27.2 Huygens's Principle: Diffraction

Wave Propagation and Huygens's Principle

Huygens's principle provides a geometric method for predicting where a wave will travel next. By treating every point on a wavefront as a tiny source of new "wavelets," you can explain how waves reflect, refract, and bend around obstacles. That bending is called diffraction, and it's central to understanding wave optics.

Propagation of Transverse Waves

Transverse waves oscillate perpendicular to the direction the wave travels. Light waves, electromagnetic waves, and waves on a string (guitar strings, jump ropes) are all transverse.

• Energy transfers through the medium, but the matter itself doesn't travel with the wave. Each particle oscillates around its equilibrium position, nudging its neighbor, which nudges the next, and so on.
• Wave speed depends on the properties of the medium. For a stretched string, the speed is:

$v = \sqrt{\frac{T}{\mu}}$

where $T$ is the tension in the string and $\mu$ is the linear mass density (mass per unit length). Higher tension means a faster wave; more mass per length means a slower one. This is why tightening a guitar string raises its pitch.

Huygens's Principle in Wave Behavior

Huygens's principle states that every point on a wavefront acts as a source of secondary wavelets. These wavelets spread out in all directions at the same speed as the original wave. The new wavefront is the surface tangent to (the "envelope" of) all those wavelets.

Think of dropping a pebble into a pond. The expanding ring is a wavefront. Huygens says you can predict the next ring by imagining tiny ripples launched from every point on the current ring.

This single idea explains several wave behaviors:

• Reflection: Secondary wavelets launched from points along a boundary combine to form the reflected wavefront, which is why the angle of incidence equals the angle of reflection.
• Refraction: When a wave enters a medium where its speed changes, the wavelets on one side of the wavefront travel slower (or faster), causing the wavefront to pivot and change direction. This is how light bends when passing through a prism.
• Diffraction: Wavelets from the edges of an obstacle or opening spread into the "shadow" region, causing the wave to bend around corners.

The Huygens-Fresnel principle extends this idea by also accounting for the amplitude and phase of each secondary wavelet, which lets you calculate actual intensity patterns.

Diffraction of Light Around Obstacles

Diffraction becomes significant when a wave encounters an obstacle or opening whose size is comparable to the wave's wavelength. Because visible light has very short wavelengths (roughly 400–700 nm), you need very small openings or obstacles to see noticeable diffraction, like a single strand of hair or a narrow slit.

Single-slit diffraction:

1. Light passes through a narrow slit of width $a$.
2. Every point across the slit acts as a source of secondary wavelets (per Huygens's principle).
3. These wavelets interfere with each other, producing a pattern on a distant screen: a broad central bright fringe flanked by alternating dark and bright fringes that get dimmer as you move outward.
4. Dark fringes (minima) occur at angles satisfying:

$\sin \theta = \frac{m\lambda}{a}$

where $m = \pm 1, \pm 2, \pm 3, \ldots$ and $\lambda$ is the wavelength. A narrower slit produces a wider central fringe, because the light spreads out more.

Double-slit diffraction (Young's experiment):

1. Light passes through two narrow slits separated by a distance $d$.
2. The two slits act as coherent sources, meaning their wavelets have a fixed phase relationship.
3. Constructive interference (bright fringes) occurs at angles where:

$d \sin \theta = m\lambda$

where $m = 0, \pm 1, \pm 2, \ldots$ The result is a series of evenly spaced bright and dark fringes. The overall pattern is also shaped by the single-slit diffraction envelope of each individual slit.

Diffraction gratings:

A diffraction grating has many equally spaced slits (hundreds or thousands per millimeter). It obeys the same condition for bright fringes:

$d \sin \theta = m\lambda$

but because there are so many slits, the bright fringes become very sharp and intense, called spectral orders ($m = 0, 1, 2, \ldots$). This is why a CD or DVD produces rainbow-like colors when light hits its surface: the closely spaced tracks act as a diffraction grating.

Types of Diffraction

• Fraunhofer diffraction applies when both the light source and the observation screen are effectively at infinity (or you use lenses to achieve parallel rays). This is the simpler case and the one described by the formulas above.
• Fresnel diffraction applies when the source or screen is close to the diffracting object. The math is more complex because you can't assume parallel rays, but Huygens's principle still governs the behavior.

Most introductory problems deal with Fraunhofer diffraction, so unless told otherwise, you can assume the far-field (Fraunhofer) setup.