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\begin{document}


\title{Zum SOA-Rauschen}

\author{H-J. W\"unsche}
\affiliation{FHI HHI}
\date{\today}

\maketitle


\section{Motivation}
%-------------------

Nach Jurais talk spielte gestern die Frage nach der Rauschzahl der langen SOAs eine grosse Rolle. Irgendwie wurde erwartet, dass die mit der L\"ange gewaltig in die H\"ohe geht. Ich denke, dass der gesaettigte hintere Teil des SOA kaum viel Rauschen beitr\"agt, weil dort eben die Nettoverst\"arkung null ist. Gero hat mir netterweise erkl\"art, was die Ingenieure unter Rauschzahl verstehen: Zunahme des Rausch-Signal-Verh\"altnisses in dB. Am einfachsten im Spektrum zu sehen als Verringerung des Abstands zwischen Signalpeak und Rauschuntergrund. 

\section{Simulationsrechnung}
%----------------------------

Das habe ich mir gleich mal mit LDSL entsprechende Spektren berechnet. 

Zun\"achst die axiale Tr\"agerdichten und Powerverteilungen:

\begin{center}
\mb{0.99}{\fig{axialN4mm.png.eps}}

\mb{0.99}{\fig{axialP4mm.png.eps}}
\end{center}

Es wird hier von rechts mit 0.75 mW eingestrahlt (Einkoppelverluste 0), die S\"attigungsintensit\"at liegt bei etwa 75 mW, die Verst\"arkung ist also etwa hundert. S\"attigung wird nach etwa 1 bis 2 mm erreicht, wobei der \"Ubergang sehr verschmiert ist. 

Und nun die optischen Spektren. Zun\"achst ihre Entwicklung innerhalb des ersten mm:

\begin{center}
\mb{0.24}{\fig{opt0005mikron}}
\mb{0.24}{\fig{opt0010mikron}}
\mb{0.24}{\fig{opt0100mikron}}
\mb{0.24}{\fig{opt01mm}}
\end{center}

Der Signalpeak w\"achst um etwa 20 dB und es wird weisses Rauschen hinzugef\"ugt (das Inputsignal ist ideal ohne noise). Das SNR bei den dargestellten L\"angen 5, 10, 100, 1000 $\mu$m betr\"agt etwa 75, 70, 65, 63 dB. Es f\"allt in den ersten hundert $\mu$m mit gro{\ss}er Verst\"arkung stark ab und beginnt zu s\"attigen. 

Nun die Entwicklung im S\"attigungsbereich des SOA:

\begin{center}
\mb{0.24}{\fig{opt01mm}}
\mb{0.24}{\fig{opt04mm}}
\mb{0.24}{\fig{opt10mm}}
\mb{0.24}{\fig{opt40mm}}
\end{center}

Zwei wichtige Effekte lassen sich hier feststellen:
\begin{itemize}
\item 
Wie vermutet stagniert das SNR, wenn man den Weissrauschpegel hinreichend entfernt von der Signalwellenl\"ange als Bezugspunkt nimmt. Also: {\bf der ges\"attigte Teil des SOA f\"ugt kein nennenswertes weisses Zusatzrauschen hinzu. }

\item 
{\bf Es entsteht farbiges Zusatzrauschen um die Signallinie herum.} Nimmt man dies als Bezugspunkt, so reduziert sich das SNR mit zunehmender L�nge: 
1 mm: 63 dB, 4 mm: 50 dB, 10 mm: 40 dB, 40 mm: 30-35 dB. Das sind aber nur pi-mal-Daumen-Werte. 
\end{itemize}


\section{Theoretische Betrachtungen}
%-----------------------------------

Um das zu untersetzen, habe ich mir folgendes \"uberlegt, wie man das im Rahmen unseres traveling-wave-models mit Rauschen theoretisch formulieren kann. Den Rest schreibe ich gleich in (Pidgin-) English, dann ist es leichter wiederzuverwerten und unsere nonGerman Kollegen k\"onnen es besser lesen. 

\subsection{Optical spectrum}
%-----------

Consider the optical amplitude ${\cal E}(z,t)$ within the SOA, $|{\cal E}|^2 = $ power. Its spectral density is
\begin{eqnarray}
S(z,\nu)=\int d\tau ~ C(z,\tau ) \exp(i\omega \tau ); ~~~~
C(z,\tau )=\braket{{\cal E}^*(z,t){\cal E}(z,t+\tau )}; ~~~~
\omega =2\pi \nu.
\end{eqnarray}
$\braket{\ldots}$: averaging (e.g. over $t$). Here and in the following, integrations are taken from $-\infty$ to $\infty$. 

\subsection{Spectrum at input}
%--------------------------

We suppose, the injected light is an ideal cw-signal with power $P_0$  and frequency $\nu_0$ superposed with white noise of constant spectral density $N_0$. Its optical spectrum is
\begin{eqnarray}
S_0(\nu)=P_0 \delta (\nu-\nu_0) + N_0.
\end{eqnarray}
Its signal-to-noise ratio is
\begin{eqnarray}
\text{SNR}(0)=\frac{P_0}{N_0 \Delta \nu}, ~~~~~~~~~~~~~~
(\Delta \nu: \text{ spectral resolution}).
\end{eqnarray}
Invers Fourier transform of $S_0$ yields the correlation function at input
\begin{eqnarray}
C(0,\tau ) = P_0 \exp(-i\omega _0\tau ) + N_0 \delta (\tau ).
\end{eqnarray}

\subsection{Propagation in the SOA}
%---------------------------------

\subsubsection{Traveling wave equation}
The traveling wave equation is
\begin{eqnarray}
-i(\partial _t+\partial _z) {\cal E}(z,t) = \beta (z,t) {\cal E}(z,t) + s(z,t), ~~~~ \text{B.C.:} ~ {\cal E}(0,t)=E_0(t).
\end{eqnarray}
$\beta $: local propagation constant of the internal waveguide, its imaginary part gives (half) the net gain. $s$: stochastic variable for the spontaneous emission into the guided wave. 

{\em Note:} the variable $z$ is 'position / group velocity'

\subsubsection{Spontaneous emission model: white noise}
%----------------------------------------------------

We assume
\begin{eqnarray} \label{spontcorr}
\braket{s(z,t)}\equiv 0 
~~~~~~ \text{ and } ~~~~~~
\braket{s^*(z,t)s(z',t')} = \rho(z,t) \delta (z-z')\delta (t-t').
\end{eqnarray}
$\rho$: noise power per spectral interval added per length due to spontaneous emission. It can be estimated by
\begin{eqnarray}
\rho \approx \beta _{sp} ~ \frac{h \nu_0}{e} ~\frac{I}{L} ~\frac{1}{\Delta \nu_{sp}}.
\end{eqnarray}
$I/L$: injection current per length, $\beta _{sp} \approx 10^{-3} \cdots 10^{-2}$: part of the spontaneous emission emitted into the guided wave, $\Delta \nu_{sp}$: effective width of the spontaneous emission spectrum. 

\subsection{Solution of the TWE}
%-------------------------------

First we transform to the relative time $T=t-z$ and take it as a new independent variable, $E(z,T)={\cal E}(z,t-z)$, yielding
\begin{eqnarray}
-iE'(z,T)=\beta (z,T+z) E(z,t) + s(z,T+z), ~~~~~~~~ \text{B.C.:} ~ E(0,T)=E_0(T).
\end{eqnarray}
The formal solution is 
\begin{eqnarray}
E(z,T)=i\int_0^zdz'e^{iB(z',T)}s(z',T+z')+e^{iB(z,T)}E_0(T); ~~~ B(z,T)=\int_0^zdz''\beta (z'',T+z'').
\end{eqnarray}

\subsection{Approximation: nonfluctuating propagation parameter}
%---------------------------------------------------------------

In general, the propagation parameter $\beta $ fluctuates also in time as a response to the noisy part of the optical wave. This effect has two different sources. 1st, a noisy modulation of the carrier density. 2nd, the direct dependence on optical power via the nonlinear gain saturation. I believe, these effects are responsible for the colored excess noise observed in the numerical calculation described above. Unfortunately, no idea at present, how to calculate analytically the impact of these effects on the optical spectra at output. 

Therefore, I neglect them henceforth and assume
\begin{eqnarray}
\beta =\beta (z)~ \Rightarrow ~ B=B(z), \text{ and } \rho =\rho (z) \text{ independent of time}.
\end{eqnarray}

\subsection{Spectrum at position $z>0$}
%--------------------------------------------------

With this formal solution and using the properties (\ref{spontcorr}), one gets for the correlation function
\begin{eqnarray} 
C(z,\tau )&=& \braket{{ E}^*(z,T){E}(z,T+\tau )} \\ &=& \delta (\tau ) \int_0^z dz' G(z')\rho (z') +G(z)C_0(\tau ),
~~~~~~~~ G(z)=\exp\left(\int_0^zdzg(z)\right).
\end{eqnarray}
$g=-2$Im$\beta $: gain coefficient. Fourier gives the spectrum
\begin{eqnarray}
\boxed{~ S(z,\nu)=\int_0^zdz'G(z')\rho (z')+G(z)N_0+G(z)P_0\delta (\nu-\nu_0) ~ }
\end{eqnarray}
Three additive contributions: ASE (independent on the input signal), amplified signal noise, and amplified signal. The signal-to-noise ratio  is
\begin{eqnarray}
\text{SNR}(z) =\frac{P_0}{N_0\Delta\nu+\int_0^z G(z,z')\Delta\nu\rho (z')}, ~~~~~~
G(z,z')=\exp\left(\int_{z'}^zdzg(z)\right).
\end{eqnarray}
The corresponding noise figure is 
\begin{eqnarray}
\text{NF}(z)= 10\lg\left(1+\int_0^zdz' G(z,z')\frac{\rho (z')}{N_0}\right).
\end{eqnarray}

These formulae hold for arbitrary stationary gain distributions $g(z)$ in the SOA. Let us specify these formula for a simple particular case. 

\subsection{Homogeneous case ($g(z), \rho (z)=$ const.)}
%---------------------------------

Looking to the output $z=L$, we find
\begin{eqnarray}
\text{NF}(L)= 10\lg\left(1+\frac{\rho L}{N_0}~\frac{e^{gL}-1}{gL}\right).
\end{eqnarray}

\mb{0.49}{ {\em Fig.1:}
Noise figure NF versus $gL$. The parameter is the ratio $\rho L/N_0$ between the noise added by spontaneous emission (without amplification) and the noise content of the input signal. 
} \hfill \mb{0.5}{\fig{nf}}

\section{Application to the UL-SOA}
%---------------------------------

Only a first portion of the UL-SOA is amplifying (about 1 mm), whereas the rest is saturated. Let us treat this situation as two different SOAs in sequence. 

The first part provides amplification, say by $e^{gL}=100 = 20$ dB, i.e. $gL\approx 5$. Let the total spontaneous emission added by this section before amplification be 10 $\mu$W. Multiplying with $e^{gL}-1/gL=99/5\approx 20$ yields 200 $\mu$W ASE at the end of the amplifying part. 

The saturated part of the SOA does not further amplify these 200 $\mu$W. It only adds further 10 $\mu$W spontaneous emission noise per mm. This remains negligible comared to the 200 $\mu$W noise from the amplifiying part as long as the saturated part keeps much smaller than 20 mm. 


\end{document}


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          \int_0^z dz'\,dz''\, \braket{e^{i[B(z'',T+\tau )-B^*(z',T)]}s^*(z',T+z')s(z'',T+z'')} + \\ \nonumber
          & & \hspace{3cm } + \braket{e^{i[B(z,T+\tau )-B^*(z,T)]}E_0^*(T)E_0(T+\tau )} .
