Within what range of defocus from the object/image plane does the Fraunhoffer approximation hold?

The Fraunhoffer apporximation of diffraction is said to hold for an image "at infinity," or to be quantitative, for any observation point $R >> \frac{a^2}{\lambda}$. But it is said that, because a lens esentially brings the point "at infinity" to the object/image plane, the Fraunhoffer approximation is also valid at the image plane, and indeed that is how the Abbe diffraction limit is derived.

However, if we want to calculate the effect of defocus (i.e. multiply the incoming wave by a phase term quadratic in distance from optical axis), this will take us away from the image plane, and the Fraunhoffer approximation will thus get worse.

Within what limits of defocus can we still trust the Fraunhoffer approximation? I know it should depend on the Numerical Apperture and focal length, but I can't come up with a good expression.

Of course, it all depends on what you are after, and what you can tolerate. But you can get an estimate by asking a similar question: over what range of defocus does the spot size not change appreciably? Or equivalently, what is length of the near field of the focused spot?

That distance can be estimated (order of magnitude!) in the following way. One picture would be worth more than all these words, but all I have right now are words. Draw a picture.

Consider the ideal spot size at the position of idea focus (Airy disk). Imagine a crude model for radiation in the near field: the rays from the disk extend perpendicularly from the disk, as if casting a shadow of a hole in a screen. The diameter of the spot is roughly $\lambda\displaystyle\frac{f}{D}$. Note that there are constant factors (1.22, $\pi$, etc) that I'm ignoring because I'm interested only in the order of magnitude

However, we know that the light is diverging in the far field due to the focusing action of the lens, and the angle of divergence is approximately the numerical aperture of the lens, or $D/f$ where $D$ is the diameter of the lens and $f$ is the focal length. Extend the far-field rays back to the focal plane so that they diverge from a point at the focal point.

We are certainly outside the range of validity of the Fraunhoffer approximation when the diverging far field rays cross the near field "shadow" rays. Let $z_0$ be the distance from the focal plane to the plane where the rays cross. The crossing occurs when $$z_0\frac{D}{f}= \lambda\frac{f}{D}$$ or $$z_0=\lambda\frac{f^2}{D^2}$$

Careful analyses put constants in this formula. The "Airy factor" of $1.22$ needs to be considered, as does a more careful definition of numerical aperture, and considerations of radius vs. diameter. For example, Born and Wolf has $$z_0=\frac{\lambda}{2\pi}\frac{f^2}{D^2} \,\,\,\,\mbox{[see correction below]}$$

You want your defocus to be much less than $z_0$.

Edit after comment

Born and Wolf does not address your question directly ... at least I'm not aware of it if it does. It does present a discussion of Fresnel diffraction and the field in the vicinity of a focal spot. This is in Sec 8.8 of the seventh (expanded) edition.

When I wrote the answer above I was depending on memory. I have the book now, and I see that I made an error: The Born and Wolf expression is in terms of the radius, not the diameter. The correct Born and Wolf expression is $$z_0 = \frac{\lambda}{2\pi}\frac{f^2}{a^2}$$ where $a$ is the radius of the aperture. See formulas (8) and (27) in Sec 8.8. And stare at Figure 8.41 for a while.

See also the extensive set of papers on the Extended Nijboer-Zernike approach to the defocus (and aberration) problem. Especially perhaps section 3. of this one.

• Thanks! Would you mind citing the page in Born and Wolf that discusses this? May 16, 2016 at 2:25
• I've provided a correction and some references in an edit to my answer. May 16, 2016 at 14:29