80 Huygens’s Principle: Diffraction

Although we often represent light as traveling along straight rays, it is fundamentally an electromagnetic wave. Like all transverse waves, light consists of oscillating electric and magnetic fields that are perpendicular to each other and to the direction in which the wave travels. Depending on the situation, it can be helpful to visualize a light wave in different ways.

Figure 80.1 shows three common representations of a transverse wave. The side view illustrates the familiar sinusoidal variation of the electric or magnetic field, while the top view shows the wavefronts—surfaces connecting points that oscillate in phase. In wave optics, the wavefront representation is often the most useful because it allows us to predict how light propagates, reflects, refracts, and bends around obstacles.

Huygens's Principle

One of the most powerful ideas in wave optics was introduced by the Dutch physicist Christiaan Huygens (1629–1695). Instead of treating light as individual rays, Huygens described light as an advancing wavefront. His simple geometric construction allows us to predict how waves propagate through space and how they behave when they encounter mirrors, lenses, or obstacles.

Imagine a straight wavefront moving through space. Every point along that wavefront continuously emits tiny secondary waves called wavelets. After a short time, each wavelet has expanded by the same distance. The envelope, or tangent surface, touching all of these wavelets marks the new position of the wavefront.

This remarkably simple construction accurately predicts the motion of water waves, sound waves, and electromagnetic waves. It also provides a straightforward explanation for many familiar optical phenomena, including reflection, refraction, interference, and diffraction. Throughout this chapter, Huygens's principle will serve as a powerful tool for understanding why light behaves as it does.

Figure 80.2 illustrates how Huygens's principle predicts the motion of a wavefront. A wavefront is a surface connecting all points of a wave that are oscillating in phase, such as the crest of a water wave or the peak of an electromagnetic wave. Instead of following individual rays, Huygens's construction follows the evolution of the entire wavefront.

Suppose a wavefront is observed at some initial time. Every point along that wavefront becomes the source of a secondary wavelet that spreads outward at the same propagation speed v. After a time t, each wavelet has traveled a distance

The new position of the wave is found by drawing a line (or, in three dimensions, a surface) that is tangent to all of these expanding wavelets. This tangent forms the new wavefront. Because every wavelet expands by the same distance, the overall shape of the wavefront is preserved unless the wave encounters a boundary or enters a different medium.

Huygens's principle applies to all types of waves, including water waves, sound waves, seismic waves, and electromagnetic waves. Throughout this chapter, it will provide a simple geometric method for understanding reflection, refraction, diffraction, and interference.

Explaining Reflection with Huygens's Principle

Huygens's principle not only predicts how waves propagate through open space but also explains the familiar law of reflection. When a plane wave strikes a flat mirror, different parts of the wavefront reach the mirror at slightly different times. The first points to touch the mirror immediately begin producing reflected wavelets, while the remaining portions of the incoming wave continue traveling until they also reach the surface.

As these reflected wavelets expand, a new reflected wavefront is formed by drawing a tangent to all of them. The direction of propagation is always perpendicular to the wavefront, so the reflected wave travels away from the surface in a new direction.

This construction naturally leads to one of the fundamental laws of optics:

Huygens's Principle and Refraction

Huygens's principle also provides a simple explanation for refraction, the bending of light as it passes from one transparent medium into another. When a wavefront reaches the boundary between two materials, different parts of the wavefront cross the interface at slightly different times. As each point enters the second medium, it begins producing new wavelets that travel at the speed of light in that material.

If the second medium has a higher index of refraction, light travels more slowly there. Consequently, the wavelets generated inside the second medium do not expand as far during the same time interval as those still traveling in the first medium. When the tangent is drawn to all of the secondary wavelets, the new wavefront is rotated relative to the original one. Because the direction of propagation is always perpendicular to the wavefront, the light ray changes direction.

This construction explains why light bends toward the normal when it enters a slower medium and away from the normal when it enters a faster medium. The same geometric argument leads directly to Snell's law, although the mathematical derivation is beyond the scope of this section.

Why Does Sound Bend Around Corners but Light Usually Does Not?

Imagine standing outside a room with an open doorway. You can easily hear someone speaking inside the room even if they are not directly visible through the doorway. In contrast, you cannot usually see objects hidden around the corner. Why do sound and light behave so differently?

The answer lies in the relationship between the wavelength of the wave and the size of the opening through which it passes.

Visible light has wavelengths between about 380 nm and 760 nm, which are millions of times smaller than the width of a typical doorway. Because the doorway is enormous compared with the wavelength, light travels through it almost exactly as predicted by geometric optics, producing sharp shadows.

Sound waves, however, have much longer wavelengths. For example, a sound with a frequency of 1000 Hz has a wavelength of approximately

which is comparable to the width of a doorway. As a result, sound spreads into the room after passing through the opening, allowing a listener to hear around corners.

Diffraction

The bending of a wave around the edges of an obstacle or after passing through an opening is called diffraction. Diffraction is a fundamental property of all waves, including water waves, sound waves, electromagnetic waves, and even matter waves.

Huygens's principle provides a simple explanation for diffraction. As a wave passes through an opening, every point across that opening becomes the source of new secondary wavelets. These wavelets spread outward beyond the edges of the opening, causing the wave to expand into the region behind the obstacle instead of continuing in a perfectly straight line.

The amount of diffraction depends strongly on the ratio of the wavelength to the size of the opening.

• If the opening is much larger than the wavelength, very little diffraction occurs and the wave travels almost straight ahead.
• If the opening is only a few wavelengths wide, noticeable spreading occurs.
• If the opening is comparable to the wavelength, the wave spreads dramatically in many directions.

The diffraction pattern produced when a laser passes through a narrow slit is one of the strongest experimental demonstrations that light behaves as a wave rather than as a collection of simple rays.

Section Summary

• Light is a transverse electromagnetic wave that can be represented by wavefronts as well as by rays. Wavefronts provide a convenient way to visualize the propagation of light in wave optics.
• Huygens's principle states that every point on a wavefront acts as a source of secondary wavelets that spread outward at the speed of the wave. The new wavefront is the surface tangent to all of these wavelets.
• Huygens's principle provides a wave-based explanation for the laws of reflection and refraction and serves as the foundation for understanding diffraction and interference.
• Diffraction is the bending and spreading of a wave as it passes around an obstacle or through an opening.
• Diffraction becomes increasingly important when the size of an opening or obstacle is comparable to the wavelength of the wave. Because visible light has extremely short wavelengths, diffraction is usually negligible in everyday situations but becomes significant in microscopes, optical instruments, and narrow apertures.