The five basic types of aberration which are due to the geometry of
lenses or mirrors, and which are applicable to systems dealing with
monochromatic light, are known as *Seidel*
aberrations, from an 1857 paper by Ludwig von Seidel.
These are the aberrations that become
evident in *third-order optics*, also known as Seidel optics.

As we know,

3 5 7 9 11 13 x x x x x x sin(x) = x - -- + -- - -- + -- - --- + --- - ... 3! 5! 7! 9! 11! 13!

and

2 4 6 8 10 12 x x x x x x cos(x) = 1 - -- + -- - -- + -- - --- + --- - ... 2! 4! 6! 8! 10! 12!

When we neglect the later terms in the series, so that we behave as if sin(x) = x, and cos(x) = 1, we obtain first-order optics, in which all lenses are perfect. When we include the x squared and x cubed terms, then we have proceeded to third-order optics, in which the aberrations resulting from the nature of real lenses, exclusive of chromatic aberration, become evident.

The five Seidel aberrations are:

**Spherical Aberration**: this is the aberration affecting rays from a point on the optical axis; because rays from this point going out in different directions pass through different parts of the lens, then, if the lens is spherical, or otherwise not the exact shape needed to bring them all to a focus, then these rays will not all be focused at the same point on the other side of the lens.**Coma**: this aberration affects rays from points off the optical axis. If spherical aberration is eliminated, different parts of the lens bring rays from the axis to the same focus. But the place where the image of an off-axis point is formed may still change when different parts of the lens are considered.**Astigmatism**: this is another aberration affecting rays from a point off the optical axis. These rays, as they head through the lens to the point in the image where they will be focused, pass through a lens that is, from their perspective, tilted. Even if neither spherical aberration nor coma prevents them from coming to a sharp focus, if we consider the rays of light that are in the plane of the tilt, and the rays of light that are in the plane perpendicular to that, these rays pass through a part of the lens with a different profile. So they may not be focused at the same distance from the lens, even if they do come to a focus in each case.**Curvature of Field**: even when light from every point in the object is brought to a sharp focus, the points at which they are brought into focus might lie on a curved surface instead of a flat plane.**Distortion**: even when all the previous aberrations have been corrected, the light from points in the object might be brought together on the image plane at the wrong distance from the optical axis, instead of being linearly proportional to the distance from the optical axis in the object. If distance increases faster than in the object, one has*pincushion*distortion, if more slowly,*barrel*distortion.

The following diagram attempts to illustrate these aberrations:

Since we know the exact laws which govern the operation of lenses and
mirrors, it is possible to determine the exact course of light through any
optical system by means of *ray tracing*. At one time, this was a
very laborious procedure, but now it can be carried out automatically by
computers, and, as well, computers are no longer expensive.

However, simply choosing optical designs at random, and then finding out whether or not they work is not likely to get one anywhere fast. Instead, one needs a sense of what kind lenses will work together in a design to compensate for each other's aberrations. Knowledge of a large number of existing optical designs is one way to address this difficulty. Third-order optics, because it breaks down the aberrations of lenses into five quantities that can be manipulated arithmetically, also plays a considerable role in finding a starting point for a new optical design.

The two kinds of chromatic aberration bring the total to seven. There are
also higher-order aberrations. Sometimes, it is so important to combat
chromatic aberration, that one tries to suppress the *secondary spectrum*
left over from an achromatic lens that brings two wavelengths together, and using
three kinds of glass, bring three wavelengths together to create an
*apochromat*. Also, given that there is chromatic aberration, a lens system
with two kinds of glass, designed to correct the five Seidel aberrations for
one wavelength of light will, even if built out of closely spaced achromats,
still have these aberrations for other wavelengths; thus, the chromatic variation
in the aberrations is another higher-order consideration. As well, going to higher-order
terms in the expansion of the sine and cosine functions leads to an additional
set of nine aberrations at the next step.

Usually, except for designing an apochromat, it isn't helpful to
think in terms of the higher-order aberrations directly; their effects were minimized
through actual ray-tracing even in the days when the calculations for it were done by
hand. (Some optical design programs do actually improve their performance by calculating
fifth-order and seventh-order aberrations as well; the classic book *Optical
Aberration Coefficients* by Buchdahl, available from Dover Publications, gives
these formulas explicitly.)

To get rid of aberration in optical design, understanding it in terms of the five Seidel aberrations is helpful. We need two things to benefit from this help. We need a simple formula for the aberrations a spherical refracting surface will cause. The formula can be approximate, as long as it gives us the right answer for when the aberrations become small. And we need a way to combine the aberrations from multiple lenses.

It turns out this is possible. One reason for that is that multiplication can sometimes be approximated by addition. So if a and b are both much smaller than one, (1+a) times (1+b), which is (1+a+b+ab), is approximately 1+a+b. So, if a and b are thought of as deviations from perfection (multiplying by 1) the deviations can be treated as simply adding even when multiplication is the correct rule. Another is that distorted images do pass through successive lenses in a simple way. This is particularly obvious in the case of the aberration of distortion: an image with only that aberration is still an image in focus on a flat plane, so it would be magnified the same way the original object would be by successive lenses. In the case of other aberrations, one optical law is the key to following them through an optical system.

**If a lens magnifies an object N times, **(small)**
front-to-back distances in the object are magnified N squared
times.**

This gives us a rule by which we can scale longitudinal aberrations of any lens in an optical system, to determine their contribution to the aberration of the system as a whole. Transverse aberrations, of course, simply scale by the magnification of the image.

Copyright (c) 2001, 2004 John J. G. Savard

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