Resolving Power of a Microscope

Because light bends around edges (diffracts) at any finite aperture, the image of even a perfect point object formed by any real optical instrument is not a point but a small diffraction pattern -- a bright central disc surrounded by faint rings. Two object points that are very close together therefore produce two overlapping diffraction patterns, and whether the instrument can show them as genuinely separate (resolved) or as a single blur depends on how much these patterns overlap. The commonly used Rayleigh criterion states that two such point images are just resolved when the central maximum of one point's diffraction pattern falls exactly on the first minimum of the other's -- any closer, and the two patterns merge into one indistinguishable blob.

For a microscope objective collecting light from an object through a half-angle $\beta$ (the half-angle of the cone of rays entering the objective from the object point), immersed in a medium of refractive index $n$, applying this criterion gives the smallest separation between two object points that the microscope can still show as distinct -- its limit of resolution -- as

where the quantity $n\sin\beta$ is called the numerical aperture (NA) of the objective. The resolving power of the microscope, defined as the reciprocal of this smallest resolvable separation, is therefore

$\text{Resolving power}=\frac{1}{d_{\min}}=\frac{2\,n\sin\beta}{1.22\,\lambda}$ …