Maximum resolution of lens with infinite aperture

Because the resolution of an objective is limited by the diffraction of its circular aperture, I don't understand why an infinite lens can not have an infinitely good resolution power.

People use Abbe relation to compute the minimum resolvable distance $$p_m$$ of an objective as a function of its numerical aperture NA:

$$p_m = \frac{1.22 \lambda}{NA} = \frac{1.22 \lambda}{n \sin(\theta_m)}$$

When the aperture goes to infinity, $$\sin \theta_m$$ goes to 1 and thus the minimum resolvable distance is : $$p_m = \frac{1.22 \lambda}{n}$$

Now, the Airy disk created by the diffraction by a circular aperture of diameter $$D$$ has a radius defined by the zero of the $$J_1$$ Bessel function : $$r_{Airy} = \frac{1.22 \lambda z}{D}$$, where $$z$$ is the distance of the observation plane.

So the radius of the Airy disk goes to zero as the circular aperture goes to infinity. Why isn't it the case for the minimum resolvable distance when the lens diameter goes to infinity ?

• How would you make an infinite lens? – mike stone Nov 4 '20 at 15:07
• more importantly, what are you looking at that needs an infinite lens – Topcode Nov 4 '20 at 16:00
• Because I don't understand the derivation of the minimum resolvable distance. – magtweezers Nov 5 '20 at 18:12

Besides a lens having $$D=\infty$$ is a rather difficult engineering task, as @mikestone indicated above, the Airy formula must break down even before we get to that challenging senior management goal...
Think of what the reason is for the finite discriminating ability be proportional to $$\lambda/D$$. While the formula expresses that the image of a single point source spreads out with an oscillating but decreasing tail it also assumes that when looking at two sources they are independent from each other and this is where the formula breaks down for large enough aperture. When sources are within a wavelength they become electromagnetically coupled and their combined image amplitude becomes strongly dependent on their relative phases. There are so-called super-resolution methods that go below Rayleigh's rule. These methods solve for the underlying interaction between the sources to be discriminated but the result is also strongly signal to noise ratio dependent while Rayleigh's rule ignores noise.