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\begin{document}
\large \sf 

\title{Feedback from an asymmetric FP with lossless mirrors}
\author{Ede W\"unsche, PHO at Physics of HU Berlin}
\date{24.05.2005}

\maketitle

\section{Summary}
%================

The reflectivity of a Fabry-P\'{e}rot resonator (FP) seems to be simple textbook matter. However, it is not easy to find explicite formulae. So I have reconsidered the following problem in the framework of traveling-wave equations (TWE) = optical field equation of the Maxwell-Bloch equations. 

Calculate the field ${\cal E}_b(t)$ reflected from a passive FP if it is illuminated with an  optical pulse ${\cal E}_\text{in}(t)$. Assume normal incidence (or a waveguide configuration), allow for mirrors having different reflectivities, and allow for absorption (scattering) losses in the FP. 

Not knowing the general properties of a lossy mirror, I consider lossless mirrors only. Then, the solution is (derivation see section \ref{sec.derivation})
%
\begin{eqnarray}
\boxed{ ~{\cal E}_b(t) = r_{1_-} \sum_{n=0}^\infty {\cal R}^n \cdot \left[{\cal E}_\text{in}(t_{n})-\frac{\cal R}{R_{1_-}}\,{\cal E}_\text{in}(t_{n+1})\right] ~ }
\hspace{1em}
\begin{matrix}
{\cal R} &=& \sqrt{R_{1_+}R_{2_-}} e^{-\alpha L} e^{i\phi} ~~~~~~~ \text{\footnotesize round-trip factor}\\
\phi &=& \arg(r_{1}r_{2}) + \omega _0 \tau - 2L \text{Re} \beta (\omega _0) \\
\alpha &=& - 2 \text{Im} \beta (\omega _0) \hspace{3em} \text{\footnotesize internal opt. losses}\\
t_n &=& t-n\tau , ~\text{\footnotesize with round-trip time }  \tau = 2L/v_g.
\end{matrix}
\end{eqnarray}
%
The parameters which enter are: $r_1,r_2$ amplitude reflectivities of the mirrors from inside the FP, $r_{1_-}$ amplitude reflectivity of the illuminated mirror from outside and $R_{\dots}=|r_{\dots}|^2$, $\omega _0$ reference frequency, $\beta (\omega _0)$ complex propagation constant at $\omega _0$ within the resonantor, $L$ resonator length, $v_g$ group velocity in the resonator. Notice that incident and reflected field are taken just at the outer side of mirror 1. Inclusion of propagation to and from this mirror (latency time) is a trivial extension. 

{\em Remarks:}
\begin{itemize}
\item
$\cal R$ does not depend on the choice of $\omega _0$ within the linear dispersion range of $\beta (\omega )$. \\ Details see end of section \ref{sec.derivation}.
\item
The round trip time $\tau $ is determined by the group velocity but not by the phase velocity. 
\item
The feedback of $T$-periodic intensities disappears in case of the general resonance condition 
%
\begin{eqnarray}
{\cal E}_\text{in}(t-T )=e^{i\psi}{\cal E}_\text{in}(t)
~~~ \text{ and } ~~~
\tau = m T, ~~~
e^{im\psi}\frac{\cal R}{R_{1_-}}=1.
\end{eqnarray}
%
$\psi$ is an arbitrary phase shift and $m$ integer. 
\item
It could be interesting to consider small deviations from resonance, $\tau =mT+\delta \tau $. In this case, the feedback becomes proportional to $\delta \tau \cdot $ actual intensity change, which can be used as control force.
\end{itemize}



\newpage
\section{Derivation} \label{sec.derivation}
%===================

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\end{document}