Nonlinear optics

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.

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