Nonlinear optics


Nonlinear optics 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 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 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 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

In these processes, the medium has a linear response to the light, but the properties of the medium are affected by other causes:
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" 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 P at time t in terms of the electric field E:
where the coefficients χ 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 and electrical field E 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 becomes an -th-rank tensor representing both the polarization-dependent nature of the parametric interaction and the 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,
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 -wave mixing. As an example, if we consider only a second-order nonlinearity, then the polarization P takes the form
If we assume that E is made up of two components at frequencies ω1 and ω2, we can write E 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 or completely absent. In the latter case, the fundamental quantum-mechanical uncertainty in the electric field initiates the process.