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\begin{document} 

{\Large \bf Is optical feedback by a Fabry-Perot suited for delayed control?}

ede \today

\figc{0.4}{fp} \hfill \mbc{0.59}{
\underline{\em sketch of a Fabry-Perot}

$r_m^\pm $: ampl. reflectivity $\pm$-wave mirror $m$\\
$t_m^\pm$: ampl. transmission $\pm$-wave mirror $m$\\
$F(t)$: incident field amplitude\\
$B(t)$: reflected field amplitude\\
$E_\pm(t)$: forw./backw. internal field at left mirror.\\
}

\begin{eqnarray}
\text{\bf cavity propagation operator: }
E_-(t)=\hat{\bf R}E_+(t)=\ds{R} E_+(t-\tau ), 
\hspace{1cm} \ds{R} = \exp(-2ic_g\beta \tau )\cdot r_2^+
\end{eqnarray}
The reflected field amplitude is governed by the equations
\begin{eqnarray}
B(t)&=&r_1^+F(t)+ t_1^-E_-(t) \\
E_-(t)&=& \hat{\bf R} [t_1^+F(t)+r_1^-E_-(t)]=\ds{R} [t_1^+F(t-\tau )+r_1^-E_-(t-\tau )].
\end{eqnarray}
Introducing $E(t)=t_1^-E_-(t)$, this is equivalent to
\begin{eqnarray} \label{fp}
\boxed{
B(t)=r_1^+F(t)+ E(t), ~~~ 
E(t)= \ds{R}\cdot [t_1^+t_1^-F(t-\tau )+r_1^-E(t-\tau )].
}
\end{eqnarray}

{\bf $\tau $-periodic fields} 

In this case, the delayed fields in Equ.\,(\ref{fp}) can be replaced by the original ones. The second equ. can be resolved for $E$ and we get
\begin{eqnarray}
B(t)=\left[r_1^++\frac{t_1^+t_1^-\ds{R}}{1-r_1^-\ds{R}}\right]F(t).
\end{eqnarray}
Now it becomes important that the reflection and transmisssion coefficients of a lossless mirror have to fulfill the energy conservation equations
\begin{eqnarray}
|r_m^\pm|^2+|t_m^\pm|^2=1 ~~~~ \text{and} ~~~~ r_m^+{t_m^-}^*+t_m^+{r_m^-}^* = 0.
\end{eqnarray}
This gives $t_1^+t_1^-=-(1-|r_1^-|^2)r_1^+/{r_1^-}^*$. Inserting this and multiplying with ${r_1^-}^*$ yields
\begin{eqnarray}
\boxed{ ~
B(t)=r_1^+\left[1-\frac{(1-|r_1^-|^2)\ds{R}/{r_1^-}^*}{1-r_1^-\ds{R}}\right]F(t).
~}
\end{eqnarray}
This is always zero for $\ds{R}={r_1^-}^*$, which allow to use it for delayed feedback control. 

Remark 1: \\
$\tau $-periodicity of the field means that the pulses have this period but in addition that the optical frequency fulfills $\omega\tau =m \cdot 2\pi  $ with an integer $m$, i.e. operating in a minimum of the FP.

Remark 2: \\
This works for arbitrary magnitudes of reflectivity, i.e. we can use e.g. a platelet of glass or transparent semiconductor.

\end{document}
