Kerr effect

The optical Kerr effect is the change in refractive index of a medium in proportion to the optical intensity passing through it. The nonlinear-index coefficient $n_2$ is small: about $2.6\times {10}^{-20}$ m²/W for fused silica and $4.5\times {10}^{-18}$ m²/W for silicon at 1550 nm. With 1 W confined to the 80 μm² effective area of standard single-mode fiber, the index rises by $3.25\times {10}^{-10}$. The effect matters because that small change acts over long lengths or at high peak intensities.

$$n(I)=n_0+n_{2}I$$

Here $n_2$ is the nonlinear-index (Kerr) coefficient, in m²/W. It arises from the third-order nonlinear susceptibility ${\chi }^{(3)}$:

$$n_2=\frac{3{\chi }^{(3)}}{4{\epsilon }_{0}c{n}_0^2}$$

Inverting this for fused silica ($n_0=1.444$) gives ${\chi }^{(3)}\approx 1.9\times {10}^{-22}$ m²/V².

Physical origin and speed. The electronic contribution to the Kerr effect is instantaneous on optical timescales (response time set by electronic polarizability, typically $<$ 10 fs). It is present in every medium, including centrosymmetric ones where the second-order ${\chi }^{(2)}$ vanishes; it is isotropic in amorphous media such as glass, while in cubic crystals such as silicon $n_2$ depends on crystal orientation. In silica fiber a delayed Raman (nuclear) response of a few tens of femtoseconds adds to it, and electrostriction contributes for long pulses or modulated CW light, so measured $n_2$ depends on the pulse duration used.

Typical values

Kerr coefficients at 1550 nm:

Published values for a given material often differ by tens of percent, and occasionally by a factor of two, depending on wavelength, pulse duration, polarization and technique. Older literature uses cm²/W (silica: $2.6\times {10}^{-16}$ cm²/W) or esu.

Measurement

Z-scan. A focused beam is translated through a thin sample while a detector behind a small aperture records transmission. Self-focusing or self-defocusing produces a peak-valley signature whose amplitude gives the on-axis nonlinear phase and therefore $n_2$; removing the aperture isolates nonlinear absorption. This is the standard bulk-material method.

Spectral broadening in fiber or waveguides. Self-phase modulation of a pulse of known peak power broadens its spectrum in a predictable way. Fitting the broadened spectrum gives the nonlinear coefficient $\gamma =2\pi n_2\mathrm{/}(\lambda A_{\text{eff}})$, from which $n_2$ follows once $A_{\text{eff}}$ is known.

Worked numbers

In standard single-mode fiber ($A_{\text{eff}}=80$ μm²) at 1550 nm, $\gamma =1.32$ W⁻¹km⁻¹, so 1 W over 1 km gives a nonlinear phase of about 1.3 rad. A silicon wire with $A_{\text{eff}}\approx 0.1$ μm² has $\gamma \approx 180$ W⁻¹m⁻¹, and 100 mW over 1 cm gives about 0.18 rad: the tight confinement and larger $n_2$ largely offset a length five orders of magnitude shorter.

The critical power for catastrophic self-focusing of a Gaussian beam in bulk material is approximately

$$P_{\text{cr}}\approx \frac{3.77{\lambda }^2}{8\pi n_0{n}_2}$$

which gives about 4.5 MW in fused silica at 1064 nm and about 2.5 MW at 800 nm. The threshold is a power, independent of beam size, so focusing more loosely does not raise it.

Applications

The Kerr effect is the basis for optical frequency combs in microresonators (Kerr combs or microcombs). High-finesse microresonators on platforms like silicon nitride, lithium niobate, or aluminum nitride pumped by a narrow-linewidth CW laser convert input power into a comb of equally spaced lines via cascaded four-wave mixing; they are used in optical clocks, ranging, and on-chip spectroscopy. In Ti:sapphire lasers the same nonlinearity forms a Kerr lens inside the crystal; an aperture or the pump-mode overlap then favors the high-intensity pulsed state, the usual route to few-cycle pulses from Ti:sapphire oscillators.

Pitfalls

Nonlinear absorption. In silicon, two-photon absorption accompanies the Kerr effect at 1550 nm. The figure of merit $n_2\mathrm{/}({\beta }_{\text{TPA}}\lambda )$ is about 0.36 with ${\beta }_{\text{TPA}}=0.8$ cm/GW, and the free carriers it generates add loss and a plasma-dispersion index change of opposite sign. Silicon nitride and silica have negligible TPA in the telecom band.

Thermal effects. Absorption heats the material and shifts the index through the thermo-optic coefficient; for CW light in resonators this slow response can exceed the Kerr shift by orders of magnitude.

Common questions

How does the Kerr effect differ from the Pockels effect?

The electro-optic (Pockels) effect changes the index in proportion to an applied field $E$ and requires a material without inversion symmetry, such as a non-centrosymmetric crystal or a poled polymer. The Kerr effect depends on $\mathrm{\mid }E{\mathrm{\mid }}^2$ and occurs in all materials. The DC Kerr effect, the quadratic response to an applied low-frequency field, is the historical form observed by Kerr in 1875 and is used in liquid Kerr cells; the optical Kerr effect is the same ${\chi }^{(3)}$ process driven by the light itself. See also Pockels effect.

Why does silica fiber show strong nonlinearity when $n_2$ is so small?

Because the light is confined to about 80 μm² and propagates over tens of kilometers with low loss. The nonlinear phase is set by $n_2$ multiplied by intensity and by interaction length, and in fiber both are large.

The Kerr effect is placed among the other nonlinear processes, with nonlinear lengths for fiber, silicon nitride and silicon waveguides, in Nonlinear Optics: An Overview.

References: Boyd, Nonlinear Optics 4th ed. 2020; Agrawal, Nonlinear Fiber Optics 6th ed. 2019; Saleh & Teich, Fundamentals of Photonics 3rd ed. 2019; M. Sheik-Bahae et al., "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760 (1990).