Editing Nonlinear optics (section)

====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>