation is also valid in the far field? Or in other words, that the Fraunhofer approximation is a subset of the Fresnel approximation? Or am I missing something else that makes the two cases significantly different? And finally, if I wanted to look at, say, $F = 1$, should I use the Fresnel approximation?

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In case someone else have this question, I finally found that Goodman (Introduction to Fourier Optics, ISBN 9780974707723) explicitly states that the Fresnel approximation is indeed valid in the far-field.

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  • $\begingroup$ Can you tell us where exactly in Goodman you found this (chapter & section)? Or elaborate more about what you have found? It would be highly appreciated. $\endgroup$ – Quantumwhisp Nov 1 '16 at 13:31
  • $\begingroup$ been a while since I've looked at this, but from a very quick glance I think I was referring to section 4.3. A couple of sentences in it goes: "If in addition to the Fresnel approximation the stronger (Fraunhofer) approximation is satisfied, then...", so essentially saying for Fraunhofer approximation you need Fresnel approximation and something more to be valid. $\endgroup$ – funklute Nov 1 '16 at 14:11

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