# Spherical aberration

Spherical aberration is a type of aberration found in optical systems that use elements with spherical surfaces. Lenses and curved mirrors are most often made with surfaces that are spherical, because this shape is easier to form than non-spherical curved surfaces. Light rays that strike a spherical surface off-centre are refracted or reflected more or less than those that strike close to the centre. This deviation reduces the quality of images produced by optical systems.

On top is a depiction of a perfect lens without spherical aberration: all incoming rays are focused in the focal point.

The bottom example depicts a real lens with spherical surfaces, which produces spherical aberration: The different rays do not meet after the lens in one focal point. The further the rays are from the optical axis, the closer to the lens they intersect the optical axis (positive spherical aberration).

(Drawing is exaggerated.)
Spherical aberration of collimated light incident on a concave spherical mirror.

## Overview

A spherical lens has an aplanatic point (i.e., no spherical aberration) only at a radius that equals the radius of the sphere divided by the index of refraction of the lens material. A typical value of refractive index for crown glass is 1.5 (see list), which indicates that only about 43% of the area (67% of diameter) of a spherical lens is useful. It is often considered to be an imperfection of telescopes and other instruments which makes their focusing less than ideal due to the spherical shape of lenses and mirrors. This is an important effect, because spherical shapes are much easier to produce than aspherical ones. In many cases, it is cheaper to use multiple spherical elements to compensate for spherical aberration than it is to use a single aspheric lens.

"Positive" spherical aberration means peripheral rays are bent too much. "Negative" spherical aberration means peripheral rays are not bent enough.

The effect is proportional to the fourth power of the diameter and inversely proportional to the third power of the focal length, so it is much more pronounced at short focal ratios, i.e., "fast" lenses.

Longitudinal sections through a focused beam with negative (top row), zero (middle row), and positive spherical aberration (bottom row). The lens is to the left.

## 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 ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$ of the front and back surfaces of the lens such that

${\displaystyle {\frac {R_{1}+R_{2}}{R_{1}-R_{2}}}={\frac {2\left(n^{2}-1\right)}{n+2}}\left({\frac {i+o}{i-o}}\right).}$ The signs of the radii follow the Cartesian sign convention.
A point source as imaged by a system with negative (top row), zero (middle row), and positive spherical aberration (bottom row). The middle column shows the focused image, columns to the left shows defocusing toward the inside, and columns to the right show defocusing toward the outside.

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.[1]

In 2018, Rafael G. González-Acuña and Héctor A. Chaparro-Romo, graduate students at the National Autonomous University of Mexico and the Monterrey Institute of Technology and Higher Education in Mexico, found a closed formula for a lens surface that eliminates spherical aberration.[2][3][4] 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.

### Coddington notation

A rather simple formalism based on ray optics, which holds for thin lenses only, is the Coddington notation.[5] In the following, n is the lens' 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, ${\displaystyle R_{1}}$ is the first lens radius, ${\displaystyle R_{2}}$ is the second lens radius, and f is the lens' 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{aligned}s&={\frac {R_{2}+R_{1}}{R_{2}-R_{1}}}\\[8pt]p&={\frac {i-o}{i+o}},\end{aligned}}}

one can write the longitudinal spherical aberration as [5]

${\displaystyle \mathrm {LSA} ={\frac {1}{8n(n-1)}}\cdot {\frac {h^{2}i^{2}}{f^{3}}}\left({\frac {n+2}{n-1}}s^{2}+2(2n+2)sp+(3n+2)(n-1)^{2}p^{2}+{\frac {n^{3}}{n-1}}\right)}$

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

${\displaystyle \mathrm {TSA} ={\frac {h}{i}}\mathrm {LSA} }$