There are two basic
kinds of lenses: *converging*, and *diverging*. A converging lens
brings all incident light-rays parallel to its optic axis together at
a point , behind the lens,
called the *focal point*, or *focus*, of the lens.
A diverging lens spreads out all incident light-rays parallel
to its optic axis so that they appear to diverge from a *virtual
focal point* in front of the lens. Here, the front side of the lens
is conventionally defined to be the side from which the light is
incident. The differing effects of a converging and a diverging lens
on incident light-rays parallel to the optic axis (*i.e.*,
emanating from a
distant object) are illustrated in Fig. 77.

Lenses, like mirrors,
suffer from *spherical aberration*, which causes
light-rays parallel to the optic axis, but a relatively long way from
the axis, to be brought to a focus, or a virtual focus, *closer* to the
lens than light-rays which are relatively close to the axis. It turns out that
spherical aberration in lenses can be completely cured by using lenses
whose bounding surfaces are *non-spherical*. However, such lenses
are more difficult, and, therefore, more expensive, to manufacture than
conventional lenses
whose bounding surfaces are spherical. Thus, the former sort of lens is only
employed in situations where the spherical aberration of a conventional
lens would be a serious problem. The usual method of curing
spherical aberration is to use *combinations* of conventional
lenses (*i.e.*, compound lenses).
In the
following, we shall make use of the *paraxial approximation*, in which
spherical aberration is completely ignored, and all light-rays parallel
to the optic axis are assumed to be brought to a focus, or a virtual
focus, at the same point . This approximation is valid as long
as the radius of the lens is small compared to the object distance
and the image distance.

The *focal length* of a lens, which is usually denoted , is
defined as the distance between the optic centre and the focal
point , as shown in Fig. 77. However, by convention, *converging* lenses have
*positive*
focal lengths, and *diverging* lenses have *negative* focal lengths. In other
words, if the focal point lies behind the lens then the focal length is
positive, and if the focal point lies in front of the lens then the focal
length is negative.

Consider a conventional lens whose bounding surfaces are *spherical*.
Let be the centre of curvature of the front
surface, and the centre of curvature of the back surface.
The radius of curvature of the front surface is the
distance between the optic centre and the point . Likewise,
the radius of curvature of the back surface is the distance
between points and . However, by convention, the
radius of curvature of a bounding surface is *positive* if its centre of
curvature lies *behind* the lens, and *negative* if its centre of
curvature lies *in front*
of the lens. Thus, in Fig. 78,
is positive and is negative.

In the paraxial approximation, it is possible to find a simple
formula relating
the focal length of a lens to the
radii of curvature, and , of its front and back bounding surfaces.
This formula is written

Suppose that a certain lens has a focal length . What happens
to the focal length if we turn the lens around, so that its front
bounding surface becomes its back bounding surface, and *vice
versa*? It is easily seen that when the lens is turned around
and
. However, the focal
length of the lens is invariant under this transformation, according
to Eq. (362). Thus, the focal length of a lens is the same for
light incident from either side. In particular, a converging
lens remains a converging lens when it is turned around, and likewise
for a diverging lens.

The most commonly occurring type of converging lens is a *bi-convex*,
or *double-convex*, lens, for which and . In this
type of lens, both bounding surfaces have a focusing effect on light-rays
passing through the lens. Another fairly common type of
converging lens is a *plano-convex* lens, for which
and . In this type of
lens, only the curved bounding surface has a focusing effect on light-rays. The plane surface has no focusing or defocusing effect.
A less common type of converging lens is a *convex-meniscus*
lens, for which and , with . In this type
of lens, the front bounding surface has a focusing effect on light-rays,
whereas the back bounding surface has a defocusing effect, but the
focusing effect of the front surface wins out.

The most commonly occurring type of diverging lens is a *bi-concave*,
or *double-concave*, lens, for which and . In this
type of lens, both bounding surfaces have a defocusing effect on light-rays
passing through the lens. Another fairly common type of
converging lens is a *plano-concave* lens, for which
and . In this type of
lens, only the curved bounding surface has a defocusing effect on light-rays. The plane surface has no focusing or defocusing effect.
A less common type of converging lens is a *concave-meniscus*
lens, for which and , with . In this type
of lens, the front bounding surface has a defocusing effect on light-rays,
whereas the back bounding surface has a focusing effect, but the
defocusing effect of the front surface wins out.

Figure 79 shows the various types of lenses mentioned above. Note that, as a general rule, converging lenses are thicker at the centre than at the edges, whereas diverging lenses are thicker at the edges than at the centre.