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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%TITLE
\title{KAGRA Frequency Stabilization Servo\\ Modeling Report}
\author{Yuta Michimura}
\date{\today}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%MAIN
\begin{document}
\maketitle

\section{Introduction}
This report is to summarize the results of KAGRA frequency stabilization servo (FSS) modeling. The modeling was done by using MATLAB Simulink based Noise Budget made by Chris Wipf~\cite{NBref}.

The main script and the model for KAGRA FSS modeling are as follows:
\begin{itemize}
  \item \url{https://granite.phys.s.u-tokyo.ac.jp/svn/LCGT/trunk/kagranoisebudget/FSS/run_FSS_NB.m}
  \item \url{https://granite.phys.s.u-tokyo.ac.jp/svn/LCGT/trunk/kagranoisebudget/FSS/KAGRA_FSS.mdl}
\end{itemize}

You will also need {\tt findNbSVNroot.m}, {\tt myzpk.m}, {\tt plotdobe.m}, and {\tt plotspectrum.m} in the same directroy to run the script.

\section{Model}
\subsection{Simulink model}
The KAGRA FSS Simulink model is shown in \zu{KAGRA_FSS}. We had to use some tricks to simulate out-of-loop stability and feedback signal with Simulink Noise Budget blocks, {\tt NbNoiseCal} and {\tt NbNoiseSink} (see yellow and cyan areas).

\begin{figure}[t]
  \begin{center}
    \includegraphics[height=140mm,angle=90]{KAGRA_FSS.pdf}
  \end{center}
  \caption{KAGRA FSS Simulink model. When the CARM loop is enabled, the MCe loop in the IMC loop is disabled.} \label{KAGRA_FSS}
\end{figure}

\clearpage
\subsection{KAGRA cavity parameters}
The round-trip lengths $L$ and finesses $\mathcal{F}$ of KAGRA cavities are summarized in the table below. These parameters are used to calculate the cavity pole

\begin{equation}
f_{\rm cp} = \frac{c}{2 L \mathcal{F}}.
\end{equation}

\begin{table}[H]
  \begin{center}
    \caption{KAGRA cavity parameters. The round-trip length for the fiber ring cavity(FRC) is multiplied by the refractive index of the fiber ($n=1.46$). The frequency reference cavity(FRC) for bKAGRA is ULE cavity.}
    \begin{tabular}{lccc}
      \hline \hline
	Cavity	&Round-trip length $L$	&Finesse $\mathcal{F}$	&Cavity pole $f_{\rm c}$ \\
      \hline
	PMC(iKAGRA)	&$0.4\unit{m}$	&200	&190 kHz \\
	PMC(bKAGRA)	&$1.95\unit{m}$	&155	&496 kHz \\
	FRC(iKAGRA)	&$1.46\times5.8\unit{m}$	&451	&39 kHz \\
	FRC(bKAGRA)	&$2\times0.1{m}$	&30000	&25 kHz \\
	IMC	&$2\times26.65\unit{m}$	&500 	&5.6 kHz \\
	Arms	&$2\times3000\unit{m}$	&1530 	&16 Hz\\
      \hline \hline
    \end{tabular}
  \end{center}
\end{table}

\subsection{Actuators}
The frequency actuators we use for the FSS are summarized in the table below.

\begin{table}[H]
  \begin{center}
    \caption{The frequency actuators for KAGRA. Note that they are rough estimates yet!! MCe actuation efficiency was measured to be $25 \unit{\mu m/V}$ by K. Arai~\cite{MCactuation}.}
    \begin{tabular}{lp{3cm}p{3.2cm}r}
      \hline \hline
	Name	&Detail	&Actuation efficiency	&Range \\
      \hline
	Laser temperature	&NPRO	&$3\unit{GHz/V}$	&$30\unit{GHz}$ \\
	Laser PZT	&NPRO	&$1\unit{MHz/V}$	&$100\unit{MHz}$ \\
	Wideband EOM	&Newport 4004	& $\disp{1.5 \times 10^{-5} \frac{if}{1\unit{Hz}} \unit{Hz/V}}$ &$0.6\unit{MHz}$ \\
	AOM	&Crystal Technology 3110-197 (with driver 1110AF-AEFO-1.5)	&$5.3\unit{MHz/V}$	&$40\unit{MHz}$ \\
	MCe	&Double pendulum	&$280\unit{MHz/V}$ at DC (on mirror)	&? \\
	ETM	&7-stage pendulum	&?	&? \\
      \hline \hline
    \end{tabular}
  \end{center}
\end{table}

\subsection{Filters}
The filters are shown in the figures below. Note that the gain scales are arbitrary.

\figFilter{FRC}
\figFilter{IMC}
\figFilter{CARM}

\subsection{Noises}
The frequency noises of each component are shown in the figure below.

\figTemplate{Noises}{Frequency noises. FRC noise shown here is unrealistic!!}


\section{Result}
\subsection{Openloop transfer functions}
The openloop transfer funtions are shown in the figures below.

\figOLTF{FRC}
\figOLTF{IMC}
\figOLTF{CARM}

\subsection{Out-of-loop frequency stability}
The out-of-loop frequency stabilities are shown in the figures below. The stability shown here is the stability when the servos for the next stages are turned off. For out-of-loop stability after the FRC servo, the IMC and the CARM loop is off. For the IMC stability, the FRC loop is on, but the CARM loop is off. When th CARM loop is on, the MCe loop in the IMC loop is disabled.

The blue line labeled "Requirement" shows the bKAGRA frequency noise requirement after IMC without CARM loop~\cite{MIFDesign}.

\figOOL{FRC}
\figOOL{IMC}
\figOOL{CARM}

\subsection{Feedback signal saturation check}
The spectra of feedback signals are shown in the figures below. Check if they are not saturating with your eyeballs.

The blue line labeled "Limit" shows the actuation range in Hz for each actuator.

\figFB{Temp}
\figFB{PZT}
\figFB{EOM}
\figFB{MCe}
\figFB{AOM}
\figFB{ETM}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%BIBLIOGRAPHY
\begin{thebibliography}{99}
\bibitem{NBref} The source code is available from \\
\url{https://svn.ligo.caltech.edu/svn/aligonoisebudget}. \\
Some instructions are given at \\
\url{https://awiki.ligo-wa.caltech.edu/aLIGO/NoiseBudget}.

\bibitem{MCactuation} Koji Arai, Shuichi Sato, Ryu Takahashi: Mode Cleaner Suspensions Transfer Functions. \\
\url{http://tamago.mtk.nao.ac.jp/tama/ifo/recycling1/suspension_tf/020322_MC/tfMC020322.pdf}

\bibitem{MIFDesign} Yoichi Aso, Yuta Michimura, Kentaro Somiya: KAGRA Main Interferometer Design Document, JGW-T1200913.\\
\url{http://gwdoc.icrr.u-tokyo.ac.jp/cgi-bin/DocDB/ShowDocument?docid=913}
\end{thebibliography}

\end{document}