27.5

Diffraction

CONCEPTS AT A GLANCE In the previous sections of this chapter we have used the principle of linear superposition to investigate some interference phenomena associated with light waves. As the Concepts-at-a-Glance chart in Figure 27-16 shows, we will now use the principle of linear superposition to explore three more topics—diffraction, resolving power, and the diffraction grating—all of which involve interference phenomena. This chart is an extension of the earlier version in Figure 27-1.

Figure 27-16 CONCEPTS AT A GLANCE The principle of linear superposition can be used to describe the topics highlighted in gold. The grooved surface of a CD behaves like a diffraction grating and separates white light into a spectrum of colors. (Dale E. Boyer/Photo Researchers, Inc.)

As we have seen in Section 17.3, diffraction is the bending of waves around obstacles or the edges of an opening. In Figure 27-17 (see page 866), which is similar to Figure 17-11, sound waves are leaving a room through an open doorway. Because the exiting sound waves bend, or diffract, around the edges of the opening, a listener outside the room can hear the sound even when standing around the corner from the doorway.

Figure 27-17 Sound bends or diffracts around the edges of a doorway, so even a person who is not standing directly in front of the opening can hear the sound. The five red points within the doorway act as sources and emit the five Huygens wavelets shown in red.

Diffraction is an interference effect, and the Dutch scientist Christian Huygens (1629–1695) developed a principle that is useful in explaining why diffraction arises. Huygens’principle describes how a wave front that exists at one instant gives rise to the wave front that exists later on. This principle states that:

> Every point on a wave front acts as a source of tiny wavelets that move forward with the same speed as the wave; the wave front at a later instant is the surface that is tangent to the wavelets.

We begin by using Huygens’ principle to explain the diffraction of sound waves in Figure 27-17. The drawing shows the top view of a plane wave front of sound approaching a doorway and identifies five points on the wave front just as it is leaving the opening. According to Huygens’ principle, each of these points acts as a source of wavelets, which are shown as red circular arcs at some moment after they are emitted. The tangent to the wavelets from points 2, 3, and 4 indicates that in front of the doorway the wave front is flat and moving straight ahead. But at the edges, points 1 and 5 are the last points that produce wavelets. Huygens’ principle suggests that in conforming to the curved shape of the wavelets near the edges, the new wave front moves into regions that it would not reach otherwise. The sound wave, then, bends or diffracts around the edges of the doorway.
Huygens’ principle applies not just to sound waves, but to all kinds of waves. For instance, light has a wave-like nature and, consequently, exhibits diffraction. Therefore, you may ask, “Since I can hear around the edges of a doorway, why can’t I also see around them?” As a matter of fact, light waves do bend around the edges of a doorway. However, the degree of bending is extremely small, so the diffraction of light is not enough to allow you to see around the corner.
As we will learn, the extent to which a wave bends around the edges of an opening is determined by the ratio , where is the wavelength of the wave and W is the width of the opening. The photographs in Figure 27-18 illustrate the effect of this ratio on the diffraction of water waves. The degree to which the waves are diffracted or bent is indicated by the two red arrows in each photograph. In part a, the ratio is small because the wavelength (as indicated by the distance between the wave fronts) is small relative to the width of the opening. The wave fronts move through the opening with little bending or diffraction into the regions around the edges. In part b, the wavelength is larger and the width of the opening is smaller. As a result, the ratio is larger, and the degree of bending becomes more pronounced, with the wave fronts penetrating more into the regions around the edges of the opening.

Figure 27-18 These photographs show water waves (horizontal lines) approaching an opening whose width W is greater in (a) than in (b). In addition, the wavelength of the waves is smaller in (a) than in (b). Therefore, the ratio increases from (a) to (b) and so does the extent of the diffraction, as the red arrows indicate. (Courtesy Education Development Center)

Based on the pictures in Figure 27-18, we might expect that light waves of wavelength will bend or diffract appreciably when they pass through an opening whose width W is small enough to make the ratio sufficiently large. This is indeed the case, as Figure 27-19 illustrates. In this picture, it is assumed that parallel rays (or plane wave fronts) of light fall on a very narrow slit and illuminate a viewing screen that is located far from the slit. Part a of the drawing shows what would happen if light were not diffracted: it would pass through the slit without bending around the edges and produce an image of the slit on the screen. Part b shows what actually happens. The light diffracts around the edges of the slit and brightens regions on the screen that are not directly opposite the slit. The diffraction pattern on the screen consists of a bright central band, accompanied by a series of narrower faint fringes that are parallel to the slit itself.

Figure 27-19 (a) If light were to pass through a very narrow slit without being diffracted, only the region on the screen directly opposite the slit would be illuminated. (b) Diffraction causes the light to bend around the edges of the slit into regions it would not otherwise reach, forming a pattern of alternating bright and dark fringes on the screen. The slit width has been exaggerated for clarity.

To help explain how the pattern of diffraction fringes arises, Figure 27-20 shows a top view of a plane wave front approaching the slit and singles out five sources of Huygens wavelets. Consider how the light from these five sources reaches the midpoint on the screen. To simplify things, the screen is assumed to be so far from the slit that the rays from each Huygens source are nearly parallel.* Then, all the wavelets travel virtually the same distance to the midpoint, arriving there in phase. As a result, constructive interference creates a bright central fringe on the screen, directly opposite the slit.

Figure 27-20 A plane wave front is incident on a single slit. This top view of the slit shows five sources of Huygens wavelets. The wavelets travel toward the midpoint of the central bright fringe on the screen, as the red rays indicate. The screen is very far from the slit.

The wavelets emitted by the Huygens sources in the slit can also interfere destructively on the screen, as Figure 27-21 illustrates. Part a shows light rays directed from each source toward the first dark fringe. The angle gives the position of this dark fringe relative to the line between the midpoint of the slit and the midpoint of the central bright fringe. Since the screen is very far from the slit, the rays from each Huygens source are nearly parallel and oriented at nearly the same angle , as in part b of the drawing. The wavelet from source 1 travels the shortest distance to the screen, while that from source 5 travels the farthest. Destructive interference creates the first dark fringe when the extra distance traveled by the wavelet from source 5 is exactly one wavelength, as the colored right triangle in the drawing indicates. Under this condition, the extra distance traveled by the wavelet from source 3 at the center of the slit is exactly one-half of a wavelength.

Figure 27-21 These drawings pertain to single-slit diffraction and show how destructive interference leads to the first dark fringe on either side of the central bright fringe. For clarity, only one of the dark fringes is shown. The screen is very far from the slit.

Therefore, wavelets from sources 1 and 3 in Figure 27-21b are exactly out of phase and interfere destructively when they reach the screen. Similarly, a wavelet that originates slightly below source 1 cancels a wavelet that originates the same distance below source 3. Thus, each wavelet from the upper half of the slit cancels a corresponding wavelet from the lower half, and no light reaches the screen. As can be seen from the colored right triangle, the angle locating the first dark fringe is given by , where W is the width of the slit.
Figure 27-22 shows the condition that leads to destructive interference at the second dark fringe on either side of the midpoint on the screen. In reaching the screen, the light from source 5 now travels a distance of two wavelengths farther than the light from source 1. Under this condition, the wavelet from source 5 travels one wavelength farther than the wavelet from source 3, and the wavelet from source 3 travels one wavelength farther than the wavelet from source 1. Therefore, each half of the slit can be treated as the entire slit was in the previous paragraph; all the wavelets from the top half interfere destructively with each other, and all the wavelets from the bottom half do likewise. As a result, no light from either half reaches the screen, and another dark fringe occurs. The colored triangle in the drawing shows that this second dark fringe occurs when . Similar arguments hold for the third- and higher-order dark fringes, with the general result being

| (27.4)

Figure 27-22 In a single-slit diffraction pattern, multiple dark fringes occur on either side of the central bright fringe. This drawing shows how destructive interference creates the second dark fringe on a very distant screen.

Between each pair of dark fringes there is a bright fringe due to constructive interference. The brightness of the fringes is related to the light intensity, just as loudness is related to sound intensity. The intensity of the light at any location on the screen is the amount of light energy per second per unit area that strikes the screen there. Figure 27-23 gives a graph of the light intensity along with a photograph of a single-slit diffraction pattern. The central bright fringe, which is approximately twice as wide as the other bright fringes, has by far the greatest intensity.

Figure 27-23 The photograph shows a single-slit diffraction pattern, with a bright and wide central fringe. The higher-order bright fringes are much less intense than the central fringe, as the graph indicates. (From Michel Cagnet, et al., Atlas of Optical Phenomena, Springer-Verlag, Berlin, 1962)

The width of the central fringe provides some indication of the extent of the diffraction, as Example 6 illustrates.

| Example 6 | | | Single-Slit Diffraction

Light passes through a slit and shines on a flat screen that is located away (see Figure 27-24). The wavelength of the light in a vacuum is . The distance between the midpoint of the central bright fringe and the first dark fringe is y. Determine the width 2y of the central bright fringe when the width of the slit is (a) and (b) .

Figure 27-24 The distance 2y is the width of the central bright fringe.

Reasoning The width of the central bright fringe is determined by two factors. One is the angle that locates the first dark fringe on either side of the midpoint. The other is the distance L between the screen and the slit. Larger values for and L lead to a wider central bright fringe. Larger values of mean greater diffraction and occur when the ratio is larger. Thus, we expect the width of the central bright fringe to be greater when the slit width W is smaller.
Solution

a. | The angle in Equation 27.4 locates the first dark fringe when : . Therefore,

According to Figure 27-24, tan , so the width 2y of the central bright fringe is

b. | Repeating the calculation in part (a) with shows that . As expected for a given wavelength, the width 2y of the central maximum in the diffraction pattern is greater when the width of the slit is smaller.

Need more practice?

Interactive LearningWare27.3
A single slit is illuminated by monochromatic light . The width of the slit is . Determine the angular width of the central bright fringe when the region between the slit and the screen is filled with (a) water and (b) air .

In the production of computer chips, it is important to minimize the effects of diffraction. As Figure 23-34 illustrates, such chips are very small and yet contain enormous numbers of electronic components. Such miniaturization is achieved using the techniques of photolithography. The patterns on the chip are created first on a “mask,” which is similar to a photographic slide. Light is then directed through the mask onto silicon wafers that have been coated with a photosensitive material. The light-activated parts of the coating can be removed chemically, to leave the ultrathin lines that correspond to the miniature patterns on the chip. As the light passes through the narrow slit-like patterns on the mask, the light spreads out due to diffraction. If excessive diffraction occurs, the light spreads out so much that sharp patterns are not formed on the photosensitive material coating the silicon wafer. Ultraminiaturization of the patterns requires the absolute minimum of diffraction, and currently this is achieved by using ultraviolet light, which has a wavelength shorter than that of visible light. The shorter the wavelength , the smaller the ratio , and the less the diffraction. Intensive research programs are under way to develop the technique of X-ray lithography. The wavelengths of X-rays are much shorter than those of ultraviolet light and, thus, will reduce diffraction even more, allowing further miniaturization.

The physics of producing computer chips using photolithography.

Another example of diffraction can be seen when light from a point source falls on an opaque disk, such as a coin (Figure 27-25). The effects of diffraction modify the dark shadow cast by the disk in several ways. First, the light waves diffracted around the circular edge of the disk interfere constructively at the center of the shadow to produce a small bright spot. There are also circular bright fringes in the shadow area. In addition, the boundary between the circular shadow and the lighted screen is not sharply defined but consists of concentric bright and dark fringes. The various fringes are analogous to those produced by a single slit and are due to interference between Huygens wavelets that originate from different points near the edge of the disk.

Figure 27-25 The diffraction pattern formed by an opaque disk consists of a small bright spot in the center of the dark shadow, circular bright fringes within the shadow, and concentric bright and dark fringes surrounding the shadow.

Check Your Understanding 4

A diffraction pattern is produced on a viewing screen by using a single slit with blue light. Does the pattern broaden or contract (become narrower) when (a) the blue light is replaced by red light and (b) the slit width is increased? (The answers are given at the end of the book.)
Background: When light passes through a single slit, the extent to which it is bent, or diffracted, depends on two parameters: the wavelength of the light and the width of the slit. This question involves 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.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.