Nonlinear optics

Nonlinear optics covers the effects that appear when a material's response to light stops being proportional to the optical field. The induced polarization then contains terms in the square and cube of the field, and these terms radiate light at new frequencies (harmonics, sums and differences of the input frequencies) and make the refractive index depend on intensity. The effects are small because optical fields are small compared with the fields that bind electrons: light at 1 GW/cm² in fused silica has a peak field of about 7 × 10⁷ V/m, roughly 1.4 × 10⁻⁴ of the atomic unit of field, 5.1 × 10¹¹ V/m. They become important with laser intensities, long interaction lengths or tight confinement. The first observation, by P. A. Franken and coworkers in 1961, was second-harmonic generation at 347.2 nm from a pulsed ruby laser at 694.3 nm focused into crystalline quartz.

The nonlinear polarization

The polarization is expanded in powers of the field:

$$P = \varepsilon_0\big(\chi^{(1)}E + \chi^{(2)}E^2 + \chi^{(3)}E^3 + \cdots\big).$$

The coefficients are the nonlinear susceptibilities. $\chi^{(1)}$ gives the ordinary refractive index; $\chi^{(2)}$ is tens of pm/V in lithium niobate, and $\chi^{(3)}$ is about 2 × 10⁻²² m²/V² in fused silica. In a medium with inversion symmetry, reversing $E$ must reverse $P$, which forces $\chi^{(2)}$ to zero. Glasses, liquids, silicon and silicon nitride therefore have no bulk second-order response, while crystals such as lithium niobate, KTP, BBO and GaAs do. Every material has a third-order response.

Second-order processes

These involve three waves, with photon energy conserved in each frequency-mixing process:

• Second-harmonic generation (SHG): $\omega + \omega \to 2\omega$, for example 1064 nm to 532 nm.
• Sum- and difference-frequency generation, which mix two input frequencies.
• Optical parametric amplification and the optical parametric oscillator, which split a pump photon into a signal and an idler and give widely tunable sources.
• Spontaneous parametric down-conversion, the standard source of entangled photon pairs.
• The Pockels effect, a refractive-index change proportional to an applied electric field, used in electro-optic modulators.

Third-order processes

The Kerr effect makes the index rise with intensity, $n = n_0 + n_2 I$. Its consequences in pulses and beams are self-phase modulation, cross-phase modulation, self-focusing and solitons. Four-wave mixing and third-harmonic generation create new frequencies. Two-photon absorption arises from the imaginary part of $\chi^{(3)}$. Stimulated Raman scattering and stimulated Brillouin scattering couple light to molecular vibrations and to acoustic waves, and supercontinuum generation combines several of these processes to broaden a pulse spectrum over an octave or more.

Intensity and length scales

The Kerr index change and the nonlinear phase it accumulates over a length $L$ are

$$\Delta n = n_2 I,\qquad \phi_\text{NL} = \frac{2\pi}{\lambda}\,n_2 I L.$$

For fused silica ($n_2$ = 2.6 × 10⁻²⁰ m²/W) at 1 GW/cm² (10¹³ W/m²), $\Delta n$ = 2.6 × 10⁻⁷, and 1 cm of glass at 1064 nm adds a nonlinear phase of only 0.015 rad. Bulk optics therefore show Kerr effects mainly with pulsed lasers of high peak power. In single-mode fiber the light stays confined to an effective area of about 80 µm² for kilometers: the nonlinear coefficient $\gamma = 2\pi n_2/(\lambda A_\text{eff})$ is 1.32 W⁻¹km⁻¹ at 1550 nm, and with 100 mW launched the phase would reach 1 rad after 7.6 km of lossless fiber (about 9.3 km at 0.2 dB/km). At low conversion, second-order efficiency grows in proportion to the input intensity.

Phase matching

Light generated at different points along the medium adds up only if the interacting waves keep step with each other. Phase matching is the condition that the wave-vector mismatch $\Delta k$ vanish. Without it, conversion oscillates over the coherence length, $L_c = \lambda/(4\Delta n)$ for SHG: 9.5 µm for 1550 nm light in lithium niobate, where the index difference between 1550 nm and 775 nm is 0.0408. Birefringent phase matching uses crystal orientation or temperature to equalize the indices; quasi-phase matching in periodically poled crystals reverses the sign of $\chi^{(2)}$ every coherence length. In fiber, four-wave mixing is strongest near the zero-dispersion wavelength, where it is phase matched.

Where it matters

Nonlinear optics supplies frequency-converted green, ultraviolet and mid-infrared sources, Pockels modulators, Kerr-lens mode locking, microresonator frequency combs and photon-pair sources. It also sets limits: fiber nonlinearity caps the launch power in long-haul links, stimulated Brillouin scattering limits narrow-linewidth power in fiber, and self-focusing constrains high-power amplifiers.

Common pitfalls are using average instead of peak intensity for pulsed sources, mixing units ($n_2$ in cm²/W is 10⁴ times the value in m²/W), and the factor of two between the coefficient $d$ and $\chi^{(2)}$ ($d = \chi^{(2)}/2$). Published $n_2$ values for one material differ by tens of percent with wavelength, pulse duration and technique.

Common questions

What is the difference between second- and third-order effects?

Second-order effects involve three waves, need a crystal without inversion symmetry and phase matching, and are used mostly for frequency conversion and electro-optic modulation. Third-order effects involve four waves, occur in every material, and dominate in glass fiber and silicon.

Does silicon have a second-order nonlinearity?

Its crystal is centrosymmetric, so bulk silicon has no $\chi^{(2)}$; strain and interfaces break the symmetry only weakly. Silicon photonics uses its large Kerr effect and contends with two-photon absorption, present wherever the photon energy exceeds half the 1.12 eV bandgap, at wavelengths below about 2.2 µm. The nonlinear optics overview article treats each process with worked numbers.

References: R. W. Boyd, Nonlinear Optics, 4th ed. (Academic Press, 2020); P. A. Franken, A. E. Hill, C. W. Peters and G. Weinreich, "Generation of optical harmonics," Phys. Rev. Lett. 7, 118 (1961); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); G. P. Agrawal, Nonlinear Fiber Optics, 6th ed. (Academic Press, 2019).