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The Aberrations of a Lens System

The early workers in optics must have been aware of the distortions in the image that a lens or lens system could produce.
However it was not until the German mathematician Ludwig von Seidel worked out
and named the various distortions that the phenomenon of lens aberration
was systematized.

In optics there are relationships that involve the sine function of some angle, say sin(α). The sine and cost functions
can be expressed as infinite series; i.e.,

where n! stands for n factorial, n*(n-1)*(n-2)…3*2*1.

The approximation that sin(α) is equal to α is called the first order approximation and corresponds to geometric
optic theory. If this approximation were exact there would be no distortion in the images produced by lenses. Ludwig von
Seidel used the approximation that sin(α) is equal to α−α^{3}/3!, which is called a third order
approximation.

Seidel classified the aberrations of a spherical lens into five categories:

Spherical Aberration:

Coma:

Astigmatism:

Curvature of Field:

Distortion:

To Seidel's five aberration there is usually added a sixth, chromatic aberration, which arises from an entirely different source
than does Seidel's five.