Nonlinear Optics: An Overview of Second- and Third-Order Effects

Scope

Nonlinear optics covers the effects that appear when a material's response to light is no longer proportional to the light's field, so that light changes the medium it travels through and beams of different frequencies can exchange energy. This article is an overview: it sets out the two orders of nonlinearity that matter in practice, what each produces, the phase-matching condition that decides whether a process is efficient, and the numbers that tell whether an effect will matter in a given device. Each effect has its own entry, linked below.

In short: second-order (${\chi }^{(2)}$) effects convert frequencies and underlie the Pockels effect, but occur only in crystals without inversion symmetry and need phase matching; third-order (${\chi }^{(3)}$) effects occur in every material, including glass and silicon, and matter wherever intensity or length is large, as in optical fiber and tightly confining waveguides.

The nonlinear polarization

Light drives the electrons of a material, and the induced polarization $P$ radiates the transmitted and reflected light. At low intensity $P$ is proportional to the field $E$. At higher intensity the response contains higher powers of the field:

$$P={\epsilon }_0({\chi }^{(1)}E+{\chi }^{(2)}{E}^2+{\chi }^{(3)}{E}^3+\cdots )\mathrm{.}$$

The linear term gives the ordinary refractive index. The $E^2$ term, fed with a field oscillating at $\omega$, contains components at $2\omega$ and at zero frequency; fed with two frequencies, it contains their sum and difference. The $E^3$ term produces components at $3\omega$ and at combinations of three frequencies, and also a term at the original frequency proportional to the intensity, which appears as an intensity-dependent refractive index.

In a material with inversion symmetry, reversing $E$ must reverse $P$, which forces ${\chi }^{(2)}$ to zero. Glass, silicon, silicon nitride and liquids therefore have no bulk second-order response; lithium niobate, KTP, BBO, GaAs and other non-centrosymmetric crystals do.

Second-order effects

All of these need a crystal without inversion symmetry, and a material that is strongly electro-optic is usually also a good frequency converter: in lithium niobate both properties come from the same polar crystal structure (see the lithium niobate materials page).

Phase matching

A frequency-conversion process is efficient only if the generated wave stays in step with the nonlinear polarization that drives it. For second-harmonic generation this requires $n(2\omega )=n(\omega )$; normal dispersion makes the index at the harmonic higher, so the two drift out of phase. Over a coherence length

$$L_c=\frac{\lambda }{4(n_{2\omega }-n_{\omega })},$$

where $\lambda$ is the fundamental wavelength, the harmonic grows; over the next $L_c$ it converts back into the fundamental.

Worked example. For 1550 nm to 775 nm in lithium niobate with both waves polarized along the extraordinary axis, the congruent-crystal Sellmeier equation gives $n_e$ = 2.1376 at 1550 nm and 2.1784 at 775 nm, a difference of 0.0408. The coherence length is 9.5 μm, so without phase matching the harmonic never builds beyond what 9.5 μm of crystal produces.

There are two remedies:

• Birefringent phase matching uses a crystal in which the fundamental and harmonic travel with different polarizations, so that the ordinary index at one frequency equals the extraordinary index at the other, tuned by the propagation angle or the temperature. See birefringence.
• Quasi-phase matching reverses the sign of ${\chi }^{(2)}$ every coherence length by periodically poling the crystal's ferroelectric domains, so the conversion resumes where it would have reversed. The poling period is $2L_c$: 19.0 μm for the example above, the reason periodically poled lithium niobate (PPLN) for 1550 nm doubling has periods near 19 μm. Quasi-phase matching lets all waves share the polarization with the largest coefficient ($d_{33}$ in lithium niobate) and works at any wavelength the poling period can be made for.

In low-conversion conditions the harmonic power grows with the square of the fundamental power and, when phase-matched, with the square of the length for a plane wave; tighter confinement raises the intensity, which is why thin-film lithium niobate waveguides reach normalized efficiencies far above those of bulk crystals.

Third-order effects

The Kerr coefficient of fused silica is small, $n_2$ ≈ 2.6 × 10⁻²⁰ m²/W: even 1 GW/cm² changes the index by only 2.6 × 10⁻⁷. The effects matter in fiber because the light is confined to a small area over a long distance. Their scale is set by the nonlinear coefficient $\gamma =2\pi n_2\mathrm{/}(\lambda A_{\text{eff}})$ and the nonlinear length $L_{\text{NL}}=1\mathrm{/}(\gamma P)$, the distance over which the Kerr phase reaches one radian.

At 100 mW in standard fiber the nonlinear length is 7.6 km, shorter than a typical span, which is why nonlinearity limits the launch power of long-haul systems. In silicon, the millimeter-scale nonlinear length makes Kerr effects usable on a chip, but two-photon absorption and the free carriers it creates add loss at the same intensities; silicon nitride has no two-photon absorption at 1550 nm, which is why microresonator frequency combs are usually made in it.

When nonlinearity matters

A process is worth considering when the relevant length (the device length, or the effective length set by loss) is comparable with or longer than the nonlinear length for third-order effects, or when a phase-matched second-order interaction has enough length and intensity. Short free-space paths through glass at continuous-wave powers are almost always linear; ultrafast pulses, focused high-power beams, long fibers and high-confinement waveguides often are not. The same effects that limit a fiber link or a high-power fiber laser (self-phase modulation, four-wave mixing, stimulated Raman and Brillouin scattering) are used deliberately in supercontinuum sources, parametric amplifiers, wavelength converters and frequency combs; see supercontinuum generation.

References: R. W. Boyd, Nonlinear Optics (4th ed., Academic Press, 2020); G. P. Agrawal, Nonlinear Fiber Optics (6th ed., Academic Press, 2019); D. E. Zelmon, D. L. Small and D. Jundt, "Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol.% magnesium oxide-doped lithium niobate," Journal of the Optical Society of America B 14, 3319 (1997).