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{\LARGE \bf  Modeling of noises -- simple laser 
}

H.-J.\ W\"unsche 15.6.2001

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\section{Intro}
I develop ideas how to check the noise performance of LDSL with a simple single mode laser.

\section{Agrawal model} %
%----------------------

According to the book of Agrawal \cite{agrawal} (chapter 6.5, noise characteristics, pp.\ 258 ff.), the Langevin equations with spontaneous emission noise can be written in the form
\begin{eqnarray}
\dot{N}&=&\frac{I}{e}-\frac{N}{\tau } -GP + F_n \\
\dot{P}&=&(G-\gamma )P+R_{sp}+F_p \\
\dot{\varphi }&=& -\omega +\frac{\alpha _H}{2}(G-\gamma )+F_\varphi .
\end{eqnarray}
$N,P = $ carrier and photon numbers, $\varphi = $ phase of the optical field, $F_k = $ stochastic Langevin forces, $\omega = $ light frequency of the stationary state without spontaneous emission  ($R_{sp}=0, F_k=0$), other quantities with their usual meaning. The ensemble averages of the stochastic forces obey the relations
\begin{eqnarray}
&&\braket{F_k(t)}=0 \hspace{15em} \braket{F_k(t)F_l(t')}=2D_{kl}\delta (t-t')  \\  
 &&
D_{PP} = R_{sp}P, ~~ D_{\varphi \varphi } =\frac{D_{PP}}{4P^2}, ~~ D_{PN}=D_{NP}=-D_{PP}, ~~ D_{NN}=\frac{N}{\tau }+D_{PP}. 
\end{eqnarray}
All other $D_{kl}$ vanish. 

The $D$-coefficients are to be calculated with stationary solutions. The spontaneous emission factor $\beta = R_{sp}\tau /N$ is a very small parameter. In lowest order the stationary solutions are
\begin{eqnarray}
\text{0th order: }&& N_0=N_{th}, P_0= \frac{I-I_{th}}{\gamma \tau }, \text{ with } G(N_{th})=\gamma , I_{th}=\frac{eN_{th}}{\tau }.\\
\text{1st order: }&&N=N_{th}+n, P=P_0+p, \text{ with } n=-\frac{\beta N_{th}}{G'\tau P_0+\beta }, p=-\frac{n}{\gamma \tau }(1+G'\tau P_0).
\end{eqnarray}

In addition to our present LDSL-treatment, noises appear not only in the field equations but also in the carrier equations. 
$D_{NN}$ has two contributions, the first one is due to shot noise\footnote{I have still to understand the derivation of it.}. The second one equals $D_{PP}$ because every generation of a photon by spontaneous emission removes one carrier. For the same reason, the nondiagonal diffusion coefficients are $-D_{PP}$.

\section{Continuous Reformulation}
%---------------------------------

I express the stochastic $F_k$ by noncorrelated stochastic $z_k$ with $\braket{z_k}=0, \braket{z_k(t)z_l(t')}=\delta _{il}\delta (t-t')$ and introduce $f=\sqrt{P}e^{i\varphi }$:
\begin{eqnarray} \label{contequ}
\dot{N}&=&\frac{I}{e}-\frac{N}{\tau } -GP + \sqrt{\frac{2N}{\tau }}z_s - \sqrt{2R_{sp}P} z_p  \\
\dot{f}&=&\frac{1}{2}(G-\gamma )(1+i\alpha _H)+\frac{R_{sp}}{2P}+ \sqrt{\frac{R_{sp}}{2P}} (z_p+iz_\varphi ) \\
\end{eqnarray}

\section{Descrete reformulation with lumped noises}
%------------------------------------------------------

I set $z_k=0$ within small time steps $dt$ and add the noises at the end in form of 
\begin{eqnarray}
N &\rightarrow& N + \sqrt{\frac{12dt\cdot 2N }{\tau }}\,\left[r_s-\frac{1}{2} \right] - \sqrt{12dt\cdot 2R_{sp}P}\,\left[r_p -\frac{1}{2} \right] \\
f &\rightarrow& f + \sqrt{\frac{12dt\cdot R_{sp}}{2P}} \left[r_p-\frac{1}{2}+i(r_\varphi-\frac{1}{2}) \right],
\end{eqnarray}
where the $r_k$ are the usual uncorrelated random numbers equally distributed in $[0,1]$.


\begin{thebibliography}{9}
\bibitem{agrawal}
G.P.Agrawal and N.K.Dutta, ``Semiconductor lasers'', second edition, 1993
\end{thebibliography}

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