Kerr Effect

What the Kerr effect actually does

Light travels slower in a dense medium than in vacuum, by a factor set by the refractive index n. The Kerr effect is the observation that n is not a fixed property of a material — it shifts when you push hard enough with an electric field. Crucially, the shift is quadratic: it depends on the field squared, not the field itself. Reverse the field and the index change stays the same.

That quadratic dependence is the whole story. A linear index change (the Pockels effect) requires a crystal with no center of symmetry, because a linear response must flip sign when the field flips. A quadratic response has no such constraint — it survives in liquids, gases, glasses, and centrosymmetric crystals, which is why the Kerr effect is universal and the Pockels effect is not.

Two flavors: DC Kerr and optical Kerr

The Kerr effect appears in two guises that share the same microscopic origin but use different fields:

	DC (electro-optic) Kerr effect	Optical (AC) Kerr effect
Driving field	External applied voltage E	The light beam's own field
What changes	Induced birefringence Δn = λKE²	Index vs. intensity n = n₀ + n₂I
Geometry	Optic axis along E; ordinary vs. extraordinary index split	Isotropic index rise wherever the beam is bright
Characteristic constant	Kerr constant K (units m/V²)	Nonlinear index n₂ (units m²/W)
Hallmark device	Kerr cell shutter, voltage-controlled retarder	Self-focusing, Kerr-lens mode locking, solitons

The DC effect is what John Kerr saw in 1875: he placed a slab of glass between crossed polarizers, applied a strong field, and watched light leak through where the field had made the glass birefringent. The optical Kerr effect is the same physics with the optical wave supplying the field — it became important only with lasers, when intensities grew large enough to bend light with light.

The DC Kerr effect and induced birefringence

Apply a transverse field E to an isotropic medium of length L. The medium becomes uniaxially birefringent, with its optic axis along the field. Light polarized parallel and perpendicular to the field travels at different indices, and the phase difference between them after length L is:

Δφ = 2π·K·L·E²

where K is the Kerr constant (m/V²)
and the induced birefringence is  Δn = n_∥ − n_⊥ = λ·K·E²

Sandwich this between crossed polarizers and you have a voltage-controlled shutter. The transmitted intensity is:

T = sin²(Δφ / 2) = sin²(π·K·L·E²)

The half-wave voltage — the voltage that swings the cell from fully dark to fully bright — occurs when Δφ = π, i.e. KLE² = ½. For a nitrobenzene cell with electrode gap d and length L, with K ≈ 2.4 × 10⁻¹² m/V² at 589 nm, gap d = 1 mm and L = 10 cm, you need a field of roughly 1.4 × 10⁶ V/m, i.e. about 1.4 kV across the gap. High voltage, but the switching is electronic and fast.

The optical Kerr effect and self-action

When the light itself is the field, the index a beam sees rises with its own intensity:

n(I) = n₀ + n₂·I

n₂ is the nonlinear index, related to χ⁽³⁾ by
n₂ = 3·χ⁽³⁾ / (4·ε₀·c·n₀²)

Two famous consequences follow:

• Self-focusing. A real beam is brighter in the middle than at the edges. The bright center sees a higher index, so it is slowed more — exactly what a converging lens does. The medium becomes a positive lens whose strength scales with power. Above a critical power (P_cr ≈ 3.77·λ²/(8π·n₀·n₂), about 4 MW in fused silica at 800 nm) self-focusing overwhelms diffraction and the beam collapses, often catastrophically damaging the glass.
• Self-phase modulation. A pulse's intensity rises and falls in time, so the index — and therefore the optical phase — is time-dependent. The instantaneous frequency shifts, broadening the spectrum. This is the engine of supercontinuum generation and a key ingredient in optical solitons, where Kerr self-phase modulation exactly balances fiber dispersion.

Where it comes from: χ⁽³⁾

The induced polarization of a material expands in powers of the field:

P = ε₀ (χ⁽¹⁾E + χ⁽²⁾E² + χ⁽³⁾E³ + …)

The linear term χ⁽¹⁾ gives the ordinary refractive index. The χ⁽²⁾ term gives the Pockels effect and second-harmonic generation — but it vanishes in any material with inversion symmetry, because χ⁽²⁾ must change sign under E → −E while a centrosymmetric material cannot. The next surviving term is χ⁽³⁾, and that is the Kerr effect. Writing the field as a probe wave plus a strong field, the χ⁽³⁾E³ term produces a contribution to the index that goes as the square of the strong field — the quadratic dependence that defines Kerr.

Two physical mechanisms contribute to χ⁽³⁾. In polar liquids like nitrobenzene, the dominant one is molecular reorientation: anisotropic molecules with permanent dipole moments align with the field, and this orientation responds on a picosecond timescale, giving the large but relatively slow DC Kerr response. In glasses and noble gases, the response is electronic — the electron cloud distorts — which is far smaller but essentially instantaneous (femtoseconds), and is what dominates the ultrafast optical Kerr effect.

Numbers that matter

Material / scenario	Kerr quantity
Nitrobenzene Kerr constant K (589 nm)	≈ 2.4 × 10⁻¹² m/V² (largest common liquid)
Water Kerr constant K	≈ 5.1 × 10⁻¹⁴ m/V²
Fused silica nonlinear index n₂	≈ 2.7 × 10⁻²⁰ m²/W
Δn in silica at 10 GW/cm²	≈ n₂·I = 2.7 × 10⁻²⁰ × 10¹⁴ ≈ 2.7 × 10⁻⁶
Self-focusing critical power in silica (800 nm)	≈ 4 MW (peak power, not energy)
Kerr-cell shutter speed (nitrobenzene)	~nanoseconds (limited by drive electronics)
Electronic Kerr response time	femtoseconds (near-instantaneous)

The index changes are minuscule — parts per million even at gigawatt intensities — yet they matter because optical phase accumulates over distance. A 2.7 × 10⁻⁶ index change builds up a full wave of extra phase over only a few centimeters, and over the thousands of kilometers of a transoceanic fiber, even milliwatt signals accumulate enough nonlinear phase to garble themselves.

Worked calculations

// DC Kerr cell: transmission through crossed polarizers
// T = sin^2(pi * K * L * E^2)
function kerrTransmission(K, L, E) {
  const dphi = 2 * Math.PI * K * L * E * E; // phase retardation
  return Math.sin(dphi / 2) ** 2;
}

// Nitrobenzene: K = 2.4e-12 m/V^2, L = 0.1 m, gap d = 1 mm
const K = 2.4e-12, L = 0.1, d = 1e-3;

// Half-wave voltage: dphi = pi  ->  K*L*E^2 = 1/2
const E_half = Math.sqrt(0.5 / (K * L));         // V/m
const V_half = E_half * d;                       // volts across gap
console.log(`Half-wave field: ${(E_half/1e6).toFixed(2)} MV/m`); // ~1.44 MV/m
console.log(`Half-wave voltage: ${(V_half/1000).toFixed(2)} kV`); // ~1.44 kV
console.log(`T at half-wave: ${kerrTransmission(K, L, E_half).toFixed(3)}`); // ~1.000

// Optical Kerr: index change from intensity
function kerrIndexShift(n2, I) { return n2 * I; }
const n2_silica = 2.7e-20;       // m^2/W
const I = 1e14;                  // 10 GW/cm^2 in W/m^2
console.log(`Delta-n in silica: ${kerrIndexShift(n2_silica, I).toExponential(2)}`); // ~2.7e-6

// Self-focusing critical power (Gaussian beam)
function criticalPower(lambda, n0, n2) {
  return 3.77 * lambda * lambda / (8 * Math.PI * n0 * n2);
}
const Pcr = criticalPower(800e-9, 1.45, n2_silica);
console.log(`Critical power: ${(Pcr/1e6).toFixed(1)} MW`); // ~4 MW

// Nonlinear phase accumulated over length (self-phase modulation core)
// phi_NL = (2*pi/lambda) * n2 * I * L
function nonlinearPhase(lambda, n2, I, Lprop) {
  return (2 * Math.PI / lambda) * n2 * I * Lprop;
}
console.log(`NL phase, silica, 1 cm @10GW/cm^2: ${nonlinearPhase(800e-9, n2_silica, I, 0.01).toFixed(2)} rad`);

Where the Kerr effect shows up

• Ultrafast lasers. Kerr-lens mode-locking uses self-focusing to favor high-intensity pulses over continuous-wave light, generating femtosecond pulses in Ti:sapphire lasers without any moving parts.
• Optical switching and shutters. Kerr cells gate light on nanosecond timescales for high-speed photography, laser Q-switching, and pulse pickers.
• Fiber communications. The Kerr effect causes self-phase modulation, cross-phase modulation, and four-wave mixing — the dominant nonlinear penalties that cap launch power in long-haul wavelength-division-multiplexed systems.
• Solitons. In fibers, Kerr self-phase modulation balanced against group-velocity dispersion produces solitons — pulses that propagate without spreading.
• Frequency combs. Microresonator (Kerr) combs convert a single laser line into thousands of evenly spaced lines via cascaded four-wave mixing, enabling chip-scale spectroscopy and optical clocks.
• Field metrology. Kerr-cell-based systems measure high-voltage transients and space-charge fields where electrodes cannot be inserted.

Common mistakes

• Confusing Kerr with Pockels. Pockels is linear (Δn ∝ E) and needs a non-centrosymmetric crystal; Kerr is quadratic (Δn ∝ E²) and works everywhere. If reversing the voltage reverses the index shift, it is Pockels, not Kerr.
• Mixing up K and n₂. The DC Kerr constant K (units m/V²) describes field-induced birefringence; the nonlinear index n₂ (units m²/W) describes intensity-induced index change. They are related but not interchangeable.
• Forgetting it is peak power, not energy, that triggers self-focusing. A long low-peak pulse can carry huge energy without reaching the critical power; a short pulse with modest energy can collapse a beam and crack the optics.
• Treating the effect as instantaneous in liquids. Molecular-reorientation Kerr in nitrobenzene has a picosecond response time; only the electronic contribution is truly femtosecond. The relevant timescale depends on the mechanism.
• Ignoring sign of dispersion for solitons. Kerr self-phase modulation only forms bright solitons together with anomalous (negative) dispersion. In the normal-dispersion regime the same nonlinearity broadens, not stabilizes, the pulse.
• Assuming the index change is large. Even at gigawatt intensities Δn is only parts per million. The Kerr effect is significant because phase accumulates over long paths, not because the index swings are big.