Discussions about resolution started in the astronomy community where there has always been a quest to see greater detail. In parallel, Ernst Abbe lead a similar quest in the microscopy community. Lord Rayleigh and Ernst Abbe came to very similar conclusions in the late 19th century: resolution is limited by an instrument’s aperture.
Before we can go into detail, it is necessary to distinguish between linear resolution and angular resolution. Linear resolution is defined as the minimum distance between objects that can be distinguished. Angular resolution is defined as the minimum angle between objects that can be resolved. For now, we will assume that the things we want to resolve are very bright compared to the surrounding area, like stars on a dark night.
Ernst Abbe’s definition of linear resolution is
R = 0.61 λ / NA,
where λ is the wavelength and NA is the Numerical Aperture of the microscope objective.
Lord Rayleigh’s definition is quite similar:
R = 1.22 λ * F/#.
By comparing the two equations, one can see that they are equivalent if
F/# ≈ 0.5 / NA.
This is a good approximation if NA < 0.25 (F/# > 2), but it degrades at larger apertures. See our Numerical Aperture and F/# page for more details.
Angular resolution is closely related to linear resolution. In fact, one can start with Lord Rayleigh’s definition of linear resolution and convert to angular by using the fact that
F/# = f / D,
where f is the focal length of the telescope and D is the diameter of the aperture. To convert to angular resolution in radians, one simply divides by f, giving
Srad = 1.22 λ / D.
The wavelength is much smaller than the aperture, so it is common to convert from radians to arcseconds (multiply by 3600 * 180/π). At the same time, let’s make use of the fact that the human eye is most sensitive at λ = 0.55 µm. This gives
S = 138 / D, where D is in mm and S is in arcseconds.
An alternative definition is Dawes’ Limit, which gives the minimum resolvable separation between two stars, in arcseconds, for a given diameter of telescope mirror.
S = 4.56 / D, where D is in inches, or
S = 116 / D where D is in mm.
Dawes vs Rayleigh
Who is right? Dawes’ criterion allows for only a 4% dip between the two peaks, while Rayleigh’s allows for a 22% dip. If your telescope is well made and seeing is good (very little atmospheric turbulence, low humidity, and a dark sky) you might be able to resolve at Dawes’ limit, but the Rayleigh criterion is a safer bet.
Figure 1: Airy disc, Dawes’ Limit, and Rayleigh’s criterion compared (click to enlarge)
While Dawes’ Limit, the Rayleigh criterion and Abbe’s criterion are great for high contrast scenes like stars on a dark night sky, most lenses are used for scenes with much lower contrast. If you have a scene more like the above photo of a moth (it’s really there, and not AI generated), you need a different way of measuring resolution.
One way to measure resolution is to determine how much a lens reduces the contrast of a scene. With a little thought, it should be obvious that the reduction in contrast depends on something like how far apart two stars are: When stars are far apart it is trivial to distinguish them, but when they get really close together, they blend into one. Similarly, large text is easier to read from a distance than small text. The way to quantify the size of text or distance between stars is “spatial frequency”, and a function that characterizes a lens in terms of the reduction of contrast as a function of spatial frequency is the Modulation Transfer Function, or MTF.
MTF is covered in detail on this page.
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