\newpage

\section{Interferometer Sensing and Control Scheme}
\label{sec-ctrl}

% \section{Interferometer Sensing and Control Scheme}

% The original LCGT sensing and control scheme was designed for BRSE
% configuration. Therefore, we have to find a new scheme to control a
% variable detuning RSE interferometer. In this section, we will examine
% two schemes for this purpose. One is a variant of the conventional
% frontal modulation scheme, where we apply two phase modulations to the
% carrier field and use the beat between the sidebands and carrier to
% extract the length information. The detuning phase is adjusted by adding
% an offset to the error signal for the SRC length.  The other one is a
% scheme utilizing a secondary laser with the orthogonal polarization to
% the main laser light. This sub-carrier beam is used to obtain the error
% signals for the SRC length and possibly other short degrees of freedom.
% This method is called ``Sub-Carrier injection''.

%\section{Length sensing scheme}

The original LCGT sensing and control scheme was designed for BRSE
configuration. Therefore, we have to find a new scheme to control a
variable detuning RSE interferometer. In this section, we will examine a
variant of the conventional frontal modulation scheme, where we apply
phase or amplitude modulations to the carrier light at two
frequencies. The lower one is called f1 and is supposed to resonate in
the SRC. The higher one is called f2 and is resonant only in the PRC.

\subsection{Sideband resonant conditions}

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{drawings/IFO-Resonances.eps}
\caption{Resonant conditions for the carrier and sidebands}
\label{fig:resonant conditions}
 \end{center}
\end{figure}

Figure\,\ref{fig:resonant conditions} shows the resonant conditions for
the carrier and the sidebands. The carrier has to resonate in the PRC
and the arm cavities. The anti-symmetric port of the Michelson is kept
dark for the carrier and f2 so that they are all reflected back to the
symmetric port. The Schnupp asymmetry and the f2 frequency are chosen so
that the MICH reflectivity for f2 is 1. With this configuration, f2 only
sees the PRC and is not affected by the motion of the SRM.  The f1
sideband has a finite transmissivity through the Michelson part and
resonates in the SRC. It is this sideband which carries the
information of the SRC length change.

Since the SRC is controlled to resonate the f1 sideband inside, the
amount of detuning for the carrier (i.e. the resonant condition of the
carrier in the SRC) is determined by the macroscopic
length of the SRC. We can also change the SRC detuning by adding an
offset to the error signal for the SRC length control.

In order to change the detuning phase, we apply an offset to the SRC
error signal to move the SRC length microscopically. For this purpose,
the resonance of the SRC to f1 should have an appropriate bandwidth to
allow us to extract a linear signal over the desired detuning range.  If
the resonance is too sharp, we won't have an enough range for detuning by
an offset. If the resonance is too wide, the SRC signal may decrease as
we don't get a large enhancement of the phase change inside SRC by
multiple bounces.

The sharpness of the SRC resonance (finesse) is determined by three
parameters, PRM reflectivity ($r_\mathrm{p}$), SRM reflectivity ($r_\mathrm{s}$) and the
Michelson reflectivity to f1 ($r_\mathrm{m1}$). Of the three parameters, two ($r_\mathrm{p}$ and
$r_\mathrm{s}$) are pre-determined to optimize the bottom line sensitivity of the
LCGT. Therefore, we are left with $r_\mathrm{m1}$ to change the finesse.

For $r_\mathrm{m1}$ to be an effective mean to change the SRC finesse, f1 has to be
resonant to PRC. When this condition is satisfied, the effective
reflectivity ($r_\mathrm{prm}$) of the cavity formed by PRM and the
Michelson part is determined by $r_\mathrm{m1}$. If $r_\mathrm{m1}$ is
close to $r_\mathrm{p}$, $r_\mathrm{prm}$ will go down.
The higher $r_\mathrm{prm}$ is, the higher the finesse of SRC.
The value of $r_\mathrm{m1}$ can be chosen by changing the Michelson
asymmetry for f1.

With these considerations in mind, we chose the macroscopic length
parameters of LCGT with the following steps:

\begin{enumerate}
 \item Choose the frequency of f2. It should not be too high because
       then we cannot use large PDs for signal detection. This limit is
       especially stringent on QPDs used for WFS. If it is too low, the
       frequency of f1 will also be very low. We do not want to use
       modulation frequencies below 10\,MHz because there, the intensity noise may not
       reach the shot noise limit. We chose 45\,MHz as the
       default f2 frequency.
 \item The PRC length ($L_\mathrm{p}$) and the Schnupp asymmetry
       $L_\mathrm{as}$ are chosen to resonate the f2 sideband inside the
       PRC. The Michelson reflectivity for f2 ($r_\mathrm{m2}$) has to
       be either 1 or -1. Depending on the sign of $r_\mathrm{m2}$, the
       f2 frequency has to be either an integral or half-integral
       multiple of the FSR of PRC. We also have to take into account the
       extra sign flip of carrier from the arm cavities to which it is
       resonant, when considering the resonant condition of the
       sidebands in the PRC.

 \item There are many possible combinations of $L_\mathrm{p}$ and
       $L_\mathrm{as}$ which satisfy the above condition. However, there
       are several constraints to the lengths. The PRC length is desired
       to be around 80\,m. Since we have 20\,m long cryo-shield parts in
       the vacuum tubes between the BS and ITMs, the PRC length has to
       be longer than this. In addition to this 20\,m, we have to
       accommodate the Schnupp asymmetry. Although it is not yet
       decided, it is likely that we have a folded PRC. 50m\, or so
       should be reserved for the folding. From the point of view of the
       construction cost, we don't want to make the PRC too long. The
       Schnupp asymmetry should not be too long because we do not want
       to introduce an excessive mismatch between the two recombining
       beams.

 \item We choose the f1 frequency to make it resonant in the PRC. For a
       given set of $L_\mathrm{p}$ and $L_\mathrm{as}$, there are
       several f1 frequencies which satisfy this condition. First of
       all, we make f1 lower than f2. Then we pick frequencies with
       Michelson reflectivity, $r_\mathrm{m1}$, close to the desired
       value. Finally, both f1 and f2 frequencies have to be integral
       multiples of the FSR of MC. Therefore, the ratio of f1 and f2
       determines the MC length. Since we do not want to make the MC too
       long for the construction cost and the back scattering problem,
       we choose a frequency which gives a short MC length. Practically,
       the MC length less than 30\,m would be reasonable.

 \item The SRC length ($L_\mathrm{s}$) is chosen to make f1 resonant in
       the SRC at a desired detuning phase of the carrier. We can
       elongate $L_\mathrm{s}$ by $c/(2 f1)$ keeping the same resonant
       condition. We chose $L_\mathrm{s}$ to be around 80\,m.

 \item We scanned the length parameters to find a set of parameters which
       satisfy all of the above constraints.
\end{enumerate}

\paragraph{Length parameters}
The table\,\ref{Candidate Length Parameters} show the candidate
parameter sets for LCGT.  The set from 1 to 4 are shown in the order of
SRC detuning range from narrow to wide. Set 5 is different from the
other four in a sense that f1 is not resonant in the PRC in the absence
of the SRM for this parameter set. In this case, the reflectivity of the
power-recycled Michelson part seen from SRM is that of an anti-resonant
cavity. Therefore the detuning range is the smallest with the parameter
set 5.

\begin{table}[htbp]
 \begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Name &f1
 &f2&$L_\mathrm{p}$&$L_\mathrm{as}$&$L_\mathrm{s}$&$L_\mathrm{MC}$ \\\hline
 Set 1&12.8571\,MHz &45MHz &81.6102\,m &6.66205\,m &69.9516\,m
		     &23.3172\,m \\\hline
Set 2&9\,MHz &45MHz &74.9481\,m &6.66205\,m &74.9481\,m
		     &16.6551\,m \\\hline
Set 3&13.5\,MHz &45MHz &83.2757\,m &3.33103\,m &61.0688\,m
		     &33.3103\,m \\\hline
Set 4&11.25\,MHz &45MHz &73.2826\,m &3.33103\,m &73.2826\,m
		     &13.3241\,m \\\hline
Set 5&14.8\,MHz &37\,MHz &81.025\,m &4.05125\,m &81.025\,m
		     &20.2562\,m \\\hline
\end{tabular}
\caption{Length parameter candidates.}
\label{Candidate Length Parameters}
 \end{center}
\end{table}

Figure\,\ref{fig: SRC Resonance Set 1} to figure\,\ref{fig: SRC Resonance Set 5}
show the power of f1 sideband in the SRC and at the reflection port
along with the phase at the pick-off port and the reflection port as the
SRC detuning phase is changed. The power is represented as the ratio to
incident power. We can see that the resonance width is changed as
$r_\mathrm{m1}$ is changed.

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{plots/Set1.eps}
\caption{SRC resonance curves for Set 1}
\label{fig: SRC Resonance Set 1}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{plots/Set2.eps}
\caption{SRC resonance curves for Set 2}
\label{fig: SRC Resonance Set 2}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{plots/Set3.eps}
\caption{SRC resonance curves for Set 3}
\label{fig: SRC Resonance Set 3}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{plots/Set4.eps}
\caption{SRC resonance curves for Set 4}
\label{fig: SRC Resonance Set 4}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{plots/Set5.eps}
\caption{SRC resonance curves for Set 5}
\label{fig: SRC Resonance Set 5}
 \end{center}
\end{figure}

\subsection{Optickle simulation}

Based on the length parameters found above, here we try to find a
suitable set of signal extraction ports and demodulation quadratures
including double demodulation. Our criteria for this search is that the
shot noise and the degeneracy of the signals have to be small enough
that the noises coupled to the gravitational wave channel through the
control loops do not contaminate the target sensitivities.

Calculations in this section are based on simulated models using the
Optickle, frequency domain interferometer simulator which can
investigate quantum effect such as radiation pressures and vacuum
injections. Here we assume to use Set 4 shown in Table\,\ref{Candidate Length Parameters},
$f_1=11.25$\,MHz $f_2=45$\,MHz since this set provides widest linear
range for detuning. Better parameter sets for each optical configuration
may be found, but here we compare the optical configurations with the
same control parameters for the simplicity. We leave the discussion for
detailed control method to the interferometer sensing and control
investigation team. Newly added parameters or already included
parameters in previous section but important for the length control are
listed in Table\,\ref{tb:param}. Other parameters are the same as
previous sections.

\begin{table}[htbp]
\begin{center}
\begin{tabular}{ll}
\hline
Item & Parameters \\
\hline\hline
Modulation frequency & $f_1=11.25$\,MHz $f_2=45$\,MHz\\
Modulation depth for BRSE & 0.15(PM)-0.1(AM)\\
Modulation depth for DRSE & 0.15(PM)-0.15(PM)\\
Loss asymmetry & $\pm$15ppm at ETM\\
Arm asymmetry & $\pm$1\% at ITM transmissivity\\
Maximum input power for photo detector & 100mW\\
Control bandwidth & 30k, 200, 50, 50, 50Hz for $L_+$, $L_-$, $l_+$, $l_-$, $l_{\rm s}$\\
Feed forward gain & 30, 30, 30 for $l_+$, $l_-$, $l_{\rm s}$\\

\hline
\end{tabular}
\caption{Default parameters assumed for the length control in this section.}
\label{tb:param}
\end{center}
\end{table}


\subsubsection{Detuning range for signal recycling cavity}

When we develop a variable bandwidth RSE, the first question will be how
difficult or how easy to obtain a linear signal for $l_{\rm s}$ to
exchange configurations from BRSE to DRSE. There are two ways to
establish a variable RSE. One way is just to acquire lock BRSE or DRSE
initially. Another way, which is probably more difficult, to exchange
bandwidth smoothly from BRSE to DRSE or DRSE to BRSE. In any case, wider
linear $l_{\rm s}$ ranges are better for control but the wider ranges
generally mean worse optical gains.

%\begin{figure}[htbp]
% \begin{minipage}{0.5\hsize}
% \begin{center}
%\includegraphics[width=7.5cm]{ls1.eps}
%\caption{$l_{\rm s}$ error signals for single demodulation. Each locking detuning phase is 
%shown as dotted lines in the plot and shown as values in the legend.}
%\label{fig:ls1}
%\end{center}
%\end{minipage}
% \begin{minipage}{0.5\hsize}
% \begin{center}
%\includegraphics[width=7.5cm]{ls2.eps}
%\caption{$l_{\rm s}$ error signals for double demodulation. Note that the signal for DRSE 
%is magnified by 10 times.}
%\label{fig:ls2}
%\end{center}	
%\end{minipage}
%\end{figure}

We assumed phase modulation\,-\,phase modulation\,(PM-PM) for DRSE and
phase modulation\,-\,amplitude modulation\,(PM-AM) for BRSE as the first
and second modulation. Firstly we hope to avoid AM because AM loses
carrier power at a modulator much more than that of PM. On the other
hand, we need to use DDM not only for $l_{\rm s}$ but also for $l_+$ or
$l_-$ to avoid degenerations from arm cavity signals\,(explained
later). For DRSE, the detuning makes unbalanced sidebands so that DDM
can produce meaningful signals even using PM-PM. However for BRSE, we
must use AM to extract useful DDM signals because only beat between AM
and PM can produce the useful DDM signals since the unbalanced sidebands
do not exist in the BRSE configuration. The shape of $l_{\rm s}$ error
signals for the single demodulation\,(SDM) and the double
demodulation\,(DDM)\ are shown in Fig.\,\ref{fig-det}.
%\,\ref{fig:ls1} and \ref{fig:ls2} respectively. 
Note that the DDM signal for DRSE is magnified by 10
times, so this slope is much smaller than other signals. And also DRSE
has another difficulty that the locking point is at the edge of the
signal for SDM. These difficulties come from that the detuning phase is
large and the $f_2$ won't be resonant anymore in SRC. In other words,
the large detuning phase burdens the locking and the signal extraction
for $l_{\rm s}$. A small detuning like vRSE(D) is fine for SDM but the
locking point is out of the linear range for DDM because the detuning
phase is too small, an opposite reason from the DRSE. This situation
might be different if the macroscopic SRC length is adjusted to be
resonant for one of the $\pm f_2$, but we do not discuss the detail for
macroscopic length change here. For BRSE, the slope of DDM is more than
10 times larger than that of DDM. Results are summarized in
Table\,\ref{tb:detuning_range}. From these results, we assume default
signals as SDM for BRSE and DDM for DRSE in the following
discussion. Using these signals makes it possible to lock both BRSE and
DRSE configurations and exchange each configurations smoothly by adding
an offset on their error signals.

  

\begin{table}[htbp]
\begin{center}
\begin{tabular}{lllll}
\hline
Configuration & BRSE & VRSE(B) & VRSE(D) & DRSE \\
locking phase & 0deg & 0deg & 3.4deg & 15.4deg \\
\hline\hline
$l_{\rm s}$ by SDM&$0\pm5$deg & $0\pm5$deg & $0\pm5$deg&$0\pm18$deg\\
&$0\pm6$mW &$0\pm7$mW & $0\pm7$mW & $0\pm6$mW\\
\hline
$l_{\rm s}$ by DDM&$0\pm6$deg&$0\pm6$deg&$10\pm6$deg (bad)&$19\pm10$deg \\
&$0\pm120$mW & $0\pm110$mW & $55\pm15$mW &$-5\pm2.5$mW\\
\hline
\end{tabular}
\caption{Linear phase range\,(upper line in each column) and linear slope range\,(lower line in each column) for $l_{\rm s}$ error signal. The linear range is shown as the center of linear range\,$\pm$\,one side linear range.}
\label{tb:detuning_range}
\end{center}
\end{table}

 
\subsubsection{Selection of signal extraction ports}

It is important to select proper signal extraction ports for the length control. An interferometer provides a lot of ports, but the number of proper ports is not so many. For example, if we use a DC readout method the DC signal of anti symmetric (AS) port is the only one choice for $L_-$, and a reflection port of the output mode cleaner\,(OMC)\footnote{It is known that the second or third candidate at pick off port with Q-phase or reflection port with Q-phase single demodulation produces very small signal and the shot noise limited sensitivity is much worse.} is also the only one choice for $l_-$.

\begin{table}[htbp]
\begin{center}
\begin{tabular}{cc||ccccc}
\hline
configuration & $L_+$ & $L_-$ & $l_+$ & $l_-$ & $l_{\rm s}$ \\
\hline\hline
BRSE & REFL $f_2$ & DC & POX $f_1$ & OMC DDM & REFL DDM\\
DRSE & REFL $f_2$ & DC & POX $f_2$ & OMC DDM & POX $f_1$\\
\hline
\end{tabular}
\caption{Selected ports for length signal sensing. Names of ports; REFL:reflection port, DC:DC readout at anti symmetric port, POX:pick off port of X-arm, OMC:OMC reflection port. $f_1$ and $f_2$ are demodulations with the first and second modulation frequencies.}
\label{tb:selected_ports}
\end{center}
\end{table}

Ports which have a low shot noise limited sensitivity should be selected
firstly. However if the ratio of two signals are similar at two ports
these two signals are degenerate and it will be difficult to obtain a
good signal separation. It sometimes happens when both two signals are
extracted from SDM, for example, $L_+$ and $l_+$ or $L_+$ and $l_{\rm
s}$. In such a case, this degeneracy appears as a huge loop noise
later. We need to select DDM signals even the shot noise limited
sensitivity is not so good when signals are degenerate. Here, we assume
to use a DC readout method for $L_-$ signal extraction because of a
disadvantage of non-stationary shot noise with RF readout and an
advantage of possibility of quantum non demolition measurement with DC
readout. Table\,\ref{tb:selected_ports} shows a list of selected
ports. These ports are almost optimized for each optical configuration,
but not perfectly optimized since the parameter space is huge.


\subsubsection{Loop coupling noise}
LCGT will have 5 length degrees of freedom\,(DOFs) for an RSE
configuration. If a plant has multiple DOFs with their control,
couplings of sensing signals exist and their shot noises are fed back as
newly induced actuation forces through the feedback loops. In other
words, if there are branches at the signal sensing (For example, $l_-$
signal exists in $L_-$ port with a ratio of 1/finesse in principle.) an
effective optical gain will be reduced through this branch of the
loop. Finally, these loop couplings appear as noises at gravitational
wave detection port.

The loop coupling noises depend on the control bandwidth. 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. Here, we assume typical control bandwidth as the unity gain frequency as 30kHz, 200Hz, 50Hz, 50Hz, 50Hz for $L_+$, $L_-$, $l_+$, $l_-$, $l_{\rm s}$ respectively and simple open-loop gain of just $f^{-1}$.

We learned from the first generation gravitational wave detector like TAMA or LIGO that a kind of feed forward noise cancellation technique is absolutely needed to realize an acceptable sensitivity for gravitational waves. We call it feed forward (FF) technique in this report and it reduces non-diagonal component in optical gain matrix. We introduce this FF technique onto $l_+, l_-, l_{\rm s}$ DOFs and this effect is shown as one parameter called FF gain. We assume typical FF gain as 30 which reduces non-diagonal components of optical gain matrix for $l_+, l_-, l_{\rm s}$ to $L_-$ by factor of 30 in all frequency. Realization of this number is though as not so difficult since Advanced LIGO assumes higher FF gain as 100. 

%\begin{figure}[htbp]
% \begin{minipage}{0.5\hsize}
% \begin{center}
%\includegraphics[width=7.5cm]{plots/601_loopnoiseFF_BRSE.eps}
%\caption{Loop coupling noise of BRSE.}
%\label{fig:601_loopnoiseFF_BRSE}
%\end{center}
%\end{minipage}
% \begin{minipage}{0.5\hsize}
% \begin{center}
%\includegraphics[width=7.5cm]{plots/601_loopnoiseFF_vBRSE.eps}
%\caption{Loop coupling noise of VRSE(B).}
%\label{fig:601_loopnoiseFF_vBRSE}
%\end{center}	
%\end{minipage}
%%\end{figure}
%%\begin{figure}[htbp]
% \begin{minipage}{0.5\hsize}
% \begin{center}
%\includegraphics[width=7.5cm]{plots/601_loopnoiseFF_vDRSE.eps}
%\caption{Loop coupling noise of VRSE(D).}
%\label{fig:601_loopnoiseFF_vDRSE}
%\end{center}
%\end{minipage}
% \begin{minipage}{0.5\hsize}
% \begin{center}
%\includegraphics[width=7.5cm]{plots/601_loopnoiseFF_DRSE.eps}
%\caption{Loop coupling noise of DRSE.}
%\label{fig:601_loopnoiseFF_DRSE}
%\end{center}	
%\end{minipage}
%\end{figure}

 Figure\,\ref{fig:601_loopnoise} show loop coupling noise of 4 configurations. If no loop coupling noise is not taken into account the total quantum noise will be the same as the original $L_-$ noise calculated from the vacuum noise divided by the optical gain at DC readout port. Other loop coupling noises contaminate this original $L_-$ noise and increase effective $L_-$ noise curve shown as dotted black line. Here only DRSE total noise is limited by the loop coupling noise for $L_+$ and $l_+$ noise. Other 3 configurations are free from loop coupling noises above 10Hz.

\subsubsection{Reduction of NS-NS binary range by loop coupling noise}
Here we summarize NS-NS binary ranges and reduction ratios of the NS-NS binary range by the loop coupling noises for 4 optical configurations in Table\,\ref{tb:NS-NS}. A reduction ratio is defined as NS-NS binary range with loop noise divided by NS-NS binary range with no RF modulation. In this calculation, the demodulation phases, detuning phases and homodyne phases\,(as the offset on end mirrors) are optimized. Demodulation phases are chosen to maximize a desired signal at each port. From these results, it appears that larger detuning phase has larger loop coupling noise.

\begin{table}[htbp]
\begin{center}
\begin{tabular}{c||ccccc}
\hline
configuration & BRSE & VRSE(B) & VRSE(D) & DRSE \\
\hline\hline
NS range, no RF(max) &260.0Mpc & 254.2Mpc & 279.8Mpc & 298.0Mpc \\
NS range, no RF(avg) &114.4Mpc & 111.8Mpc & 123.1Mpc & 131.1Mpc \\

\hline
NS range for loop noise, no FF(max) &142.1Mpc & 129.2Mpc & 61.0Mpc & 33.4Mpc \\
loop noise, no FF/no RF& 54.7\% & 50.8\% & 21.8\% & 11.2\% \\
\hline
NS range for loop noise with FF(max) &258.3Mpc & 251.4Mpc & 274.8Mpc & 266.4Mpc \\
loop noise with FF/no RF& 99.3\% & 99.9\% & 98.2\% & 89.4\% \\
\hline
\end{tabular}
\caption{NS-NS binary range and the reduction ratio.}
\label{tb:NS-NS}
\end{center}
\end{table}

It is useful to check how safe or robust these configurations if we change the parameters which determine the control loop coupling noises. We will see NS-NS ranges and the reduction with swept parameters then. 
Figure\,\ref{fig:UGF1} and \ref{fig:UGF2} show dependence of the NS-NS binary range and the reduction ratio for each optical configuration on the unity gain frequency as the control bandwidth. These results tell us that closer to BRSE is more robust for the increase of control band width. Figure\,\ref{fig:FF1} and \ref{fig:FF2} show dependence of the NS-NS binary range and the reduction ratio on the feed forward gain. These results show also that closer to BRSE is more robust for the reduction of FF gain.

\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
 \begin{center}
\includegraphics[width=7cm]{UGF1.eps}
\caption{Dependence of the NS-NS binary range on the unity gain frequency.}
\label{fig:UGF1}
\end{center}
\end{minipage}
 \begin{minipage}{0.5\hsize}
 \begin{center}
\includegraphics[width=7cm]{UGF2.eps}
\caption{Dependence of the reduction of the NS-NS binary range on the unity gain frequency.}
\label{fig:UGF2}
\end{center}	
\end{minipage}
%\end{figure}
%\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
 \begin{center}
\includegraphics[width=7cm]{FF1.eps}
\caption{Dependence of the NS-NS binary range on the feed forward gain.}
\label{fig:FF1}
\end{center}
\end{minipage}
 \begin{minipage}{0.5\hsize}
 \begin{center}
\includegraphics[width=7cm]{FF2.eps}
\caption{Dependence of the reduction of the NS-NS binary range on the feed forward gain.}
\label{fig:FF2}
\end{center}	
\end{minipage}
%\end{figure}
%\begin{figure}[htbp]
 \begin{minipage}{0.5\hsize}
 \begin{center}
\includegraphics[width=7cm]{Eoffset1.eps}
\caption{Optimization of the NS-NS binary range for offsets on the end mirrors differentially as a homodyne phase.}
\label{fig:Eoffset1}
\end{center}	
\end{minipage}
 \begin{minipage}{0.5\hsize}
 \begin{center}
\includegraphics[width=7cm]{Eoffset2.eps}
\caption{Dependence of the reduction of the NS-NS binary range on offsets on the end mirrors differentially as a homodyne phase.}
\label{fig:Eoffset2}
\end{center}	
\end{minipage}
\end{figure}


At the last, we see an optimization for the homodyne phases on the
simulation models. In the previous section, homodyne phase was optimized
analytically. Here setting of the homodyne phase is realized by changing
a microscopic offset on two end mirrors differentially. It should be
investigated that the possible offset can be applied actually since too
large offset may introduce bad asymmetries on the interferometer and it
causes a reduction of optical gain or an increase of loop coupling
noise. Fig\,\ref{fig:Eoffset1} and \ref{fig:Eoffset2} show relationships
between an ETM offset and a NS-NS range. All NS-NS range is optimized
around $10^{-12}$\,m which is thought that it is not so difficult to be
realized.


\section{Sub-carrier injection scheme}
\label{sec-sc}

A rather different scheme for the extraction of the SRC length
information with a variable detuning capability was proposed in the
working group. The scheme is called ``sub-carrier injection'' because it
makes use of the injection of a secondary laser with the orthogonal
polarization to the main laser.  Since the study on this scheme is still
in its early stage, we only present the basic ideas of the scheme 
briefly in this section.

The conceptual design of this scheme is shown in figure\,\ref{fig:
Sub-Carrier setup}. We use an auxiliary laser with the orthogonal
polarization to the main laser to sense the SRC length.
This laser is phase locked to the main laser with a frequency offset.
Then the sub-carrier is phase modulated to add sidebands and combined
with the main laser beam through a PBS before the second MC.
We add two sidebands to the sub-carrier. One is not resonant to the MC
and used to lock the sub-carrier to the MC. The other one goes through
the MC but not resonant in the main interferometer. Therefore, it serves
as the local oscillator for the sub-carrier, which is resonant in the
dual-recycled Michelson, to sense the motion of SRM.

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{drawings/Sub-Carrier-Setup.eps}
\caption{Conceptual setup of the sub-carrier injection scheme}
\label{fig: Sub-Carrier setup}
 \end{center}
\end{figure}

For the sub-carrier to be resonant in the MC, its frequency offset from
the main-carrier ($f_\mathrm{sc}$) must be apart from the main carrier
by $(n+1/2)f_\mathrm{MC}$, where $n$ is an integer and $f_\mathrm{MC}$
is the FSR of the MC. This constraint gives us a set of discrete
frequencies for possible $f_\mathrm{sc}$. For each possible
$f_\mathrm{sc}$, we examined the detuning phase of the SRC to the main
carrier when the sub-carrier is made resonant. We also checked the power
build up of the sub-carrier inside the SRC. Figure\,\ref{fig:
Sub-Carrier Resonances} shows the result. We picked only the
$f_\mathrm{sc}$ candidates with more than 4 times of power enhancement
of the sub-carrier in the SRC and plotted them. The horizontal axis of
the figure is the detuning phase of the SRC and the vertical axis is the
power enhancement factor.

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=10cm]{plots/Sub-Carrier-Resonances.eps}
\caption{Scan of sub-carrier frequency.}
\label{fig: Sub-Carrier Resonances}
 \end{center}
\end{figure}

As we can see from the figure, the detuning phase of the SRC can be
chosen relatively freely by changing $f_\mathrm{sc}$, although it is
discrete. To make figure\,\ref{fig: Sub-Carrier Resonances}, we scanned
$f_\mathrm{sc}$ for $\pm 1$\,GHz.

The PLL part of this scheme includes a double-pass AOM to shift the
frequency of either the sub-carrier or the main-carrier. This is to make
the beat frequency not too high (like 1\,GHz). The error signal from the
MC is fed back to the frequency of the AOM driver to resonate the
sub-carrier in the MC. We will use wave front sensors (WFSs) to monitor
the mis-alignment between the sub-carrier beam and the MC. The WFS
signals will be fed back to the sub-carrier's steering mirrors to
align the sub-carrier and the main-carrier beams.
