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\subsection{Relationship between $\chi^{(3)}$ and to non-linear pulse propagation constants}
At low field intensities, there are familiar relations between the refractive index, $n_0$, the absorptivity coefficient, $\alpha$, the linear susceptibility, $\chi$, and the complex wave-vector, $k$:
\begin{equation}
\begin{split}
k &= \frac{\omega n_0}{c} + i\frac{\alpha}{2} \\
&= \frac{\omega}{c} \sqrt{1 + \kappa \chi},
\end{split}
\end{equation}
where $\kappa= 1$ in the SI unit system and $\kappa=4\pi$ in the cgs unit system.
$E_i$ is the electric field amplitude of pulse $m$, so $I_i = \frac{nc}{8\pi} |E_i|^2$.
For higher fluences, there are three important phenomenologies, each based on a different optical constant:
\begin{gather}
k = \frac{\omega n^*}{c} + i \frac{\alpha^*}{2} \\
\alpha^* = \alpha + \beta I \\
n^* = n_0 + n_2 I \\
\chi^* = \left[ \chi^{(1)} + D \chi^{(3)} \left| E_2 \right|^2 \right]
\end{gather}
where $D$ is the permutation degeneracy factor:\cite{Maker1965} if a weak probe pulse is distorted by perturbations from a separate pump (as in transient absorption measurements), $D=6$, while if a lone probe pulse is intense enough to cause its own distortions (i.e. $E_2 = E_1$, as in $z$-scan measurements), $D=3$.
%where $M_2 = \frac{1 - e^{-\alpha_0(\omega_2) \ell}}{\alpha_0(\omega_2) \ell}$ accounts for absorptive effects of the pump beam $E_2$.
For small perturbations ($\left|\kappa D \chi^{(3)}|E_2|^2 \right| \ll |1+\kappa \chi^{(1)}|$), we can use a first-order Taylor expansion of $\mathbf{n}^*$ about $1 + \kappa \chi^{(1)}$ to write
\begin{equation}
\begin{split}
\mathbf{n}^* &= \sqrt{1 + \kappa \left( \chi^{(1)} + D \chi^{(3)} |E_2|^2 \right)} \\
&\approx \mathbf{n} + \frac{1}{2 \mathbf{n}} \kappa D \chi^{(3)}|E_2|^2
\end{split}
\end{equation}
and in the common case (as is herein) where $\frac{\alpha c}{2\omega} \ll n_0$, $\mathbf{n}^{-1} \approx n_0^{-1}$.
Ignoring terms independent of $I$, we arrive at
\begin{equation}
\begin{split}
\left(n_2 + i\frac{c \beta}{2\omega}\right)I &= \frac{\kappa D \chi^{(3)}}{2 n_0} |E_2|^2 \\
&= \frac{4 \pi \kappa D \chi^{(3)}}{n_0^2 c} I
\end{split}
\end{equation}
Eqns. \ref{eq:beta_to_chi} and \ref{eq:n2_to_chi} then follow with Eqn. \ref{eq:hyperpolarizability}.
%Relations between non-linear constants and $\chi^{(3)}$ are summarized by:
%\begin{gather}
% \beta = \frac{8 \pi \kappa D \omega}{n_0^2 c^2} \text{Im}\left[ \chi^{(3)} \right] \\
% n_2 = \frac{4 \pi \kappa D }{n_0^2 c} \text{Re}\left[ \chi^{(3)} \right]
%\end{gather}
%From this it is easily seen that $\beta_\text{TA} = 2 \beta_\text{z-scan}$.
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