Wave Interference

Superposition principle

When two or more waves are at the same point, the total displacement is the sum of the individual displacements:

y_total(x, t) = y_1(x, t) + y_2(x, t) + ...

For sinusoidal waves of same frequency:

y_1 = A·sin(kx − ωt)
y_2 = A·sin(kx − ωt + φ)
y_total = 2A·cos(φ/2)·sin(kx − ωt + φ/2)

The 2A·cos(φ/2) factor is the resulting amplitude — depends on phase difference φ.

Constructive and destructive interference

Phase difference	Path difference	Amplitude result	Type
0 (same phase)	0, λ, 2λ, ...	2A (max)	Constructive
π/3	λ/6	√3 · A	Partial
π/2	λ/4	√2 · A	Partial
π (opposite)	λ/2, 3λ/2, ...	0	Destructive
2π	λ	2A	Constructive

Young's double-slit experiment

Light from a single source passes through two narrow slits separated by d. On a screen distance L away, bright and dark fringes appear:

Bright fringes at: y = m · λ · L / d  (m = 0, ±1, ±2, ...)
Dark fringes at: y = (m + ½) · λ · L / d

Adjacent bright fringes are spaced Δy = λL/d apart.

For 633 nm laser, d = 0.5 mm, L = 1 m: fringe spacing = 1.27 mm — easily visible.

Thin-film interference (briefly)

Light reflects from front and back of a thin film. Path difference = 2t (thickness). Plus phase shifts at boundaries (depending on n's). For constructive reflection at thin soap film (in air, water-air-water):

2nt = (m + ½) · λ  (constructive reflection)

Different t → different colors reflect strongly. Soap bubbles, oil slicks on water.

Real-world interference

System	How interference is used/observed
Soap bubble colors	Thin-film interference; thickness varies → different colors
Oil slick on water	Thin-film interference
CD/DVD rainbow surface	Diffraction grating + interference of different orders
Noise-canceling headphones	Speaker generates anti-phase wave; sum near zero
Holograms	Reference + object beam interference recorded
Michelson interferometer	Tiny path-length changes detected via fringe shifts
LIGO (gravitational waves)	Mirror movement < 10⁻¹⁸ m detected via interferometry
Anti-reflection coatings (camera lenses)	Coating thickness designed for destructive reflection
Acoustic noise cancellation	Same idea as headphones, on industrial scales

JavaScript — interference calculations

// Sum of two sine waves
function combinedAmplitude(A1, A2, phaseDiff) {
  return Math.sqrt(A1*A1 + A2*A2 + 2*A1*A2*Math.cos(phaseDiff));
}

console.log(`Same phase: ${combinedAmplitude(1, 1, 0)}`);     // 2 (constructive)
console.log(`Opposite: ${combinedAmplitude(1, 1, Math.PI)}`); // 0 (destructive)
console.log(`90° phase: ${combinedAmplitude(1, 1, Math.PI/2)}`); // √2 ≈ 1.414

// Phase difference from path difference
function phaseFromPath(pathDifference, wavelength) {
  return 2 * Math.PI * pathDifference / wavelength;
}

// Path difference of λ/4 → π/2 phase
console.log(`λ/4 path: ${phaseFromPath(0.25, 1)} rad`);  // π/2

// Young's double slit fringe positions
function youngsBrightFringes(slitSeparation, wavelength, screenDistance, num = 5) {
  return Array.from({length: num*2 + 1}, (_, i) => {
    const m = i - num;
    return { m, y: m * wavelength * screenDistance / slitSeparation };
  });
}

// 633 nm laser, 0.5 mm slits, 1 m screen
const fringes = youngsBrightFringes(0.5e-3, 633e-9, 1, 3);
console.log(fringes.map(f => ({ m: f.m, y_mm: (f.y * 1000).toFixed(2) })));
// Spacing 1.27 mm

// Anti-reflection coating thickness
function antiReflectionThickness(wavelength_target, n_coating, n_substrate, n_air = 1) {
  // Quarter-wavelength: t = λ/(4n)
  // Plus n_coating must equal √(n_air · n_substrate) for full cancellation
  return wavelength_target / (4 * n_coating);
}

// AR coating for 550 nm (peak vision sensitivity), MgF₂ (n=1.38)
console.log(`AR coating thickness: ${(antiReflectionThickness(550e-9, 1.38, 1.5) * 1e9).toFixed(0)} nm`);

// Michelson interferometer fringe shift from arm length change
function michelsonFringeShift(armLengthChange, wavelength) {
  // Round trip → 2·ΔL/λ fringes
  return 2 * armLengthChange / wavelength;
}

// LIGO sensitivity: 10⁻¹⁸ m mirror displacement at 1064 nm
console.log(`LIGO 10⁻¹⁸ m → fringe shift: ${(michelsonFringeShift(1e-18, 1064e-9)).toExponential(2)}`);
// ~10⁻⁹ — a billionth of a fringe (detectable with photon-counting techniques)

// Coherence length (rough)
function coherenceLength(linewidth_Hz, c = 3e8) {
  return c / linewidth_Hz;
}

// Laser linewidth ~ 1 MHz
console.log(`1 MHz laser coherence: ${coherenceLength(1e6).toFixed(0)} m`); // 300

// Sun (broad spectrum, ~10¹⁴ Hz linewidth)
console.log(`Sun coherence: ${(coherenceLength(1e14) * 1e6).toFixed(2)} µm`); // ~3 µm

Where interference matters

• Optical instruments. Anti-reflection coatings, optical filters, multilayer mirrors, dielectric stacks.
• Spectroscopy. Diffraction grating spectrometers split light into colors via interference.
• Holography. Records 3D images via interference of reference and object beams.
• Interferometric sensors. Michelson, Mach-Zehnder for ultra-precise distance, refractive index, temperature measurements.
• Gravitational wave detection. LIGO and Virgo use kilometer-scale interferometers.
• Acoustics. Noise cancellation, room acoustics, speaker arrays.
• Quantum mechanics. Wave-particle duality demonstrated via interference of electrons, atoms, molecules.

Common mistakes

• Forgetting coherence requirement. Two random light sources don't produce stable interference. Need same source (or two phase-locked sources).
• Adding intensities instead of amplitudes. For coherent waves, ADD AMPLITUDES (with phase), then square. For incoherent, just add intensities.
• Wrong sign for phase difference. Sign matters in interference equations; constructive vs destructive depends on it.
• Forgetting phase shift on reflection. When reflecting off a denser medium, light gets a 180° phase shift. Critical for thin-film interference.
• Confusing interference and diffraction. Interference involves multiple sources. Diffraction is the bending of waves at edges/apertures. They're related but not identical.
• Treating laser as monochromatic. Even lasers have finite linewidth (coherence length). For very long path differences, interference washes out.