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\begin{document}

{\large \bf TWE for SOA with 2 input waves}
%
\begin{eqnarray*}
\bullet \text{ optical field}  ~ E(z,t)= f(z,t) e^{i(\omega_0 t-k_0z)} \text{ only forward.}
\end{eqnarray*}
$\bullet$ dimensionless form of TWE:
\begin{eqnarray*}
2(\partial _t+\partial _z)f&=&\left[\frac{g'n(1-i\alpha )}{1+\varepsilon f^*f}-1\right] f \\
\partial_t{n}&=&j-n \left[\frac{1}{\tau}+\frac{g'f^*f}{1+\varepsilon f^*f}\right]
\\
\text{RB: } f(t,0)&=&e^{i\omega t}\left(f_{00}(t)+f_{10}(t)e^{i\delta t}\right)
\end{eqnarray*}
$\bullet$ separation into partial waves:
\begin{eqnarray*}
&& \text{\underline{Ansatz:} ~ } f(t,z)=e^{i\omega (t-z)}\sum_m f_m(t,z) e^{im\delta (t-z)}.\\
&& \hspace{6em} \text{with } f_m(t,z) \text{ slow compared to } 1/\delta \\
&&\text{yields for photon density } s=f^*f=\bar{s}+s \\
&& \text{stationary contribution} ~ \bar{s}=\sum_mf_m^*f_m \\
&& \text{beating contribution}    ~ s=\sum_{m,n}^{(m\ne n)} f_m^*f_n e^{i(n-m)(t-z)}. ~~~~~~~~~~~~~
\end{eqnarray*}
$\bullet$ restriction to high-$\delta $-limit: \\ $~~$ negligible beating contributions to $n$
\newpage
\begin{eqnarray*}
&& \partial_t{n}=j-n \left[\frac{1}{\tau}+\frac{g'\bar{s}}{1+\varepsilon\bar{s}}\right].
\end{eqnarray*}
$\bullet$ $f$-equ: linearisation with respect to beating term $s$ 
\begin{eqnarray*} 
&& 2(\partial _t+\partial _z)f_k=-f_k+(1-i\alpha )\frac{g'n\bar{s}}{1+\varepsilon\bar{s}}\times \\ 
&& \hspace{5em} \times \left(f_k-\frac{\varepsilon }{1+\varepsilon\bar{s}}\sum_{m,l}^{m\ne k} f_mf_l^*f_{l+k-m}\right).
\end{eqnarray*}
$\bullet$ the nonlinear term mixes the 3 partial waves $m,l,l+k-m$ with partial wave $k$, this is 4-wave-mixing.

$\bullet$ \underline{assumption:} the two input waves $k=0,1$ remain dominant, hence
\begin{eqnarray*}
2(\partial _t+\partial _z)f_k=\left[g'n(1-i\alpha )\frac{1+\varepsilon {s}_k}{(1+\varepsilon \bar{s})^2}-1\right]f_k
 \hspace{1em} (k=0,1)
\end{eqnarray*}
or for the intensities $s_k=f_k^*f_k$

\kasten{ ~ \\[-5.5ex]
\begin{eqnarray*} 
(\partial _t+\partial _z)s_k
%=\left[g'n\frac{1+\varepsilon s_k}{(1+\varepsilon s)^2}-1\right]s_k
=\left[\frac{g'n}{1+\varepsilon s}-1\right]s_k-\frac{\varepsilon s_0s_1}{(1+\varepsilon s)^2}
\hspace{1em} (k=0,1)
\end{eqnarray*}}

$\bullet$ 4wm reduces each intensity by the same amount \\
$~~~ \Rightarrow$ the lower one is the loser.

\newpage

$\bullet $ transformation to a moving frame with 
$\tau =t-z$, \hfill $p_k(\tau ,z)=s_k(\tau +z,z)$,  $\rho (\tau ,z)=n(\tau +z,z)$:
\begin{eqnarray*}
&&\partial _zp_k(\tau ,z)=\left[g'\rho(\tau ,z)\frac{1+\varepsilon p_k(\tau ,z)}{(1+\varepsilon p(\tau ,z))^2}-1\right]p_k(\tau ,z)\\
&& \hspace{6em} \text{Randbedingung: } p_k(\tau ,0)=|f_{k0}(\tau )|^2
\\
&&\partial_\tau {\rho(\tau ,z) }=j-\rho(\tau ,z)  \left[\frac{1}{\tau _c}+\frac{g'p(\tau ,z)}{1+\varepsilon p(\tau ,z)}\right]\\
&& \hspace{4em} \text{''Anfangs''bedingung: } \rho (0 ,z)=n(t,z)\Big|_{t=z}
\end{eqnarray*}
-- two ordinary differential equations only,
\\
-- photons: $\tau $ a parameter, ~~ carriers: $z$ a parameter
\\
-- satanic detail: $n(t=z,z)$ to be known.

$\bullet$ stationary: $\rho =1/g',$ $p_0+p_1=j-\rho /\tau _c,$ \\
\hspace*{5em}   and  $p_0\cdot p_1=0$, i.e. bistable.

$\bullet$ quasistationary approximation: $|\partial _\tau \rho | \ll j $ \\
$~~~$ numerical solution.

\newpage

{\bf wave mixing: modeling in time domain}

I propose 

$\bullet$ to withdraw the present form of gain saturation and

$\bullet$ instead to add to $\beta $ the nonlinear contribution
\begin{eqnarray*} \label{equ:betnl}
\delta \beta ^{nl}= -\Gamma\sum_{m=\text{ch,shb}} \frac{g'_m}{2}\,(N-N_m)\left(i+\alpha _m\right)\varepsilon _mS_m.
\end{eqnarray*}
with the averaged optical intensity 
\begin{eqnarray*} \label{equ:sm}
S_m=\int_{-\infty}^t\frac{dt'}{\tau _m} e^{-(t-t')/\tau _m}S(t').
\end{eqnarray*}

the cpp-contribution comes from the linear $\beta (N)$  

{\bf problem for large detuning: gain dispersion}

If a broad spectrum contributes to $S(t)$, the coefficients in front of $S_m$ depend on the individual frequencies of the partial waves.

\end{document}