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.
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