Editing Nonlinear optics (section)

====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.