
\newpage

\section{Optical parameter candidates}

LCGT has an optical configuration of Resonant-Sideband Extraction (RSE).
RSE configuration has been adopted because of its advantages in capability 
of high laser power in arm cavities and flexibility in observation band.
In RSE configuration, we have choices in several optical parameters, 
such as the observation frequency band, the finesse of the arm cavities, and so on.
In this section, we review the baseline design of LCGT with a RSE configuration, 
and define three candidate optical parameter sets, which are evaluated in the following sections.



\begin{figure}[b]
\begin{center}
 \includegraphics*[width=15cm]{ifo_config.eps}
 \caption{Various interferometer configurations:
a dual-recycled Michelson (DRMI) configuration as GEO,
a dual-recycled Fabry-Perot-Michelson (DRFPMI) configuration, 
a power-recycled Fabry-Perot-Michelson (PRFPMI) configuration as LIGO, VIRGO, and TAMA, 
RSE configuration,
and an extreme RSE (ExRSE) configuration which is a RSE configuration without a power recycling mirror.}
\label{fig-comp}
\end{center}
\end{figure}

\subsection{Basic configuration and constraints}

The basic concept of LCGT is (1) to have a baseline length of 3\,km,
(2) with cryogenic test-mass mirrors to suppress thermal noises, and 
(3) placed at Kamioka underground site for smaller seismic disturbances.
The optical configuration of LCGT is designed starting from these boundary conditions. 

%Main parameters of LCGT is summarized in Tab.\,\ref{tab-parameters}.
LCGT has an optical configuration of Resonant-Sideband Extraction (RSE).
RSE configuration has adopted because of its advantages in capability 
of high laser power in arm cavities and flexibility in observation band
(Figs.\,\ref{fig-comp}, \ref{fig-RSE} and Table\,\ref{tab-comp}).
So as to enhance interaction with incident gravitational waves (GWs), 
and thus, to increase the detector sensitivity to GWs,
an interferometric detector should have high laser power in its arm cavities.
In a practical detector, the laser power in the arm cavities
is limited by optical losses in the interferometer with a given input laser power.
An RSE configuration minimizes the optical losses originate in absorption in substrates
of optical components (input test masses and the beam splitter), and contrast defect 
in recombination of beams from two arms on the beamsplitter, 
by storing most of the laser beam in the arm cavities.
Absorption in substrates of input test masses is also a critical issue in LCGT;
too much transmission light through the input test mass prevent us to cool it down
to cryogenic temperature of 20\,K%
\footnote{Capability of the cryocooler for one input test mass is 1\,W.
Assuming that heat absorption of sapphire substrate is 20\,ppm/cm, transmission power
through the input test mass (ITM) is required to be less than 1.7\,kW (without any safety margin).}.
An RSE configuration enables high power in arm cavities with moderate absorption in ITM.

An RSE configuration has another advantage in flexible observation frequency band.
Incident GWs would generate signal sideband components by modulating the laser beam 
stored in the arm cavities.
The signal-extraction cavity (SEC), which is formed by the ITMs and the signal-extraction 
mirror (SEM) at the detection port of the interferometer, extract these signal sidebands from
the arm cavities. 
By adjusting the signal-band gain (determined by the finesse of the SEC) and the detuning phase
(determined by the round-trip phase of the signal sidebands in the SEC), 
the observation frequency bandwidth and center frequency can be tuned for 
target gravitational waves. 
Figure\,\ref{fig-comp} and Table\,\ref{tab-comp} shows comparison of feasibility to realize high sensitivity 
for several optical configurations.
From these considerations, we selected RSE as the baseline 
interferometer configuration of LCGT.



\begin{table}[t]
\begin{center}
\caption{Comparison between possible optical configurations for LCGT.
This table shows technical feasibility scores; score of 5 represents most feasible.
LCGT will adopt a RSE configuration as a baseline design.}
\label{tab-comp}
%\footnotesize %\vspace*{1mm}
\small
\begin{tabular}{l c c l l l}
\hline 
  Config. & BS Pow$^{1)}$ & ITM Pow$^{2)}$ &  Obs. Band & Control & Special requirements  \\
\hline 
  DRMI      & 1       & ---     & 5 (Tunable) & 5 (3 DoF)  & Cryogenic BS   \\
  DRFPMI     & 2       & 2       & 5 (Tunable)   & 4 (5 DoF)  & High power recycling \\
  PRFPMI     & 3       & 3       & 3 (Fixed)   & 5 (4 DoF)  & Powerful cryocooler \\
  {\bf RSE} & {\bf 4} & {\bf 4} & {\bf 5 (Tunable)} & {\bf 4 (5 DoF)} \\
  ExRSE     & 5       & 5       & 5 (Tunable) & 3 (4 DoF)& High optical gains \\
\hline
\multicolumn{4}{l}{$^{1)}$ Power on the beamsplitter} \\
\multicolumn{4}{l}{$^{2)}$ Power transmit through a ITM (Input Test Mass)} 
\end{tabular}
\end{center}
\end{table}





In an RSE configuration, we have choices in several optical parameters:
%Here the setup includes 
%laser power $I_0$, 
readout phase $\zeta$, ITM power transmittance $T$ (or finesse ${\cal F}$), detune phase $\phi$ of SEC, 
and SEM power transmittance $T_s$. 
The optimal ${\cal F}$ and $T_s$ to maximize the detector sensitivity are quite different 
between a broadband RSE (BRSE, $\phi=90$~deg) and a detuned RSE 
(DRSE, $\phi$ is optimized for target GW sources) configurations. 
As it is not easy to replace the mirrors during the observation, 
we should choose the right setup from the optimized BRSE, optimized DRSE, 
or one (VRSE) in the middle that performs 
well in both configurations by changing $\phi$.
In the following parts, we define the candidates for the interferometer setup of LCGT, 
referring mainly to its observable distance for neutron star inspirals.


\subsection{Broad band configuration}

At first, we estimate the detector sensitivity with a broadband RSE configuration.
Table~\ref{tab-parameters}, \ref{table:classic} shows the mechanical and optical parameters of LCGT 
that are used to calculate its sensitivity%
\footnote{Some of them, for example the beam radius or the suspension wire length, could be modified to reduce thermal noise, 
but there is a counterpart and the reduction is not so remarkable. }.
Thermal noise levels are calculated using 
those well-known equations from Ref.\,{\cite{Levin,Harry,Gaby}} that are used for other 
gravitational-wave detectors, except for substrate thermoelastic noise and coating thermo-optic noise, 
the equations of which are different in the low temperature~\cite{LCGTdocYamamoto}. The readout scheme 
is set to be DC readout while RF readout is still an option for the BRSE operation. The readout phase $\zeta$ 
can be tuned by changing amount of the offset in the arm cavity. Here we follow the convention used in Ref.\,\cite{BC}; 
90\,deg is the phase quadrature for BRSE, where the signal response is maximized. 
The laser power in the power-recycling cavity (PRC) is set 825\,W 
(75\,W input power to the interferometer and with a power-recycling gain of 11). 
%The cryogenic cooling can absorb the heat of 
%1\,W per mirror and it limits the laser power on the ITM to be 825\,W or less. 
%This is one of the remarkable differences from other second-generation detectors in choosing the setup.


\begin{figure}[p]
\begin{center}
 \includegraphics*[width=12cm]{RSE_ifo.eps}
 \caption{Baseline interferometer configuration of LCGT: a RSE configuration.}
 \label{fig-RSE}
\end{center}
\end{figure}

\begin{table}[p]
\begin{center}
\caption{Main parameters of LCGT}
\label{tab-parameters}
%\footnotesize %\vspace*{1mm}
\small
\begin{tabular}{l l l l}
\hline 
  & \multicolumn{1}{l}{Item} & Parameters  \\
\hline 
  \multicolumn{2}{l}{Interferometer} \\
   & Optical configuration & Resonant-Sideband Extraction (RSE) \\
   & Baseline Length	& 	3\,km 	\\
   & Input Laser power  & 75\,W (power of carrier beam) \\
   & Beam spot size     & 30\,mm \\
 \hline 
  \multicolumn{2}{l}{Mirror} \\
  & Substrate & Sapphire, Optical absorption 20\,ppm/cm \\
  & Temperature & 20\,K  \\ 
  & Mechanical loss  & $1\times10^{-8}$ \\
  & Dimensions & Mass 30\,kg,  Radius 125\,mm, Thickness 150\,mm \\
\hline
  \multicolumn{2}{l}{Main Suspension} \\
  & Material  & Sapphire fibers \\ 
  & Number of fibers & 4 \\
  & Temperature & 16\,K  \\
  & Mechanical loss  & $2\times10^{-7}$ \\
  & Fiber Dimensions  & Length 400\,mm, Diameter 1.8\,mm   \\
\hline
\end{tabular}
\end{center}
\end{table}


\begin{table}[p]
\begin{center}
\caption{Parameters assumed in the sensitivity estimation.}
\label{table:classic}
%\footnotesize %\vspace*{1mm}
\small
\begin{tabular}{l l l  l l l}
\hline 
  & \multicolumn{1}{l}{Item} & Parameters  \\
\hline 
  \multicolumn{2}{l}{Mirror coating} \\
   & Silica loss & $ 1\times 10^{-4} $  \\
   & Tantalum loss& $ 4 \times 10^{-4} $ \\
   & Number of coating layers& 9/18 \\
  \multicolumn{2}{l}{Optical losses} \\
   & Optical loss of a mirror&20~ppm \\
   & ETM transmittance&30~ppm\\
   & Optical loss in SRC&2~\% \\
   & Quantum efficiency&90~\% \\
\hline
\end{tabular}
\end{center}
\end{table}




\begin{figure}[htbp]
\begin{center}
 \includegraphics*[height=8cm]{sensitivity.eps}
%\hspace{0.2cm}
% \includegraphics*[height=10cm]{BRSE-DRSE.eps}
\caption{Calculated noise budget of LCGT with the BRSE 
configuration, the parameters of which is chosen to obtain the highest 
inspiral range without detuning. The readout phase is set 90~deg.}
\label{fig:sensitivity1}
\end{center}
\end{figure}


%\begin{table}[b]
%\begin{center}
%\begin{tabularx}{12cm}{|c|c||c|c|}\hhline{|*{4}{-}|}
%mirror radius&12.5~cm&mirror thickness&15~cm\\ \hhline{|*{4}{-}|}
%mirror temperature&20~K&sapphire (bulk) loss&1e-8\\ \hhline{|*{4}{-}|}
%beam radius&3~cm&number of coating layers&9/18\\ \hhline{|*{4}{-}|}
%silica loss (coating)&1e-4&tantala loss&4e-4\\ \hhline{|*{4}{-}|}
%number of wires&4&sapphire (suspension) loss&2e-7\\ \hhline{|*{4}{-}|}
%wire length&40~cm&wire diamter&1.8~mm\\ \hhline{|*{4}{-}|}
%wire temperature&16~K&laser power in PRC&825~W\\ \hhline{|*{4}{-}|}
%optical loss of a mirror&20~ppm&ETM transmittance&30~ppm\\ \hhline{|*{4}{-}|}
%optical loss in SRC&2~\%&quantum efficiency&90~\%\\ \hhline{|*{4}{-}|}
%\end{tabularx}
%\caption{Parameters that are used to derive LCGT sensitivity.}\label{table:classic}
%\end{center}
%\end{table}

%The left panel of 
Figure\,\ref{fig:sensitivity1} shows the estimated noise budget of LCGT with a broadband configuration:
%Here the setup is BRSE; 
${\cal F}=1550$ and $T_s=0.23$, which gives the highest inspiral range without 
detuning under the condition that the finesse be 1550 or less%
\footnote{Higher finesse will have difficulty in lock acquisition of the cavity.  }.
The readout phase is set to be 90 deg. 
We can see that the sensitivity is not limited by classical noise at most of the frequencies, 
thanks to low seismic noise in underground and low thermal noise with the cryogenic cooling. 


\begin{figure}[htbp]
\begin{center}
% \includegraphics*[height=5.5cm]{sensitivity.eps}
%\hspace{0.2cm}
 \includegraphics*[height=6cm]{BRSE-DRSE.eps}
 \caption{Inspiral range with different finesse (represented by the 
ITM power transmittance $T$) and SRM power transmittance $T_s$. Signal-
to-noise ratio is set 8 and the optimal direction (azimuthal incident 
and best polarization) is assumed. The readout phase for both BRSE and 
DRSE, and also the detune phase for DRSE is chosen to obtain the highest 
inspiral range with the given $T$ and $T_s$. The green crosses indicate 
the inspiral ranges of BRSE and DRSE with the setup we finally chose.
 \label{fig:sensitivity2}}
\end{center}
\end{figure}


\subsection{Detuned configuration}

With detuning, quantum noise can be reduced to raise the inspiral range 
(IR, the observable range for neutron-star binaries).
Figure\,~\ref{fig:sensitivity2} shows inspiral range of BRSE and DRSE with different ${\cal F}$ and $T_s$. 
The detune phase of DRSE is chosen to maximize the inspiral range for each ${\cal F}$ and $T_s$.
 For BRSE, the more the finesse, the higher the range; $T_s$ is to be chosen to optimize the bandwidth for each finesse. 
 For DRSE, decent finesse is rather desirable since there is more room to reduce quantum noise with the optical spring. 
 The highest region of DRSE plots could be damped to be flat without the reduction of thermal noise. 
 The detuning is effective thanks to the cryogenic cooling.

The highest inspiral range is realized with ${\cal F}=690$ and $T_s=0.08$.
The estimated sensitivity curve is shown as the green curve in Fig.\,\ref{fig:candidates}.
The sensitivity curve of this setup is so narrow-band that gravitational-wave signals 
from other targets like supernovae or LMXB will be hard to observe
(detailed in Section\,4). 


\subsection{Variable tuning configuration}

As is proposed for Advanced LIGO~\cite{AdLIGOcdr}, choosing a setup not to maximize 
the inspiral range for either BRSE or DRSE alone but to make the best use of both configurations 
would be a reasonable way for LCGT. 
In fact, there are many choices in parameter selection. 
The importance of the inspiral range would not be equal between the BRSE and DRSE configurations. 
%It would be also a key point if the inspiral range is comparable to other second-generation detectors.

Taking into account a wide spectrum of factors, we finally picked up a set of parameters for the variable detuning: 
${\cal F}=1550$ with $T_s=0.12$. 
One can see that the inspiral range for DRSE does not depend much on the selection of 
$T$ and $T_s$ while it does for BRSE, which is one reason to choose parameters close to the best for BRSE. 
In addition, the sensitivity at high frequencies with DRSE can be better if the finesse is higher. 
As is shown in Fig.~\ref{fig:candidates}, the VRSE(D) sensitivity is almost as broad as the BRSE sensitivity. 
The configuration will be chosen between VRSE(D) that is operated with $\phi=86.5$~deg and $\zeta=134.7$~deg, 
and VRSE(B) that is operated with $\zeta=121.8$~deg to maximize the inspiral range% 
\footnote{There are the other options to optimize the detune phase and the readout phase for a particular 
kind of gravitational-wave source. 
For example, the inspiral range for the 60 solar-mass black-hole binaries can be 30~\% higher than with the 
VRSE (D) configuration if we lower the power to 10~W and use $\phi=80$~deg and $\zeta=140$~deg. 
It will be also an option to optimize the phases for an intermediate operation like a low-power operation 
or a room-temperature operation. Considering these options, we should better prepare for the phase parameters 
as tunable as possible, while it brings more control noise that may limit the sensitivity. 
%We shall discuss this issue more in Sec.~\ref{sec:control}
}.


\subsection{Summary of candidate parameter sets}

Table~\ref{table:summary_candidates} 
shows brief summary of the setups. In the following sections, we will compare these 
3 candidates from various points of view to determine the interferometer configuration of LCGT.

\begin{figure}[t]
\begin{center}
 \includegraphics*[height=6.5cm]{candidates.eps}
 \caption{Sensitivity curves of broadband RSE (BRSE), detuned RSE (DRSE), and variable RSE (VRSE). 
 The setup parameters are shown in Table~\ref{table:summary_candidates}.\label{fig:candidates}}
\end{center}
\end{figure}




\begin{table}[hbtp]
\caption{Setup parameters, inspiral range (IR), and strain sensitivity at 1\,kHz
of each configuration.
The first number in each IR column is with the assumption that wave come 
from the optimal direction for LCGT, and each number in parenthesis is
with the average over whole sky.
\label{table:summary_candidates}}
\begin{center}
\begin{tabularx}{13cm}{ l l c c c c c }\hhline{|*{7}{-}|}
   &$T$ (Finesse)&$T_s$&$\phi$&$\zeta$&IR& Sensitivity\\ % \hhline{|*{7}{-}|}
    &             &      & [deg] & [deg] & [Mpc] & [${\rm Hz}^{-1/2}$] \\
\hhline{|*{7}{-}|}
BRSE&0.004 (1550)&0.23&90&127.6& 259 (114) & $1.4 \times 10^{-23} $ \\ \hhline{|*{7}{-}|}
DRSE&0.009 (690)&0.08&74.6&103.8& 299 (132)  & $2.1\times 10^{-22} $\\ \hhline{|*{7}{-}|}
VRSE (B)&0.004 (1550)&0.15&90&121.8& 255 (112)  & $1.1 \times 10^{-23} $\\ % \hhline{|*{7}{-}|}
VRSE (D)&0.004 (1550)&0.15&86.5&134.7& 281 (123) & $1.5 \times 10^{-23}$ \\ \hhline{|*{7}{-}|}
\end{tabularx}
\end{center}
\end{table}


\clearpage

%\end{document}

