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 $\delta/l_1>\lambda/d$ .
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.