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As I was reading through Luigi Picasso's Lectures in QM, I came across this paragraph where he explains the reason behind the resolution power of the optical instruments. He writes:

Let us now examine how the microscope works. First of all, let us recall that any optical instrument has a finite resolution power: this means that, given two points P1 and P2 a distance δ apart from each other (Fig. 3.9), there exists a δmin such that, for δ < δmin the instrument (lens) is not able to distinguish them. Why does this happen? A lens ‘cuts’ a portion of the wave front of the incident light, the size of this portion being equal to that of the lens – much as a hole in a screen – and so gives rise to the phenomenon of diffraction: we could say a lens is a “hole of glass in the vacuum”. So the images of points are not points, but diffraction patterns produced by the lens, i.e. small spots. Since, due to diffraction, any ray incident on the lens gives rise to a cone of rays, whose aperture is sin φ $\simeq$ λ/d (Fig. 3.9), d being the diameter of the lens, it follows that the image of a point is a spot Fig. 3.9 whose linear size is $l_2 \sin\phi=l_2\lambda /d$ . The images of points P1 and P2 will be distinguished if the relative spots do not overlap: this happens if $l_2\lambda /d < \delta \ l_2/l_1$, i.e. if $\delta/l_1>\lambda/d$ .

enter image description here

I am unable to understand how the lens acts as a hole in vacuum and how the further math is developed in the above paragraph.

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  • $\begingroup$ The figure is a bit confusing, diagramming a wave effect using a ray model. $\endgroup$
    – John Doty
    Sep 15, 2023 at 13:20
  • $\begingroup$ Do you know the interference pattern of a hole, than in the picture the lens is very large, imagine it to be really very small or with some hle before it, than both Points would have a interference pattern, id the two maxima overlap you can not distinguish between the two points. The calculation take the diameter of lens equals d, so you can not see it in the illustration. $\endgroup$
    – trula
    Sep 15, 2023 at 13:27

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He is just making the analogy between the diffraction of an aperture and the diffraction due to a finite sized lens. The former is assumed to be already familiar for the reader. I guess that it is not the case for you, which is why the explanation is not particularly enlightening.

Perhaps start by looking into more basic examples of diffraction like through an aperture. In practice, with circular symmetry, patterns such as the Airy disk is especially relevant.

Hope this helps.

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