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 10 V/m (and thus comparable to the atomic electric field of ~10 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.
It must be pointed out that 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.