Mirrors
The angle of specular reflection of light rays from an interface is independent of the refractive index of the materials on each side of the interface, as the reflected light is not entering the second material. Obeying the Fermat principle of least time, the angle of reflection, as measured from the normal to the surface, is equal to the angle of incidence. Also, as with Snell’s law of refraction, the reflected ray must lie in the same plane as the incident ray and the normal to the surface (see Fig 1-3).
Reflection From a Plane Mirror
The image of a real object in front of a mirror is located equally far behind the mirror, erect, and virtual. Looking into a mirror, you see an image that is laterally inverted—that is, what appears to be your right hand in the mirror is the virtual image of your left hand. To see yourself from head to toe in a plane mirror, you need the mirror to extend from the top of your head only halfway to the floor (Fig 1-53).
Figure 1-53 A half-length mirror gives a full-length view. (Illustration developed b y Edmond H. Thall, MD, and Kevin M. Miller, MD, and rendered b y C. H. Wooley.)
Spherically Curved Mirrors
The focal length (f) of a mirror is half its radius (r) of curvature. The power of a mirror is 1/focal length (1/f). Convex mirrors add negative vergence (like minus lenses). Looking in the convex rearview mirror of an automobile, you see a virtual, erect, minified image of the car behind you. Concave
mirrors add positive vergence (like plus lenses). When you look in a convex cosmetic mirror, the image you see depends on how far you are from the mirror. If you are within the focal length, you will see a virtual, erect, magnified image of yourself. As you move farther away, rays cross, and the image becomes real, inverted, and magnified, and then it becomes minified (Fig 1-54).
Figure 1-54 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 b y Kevin M. Miller, MD, and rendered b y C.H. Wooley.)
Reversal of the Image Space
The basic vergence relationship, U + P = V, can be applied to mirrors, except that the mirror folds the optical path, reversing the image space. In our diagrams, light travels from left to right as it approaches the mirror and from right to left after reflection. Converging image rays have positive vergence and will form a real image to the left of the mirror, and diverging image rays with negative vergence will appear to come from a virtual image to the right of the mirror.
The Central Ray for Mirrors
The central ray for mirrors (see Fig 1-54), which passes through the center of curvature of the mirror, is as useful as the central ray for lenses, for if the image location is determined by vergence calculation, the central ray then immediately indicates the orientation and size of the image. Note that in using the ratio of image distance to object distance to calculate the size of the image, the image and object distances are measured from the center of curvature of the mirror, where we find the similar triangles to compare, just as we did for lenses.
Vergence Calculations for Mirrors
Plane mirrors create upright virtual images from real objects, with the virtual image located as far behind the mirror as the real image is in front. For example, light from an object 1 m to the left of a plane mirror has a vergence of –1 D at the mirror. On reflection, the vergence will still be –1 D, but by tracing imaginary extensions of the reflected image rays to the far side of the mirror (into virtual image space), we see the virtual image is located 1 m to the right of the mirror.
Concave mirror
A concave mirror adds positive vergence to incident light. It therefore has positive, or converging, power. If parallel rays strike the mirror, they will be reflected and converged toward a focal point halfway to the center of curvature. Note that the focal point of a concave mirror is not unique, for any central ray can serve as an optical axis. Note further that the primary and secondary focal points of a concave mirror are the same point. A central ray is constructed to pass through an object and the center of curvature.
As an example, we are given an object 1 m to the left of a concave mirror with a radius of curvature of 50 cm. Where is the image? The power of the mirror is equal to 1/f, where f = −r/2, so the power is 4 D. Using U + P = V, we have –1 + 4 = 3. Therefore, the image is located 1/3 m, or 33 cm, to the left of the mirror, in real image space. It is also minified and inverted.
Convex mirror
A convex mirror adds minus vergence to incident light. It therefore has negative, diverging power. The primary and secondary focal points, which coincide, are virtual focal points located halfway back to the center of curvature.
Using a convex, rather than a concave, mirror in the previous example, with the same radius of curvature, the power of the mirror will be –4 D:
U + Pm = V