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Geometrical optics: summary
Geometrical optics deals with light (and more generally with waves) in situations where it is possible to ignore the wave character of the phenomenon. This usually means that the wavelengths involved are very much smaller than the dimensions of anything with which the waves interact. Later we will see that geometrical optics is a limiting case of wave optics. The usual value in limiting cases is their simplicity, and geometrical optics shares this asset, with reservations.
All of geometrical optics may be deduced from three simple empirical rules:
1. Light travels in straight lines in homogeneous media.
2. The angle at which light is reflected from a surface is equal to the angle at which it is incident.
3. When light passes from one medium to another, its path is described by the equation n1 sin θ1 = n2 sin θ2.
Figure 1.1 summarizes these rules and defines the various angles. A â ray â is a line along the path the light follows. We think of this as a very narrow beam of light.
Figure 1.1: Reflection and refraction.
We may regard rule 3, which describes refraction, as defining the relative index of refraction: n2/n1.
If we follow custom and define the index of vacuum to be nvac = 1, then n1 and n2 are the indices of each medium relative to vacuum, and are so listed in handbooks. Later we will see that this ability to characterize each medium by a single number is extremely important. Among other consequences, it will lead us to regard nk as the ratio of the speed of light in vacuum (c) to (ck), the speed of light in medium k: nk = c/ck . This in turn allows us to deduce the three rules from the more general Fermatâs principle. The main asset of this principle is the esthetic one of unification. It is not essential to the conclusions of geometrical optics, although occasional simplifications are possible. But the three rules suffice.
A further statement limits the kinds of media usually considered. This is the reciprocity principle, which requires that if light can follow a certain path from A to B, then it can follow the same path from B to A. Some media do not support this principle, but they seldom occur in questions pertaining to geometrical optics.
If we apply our rules directly to plane surfaces, they describe the behavior of mirrors and prisms (1.1â1.3)1. Rule 3 also predicts the phenomenon of total internal reflection. When light passes from a medium with a larger index to one with a smaller index (as, for instance, from glass to air), the ray is bent toward the surface. That is, θ2 > θ1. Eventually, the ray emerging from the âdenserâ medium (the one with larger index) lies parallel to the surface. This occurs at the critical angle: n1 sin θc = n2 sin(90°) = n2. If θ1 is bigger than θc, no light emerges. In this case, all incident light is contained in the reflected ray so that, from inside, the surface appears to be a perfect mirror. Such mirrors are important in various optical instruments. Familiar examples are the right-angle prisms in binoculars and the light pipes which illumine hard-to-reach places.
Figure 1.2: Total internal reflection.
Figure 1.3: Applications of total internal reflection.
Notice that all light incident from the outside of a totally reflecting surface will enter the surface, but it will not reach the region for which θ1 > θc. (1.4)
In applying rule 3, we find a new empirical fact: n is different for different colors of light. Later we will study the source of this dispersion in some detail, but in geometrical optics we merely use it, for example, when we separate colors with a prism. Or, we may compensate for it, as in making achromatic lenses of two kinds of glass, in which the dispersion of one compensates for that of the other. (1.5)
Rule 2 for reflection governs the behavior of instruments with curved mirrors, such as the astronomical telescope. Since the manipulation is similar to that of lenses, we can summarize the operations of the two devices together.
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