# $\chi^{(3)}$ and Third Harmonic Generation

We’ve made it to third order nonlinear phenomena! Everyone, pat yourselves on the back because the second order phenomena in nonlinear optics can be somewhat difficult to swallow, so if you were successful in that then this should be a breeze. We will start off, as always, with a general description of third order polarization (which is of course represented by an equation), and then get into the third order counterparts of second order phenomena.

GENERAL DESCRIPTION

The general description for nonlinear polarization in the frequency domain is given as:

\begin{equation} P_i^3(\omega) = \epsilon_0\int_{-\infty}^{\infty}\frac{d\omega_1}{2\pi}\int_{-\infty}^{\infty}\frac{d\omega_2}{2\pi}\int_{-\infty}^{\infty}\frac{d\omega_3}{2\pi}\chi^3_{ijkl}(\omega_1,\omega_2,\omega_3)E_j(\omega_1)E_k(\omega_2)E_l(\omega_3)2\pi\delta(\omega-\omega_1-\omega_2-\omega_3)\end{equation}

where we have used Einstein summation notation to imply summation over the repeated indices, and the $\delta$-function places the necessary restrictions on which frequency components can combine to create the nonlinear polarization.

$\chi^{(3)}$ IN ISOTROPIC MEDIA

In centrosymmetric isotropic media, only 3 of the 21 nonzero elements of $\chi_{ijkl}^{(3)}$ are independent. The third-order nonlinear susceptibility tensor for this type of media can thus be written as:

\begin{equation} \chi^{(3)}_{ijkl} = \chi_{1122}^{(3)}\delta_{ij}\delta_{kl} + \chi^{(3)}_{1212}\delta_{ik}\delta_{jl} + \chi^{(3)}_{1221}\delta_{il}\delta_{jk} \end{equation}

where $\delta_{ij}$ is 1 if $i=j$ and is 0 if $i\neq j$.

THIRD HARMONIC GENERATION

Understanding of third harmonic generation (THG) follows more easily if you have a handle on second harmonic generation. Like second harmonic generation, the third harmonic generation requires phase-matching in order to have high efficiency of conversion into the third order harmonic wave. The polarization responsible for THG is given as:

\begin{equation} P_i^{3\omega} = \frac{\epsilon_0}{4}\chi_{ijkl}^{(3)}(-3\omega;\omega,\omega,\omega)E_j^{\omega}E_k^{\omega}E_l^{\omega} \end{equation}

where Einstein summation notation is again used to imply summation over the repeated indices. Then, for isotropic media, we can use the results we got in the last part for the form of the susceptibility tensor to write the polarization as:

\begin{align} P_i^{3\omega} & = \frac{\epsilon_0}{4}(\chi_{1122}^{(3)}\delta_{ij}\delta_{kl}+\chi_{1212}^{(3)}\delta_{ik}\delta_{jl}+\chi_{1221}^{(3)}\delta_{il}\delta_{jk})E_j^{\omega}E_k^{\omega}E_l^{\omega} \\ & = \frac{\epsilon_0}{4}(\chi_{1122}^{(3)}E_i^{\omega}E_k^{\omega}E_k^{\omega}+\chi_{1212}^{(3)}E_i^{\omega}E_l^{\omega}E_l^{\omega}+\chi_{1221}^{(3)}E_j^{\omega}E_k^{\omega}E_k^{\omega}) \\ & = \frac{\epsilon_0}{4}(\chi_{1122}^{(3)}+\chi_{1212}^{(3)}+\chi_{1221}^{(3)})(\bar{E}^\omega \cdot \bar{E}^\omega)E_i^\omega\end{align}

We can then use the relation that $\chi_{1111}^{(3)} = \chi^{(3)}_{1122}+\chi_{1212}^{(3)} + \chi_{1221}^{(3)}$ to write the third order polarization more succinctly as:

\begin{equation} P_i^{(3\omega)} = \frac{\epsilon_0}{4}\chi^{(3)}_{1111}(-3\omega;\omega,\omega,\omega)(\bar{E^\omega}\cdot \bar{E^\omega})E^\omega_i \end{equation}

Interestingly, the strength of the resulting nonlinear material polarization depends on the polarization of the input light field. For example, circularly polarized light cannot produce nonlinear polarization in an isotropic medium.

The conversion efficiency into the third harmonic using a single nonlinear material is actually quite low. As such, in practice, a sequence of two crystals are often used: one crystal is used to produce the second harmonic, and the output of the crystal combined with the fundamental is used with another crystal that can do sum frequency generation. Efficiencies up to 80% have been achieved with this method.