Interference

Understanding how coherent waves combine to produce interference patterns

SPEC 3.3.2.1 • WAVES

Path Difference

Definition

Path difference is the difference in distance travelled by two waves from their sources to a given point.

When two waves meet at a point, they may have travelled different distances. This difference determines whether they interfere constructively or destructively.

Key Concept

Path difference is usually expressed in terms of wavelength (λ). A path difference of 2λ means one wave has travelled two whole wavelengths further than the other.

Constructive & Destructive Interference

The type of interference depends on whether waves arrive in phase or out of phase:

Constructive

Waves arrive in phase

Path difference = nλ

(where n = 0, 1, 2, 3...)

Amplitudes ADD → Bright fringe

Destructive

Waves arrive out of phase

Path difference = (n + ½)λ

(where n = 0, 1, 2, 3...)

Amplitudes CANCEL → Dark fringe

Coherence & Monochromatic Light

Coherent Sources

Sources that have:

The same frequency

A constant phase difference

Required for a stable, observable interference pattern.

Monochromatic

Light of a single wavelength (and therefore single frequency).

Examples: laser light, sodium lamp light.

Monochromatic light is needed for clear, sharp fringes.

Why Coherence Matters

Two separate light bulbs are NOT coherent - they emit light with random phase differences that change billions of times per second. The interference pattern shifts too fast to see.

Common Misconception

Coherent does NOT mean "in phase". Two coherent sources can have ANY phase difference - as long as that difference stays CONSTANT.

Achieving Coherence

There are several ways to produce coherent sources for interference experiments:

Methods for Coherent Light

Use a single source split into two - e.g., double slit. Light from the same source guarantees same frequency and constant phase relationship.

Use a laser - produces coherent, monochromatic light directly. Lasers emit light where all photons are in phase.

For Sound Waves

Two speakers connected to the same signal generator - ensures both emit the same frequency sound wave at the same time, making them coherent sources.

Why Single Source?

When light passes through both slits of a double-slit, it originates from ONE source. This guarantees the waves at both slits have the same frequency and are coherent with each other.

Young's Double-Slit Experiment

In 1801, Thomas Young demonstrated the wave nature of light:

Why This Works

The single slit ensures light reaching both slits comes from the same source - making them coherent. The two slits then act as two coherent sources.

Brightness decreases away from centre due to diffraction envelope

Why Does Brightness Decrease?

The fringes get dimmer further from the centre because of single-slit diffraction. Each slit acts as a diffraction source, creating a diffraction envelope that modulates the intensity of the interference pattern. The central diffraction maximum is brightest, so fringes near the centre are brighter.

How Interference Fringes Form

At different points on the screen, the path difference from the two slits varies:

PositionPath DifferenceResult

Central maximum0Bright

1st bright fringeλBright

1st dark fringe½λDark

2nd bright fringe2λBright

Worked Example01

Two coherent sources S₁ and S₂ emit waves of wavelength 0.15 m. At point P, the distance from S₁ is 4.50 m and from S₂ is 4.95 m. Determine whether constructive or destructive interference occurs at P.

1

Path difference = 4.95 − 4.50 = 0.45 m

2

In wavelengths: 0.45 ÷ 0.15 = 3λ

3

3λ is a whole number of wavelengths (n = 3)

Answer: Constructive interference (path diff = nλ)

Fringe Spacing Formula

Formula

w = \frac{λ D}{s}

wFringe spacing (m)λWavelength (m)DDistance from slits to screen (m)sSlit separation (m)

Understanding the Formula

Fringe spacing increases with: larger λ, larger D

Fringe spacing decreases with: larger s

Important Condition

This formula only applies when D >> s (small angle approximation).

Worked Example02

Light of wavelength 589 nm passes through two slits separated by 0.25 mm. The screen is 1.8 m away. Calculate the fringe spacing.

1

Convert: λ = 589 × 10⁻⁹ m, s = 0.25 × 10⁻³ m

2

Apply: w = λD/s

3

w = (589 × 10⁻⁹ × 1.8)/(0.25 × 10⁻³)

4

w = 4.24 × 10⁻³ m

Answer: w = 4.2 mm

Worked Example03

Slit separation is 0.40 mm, screen distance is 2.0 m, fringe spacing is 2.8 mm. Calculate the wavelength.

1

Rearrange: λ = ws/D

2

λ = (2.8 × 10⁻³ × 0.40 × 10⁻³)/2.0

3

λ = 5.6 × 10⁻⁷ m = 560 nm

Answer: λ = 560 nm (green light)

White Light Interference

When white light (not monochromatic) is used:

Central Fringe

White - all wavelengths have path diff = 0, so all colours constructively interfere at same point.

Other Fringes

Coloured - since w ∝ λ, red light (longer λ) spreads more than violet (shorter λ).

Worked Example04

Fringe spacing is 3.0 mm. What happens if: (a) D doubles, (b) s halves, (c) λ changes from 450 nm to 650 nm?

a)

w ∝ D: D doubles → w = 6.0 mm

b)

w ∝ 1/s: s halves → w = 6.0 mm

c)

w ∝ λ: ratio = 650/450 = 1.44 → w = 4.3 mm

Laser Safety

Lasers produce coherent, monochromatic light - ideal for interference but dangerous.

Danger of Lasers

Lasers are highly luminous over a very small area - concentrated energy

The major risk is eye injury - permanent retinal damage

With high-powered lasers, the body's natural aversion reflex (looking away from bright light) is too slow to prevent damage to the retina

Even reflected beams can be dangerous

Safety Precautions

Do not point laser in people's eyes

Wear protective clothing and eyewear as necessary

Use internationally agreed signage outside and inside laboratories

Stick to institutional safety requirements

Exam Note

You must be aware of laser safety but will NOT be asked how a laser works.

Interference of Sound Waves

Interference can be observed in all types of waves, including sound waves.

Experimental Setup

Two speakers connected to the same signal generator (coherent sources)

The same sound must be played at the same time in each speaker

Moving the Microphone

By moving the microphone from one side to the other, we can measure constructive and destructive interference by whether the sound is loud (constructive) or quiet (destructive).

Understanding EM Radiation

Scientists have changed and updated their understanding of the nature of electromagnetic radiation over time.

Newton's Corpuscles (~1670s)

Isaac Newton theorised light as made up of "corpuscles" - little packets of light (particle theory).

Huygens' Waves (1600s)

Huygens stated his wavefront theory - light propagates as waves spreading out from a source.

Young's Double-Slit Experiment (1801)

Young, with his double-slit experiment, provided evidence of Huygens' wavefront principle. The interference pattern could only be explained if light behaved as a wave.

Understanding EM Radiation (cont.)

Einstein's Photons (1905)

Einstein provided evidence, through the photoelectric effect, supporting Newton's corpuscular theory. This was the beginnings of quantum theory - light comes in discrete packets called photons.

Compton's Scattering Experiment (1923)

Further on in the 20th century, Compton scattering (a photon scattering off an electron) provided more evidence for the quantisation of light.

Modern Understanding: Wave-Particle Duality

Light exhibits both wave and particle properties:

Wave behaviour: interference, diffraction

Particle behaviour: photoelectric effect, Compton scattering

Exam Note

You should appreciate how scientific understanding of electromagnetic radiation has changed over time.

Practice

Question: Two coherent sources emit waves of wavelength 2.5 cm. At point X, waves travel 45.0 cm from source 1 and 48.75 cm from source 2. Is X a maximum or minimum?

1

Path diff = 48.75 − 45.0 = 3.75 cm

2

In wavelengths: 3.75 ÷ 2.5 = 1.5λ

3

1.5λ = (1 + ½)λ → destructive

Answer: Minimum (destructive)

Practice

Question: λ = 632 nm, s = 0.30 mm, D = 1.5 m. Find: (a) fringe spacing, (b) distance to 4th bright fringe.

a)

w = λD/s = (632×10⁻⁹ × 1.5)/(0.30×10⁻³) = 3.2 mm

b)

Distance = 4w = 4 × 3.16 = 12.6 mm

Summary

Key Formulas & Definitions

Term/ConditionDefinition/Formula

Coherent sourcesSame frequency, constant phase difference

MonochromaticSingle wavelength light

ConstructivePath diff = nλ (bright)

DestructivePath diff = (n+½)λ (dark)

Fringe spacingw = λD/s

White Light

Central white fringe; coloured fringes with red outside, violet inside.

Historical

Newton → Huygens → Young → Einstein → wave-particle duality.