# Topic 28. Aberrations

The graphical methods that we have used to construct images are based on approximations that are never exactly true. Images formed by real lenses and mirrors therefore are never exactly identical to the predictions of the simple paraxial ray methods that we have developed. (This discrepancy is due to the limitations of the paraxial ray approximation, and is present even when the algebraic form of the ray tracing rules is used.) The differences between the prediction of the paraxial ray method and the actual image are called aberrations.  These aberrations are usually divided into 5 broad classes:

1. Spherical aberration. Although our ray-tracing model assumed that all rays parallel to the axis are reflected or refracted so that they pass through a single focal point that was a characteristic of the lens or mirror, this is actually not the case. This assumption gets worse as the distance of the ray from the axis increases. The difference between the focal points for rays that are close to the axis and for rays that strike the lens near its edge is called spherical aberration. Positive spherical aberration means that rays near the edge of the lens have an effective focal point that is closer to the lens than rays that strike the lens near the axis. Negative spherical aberration means that rays near the edge of the lens have an effective focal point that is at a greater distance from the lens than rays that strike the lens near the axis.

Since the effective focal point determines the position of the image for any object, the fact there are several different focal points for the different kinds of rays means that there are several “ghost” images formed by the lens – the “real” one determined by the location of the paraxial-ray focal point, and a number of other images that are formed at different distances depending on where the rays that form them struck the lens. A screen placed at the point where we expect the paraxial ray image therefore also shows these other images not quite in focus, since they are really being formed elsewhere.

Spherical aberration obviously increases with the diameter of the lens, and it can be minimized by limiting the opening of the lens so that only rays in the paraxial region can pass through it.

2. Coma.  This aberration affects rays that come from an object (or a part of an object) that is not at the center of the lens. The aberration is caused by the fact that the magnification of a lens (that is the size of the image for a given object) is different for rays that strikes the lens in the paraxial region and for rays that strike the lens near its edge.  When coma is positive, the image of an object produced by off-axis rays is slightly larger than the image produced by paraxial rays; when coma is negative, the image produced by the off-axis rays is slightly smaller.

Unlike spherical aberration which affects where in space the different images are formed, coma results in several different images of different sizes all of which are in focus on the same screen. The result is a bright image formed by the paraxial rays and a series of smaller (or larger) images formed by the rays that hit the lens far away from the axis. The result often looks a bit like a comet – a clear image with a fuzzy tail oriented along the screen and composed of weaker images formed by the off-axis rays.

There is a defect in the eye which produces an image that looks like coma, but is actually caused by a different problem.

3. Astigmatism.  This is also an off-axis effect. When an object point is quite far off of the central axis of the lens, the effective radius of curvature (and hence the effective focal length of the lens) in one direction is not the same as in a perpendicular direction. Think of circles of constant longitude on the Earth, which always have the same circumference at any latitude and circles of constant latitude, whose circumference decreases as you move away from the equator. Again, the result is to produce a slight difference in the focal length for rays that are in the longitudinal plan of the lens and for rays that leave this plane. Thus a point on an off-axis object has two image points – one for the rays that strike the lens in the plane of the object and the central axis of the lens and another for rays that strike the lens perpendicular to this plane. At either of these two image points, the rays forming the other image are somewhat out of focus, and often form a small line segment. At intermediate points between the two image points the rays combine to form an image that sometimes looks like a small + sign.

There is also a defect in the eye called astigmatism. It produces a similar effect but is caused by a related, but different problem.

4. Curvature of Field. Even if all of the effects discussed above could be eliminated, this effect would remain. It arises because the image plane is not really a plane but a spherical surface whose center is located at the same distance from the lens as the image plane. In the paraxial approximation, the rays passing through the lens strike this image surface near its center where the curvature of this surface is not great, and the approximation that it is a flat plane is not too bad. However, this approximation gets worse as the rays deviate from the center of the lens. The effect of this field curvature is to produce a blurring the edges of the image.

It is possible to correct for this effect using a combination of a positive and a negative lens that are very close together, and this is usually done in camera lenses.

5. Distortion. This effect is caused by a variation in the magnification of the image across the field of view. That is, some parts of the image are magnified more than others, so that the overall effect is to produce a distorted version of the object, which is still in focus everywhere. This aberration does not affect the focus of the image, but it does affect the accuracy with which the image represents an accurate representation of the object.  Distortion is very noticeable at the edges of a convex mirror, for example, where the magnification is very different from the magnification in the paraxial region.

5. Chromatic aberration. This is caused by a variation in the index of refraction of a lens with the wavelength of the incident light. It is in addition to all of the other aberrations discussed above, which are caused by the inadequacy of the paraxial ray approximation and have nothing to do with the color of the incident light. It is only present in lenses – mirrors obviously have no chromatic aberration because they operate by reflection rather than refraction.

The most noticeable effect of chromatic aberration is that images of objects illuminated by white light show colored bands around the edges because the different colors in the white light focus at different points in the image plane. This effect is in addition to all of the other aberrations discussed above.

It is possible to correct for chromatic aberration by using two lenses made of different materials so that variation in the index of refraction of one lens is cancelled by the opposite variation of the other one. Such doublets are called “achromats” and are commonly used in camera lenses.