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\title{
%\vspace*{-6ex}
Derivation of Lang-Kobayashi Equations: \\The coupling rates
\author{Hans-J\"urgen W\"unsche
\\ \small
Institute of Physics, Humboldt University in Berlin,
Invalidenstrasse 110, 10115 Berlin
\date{February 13, 2005}
}
}

\begin{document}

\maketitle

\begin{abstract}
This is an addition to the pamphlet ''Derivation of Lang-Kobayashi Equations'' from July 7, 2000. I correct som sign errors and express the coupling rate in terms of the derivation of the inverse reflectivity of the solitary laser with respect to $\omega $.
\end{abstract}

I start with Eq. (12) of the old paper:
\begin{eqnarray} \nonumber
-i\dot{f}=- v_g \beta 'n f +i v_g \Phi_0^+(0)a(t). ~~~\mbc{0.5}{I corrected the sign of the $\beta '$-term and introduced $v_g$ to have correct dimensions.}
\hspace{3em}\text{(old 12)}
\end{eqnarray}
Now, I consider reflection of a harmonic wave. I set $\Psi^-(0,t)=a(t)=a_\text{in}e^{i\Omega t}$ for the input field and regard $n$ as a constant. Then, the reflected amplitude will oscillate in the same manner,  $\Psi^+(0,t)=f(t)\Phi^+_0(0)=a_\text{out}e^{i\Omega t}$. Inserting $\dot{f}=i\Omega f$ into (old 12), multiplying with $\Phi^+_0(0)$, dividing by $a_\text{out}e^{i\Omega t}$, yields
%
\begin{eqnarray} \label{refl}
\Omega +v_g\beta 'n = i v_g \Phi^+_0(0)^2 \frac{a_\text{in}}{a_\text{out}}
\hspace{0em} \text{ bzw. }
\frac{a_\text{out}}{a_\text{in}} = \boxed{~ r(\Omega,n) = \frac{i v_g \Phi^+_0(0)^2}{\Omega +v_g\beta 'n}}
~~ \mbc{0.25}{\footnotesize reflectivity of the DFB in this single-mode approximation.}
\end{eqnarray}
%
%
\begin{eqnarray} \label{invrefl}
\text{Consequence: } i v_g \Phi^+_0(0)^2 = \left(\frac{\partial q}{\partial \omega }\right)^{-1},
\end{eqnarray}
%
where $q=1/r$ is the inverse of the DFB reflectivity in the given approximation. 

Next I turn to delayed feedback by setting $a(t)=K\Psi^+(0,t-\tau )=K\Phi^+_0(0)f(t-\tau )$ in Eq.\ (old 12). Multiplying with $i$ and using the equations derived before, one gets immediately
%
\begin{eqnarray}
\dot{f}=- i v_g \beta 'n f + \eta f(t-\tau ) ~~~ \text{ with } ~~~ \boxed{~ \eta = \frac{iK}{\partial _\omega  q} ~ }
\end{eqnarray}
%
Up to the factor $i$, this is the LK coupling coefficient which we ady used in the coupled laser paper. Is the $i$ correct or not?

\end{document}
