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\begin{document}

%{{{ Title

\title{LCGT Main Interferometer Design}
\author{LCGT Main Interferometer Working Group}
\maketitle
\tableofcontents

%}}}

%{{{ \chapter{Design for bLCGT}

\chapter{Design for bLCGT}
We first present a preliminary design of the main interferometer for
bLCGT. 

%{{{ \section{Definition and Scope}

\section{Definition and Scope}

The main interferometer (MIF) is a subsystem which is responsible for
converting gravitational waves into electronic signals on
a photodetector. It includes two arm cavities, the power recycling cavity
and the signal recycling cavity. The MIF subsystem also provides
specifications for output mode cleaner, which will be manufactured by
the input/output optics group. In the LCGT's subsystem structure, this
subsystem also includes the control system to keep the interferometer at
the optimal operating point.

This subsystem provides no hardware because all the hardware which
constitutes the main interferometer is provided by other subsystems,
e.g. mirrors are provided by the mirror subsystem. What this subsystem
provides is a design of the interferometer, which is transferred to other
subsystems as specifications for their components. We are also mostly
responsible for the commissioning of the interferometer once components
are installed.

Here is a list of tasks we are assigned to:
 \begin{itemize}
  \item Determine the optical parameters of the main interferometer,
	except for the ones already determined by the detector
	configuration group.
  \item Determine the detailed optical layout of the main
	interferometer.
  \item Design robust length and alignment sensing schemes which do not
	contaminate the target sensitivity. 
  \item Design a robust lock acquisition scheme.
  \item Make a commissioning plan to achieve the target sensitivity as
	soon as possible.
 \end{itemize}


%}}}

%{{{ \section{Requirements}

\section{Requirements}
\begin{itemize}
 \item The main interferometer has to be able to achieve the target
sensitivity of bLCGT (Figure\,\ref{bLCGT Target Sensitivity}). The
       target sensitivity of bLCGT is given by the
fundamental noises estimated by the detector configuration group. All
other technical noises has to be well below the fundamental noises.

 \item bLCGT has two operation modes: BRSE and DRSE. The main interferer
configuration should allow us to switch between the two modes within a
       reasonable amount of time.

 \item The control schemes of the LCGT has to be robust enough to ensure
       stable operation of the interferometer in the environmental
       disturbances of Kamioka mine.

\end{itemize}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=14cm]{figures/TargetSens.eps}
\caption{bLCGT Target Sensitivity}
\label{bLCGT Target Sensitivity}
 \end{center}
\end{figure}



%}}}

%{{{ \section{Interface}

\section{Interface}

%{{{ \subsection{Mirror Subsystem}

\subsection{Mirror subsystem}
Requirements to the mirror subsystem from MIF are listed in the
tables\,\ref{Input Test Mass Requirements} - \ref{SR3 Requirements}.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
   HR Reflectivity@1064\,nm &99.6$\pm$0.01\%& Power reflectivity\\\hline
   HR Reflectivity Mismatch @1064\,nm &$<$0.05\%&
	   $|R_\mr{ITMX}-R_\mr{ITMY}|$\\\hline
HR Reflectivity@532\,nm &80$\pm$10\%&ITM reflectivity must be smaller than ETM\\\hline
   HR loss@1064\,nm &$<$45\,ppm&Including diffraction and absorption \\\hline
   HR loss@532\,nm &$<$1\%&Including diffraction and absorption \\\hline
   AR Reflectivity@1064\,nm &200(+100/-0)\,ppm& \\\hline
   AR Reflectivity@532\,nm &$<$5\%& \\\hline
   HR ROC&$>$500\,km& As large as possible\\\hline
   HR ROC Mismatch between ITMX and ITMY&$<$10\,km& \\\hline
   AR ROC&$>$500\,km& As large as possible\\\hline
   AR Wedge Angle&0.2$\pm$0.002\,deg&\\\hline
  \end{tabular}
\caption{Input Test Mass Requirements}
\label{Input Test Mass Requirements}
 \end{center}
\end{table}

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
   HR Reflectivity@1064\,nm &$>$99.9945\%& Power reflectivity\\\hline
   HR Transmissivity@1064\,nm &$>$1\,ppm&\\\hline
   HR Reflectivity@532\,nm &80$\pm$10\%& ETM reflectivity must be
	   larger than ITM\\\hline
   HR loss@1064\,nm &$<$45\,ppm&Including diffraction and absorption \\\hline
   HR loss@532\,nm &$<$1\%&Including diffraction and absorption \\\hline
   AR Reflectivity@1064\,nm &$<$1000\,ppm& \\\hline
   AR Reflectivity@532\,nm &No requirement& \\\hline
   HR ROC&7000$\pm$100\,m& \\\hline
   HR ROC Mismatch between ITMX and ITMY&$<$7\,m& \\\hline
   AR ROC&$>$100\,km& As large as possible\\\hline
   AR Wedge Angle&0\,deg&\\\hline
  \end{tabular}
\caption{End Test Mass Requirements}
\label{End Test Mass Requirements}
 \end{center}
\end{table}

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
   HR Reflectivity@1064\,nm &50$\pm$0.5\%& S Polarization\\\hline
   HR Reflectivity@532\,nm &$<$10\%& As small as possible\\\hline
   AR Reflectivity@1064\,nm &$<$50\,ppm& \\\hline
   AR Reflectivity@532\,nm &$<$10\%& \\\hline
   HR ROC&$>$500\,km&As large as possible \\\hline
   AR ROC&$>$500\,km&As large as possible\\\hline
   AR Wedge Angle&0.383$\pm$0.002\,deg&\\\hline
  \end{tabular}
\caption{Beam Splitter Requirements}
\label{Beam Splitter Requirements}
 \end{center}
\end{table}

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
   HR Reflectivity@1064\,nm &90+0/-0.1\%& \\\hline
   HR Loss@1064\,nm &$<$100\,ppm& \\\hline
   AR Reflectivity@1064\,nm &$<$100\,ppm& \\\hline
   HR ROC&300.6$\pm$1\,m& \\\hline
   AR ROC&$>$100\,km&As large as possible\\\hline
   AR Wedge Angle&2.0$\pm$0.1\,deg&\\\hline
  \end{tabular}
\caption{PRM Requirements}
\label{PRM Requirements}
 \end{center}
\end{table}

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
   HR Reflectivity@1064\,nm &85.0$\pm$1\%& \\\hline
   HR Loss@1064\,nm &$<$100\,ppm& \\\hline
   AR Reflectivity@1064\,nm &$<$100\,ppm& \\\hline
   HR ROC&300.6$\pm$1\,m& \\\hline
   AR ROC&$>$100\,km&As large as possible\\\hline
   AR Wedge Angle&2.0$\pm$0.1\,deg&\\\hline
  \end{tabular}
\caption{SRM Requirements}
\label{SRM Requirements}
 \end{center}
\end{table}

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
   HR Reflectivity@1064\,nm &99.95$\pm$0.01\%& \\\hline
   HR Reflectivity@532\,nm &$<$10\%&As small as possible \\\hline
   HR Loss@1064\,nm &$<$100\,ppm& \\\hline
   HR Loss@532\,nm &$<$1\%& \\\hline
   AR Reflectivity@1064\,nm &$<$500\,ppm& \\\hline
   AR Reflectivity@532\,nm &$<$10\%& \\\hline
   HR ROC&-3.251$\pm$0.01\,m& \\\hline
   AR ROC&$>$50\,km&As large as possible\\\hline
   AR Wedge Angle&2.0$\pm$0.1\,deg&\\\hline
  \end{tabular}
\caption{PR2 Requirements}
\label{PR2 Requirements}
 \end{center}
\end{table}

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
   HR Reflectivity@1064\,nm &$>$99.99\%& \\\hline
   HR Reflectivity@532\,nm &$>$80\%&As large as possible \\\hline
   HR Loss@1064\,nm &$<$100\,ppm& \\\hline
   HR Loss@532\,nm &$<$1\%& \\\hline
   AR Reflectivity@1064\,nm &$<$1000\,ppm& \\\hline
   AR Reflectivity@532\,nm &No requirement& \\\hline
   HR ROC&27.36$\pm$0.1\,m& \\\hline
   AR ROC&$>$10\,km&As large as possible\\\hline
   AR Wedge Angle&2.0$\pm$0.1\,deg&\\\hline
  \end{tabular}
\caption{PR3 Requirements}
\label{PR3 Requirements}
 \end{center}
\end{table}

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
   HR Reflectivity@1064\,nm &99.95$\pm$0.01\%& \\\hline
   HR Reflectivity@532\,nm &$<$10\%&As small as possible \\\hline
   HR Loss@1064\,nm &$<$100\,ppm& \\\hline
   HR Loss@532\,nm &$<$1\%& \\\hline
   AR Reflectivity@1064\,nm &$<$500\,ppm& \\\hline
   AR Reflectivity@532\,nm &$<$10\%& \\\hline
   HR ROC&-3.251$\pm$0.01\,m& \\\hline
   AR ROC&$>$50\,km&As large as possible\\\hline
   AR Wedge Angle&2.0$\pm$0.1\,deg&\\\hline
  \end{tabular}
\caption{SR2 Requirements}
\label{SR2 Requirements}
 \end{center}
\end{table}

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
   HR Reflectivity@1064\,nm &$>$99.99\%& \\\hline
   HR Reflectivity@532\,nm &$>$80\%&As large as possible \\\hline
   HR Loss@1064\,nm &$<$100\,ppm& \\\hline
   HR Loss@532\,nm &$<$1\%& \\\hline
   AR Reflectivity@1064\,nm &$<$1000\,ppm& \\\hline
   AR Reflectivity@532\,nm &No requirement& \\\hline
   HR ROC&27.36$\pm$0.1\,m& \\\hline
   AR ROC&$>$10\,km&As large as possible\\\hline
   AR Wedge Angle&2.0$\pm$0.1\,deg&\\\hline
  \end{tabular}
\caption{SR3 Requirements}
\label{SR3 Requirements}
 \end{center}
\end{table}

%}}}

%{{{ \subsection{Input/Output Optics Subsystem}
\subsection{Input/Output Optics Subsystem}
Interface items with the input/output optics subsystem are listed in
Table\,\ref{Interface Items with the Input/Output Optics Subsystem}.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|p{12em}|p{15em}|}
\hline
   {\bf Item} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height
   0.8pt}
Laser Power&85\,W at PRM input &Carrier Power \\ \hline
Modulation Frequencies &16.875\,MHz, 45\,MHz, 39.375\,MHz, 56.25\,MHz&see section\,\ref{bLCGT LSC}\\\hline
Modulation Depth&max 0.3&\\\hline
Mode Cleaner Length& & \\ \hline
Laser Frequency Noise   &TBD & \\ \hline
Laser Intensity Noise   &TBD & \\ \hline
Frequency Actuation&TBD & \\ \hline
Output Mode Cleaner  &TBD & \\ \hline
Mode Matching Telescopes&TBD & \\ \hline
Beam Reducing Telescopes&POX, POY, TRX, TRY & \\ \hline
Green Lasers&TBD & \\ \hline
  \end{tabular}
\caption{Interface Items with the Input/Output Optics Subsystem}
\label{Interface Items with the Input/Output Optics Subsystem}
 \end{center}
\end{table}

%}}}

%{{{ \subsection{Suspension Subsystem}
\subsection{Suspension Subsystem}
Interface items with the suspension subsystem are listed in
Table\,\ref{Interface Items with the Suspension Subsystem}.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|p{12em}|p{17em}|}
\hline
   {\bf Item} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
Vibration Isolation Performance for Type A SAS &$3\times
       10^{-20}\mr{m/\sqrt{Hz}}$@10\,Hz&see section\,\ref{Displacement
	   Noise Requirement}\\\hline
Vibration Isolation Performance for Type B SAS &$1.5\times
       10^{-17}\mr{m/\sqrt{Hz}}$@10\,Hz&see section\,\ref{Displacement
	   Noise Requirement}\\\hline
 Adjustable Range of Mirror Locations&TBD (in the order of a few cm)
       &After the installation of the mirrors, the locations must be
	   adjusted to compensate for ROC errors. \\ \hline
Actuation Range&TBD&\\\hline
Local Sensors&TBD & \\ \hline
  \end{tabular}
\caption{Interface Items with the Suspension Subsystem}
\label{Interface Items with the Suspension Subsystem}
 \end{center}
\end{table}

%}}}

%{{{ \subsection{Digital Subsystem}
\subsection{Digital Subsystem}
Interface items with the digital subsystem are listed in
Table\,\ref{Interface Items with the Digital Subsystem}.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Item} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
Sampling Frequency&max 64\,kHz&\\\hline
ADC dynamic range&80\,dB&This requirement comes from lock acquisition\\\hline
  \end{tabular}
\caption{Interface Items with the Digital Subsystem}
\label{Interface Items with the Digital Subsystem}
 \end{center}
\end{table}

%}}}

%{{{ \subsection{Analog Electronics Subsystem}
\subsection{Analog Electronics Subsystem}
Interface items with the analog electronics subsystem are listed in
Table\,\ref{Interface Items with the Analog Electronics Subsystem}.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Item} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
Maximum Power on PDs &$>$100\,mW&\\\hline
Maximum RF PD Frequency for observation&60\,MHz&\\\hline
Maximum RF PD Frequency for lock acquisition&135\,MHz&\\\hline
Maximum RF QPD Frequency&60\,MHz&\\\hline
  \end{tabular}
\caption{Interface Items with the Analog Electronics Subsystem}
\label{Interface Items with the Analog Electronics Subsystem}
 \end{center}
\end{table}

%}}}

%{{{ \subsection{Vacuum Subsystem}
\subsection{Vacuum Subsystem}
Interface items with the vacuum subsystem are listed in
Table\,\ref{Interface Items with the Vacuum Subsystem}.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Item} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
Optical Layout &&see section\,\ref{Optical Layout}\\\hline
View Ports&TBD & \\ \hline
  \end{tabular}
\caption{Interface Items with the Vacuum Subsystem}
\label{Interface Items with the Vacuum Subsystem}
 \end{center}
\end{table}

%{{{ \subsection{Cryogenic}
\subsection{Cryogenic}
Interface items with the cryogenic subsystem are listed in
Table\,\ref{Interface Items with the Cryogenic Subsystem}.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Item} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
Length of radiation shield &20\,m&\\\hline
Location of holes in the radiation shield baffles for POX and POY&&see section\,\ref{Optical Layout}\\\hline
  \end{tabular}
\caption{Interface Items with the Cryogenic Subsystem}
\label{Interface Items with the Cryogenic Subsystem}
 \end{center}
\end{table}

%}}}

%}}}

%}}}

%{{{ \section{Optical Configuration}

\section{Optical Configuration}

%{{{ \subsection{Overview}

\subsection{Overview}
The main interferometer part of bLCGT is a dual recycled Fabry-Perot
Michelson interferometer operating in a resonant-sideband extraction
(RSE) mode.
The schematic view of the main interferometer and the naming convention of
the interferometer components are shown in Figure \ref{IFO diagram}.

\begin{figure}[htbp]
 \begin{center}
 \includegraphics[width=13cm]{figures/IFO-diagram.eps} 
\caption{Schematic of the main interferometer and the naming convention of IFO parameters}
\label{IFO diagram}
 \end{center}
\end{figure}


%}}}

%{{{ \subsection{Arm Cavity}

\subsection{Arm Cavity}
The arm cavity length is constrained to less than 3\,km by the size of
the mountain. So we chose the largest one. 
The reflectivities of the mirrors are set by the detector configuration
group to optimize the sensitivity under the constraint that the heat
absorption at ITM should not exceed the cooling power of the cryogenic
system. This constraint led us to choose a relatively high finesse and a
high RSE gain.

The radius of curvature of the mirrors are chosen by
considering the following factors.

\begin{itemize}
 \item The beam spot size should not be too large to induce more than
       1\,ppm of diffraction loss.
 \item The beam spot size should not be too small to make the coating
       thermal noise too large.
 \item The g-factor of the cavity should be stable enough, i.e.
       higher order spatial modes should not resonate in the arm cavity
       at the same time as the fundamental mode. 
 \item The parametric instability must be avoided.
\end{itemize}

Arm cavity parameters are listed in Table\,\ref{Arm cavity parameters}.
Detailed reasoning behind the choice of the parameters is given in appendix\,\ref{Apendix: Recycling Cavity Length}.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|l|c|c|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\hline
   Arm cavity length&3000.00\,m & \\\hline
   ITM Reflectivity&99.6\% & \\ 
   ITM Radius of Curvature& flat ($>$500\,km)& \\ 
   ITM Beam Size&3.43\,cm & $1/e^2$ radius\\ \hline
   ETM Reflectivity&99.9945\%& \\ 
   ETM Radius of Curvature& 7000\,m& \\ 
   ETM Beam Size&4.53\,cm & $1/e^2$ radius\\ \hline
   g-factor&0.572 &$g_1\cdot g_2$ \\ 
   Round Trip Loss&$<$100\,ppm& \\ 
   Finesse&1546 & \\ \hline
  \end{tabular}
\caption{Arm cavity parameters}
\label{Arm cavity parameters}
 \end{center}
\end{table}


%}}}

%{{{ \subsection{Recycling Cavities}
\subsection{Recycling Cavities}

The reflectivities of the recycling mirrors are determined by the
detector configuration group to optimize the power and signal recycling gain.

The length parameters of the recycling cavities (RCs) are set to resonate the
RF sidebands used for the control signal extraction, which is explained
in section\,\ref{bLCGT LSC}. There are two RF sidebands entering the
power recycling cavity (PRC), called f1 and f2. The Schnupp asymmetry of the
Michelson part (MICH) is chosen to perfectly reflect the f2 sideband by
MICH, so that f2 does not see the signal recycling cavity (SRC). The f1 sideband
transmits through MICH and resonates in the compound cavity formed by PRC
and SRC through MICH.

The actual length of the recycling cavities have to be fine adjusted to fully
resonate the f1 and f2 sidebands, because f1 and f2 get non-zero phase
shift when reflected by the arm cavities. This is achieved by fine tuning the f1
and f2 frequencies so that the reflection phases of those sidebands to
the arm cavities have the ratio of f1:f2. The mode cleaner length and
the recycling cavity lengths are adjusted accordingly.

The average length of MICH (the average distance between BS and ITMs) is
set to 25\,m. Out of this 25\,m, 20\,m is used for the cryogenic
radiation shield, which protects the cryogenic ITMs from 300\,K thermal
radiation of the BS chamber. The remaining 5\.m is used to absorb the
Schnupp asymmetry.

The recycling cavities are both folded by two additional mirrors to
allow focusing of the beam inside the cavities. This configuration makes
the recycling cavities stable in terms of spatial modes. However, the
recycling cavities cannot be too stable, because then TEM10 or TEM01
modes generated by the misalignment of the arm cavities will be
suppressed in the RCs. Those higher order modes are necessary for the
alignment sensing by the wavefront sensing (WFS) scheme.  The one-way
Gouy phase change of the TEM00 beam resonating in either recycling
cavities is set to 20\,deg. as a compromise between the stability of the
RCs and the WFS signal strength. The ROCs of the folding mirrors are set
to realize the Gouy phase shift.

The geometry of the folding part is mainly determined by the constraints
from the vacuum system, such as the minimum separation between vacuum
chambers. In order to minimize the astigmatism, we chose a configuration
which minimizes the folding angle under the constraints. The beam from
PR3 to BS has a large radius (3.5\,cm). Therefore, PR2 has to be located
far enough from the beam to avoid beam clipping.  On the other hand, we
want to put PR2 as close to the beam as possible to minimize the
astigmatism. As a result, we located the edge of PR2 at 4 times the beam
radius away from the beam.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|l|c|c|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\hline
   Power Recycling Cavity Length&66.591\,m & \\
   Signal Recycling Cavity Length&66.591\,m & \\
   Michelson Average Length&25.0\,m& \\ 
   Michelson Asymmetry&3.330\,m & \\ \hline
   PRM Reflectivity&90\%& \\ 
   PRM ROC&300.624\,m & \\ 
   PRM Beam Size&4.05\,mm &$1/e^2$ radius \\ \hline
   PR2 ROC&-3.251\,m & \\ 
   PR2 Beam Size&4.05\,mm &$1/e^2$ radius \\ \hline
   PR3 ROC&27.36\,m & \\ 
   PR3 Beam Size&34.25\,mm &$1/e^2$ radius \\ \hline
   PRM Reflectivity&84.64\% & \\ 
   SRM ROC&300.624\,m & \\ 
   SRM Beam Size&4.05\,mm &$1/e^2$ radius \\ \hline
   SR2 ROC&-3.251\,m & \\ 
   SR2 Beam Size&4.05\,mm &$1/e^2$ radius \\ \hline
   SR3 ROC&27.36\,m & \\ 
   SR3 Beam Size&34.25\,mm &$1/e^2$ radius \\ \hline
  \end{tabular}
\caption{Recycling cavity parameters}
\label{Recycling cavity parameters}
 \end{center}
\end{table}

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|l|c|c|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\hline
   Lp1&14.761\,m & Distance between PRM and PR2\\ 
   Lp2&12.067\,m & Distance between PR2 and PR3\\ 
   Lp3&14.764\,m & Distance between PR3 and BS\\ \hline
   Ls1&14.761\,m & Distance between SRM and SR2\\ 
   Ls2&12.067\,m & Distance between SR2 and SR3\\
   Ls3&14.764\,m & Distance between SR3 and BS\\\hline
   Folding Angle&0.6293\,deg &The incident angle to the folding mirrors.\\ \hline
  \end{tabular}
\caption{Folding parameters. See Figure \ref{IFO diagram} for the
  meaning of the parameters.}
\label{Folding parameters}
 \end{center}
\end{table}


%}}}

%}}}

%{{{ \section{Optical Layout}

\section{Optical Layout}
\label{Optical Layout}

The exact locations and the orientations of the interferometer mirrors have to be determined
to satisfy the following criteria:
\begin{itemize}
 \item X-arm and Y-arm are orthogonal.
 \item Beams hit the mirrors at the center.
 \item Recycling cavity lengths and the Schnupp asymmetry match the
       designed values including the optical distance of the
       transmissive optics. 
\end{itemize} 

Since some optics have AR wedge, it is not a trivial task to trace the
beams through the interferometer and find a configuration which satisfy
the above conditions. Doing it manually is also an error-prone process.
We also have to track unwanted reflections from the AR surfaces to
appropriately damp the stray beams. This is also a daunting task.
Therefore, we developed a Python library, called gtrace, for tracing
beam paths and the evolution of Gaussian beam parameters through the
interferometer.  Detailed optical layout of LCGT main interferometer is
automatically generated by a Python code given a set of interferometer
parameters (distance between the mirrors, mirror properties etc) and
constraints.

The optical layout generation code and the generated CAD files are
available in the LCGT svn\,\cite{LCGT svn layout}.

\subsection{Tunnel Slope}
One thing to be kept in mind here is the tunnel slope. Since the 3\,km
tunnels of LCGT are slightly tilted for water drainage, the two arms are
not on a level plane. We decided that all the main interferometer optics
be placed on the plane defined by the two arms. This plane is tilted
with respect to the local gravitationally level plane. The optical
layout generated by the above code is drawn on this tilted plane.  In
the actual construction and the installation of the vacuum chambers, the
optical layout has to be projected from the tilted plane to the
reference plane used for the construction.

%}}}

%{{{ \section{Length Sensing and Control Scheme}

\section{Length Sensing and Control Scheme}
\label{bLCGT LSC}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=15cm]{figures/SB-Resonance-Signal-Ports.eps}
\caption{RF sideband resonant conditions and signal ports. POP is drawn
  at the transmission of PR3 to avoid congestion of the
  diagram. However, POP is actually planned to be picked up from the back of PR2
  because the beam size is much smaller there. }
\label{Signal Ports}
 \end{center}
\end{figure}

\subsection{Overview}
The length degrees of freedom to be controlled are DARM, CARM, MICH,
PRCL and SRCL (see appendix \ref{Appendix Terminology} for the
definitions of the acronyms). DARM is sensed at the AS port by the DC
readout scheme.  Other degrees of freedom are sensed by a variant of
frontal modulation scheme. The input laser beam is phase or amplitude
modulated to generate RF sidebands. There are two RF sidebands, which
resonate in the central part (PRC, MICH, SRC) of the interferometer.
The sideband resonant conditions are shown in Figure\,\ref{Signal
Ports}.  The f1 sideband resonates in the compound cavity of
PRC-SRC. The MICH reflectivity to the f2 sideband is chosen to be almost
100\%. Consequently, f2 only resonates in PRC. Optionally, we may add
another RF sideband, f3, which does not enter the interferometer at
all. f3 is called non-resonant sideband (NRS).

When operated in DRSE configuration, the detuning of the SRC is done by
adding offset to the error signal of SRCL. The required detuning of SRC
is 3.5\,degree in terms of the one-way phase shift of SRCL, which
corresponds to 10\,nm shift of SRM position. The f1 sideband frequency
is chosen to make the resonance of f1 to SRC not too sharp, so that the
detuned SRC can still produce a reasonable error signal using f1. The
SRCL error signal is plotted as a function of SRM position in
Figure\,\ref{SRCL Sweep}. The operating point of DRSE has enough slope
to produce strong error signal. However, the non-linearity of the error
signal is stronger at the DRSE operating point. Discussions on this
non-linearity and other issues with the offset detuning of SRC can be
found in Appendix\,\ref{Reasoning Behind Parameter Choices}.

The selected RF modulation frequencies are listed in Table\,\ref{RF
Sideband Frequencies}. The mode cleaner has to transmit the RF
sidebands. For this reason, the FSR of the MC is chosen to be f2/8.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c|c|c|}
\hline
f1&16.880962\,MHz&$3\times f_\mr{MC}$, PM \\ \hline
f2&45.015898\,MHz&$8\times f_\mr{MC}$, PM \\ \hline
f3&39.388910\,MHz and 56.269873\,MHz&$7\times f_\mr{MC}$ and $10\times f_\mr{MC}$, AM \\
   \hline
$f_\mr{MC}$&5.626987\,MHz&MC FSR \\ \hline
$L_\mr{MC}$&26.6388\,m&MC Length \\ \hline
  \end{tabular}
\caption{RF Sideband Frequencies}
\label{RF Sideband Frequencies}
 \end{center}
\end{table}

\paragraph{Simulation Codes}
The calculations shown in the following sections are all done by using a simulation tool
called Optickle\,\cite{Optickle}.
The interferometer model and simulation codes are available in the LCGT
svn at \cite{LSC Code}.

\subsection{Signal Name Convention}
In this document, signal names follow the convention described here.  A
signal name consists of a port name followed by an indicator of
demodulation scheme connected with an underscore (\_). For example,
``REFL\_1I'' means a signal detected at the reflection port and
demodulated at the f1 frequency in I-phase. Another example is
``AS\_DC'', which means a DC signal detected at the AS port.

Double demodulation may be used if we use the f3 sideband. In this case,
a signal name looks like ``REFL\_1DmQ''. This means a signal detected at
the REFL port demodulated at f3-f1 frequency in Q-phase. Double
demodulation is always between f3 and one of the other SBs. Therefore,
only one number is specified. The letter ``D'' means double
demodulation, ``m'' means f3 minus f1. In the case of f3 plus f1, this
letter will be ``p''.  

``REFL\_1DmQ'' is not a true double-demodulation, where the signal should
be demodulated twice at f3 and f1 frequencies. However, it carries
similar information as the true double-demodulation.



\subsection{Signal Extraction Ports}

The default length sensing scheme for bLCGT uses two phase modulated RF
sidebands, f1 and f2. Beat signals of these sidebands with the carrier
are detected at REFL and POP to extract necessary error signals. The
signal sensing matrix for this scheme in the case of BRSE operation is
shown in Table\,\ref{Sensing Matrix LSC SDM}.

CARM signal produces large non-diagonal elements to the PRCL and
SRCL. This is because the phase change of the carrier by CARM is usually
much larger than that of the RF sidebands by PRCL or SRCL. However, CARM
loop can have a very large control gain because of the very fast nature
of the laser frequency feedback. We can rely on this fact to suppress
the interference of CARM to PRCL and SRCL.

In REFL\_1Q, the mixture of PRCL and SRCL to MICH is large. This does
not happen when there is no asymmetry in the interferometer. When
calculating the sensing matrix, we introduced 1\% asymmetry\footnote{Not
the Schnupp asymmetry.} between the two arms. This asymmetry makes the
CARM, PRCL and SRCL signals no longer be at exactly the orthogonal
quadrature to the MICH signal. In the case of Table\,\ref{Sensing Matrix
LSC SDM}, we chose a demodulation phase which minimizes the coupling of
CARM. In return, the PRCL and SRCL signals became comparable to
MICH. However, these signals cab be separated because the lower right
$3\times 3$ matrix of Table\,\ref{Sensing Matrix LSC SDM} is invertible.


\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c@{\ \vrule width 0.8pt\ }c|c|c|c|c|}
\hline
&   {\bf DARM} &{\bf CARM} &{\bf MICH} &{\bf PRCL}&{\bf
   SRCL}\\\noalign{\hrule height 0.8pt}
   & & & & & \\[-10pt] 
{\bf AS\_DC}&1&$4.2\times10^{-5}$&$1.0\times10^{-3}$&$4.8\times10^{-6}$&$4.7\times10^{-6}$\\\hline
{\bf
   REFL\_1I}&$5.4\times10^{-3}$&1&$4.3\times10^{-5}$&$6.5\times10^{-3}$&$4.3\times10^{-3}$\\\hline
{\bf REFL\_1Q}&$5.0\times10^{-3}$&$1.3\times10^{-2}$&1&1.02&0.67\\\hline
{\bf
   POP\_2I}&$2.3\times10^{-2}$&4.3&$1.0\times10^{-2}$&1&$2.5\times10^{-4}$\\\hline
{\bf POP\_1I}&$8.7\times10^{-2}$&16.2&$3.1\times10^{-2}$&2.1&1\\\hline
  \end{tabular}
\caption{Normalized Sensing Matrix of LSC in the case of BRSE. Each row is normalized by
  the diagonal element. The interferometer response was evaluated at
  100\,Hz to create this matrix.}
\label{Sensing Matrix LSC SDM}
 \end{center}
\end{table}
The sensing matrix for DRSE operation is shown in Table\,\ref{Sensing
Matrix LSC SDM DRSE}.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c@{\ \vrule width 0.8pt\ }c|c|c|c|c|}
\hline
&   {\bf DARM} &{\bf CARM} &{\bf MICH} &{\bf PRCL}&{\bf
   SRCL}\\\noalign{\hrule height 0.8pt}
   & & & & & \\[-10pt] 
{\bf AS\_DC}&1&$4.1\times10^{-5}$&$1.0\times10^{-3}$&$4.5\times10^{-6}$&$7.6\times10^{-6}$\\\hline
{\bf
   REFL\_1I}&$1.2\times10^{-2}$&1&$1.3\times10^{-4}$&$1.2\times10^{-2}$&$1.4\times10^{-3}$\\\hline
{\bf REFL\_1Q}&$2.8\times10^{-2}$&$9.9\times10^{-3}$&1&0.39&0.18\\\hline
{\bf
   POP\_2I}&$2.7\times10^{-2}$&4.3&$1.0\times10^{-2}$&1&$8.5\times10^{-5}$\\\hline
{\bf POP\_1I}&$1.7\times10^{-1}$&35&$3.1\times10^{-2}$&2.0&1\\\hline
  \end{tabular}
\caption{Normalized Sensing Matrix of LSC in the case of DRSE. Each row is normalized by
  the diagonal element. The interferometer response was evaluated at
  100\,Hz to create this matrix.}
\label{Sensing Matrix LSC SDM DRSE}
 \end{center}
\end{table}

If we use the f3 sideband, which is an AM non-resonant sideband, the
sensing matrix looks like Table\,\ref{Sensing Matrix LSC NRS} (BRSE). The large
CARM interference disappeared in this case. So we don't have to rely on
the gain hierarchy. However, in this configuration, we have to introduce
a Mach-Zehnder interferometer at the modulation stage to separate the AM
generation path and the PM generation path. This is necessary to avoid
the generation of sub-sidebands at the double demodulation
frequencies. This addition of Mach-Zehnder may introduce additional
noise.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c@{\ \vrule width 0.8pt\ }c|c|c|c|c|}
\hline
&   {\bf DARM} &{\bf CARM} &{\bf MICH} &{\bf PRCL}&{\bf
   SRCL}\\\noalign{\hrule height 0.8pt}
   & & & & & \\[-10pt] 
{\bf AS\_DC}&1&$4.1\times10^{-5}$&$1.0\times10^{-3}$&$4.8\times10^{-6}$&$4.7\times10^{-6}$\\\hline
{\bf
   REFL\_1I}&$5.4\times10^{-3}$&1&$3.9\times10^{-5}$&$5.4\times10^{-3}$&$4.5\times10^{-3}$\\\hline
{\bf REFL\_1DmQ}&$4.8\times10^{-3}$&$2.5\times10^{-3}$&1&0.7&$1.3\times10^{-3}$\\\hline
{\bf
   REFL\_2DmI}&$1.83\times10^{-3}$&$8.3\times10^{-2}$&0.18&1&0.32\\\hline
{\bf REFL\_1DmI}&$2.5\times10^{-4}$&$1.5\times10^{-2}$&$2.4\times10^{-2}$&1.7&1\\\hline
  \end{tabular}
\caption{Normalized Sensing Matrix of LSC for BRSE using the f3 sideband. Each row is normalized by
  the diagonal element. The interferometer response was evaluated at
  100\,Hz to create this matrix.}
\label{Sensing Matrix LSC NRS}
 \end{center}
\end{table}

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{figures/SweepSRCL.eps}
\caption{SRCL error signal as a function of SRM position. The green
  vertical line indicates where the SRM should be when operated in DRSE configuration. }
\label{SRCL Sweep}
 \end{center}
\end{figure}


\subsection{Loop Noise}
In general, the auxiliary degrees of freedom have larger shot noise than
DARM. By using those signals for mirror control, we are effectively
injecting noise into the interferometer. Especially, MICH has an
unavoidable coupling to DARM by about 1/Finesse. Therefore, the shot
noise of the MICH error signal appears to DARM attenuated by that
factor. This is noise coupling mechanism is called loop noise.

The shot noise coupling by the control loops was calculated by assuming
the unity gain frequencies shown in Table\,\ref{Control Loop UGFs}.
The MICH UGF for DRSE is lowered to 10\,Hz to further reduce the noise coupling.

\begin{table}[htbp]
 \begin{center}
  \begin{tabular}{|c|c|c|}
\hline
   &{\bf BRSE} &{\bf DRSE} \\\noalign{\hrule height 0.8pt}
 DARM  &200\,Hz& 200\,Hz\\\hline
 CARM  &10\,kHz& 10\,kHz\\\hline
 MICH  &50\,Hz& 10\,Hz\\\hline
  PRCL  &50\,Hz& 50\,Hz\\\hline
 SRCL  &50\,Hz& 50\,Hz\\\hline
  \end{tabular}
\caption{Control Loop UGFs}
\label{Control Loop UGFs}
 \end{center}
\end{table}

The loop noise contributions from the auxiliary degrees of freedom are shown
for the BRSE case in Figure\,\ref{Loop Noise Coupling: BRSE, SDM}. The
curve labeled DARM is the target quantum noise level. It is clear that
loop noise couplings from other degrees of freedom are larger than the
target level.

The loop noise coupling can be reduced by a technique called feed
forward\,\cite{LoopNoise}.  Its working principle is the
following. Taking PRCL as an example, we can measure the transfer
function from the motion of PRM to the DARM signal. Then we assume that
the error signal of PRCL is dominated by shot noise (or any sensing
noise). This means that PRM is moved according to the shot noise by the
feedback. From the feedback signal, we know exactly how much the PRM is
erroneously moved. Therefore, we can predict how much noise is injected
from this PRM motion to DARM with the knowledge of the above mentioned
transfer function. By feeding forward this information to DARM, we can
subtract the loop injected noise. The quality of feed forward is
measured by the accuracy of the subtraction. Feed forward gain is defined 
as the inverse of the accuracy. If the accuracy is 1\%, the feed forward
gain is 100.

In Figure\,\ref{Loop Noise Coupling FF:
BRSE, SDM}, the loop noise couplings are shown when the feed forward is
applied with the gain of 100 to MICH, PRCL and SRC. With the feed
forward, the loop noise couplings are well below the DARM quantum noise.
The loop noise coupling with the feed forward for the case of DRSE is
shown in Figure\,\ref{Loop Noise Coupling FF: DRSE, SDM}. Again, the
loop noise is not a problem.


\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{figures/LoopNoise_SDM_BRSE.pdf}
\caption{Loop Noise Coupling: BRSE}
\label{Loop Noise Coupling: BRSE, SDM}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{figures/LoopNoise_SDM_BRSE_FF.pdf}
\caption{Loop Noise Coupling with Feed Forward: BRSE}
\label{Loop Noise Coupling FF: BRSE, SDM}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{figures/LoopNoise_SDM_DRSE_FF.pdf}
\caption{Loop Noise Coupling with Feed Forward: DRSE}
\label{Loop Noise Coupling FF: DRSE, SDM}
 \end{center}
\end{figure}

\subsection{Displacement Noise Requirement}
\label{Displacement Noise Requirement}
One caveat of feed forward is that it can increase the displacement
noise coupling of the auxiliary degrees of freedom to DARM. Feed forward
assumes that whatever you see in the error signal of an auxiliary degree
of freedom, say PRC, is sensing noise, i.e. not a real motion of the
mirror. This assumption is not valid in some frequencies. If the error
signal reflects real motion of the mirror, this motion is suppressed by
the feedback. Feeding forward this error signal to DARM means that you
are trying to cancel the motion of the mirror (PRM) which is already
suppressed by the (PRCL) feedback. The net result of this is the injection of
the displacement noise of the auxiliary degrees of freedom into DARM.

One can calculate the transfer functions from the motion of auxiliary
degrees of freedom to DARM with the feedback and the feed forward
engaged. By requiring the displacement noise couplings to be a factor of
10 below the quantum noise of DARM, we can deduce the requirements to
the displacement noise for each degrees of freedom. The calculated
displacement noise requirements are shown in Figure\,\ref{Displacement
Noise Requirent for Each Mirror: BRSE, SDM} (BRSE) and
Figure\,\ref{Displacement Noise Requirent for Each Mirror: DRSE, SDM}.
The line labeled ``SAS noise'' in the plots is a simplified noise model
of Type A SAS. The Type A SAS performance satisfies the displacement noise
requirements of all mirror.

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{figures/DispNoiseReq_SDM_BRSE_FF.pdf}
\caption{Displacement Noise Requirement for Each Mirror: BRSE, SDM}
\label{Displacement Noise Requirent for Each Mirror: BRSE, SDM}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{figures/DispNoiseReq_SDM_BRSE_FF.pdf}
\caption{Displacement Noise Requirement for Each Mirror: DRSE, SDM}
\label{Displacement Noise Requirent for Each Mirror: DRSE, SDM}
 \end{center}
\end{figure}


%}}}

%{{{ \section{Alignment Sensing and Control Scheme}
\section{Alignment Sensing and Control Scheme}
\section{Overview}
The alignment sensing and control (ASC) scheme of bLCGT has not yet
fully determined. We plan to use a combination of the wave front sensing
(WFS) scheme and optical levers. Currently we are studying the sensing
matrix of WFS with an Optickle model\footnote{In particular, we are using the
Pickle package developed by Lisa Barsotti.}.
A preliminary sensing matrix is shown in Figure\,\ref{Preliminary sensing matrix for ASC}.

The WFS signals are sensed in the soft-hard mode basis in terms of the
radiation pressure\,\cite{Sidles-Sigg}. CS, CH, DS and DH in the sensing
matrix mean, ``Common Soft'', ``Common Hard'', ``Differential Soft'' and
``Differential Hard'' respectively.

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=17cm]{figures/ASC_SensMatrix_bLCGT.pdf}
\caption{Preliminary sensing matrix for ASC. A and B in the signal name
  denotes two different Gouy phases.}
\label{Preliminary sensing matrix for ASC}
 \end{center}
\end{figure}



%}}}

%{{{ \section{Lock Acquisition Scheme}
\section{Lock Acquisition Scheme}

%{{{ \subsection{Overview}
\subsection{Overview}
Quick and robust lock acquisition is important for maintaining high duty
cycle. The lock acquisition procedure of bLCGT proposed here consists of
three stages. First, the arm cavities are locked by green lasers at
off-resonant positions for the main laser carrier. Then the central part of the
interferometer is locked either by using the third harmonics
demodulation signals  or non-resonant sideband. Finally, the arm
cavities are brought to full resonance to the main laser by changing the
relative frequency of the green lasers to the main laser. After all the
degrees of freedom are brought to the operation points, the error
signals are switched to the ones with good shot noise. 

The control signals for the central part can be disturbed by the arm
cavities if one of the RF sidebands accidentally resonates in the arm
cavities. Free hanging mirrors move around and randomly pass by the
resonances of the RF sidebands. This makes the lock acquisition very
difficult and an non-deterministic process. For this reason, we will
pre-lock the arm cavities at a off-resonant position. The pre-lock
position is off-resonant to the carrier because if pre-locked at the
full resonance, a huge increase of the carrier power induces a radiation
pressure thrust to the mirrors when the PRC is locked. To avoid this
shock, the arms are first locked to off-resonance.

After locking the central part, the arm offset is slowly reduced to
bring them to the full resonance. During this process, the error signals
to lock the central part may be affected. Especially, the single
demodulation signals are strongly affected by the large change of the
carrier power and phase. Therefore, these signals are not suitable for
the lock acquisition. We will use either the third harmonics
demodulation signals or the double-demodulation signals with NRS for the
lock of the central part during the lock acquisition.

%}}}

%{{{ \subsection{Green Laser Pre-Lock}
\subsection{Green Laser Pre-Lock}
\label{Green Laser Pre-Lock}

In order to lock the arms at off-resonance of the carrier, we will use 
phase locked green lasers. Two frequency doubled 532\,nm lasers are
used. The seed lasers (1064\,nm) for those green lasers are phase locked
to the main laser carrier with a PLL. Using this PLL, we can sweep the
relative frequency of the green lasers to the main laser.

The arm cavity mirrors are dichroic coated to have some reflectivities
to 532\,nm, forming a low finesse cavity. Each arm cavity is locked to a
green laser by the usual PDH scheme. By sweeping the green laser
frequency relative to the main laser, the resonant condition of the arm
cavities to the main carrier can be changed smoothly.

The injection points of the green lasers are shown in Figure\,\ref{Green Lock Schematic}.
The green laser beams are injected from the back of PR2 (for X-arm) and
SR2 (for Y-arm). PR2 is chosen over PR3 because the beam size at PR2 is
about 4\,mm, which is more manageable compared to 3.5\,cm at PR3.


PR2 and PR3 are dichroic coated to have a good transmittance
to 532\,nm. PR3 and SR3 should have high reflectivities to the green
beam, whereas BS should have a high transmittance. Therefore, the beam
injected from PR2 mainly reaches the X-arm and the one from SR2 sees
only the Y-arm. Of course, the beam separation is not perfect,
especially considering that it is difficult to put high spec coatings
for 532\,nm keeping the coating performance for 1064\,nm.
Since we do not want to compromise on the performance of the coatings
for 1064\,nm, we relaxed the requirements for 532\,nm, so that the
coating company can optimize the coatings mainly for 1064\,nm.
The problem of the mixture of the light coming back from X-arm and Y-arm
can be mitigated by frequency shifting the two green lasers by 100\,MHz
or so. 



\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=13cm]{figures/GreenLock.eps}
\caption{Conceptual configuration of the green laser pre-lock. Green
  lasers are injected from the back of PR2 and SR2. Two green lasers are
  phase locked to the main laser with a frequency offset of about 100\,MHz.}
\label{Green Lock Schematic}
 \end{center}
\end{figure}

%}}}

%{{{ \subsection{Third Harmonics Demodulation}
\subsection{Third Harmonics Demodulation}
\label{Third Harmonics Demodulation}

One way to get error signals of the central part insensitive to the
arm resonance is a method called third harmonics demodulation (THD).
In this scheme REFL port signal is demodulated at the third harmonics
frequencies of the RF sidebands. The signal is produced by the beat
between the second harmonics of the upper (lower) RF sideband and the first
order lower (upper) RF sideband. Since both sidebands are not resonant
to the arm cavities, when arm cavities are sufficiently close to the
carrier resonance, these sidebands are not affected by the arm cavity
motion.

Although the first and second harmonics of the RF sidebands are not
resonant to the arm cavity, there is a contribution from the the third
harmonics of the RF sideband to the THD signal. This is a beat between
the carrier and the third harmonics of the RF sideband. Therefore, there
is still inevitable coupling of CARM and DARM motion to the THD signals.

Figures \ref{THD MICH error signal with various CARM offset} to \ref{THD
SRCL error signal with various CARM offset} show the error signals of the
central part plotted by varying the CARM offset from 2\,nm to 0. The
signals are almost insensitive to the CARM offset. However, MICH signal
is largely affected by CARM when the offset is close to zero.

An advantage of THD is its simplicity. No additional modulator or
Mach-Zehnder is necessary. We can try it by just adding PDs capable of
detecting the third order harmonics. The PD for $3\times f2 = 135$\,MHz
may be challenging. However, since this PD is used only for lock
acquisition, the noise requirement is not severe.

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=10cm]{figures/THD_MICH_CARM_Sweep.pdf}
\caption{THD MICH error signal with various CARM offset. Signal port is
  REFL, demodulated at $3\times f1$ in Q-phase.}
\label{THD MICH error signal with various CARM offset}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=10cm]{figures/THD_PRCL_CARM_Sweep.pdf}
\caption{THD PRCL error signal with various CARM offset. Signal port is
  REFL, demodulated at $3\times f2$ in I-phase.}
\label{THD PRCL error signal with various CARM offset}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=10cm]{figures/THD_SRCL_CARM_Sweep.pdf}
\caption{THD SRCL error signal with various CARM offset. Signal port is
  REFL, demodulated at $3\times f1$ in I-phase.}
\label{THD SRCL error signal with various CARM offset}
 \end{center}
\end{figure}

%}}}

%{{{ \subsection{Non-Resonant Sideband for Lock Acquisition}

\subsection{Non-Resonant Sideband for Lock Acquisition}
\label{Non-Resonant Sideband for Lock Acquisition}

Another way to produce robust signals for the central part during lock
acquisition is the use of a non-resonant sideband (NRS).
The NRS is chosen not to be resonant to any part of the
interferometer. So it serves as a stable local oscillator for signal
generation.

The NRS error signals of the central part with changing CARM offset are
shown in Figures\,\ref{NRS MICH error signal with various CARM offset}
to \ref{NRS SRCL error signal with various CARM offset}. As expected,
the signals are not affected by CARM at all. We used $f3 = 7\times
f_\mr{MC}$ as the NRS frequency for those plots.

The NRS scheme requires additional AM modulator to be introduced in the
modulation stage. To avoid the generation of sub-sidebands, which
interferes with the double demodulation, we also have to use a
Mach-Zehnder to separate the AM path from the PM. This is a disadvantage
of the NRS method. Since the NRS is only used in the lock acquisition
phase, we do not need a large AM. We may also be able to close the AM
path after the interferometer is locked, so that the Mach-Zehnder may
not introduce excess noise to the laser.

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=10cm]{figures/NRS_MICH_CARM_Sweep.pdf}
\caption{NRS MICH error signal with various CARM offset. Signal port is
  REFL, demodulated at $|f3-f1|$ in Q-phase.}
\label{NRS MICH error signal with various CARM offset}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=10cm]{figures/NRS_PRCL_CARM_Sweep.pdf}
\caption{NRS PRCL error signal with various CARM offset. Signal port is
  REFL, demodulated at $|f3-f2|$ in I-phase.}
\label{NRS PRCL error signal with various CARM offset}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=10cm]{figures/NRS_SRCL_CARM_Sweep.pdf}
\caption{NRS SRCL error signal with various CARM offset. Signal port is
  REFL, demodulated at $|f3-f1|$ in I-phase.}
\label{NRS SRCL error signal with various CARM offset}
 \end{center}
\end{figure}

%}}}

%}}}

%{{{ \section{Schedule}
\section{Schedule}
\label{Schedule}

The schedule of the main interferometer subsystem is shown in
Tables\,\ref{Schedule during the excavation and vacuum installation
period},\ref{Schedule: FPMI, RSE1 and RSE2} and  \ref{Schedule: Cryogenic RSE and Observation}.

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=16cm]{figures/Schedule_ExcavationPeriod.pdf}
\caption{Schedule during the excavation and vacuum installation period}
\label{Schedule during the excavation and vacuum installation period}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=16cm]{figures/Schedule_FPMItoRSE2.pdf}
\caption{Schedule: FPMI, RSE1 and RSE2}
\label{Schedule: FPMI, RSE1 and RSE2}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=16cm]{figures/Schedule_CRSEtoObs.pdf}
\caption{Schedule: Cryogenic RSE and Observation}
\label{Schedule: Cryogenic RSE and Observation}
 \end{center}
\end{figure}

%}}}

%{{{ \section{Prototype Test Plan}
\section{Prototype Test Plan}
\label{Prototype Test Plan}
We do not plan to do a prototype experiment for the main
interferometer. However, there have been several experimental tests of
RSE interferometers. Particularly Japanese researchers have been
involved heavily in the experiments at NAOJ and at Caltech 40m lab.

Instead of real experiment, we will use computer simulations of the
interferometer to predict the behavior of the interferometer and develop
suitable control schemes.

Since the interferometer control will mostly be done by the digital
system in LCGT, we can develop and test the interferometer control system
without a real interferometer during the tunnel excavation period.  The
basic idea is to create a simulated plant of the LCGT interferometer in
a computer. This plant accepts usual interferometer inputs, such as
mirror actuator drives, laser frequency control, seismic noise etc. In
response to those inputs, the simulated plant produces sensing
signals. The simulation engine can be a simple linearized model of the
interferometer around the operation point\footnote{Optickle can be used
to generate such a model} or a more complicated time domain simulation
codes, such as e2e.  The simulated plant is connected to the digital
controller to simulate the full closed loop system. The digital control
system developed in this way can be used for the real interferometer by
simply changing the connection to the simulated plant to real ADC/DACs.
This way, we could save some commissioning time.

%}}}

%{{{ \section{First Article Test Plan}
\section{First Article Test Plan}
Commissioning is the first article test for the main interferometer.

%}}}

%{{{ \section{Installation/Adjustment Procedure}
\section{Installation/Adjustment Procedure}

The physical installation of the main interferometer components will be done
by other subsystems, such as suspension and mirror. After the initial
installation of the mirrors, the distance between them (arm length,
PRCL, SRCL, Schnupp asymmetry) have to be checked.

The cavity lengths are checked by measuring the FSR of the cavities. The
Schnupp asymmetry can be measured by locking the arms one by one using
the REFL port PDH signal, and measuring the difference of the optimal demodulation
phases. The g-factor of the arm cavities can be measured by injecting a
slightly mis-aligned secondary laser and check the frequency separation
between resonances of the TEM00 mode and the TEM10 or TEM01 modes.
The finesse of the arm cavities must also be measured.

After those measurements, the MC length should be fine adjusted to set
the RF sideband frequencies at a desirable location in the FSR of the
arm cavities. Then the length of PRC and SRC will be adjusted to
resonate these sidebands. The ROC error of the PR3 (SR3) also has to be compensated
by adjusting the distance between PR2 and PR3 (SR2 and SR3), keeping the overall PRC
(SRC) length the same.

Commissioning is almost a synonym of adjustments to the
interferometer. Therefore, the whole commissioning process is the
adjustment process.

%}}}

%{{{ \section{Risk assessment}
\section{Risk assessment}
\begin{itemize}
 \item The arm cavity loss may be higher. In this case, the PRC becomes
       under-coupled. This is not acceptable. We may have to prepare
       several PRMs with slightly different reflectivities.
 \item The arm cavity loss might be smaller than expected. In this case,
       the carrier light power at REFL will increase. This will result
       in the increased shot noise for double demodulation and THD
       signals. However, as long as we use single demodulation signals
       for observation, this should not be a problem.
 \item The reflectivity mismatch of the two arm cavities may be too
       large. In this case, we have to add a large DARM offset to create
       strong enough DC offset light at the AS port and make the
       homodyne phase to the desired value.
 \item The non-linearity of the SRC error signal when detuned may produce
       unwanted up converted noise. Although the up conversion noise was
       estimated to be not a problem from the predicted seismic noise
       and shot noise, it may still create wide side lobes around sharp
       noise lines.
 \item By detuning the SRC, the f1 sideband gets some phase
       rotation. This means the f1 no longer acts as a pure PM for the
       carrier. The result is large offsets in MICH and CARM (if f1 is
       used for CARM). This offset (in the order of nm for MICH and
       10\,pm for CARM) has to be canceled by electric offset.
 \item The design of the output mode cleaner has not yet studied in
       detail. Especially, the control schemes (length and alignment)
       have not decided yet.
 \item In the current folding design, the mode of the PRC is very
       sensitive to the ROC of PR3. We may have to change the distance
       between PR2 and PR3 by more than 10\,cm if the ROC error of PR3
       is large (about 1\%).



\end{itemize}
%}}}

%}}}

%{{{ \chapter{Design for iLCGT}
\chapter{Design for iLCGT}
%{{{ \section{Definition and scope of the Main Interferometer subsystem}
\section{Definition and scope of the Main Interferometer subsystem}

The definition of the subsystem is the same as bLCGT.

%}}}

%{{{ \section{Requirements}
\section{Requirements}
The requirement for the iLCGT main interferometer is to be able to
operate a 3\,km Fabry-Perot Michelson interferometer stably. The
sensitivity is not specified as a requirement.

%}}}

%{{{ \section{Interface}
\section{Interface}
Interface items of iLCGT are almost the same as bLCGT.
The only significant difference is the wedge angle of the ITMs.
The requirements to the ITMs are listed in Table\,\ref{Input Test Mass Requirements for iLCGT}.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|p{13em}|c|p{17em}|}
\hline
   {\bf Parameter Name} &{\bf Value} &{\bf Comments} \\\noalign{\hrule height 0.8pt}
   HR Reflectivity@1064\,nm &99.6$\pm$0.01\%& Power reflectivity\\\hline
   HR Reflectivity Mismatch @1064\,nm &$<$0.05\%&
	   $|R_\mr{ITMX}-R_\mr{ITMY}|$\\\hline
HR Reflectivity@532\,nm &80$\pm$10\%&ITM reflectivity must be smaller than ETM\\\hline
   HR loss@1064\,nm &$<$45\,ppm&Including diffraction and absorption \\\hline
   HR loss@532\,nm &$<$1\%&Including diffraction and absorption \\\hline
   AR Reflectivity@1064\,nm &200(+100/-0)\,ppm& \\\hline
   AR Reflectivity@532\,nm &$<$5\%& \\\hline
   HR ROC&$>$500\,km& As large as possible\\\hline
   HR ROC Mismatch between ITMX and ITMY&$<$10\,km& \\\hline
   AR ROC&$>$500\,km& As large as possible\\\hline
   AR Wedge Angle&0.335$\pm$0.002\,deg&\\\hline
  \end{tabular}
\caption{Input Test Mass Requirements for iLCGT}
\label{Input Test Mass Requirements for iLCGT}
 \end{center}
\end{table}

 %}}}

%{{{ \section{Optical Configuration}
\section{Optical Configuration}

The optical configuration of iLCGT is a Fabry-Perot Michelson
interferometer without recycling. Since iLCGT is on the pathway to
bLCGT, the optical parameters are mostly the same as bLCGT. The PRM is
replaced with a mirror blank with the HR coating replaced by
AR. The mirror blank has the same ROC as the PRM of bLCGT to
keep the spatial mode matching from the MC to the PRC.
SRM will not be installed at this stage.

The arm cavity mirrors are made of fused silica instead of sapphire.
All the other parameters are exactly the same as the bLCGT design.


%}}}

%{{{ \section{Optical Layout}
\section{Optical Layout}

The optical layout of iLCGT is also generated by the python code.
Since the ITM substrate is silica in iLCGT, the index of refraction is
different from the sapphire mirror of bLCGT. To compensate for this, the
wedge angle of the ITMs for iLCGT is different from the ones for bLCGT. 

%}}}

%{{{ \section{Length Sensing and Control Scheme}
\section{Length Sensing and Control Scheme}
All the degrees of freedom (DARM, CARM, MICH) are sensed by RF readout schemes.
The sensing matrix is shown in Table\,\ref{Sensing Matrix LSC iLCGT}.
DC readout for DARM may be tested with iLCGT, but it is not the default
plan.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c@{\ \vrule width 0.8pt\ }c|c|c|}
\hline
&   {\bf DARM} &{\bf CARM} &{\bf MICH} \\\noalign{\hrule height 0.8pt}
   & & & \\[-10pt] 
{\bf AS\_1I}&1&$1.7\times10^{-3}$&$1.0\times10^{-3}$\\\hline
{\bf
   REFL\_1I}&$8.8\times10^{-3}$&1&$1.3\times10^{-4}$\\\hline
{\bf REFL\_1Q}&$4.5\times10^{-3}$&$5.5\times10^{-5}$&1\\\hline
  \end{tabular}
\caption{Normalized Sensing Matrix of LSC for iLCGT. Each row is normalized by
  the diagonal element. The interferometer response was evaluated at
  100\,Hz to create this matrix.}
\label{Sensing Matrix LSC iLCGT}
 \end{center}
\end{table}

%}}}

%{{{ \section{Alignment Sensing and Control Scheme}
\section{Alignment Sensing and Control Scheme}
We will use WFS for the alignment sensing of iLCGT. The sensing matrix
is shown in Table\,\ref{Sensing Matrix ASC iLCGT}. It can be
diagonalized as shown in Table\,\ref{Sensing Matrix ASC iLCGT Diagonal}.
Note that this diagonalization does not take into account the shot noise
of different ports. In general, when taking a linear combination of
signals from various ports, we should avoid using ports with poor shot
noise, even at the expense of some signal mixing. This is still in our TO DO list.


\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c@{\ \vrule width 0.8pt\ }c|c|c|c|}
\hline
&   {\bf CSOFT} &{\bf CHARD} &{\bf DSOFT} &{\bf DHARD}\\\noalign{\hrule height 0.8pt}
   & & & & \\[-10pt] 
{\bf REFL\_2IB}&$-6.1\times10^{1}$&$-2.8$&$-1.5\times10^{-3}$&$1.1\times10^{-2}$\\\hline
{\bf
   REFL\_2IA}&$-3.2$&$-2.2\times10^{1}$&$-2.9\times10^{-2}$&$-1.4\times10^{-3}$\\\hline
{\bf
   AS\_1QB}&$-8.0\times10^{-2}$&$-1.3\times10^{-2}$&$2.7\times10^{4}$&$4.4\times10^{3}$\\\hline

{\bf REFL\_1QA}&$2.7\times10^{-3}$&$-4.7\times10^{-4}$&$-1.4\times10^{1}$&$1.8\times10^{1}$\\\hline

  \end{tabular}
\caption{Sensing Matrix of ASC for iLCGT [W/rad]. }
\label{Sensing Matrix ASC iLCGT}
 \end{center}
\end{table}

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c@{\ \vrule width 0.8pt\ }c|c|c|c|}
\hline
&   {\bf CSOFT} &{\bf CHARD} &{\bf DSOFT} &{\bf DHARD}\\\noalign{\hrule height 0.8pt}
   & & & & \\[-10pt] 
{\bf REFL\_2IB}&1&0&$-8.7\times10^{-5}$&$-1.8\times10^{-4}$\\\hline
{\bf
   REFL\_2IA}&0&1&$1.3\times10^{-3}$&$8.7\times10^{-5}$\\\hline
{\bf
   AS\_1QB}&$-2.5\times10^{-5}$&$3.4\times10^{-6}$&1&0\\\hline

{\bf REFL\_1QA}&$1.3\times10^{-4}$&$-2.4\times10^{-5}$&0&1\\\hline

  \end{tabular}
\caption{Diagonalized Sensing Matrix of ASC for iLCGT. The each row is
  normalized by the diagonal element. }
\label{Sensing Matrix ASC iLCGT Diagonal}
 \end{center}
\end{table}

%}}}

%{{{ \section{Lock Acquisition Scheme}
\section{Lock Acquisition Scheme}
We plan to test the green laser pre-lock in iLCGT.

%}}}

%{{{ \section{Schedule}
\section{Schedule}
See section\,\ref{Schedule}.
%}}}

%{{{ \section{Prototype Test Plan}
\section{Prototype Test Plan}
See section\,\ref{Prototype Test Plan}.
%}}}


%{{{ \section{First Article Test Plan}
\section{First Article Test Plan}
Commissioning is the first article test for the main interferometer.
%}}}

%{{{ \section{Installation/Adjustment Procedure}
\section{Installation/Adjustment Procedure}
%}}}

%{{{ \section{Risk assessment}
\section{Risk assessment}
\begin{itemize}
 \item Commissioning time is extremely limited.
\end{itemize}
%}}}

%}}}

%{{{ Appendix
\appendix

%{{{ \chapter{Reasoning Behind Parameter Choices}
\chapter{Reasoning Behind Parameter Choices}
\label{Reasoning Behind Parameter Choices}

This section is to be written.
%}}}

%{{{ \section{Arm Cavity Parameters}

\section{Arm Cavity Parameters}
\label{Apendix: Arm Cavity Parameters}
%}}}

%{{{ \section{Recycling Cavity Length}
\section{Recycling Cavity Length}
\label{Apendix: Recycling Cavity Length}


%}}}

%{{{ \section{Length Sensing Scheme}
\section{Length Sensing Scheme}
\label{Apendix: Length Sensing Scheme}

\subsection{Alternative Schemes}

%}}}


\chapter{Terminology}
\label{Appendix Terminology}
\begin{table}[hp]
 \begin{center}
\caption{Terminology}
\label{Terminology Table}
  \begin{tabular}{|p{7em}|c|p{15em}|}
\hline
   DARM&Differential Arm Length& \\\hline
   CARM&Common Arm Length&\\\hline
   MICH&Michelson Part&L shaped part of the interferometer formed by BS
	   and two ITMs\\\hline
   PRC&Power Recycling Cavity&\\\hline
   PRCL&Power Recycling Cavity Length&\\\hline
   SRC&Signal Recycling Cavity&\\\hline
   SRCL&Signal Recycling Cavity Length&\\\hline
   AS&Anti-symmetric port& \\ \hline
   REFL&Reflection port& \\ \hline
   POP&Pick-off in the Power Recycling Cavity & \\ \hline
   POX&Pick-off at the ITMX & \\ \hline
   POY&Pick-off at the ITMY & \\ \hline
   Auxiliary DOF& &Length degrees of freedom other than DARM \\ \hline
   AM&Amplitude Modulation & \\ \hline
   PM&Phase Modulation & \\ \hline
   MC&Mode Cleaner & \\ \hline
   MZ&Mach-Zehnder & \\ \hline
  \end{tabular}
 \end{center}
\end{table}

\chapter{Contributors}

Followings are the people contributed to the discussion of the main
interferometer design.

Yoichi Aso (chair), Kentaro Somiya, Osamu Miyakawa, Yuta Michimura,
Kazuhiro Agatsuma, Erina Nishida, Chen Dan, Daisuke Tatsumi, Tomotada
Akutsu, Kiwamu Izumi, Koji Arai, Kazuhiro Yamamoto, Hiroaki Yamamoto, Masaki Ando.

%}}}

%{{{ References

\begin{thebibliography}{00}
 \bibitem{LCGT svn layout}
	 https://granite.phys.s.u-tokyo.ac.jp/svn/LCGT/trunk/mif/OptLayout/
 \bibitem{Optickle} M. Evans, LIGO Document T070260.
 \bibitem{LSC Code}
	 https://granite.phys.s.u-tokyo.ac.jp/svn/LCGT/trunk/mif/IFOmodel
 \bibitem{LoopNoise}K. Somiya and O. Miyakawa, {\it Applied Optics},
	 {\bf 23} (2010) 4335 
 \bibitem{Sidles-Sigg} J. Sidles and D. Sigg, {\it Phys. Lett. A}, {\bf
	 354} (2006) 167
\end{thebibliography}
\end{document}


%}}}
