# Lecture 18: Wave Optics: Interference

## Physics II — Electromagnetism, Optics & Modern Physics

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Explain the principle of superposition and the conditions for constructive and destructive interference
2. Describe and analyze Young's double-slit experiment
3. Calculate the positions of bright and dark fringes in double-slit interference
4. Analyze thin-film interference and determine conditions for constructive/destructive outcomes
5. Explain the role of coherence in producing observable interference patterns

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## Lecture Content

### I. Wave Nature of Light and Superposition

Light exhibits both wave and particle properties; this lecture focuses on its wave behavior. The **superposition principle** states that when two or more waves overlap, the resultant displacement at any point is the algebraic sum of the individual displacements.

**Interference** is the combination of two or more waves to produce a resultant wave. **Constructive interference** occurs when waves are in phase (crests align with crests), causing the amplitude to increase. The condition for constructive interference is a path difference equal to m lambda (m = 0, 1, 2, ...), corresponding to a phase difference of 2m pi. **Destructive interference** occurs when waves are out of phase (crests align with troughs), causing the amplitude to decrease or cancel entirely. The condition is a path difference of (m + 1/2) lambda, corresponding to a phase difference of (2m + 1) pi.

For interference patterns to be observable, the sources must be **coherent**: they must have the same frequency and maintain a constant phase relationship. Lasers are highly coherent, while incandescent bulbs are incoherent.

### II. Young's Double-Slit Experiment (1801)

Thomas Young demonstrated the wave nature of light by passing monochromatic light through two narrow, closely spaced slits. Each slit acts as a source of coherent cylindrical waves (by Huygens' principle), and the waves from the two slits overlap on a distant screen, producing an **interference pattern** of alternating bright and dark fringes.

In the standard setup, the slits are separated by distance d, the screen is at distance L (with L >> d), and the light has wavelength lambda. The **path difference** between the waves arriving at a point on the screen at angle theta is Delta r = d sin(theta).

**Bright fringes** (constructive interference) appear where d sin(theta) = m lambda, with m = 0, +/-1, +/-2, ... being the order number. The central maximum occurs at m = 0 (theta = 0). On the screen, the positions of bright fringes are y_m = m lambda L / d for small angles. **Dark fringes** (destructive interference) appear where d sin(theta) = (m + 1/2) lambda, at positions y = (m + 1/2) lambda L / d.

The **fringe spacing** is Delta y = lambda L / d. The fringes are more widely spaced when the wavelength is longer, the screen is farther away, or the slit separation is smaller.

<image>A diagram of Young's double-slit experiment. Monochromatic light from the left passes through two narrow slits (S_1 and S_2) separated by distance d. Semicircular wavefronts emerge from each slit and overlap. On a screen at distance L to the right, an interference pattern of alternating bright and dark fringes is shown. The path difference Delta r = d sin(theta) is illustrated with a right triangle formed by the two slit-to-screen rays and a line perpendicular to one ray from the other slit. The central bright fringe (m = 0) and first-order fringes (m = +/-1) are labeled. The intensity pattern (sinusoidal envelope) is shown alongside the screen.</image>

### III. Intensity in Double-Slit Interference

The electric fields from the two slits add as vectors (phasors). For two sources of equal amplitude E_0, the total field is E_total = 2E_0 cos(delta/2), where delta = 2 pi d sin(theta) / lambda is the phase difference. Since intensity is proportional to the square of the electric field, I = I_max cos^2(delta/2) = I_max cos^2(pi d sin(theta) / lambda), where I_max = 4 I_0 and I_0 is the intensity from a single slit.

The resulting pattern is a series of equally spaced bright fringes with cosine-squared intensity. Energy is conserved: no energy is destroyed in interference; rather, energy is redistributed from the dark fringes to the bright fringes. When N slits are used instead of two (a diffraction grating), the principal maxima become much sharper and brighter, with intensity proportional to N^2.

### IV. Thin-Film Interference

Interference also occurs between light reflected from the top and bottom surfaces of a thin transparent film. Two effects determine the phase relationship between the two reflected beams.

First, the **path difference** arises because light traveling through the film traverses an extra optical path of 2nt, where n is the film's index of refraction and t is its thickness (at near-normal incidence). Second, a **phase change on reflection** may occur: reflection from a higher-n medium introduces a pi phase shift (equivalent to a half-wavelength shift), while reflection from a lower-n medium produces no phase shift. The net phase shift depends on how many of the two reflections involve a transition to a higher-n medium.

For **constructive interference** (bright reflection) when there is one phase-shifting reflection, the condition is 2nt = (m + 1/2) lambda. When there are zero or two phase-shifting reflections, the condition is 2nt = m lambda. The conditions for **destructive interference** are the opposites.

Thin-film interference explains many everyday phenomena. Oil on water displays rainbow colors because different thicknesses satisfy the constructive interference condition for different wavelengths. Soap bubbles show colorful patterns from varying film thickness across their surface. Anti-reflection coatings are designed so that reflected waves cancel, using a film thickness of t = lambda/(4n). Newton's rings are circular interference fringes formed between a curved lens surface and a flat glass plate.

<image>A thin-film interference diagram. A thin film of thickness t and index n_film is sandwiched between medium 1 (above, index n_1) and medium 2 (below, index n_2). An incident ray hits the top surface at near-normal incidence. Ray 1 reflects off the top surface (with or without a phase shift, depending on whether n_film > n_1). Ray 2 transmits into the film, reflects off the bottom surface, and exits back through the top surface. The extra optical path for Ray 2 is 2n_film*t. The two reflected rays interfere. A callout box summarizes: phase shift of pi occurs upon reflection from a higher-n medium; no shift from a lower-n medium. Below, the conditions for constructive and destructive interference are listed for the case of one phase-shifting reflection.</image>

### V. Applications of Thin-Film Interference

**Anti-reflection coatings** use a quarter-wavelength film (t = lambda/(4n_film)) to create destructive interference for reflected light, maximizing transmission. They are widely used on camera lenses, eyeglasses, and solar cells. For equal reflection amplitudes, the ideal film index is n_film = sqrt(n_glass).

**Highly reflective coatings (dielectric mirrors)** use alternating layers of high-n and low-n materials, each a quarter wavelength thick, to produce constructive interference of all reflected beams. These can achieve reflectivities exceeding 99.9% at a specific wavelength.

The **Michelson interferometer** splits a beam into two paths, reflects them back, and recombines them. It is exquisitely sensitive to path length differences on the order of lambda/2 and is used to measure tiny displacements, indices of refraction, and in the historic Michelson-Morley experiment that found no evidence for the luminiferous ether.

### VI. Coherence and Practical Considerations

**Temporal coherence** describes how long a wave train maintains a constant phase, and it is related to the bandwidth of the source. A narrow bandwidth gives a long coherence length: L_c = c / Delta f, which is approximately lambda^2 / Delta lambda. Lasers have very long coherence lengths (meters to kilometers), while white light has a very short coherence length (roughly 1 micrometer).

**Spatial coherence** describes how uniform the phase is across the wavefront. Point sources have high spatial coherence, while extended sources have low spatial coherence.

For thin-film interference with white light, only a few orders of interference are visible before the pattern washes out, and different colors appear at different thicknesses, producing the colorful patterns seen on soap bubbles and oil films. For double-slit interference, the source must be coherent and nearly monochromatic to produce clear fringes. White light produces a central white fringe flanked by colored fringes on either side.