8 Wave Optics

Light’s wave nature

Light is an electromagnetic wave: its electric and magnetic fields oscillate perpendicular to one another and to the direction of travel. In a vacuum, the speed of light, frequency, and wavelength are related by

When light enters a material, its speed and wavelength change, but its frequency remains the same. The wave model is especially useful when light interacts with openings or objects comparable in size to its wavelength. It explains interference, diffraction, and polarization—effects that a simple ray model cannot describe.

Interference and double-slit patterns

Interference occurs when overlapping light waves combine according to superposition: their electric fields add, and the resulting intensity depends on their phase difference. Constructive interference happens when waves arrive in phase, so they reinforce and form a bright region. Destructive interference happens when waves arrive half a cycle out of phase, so they cancel and form a dark region. Stable fringes require coherent waves, which have the same frequency and a steady phase relationship.

In Young’s double-slit experiment, the two slits act as coherent sources. For slit separation $d$ and observation angle $\theta$, the path difference is approximately

Bright fringes occur when the path difference is an integer multiple of the wavelength:

Dark fringes occur halfway between these conditions:

For a screen much farther away than the slit separation, the spacing between adjacent bright fringes is approximately

Here, $D$ is the distance to the screen. For example, if $\lambda=500\,\text{nm}$, $D=2.0\,\text{m}$, and $d=0.25\,\text{mm}$, then $\Delta y\approx4.0\,\text{mm}$. A longer wavelength or a more distant screen spreads the fringes farther apart; increasing the slit separation brings them closer together.

Takeaway: Path difference determines whether overlapping waves reinforce or cancel, while wavelength, screen distance, and slit separation determine fringe spacing.

Diffraction through openings

Diffraction is the spreading of a wave as it passes through an opening or around an obstacle. It becomes more pronounced when the opening or obstacle is comparable in size to the wavelength. This is why sound bends around a doorway more noticeably than visible light: sound has a much larger wavelength.

For a single slit of width $a$, light from different parts of the slit interferes. Dark fringes, or minima, occur at angles satisfying

The central bright maximum lies between the first minima and is wider and brighter than the side maxima. Narrowing the slit widens the diffraction pattern; widening the slit narrows it.

A double slit also has diffraction because each slit has a finite width. The fine interference fringes sit inside a broader single-slit diffraction envelope. Some interference fringes may be absent when an interference maximum coincides with a diffraction minimum.

Takeaway: Slit separation controls the fine double-slit fringe spacing, while slit width affects the broader diffraction envelope.

Polarization and light intensity

Polarization describes the direction in which a transverse wave oscillates. For light, this direction is defined by its electric field. An ideal polarizing filter transmits the electric-field component parallel to its transmission axis and blocks the perpendicular component. Sound in air is longitudinal, so it cannot be polarized in this way.

For polarized light passing through a filter at angle $\theta$ to the light’s polarization, Malus’s law gives the transmitted intensity:

Here, $I_0$ is the incident intensity. Parallel axes transmit the most light, while crossed axes ideally transmit none. Unpolarized light passing through an ideal first polarizer emerges polarized with half its original intensity; a second polarizer then reduces that intensity according to Malus’s law. Polarized sunglasses reduce glare because light reflected from horizontal surfaces is often partially polarized.

Takeaway: A polarizer selects an electric-field direction, and the angle between the selected direction and the light’s polarization determines the transmitted intensity.

• Wave optics