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**Definition/Summary**Light propagating through a vacuum will obey the principle of superposition, however this is not generally true for light propagating through gaseous or condensed media. As light propagates through transparent media, it induces a dipole moment on any atoms present in the propagating electromagnetic field. At sufficiently high field strengths, the induced dipole moment is no longer proportional to the applied field - this is the origin of the term "nonlinear" in the context of nonlinear optics.

Nonlinear optical processes are used in a huge variety of applications including, but not limited to, frequency generation, optical switching, sensing, microscopy and quantum optics.

**Equations**[tex]\mathbf{P} = \epsilon_0(\chi^{(1)}\mathbf{E}+\chi^{(2)}\mathbf{E.E}+\chi^{(3)}\mathbf{E.E.E}+...)[/tex]

[itex]\mathbf{P}[/itex] - Polarisation (induced dipole per unit volume).

[itex]\mathbf{E}[/itex] - Applied electric field.

[itex]\epsilon_0[/itex] - The permittivity of free space.

[itex]\chi^{(1)}[/itex] - Linear susceptibility.

[itex]\chi^{(n)}[/itex] - nth order nonlinear susceptibility.

[tex]\nabla^2\mathbf{E}=\mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2} + \mu_0\frac{\partial^2\mathbf{P_{NL}}}{\partial t^2}[/tex]

[itex]\mu_0[/itex] - The permeability of free space.

[itex]P_{NL}[/itex] - The nonlinear polarisation.

**Extended explanation**The dipole per unit volume (confusingly called the

*polarisation*), can be generally expressed as follows;

[tex]\mathbf{P} = \epsilon_0 (V(\mathbf{r})\mathbf{E})[/tex]

where V(

**r**) is the restoring force acting on the polarised medium as a function of electron displacement from the nucleus. If V(

**r**) is perfectly linear, then;

[tex]\mathbf{P}= \epsilon_0 ((1+\epsilon) \mathbf{E})[/tex]

where [itex]\epsilon[/itex] is the permittivity of the medium, and is related to the refractive index at optical frequencies;

[tex]\epsilon = n^2[/tex]

At low field strengths, a linear approximation of V(

**r**) is suitable and we only need characterise an optical medium by its refractive index. V(

**r**) however is not linear in the general case, however the expression V(

**r**)

**E**can be expanded as a Taylor series;

[tex]\mathbf{P}= \epsilon_0(\chi^{(1)}\mathbf{E}+\chi^{(2)}\mathbf{E.E}+\chi^{(3)}\mathbf{E.E.E}+...)[/tex]

where the symbol [itex]\chi^{(1)}[/itex] denotes the linear susceptibility and [itex]\chi^{(n)}[/itex] denotes the nth order nonlinear susceptibility where [itex]\chi^{(n+1)}<<\chi^{(n)}<<\chi^{(n-1)}[/itex]. If V(

**r**) is a symmetric function, then the even ordered nonlinear susceptibilities are zero. Note that the susceptibilities are tensors in the general case.

The wave-equation in the presence of a nonlinear polarisation is given by;

[tex]\nabla^2\mathbf{E}=\mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2} + \mu_0\frac{\partial^2\mathbf{P_{NL}}}{\partial t^2}[/tex]

where [itex]P_{NL}[/itex] is the nonlinear polarisation, that is the polarisation without the linear term.

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