Topic 20. Spherical Mirrors, Part 1.

The principle that the angle of incidence is equal to the angle of reflection is also applicable to a mirror that is curved. The most common example is a spherical mirror, although other shapes (such as a mirror in the shape of a parabola) are also used. Since a line along the radius from the center of the sphere (shown as “C” in the figure below) to the circumference is always perpendicular to the circumference, every radius line can be used to construct the normal to any point on the sphere. When an incoming ray (such as “I”) is incident at some point on the sphere, it is reflected as ray R using the usual rule that the angle of incidence  is equal to the angle of reflection. The reflected ray appears to come from some point that is inside the sphere as shown in the figure. This type of mirror is called convex, because the mirror surface is curved outward toward the object.

Although the direction of the radius vector changes from point to point, all rays that are parallel to ray I are reflected in such a way that the reflected rays appear to come from a single point. This is called the focal point (“F” in the figure) and is midway between the center of the sphere and the circumference as shown in the figure. Unlike a plane mirror, where all of the rays really did appear to come from a single point, the situation here is only an approximation that is reasonably correct provided the rays do not strike the sphere too far away from the axis.

Since the angle of incidence and the angle of reflection are always equal, the law of reflection is symmetric, and a ray that was incident along direction R would be reflected along direction I. Thus we can formulate the 3 rules that describe the reflection of light rays from a spherical mirror:

1. A light ray that is incident along a direction parallel to the axis of the mirror is reflected so that it appears to come from the focal point – a point that is halfway between the center of the mirror and the circumference along the axis.

2. A light ray that is incident along a direction that is pointed directly at the focal point is reflected parallel to the axis. This is the same as the previous rule with the roles of the incident and reflected rays interchanged.

3. A ray that is incident along the radius vector is reflected back on itself. Such a ray has an angle of incidence of 0 by definition, and its angle of reflection is therefore 0 as well. This in the only one of the 3 rules that is exactly true.

In addition to the three common rules, there is a 4th rule that is often useful. Since the mirror is always symmetric about its central axis, a ray that strikes the mirror at its center (at any angle of incidence) is reflected symmetrically backwards by the same angle below the axis.

It is important to remember that rules 1 and 2 are approximations that are valid only when the incident rays are not too far from the axis. This is called the paraxial ray approximation. This approximation is an important limitation for reflections and refractions at curved surfaces that are spherical. (It is possible to remove this limitation in some cases using non-spherical shapes, but these shapes are usually too expensive to fabricate and are used only in very special situations.)

The validity of the paraxial ray approximation is determined by how far the incident ray is from the axis. The scale factor is the ratio of the distance of the ray from the axis divided by the focal length, and the paraxial ray approximation requires that this ratio be small compared to 1. This ratio often appears as its reciprocal: the ratio of the focal length to the size of the object (or to the diameter of the lens or the mirror), and the paraxial ray approximation implies that this reciprocal ratio be much larger than 1. This reciprocal ratio of the focal length to the size of the object or the diameter of the lens or the mirror is called the f-number, and it plays an important role in many other aspects of the formation of images by lenses and mirrors. Using this notation, the paraxial ray approximation is equivalent to the statement that this simple model of constructing the image produced by a lens or mirror is valid only when the f-number is large, and the approximation becomes increasingly poor as the f-number decreases.

The following diagram shows the rules governing the reflection of rays from a convex, that is one whose curvature is outward towards the source of the rays. Note that the fact that the angle of incidence always equals the angle of reflection implies that rays R and I can be interchanged.

The following diagram shows the same rules applied to a concave mirror, that is one whose curvature is inward away from the source of the rays. A ray that travels parallel to the axis is reflected through the focal point, as shown. As above, the fact that the angle of incidence always equals the angle of reflection implies that this ray can also be reversed. That is, a ray striking the mirror from the focal point is reflected parallel to the axis. As above, a ray that strikes the mirror along the radius of curvature is simply reflected back along itself, since any such ray strikes the mirror with an angle of incidence of exactly 0.

This figure also shows  the other two rules: a ray directed along the center of curvature is simply reflected back on itself, since the angle of incidence and reflection are both 0. Likewise, a ray that strikes the mirror exactly on its symmetry axis is reflected below the axis by the same angle.