The effect of smooth mirrors on the paths of light rays is governed by the law of reflection. Sometimes referred to as the law of specular reflection, this law states that the angle of reflection, between the reflected ray and the surface normal (line perpendicular to the reflecting surface) at the point of reflection, equals the angle of incidence, between the incident ray and the surface normal.
Plane mirrors simply reverse the direction of propagation of light, without altering vergence. This effect is often described as a “reversal of the image space.” Curved mirrors, in addition to reversing the image space, also add or subtract vergence.
For paraxial rays, we can draw a simple geometric construction, akin to the derivation of the vergence equation and the lensmaker’s equation, to demonstrate a similar vergence equation for mirrors (Appendix 1.3). We use the following conventions:
we assume the incident light travels from left to right in the usual way;
r is the (signed!) radius of curvature of the mirror, with the usual sign conventions (a mirror concave to the left has a negative number as its radius, so the power P of such a mirror is a positive number; a mirror convex to the left has a positive number as its radius, so the power P is a negative number);
“ν” is the distance from the mirror to the image in the reversed image space—rays converging from right to left are considered to have positive vergence; rays diverging from right to left are considered to have negative vergence. We use quotation marks around the letter “ν” to remind you of the change in sign convention necessary when you use the vergence equation for mirrors rather than for lenses.
Notice that the refractive index does not appear in these equations—the law of reflection does not include a correction for refractive index.
So, for example, parallel rays moving from left to right, striking a mirror concave to the left, emerge traveling right to left with positive vergence. They converge on the focal point of the mirror at half the radius of the mirror, to the left of the mirror. Conversely, parallel rays moving from left to right that strike a mirror convex to the left emerge diverging from right to left, with an apparent focus (a virtual image) at distance r/2 to the right of the mirror.
It is often useful to clarify these unusual sign conventions by ray tracing. For mirrors, we recognize 3 special rays: (1) the ray through the center of curvature of the mirror (not the vertex on the optic axis, as for lenses) is the undeviated ray; (2) the ray that strikes the vertex is reflected at an angle equal to the angle of incidence; and (3) the ray that propagates parallel to the optic axis passes through the focal point of the mirror, which is halfway between the mirror surface and the center of curvature.
Only a couple of examples occur commonly in practice: a large-radius concave mirror, such as those used as cosmetic or shaving mirrors, with source object closer than the focal point, and a small-radius convex mirror, such as those used in rear-view mirrors on automobiles (Figure 1-34). The cornea is used as a small-radius reflecting surface to measure the corneal curvature (keratometry).
Figure 1-34 Ray tracing for concave (A) and convex (B) mirrors. The central ray for mirrors is different from the central ray for lenses in that it passes through the center of curvature (“C”) of the mirror, not through the center of the mirror.
(Illustration developed by Kevin M. Miller, MD, and rendered by C. H. Wooley.)
Excerpted from BCSC 2020-2021 series : Section 3 - Clinical Optics. For more information and to purchase the entire series, please visit https://www.aao.org/bcsc.