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My biology textbook says:

The general rule is that the limit of resolution is about one half the wavelength of the radiation used to view the specimen.

This means anything smaller than half the wavelength won’t be resolved. Why exactly is this the case?

I gathered diffraction has a role to play.

From my research I came across many interesting concepts like Huygen’s Principle which completely changed my view on diffraction and Abbe’s limit which is:

d = 0.66λ/N.A.

I do not pretend to understand the intricacies of the resolution equation but this proves that as wavelength increases the limit of resolution does as well.

Rayleigh’s criterion made me understand the concept of resolution better and was extremely helpful in understanding why longer wavelengths give rise to lower resolution.

enter image description here

But so far I don’t really see or understand why the limit of resolution should be half the wavelength? Am I missing something?

Edit: So I googled common microscope apertures and found that they are mostly in the range of 1.0 to 1.35 and so this would very roughly approximate the constant 0.66/N.A to 0.5. So indeed the maths makes it work out. So I guess perhaps an explanation or derivation of Abbe’s equation would suffice. Can’t seem to find a derivation of it on the web!

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    $\begingroup$ If you find your answer, feel free to answer your own question. $\endgroup$ Sep 10 '20 at 16:41
  • $\begingroup$ I would have thought the resolution was limited to the wavelength, which is basic wave uncertainty. I am interested to see if there is a good explanation for getting twice that resolution. Is this only true for EM imaging or does it apply to sound as well? $\endgroup$
    – Winston
    Sep 10 '20 at 22:02
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The distance between the central maxima and the first minima is d = 0.66λ/N.A. So this is essentially the limit of resolution as an object is just resolved at the point where the central maxima of both diffraction spectra coincides with the first minima of the other:

enter image description here

The lower case sigma in the above diagram is equivalent to d in Abbe’s equation.

If the diffraction spectra overlap too much the distance between the central maximas would be less than the distance between the central maxima and the first minima, therefore the object would be unresolved.

I came upon this fact from this gem of a video: https://youtu.be/sTa-Hn_eisw

Around the 28:30 mark

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