Huygens' Principle

The construction in one paragraph

Take any wavefront at time t — a surface where the wave has the same phase. Pick a fine grid of points on it. From each point, draw a spherical wavelet of radius v·Δt (where v is the wave speed). The forward-tangent surface to all those wavelets is the new wavefront at time t + Δt. Repeat. That's the whole rule.

Huygens published this in Traité de la Lumière in 1678, about 200 years before Maxwell wrote down the equations that justify it. He used it to derive both the law of reflection and Snell's law of refraction with nothing but a ruler and compass.

Worked example — deriving Snell's law

A planar wavefront in medium 1 (speed v₁) hits a boundary at angle θ₁ from the normal. Consider two points on the wavefront, A and B. A reaches the interface at time 0; B reaches it Δt = (AB · sin θ₁) / v₁ later.

From A, fire a wavelet into medium 2 (speed v₂). In the time Δt before B arrives, this wavelet expands to radius v₂·Δt. Meanwhile, B's wavelet has just been born — radius 0. The new wavefront on the medium-2 side is the line tangent to A's wavelet that passes through B.

AB · sin θ₁ = v₁ · Δt
AB · sin θ₂ = v₂ · Δt
⇒  sin θ₁ / sin θ₂ = v₁ / v₂ = n₂ / n₁  (Snell)

Two right triangles sharing the hypotenuse AB. No calculus, no Maxwell — just the speed-ratio interpretation of n.

Huygens-Fresnel — making it quantitative

Pure Huygens predicts the right wavefront shape but says nothing about intensity. It also predicts a backward-traveling wave that isn't seen. Fresnel (1815) fixed both: treat each secondary wavelet as a complex amplitude with phase k·r and a 1/r falloff, plus an obliquity factor K(χ) = (1 + cos χ)/2 that goes to 1 forward and 0 backward.

U(P) = (1/iλ) ∬ U(Q) · K(χ) · exp(ikr) / r  dS_Q

(Huygens-Fresnel integral; sum over wavefront points Q, with r = |P − Q|)

This integral reproduces Fraunhofer single-slit sinc² fringes, the Airy pattern of a circular aperture, and Fresnel zone plates. Kirchhoff (1882) showed it follows rigorously from the scalar wave equation with a particular boundary condition.

Pure Huygens vs Huygens-Fresnel vs Kirchhoff

Form	Year	Predicts	Limitation
Pure Huygens (geometric envelope)	1678	Wavefront shape; reflection; refraction	No intensity; spurious backward wave
Huygens-Fresnel (phasor sum + obliquity)	1815	Diffraction patterns; intensities; phase	Heuristic — obliquity factor justified ad hoc
Kirchhoff integral	1882	Same as Huygens-Fresnel, derived from wave equation	Boundary conditions inconsistent for shadows
Rayleigh-Sommerfeld	1896	Same predictions, consistent boundary conditions	Scalar — ignores polarization
Stratton-Chu (vector)	1939	Full electromagnetic diffraction	Heavy machinery; rarely needed below the Abbe limit
FDTD numerical	1966+	Anything Maxwell does, including near field	Compute cost ∝ (volume/λ³); slow at large scales

Diffraction — the qualitative story

When a wavefront hits an aperture of width a, only the wavelets emitted from the open portion contribute to the downstream field. Beyond the aperture each of those wavelets spreads spherically, so the wave fans into the geometric shadow.

The angular width of the spread scales as θ ≈ λ/a. For visible light (λ ≈ 0.5 µm) through a 1 mm slit: θ ≈ 5 × 10⁻⁴ rad — invisible unless the screen is far away. For audible sound (λ ≈ 1 m) through a doorway (a ≈ 1 m): θ ≈ 1 rad — sound floods the next room. Same equation, different λ/a regime.

Worked numbers — when does diffraction matter?

Setup	λ	Aperture a	Spread angle θ ≈ λ/a
Green laser through pinhole	532 nm	50 µm	0.6° — clearly visible
Camera lens at f/22	550 nm	~2.3 mm	0.014° — softens fine detail
Telescope (Hubble)	550 nm	2.4 m	0.058 arcsec (1.22 λ/D, circular)
FM radio around a hill	3 m (100 MHz)	30 m hill	5.7° — wraps reasonably
Acoustic doorway (500 Hz)	0.68 m	0.8 m doorway	49° — sound floods
X-ray crystal plane	0.15 nm	0.3 nm spacing	30° (Bragg) — sharp peaks

JavaScript — propagating a wavefront step by step

// Discrete Huygens: sample the wavefront, fire wavelets, take the envelope.
// Toy 2D version — show how a plane wave through a slit develops curvature.

function propagate(sources, target, lambda, dt = 1, c = 1) {
  // Each source emits a spherical wavelet (1/r amplitude, k·r phase).
  // Sum phasors at each target point; return |U|² (intensity).
  const k = 2 * Math.PI / lambda;
  return target.map(p => {
    let re = 0, im = 0;
    for (const s of sources) {
      const dx = p.x - s.x, dy = p.y - s.y;
      const r = Math.hypot(dx, dy);
      if (r < 1e-9) continue;
      // Obliquity factor — forward only
      const cosChi = dy / r;  // assuming downstream is +y
      const K = 0.5 * (1 + cosChi);
      const amp = K * s.A / Math.sqrt(r);  // 2D: 1/√r, not 1/r
      re += amp * Math.cos(k * r);
      im += amp * Math.sin(k * r);
    }
    return { x: p.x, y: p.y, I: re * re + im * im };
  });
}

// Plane wave at y=0 chopped by a slit from x=-a/2 to x=+a/2
function planeWaveThroughSlit(a, samplesAcrossSlit = 200) {
  const sources = [];
  for (let i = 0; i < samplesAcrossSlit; i++) {
    const x = -a/2 + a * (i + 0.5) / samplesAcrossSlit;
    sources.push({ x, y: 0, A: 1 });
  }
  return sources;
}

// Screen 1000 wavelengths downstream
function screen(L, half = 50, samples = 401) {
  const pts = [];
  for (let i = 0; i < samples; i++) {
    pts.push({ x: -half + 2 * half * i / (samples - 1), y: L });
  }
  return pts;
}

const lambda = 1;
const a = 5;        // slit 5 λ wide
const sources = planeWaveThroughSlit(a, 200);
const target = screen(200, 60, 121);
const intensity = propagate(sources, target, lambda);

// First zero predicted at x ≈ L · λ / a = 200·1/5 = 40
const minIdx = intensity.findIndex((p, i) =>
  i > 60 && intensity[i].I < intensity[i+1]?.I && intensity[i].I < intensity[i-1].I);
console.log(`First minimum at x ≈ ${intensity[minIdx]?.x.toFixed(1)} (predicted ≈ 40)`);
// Confirms the Huygens-Fresnel sum reproduces single-slit diffraction.

Where Huygens lives today

• Scalar diffraction. Every Fresnel and Fraunhofer integral is a Huygens-Fresnel sum.
• Antenna engineering. Aperture antennas (horn antennas, parabolic dishes) are designed by summing wavelets from the aperture plane.
• Seismic wavefront construction. Geophysicists propagate seismic fronts through Earth by firing wavelets from each grid point — a discrete Huygens.
• FDTD numerics. Yee-cell updates in finite-difference time-domain electromagnetics are a discretized Huygens: each cell radiates into its neighbors each timestep.
• Beam propagation methods. Photonic waveguide simulation propagates the field plane by plane via angular-spectrum Huygens-Fresnel.
• Pedagogy. Teaching reflection, refraction, and diffraction before students see calculus — still the cleanest visual.

Common mistakes

• Treating the wavelets as real emitters. They're a calculation device — there are no physical Huygens sources sitting on a wavefront. The construction works because the wave equation is linear.
• Forgetting the obliquity factor. Without (1 + cos χ)/2, you predict a backward wave that doesn't exist. Pure Huygens needs Fresnel's amendment to be physical.
• Using Huygens-Fresnel inside one wavelength of an edge. Near-field diffraction (Sommerfeld 1896) needs careful boundary conditions; the simple scalar form gets shadow edges slightly wrong.
• Confusing 2D and 3D amplitude falloff. 3D spherical wavelets fall as 1/r; 2D cylindrical wavelets fall as 1/√r. Wrong dimension → wrong intensity.
• Applying it to non-linear media. Huygens needs superposition. In nonlinear optics (Kerr media, harmonic generation), wavelets don't simply add — the construction fails.
• Ignoring polarization. Scalar Huygens-Fresnel can't predict polarization-dependent diffraction (Brewster, near-field effects). For that you need a vector formulation (Stratton-Chu).