The resolving power of a
lens is ultimately limited by
diffraction (see
Point Spread Function,
Airy disc). The lens'
aperture is analogous to a two-dimensional version of the
single-slit experiment.
Light passing through the lens
interferes with itself creating a ring-shaped diffraction pattern, known as the
Airy pattern, if the
phase of the transmitted light is taken to be spherical over the exit aperture. The result is a
blurring of the image. An
empirical diffraction limit is given by the
Rayleigh criterion invented by
Lord Rayleigh:
where
θ is the angular resolution,
λ is the
wavelength of light,and
D is the
diameter of the lens.
The factor 1.22 is derived from a calculation of the position of the first dark ring surrounding the central
Airy disc of the
diffraction pattern. If one considers diffraction through a circular aperture, then the calculation involves a
Bessel function -- 1.220 is approximately the first zero of the Bessel function of the first kind, of order one (i.e.
J1), divided by
π (3.14159). This factor is used to approximate the ability of the human
eye to distinguish two separate point sources depending on the
overlap of their Airy discs: the minimum of one point source is located at the maximum of the other. Modern
telescopes and
microscopes with video
sensors may be slightly better than the human eye in their ability to discern overlap of Airy discs. Thus it is worth bearing in mind that the Rayleigh criterion is an empirical estimate of resolution based on the assumption of a human observer, and may slightly underestimate the resolving power of a particular optical train. For specialized imaging, foreknowledge of some characteristics of the image can also improve on technical resolution limits through computerized
image processing.
For an ideal lens of
focal length f, the Rayleigh criterion yields a minimum
spatial resolution, Δ
l:

.
This is the size of smallest object that the lens can resolve, and also the
radius of the smallest spot that a
collimated beam of
light can be focused to. The size is proportional to wavelength,
λ, and thus, for example,
blue light can be focused to a smaller spot than
red light.