Spherical aberration

Correction

In lens systems, aberrations can be minimized using combinations of convex and concave lenses, or by using aspheric lenses or aplanatic lenses.

Lens systems with aberration correction are usually designed by numerical ray tracing. For simple designs, one can sometimes analytically calculate parameters that minimize spherical aberration. For example, in a design consisting of a single lens with spherical surfaces and a given object distance o, image distance i, and refractive index n, one can minimize spherical aberration by adjusting the radii of curvature R₁ and R₂ of the front and back surfaces of the lens such that $${\displaystyle \frac{R_1+R_2}{R_1-R_2}=\frac{2\left(n^2-1\right)}{n+2}\left(\frac{i+o}{i-o}\right),}$$ where the signs in this formula follow the Cartesian sign convention, in which a radius of curvature is positive if the center of curvature is to the right of the surface and negative if it is to the left. Similarly the object and image distances are positive if the object or image is to the right of the lens and negative if they are to the left.

For small telescopes using spherical mirrors with focal ratios shorter than f/10, light from a distant point source (such as a star) is not all focused at the same point. Particularly, light striking the inner part of the mirror focuses farther from the mirror than light striking the outer part. As a result, the image cannot be focused as sharply as if the aberration were not present. Because of spherical aberration, telescopes with focal ratio less than f/10 are usually made with non-spherical mirrors or with correcting lenses.

Spherical aberration can be eliminated by making lenses with an aspheric surface. Descartes showed that lenses whose surfaces are well-chosen Cartesian ovals (revolved around the central symmetry axis) can perfectly image light from a point on the axis or from infinity in the direction of the axis. Such a design yields completely aberration-free focusing of light from a distant source.

In 2018, researchers found a closed formula for a lens surface that eliminates spherical aberration. Their equation can be applied to specify a shape for one surface of a lens, where the other surface has any given shape.

Estimation of the aberrated spot diameter

Many ways to estimate the diameter of the focused spot due to spherical aberration are based on ray optics. Ray optics, however, does not consider that light is an electromagnetic wave. Therefore, the results can be wrong due to interference effects arisen from the wave nature of light.

Coddington notation

A rather simple formalism based on ray optics, which holds for thin lenses only, is the Coddington notation. In the following, n is the lens's refractive index, o is the object distance, i is the image distance, h is the distance from the optical axis at which the outermost ray enters the lens, R₁ is the first lens radius, R₂ is the second lens radius, and f is the lens's focal length. The distance h can be understood as half of the clear aperture.

By using the Coddington factors for shape, s, and position, p, $${\displaystyle \begin{array}{cc}s& =\frac{R_2+R_1}{R_2-R_1}\\ p& =\frac{i-o}{i+o},\end{array}}$$ one can write the longitudinal spherical aberration as $${\displaystyle \mathrm{L}\mathrm{S}\mathrm{A}=\frac{1}{8n(n-1)}\cdot \frac{h^2{i}^2}{f^3}\left(\frac{n+2}{n-1}{s}^2+2(2n+2)sp+(3n+2)(n-1{)}^2{p}^2+\frac{n^3}{n-1}\right).}$$

If the focal length f is very much larger than the longitudinal spherical aberration LSA, then the transverse spherical aberration, TSA, which corresponds to the diameter of the focal spot, is given by $${\displaystyle \mathrm{T}\mathrm{S}\mathrm{A}=\frac{h}{i}\mathrm{L}\mathrm{S}\mathrm{A}.}$$