D. Refraction in prisms
Glass prisms are often used to bend light in a given direction as well as to bend it back again (retroreflection). The process of refraction in prisms is understood easily with the use of light rays and Snell’s law. Look at Figure 3-11a. When a light ray enters a prism at one face and exits at another, the exiting ray is deviated from its original direction. The prism shown is isosceles in cross section with apex angle A = 30° and refractive index n = 1.50. The incident angle q and the angle of deviation d are shown on the diagram.
Figure 3-11b shows how the angle of deviation d changes as the angle q of the incident ray changes. The specific curve shown is for the prism described in Figure 3-11a. Note that d goes through a minimum value, about 23° for this specific prism. Each prism material has its own unique minimum angle of deviation.
(a)
(b)
Figure 3-11 Refraction of light through a prism
1. Minimum angle of deviation. It turns out that we can determine the refractive index of a transparent material by shaping it in the form of an isosceles prism and then measuring its minimum angle of deviation. With reference to Figure 3-11a, the relationship between the refractive index n, the prism apex angle A, and the minimum angle of deviation dm is given by
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(3-4) |
where both A and dm are measured in degrees.
The derivation of Equation 3-4 is straightforward, but a bit tedious. Details of the derivation—making use of Snell’s law and geometric relations between angles at each refracting surface—can be found in most standard texts on geometrical optics. (See suggested references at the end of the module.) Let’s show how one can use Equation 3-4 in Example 4 to determine the index of refraction of an unknown glass shaped in the form of a prism.
Example 4
A glass of unknown index of refraction is shaped in the form of an isosceles prism with an apex angle of 25°. In the laboratory, with the help of a laser beam and a prism table, the minimum angle of deviation for this prism is measured carefully to be 15.8°. What is the refractive index of this glass material?
Solution: Given that dm = 15.8° and A = 25°, we use Equation 3-4 to calculate the refractive index.
(Comparing this value with refractive indexes given in Table 3-1, the unknown glass is probably flint glass.)
2. Dispersion of light. Table 3-1 lists indexes of refraction for various substances independent of the wavelength of the light. In fact, the refractive index is slightly wavelength dependent. For example, the index of refraction for flint glass is about 1% higher for blue light than for red light. The variation of refractive index n with wavelength l is called dispersion. Figure 3-12a shows a normal dispersion curve of nl versus l for different types of optical glass. Figure 3-12b shows the separation of the individual colors in white light—400 nm to 700 nm—after passing through a prism. Note that nl decreases from short to long wavelengths, thus causing the red light to be less deviated than the blue light as it passes through a prism. This type of dispersion that accounts for the colors seen in a rainbow, the “prism” there being the individual raindrops.
(a) Refraction by a prism
(b) Optical glass dispersion curves
Figure 3-12 Typical dispersion curves and separation of white light after refraction by a prism
3. Special applications of prisms. Prisms that depend on total internal reflection are commonly used in optical systems, both to change direction of light travel and to change the orientation of an image. While mirrors can be used to achieve similar ends, the reflecting faces of a prism are easier to keep free of contamination and the process of total internal reflection is capable of higher reflectivity. Some common prisms in use today are shown in Figure 3-13, with details of light redirection and image reorientation shown for each one. If, for example, the Dove prism in Figure 3-13b is rotated about its long axis, the image will also be rotated.
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(a) Right-angle prism
(b) Dove prism
(c) Penta prism
(d) Porro prism
Figure 3-13 Image manipulation with refracting prisms
The Porro prism, consisting of two right-angle prisms, is used in binoculars, for example, to produce erect final images and, at the same time, permit the distance between the object-viewing lenses to be greater than the normal eye-to-eye distance, thereby enhancing the stereoscopic effect produced by ordinary binocular vision.
 
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