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\title {On Wavelength Asymmetry of Extinction-Ratio Improvement \\ by Two-Wave Competition in ULSOAs}
\author{H.-J. W\"unsche \\
Humboldt Universit\"at zu Berlin, Institut f\"ur Physik\\
Newtonstr.15 12489 Berlin, Germany}
\date{25.08.2005}
%\begin{abstract} \end{abstract}
\maketitle
Experiments have shown: the improvement of the extinction ratio (ER) by two-wave competition (TWC) exhibits a strong wavelength asymmetry. It ''is a prerequisite for achieving an ER improvement, to set the CW beam to the long wavelength side of the signal'' \cite{bramann05}. This asymmetry has qualitatively been atributed to the asymmetry of four-wave mixing (FWM). Here I shall check this hypothesis quantitatively on base of the theory presented in \cite{bramann05}.
I consider copropagation of a signal and a CW wave with intensities $S_s$ and $S_\text{cw}$, respectively, in the saturated SOA. The asymmetry of FWM is generally a result of the interferences between fast intraband contributions, represented by the gain saturation coefficient $\varepsilon $, and contributions from carrier population pulsations (CPP). In the appendix of pamphlet \cite{twcnotes050823}, I have already shown how the CPP contributions enter the evolution equation for the intensity ratio. Here, I rewrite these equations with an adapted notation.
%
\begin{eqnarray} \label{propagation1}
\frac{\partial }{\partial z} \ln(\frac{S_s}{S_\text{cw}}) = q_s S_s-q_\text{cw}S_\text{cw},
\end{eqnarray}
%
%
\begin{eqnarray} \label{q1}
q_{s,\text{cw}} = g \left(\varepsilon \mp \frac{g'|\alpha _H| \lambda _0^2}{2\pi n_g (\lambda _\text{cw}-\lambda _s) } \right),
\end{eqnarray}
%
where the upper sign belongs to $q_s$.
The crucial points for the ER are the ''zeros'' of the signal bit sequence. Thus, I can confine to a case with $S_s \ll S_\text{cw}$. Thus, we can neglect the signal intensity on the r.h.s. of Eq.\,(\ref{propagation1}). Furthermore, saturation conditions require $S_s+S_\text{cw}\approx S_\text{cw}=S_\text{sat}$. Thus, we can approximate the cw intensity by the saturation intensity, yielding
%
\begin{eqnarray} \label{propagation2}
\frac{\partial }{\partial z} \ln(\frac{S_s}{S_\text{sat}}) = -q_\text{cw}S_\text{sat},
\end{eqnarray}
%
Obviously, the quantity $q_\text{cw}$ governs the decay of the signal intensity. The right hand side is constant, hence the solution is
%
\begin{eqnarray}
S_s(z) \sim \exp (- z/L_s)
\end{eqnarray}
%
with the decay length
%
\begin{eqnarray}
L_s= \frac{L_\text{TWC}}{1+\Delta _ \text{asym}/(\lambda _\text{cw}-\lambda _s)}.
\end{eqnarray}
%
Here $L_\text{TWC}$ is the large-detuning limit of the characteristic decay length as given by formula (13) of our paper \cite{bramann05}, and
%
\begin{eqnarray}
\Delta _\text{asym} = \frac{g'|\alpha _H| \lambda _0^2}{2\pi n_g\varepsilon }
\end{eqnarray}
%
is a critical detuning below which the wavelength asymmetry becomes essential. With the parameters of \cite{bramann05}, it is $\Delta_\text{asym}=3.86$ nm.
The effectiveness of ER improvement increases with the inverse of the signal decay length $L_s$. Thus, without explicitely calculating the ER, the plot of $L_\text{TWC}/L_s$ in Fig.1 allowes a theoretical estimate how the ER improvement should depend on the detuning. Indeed, it is quite similar to Fig.8 (a) of \cite{bramann05}. Of course, the agreement is not complete because the approximations hold only in the very last part of the SOA and the parameters are only estimates.
\begin{center}
\includegraphics*[width=0.9 \columnwidth]{lsversusdeltalambda}
\end{center}
Fig.1: {\em Plot of the inverse decay length versus wavelength detuning.}
{\bf Conclusion}
The basic features of the measured wavelength asymmetry of the ER improvement by TWC are well described by the theory developed in \cite{bramann05}.
\begin{thebibliography}{10}
\bibitem{bramann05}
Gero Bramann, Hans-J\"{u}rgen W\"{u}nsche, Ulrike Busolt, Christian Schmidt, Michael Schlak, Bernd Sartorius, and Hans-Peter Nolting,
''Two-Wave Competition in Ultra Long Semiconductor Optical Amplifiers'',
to appear in IEEE Journal Quantum Electronics, October 2005
\bibitem{twcnotes050823}
H.-J. W\"unsche,
''Notes on Two-Wave Competition by Four-Wave Mixing'',
http://photonik.physik.hu-berlin.de/ede/pamphlets/twcnote.pdf
\end{thebibliography}
\end{document}