Rayleigh criterion: angular or spatial resolution?

Question

The Rayleigh criterion sets the fundamental limit for what features an optical system can resolve. Formally, the criterion is defined for some wavelength $\lambda$ in the case of a uniformly-illuminated aperture (consider one that is circular) of diameter $d$. An airy disk is formed in the far field (or at the focus of a lens), with first intensity minimum described by sin($\theta_{min}$)$ = 1.22\frac{\lambda}{d}$, where $\theta_{min}$ is the angle from the optic axis to the radial location of the intensity distribution in the observation plane, from the perspective of aperture center. Two adjacent apertures can just be resolved if the peak of Airy pattern one falls over the minimum of the other. The wiki article outlines this nicely: https://en.wikipedia.org/wiki/Airy_disk.

The criterion for resolution is very general: we are only talking about an aperture thickness and wavelength. However, we can define an optimal spatial resolution too: sin($\theta_{min}$) $= x_{min}/R$, where $R$ is the distance from the aperture center to a point on the observation plane, and $x_{min}$ is a radial distance in the observation plane from the optic axis on that plane. So our fundamental resolution limit is governed by $\theta_{min}$, but we can have a varying $x_{min}$ depending on our distance to the observation plane, $R$. What then is the physical significance of $x_{min}$, if the Airy disk is the same (when plotted against $\theta$) regardless of $R$ (assuming far field)? Furthermore, when I use a lens of focal length $f$, we can say $R=f$, but how does a lens with higher optical power improve system resolution when the fundamental limit has been set in terms of the angular resolution?

Answer 1

The spatial resolution, $x_{min} = R\sin\theta_{min}$ is the radius of the Airy disk projected on the photo sensor (or film) of a camera. If the size of a pixel is bigger than $x_{min}$, it will be the resolution bottleneck (rather than the fundamental limit).

Note that there are also other factors that may reduce the resolution in respect to the diffraction/fundamental limit. Among the most important ones are a) aberrations, resulting mainly from the imperfection/quality of the optical system, and b) refraction of the light in the atmosphere (perhaps less crucial for commercial cameras, but certainly critical for telescopes).

Finally, the principal reason why the fundamental limit is given in angular rather than length units is because the angular size of a very large but distance object may easily be smaller than the angular size of a small but nearby one. E.g., stars are huge objects that typically cannot be resolved even by powerful telescopes, whereas taking detailed photos of a bee is possible with a hand-held camera or even a smartphone.