Nonlinear optics (NLO) is a branch of optics that studies the case when optical properties of matter depend on the intensity of the input light. Nonlinear phenomena become relevant only when the input light is very intense. Typically, in order to observe nonlinear phenomena, an intensity of the electromagnetic field of light larger than 108 V/m (and thus comparable to the atomic electric field of ~1011 V/m) is required. In this case, the polarization density P responds non-linearly to the electric field E of light. In order to obtain an electromagnetic field that is sufficiently intense, laser sources must be used. In nonlinear optics, the superposition principle no longer holds, and the polarization of the material is no longer linear in the electric field intensity. Instead, in the perturbative limit, it can be expressed by a polynomial sum of order n.
Many different physical mechanisms can cause nonlinearities in the optical behaviour of a material, i.e. the motion of bound electrons, field-induced vibrational or orientational motions, optically-induced acoustic waves and thermal effects. The motion of bound electrons, in particular, has a very short response timescale, so it is of particular relevance in the context of ultrafast nonlinear optics. The simplest way to picture this behaviour in a semiclassical way is to use a phenomenological model: an anharmonic oscillator can model the forced oscillations of a bound electron inside the medium. In this picture, the binding interaction between the ion core and the electron is the Coulomb force and nonlinearities appear as changes in the elastic constant of the system (which behaves similarly to a mass attached to a spring) when the stretching or compression of the oscillator is large enough.
Maxwell's equations are linear in vacuum, so, nonlinear processes only occur in media. However, the theory of quantum electrodynamics (QED) predicts that, above the Schwinger limit, vacuum itself can behave in a nonlinear way.
The description of nonlinear optics usually presented in textbooks is the perturbative regime, which is valid when the input intensity remains below 1014 W/cm2, which implies that the electric field is well below the intensity of interatomic fields. This approach allows to use a Taylor series to write down the polarization density as a polynomial sum. It is also possible to study the laser-matter interaction at a much higher intensity of light: this field is referred to as nonperturbational nonlinear optics or extreme nonlinear optics and investigates the generation of extremely high-order harmonics, attosecond pulse generation and relativistic nonlinear effects.
History The first nonlinear optical effect to be predicted was two-photon absorption, by Maria Goeppert Mayer for her PhD in 1931, but it remained an unexplored theoretical curiosity until 1961 and the almost simultaneous observation of two-photon absorption at Bell Labs and the discovery of second-harmonic generation by Peter Franken et al. at University of Michigan, both shortly after the construction of the first laser by Theodore Maiman. However, some nonlinear effects were discovered before the development of the laser. The theoretical basis for many nonlinear processes was first described in Bloembergen's monograph "Nonlinear Optics".
Nonlinear optical processes Nonlinear optics explains nonlinear response of properties such as frequency, polarization, phase or path of incident light. These nonlinear interactions give rise to a host of optical phenomena:
Frequency-mixing processes Second-harmonic generation (SHG), or frequency doubling, generation of light with a doubled frequency (half the wavelength), two photons are destroyed, creating a single photon at two times the frequency. Third-harmonic generation (THG), generation of light with a tripled frequency (one-third the wavelength), three photons are destroyed, creating a single photon at three times the frequency. High-harmonic generation (HHG), generation of light with frequencies much greater than the original (typically 100 to 1000 times greater). Sum-frequency generation (SFG), generation of light with a frequency that is the sum of two other frequencies (SHG is a special case of this). Difference-frequency generation (DFG), generation of light with a frequency that is the difference between two other frequencies. Optical parametric amplification (OPA), amplification of a signal input in the presence of a higher-frequency pump wave, at the same time generating an idler wave (can be considered as DFG). Optical parametric oscillation (OPO), generation of a signal and idler wave using a parametric amplifier in a resonator (with no signal input). Optical parametric generation (OPG), like parametric oscillation but without a resonator, using a very high gain instead. Half-harmonic generation, the special case of OPO or OPG when the signal and idler degenerate in one single frequency, Spontaneous parametric down-conversion (SPDC), the amplification of the vacuum fluctuations in the low-gain regime. Optical rectification (OR), generation of quasi-static electric fields. Nonlinear light-matter interaction with free electrons and plasmas.
Other nonlinear processes Optical Kerr effect, intensity-dependent refractive index (a effect). Self-focusing, an effect due to the optical Kerr effect (and possibly higher-order nonlinearities) caused by the spatial variation in the intensity creating a spatial variation in the refractive index. Kerr-lens modelocking (KLM), the use of self-focusing as a mechanism to mode-lock laser. Self-phase modulation (SPM), an effect due to the optical Kerr effect (and possibly higher-order nonlinearities) caused by the temporal variation in the intensity creating a temporal variation in the refractive index. Optical solitons, an equilibrium solution for either an optical pulse (temporal soliton) or spatial mode (spatial soliton) that does not change during propagation due to a balance between dispersion and the Kerr effect (e.g. self-phase modulation for temporal and self-focusing for spatial solitons). Self-diffraction, splitting of beams in a multi-wave mixing process with potential energy transfer. Cross-phase modulation (XPM), where one wavelength of light can affect the phase of another wavelength of light through the optical Kerr effect. Four-wave mixing (FWM), can also arise from other nonlinearities. Cross-polarized wave generation (XPW), a effect in which a wave with polarization vector perpendicular to the input one is generated. Modulational instability. Raman amplification Optical phase conjugation. Stimulated Brillouin scattering, interaction of photons with acoustic phonons Multi-photon absorption, simultaneous absorption of two or more photons, transferring the energy to a single electron. Multiple photoionisation, near-simultaneous removal of many bound electrons by one photon. Chaos in optical systems.
Related processes In these processes, the medium has a linear response to the light, but the properties of the medium are affected by other causes: Pockels effect, the refractive index is affected by a static electric field; used in electro-optic modulators. Acousto-optics, the refractive index is affected by acoustic waves (ultrasound); used in acousto-optic modulators. Raman scattering, interaction of photons with optical phonons.
Parametric processes Nonlinear effects fall into two qualitatively different categories, parametric and non-parametric effects. A parametric non-linearity is an interaction in which the quantum state of the nonlinear material is not changed by the interaction with the optical field. As a consequence of this, the process is "instantaneous". Energy and momentum are conserved in the optical field, making phase matching important and polarization-dependent.
Theory Parametric and "instantaneous" (i.e. material must be lossless and dispersionless through the Kramers–Kronig relations) nonlinear optical phenomena, in which the optical fields are not too large, can be described by a Taylor series expansion of the dielectric polarization density (electric dipole moment per unit volume) P(t) at time t in terms of the electric field E(t):
where the coefficients χ(n) are the n-th-order susceptibilities of the medium, and the presence of such a term is generally referred to as an n-th-order nonlinearity. Note that the polarization density P(t) and electrical field E(t) are considered as scalar quantities for simplicity. In a full treatment of nonlinear optics, both the polarization density and the field must be vectors, while χ(n) becomes an (n + 1)-th-rank tensor representing both the polarization-dependent nature of the parametric interaction and the symmetries (or lack of symmetries) of the nonlinear material.
Wave equation in a nonlinear material Central to the study of electromagnetic waves is the wave equation. Starting with Maxwell's equations in an isotropic medium, containing no free charge, it can be shown that
where PNL is the nonlinear part of the polarization density, and n is the refractive index, which comes from the linear term in P.
Note that one can normally use the vector identity
and Gauss's law (assuming no free charges, ),
to obtain the more familiar wave equation
For a nonlinear medium, Gauss's law does not imply that the identity
is true in general, even for an isotropic medium. However, even when this term is not identically 0, it is often negligibly small and thus in practice is usually ignored, giving us the standard nonlinear wave equation:
Nonlinearities as a wave-mixing process The nonlinear wave equation is an inhomogeneous differential equation. The general solution comes from the study of partial differential equations and can be obtained by the use of a Green's function. Physically, one gets the electromagnetic wave solutions to the homogeneous part of the wave equation:
and the inhomogeneous term
acts as a driver/source of the electromagnetic waves. One of the consequences of this is a nonlinear interaction that results in energy being mixed or coupled between different frequencies, which is often called a "wave mixing".
In general, an n-th order nonlinearity will lead to (n + 1)-wave mixing. As an example, if we consider only a second-order nonlinearity (three-wave mixing), then the polarization P takes the form
If we assume that E(t) is made up of two components at frequencies ω1 and ω2, we can write E(t) as
and using Euler's formula to convert to exponentials,
where "c.c." stands for complex conjugate. Plugging this into the expression for P gives
which has frequency components at 2ω1, 2ω2, ω1 + ω2, ω1 − ω2, and 0. These three-wave mixing processes correspond to the nonlinear effects known as second-harmonic generation, sum-frequency generation, difference-frequency generation and optical rectification respectively.
Note: Parametric generation and amplification is a variation of difference-frequency generation, where the lower frequency of one of the two generating fields is much weaker (parametric amplification) or completely absent (parametric generation). In the latter case, the fundamental quantum-mechanical uncertainty in the electric field initiates the process.
Phase matching
The above ignores the position dependence of the electrical fields. In a typical situation, the electrical fields are traveling waves described by
at position , with the wave vector , where is the velocity of light in vacuum, and is the index of refraction of the medium at angular frequency . Thus, the second-order polarization at angular frequency is
At each position within the nonlinear medium, the oscillating second-order polarization radiates at angular frequency and a corresponding wave vector . Constructive interference, and therefore a high-intensity field, will occur only if
This equation is known as the phase-matching condition.