I am interrested whether one can derive a formula for the point resolution (like Abbe did) of an optical system from the Rayleigh criterion (without the use of small angle approximation i.e. $\rm{sin}(\alpha)=\rm{tan}(\alpha)$ which is not really suitable e.g. for microscopy).
And if so whether formula is directly comparable to the Abbe limit for point (or rather line) resolution.
The Rayleigh criterion is given as: $$\theta_{min}=1.22\frac{\lambda}{D}$$ where $\theta_{min}$ is the smallest resolvable angle, $\lambda$ is the wavelength of the used lightsource and $D$ is the diameter of the used aperture (or of the used lens).
And the Abbe limit is given as:
$$d=\frac{\lambda}{2\,n\,\rm{sin}(\alpha)}=\frac{\lambda}{2\,\rm{NA}}$$
where $d$ is the smallest resolvable distance, $n$ is the refractive index of the medium between the object and the optical system, $\alpha$ is the biggest scattering angle (incident on the optical system) and $\rm{NA}$ is the numerical aperture.