
Introduction to Seidel aberrations
The Seidel aberrations are named after Philipp Ludwig von Seidel (1821-1896) who pioneered the mathematical study of the aberrations of imaging systems. The five Seidel aberrations are known as Spherical aberration, Coma, Astigmatism, Field curvature, and Distortion. These are the primary aberrations of an axially symmetric lens system and are calculated by well-known formulas in the case of spherical lens surfaces.
Spherical aberration
This feature of the performance of any lens, namely that rays of light (of one particular color) initially parallel to the lens axis, are bent by the lens so that the rays from the margin of the lens cut the axis in points other than those in which it is cut by rays from inner zones, is the "spherical aberration" of the lens.
- Arthur Cox, Optics - The technique of definition
Spherical aberration is independent of field: the image of a point source is blurred equally no matter where in the field of view it occurs. The blurring can not be removed by focusing: there is no plane where the image of a point source is a point. Spherical aberration deflects rays depending on their distance from the axis in the exit pupil. If the rays from the edge of the exit pupil cross the axis before the paraxial focus the spherical aberration is called "undercorrected", in the opposite case it is called "overcorrected".
Coma
Coma comes into play only when a point away from the lens axis is sending light into the lens...
...The light patch fades away in intensity from the head of the patch like the tail of a comet, hence the name.
- Arthur Cox, Optics - The technique of definition
Coma is a combination of blurring and displacement of the image of a point source, Because the size of the coma spot is proportional to the first power of the distance of the point source from the optical axis, it can be quite important even near the optical axis. In the case of a Newtonian telescope, for example, coma severely limits the field of view. The displacement of the image by coma is troublesome for quantitative measurements of object position.
![]() |
The characteristic shape of coma. The image of the original point (the dot) is flared in the radial direction, either toward or away from the optical axis depending on the sign of the aberration. The shape is a cone subtending an angle of 60°, capped by a circular arc. |
One measure of coma is the so-called "offense against the sine condition".
Astigmatism
The characteristic effect of astigmatism is that one set of lines on a photographic plate is sharply in focus, and at the same time another set, at right angles to the first, is out of focus.
- Arthur Cox, Optics - The technique of definition
Astigmatism leads to a radial blurring of the spot, by an amount depending on the distance of the ideal image point from the optical axis. Because the displacement is radial, radial lines on the object are imaged as radial lines on the original focal plane. Circles centered on the optical axis are also imaged perfectly, but a separate focal distance is required for each circle. The displacement of the plane of best focus is proportional to the area of the circle. A zero radius circle is in focus with the radial lines.
![]() |
The original focal plane showing sharp image of radial lines. The circle is blurred because of the radial blurring introduced by astigmatism. |
![]() |
Focal plane of the circle (and only that circle). The radial lines are blurred because we have introduced defocus to make the circle sharp. |
Note that the aberration called astigmatism is not the astigmatism of the eye that is corrected with eyeglasses. Astigmatism of the eye refers to an asymmetry of the eye such that vertical lines are imaged in a different plane than horizontal lines even at the center of the field of view. This asymmetry is called linear astigmatism.
Field curvature
When there is only curvature of field in the lens a sharp image of a point in the field is formed, but in place of such image points lying on a plane, the flat surface of a plate or film, they lie on a curved surface.
- Arthur Cox, Optics - The technique of definition
The field curvature aberration results in the point images forming on a surface whose radius of curvature is given by the Petzval sum.
Distortion
In dealing with distortion attention is directed to the truth of the definition as reflected in the faithful reproduction of the shape of the object being photographed.
- Arthur Cox, Optics - The technique of definition
Distortion causes a nonuniform displacement of image points without introducing any blurring. This can be imagined as a magnification that depends on distance from the optical axis. If the magnification increases the further the image point is from the optical axis, we say there is pincushion distortion. If the magnification decreases the further the image point is from the optical axis, we say there is barrel distortion. These names come from the appearance of the image of a rectilinear grid.
Definitions
The Seidel aberration coefficients are denoted
S1 | Spherical aberration |
S2 | Coma |
S3 | Astigmatism |
S4 | Field curvature |
S5 | Distortion |
and are defined to be equal to eight times the wavefront distortion associated with the aberration. See, e.g., Kidger's Fundamental Optical Design [Kidger2002]. The Seidel aberrations all have units of length. The usual criterion for diffraction-limited performance is that the total wavefront distortion should be less than one-quarter wave, which means that the sum of the Seidel aberration coefficients should be less than two waves.
The Seidel aberrations have the notable property that the total aberration of the lens is the sum of the aberrations due to the surfaces. The surfaces contribute aberrations independently (however, the contributions depend on the details of the marginal and chief ray at each surface). In general one would like the aberrations of individual surfaces not to be too large because otherwise there will be too delicate a balance in the sum and because higher-order aberrations will proliferate.