Diffraction and Interference

Diffraction and Interference

Diffraction is the spreading of waves as they pass through a gap or around an obstacle. Interference is the superposition of two or more coherent waves, producing regions of reinforcement and cancellation.

Conditions for Observable Diffraction

Diffraction is most significant when the gap width is comparable to the wavelength of the wave. For visible light (λ ≈ 400–700 nm), this requires very narrow slits.

Coherence

For a stable, observable interference pattern, the sources must be coherent:

• Same frequency (and therefore same wavelength)
• Constant phase difference between them

Two separate light sources are never coherent because they emit photons randomly. Instead, we use a single source split into two (e.g., Young's double slit).

Young's Double Slit Experiment

Thomas Young (1801) demonstrated the wave nature of light by passing monochromatic light through two narrow, closely spaced slits.

Setup: Single slit (to create a coherent line source) → double slit → screen

The two slits act as coherent sources. Waves from the two slits travel different path lengths to reach a point on the screen.

Constructive interference (bright fringe) occurs when the path difference = nλ (n = 0, 1, 2, ...)

Destructive interference (dark fringe) occurs when the path difference = (n + ½)λ

The Double Slit Equation

For small angles (which is valid when D >> a):

fringe spacing w = λD / a

where:

• w = distance between adjacent bright fringes (m)
• λ = wavelength (m)
• D = distance from slits to screen (m)
• a = slit separation (m)

Worked Example

Monochromatic light of wavelength 589 nm passes through a double slit with slit separation 0.50 mm. The screen is 1.2 m away. Calculate the fringe spacing.

w = λD/a = (589 × 10⁻⁹ × 1.2) / (0.50 × 10⁻³)

w = 7.068 × 10⁻⁷ / 5.0 × 10⁻⁴ = 1.41 × 10⁻³ m = 1.41 mm

White Light Fringes

With white light:

• The central maximum is white (all wavelengths have zero path difference)
• Other fringes show spectral colours, with violet closest to the centre (smallest λ, smallest w) and red furthest (largest λ)
• After a few orders, fringes overlap and the pattern becomes washed out

Single Slit Diffraction

A single slit of width b produces a diffraction pattern with:

• A wide, bright central maximum
• Narrower, dimmer subsidiary maxima on each side
• Dark minima at angles where: b sin θ = nλ (n = 1, 2, 3, ...)

The central maximum is twice as wide as the subsidiary maxima. The intensity of subsidiary maxima falls off rapidly.

The Diffraction Grating

A diffraction grating has many equally spaced slits (typically 300–1000 per mm). It produces much sharper and brighter maxima than a double slit.

The grating equation: d sin θ = nλ

where:

• d = slit spacing = 1/(number of slits per metre)
• θ = angle of the nth order maximum from the central maximum
• n = order number (0, 1, 2, ...)
• λ = wavelength

Maximum Number of Orders

Since sin θ cannot exceed 1: n_max = d/λ (rounded down to nearest integer)

Worked Example — Diffraction Grating

A grating has 600 lines per mm. Light of wavelength 520 nm is incident normally. Find the angle of the second-order maximum.

d = 1/600000 = 1.667 × 10⁻⁶ m

d sin θ = nλ → sin θ = nλ/d = (2 × 520 × 10⁻⁹) / (1.667 × 10⁻⁶)

sin θ = 0.6240 → θ = 38.6°

Maximum order: n_max = d/λ = 1.667 × 10⁻⁶ / 520 × 10⁻⁹ = 3.2 → n_max = 3

Comparing Double Slit and Diffraction Grating

Feature	Double slit	Diffraction grating
Number of slits	2	Hundreds or thousands
Maxima	Broad, low intensity	Sharp, bright
Resolution	Poor — fringes merge	High — wavelengths clearly separated
Use	Demonstrate interference	Precise wavelength measurement

Applications of Diffraction Gratings

• Spectroscopy: identifying elements from emission/absorption spectra
• Measuring wavelength: using d sin θ = nλ with a known grating
• X-ray crystallography: atomic planes in crystals act as a 3D diffraction grating (Bragg's law: 2d sin θ = nλ)
• CDs/DVDs: the track spacing acts as a reflection grating, producing rainbow colours

Path Difference and Phase Difference

The relationship between path difference and phase difference:

Phase difference = (2π/λ) × path difference

• Path difference of λ → phase difference of 2π (in phase)
• Path difference of λ/2 → phase difference of π (antiphase)

Required Practical: Measuring Wavelength with a Diffraction Grating

1. Set up a laser, grating, and screen

2. Measure the distance D from grating to screen

3. Measure the distance x from central maximum to nth order maximum

4. Calculate θ = arctan(x/D)

5. Use d sin θ = nλ to find λ

6. Repeat for multiple orders and average

Safety: Never look directly into the laser beam; use a Class 2 laser; display warning signs.