# What are nonlinear optics

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

$$\mathbf{P} = \epsilon_0(\chi^{(1)}\mathbf{E}+\chi^{(2)}\mathbf{E.E}+\chi^{(3)}\mathbf{E.E.E}+...)$$

$\mathbf{P}$ - Polarisation (induced dipole per unit volume).
$\mathbf{E}$ - Applied electric field.
$\epsilon_0$ - The permittivity of free space.
$\chi^{(1)}$ - Linear susceptibility.
$\chi^{(n)}$ - nth order nonlinear susceptibility.

$$\nabla^2\mathbf{E}=\mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2} + \mu_0\frac{\partial^2\mathbf{P_{NL}}}{\partial t^2}$$

$\mu_0$ - The permeability of free space.
$P_{NL}$ - The nonlinear polarisation.

Extended explanation

The dipole per unit volume (confusingly called the polarisation), can be generally expressed as follows;

$$\mathbf{P} = \epsilon_0 (V(\mathbf{r})\mathbf{E})$$

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;

$$\mathbf{P}= \epsilon_0 ((1+\epsilon) \mathbf{E})$$

where $\epsilon$ is the permittivity of the medium, and is related to the refractive index at optical frequencies;

$$\epsilon = n^2$$

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;

$$\mathbf{P}= \epsilon_0(\chi^{(1)}\mathbf{E}+\chi^{(2)}\mathbf{E.E}+\chi^{(3)}\mathbf{E.E.E}+...)$$

where the symbol $\chi^{(1)}$ denotes the linear susceptibility and $\chi^{(n)}$ denotes the nth order nonlinear susceptibility where $\chi^{(n+1)}<<\chi^{(n)}<<\chi^{(n-1)}$. 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;

$$\nabla^2\mathbf{E}=\mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2} + \mu_0\frac{\partial^2\mathbf{P_{NL}}}{\partial t^2}$$

where $P_{NL}$ is the nonlinear polarisation, that is the polarisation without the linear term.

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