Topic
22. Thin Lenses, part 1. Types of Lenses
In addition to using mirrors to form
images, it is also possible to form images using lenses. Although many
different types of lenses exist, we will consider in detail only lenses whose
surfaces are spherical. In addition, the lenses we will study are made from
glass or some other similar material whose index of refraction is greater than
the index of refraction of the air that surrounds them. (The opposite situation
– a spherical bubble of air in the center of a liquid, for example, can be
handled by the same general methods, but we will not discuss these
configurations.) We will also not discuss in detail lenses that have more
complicated shapes, such as those whose surfaces are in the shapes of ellipses
or parabolas.
The lenses we will study fall into two
broad categories: converging, or positive lenses and diverging, or negative
ones. Each category has a number of different configurations, but all of the
shapes in either category share a common characteristic. Converging, or
positive lenses are always thicker in the middle than they are at the edges,
while diverging, or negative lenses are always thinner in the middle than they
are at the edges.
These
are some examples of converging, or positive lenses:
The
lens on the left is called a positive meniscus (or “fish-eye”) lens, the middle
one is a plano-convex lens, and the one on the right is a double-convex lens. Although
all of these shapes have the same basic optical properties, the first two
produce better images when the field of view is very large or when the object
is very far away compared to the size of the lens. The ray-tracing examples
will always use the symmetric configuration on the right, but the same
ray-tracing rules will work for the other two shapes as well.
Here
are some examples of diverging, or negative lenses:
As above, the lens on the left is a negative meniscus lens, the one in the middle is a plano-concave shape and the one on the right is a double-concave shape. The choice among these shapes is governed by the same considerations as for a positive lens as described above. We will use the symmetric concave shape in the ray-tracing examples.
In each case, the properties of the lens are completely specified by its focal length. However, unlike the case of a mirror where the focal length is a simple geometrical parameter, the focal length of a lens depends both on its shape and on the index of refraction of the material it is made of. For the symmetric convex and concave shapes shown above, the approximate focal length is determined from the equation:
where R is the radius of curvature of either surface and n is the index of refraction of the glass (or whatever material the lens is made of). The magnitude of the focal length is determined by this relationship for either a double convex or a double concave lens; the sign of the focal length is positive for a converging (convex) lens and negative for a diverging (concave) shape.
Since all lenses have light rays traveling through the lens, all lenses have two focal points: one on each side of the lens. The distance from either focal point to the center of the lens is given by the expression above.
It is very common for relationships involving the focal length to depend not on the focal length itself but on its reciprocal. This reciprocal appears so often in optical design that it is called the “strength” or “power” of the lens and it is measured in diopters. The strength of a lens in diopters is the reciprocal of its focal length in meters. A lens with a focal length of 50 cm (0.5 m) would have a strength of 2 diopters, a focal length of 25 cm would be a strength 4 diopters, etc. Note that the focal length must be converted to meters to compute the strength of the lens. The sign of the strength of the lens in diopters is the same as the sign of the focal length in meters: positive for convex or converging lenses and negative for diverging ones.