Resolution limit [Instrumental optics and microscopy]

With the objective \(50\times /0.80\) (i.e. of object-side numerical aperture \(0.80\)), the object-space spatial frequency cutoff for a coherent illumination and a wavelength of \(0.55~\mu m\) (in the middle of the green spectrum) is \(0.80/(0.55~\mu m) \simeq 1450~mm^{-1}\). For a completely incoherent illumination it reaches the double, i.e. \(\sim 2900~mm^{-1}\). Sinusoidal test patterns that would allow to directly test those limits should therefore have periods of respectively \(\sim 0.70~\mu m\) et \(\sim 0.35~\mu m\), which is totally unfeasible. For a completely incoherent illumination, the resolution limit in the sense of Rayleigh criterion is, by definition, equal to the radius of the Airy pattern in the intermediate image given by the objective and the tube lens (since the eyepiece doesn't bring any additional limitation):

r ’ = \frac{1,22 λ}{(2 {NA}_{im})}

where \(NA_{im}\) is the image-side numerical aperture of the system objective \(+\) tube lens. To bring this value back to the object space, one needs to use the Abbe sine condition applied to the optical conjugation object/intermediate image of the system objective \(+\) tube lens, which is supposed to be aplanetic; if \(NA_{obj}\) is the object-side numerical aperture of this system:

\frac{{NA}_{obj}}{{NA}_{im}} = | g_{y} |

and we obtain in the object space:

r = \frac{1,22 λ}{2 {NA}_{obj}}

The numerical application gives \(0.42~\mu m\) for the objective \(50×/0.80\) in the visible, in good agreement with the indication in the manufacturer datasheet.