§ 10. Interference of Light Wavefronts (Young's Double-Slit) (Grade 11)
Superposition of two spherical light wavefronts, maxima & minima. When two coherent light waves overlap, the spatial intensity distribution is governed by the phase difference between their wavefronts.
Theory Summary & Scientific Mechanism
Thomas Young’s double-slit experiment (1801) established the wave nature of light. When a plane monochromatic wavefront illuminates two narrow slits S₁ and S₂ separated by distance d, Huygens’ principle turns the slits into two coherent cylindrical wave sources with identical frequency and locked phase.
At position x on a screen at distance L ≫ d, the two waves arrive with path difference Δ = r₂ - r₁ = dsin θ ≈ d(x)/(L) and phase difference Δφ = (2π)/(λ)Δ. Superposition redistributes energy into fringes of intensity I(x) = 4I₀ cos ² ≤ft((π d · x)/(λ L) ): constructive maxima (Δ = mλ) reach 4I₀, while destructive minima (Δ = (m + (1)/(2))λ) cancel to zero (I = 0).
The fringe spacing between adjacent bright bands is Δ x = (λ L)/(d). Increasing wavelength λ (shifting from violet toward red light) widens the interference fringes, whereas increasing slit separation d packs the fringes closer together.
Everyday Analogy
Two pebbles dropped side by side in a pond create overlapping ripples: where crest meets crest, waves double up; where crest meets trough, they cancel out! With light, this creates bright and dark stripes on a screen.
Interactive Question & Derivation
In Young's experiment, two coherent light wavefronts from slits S₁ and S₂ arrive at a point on the screen with an optical path difference Δ d = 1.5λ (three half-wavelengths). What is observed at this point?
- A: An interference maximum (I = 4I₀), because the waves reinforce each other
- B: An interference minimum (dark fringe, I = 0), because the crest of one wavefront meets the trough of the other in anti-phase
- C: Uniform illumination I = 2I₀
Study Hint: Check Young’s fringe-spacing formula Δ y = (λ L)/(d): is slit separation d in the numerator or the denominator?
Proof & Explanation: When the path difference equals an odd number of half-wavelengths Δ d = (m + (1)/(2))λ, the phase difference is Δφ = π, 3π, … At this point, the crest of the wavefront from the first slit overlaps the trough from the second slit, resulting in destructive interference (I = 0).
Key Equations & 3D Simulation Parameters
Δ d = mλ (maximum), Δ d = ≤ft(m + (1)/(2) )λ (minimum)
- Light wavelength λ: 400 … 700 nm (default: 520 nm)
- Slit separation d: 1 … 3.6 mm (default: 2.2 mm)
- Distance to screen L: 4.5 … 7.5 m (default: 6 m)
- Observation point on screen z: -3.2 … 3.2 mm (default: 0.8 mm)