\newpage

\section{Technical Feasibility}

In this section, we examine and compare the technical feasibility 
to realize the target sensitivities with candidate interferometer configurations:
fixed BRSE (denoted BRSE), fixed DRSE (denoted DRSE), variable RSE operated 
in broadband (denoted VRSE(B)) and variable RSE operated detuned (denoted VRSE(D)).
For concise comparison of the technical feasibility, the four configurations are
compared for above mentioned aspects with  scores (Table\,\ref{Table-tech_feas}). 
The scale is from 1 to 10. Larger number means the feasibility is high or the
problem is less serious. The scores are normalized column wise, i.e. the
best configuration(s) in a column are always given 10.


\begin{table}[bp]
\caption{Comparison of the technical feasibility. BRSE: fixed BRSE,
  DRSE: fixed DRSE, VRSE(B): variable RSE operated in broadband,
  VRSE(D): variable RSE operated detuned. The scores are in
  the scale from 1 to 10. Larger number is better.}
\label{Table-tech_feas}
 \begin{center}
  \begin{tabular}{ c c c c c }\hline
   & Control &Mirror loss &Calibration & Commissioning time  \\\hline
 BRSE     &10 &7  &10 &10   \\ % \hline
 DRSE     &6  &10 &7  &9   \\ % \hline
 VRSE(B)  &9  &7  &9 &10   \\ % \hline
 VRSE(D)  &8.5  &7  &8  &9  \\  \hline
  \end{tabular}
 \end{center}
\end{table}

\subsection{Interferometer control}

In order to keep an interferometer at its operation point and its best sensitivity,
the length between the suspended mirrors must be controlled with precision much 
better than the laser wavelength.
In an RSE interferometer configuration, 5 degrees of length freedom (DoF) should be controlled:
$ L_-$, $ L_+$ (differential and common length changes in arm cavities),
$ l_-$, $ l_+$ (differential and common changes in the lengths between the beamsplitter and input test masses), and
$ l_{\rm s}$ (length of SEC, the signal-extraction cavity).
Controlling those degrees of freedom involves two key issues:
The first one is the length sensing, which means extraction of length-change information 
(error signals) from the interferometer using modulation-demodulation schemes.
Signals with good separation from other DoF, and with good signal-to-noise ratio 
(SNR, mainly limited by shot noises) are required.
The second issue is control noises.
The signals for the central part of the interferometer 
($ l_-$, $ l_+$, and $ l_{\rm s}$) have worse SNR in general,
because of smaller optical enhancement in these DoF with usual
interferometer design.
Thus, poor design of control loops for these DoF would spoil the interferometer's 
sensitivity for GWs in the $ L_-$ DoF.


\vspace*{2mm}
\noindent
{\bf Length sensing}

The frontal modulation scheme is widely used in the current interferometric
gravitational wave detectors. 
It has a relatively simple optical configuration requiring only a few modulators 
in front of the interferometer%
\footnote{
There are two candidates for the sensing scheme of the LCGT. One is a
variant of the conventional frontal modulation scheme. The other one is
called sub-carrier injection method. In this working group, we focused
mostly on the frontal modulation scheme. Therefore, the technical
feasibility is discussed assuming the use of this sensing scheme in
this section. The basic ideas of the sub-carrier injection technique are
explained in Section\,\ref{sec-sc}.}.
In order to extract signals for 5 length DoF control, 
the incident laser beam (the carrier beam) is modulated 
at two frequencies in its phase (phase modulation, PM) 
and amplitude (amplitude modulation, AM);
the lower-frequency one is called $ f_1$,
and the higher one is called $ f_2 $.
These frequencies are selected by resonance conditions of these modulation-introduced
sidebands in the interferometer and other constraints%
\footnote{For sufficient response of the photo detectors,
the modulation frequency should be roughly less than 50\,MHz.
In addition, so as to reject the junk lights from modulators,
a mode cleaner (MC) should be placed between the modulators and the main interferometer;
the modulation sidebands have to pass through the MC.
Thus, the choice of modulation frequencies are limited to discrete numbers,
depending on the FSR (free-spectrum range) which is inverse proportional to the MC cavity length.};
$f_1$ sideband is supposed to resonate in the SEC (signal-extraction cavity),
and $f_1$ sideband is supposed to resonate only in PRC (power-recycling cavity).
Though, there are wide range of selections in these frequencies and interferometer
length parameters, we choose the parameter set%
\footnote{Though better parameter sets may be found for each configuration, 
here we compare the optical configurations with the same control parameters for the simplicity.}
shown in Table\,\ref{tb-param0}.
We assumed a phase modulation\,-\,phase modulation\,(PM-PM) pair for DRSE and
phase modulation\,-\,amplitude modulation\,(PM-AM) for BRSE as the first
and second modulations.
With these modulations and signal ports shown in Table\,\ref{tb-param0},
error signals with sufficient separation and SNR can be extracted
(details are described in Section \ref{sec-ctrl}). 

\begin{table}[tbp]
\caption{Default parameters assumed for the length control investigations.
Signal ports are described as AS for the anti-symmetric port, REFL for the reflection port, and PO for a pick-off port.
DC and DDM represent homodyne signal detection and double demodulation, respectively.}
\label{tb-param0}
\begin{center}
\begin{tabular}{l l |c |c}
\hline
Item & & BRSE & DRSE \\

\hline
Modulation frequency & & \multicolumn{2}{l}{ $f_1=11.25$\,MHz, $f_2=45$\,MHz}  \\
Cavity length        & & \multicolumn{2}{l}{PRC 73.3\,m,  SEC 73.3\,m} \\
Length of MC         & & \multicolumn{2}{l}{13.3\,m (or its integer multiples)} \\
\hline 
Signal extraction & \\
  \multicolumn{2}{c|}{$ L_- $}  & \multicolumn{2}{c}{AS DC} \\ 
  \multicolumn{2}{c|}{$ L_+ $}  & \multicolumn{2}{c}{REFL $f_2$} \\ 
  \multicolumn{2}{c|}{$ l_- $}  & \multicolumn{2}{c}{AS DDM} \\ 
  \multicolumn{2}{c|}{$ l_+ $}  & PO $ f_1$ & PO $ f_2$ \\ 
  \multicolumn{2}{c|}{$ l_{\rm s} $}  & REFL DDM & PO $f_1$ \\ 
Modulation type & & $f_1$-PM, $f_2$-AM  & $f_1$-PM, $f_2$-PM \\
\hline 
Control bandwidth$^*$ & & \multicolumn{2}{l}{$L_-$: 200\,Hz, $L_+$: 30\,kHz}   \\
                  & & \multicolumn{2}{l}{$l_+$, $l_-$, and $l_{\rm s}$: 50\,Hz} \\
Feed forward gain & & \multicolumn{2}{l}{$l_+$, $l_-$, and $l_{\rm s}$: 30} \\
\hline
\multicolumn{4}{l}{$^*$Simple open loop gain of $f^{-1}$ is assumed.}

\end{tabular}
\end{center}
\end{table}



\vspace*{2mm}
\noindent
{\bf Variable detuning}

In VRSE configuration, it is required to change the round-trip phase of the laser beam in the SEC.
The amount of detuning is adjusted by changing the operation point of $l_{\rm s}$, 
with an additional offset in the  $l_{\rm s}$ control loop.
A detuning phase range of 3.4\,deg (59\,mrad) is required for the candidate VRSE configuration.
Since it requires only an addition of an offset electrically, 
the complexity of the control scheme is the same for all the three
configurations (BRSE, DRSE and VRSE).  
%In contrast, the sub-carrier injection requires a lot
%of additional optics and control loops.
On the other hand, the linearity of the error signal within the tuning range is critical
in detuning the phase with an offset to the SEC error signal ($l_{\rm s}$).
The shape of $l_{\rm s}$ error signals for the single demodulation\,(SDM) in
the pick-off port, and the double demodulation\,(DDM)\ in the reflection port
are shown in Fig.\,\ref{fig-det}. 
In SDM, the detuning phases required for VRSE(B and D) are well within the linear range%
\footnote{On the other hand, locking point is out of the linear range for DDM 
because the detuning phase is too small. This situation might be different if the 
macroscopic SRC length is adjusted to be resonant for one of the $\pm f_2$.}.


\begin{figure}[tbp]
\begin{center}
\includegraphics[width=7cm]{ls1.eps}
\hspace*{2mm}
\includegraphics[width=7cm]{ls2.eps}
\caption{$l_{\rm s}$ error signals with single demodulation ({\it Left})
and with double demodulation ({\it Right}, 
the signal for DRSE is magnified by 10 times).}
\label{fig-det}
\end{center}	
\end{figure}



\vspace*{2mm}
\noindent
{\bf Noises from control loops}

LCGT has 5 length DoF to be controlled. 
If a plant has multiple DoF to be controlled, 
the shot noises of auxiliary degrees of freedom can contaminate the signal of the main 
degree of freedom (L- in our case) through inter-couplings between the sensing signals and the feedback loops.
The loop coupling noises depend on the control bandwidth. 
Here, we assume typical control bandwidth summarized in Table\,\ref{tb-param0}%
\footnote{LCGT will be placed in the Kamioka mine which has a very quiet seismic noise environment, 
so the control bandwidth is not necessary so high. }.
In addition, a feed forward (FF) noise cancellation technique onto $l_+, l_-, and l_{\rm s}$ DoF
is absolutely necessary to achieve the target sensitivity for gravitational waves. 
The amount of the canceling is characterized by a parameter called FF gain;
we assume the default FF gain to be 30%
\footnote{This number is thought not to be so difficult. Advanced LIGO assumes higher FF gain of 100. }.

Figure\,\ref{fig:601_loopnoise} shows loop coupling noises of 4 configurations. 
The red curves represent the sensitivity level without loop coupling noises. 
Loop coupling noises contaminate this original noise level, 
which is shown as dotted black curves. 
Only the DRSE noise is contaminated by the loop coupling noise. 
In this case the culprits are  $L_+$ and $l_+$ noises%
\footnote{The main reason why $l_+$ and $L_+$ noises are large is that the fineness 
of arms for DRSE configuration is lower than that of other configurations. 
DRSE itself is not guilty.}. 
Other 3 configurations are free from loop coupling noises above 10\,Hz.
The effects of loop coupling noises are also evaluated quantitatively in terms of observable range for 
neutron-star binaries.
Reduction ratios of the range by the loop coupling noises are summarized in Table\,\ref{tb:NS-NS0}. 


\begin{figure}[p]
\begin{center}
\includegraphics[width=7.5cm]{601_loopnoiseFF_BRSE.eps}
\hspace*{2mm}
\includegraphics[width=7.5cm]{601_loopnoiseFF_DRSE.eps} \\
%\space*{2mm}
\includegraphics[width=7.5cm]{601_loopnoiseFF_vBRSE.eps} 
\hspace*{2mm}
\includegraphics[width=7.5cm]{601_loopnoiseFF_vDRSE.eps}
\caption{Loop coupling noise of BRSE and DRSE ({\it  Upper left and right}),
and VRSE (B and D, {\it Lower left and right}) .}
\label{fig:601_loopnoise}
\end{center}	
\end{figure}

\begin{table}[bp]
\begin{center}
\caption{Observable ranges for neutron-star binaries (optimal direction, SNR=8) with the candidate
configurations, and ranges with loop coupling noises.}
\label{tb:NS-NS0}
\begin{tabular}{c ccccc}
\hline
configuration & BRSE & VRSE(B) & VRSE(D) & DRSE \\
\hline
NS range, original  &260.0\,Mpc & 254.2\,Mpc & 279.8\,Mpc & 298.0\,Mpc \\
  (Averaged range)  &(114.4\,Mpc) & (111.8\,Mpc) & (123.1\,Mpc) & (131.1\,Mpc) \\
\hline
NS range for loop noise  &258.3Mpc & 251.4Mpc & 274.8Mpc & 266.4Mpc \\
Ratio of the ranges & 99.3\% & 99.9\% & 98.2\% & 89.4\% \\
\hline
\end{tabular}
\end{center}
\end{table}


For the detuned case, the DRSE configuration has the largest susceptibility
to the control loop noises. 
Therefore the actually achievable NS-NS ranges with the loop noise are comparable 
for the DRSE and VRSE(D) with default FF gain.
In other words, DRSE will require higher FF gain to take advantage of its tuned sensitivity
(detailed in Section\,\ref{sec-ctrl}).
This result diminishes the advantage of the fixed DRSE.
For the broadband case, the immunity to the loop noise is the same for
the BRSE and VRSE(B) cases.




\subsection{Mirror loss}

The loss of the test mass mirrors affects the sensitivity of an
interferometer mainly through the degradation of the power recycling gain. 
Higher finesse cavities are more susceptible to additional losses.
Among the four candidate configurations, only the DRSE has a different
finesse. Since it has a much lower finesse than the others, the DRSE is
less affected by unexpected loss increase in the mirrors.

The optical gain of a Fabry-Perot cavity is roughly described as 1/$\cal L$,
where $\cal L$ is a round-trip optical loss in the cavity.
Here we consider that the arm cavity is formed by ETM (end test mass), and
a compound mirror with ITM (input test mass) and PRM (power-recycling mirror).
In BRSE and VRSE configurations, the total power inside the arm cavities
will be 814\,kW, which is larger than the incident laser power by $ 1.1\times 10^{4} $ times.
This ratio corresponds to a requirement for a round-trip optical loss $\cal L$ of 92\,ppm 
in reflection at ETM and a compound mirror with by ITM and PRM.
In the DRSE case, the power ratio is  $ 4.8\times 10^{3} $, 
which corresponds to a loss requirement of 200\,ppm.


%Note that the decrease of the power recycling gain can be compensated by
%the laser power 



\subsection{Calibration}


The optical gain of DRSE interferometer shows a complex frequency
response compared to BRSE. The shape of the response depends on the
easy-to-change parameters such as detuning phase, homodyne phase, laser
power etc.  Therefore, we have to monitor the loop gain at several
frequencies during the observation to keep track of the optical gain
changes.  Especially in the case of VRSE(D), the detuning phase can vary
with the laser power fluctuation. Therefore, the calibration has to be
done more carefully.

In contrast, the optical gain of the BRSE is characterized only by the
overall gain and the cavity pole frequency. Since the cavity pole should
not change much during the operation, we only have to monitor the
overall gain. This is much simpler than DRSE.



