10.2 Huygens's Principle: Diffraction

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Learning Objectives

By the end of this section, you will be able to do the following:

Discuss the propagation of transverse waves
Discuss Huygens’s principle
Explain the bending of light

The information presented in this section supports the following AP® learning objectives and science practices:

6.C.4.1 The student is able to predict and explain, using representations and models, the ability or inability of waves to transfer energy around corners and behind obstacles in terms of the diffraction property of waves in situations involving various kinds of wave phenomena, including sound and light. (S.P. 6.4, 7.2)

Figure 10.4 shows how a transverse wave looks as viewed from above and from the side. A light wave can be imagined to propagate like this, although we do not actually see it wiggling through space. From above, we view the wavefronts, or wave crests, as we would by looking down on the ocean waves. The side view would be a graph of the electric or magnetic field. The view from above is perhaps the most useful in developing concepts about wave optics.

The figure contains three images. The first image, labeled view from above, represents a wave viewed from above as a series of thin, straight strips arranged adjacent to each other across the page. The color of the strips changes gradually from a darker blue near the crests of the waves to white near the troughs of the waves. A single black horizontal arrow points from left to right across the image. The second image, labeled view from side, shows a typical sine curve oscillating above and below a black a

Figure 10.4 A transverse wave, such as an electromagnetic wave like light, as viewed from above and from the side. The direction of propagation is perpendicular to the wavefronts, or wave crests, and is represented by an arrow like a ray.

The Dutch scientist Christiaan Huygens (1629–1695) developed a useful technique for determining in detail how and where waves propagate. Starting from some known position, Huygens’s principle states that every point on a wavefront is a source of wavelets that spread out in the forward direction at the same speed as the wave itself. The new wavefront is a line tangent to all of the wavelets.

Figure 10.5 shows how Huygens’s principle is applied. A wavefront is the long edge that moves, for example, the crest or the trough. Each point on the wavefront emits a semicircular wave that moves at the propagation speed $v .$ These are drawn at a time $t$ later, so that they have moved a distance $s = vt .$ The new wavefront is a line tangent to the wavelets and is where we would expect the wave to be a time $t$ later. Huygens’s principle works for all types of waves, including water waves, sound waves, and light waves. We will find it useful not only in describing how light waves propagate, but also in explaining the laws of reflection and refraction. In addition, we will see that Huygens’s principle tells us how and where light rays interfere.

This figure shows two straight vertical lines, with the left line labeled old wavefront and the right line labeled new wavefront. In the center of the image, a horizontal black arrow crosses both lines and points to the right. The old wavefront line passes through eight evenly spaced dots, with four dots above the black arrow and four dots below the black arrow. Each dot serves as the center of a corresponding semicircle, and all eight semicircles are the same size. The point on each semicircle that is on

Figure 10.5 Huygens’s principle applied to a straight wavefront. Each point on the wavefront emits a semicircular wavelet that moves a distance

s = v t .

The new wavefront is a line tangent to the wavelets.

Figure 10.6 shows how a mirror reflects an incoming wave at an angle equal to the incident angle, verifying the law of reflection. As the wavefront strikes the mirror, wavelets are first emitted from the left part of the mirror and then the right. The wavelets closer to the left have had time to travel farther, producing a wavefront traveling in the direction shown.

The figure shows a grid pattern made of dots. The overall grid pattern would be square were its upper-right four dots not cut off by a gray solid rectangle oriented at forty five degrees counterclockwise from the vertical. Semicircles representing wavelets are centered on each dot. Arrows indicate that the wavelets approach the angled surface from the left and then reflect downward.

Figure 10.6 Huygens’s principle applied to a straight wavefront striking a mirror. The wavelets shown were emitted as each point on the wavefront struck the mirror. The tangent to these wavelets shows that the new wavefront has been reflected at an angle equal to the incident angle. The direction of propagation is perpendicular to the wavefront, as shown by the downward-pointing arrows.

The law of refraction can be explained by applying Huygens’s principle to a wavefront passing from one medium to another (see Figure 10.7). Each wavelet in the figure was emitted when the wavefront crossed the interface between the media. Since the speed of light is smaller in the second medium, the waves do not travel as far in a given time, and the new wavefront changes direction as shown. This explains why a ray changes direction to become closer to the perpendicular when light slows down. Snell’s law can be derived from the geometry in Figure 10.7, but this is left as an exercise for ambitious readers.

The figure shows two media separated by a horizontal line labeled surface. The upper medium is labeled medium one and the lower medium is labeled medium two. A vertical dotted line cuts through both media and is perpendicular to the surface. The point where the dotted line crosses the surface between the media will be called the point of contact. In medium one, a ray pointing down and to the right makes an abrupt turn at the point of contact. The path of the ray makes an angle theta sub one with the dotte

Figure 10.7 Huygens’s principle applied to a straight wavefront traveling from one medium to another where its speed is less. The ray bends toward the perpendicular, since the wavelets have a lower speed in the second medium.

What happens when a wave passes through an opening, such as light shining through an open door into a dark room? For light, we expect to see a sharp shadow of the doorway on the floor of the room, and we expect no light to bend around corners into other parts of the room. When sound passes through a door, we expect to hear it everywhere in the room and, thus, expect that sound spreads out when passing through such an opening (see Figure 10.8). What is the difference between the behavior of sound waves and light waves in this case? The answer is that light has very short wavelengths and acts like a ray. Sound has wavelengths on the order of the size of the door and bends around corners—for frequency of 1,000 Hz, $λ = c / f = (330 m / s) / (1,000 s^{- 1}) = 0 . 33 m,$ about three times smaller than the width of the doorway.

Part a of the figure is a view from above of a diagram of a wall in which is cut an open door. The wall extends from the bottom of the diagram to the top, and the door forms a gap in the wall. The door itself is opened to the left and is positioned about forty five degrees from the wall on which it pivots. From the left comes a bright light, which is labeled small lambda, and the door and wall create sharp shadows by blocking this light. The edges of these shadows are labeled straight-edge shadows. Some o

Figure 10.8 (a) Light passing through a doorway makes a sharp outline on the floor. Since light’s wavelength is very small compared with the size of the door, it acts like a ray. (b) Sound waves bend into all parts of the room, a wave effect, because their wavelength is similar to the size of the door.

If we pass light through smaller openings, often called slits, we can use Huygens’s principle to see that light bends as sound does (see Figure 10.9). The bending of a wave around the edges of an opening or an obstacle is called diffraction. Diffraction is a wave characteristic and occurs for all types of waves. If diffraction is observed for some phenomenon, it is evidence that the phenomenon is a wave. Thus the horizontal diffraction of the laser beam after it passes through slits in Figure 10.3 is evidence that light is a wave.

Three related diagrams showing how waves spread out when passing through various-size openings. The first diagram shows wavefronts passing through an opening that is wide compared to the distance between successive wavefronts. The wavefronts that emerge on the other side of the opening have minor bending along the edges. The second diagram shows wavefronts passing through a smaller opening. The waves experience more bending. The third diagram shows wavefronts passing through an opening that has a size sim

Figure 10.9 Huygens’s principle applied to a straight wavefront striking an opening. The edges of the wavefront bend after passing through the opening, a process called diffraction. The amount of bending is more extreme for a small opening, consistent with the fact that wave characteristics are most noticeable for interactions with objects about the same size as the wavelength.

Making Connections: Diffraction

Diffraction of light waves passing though openings is illustrated in Figure 10.9. But the phenomenon of diffraction occurs in all waves, including sound and water waves. We are able to hear sounds from nearby rooms as a result of diffraction of sound waves around obstacles and corners. The diffraction of water waves can be visually seen when waves bend around boats.

As shown in Figure 10.8, the wavelengths of the different types of waves affect their behavior and diffraction. In fact, no observable diffraction occurs if the wave’s wavelength is much smaller than the obstacle or slit. For example, light waves diffract around extremely small objects but cannot diffract around large obstacles, as their wavelength is very small. On the other hand, sound waves have long wavelengths and hence can diffract around large objects.

10.2 Huygens's Principle: Diffraction

10.2 Huygens's Principle: Diffraction

Learning Objectives

Learning Objectives

Making Connections: Diffraction

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