27.7

The Diffraction Grating

Diffraction patterns of bright and dark fringes occur when monochromatic light passes through a single or double slit. Fringe patterns also result when light passes through more than two slits, and an arrangement consisting of a large number of parallel, closely spaced slits called a diffraction grating, has proved very useful. Gratings with as many as 40 000 slits per centimeter can be made, depending on the production method. In one method a diamond-tipped cutting tool is used to inscribe closely spaced parallel lines on a glass plate, the spaces between the lines serving as the slits. In fact, the number of slits per centimeter is often quoted as the number of lines per centimeter.

The physics of the diffraction grating.

Figure 27-32 illustrates how light travels to a distant viewing screen from each of five slits in a grating and forms the central bright fringe and the first-order bright fringes on either side. Higher-order bright fringes are also formed but are not shown in the drawing. Each bright fringe is located by an angle relative to the central fringe. These bright fringes are sometimes called the principal fringes or principal maxima, since they are places where the light intensity is a maximum. The term “principal” distinguishes them from other, much less bright fringes that are referred to as secondary fringes or secondary maxima.

Figure 27-32 When light passes through a diffraction grating, a central bright fringe () and higher-order bright fringes form when the light falls on a distant viewing screen.

Constructive interference creates the principal fringes. To show how, we assume the screen is far from the grating, so that the rays remain nearly parallel while the light travels toward the screen, as in Figure 27-33. In reaching the place on the screen where the first-order maximum is located, light from slit 2 travels a distance of one wavelength farther than light from slit 1. Similarly, light from slit 3 travels one wavelength farther than light from slit 2, and so forth, as emphasized by the four colored right triangles on the right-hand side of the drawing. For the first-order maximum, the enlarged view of one of these right triangles shows that constructive interference occurs when , where d is the separation between the slits. The second-order maximum forms when the extra distance traveled by light from adjacent slits is two wavelengths, so that . The general result is

| (27.7)

Figure 27-33 The conditions shown here lead to the first- and second-order intensity maxima in the diffraction pattern produced by the diffraction grating on the right.

The separation d between the slits can be calculated from the number of slits per centimeter of grating; for instance, a grating with 2500 slits per centimeter has a slit separation of . Equation 27.7 is identical to Equation 27.1 for the double slit. A grating, however, produces bright fringes that are much narrower or sharper than those from a double slit, as the intensity patterns in Figure 27-34 reveal. Note also from the drawing that between the principal maxima of a diffraction grating there are secondary maxima with much smaller intensities.

Figure 27-34 The bright fringes produced by a diffraction grating are much narrower than those produced by a double slit. Note the three small secondary bright fringes between the principal bright fringes of the grating. For a large number of slits, these secondary fringes become very small.

The next example illustrates the ability of a grating to separate the components in a mixture of colors.

| Example 9 | | | Separating Colors with a Diffraction Grating

A mixture of violet light ( in vacuum) and red light ( in vacuum) falls on a grating that contains . For each wavelength, find the angle that locates the first-order maxima. Reasoning Before Equation 27.7 can be used here, a value for the separation d between the slits is needed: . For violet light, the angle for the first-order maxima is given by , with an analogous equation applying for the red light.
Solution For violet light, the angle locating the first-order maxima is

For red light, a similar calculation with shows that . Because and are different, separate first-order bright fringes are seen for violet and red light on a viewing screen.

If the light in Example 9 had been sunlight, the angles for the first-order maxima would cover all values in the range between 24° and 41°, since sunlight contains all colors or wavelengths between violet and red. Consequently, a rainbow-like dispersion of the colors would be observed to either side of the central fringe on a screen, as can be seen in Figure 27-35. This drawing shows that the spectrum of colors associated with the order is completely separate from that of the order. For higher orders, however, the spectra from adjacent orders may overlap (see problems 43 and 55). The central maximum is white because all the colors overlap there.

Figure 27-35 When sunlight falls on a diffraction grating, a rainbow of colors is produced at each principal maximum . The central maximum , however, is white but is not shown in the drawing.

The physics of a grating spectroscope. An instrument designed to measure the angles at which the principal maxima of a grating occur is called a grating spectroscope. With a measured value of the angle, calculations such as those in Example 9 can be turned around to provide the corresponding value of the wavelength. As we will point out in Chapter 30, the atoms in a hot gas emit discrete wavelengths, and determining the values of these wavelengths is one important technique used to identify the atoms. Figure 27-36 shows the principle of a grating spectroscope. The slit that admits light from the source (e.g., a hot gas) is located at the focal point of the collimating lens, so the light rays striking the grating are parallel. The telescope is used to detect the bright fringes and, hence, to measure the angle .

Figure 27-36 A grating spectroscope.

Check Your Understanding 5

The drawing shows a top view of a diffraction grating and the mth-order principal maxima that are obtained with red and blue light. Red light has the longer wavelength. (a) Which principal maximum is associated with blue light? (b) If the number of slits per centimeter in the grating were increased, would these two principal maxima move away from the central maximum, move toward the central maximum, or remain in the same place? (The answers are given at the end of the book.)

Background: The angle that specifies the mth principal maximum of a diffraction grating depends on two parameters–the wavelength of the light and the separation d between the slits. This question deals with both of these parameters.
For similar questions (including calculational counterparts), consult Self-Assessment Test 27.2, which is described at the end of Section 27.9.

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