The ray-tracing methods that we have
developed can also be applied to a number of lenses that are close together.
These combinations are called compound lenses, and are very commonly found in cameras
and other optical instruments.
In order to apply the standard
ray-tracing method to the combination, we start by finding the image that would
be produced by the first lens acting alone. For example, consider the compound
lens shown below.
In
order to find the location of the image produced by the combination of the two
lenses, start by using the usual ray-tracing rules to find the image produced
by the first lens. The image is located using the usual two rays: one that
strikes the lens parallel to the axis and emerges directed towards the focal
point and the second which strikes the lens at its center and continues without
any change in direction.
Once
the image has been found, we can add any number of additional rays from the
object to the image without the need for any rules, because every ray that
leaves a point on the object and strikes the lens is refracted so that it
converges on the corresponding point of the image. Although we can choose any
additional rays, it is most convenient to choose rays that will make the next
step easiest. Thus we choose rays in this step that will pass through the
center of the second lens and will strike the second lens parallel to the axis.
Note the two additional rays have been chosen with this principle in mind.
We
then apply these rays to the second lens and use the usual rules to compute the
location of the second image. This job was made easy by choosing additional
rays which would satisfy the usual rules with respect to the second lens. That
is, one of our additional rays passes through the center of the second lens and
therefore emerges unchanged and the second ray strikes the second lens parallel
to the axis and therefore emerges directed towards the focal point of the
second lens as shown.
Note
that the combination of the two lenses produces a real, inverted image that is
closer to the first lens than the image that would have been produced by the
first lens alone. In other words, the strength of the combination is greater
than the strength of either lens. (see below)
Although
it is usually easier to consider the effect of the first lens and then add the
effect of the second lens, this is not always convenient. For example, consider
the case of a person who is myopic (near-sighted). As we will see later,
the lens in a myopic eye focuses a relatively distant object in front of the
retina (rather than on it) as shown below. The green rays show how the image of
the red arrow is formed by the lens in the eye acting alone. (The image is
real, inverted and smaller than the object – the usual situation for this
configuration.)
Since
the image is formed in front of the retina, the rays which reach the retina are
not in focus. That is, they do not converge to a single point on the retina but
the different rays strike at different places as shown. The result is to form
several spatially-separated copies of each point of the object which combined
to form a blurred image.
The
solution is to add a negative lens in front of the eye, which moves the red
image formed in front of the retina by the lens of the eye alone so that it is
formed exactly on the retina as is shown by the purple image in the figure
below.
The
position of the object and the configuration of the eye are fixed parameters, so
that there are only two adjustable parameters – the focal length of the
negative lens and where it will be located. If the person chooses conventional
eye-glasses (not contact lenses), then the position of the negative lens is
fixed by the shape of the person’s face, so that the only adjustable parameter
is the focal length of the negative lens. This focal length can be determined
using a single ray – the purple ray shown below.
This
purple ray leaves the object directed towards the far focal point of the
negative lens we are planning to add. It is refracted by the negative lens and
emerges parallel to the axis. It then strikes the lens of the eye parallel to
the axis and is refracted through its focal point and then strikes the retina
as shown. The focal point of the first
lens is then determined by simply extending the initial direction of the purple
ray and noting where it meets the axis. The distance from this point back to
the lens is then the required focal length of the eye-glass lens.
Another
way of coming to the same result is to note that the effect of the negative
lens is to create a virtual image of the initial object, and to then compute
the final image on the retina using this virtual image as an intermediate
object. This construction is shown below:
The
virtual image produced by the negative lens is shown as “v” in the figure
above. Its position is found using two
purple rays: a ray headed for the far focal point of the first lens which
emerges parallel to the axis (this ray is the same one that was used above) and
a ray through the center of the negative lens which continues unchanged. These
two rays appear to diverge from point “v” so that this is the location of the
virtual image produced by the negative lens.
Using
this virtual image as an object, the image on the retina can be found using the
normal rules: a ray emerging from this
virtual image parallel to the axis is refracted by the lens in the eye so that
it passes through the focal point, and a ray that passes through the center of
the lens in the eye continues in the same direction. These rays are shown in
the figure above, and they confirm that the final image is located on the retina
as desired.
Another
way of describing the effect of the negative lens used in these eye glasses is
to note that it works by creating a virtual copy of the object that is closer
to the eye than the real object is. The person could accomplish the same thing
(without eye glasses) by moving her eyes closer to the real object, and many
near-sighted people do just that – they bring relatively distant objects into
focus by moving them closer to their face.
Note
that this combination of the negative and the positive lens is still positive –
the final image is still real and inverted. Only its position has been changed.
The size of the final image in this case is about the same with or without the
negative lens, but this may not be true if the negative lens is quite strong.
People who are very near-sighted see all distant objects as a blur, and usually
have to wear glasses all of the time.
They get used to the change in the apparent size of objects that is
caused by the strong negative lenses; in fact since they can’t see clearly
without their glasses on, they have no basis for comparing the difference in
the apparent size of an object with and without the negative lenses.
These
conclusions about the type and size of the final image are not universally
true. The image produced by a pair of lenses can be real or virtual, erect or
inverted, and need not be the same as the image that would be produced by
either lens alone.
There
are a number of special cases, where it is easy to compute the effect of a
compound lens without all of this work.
1. If the two lenses are touching each
other then the effect of the pair is equivalent to the effect that would be
produced by a single lens whose power was the algebraic sum of the two powers.
(Recall that the power of a lens is measured in diopters and is the reciprocal
of the focal length in meters.) Thus in our first example above, the power of
the two lens combination is the sum of two positive quantities, which is a
numerically larger value. The equivalent focal length of the combination is
smaller than the focal length of either lens by itself, so that the resulting
image is closer to the first lens as can be seen from the figures. Likewise,
the algebraic sum of the strengths of the two lenses in the second example is
smaller than the strength of the lens in the eye alone (since the lens we have
added has a negative focal length and therefore a negative strength). The
result is weaker lens – one whose focal length is greater than the focal length
of the lens in the eye alone. The image therefore moves further away from the
lens, as is required in this case to put the image on the retina.
2. If the two lenses are separated by
a distance d which is not too much greater than the sum of the two focal
lengths, then this combination behaves as a single lens whose power is
approximately equal to
where
f1 and f2 are the focal lengths of the two lenses by
themselves. If d is much smaller than the focal length of the second lens, then
it can be ignored, and this expression reduces to the simple sum of strengths
relationship described in the previous paragraph. Note that the equivalent
focal length of two positive lenses need not be positive.
The
effect of wearing eye-glasses can be estimated using the first rule for contact
lenses; although the first rule is approximately correct even for normal
(non-contact) configurations, the second rule usually gives a more accurate
result for this configuration.
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