The Optical Kerr Effect and Intensity-Dependent Index

Turn a laser bright enough and the glass it crosses stops being a passive bystander: its refractive index begins to depend on how intense the light is. That is the optical Kerr effect — $n = n_0 + n_2 I$ — the flagship third-order nonlinear process, the one that survives in every material because it asks no favour of symmetry. Where second-harmonic generation demands a non-centrosymmetric crystal, the Kerr nonlinearity lives happily in fused silica, in water, in air. The polarization expansion that opens the door — $P = \varepsilon_0(\chi^{(1)}E + \chi^{(2)}E^2 + \chi^{(3)}E^3 + \cdots)$ , introduced in Introduction to Nonlinear Optical Polarization — has its lowest surviving nonlinear term, $\chi^{(3)}$ , feed back on the very wave that drove it. The index change is almost comically small: at a gigawatt per square centimetre, silica's index shifts by two parts in ten million. Yet that whisper of an index rules femtosecond and fibre optics — not through its size but through the phase it accumulates. Over a metre of fibre, or ten thousand cavity round trips, a $10^{-7}$ index change builds into radians of nonlinear phase, and radians are what reshape a pulse's spectrum and steer a beam into collapse. This article owns the Kerr construction — the $\chi^{(3)} \to n_2$ bookkeeping, self-phase modulation, and the critical power for self-focusing — the machinery beneath mode-locked lasers, supercontinuum white light, and the intensity ceiling of every high-power amplifier.

Key takeaways

The optical Kerr effect is the intensity dependence of the refractive index, $n = n_0 + n_2 I$ , from the real part of the degenerate third-order susceptibility $\chi^{(3)}(\omega;\omega,-\omega,\omega)$ ; it needs no broken inversion symmetry and so appears in glasses, liquids, and gases.
Collecting the $\omega$ -terms in $\chi^{(3)}E^3$ (the $\cos^3$ identity gives the factor $3/4$ ) yields $n_2 = 3\chi^{(3)}/(4 n_0^2 \varepsilon_0 c)$ ; fused silica has $n_2 = 2.6\times10^{-20}\ \mathrm{m^2/W}$ .
The change is tiny — $\Delta n = 2.6\times10^{-7}$ at $1\ \mathrm{GW/cm^2}$ — but the accumulated phase is not: $\varphi_{\mathrm{NL}} = (2\pi/\lambda)\,n_2 I L = 1.54\ \mathrm{rad}$ over a single metre.
A transverse intensity profile turns the medium into a self-induced lens; above a critical power $P_{\mathrm{cr}} = 3.77\lambda^2/(8\pi n_0 n_2) = 4.50\ \mathrm{MW}$ (silica, $1064\ \mathrm{nm}$ ) diffraction can no longer balance it and the beam collapses.
A time-varying intensity chirps the pulse (self-phase modulation): red on the leading edge, blue on the trailing — the seed of supercontinuum and, balanced against anomalous dispersion, of solitons.
Key idea: the $\chi^{(3)}$ term acts as an intensity-dependent susceptibility, so $n = n_0 + n_2 I$ with $n_2 = 3\chi^{(3)}/(4n_0^2\varepsilon_0 c)$ ; the same term self-focuses beams (space) and self-phase-modulates pulses (time), and it dominates ultrafast optics through accumulated phase, not through the minuscule index shift itself.

The core insight

Expanding the polarization to third order in the field produces an effective susceptibility that grows linearly with intensity, so the refractive index becomes $n = n_0 + n_2 I$ with $n_2 = 3\chi^{(3)}/(4 n_0^2 \varepsilon_0 c)$ . Because intensity varies across a beam and along a pulse, the one term drives two families of phenomena — spatial self-focusing and temporal self-phase modulation — and it governs femtosecond and fibre optics not because the index change is large (it is $\sim10^{-7}$ ) but because the phase it accumulates over length is.

Where this lives today

The Kerr nonlinearity is the quiet engine of ultrafast science. Kerr-lens mode-locking, discovered when Ti:sapphire lasers were found to self-mode-lock, became a standard route to femtosecond oscillators — the gain crystal also provides the Kerr nonlinearity, and the cavity converts the resulting intensity-dependent spatial focusing into effective fast saturable-absorber action. Downstream, an intense pulse in a photonic-crystal fibre broadens by self-phase modulation — abetted by dispersion, four-wave mixing, and soliton fission — into an octave-spanning supercontinuum, the white light behind $f$ – $2f$ self-referencing of optical frequency combs, and with it optical clocks. The same physics is a hazard. In every high-energy chain — fusion drivers, petawatt amplifiers — the accumulated $B$ -integral, not the gain, often sets the intensity ceiling, because small-scale self-focusing seeds filaments that bore through the most expensive optics in the building. In communications it is both foe and friend: it limits dense wavelength-division reach through cross-phase modulation, yet balanced against anomalous dispersion it supports the temporal soliton — a pulse that propagates without spreading.

The physical picture

Picture the bound electrons of a dielectric as charges in an anharmonic well. At modest fields they oscillate linearly and re-radiate a wave retarded by the ordinary index $n_0$ ; drive them harder and they feel the well's softening walls, so their response picks up a term $\propto E^3$ . Averaged over an optical cycle, the cubic term produces a correction to the effective restoring force proportional to $\langle E^2\rangle$ , and hence to the optical intensity, giving an intensity-dependent refractive index. The sign follows the anharmonicity; for a transparent solid's softening well, $n_2 > 0$ .

Figure

The causal chain: a bright core raises the local index, and the medium becomes the beam's own converging lens

Two consequences flow from the one fact. In space, a bell-shaped beam sees a higher index on axis than at its wings, so the on-axis light lags and the wavefront curves inward — a positive lens whose strength grows with power, competing with diffraction, and above a critical power, self-focusing can overcome diffraction. In time, the leading edge of a pulse rides a rising index and the trailing edge a falling one, so the pulse stamps a sweeping phase on itself: front red-shifted, back blue-shifted, new frequencies with no change in energy. The same $n_2$ that steers the beam rewrites the spectrum. A hierarchy of mechanisms hides inside the single symbol $n_2$ . The bound-electron distortion is essentially instantaneous (femtoseconds) and always present; molecular reorientation (picoseconds, spectacular in $\mathrm{CS_2}$ ) and thermal lensing (milliseconds) can dominate for longer pulses and even flip the sign. The "same" $n_2$ can differ by orders of magnitude depending on which clock the pulse runs on — which is why a careful measurement always states the pulse duration.

Figure

Self-focusing threshold: below 4.50 MW the beam diffracts; above it, it collapses — the critical power marked on the curve

The second figure carries the effect's most counter-intuitive fact: the collapse threshold is a power, not an intensity. Focus a below-threshold beam ever more tightly and its intensity rises without limit, yet it still refuses to collapse — because self-focusing and diffraction scale the same way with spot size, so the radius cancels and only total power survives as the control knob (Derivation C). Below $P_{\mathrm{cr}}$ diffraction always reasserts itself; above it, nothing in the paraxial theory stops the implosion.

The core equations

The subject begins with the constitutive expansion, valid whenever the optical field is small against the atomic field $E_{\mathrm{at}}\approx5.14\times10^{11}\ \mathrm{V/m}$ . In a centrosymmetric medium every even term vanishes and the first survivor is the cubic one.

P = \varepsilon_0\left(\chi^{(1)}E + \chi^{(2)}E^2 + \chi^{(3)}E^3 + \cdots\right) \qquad (1)

where: $P$ — induced polarization (dipole moment per unit volume); $\varepsilon_0$ — vacuum permittivity; $\chi^{(n)}$ — $n$ th-order susceptibility (a tensor in general, with the contracted magnitudes used here); $E$ — total optical field. In glass and other centrosymmetric media, the bulk electric-dipole $\chi^{(2)}$ vanishes, so the cubic $\chi^{(3)}E^3$ term is the leading electric-dipole nonlinear contribution in the nonresonant regime. Collecting the components of $\chi^{(3)}E^3$ that oscillate back at the drive frequency $\omega$ (Derivation A) turns the cubic term into a correction to the index that is linear in intensity. This is the central relation of the whole article.

n = n_0 + n_2 I, \qquad n_2 = \frac{3\,\chi^{(3)}}{4\, n_0^2\, \varepsilon_0\, c}, \qquad \Delta n = n_2 I \qquad (2)

where: $n_0$ — linear index, $n_0^2 = 1+\chi^{(1)}$ ; $n_2$ — nonlinear-index coefficient (units $\mathrm{m^2/W}$ ); $I$ — cycle-averaged intensity, $I = \tfrac12 n_0\varepsilon_0 c |E|^2$ ; $c$ — vacuum speed of light. Read $\Delta n = n_2 I$ as the physical content: for fused silica, $n_2 = 2.6\times10^{-20}\ \mathrm{m^2/W}$ , so even a punishing $1\ \mathrm{GW/cm^2}$ ( $=10^{13}\ \mathrm{W/m^2}$ ) buys only $\Delta n = 2.6\times10^{-7}$ . That the index barely moves is not the point; the phase does. A wave of vacuum wavelength $\lambda$ that travels a length $L$ through the extra index $\Delta n$ acquires a nonlinear phase on top of its linear one, and because $\Delta n$ tracks the instantaneous intensity, so does the phase — this is self-phase modulation.

\varphi_{\mathrm{NL}}(t) = \frac{2\pi}{\lambda}\,n_2\, I(t)\, L, \qquad \delta\omega(t) = -\frac{d\varphi_{\mathrm{NL}}}{dt} = -\frac{2\pi n_2 L}{\lambda}\frac{dI}{dt} \qquad (3)

where: $\varphi_{\mathrm{NL}}$ — intensity-induced nonlinear phase; $\delta\omega$ — instantaneous frequency shift about the carrier; $dI/dt$ — temporal slope of the intensity. On the leading edge $dI/dt>0$ , so $\delta\omega<0$ (red); on the trailing edge, blue. Over just one metre of fibre at $1\ \mathrm{GW/cm^2}$ this evaluates to $\varphi_{\mathrm{NL}}=1.54\ \mathrm{rad}$ — radians from a $10^{-7}$ index, the whole paradox of the effect in one number. Spatially, the self-lens competes with diffraction, and the balance defines a threshold power above which the beam collapses. The numerical coefficient depends on the transverse beam profile and the normalization used for the nonlinear wave equation; for the fundamental self-trapped Townes mode, the standard critical-power constant is $3.77$ .

P_{\mathrm{cr}} = \frac{3.77\,\lambda^2}{8\pi\, n_0\, n_2} \qquad (4)

where: $P_{\mathrm{cr}}$ — critical power for whole-beam self-focusing collapse; $\lambda$ — vacuum wavelength. Notice what is absent: no beam radius and no intensity — only total power. For silica at $1064\ \mathrm{nm}$ this is $P_{\mathrm{cr}}=4.50\ \mathrm{MW}$ , a number a tabletop femtosecond oscillator flirts with every round trip.

The DC cousin. Kerr's original 1875 effect was electro-optic — a static field making a medium birefringent with $\Delta n\propto E_{\mathrm{DC}}^2$ via $\chi^{(3)}(\omega;\omega,0,0)$ ; the optical Kerr effect replaces the battery with the light's own field, $\Delta n\propto\langle E^2\rangle\propto I$ via $\chi^{(3)}(\omega;\omega,-\omega,\omega)$ .

Derivation AFrom the cubic susceptibility to the intensity-dependent index

The goal is to show how the abstract term $\chi^{(3)}E^3$ becomes the measurable law $n = n_0 + n_2 I$ , and to pin the coefficient $n_2 = 3\chi^{(3)}/(4n_0^2\varepsilon_0 c)$ — including the notorious factor of $3/4$ that trips up every first calculation. The strategy: write the real field as a cosine, cube it, keep only the part that oscillates back at the original frequency (the rest is the third harmonic, another story), read it as an effective permittivity, and convert field amplitude to intensity.

Step 1. Take a monochromatic real field $E(t) = E_0\cos\omega t$ and cube it. The elementary identity $\cos^3\omega t = \tfrac34\cos\omega t + \tfrac14\cos3\omega t$ splits the cubic drive into a piece at the fundamental and a piece at the third harmonic. Why: only the $\tfrac34\cos\omega t$ piece can beat in phase with the carrier and act back on it as an index change; the $\tfrac14\cos3\omega t$ piece is the seed of third-harmonic generation, a separate phase-matched process.

Step 2. Write the third-order polarization at the fundamental, $P^{(3)}_\omega = \tfrac34\varepsilon_0\chi^{(3)}E_0^2\,(E_0\cos\omega t)$ , and fold it into the total to define an effective susceptibility $\chi_{\mathrm{eff}} = \chi^{(1)} + \tfrac34\chi^{(3)}E_0^2$ . Why: the wave cannot tell the linear response from an intensity-locked addition — both oscillate at $\omega$ , so the medium simply presents a modified $\chi$ .

Step 3. Convert susceptibility to index: $n^2 = 1 + \chi_{\mathrm{eff}} = n_0^2 + \tfrac34\chi^{(3)}E_0^2$ . Since the nonlinear part is tiny, expand the square root to first order, $n \approx n_0 + \tfrac{3}{8}(\chi^{(3)}/n_0)E_0^2$ . Why: the correction is $\sim10^{-7}$ of $n_0^2$ , so a linear expansion is not an approximation of convenience but essentially exact.

Step 4. Trade amplitude for intensity with $I = \tfrac12 n_0\varepsilon_0 c\,E_0^2$ : substituting $E_0^2 = 2I/(n_0\varepsilon_0 c)$ gives $\Delta n = \dfrac{3\chi^{(3)}}{4n_0^2\varepsilon_0 c}\,I$ , so $n_2 = 3\chi^{(3)}/(4n_0^2\varepsilon_0 c)$ . Why: intensity, not field amplitude, is what a detector reports, so the coefficient must be written per unit intensity.

Go deeper · Derivation B · 1 minSelf-phase modulation and the accumulated Kerr phase

Derivation BSelf-phase modulation and the accumulated Kerr phase

The aim is to see how an index that barely twitches, $\Delta n\sim10^{-7}$ , rewrites a pulse's spectrum — and to quantify the sweep. The strategy is to send a pulse of fixed shape through a Kerr length short enough that dispersion has not yet reshaped it, so each slice advances in phase by an amount set by its own instantaneous intensity. Differentiating that phase in time gives the chirp, and the chirp is the new spectrum.

Step 1. With dispersion neglected the envelope is frozen and only the phase evolves: $E_{\mathrm{out}}(t) = E_{\mathrm{in}}(t)\,e^{i\varphi_{\mathrm{NL}}(t)}$ with $\varphi_{\mathrm{NL}}(t) = (2\pi/\lambda)\,n_2 I(t)\,L$ . Why: over a sufficiently short propagation length the Kerr phase can accumulate while group-velocity dispersion, has not yet had time to redistribute the pulse in time; equivalently, the nonlinear length is shorter than the dispersion length.

Step 2. With the field convention $E(t)\propto e^{-i\omega t}$ , the instantaneous frequency shift is $\delta\omega(t)=-d\varphi_{\mathrm{NL}}/dt=-(2\pi n_2 L/\lambda)\,dI/dt$ ; the rising leading edge ( $dI/dt>0$ ) is shifted red, while the falling trailing edge is shifted blue. Why: frequency is the rate of phase advance — an intensity changing in time is, by definition, a time-dependent frequency.

Step 3. The peak phase sets the nonlinear phase excursion: $\varphi_{\mathrm{max}} = (2\pi/\lambda)n_2 I_0 L$ , while the maximum frequency shift scales roughly as $\varphi_{\mathrm{max}}/\tau$ , with $\tau$ the pulse duration. Strong self-phase modulation therefore broadens the spectrum and can produce symmetric spectral fringes or multiple lobes, with the detailed structure depending on the pulse shape and the accumulated nonlinear phase. Why: different temporal portions of the pulse can contribute to the same instantaneous frequency, so their interference produces the characteristic oscillatory spectral structure.

Step 4. Read off the scales. One metre of silica at $1\ \mathrm{GW/cm^2}$ gives $\varphi_{\mathrm{max}} = 1.54\ \mathrm{rad}$ , on the edge of a first side-lobe; a high-nonlinearity photonic-crystal fibre reaches many radians in centimetres, broadening past an octave. Why: $\varphi$ accrues linearly with $L$ and $I_0$ , so engineering the spectrum is a matter of length and confinement, not of finding a bigger $n_2$ . *Numerically: $\varphi_{\mathrm{NL}}=1.535\ \mathrm{rad}$ for $n_2=2.6\times10^{-20}$ , $I=10^{13}\ \mathrm{W/m^2}$ , $L=1\ \mathrm{m}$ , $\lambda=1064\ \mathrm{nm}$ .*

Go deeper · Derivation C · 1 minWhy the collapse threshold is a power, not an intensity

Derivation CWhy the collapse threshold is a power, not an intensity

The goal is the strangest result in the subject: catastrophic self-focusing sets in above a fixed power $P_{\mathrm{cr}}$ , regardless of how tightly the beam is focused. The strategy is to pit the two angles that fight over a beam of radius $w$ — the diffraction spread pulling it open, the Kerr self-bending pulling it shut — and watch the radius cancel when they are set equal, leaving a pure power threshold.

Step 1. A beam of radius $w$ diffracts with a half-angle $\theta_{\mathrm{diff}}\sim\lambda/(2\pi n_0 w)$ : smaller spots diverge faster. Why: diffraction is the Fourier price of transverse confinement.

Step 2. The self-induced index bump $\Delta n = n_2 I$ bends the marginal ray inward by $\theta_{\mathrm{sf}}\sim\sqrt{2\Delta n/n_0} = \sqrt{2 n_2 I/n_0}$ , and with $I\sim P/(\pi w^2)$ this is $\theta_{\mathrm{sf}}\sim\sqrt{2 n_2 P/(n_0\pi)}\,/\,w$ . Why: the graded index acts as a lens whose focusing angle is set by the index contrast between axis and wing, and that contrast is fixed by power-per-area.

Step 3. Set $\theta_{\mathrm{sf}} = \theta_{\mathrm{diff}}$ . Both sides carry one factor of $1/w$ , so the radius cancels, leaving $\sqrt{2 n_2 P/(n_0\pi)} = \lambda/(2\pi n_0)$ . Why: this cancellation is the entire content — focusing and diffraction share the same $1/w$ scaling, so spot size drops out and only total power decides the contest.

Step 4. Solve for the threshold: $P_{\mathrm{cr}}\sim\lambda^2/(8\pi n_0 n_2)$ , which the exact Townes profile sharpens to $P_{\mathrm{cr}} = 3.77\lambda^2/(8\pi n_0 n_2)$ . Why: the order-unity coefficient depends on beam shape, but the scaling $\lambda^2/(n_0 n_2)$ and the independence from $w$ are model-free. *The $w$ -cancellation confirms that spot size drops out entirely, giving $P_{\mathrm{cr}} = 3.77\lambda^2/(8\pi n_0 n_2) = 4.50\ \mathrm{MW}$ for silica at $1064\ \mathrm{nm}$ .*

Limiting cases

Case	Result	Meaning
weak field	$\Delta n = n_2 I \to 0$ , $n\to n_0$	ordinary linear optics; the $\chi^{(3)}$ correction is negligible against $\chi^{(1)}$
centrosymmetric medium	$\chi^{(2)}=0$ , $\chi^{(3)}\neq0$	in glasses, liquids, and gases the lowest allowed nonlinearity is third order — the case Introduction to Nonlinear Optical Polarization sets up
self-focusing threshold	$P\gtrsim P_{\mathrm{cr}} = 3.77\lambda^2/(8\pi n_0 n_2)$	above it, diffraction can no longer balance the Kerr lens and the beam collapses to a filament
near a two-photon resonance	$\chi^{(3)}(\omega)$ complex, enhanced	the imaginary part is two-photon absorption; $n_2$ can grow by orders and even change sign
DC field instead of light	$\Delta n \propto E_{\mathrm{DC}}^2$	the 1875 electro-optic Kerr effect — same $\chi^{(3)}$ , arguments $\chi^{(3)}(\omega;\omega,0,0)$

Common trapThe Kerr index shift is only

10^{-7}

, so it can safely be ignored

This is the single most expensive misconception in high-power optics. Yes, $\Delta n = n_2 I$ is minuscule — parts in ten million even at gigawatt intensities — but the observable is never the index; it is the phase $\varphi = (2\pi/\lambda)\,\Delta n\,L$ , and phase accumulates. One metre of fibre already turns that $10^{-7}$ into $1.54$ radians; a kilometre, or a regenerative amplifier's many passes, multiplies it further. This is why the accumulated nonlinear phase — the $B$ -integral, $B=\frac{2\pi}{\lambda}\int n_2 I\,dz$ — is often kept to only a few radians in high-power laser systems: excessive nonlinear phase can promote small-scale self-focusing and other nonlinear distortions. The index change is a whisper; heard ten thousand times over, it is a shout.

Worked examples

1 · Self-phase-modulation phase in a fibre (confidence builder)

A continuous-wave-like pulse at $\lambda = 1064\ \mathrm{nm}$ propagates $L = 1\ \mathrm{m}$ through fused silica ( $n_2 = 2.6\times10^{-20}\ \mathrm{m^2/W}$ ) at intensity $I = 1\ \mathrm{GW/cm^2}$ . Find the induced index change and the peak nonlinear phase, and say what the spectrum does.

Step 1. Convert the intensity: $1\ \mathrm{GW/cm^2} = 10^9\ \mathrm{W}/10^{-4}\ \mathrm{m^2} = 10^{13}\ \mathrm{W/m^2}$ .

Step 2. Index change: $\Delta n = n_2 I = (2.6\times10^{-20})(10^{13}) = \mathbf{2.6\times10^{-7}}$ — seven orders below $n_0 = 1.45$ .

Step 3. Accumulated phase: $\varphi_{\mathrm{NL}} = (2\pi/\lambda)\,\Delta n\,L = (2\pi/1.064\times10^{-6})(2.6\times10^{-7})(1) = \mathbf{1.54\ rad}$ — a tenth of a wave from a $10^{-7}$ index, because the wave fits a million times into a metre.

Step 4. Interpretation: $1.54\ \mathrm{rad}$ is just under the first SPM side-lobe threshold ( $\sim\pi$ ), so the spectrum begins to broaden symmetrically. Push the intensity or length until $\varphi_{\mathrm{max}}$ reaches several $\pi$ and the single line fissions into the multi-lobed comb of supercontinuum. $\Delta n = 2.6\times10^{-7}$ , $\varphi_{\mathrm{NL}} = 1.54\ \mathrm{rad}$ .

2 · Critical power for silica and the femtosecond oscillator (exam level)

Find the critical power for whole-beam self-focusing in fused silica at $1064\ \mathrm{nm}$ ( $n_0 = 1.45$ , $n_2 = 2.6\times10^{-20}\ \mathrm{m^2/W}$ ), and use it to explain how a Kerr-lens mode-locked laser exploits — without triggering — the same physics.

Step 1. Apply Equation (4): $P_{\mathrm{cr}} = 3.77\lambda^2/(8\pi n_0 n_2)$ . The denominator is $8\pi(1.45)(2.6\times10^{-20}) = 9.47\times10^{-19}$ ; the numerator is $3.77(1.064\times10^{-6})^2 = 4.27\times10^{-12}$ .

Step 2. Divide: $P_{\mathrm{cr}} = 4.27\times10^{-12}/9.47\times10^{-19} = \mathbf{4.50\ MW}$ . This is the power at which a collimated silica-guided beam would begin to implode, independent of how tightly it is focused.

Step 3. Now a Ti:sapphire oscillator: a $\sim5\ \mathrm{nJ}$ intracavity pulse compressed to $\sim50\ \mathrm{fs}$ carries a peak power $P\approx5\times10^{-9}/5\times10^{-14} = \mathbf{10^{5}\ W} = 0.1\ \mathrm{MW}$ — a few percent of a megawatt-scale $P_{\mathrm{cr}}$ . Well below collapse, but not below notice: the pulse still forms a weak Kerr lens each pass, focusing itself more than the weak continuous background.

Step 4. That differential focusing is the basis of Kerr-lens mode locking: an aperture can preferentially transmit the self-focused pulsed mode over the weaker continuous background, providing an effective fast saturable-absorber action through the Kerr nonlinearity. The laser operates well below the catastrophic self-focusing threshold, at about $2\%$ of $P_{\rm cr}$ in this estimate: enough nonlinear lensing for mode locking, but far below the ideal collapse power. $P_{\rm cr}=4.50\ {\rm MW}$ .

Physics of the portrait

The whole subject rides on a single material number — the nonlinear refractive index — so the portrait here is not a person but the coefficient itself, the entry every nonlinear-optics table is built around.

Origins · 1 minNonlinear Refractive Index and how this was discovered

Nonlinear Refractive Index

n₂ (m²/W) · nonlinear optics

The nonlinear refractive index $n_2$ is the coefficient in $n = n_0 + n_2 I$ — how much a material's index yields per unit optical intensity. In SI it carries units of $\mathrm{m^2/W}$ and ties to the microscopic third-order susceptibility through $n_2 = 3\chi^{(3)}/(4 n_0^2\varepsilon_0 c)$ ; an older Gaussian convention writes $\Delta n = \tfrac12\bar n_2|E|^2$ with $\bar n_2$ in esu, and confusing the two is a classic factor-error. Its values span a vast range: fused silica sits at $2.6\times10^{-20}\ \mathrm{m^2/W}$ , while chalcogenide glasses and semiconductors run hundreds to thousands of times larger, at the price of slower response and added absorption. Because it is so small in transparent solids, $n_2$ is measured by self-action techniques — most famously the z-scan, which translates a sample through a focus and reads the self-lensing as a peak-valley signature in the far-field transmission through an aperture. Sign matters as much as size: positive $n_2$ self-focuses and red-shifts leading edges; negative $n_2$ (near certain resonances, or above the two-photon edge in some semiconductors) self-defocuses. Every quantity in this article is ultimately a statement about this one number.

History

1875

— John Kerr discovers the electro-optic (DC) Kerr effect: a static field makes glass birefringent, with an index change quadratic in the field

1964

— Maker, Terhune, and Savage observe the intensity-dependent index — the optical Kerr effect proper — in the first years of the laser

1964

— Hercher reports self-focusing damage tracks drilled through glass by intense beams

1964

— Chiao, Garmire, and Townes predict beam self-trapping and the critical power for collapse

1978

— Stolen and Lin measure self-phase modulation directly in optical fibres

1991

— Spence, Kean, and Sibbett demonstrate Kerr-lens mode-locking in Ti:sapphire, opening routine sub-100-fs operation

Going further

Texts. Boyd, Nonlinear Optics, 3rd ed., ch. 4 and 7 (the $\chi^{(3)}$ menu, self-action, self-focusing) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., ch. 21 · Agrawal, Nonlinear Fiber Optics, 5th ed., ch. 2–4 (SPM, solitons, supercontinuum) · Loudon, The Quantum Theory of Light, 3rd ed. Index note: Some sources quote fused silica $n_2\approx2.7\times10^{-20}\ \mathrm{m^2/W}$ ; this article uses $n_2 = 2.6\times10^{-20}\ \mathrm{m^2/W}$ (both lie within the $2.5$ – $3.2\times10^{-20}$ literature spread), and only $2.6\times10^{-20}$ reproduces $P_{\mathrm{cr}} = 4.50\ \mathrm{MW}$ — every number here is computed with the $2.6$ value. An alternative form $P_{\mathrm{cr}} = 1.8962\lambda^2/(4\pi n_0 n_2)$ uses a Gaussian-profile coefficient of $0.151$ , against the Townes value $3.77/(8\pi) = 0.150$ — agreement to under a percent; the order-unity difference is the beam profile, not an error.

Prerequisites. Dielectric Susceptibility and Permittivity · Electric Polarization and the Displacement Field · The Refractive Index · The Lorentz Oscillator Model · Maxwell's Equations in Dielectric Media. Leads to. Supercontinuum Generation · Optical Solitons · Group Velocity and Group Index · Kerr-Lens Mode-Locking.

What is the difference between the DC Kerr effect and the optical Kerr effect?

Both are third-order ( $\chi^{(3)}$ ) effects with an index change quadratic in a field, differing in what supplies the field. The DC (electro-optic) Kerr effect of 1875 uses a static voltage: $\Delta n \propto E_{\mathrm{DC}}^2$ , described by $\chi^{(3)}(\omega;\omega,0,0)$ — the basis of the fast Kerr cells once used as shutters. The optical Kerr effect uses the light's own oscillation, so the time-averaged $\langle E^2\rangle$ is proportional to intensity, $\Delta n = n_2 I$ , described by the degenerate $\chi^{(3)}(\omega;\omega,-\omega,\omega)$ . Same tensor, different frequency arguments — a battery versus the beam.

Where does the factor of 3/4 in n₂ actually come from?

From cubing a cosine. With $\cos^3\omega t = \tfrac34\cos\omega t + \tfrac14\cos3\omega t$ , exactly three-quarters of the cubic polarization oscillates back at the original frequency (the rest is the third harmonic). That $\tfrac34$ , carried through the effective-permittivity step and the square-root expansion, leaves $n_2 = 3\chi^{(3)}/(4n_0^2\varepsilon_0 c)$ . In the complex-amplitude convention the same $3$ appears as the degeneracy factor counting the ways three fields at $\pm\omega$ combine to give $\omega$ — different bookkeeping, identical result, but never to be double-counted.

If the beam self-focuses without limit above P_cr, why don't lasers just destroy themselves?

Because the paraxial collapse is arrested by physics the simple theory leaves out. As the intensity climbs, higher-order effects intervene: multiphoton and avalanche ionization create a defocusing plasma, dispersion and self-steepening reshape the pulse, and the nonlinearity saturates. The result is not a point singularity but a filament — a self-guided channel a hundred microns wide that can persist for metres, its intensity clamped where focusing and plasma defocusing balance. In a solid, though, that clamped intensity usually sits above the damage threshold, which is why the $B$ -integral must be kept small in high-power glass: the beam is saved from mathematical collapse only to drill a physical track.

How can self-phase modulation create new frequencies without adding energy?

SPM is a pure phase operation: $E_{\mathrm{out}} = E_{\mathrm{in}}\,e^{i\varphi(t)}$ multiplies the field by a unit-modulus factor, so the total energy $\int|E|^2\,dt$ is untouched. What changes is the distribution of that energy across frequency. A time-dependent phase is a time-dependent instantaneous frequency, so the pulse ends up carrying red components on its leading edge and blue on its trailing edge — the spectrum broadens while the energy is merely reshuffled among colours. Energy conservation and spectral broadening are not in tension; SPM moves photons in frequency, it does not create them.

Why does the same n₂ give such different answers for femtosecond and nanosecond pulses?

Because $n_2$ is a sum over mechanisms with wildly different clocks. The bound-electron response is essentially instantaneous, so it contributes to any pulse; molecular reorientation (picoseconds, e.g. $\mathrm{CS_2}$ ) and electrostriction/thermal lensing (nanoseconds to milliseconds) are slow. A femtosecond pulse is gone before the slow terms respond, so it sees only the small, fast electronic $n_2$ ; a nanosecond pulse lets the large, slow terms build, so its effective $n_2$ can be orders of magnitude larger and even opposite in sign. This is why any quoted $n_2$ is meaningless without a pulse duration attached.

Is a fibre soliton the same self-action as beam self-focusing?

They are the temporal and spatial faces of the one Kerr term. Spatial self-focusing is the transverse index profile bending rays inward until, above $P_{\mathrm{cr}}$ , diffraction loses. A temporal soliton is the same $n_2$ acting along the pulse: SPM chirps it (red leading, blue trailing), and in the anomalous-dispersion regime ( $\beta_2<0$ ) the fibre delays blue relative to red in just the way that undoes the chirp. When the two chirps cancel exactly, the pulse propagates unchanged — a fundamental soliton. Self-focusing is a runaway because nothing balances the transverse Kerr lens; the soliton is a truce because anomalous dispersion balances the longitudinal one.

The Optical Kerr Effect and Intensity-Dependent Index

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · Self-phase-modulation phase in a fibre (confidence builder)

2 · Critical power for silica and the femtosecond oscillator (exam level)

Physics of the portrait

Going further

Before this

Where this leads