Editing Nonlinear optics (section)

===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 [[Perturbation theory|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''):
:<math>\mathbf{P}(t) = \varepsilon_0 \left( \chi^{(1)} \mathbf{E}(t) + \chi^{(2)} \mathbf{E}^2(t) + \chi^{(3)} \mathbf{E}^3(t) + \ldots \right),</math>
where the coefficients χ<sup>(''n'')</sup> are the ''n''-th-order [[Electric susceptibility|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 for simplicity. In general, χ<sup>(''n'')</sup> is an (''n''&nbsp;+&nbsp;1)-th-rank [[tensor]] representing both the [[Polarization (waves)|polarization]]-dependent nature of the parametric interaction and the [[Crystal symmetry|symmetries]] (or lack) of the nonlinear material.

====Wave equation in a nonlinear material====
Central to the study of electromagnetic waves is the [[Electromagnetic wave equation|wave equation]].  Starting with [[Maxwell's equations]] in an isotropic space, containing no free charge, it can be shown that
:<math>
\nabla \times \nabla \times \mathbf{E} + \frac{n^2}{c^2}\frac{\partial^2}{\partial t^2}\mathbf{E}
= -\frac{1}{\varepsilon_0 c^2}\frac{\partial^2}{\partial t^2}\mathbf{P}^\text{NL},
</math>
where '''P'''<sup>NL</sup> 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
:<math>\nabla \times \left( \nabla \times \mathbf{V} \right) = \nabla \left( \nabla \cdot \mathbf{V} \right) - \nabla^2 \mathbf{V}</math>
and [[Gauss's law]] (assuming no free charges, <math>\rho_\text{free} = 0</math>),
:<math>\nabla\cdot\mathbf{D} = 0,</math>
to obtain the more familiar [[Electromagnetic wave equation|wave equation]]
:<math>
\nabla^2 \mathbf{E} - \frac{n^2}{c^2}\frac{\partial^2}{\partial t^2}\mathbf{E} = \mathbf{0}.
</math>
For a nonlinear medium, [[Gauss's law]] does not imply that the identity
:<math>\nabla\cdot\mathbf{E} = 0</math>
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:
:<math>
\nabla^2 \mathbf{E} - \frac{n^2}{c^2}\frac{\partial^2}{\partial t^2}\mathbf{E}
= \frac{1}{\varepsilon_0 c^2}\frac{\partial^2}{\partial t^2}\mathbf{P}^\text{NL}.
</math>

====Nonlinearities as a wave-mixing process====
The nonlinear wave equation is an inhomogeneous differential equation.  The general solution comes from the study of [[ordinary differential equations]] and can be obtained by the use of a [[Green's function]].  Physically one gets the normal [[electromagnetic wave]] solutions to the homogeneous part of the wave equation:
:<math>\nabla^2 \mathbf{E} - \frac{n^2}{c^2}\frac{\partial^2}{\partial t^2}\mathbf{E} = \mathbf{0},</math>
and the inhomogeneous term
:<math>\frac{1}{\varepsilon_0 c^2}\frac{\partial^2}{\partial t^2}\mathbf{P}^\text{NL}</math>
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''&nbsp;+&nbsp;1)-wave mixing. As an example, if we consider only a second-order nonlinearity (three-wave mixing), then the polarization '''P''' takes the form
:<math>\mathbf{P}^\text{NL} = \varepsilon_0 \chi^{(2)} \mathbf{E}^2(t).</math>
If we assume that ''E''(''t'') is made up of two components at frequencies ''ω''<sub>1</sub> and  ''ω''<sub>2</sub>, we can write ''E''(''t'') as
:<math>\mathbf{E}(t) = E_1\cos(\omega_1t) + E_2\cos(\omega_2t),</math>
and using [[Euler's formula]] to convert to exponentials,
:<math>\mathbf{E}(t) = \frac{1}{2}E_1 e^{-i\omega_1 t} + \frac{1}{2}E_2 e^{-i\omega_2 t} + \text{c.c.},</math>
where "c.c." stands for [[complex conjugate]].  Plugging this into the expression for '''P''' gives
:<math>\begin{align}
     \mathbf{P}^\text{NL}
  &= \varepsilon_0 \chi^{(2)} \mathbf{E}^2(t) \\[3pt]
  &= \frac{\varepsilon_0}{4} \chi^{(2)} \left[{E_1}^2 e^{-i2\omega_1 t} + {E_2}^2 e^{-i2\omega_2 t} + 2E_1 E_2 e^{-i(\omega_1 + \omega_2)t} + 2E_1 {E_2}^* e^{-i(\omega_1 - \omega_2)t} + \left(\left|E_1\right|^2 + \left|E_2\right|^2\right)e^{0} + \text{c.c.}\right],
\end{align}</math>
which has frequency components at 2''ω''<sub>1</sub>, 2''ω''<sub>2</sub>, ''ω''<sub>1</sub>&nbsp;+&nbsp;''ω''<sub>2</sub>, ''ω''<sub>1</sub>&nbsp;−&nbsp;''ω''<sub>2</sub>, 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.

<!-- The following note is taken (with permission) from Han-Kwang Nienhuys's PhD thesis "Femtosecond mid-infrared spectroscopy of water" (2002). -->
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 mechanics|quantum-mechanical]] uncertainty in the electric field initiates the process.