Thin-Film Interference

The setup

Light hits a thin film of refractive index n and thickness t. Two reflections happen:

1. From top surface (air-film boundary).
2. From bottom surface (film-substrate boundary).

The two reflected beams interfere. Constructive or destructive depends on:

• Geometric path difference: 2·n·t·cos θ_t (where θ_t is the angle inside the film).
• Phase shifts at the boundaries: 180° flip when reflecting off a denser medium.

Phase shifts at reflection

Reflecting from	Phase shift
Less-dense medium (e.g., air-water from air side)	180° flip
More-dense medium (e.g., air-water from water side)	No shift
Same medium (n equal)	No shift

Constructive/destructive conditions

For normal incidence, with N phase flips total at the two reflections:

Number of flips	Constructive (bright reflection)	Destructive (dim)
0 (e.g., glass on glass)	2·n·t = m·λ	2·n·t = (m + ½)·λ
1 (e.g., soap bubble)	2·n·t = (m + ½)·λ	2·n·t = m·λ
2 (e.g., oil on water if n_oil < n_water)	2·n·t = m·λ	2·n·t = (m + ½)·λ

Real-world thin films

System	Configuration	Why colors appear
Soap bubble	Air-soap-air	1 flip; varying thickness → varying colors
Oil slick on water	Air-oil-water	2 flips (if n_oil < n_water); thickness gradient → rainbow
Peacock/butterfly	Multilayer biological film	Structural; viewing angle changes color
Anti-reflective lens coatings	MgF₂ on glass	Designed thickness for AR at 550 nm
Dielectric mirror	Many alternating thin layers	Each reflects at a wavelength; sum is high reflectivity
Newton's rings	Air gap between lens and flat	Varying air-gap thickness → concentric rings
Iridescent shells	Layered nacre	Multilayer interference

JavaScript — thin-film calculations

// For thin film at normal incidence
// 1 phase flip (e.g., soap bubble): constructive at 2nt = (m + 0.5)·λ
function constructiveWavelengthSoap(thickness, n_film, m_max = 5) {
  const results = [];
  for (let m = 0; m < m_max; m++) {
    const wavelength = (2 * n_film * thickness) / (m + 0.5);
    if (wavelength > 380e-9 && wavelength < 750e-9) {
      results.push({ m, wavelength_nm: wavelength * 1e9 });
    }
  }
  return results;
}

// Soap film, n = 1.33, t = 200 nm
console.log(constructiveWavelengthSoap(200e-9, 1.33, 3));
// m=0: 2·1.33·200/0.5 = 1064 nm (IR — invisible)
// m=1: 1064/3 = 355 nm (UV — invisible)
// Hmm — this thickness shows no visible color at first orders

// Better example: soap film of t = 100 nm
console.log(constructiveWavelengthSoap(100e-9, 1.33, 3));
// m=0: 532 nm (green)
// m=1: 178 nm (UV)

// 2 flips (air-oil-water, n_oil=1.5, n_water=1.33): constructive at 2nt = m·λ
function constructiveWavelengthOil(thickness, n_film, m_max = 5) {
  const results = [];
  for (let m = 1; m < m_max; m++) {
    const wavelength = (2 * n_film * thickness) / m;
    if (wavelength > 380e-9 && wavelength < 750e-9) {
      results.push({ m, wavelength_nm: wavelength * 1e9 });
    }
  }
  return results;
}

// Oil 200 nm thick, n = 1.5
console.log(constructiveWavelengthOil(200e-9, 1.5, 5));
// m=1: 600 nm (orange/red)
// m=2: 300 nm (UV)

// Anti-reflection coating: design thickness for given target wavelength
function ARThickness(targetWavelength, n_coating) {
  // Quarter-wave: t = λ / (4n)
  return targetWavelength / (4 * n_coating);
}

// MgF₂ (n = 1.38) AR for 550 nm (peak human vision)
console.log(`MgF₂ AR thickness: ${(ARThickness(550e-9, 1.38) * 1e9).toFixed(1)} nm`); // ~99.6 nm

// Color of soap film vs thickness
function colorOfFilm(thickness_nm, n_film = 1.33) {
  // Find which visible wavelength has constructive at 2nt = (m+0.5)·λ
  const t = thickness_nm * 1e-9;
  const targets = [];
  for (let m = 0; m < 5; m++) {
    const wavelength = (2 * n_film * t) / (m + 0.5);
    if (wavelength > 380e-9 && wavelength < 750e-9) {
      targets.push({ m, wavelength_nm: wavelength * 1e9 });
    }
  }
  return targets;
}

// Sweep thicknesses
[100, 150, 200, 250, 300, 350, 400].forEach(t => {
  console.log(`${t} nm soap: ${colorOfFilm(t).map(x => x.wavelength_nm.toFixed(0) + ' nm').join(', ')}`);
});

// Newton's rings: ring radii on a curved lens
function newtonsRingRadius(R_lens, m, wavelength) {
  // Air gap at distance r: t ≈ r²/(2R). Constructive at 2t = (m+½)λ → r = √((m+½)·λ·R)
  return Math.sqrt((m + 0.5) * wavelength * R_lens);
}

// 1 m radius lens, 633 nm laser
for (let m = 0; m <= 5; m++) {
  console.log(`Ring m=${m}: r = ${(newtonsRingRadius(1, m, 633e-9) * 1000).toFixed(2)} mm`);
}

Where thin-film interference matters

• Anti-reflection coatings. Camera lenses, eyeglasses, solar cells — single-layer or multilayer designs.
• Dielectric mirrors. High-reflectivity at specific wavelengths; used in lasers, telescopes.
• Optical filters. Interference filters pass narrow bandwidths.
• Color in nature. Peacocks, butterflies, beetles, abalone — structural coloration via thin films.
• Soap bubbles and oil slicks. Most familiar examples; demonstrate physics directly.
• Industrial inspection. Thickness measurement of coatings via interference; Newton's rings for lens shape.
• Photonic devices. Laser diodes, photonic crystals, multilayer reflectors.

Common mistakes

• Forgetting phase shifts. Reflection off denser medium → 180° phase flip. Forgetting these flips gets the constructive/destructive conditions wrong.
• Confusing reflection vs transmission interference. Constructive REFLECTION condition is destructive TRANSMISSION condition (energy conservation).
• Treating the geometric path difference alone. Path difference = 2nt for normal incidence. For oblique incidence, use 2nt·cos θ_t.
• Using vacuum wavelength inside the film. Light wavelength in the film is λ/n (slower → shorter λ). Path difference uses the wavelength IN the film.
• Assuming a single color. Most films show multiple wavelengths simultaneously (different m's). Color is a mix; what you see depends on which wavelengths add constructively.
• Forgetting incidence angle dependence. Oblique angles change the path difference, so the colors observed depend on viewing angle (peacock feathers, etc.).