Evanescent Wave

Where the imaginary wavevector comes from

Start with Snell's law at a dense-to-rare interface (n₁ > n₂):

n₁ sin θ₁ = n₂ sin θ₂

Above the critical angle θ_c = arcsin(n₂/n₁), the right-hand side would need sin θ₂ > 1 — impossible for a real angle. Geometric optics declares the wave fully reflected. But Maxwell's equations don't quit at the surface; they demand a transmitted solution that matches boundary conditions everywhere along the interface.

Write the transmitted field as a plane wave with wavevector k_t = (k_x, 0, k_z). Tangential matching forces k_x = (2π/λ₀)·n₁·sin θ₁. The dispersion relation in medium 2 demands k_x² + k_z² = (2π·n₂/λ₀)². Solving:

k_z = ±(2π/λ₀)·√(n₂² - n₁²sin²θ₁)

When sin θ₁ > n₂/n₁ the square root flips sign and k_z = ±iκ. Picking the physical (decaying, not growing) root:

E_t(z, t) = E₀ · exp(-κz) · exp(i(k_x x - ωt))   with κ = (2π/λ₀)·√(n₁²sin²θ₁ - n₂²)

This is the evanescent wave: a wave that oscillates along the surface and decays exponentially away from it.

Penetration depth — concrete numbers

The 1/e penetration depth d_p = 1/κ. Plugging in for visible-band glass-air interfaces:

Wavelength λ₀	n₁ (BK7)	θ₁	Penetration d_p
633 nm (HeNe)	1.515	45° (just past θ_c = 41.3°)	≈ 197 nm
633 nm	1.515	50°	≈ 130 nm
633 nm	1.515	60°	≈ 92 nm
1550 nm (telecom)	1.500	50°	≈ 330 nm
400 nm (blue)	1.530	50°	≈ 80 nm

A rule of thumb worth memorising: d_p ≈ λ/2π when you're well past the critical angle. For visible light that's about 100 nm — small enough that the field is essentially confined to the surface skin.

Frustrated total internal reflection — light tunnelling across a gap

If a second dense medium (a parallel prism, say) is brought within the evanescent tail of the first, the field at the second surface is still non-zero. The boundary conditions there allow it to launch a propagating wave back into glass. Power transmitted across the air gap scales as:

T(d) ≈ T₀ · exp(-2κd)

Halving κd doubles the transmitted intensity. Two prisms separated by 50 nm at θ = 50° (d_p = 130 nm) leak about exp(-2·50/130) ≈ 46% of the incident power. Pull them to 500 nm apart and transmission drops to exp(-2·500/130) ≈ 0.046%. Mechanically squeezing a flexible film against a prism gives a touch sensor — exactly how some smartphone fingerprint scanners read ridges.

Real-world evanescent-wave systems

System	Role of evanescent field
Optical fibre	Field tail in cladding sets bend losses and coupling between adjacent cores
FTIR spectroscopy (ATR)	Sample placed on ZnSe/Ge crystal; evanescent field probes ~1 µm into sample
NSOM / SNOM	Sub-λ tip in near field — resolution set by tip diameter, not diffraction
TIRF microscopy	Only fluorophores within ~100 nm of coverslip are excited — clean basal-membrane imaging
Surface plasmon resonance	Evanescent field drives plasmon at thin Au layer; angle shift reports binding
Fingerprint scanners	Ridges contact glass → FTIR couples light away → dark stripes
Prism / grating couplers	Match phase of evanescent field to a waveguide mode to inject light
Quantum-optics analog	Photon barrier tunnelling — pedagogical model for QM tunneling

JavaScript — evanescent-wave calculations

// Critical angle
function thetaCritical(n1, n2) {
  return Math.asin(n2 / n1) * 180 / Math.PI;
}

// Decay constant κ (1/m) for an angle past critical
function kappa(n1, n2, theta_deg, lambda0_m) {
  const th = theta_deg * Math.PI / 180;
  const arg = n1*n1 * Math.sin(th)**2 - n2*n2;
  if (arg <= 0) return 0;  // below critical: no evanescent
  return (2 * Math.PI / lambda0_m) * Math.sqrt(arg);
}

// Penetration depth (1/e amplitude)
function penetrationDepth(n1, n2, theta_deg, lambda0_m) {
  const k = kappa(n1, n2, theta_deg, lambda0_m);
  return k > 0 ? 1 / k : Infinity;
}

const lambda = 633e-9;
console.log(`θ_c (BK7-air): ${thetaCritical(1.515, 1).toFixed(2)}°`);  // 41.3°

for (const theta of [45, 50, 60, 75]) {
  const dp = penetrationDepth(1.515, 1, theta, lambda);
  console.log(`θ=${theta}°  d_p = ${(dp*1e9).toFixed(0)} nm`);
}
// 45° → ~197 nm, 50° → 130, 60° → 92, 75° → 75

// FTIR — fraction transmitted across an air gap of width d
function ftirTransmission(n1, n2, theta_deg, lambda0_m, gap_m, T0 = 1) {
  const k = kappa(n1, n2, theta_deg, lambda0_m);
  return T0 * Math.exp(-2 * k * gap_m);
}

// Two BK7 prisms separated by 50 nm at 50° / 633 nm
console.log(`FTIR @ 50 nm: ${(ftirTransmission(1.515, 1, 50, lambda, 50e-9) * 100).toFixed(1)}%`);
// ≈ 46%
console.log(`FTIR @ 500 nm: ${(ftirTransmission(1.515, 1, 50, lambda, 500e-9) * 100).toFixed(3)}%`);
// ≈ 0.04%

// TIRF excitation profile — fraction of fluor at depth z
function tirfIntensity(z_m, dp_m) {
  return Math.exp(-2 * z_m / dp_m); // intensity ∝ |E|² → 2× decay
}

const dp = penetrationDepth(1.515, 1.33, 65, lambda); // glass-water 65°
console.log(`TIRF @ 100 nm above surface: ${(tirfIntensity(100e-9, dp) * 100).toFixed(1)}%`);
// ~3% — strong surface selectivity

Worked example — designing an FTIR beam splitter

You need a variable beam splitter: tune transmission from 5% to 95% by changing a gap. Wavelength 1064 nm (Nd:YAG), prism material fused silica (n = 1.45), incidence 60°.

1. θ_c = arcsin(1/1.45) = 43.6°. We're well past critical.
2. κ = (2π / 1.064 µm)·√(1.45²·sin²60° - 1²) = 5.91·√(1.578 - 1) = 5.91·0.760 ≈ 4.49 µm⁻¹.
3. d_p = 1/κ ≈ 223 nm.
4. For T = 5% → exp(-2κd) = 0.05 → d = -ln(0.05)/(2κ) = 3.00/8.98 ≈ 334 nm.
5. For T = 95% → d = -ln(0.95)/(2κ) ≈ 5.7 nm. (Difficult — surfaces would essentially be in contact.)

A piezo actuator with 1 µm travel comfortably spans this range. Linearity is exponential in gap, so the mechanical motion is small for the most useful (~50%) regime.

Where evanescent waves matter

• Surface-only sensing. When you want signal only from molecules within ~100 nm of an interface — biosensors, membrane biology, electrochemistry.
• Sub-diffraction microscopy. NSOM and tip-enhanced spectroscopies use evanescent coupling at probe tips.
• Variable couplers. Beam splitting, optical switching, fingerprint and touch sensing — all controlled by sub-wavelength gaps.
• Integrated photonics. Directional couplers between waveguides rely on overlap of evanescent tails.
• Spectroscopy. Attenuated total reflection (ATR-FTIR) lets you probe absorbance of opaque or aqueous samples without transmission cells.
• Plasmonics. Surface plasmons are bound modes — entirely evanescent away from the metal-dielectric interface.
• Quantum analogy. Excellent classical visualisation of tunnelling for teaching and for testing tunnelling-time controversies.

Common mistakes

• Saying TIR is no longer total because the field exists in the rare medium. It is total — the time-averaged Poynting flux normal to the surface is zero. The field is non-zero but doesn't carry net energy out.
• Confusing d_p with where the wave "ends". The wave only decays; it never truly reaches zero. d_p is the 1/e amplitude point. Intensity (|E|²) decays twice as fast.
• Treating the evanescent wave as a separate object. It's the transmitted Maxwell solution above the critical angle. Same boundary conditions, same physics — just an imaginary k_z.
• Forgetting polarisation matters. s- and p-polarised evanescent fields differ in amplitude and direction. Plasmons couple only to p (TM).
• Assuming exponential decay is wavelength-independent. κ scales with 1/λ₀ — IR waves penetrate further than UV at the same geometry.
• Ignoring the in-plane wavevector. The wave does propagate along the surface with k_x > k_0 — it's a "fast" surface wave that can launch surface plasmons and waveguide modes.