27.1 The Wave Aspect of Light: Interference

Wave Behavior of Light

Light behaves as both a particle and a wave. Its wave properties, including wavelength, frequency, and speed, explain how it interacts with objects through reflection, refraction, diffraction, and interference.

Understanding light's wave nature is the foundation for this unit. A key concept here is the index of refraction, which relates light's speed in a vacuum to its speed in a medium. It explains why light bends and why its wavelength changes when moving between materials.

Wave Properties of Light

Light waves are characterized by four quantities: wavelength ($\lambda$), frequency ($f$), speed ($v$), and amplitude. These are connected by the fundamental wave equation:

$v = \lambda f$

The speed of light depends on the medium it travels through:

• In a vacuum, light travels at $c = 3 \times 10^8$ m/s.
• In any other medium (water, glass, etc.), light slows down according to $v = c/n$, where $n$ is the index of refraction of that medium.

When light enters a new medium, its frequency stays the same but its wavelength changes. The wavelength in a medium is:

$\lambda_n = \lambda_0 / n$

where $\lambda_0$ is the wavelength in vacuum. Since $n$ is always ≥ 1, the wavelength in a medium is always shorter than (or equal to) the wavelength in vacuum.

Light Interactions with Objects

How light behaves when it hits something depends on the size of that object compared to light's wavelength.

• Ray behavior (geometric optics) applies when the object is much larger than the wavelength. Light travels in straight lines, and you can explain reflection and refraction using simple ray diagrams. Think mirrors and lenses.
• Wave behavior (physical optics) shows up when the object is comparable to or smaller than the wavelength. Two major wave phenomena emerge here:

Diffraction is the bending of light around edges or through small openings. A single slit produces a pattern of alternating bright and dark fringes. A double slit produces a more distinct interference pattern (more on this below).

Interference is what happens when two or more waves overlap (superpose):

• Constructive interference occurs when waves are in phase (crests align with crests). The waves reinforce each other, producing bright fringes.
• Destructive interference occurs when waves are out of phase (crests align with troughs). The waves cancel, producing dark fringes.

Wave Characteristics and Interference

A few terms you need to know for understanding interference:

• Phase describes where a wave is in its cycle at a given moment. Two waves that are "in phase" have their peaks and troughs aligned. The phase difference between waves determines whether interference is constructive or destructive.
• Coherence is the degree to which two waves maintain a constant phase relationship over time. You need coherent light sources to produce stable, observable interference patterns. Two random light bulbs won't do it; their phases shift too rapidly.
• Polarization describes the direction in which a light wave's electric field oscillates. While not central to interference calculations, it matters because only waves oscillating in the same direction can fully interfere with each other.

Young's double-slit experiment (Thomas Young, 1801) is the classic demonstration of light's wave nature. By shining coherent light through two narrow slits, Young observed an interference pattern of bright and dark bands on a screen. This was powerful evidence that light behaves as a wave, since particles would simply form two bright lines behind the slits.

Interaction of Light with Matter

Index of Refraction Calculations

The index of refraction ($n$) is defined as:

$n = c / v$

It tells you how much slower light travels in a medium compared to vacuum. For example, water has $n \approx 1.33$, meaning light travels about 1.33 times slower in water than in vacuum.

Here's how to use it in calculations:

• Speed in a medium: $v = c / n$. A higher $n$ means a lower speed. Light moving from air ($n \approx 1.00$) into water ($n \approx 1.33$) slows down.
• Wavelength in a medium: $\lambda_n = \lambda_0 / n$. A higher $n$ means a shorter wavelength. If light with $\lambda_0 = 600$ nm enters water, its wavelength becomes $600 / 1.33 \approx 451$ nm.
• Frequency does not change when light moves between media. This is a point that trips people up. The frequency is set by the source and stays constant.

Snell's Law describes how light bends at the boundary between two media:

$n_1 \sin \theta_1 = n_2 \sin \theta_2$

where $\theta_1$ is the angle of incidence and $\theta_2$ is the angle of refraction, both measured from the normal (the line perpendicular to the surface). When light enters a medium with a higher index of refraction, it bends toward the normal. When it enters a medium with a lower index, it bends away.