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\begin{document}

%{{{ Title

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

%}}}

%{{{ \chapter{Introduction}

\chapter{Introduction}
In this document, we will describe the design of the KAGRA
interferometer. KAGRA will be developed in two phases. The first phase
of KAGRA is called initial KAGRA or iKAGRA. This is a reduced version of
the full configuration KAGRA, which is referred to as the baseline KAGRA or
bKAGRA. The main purpose of iKAGRA is to gain experience in operating a
large interferometer in the underground environment and to identify
potential facility problems as early as possible. Therefore, the
design of iKAGRA is derived from bKAGRA as a natural pass point to the
full configuration. For this reason, we will mainly focus on design of bKAGRA
interferometer. At the end of this document, the iKAGRA design is
explained by pointing out the difference between the bKAGRA configuration.

%{{{ \section{Design requirements}

\section{Requirements}
The bKAGRA interferometer has to meet the requirements listed below.

\begin{itemize}
 \item The main interferometer has to be able to achieve the target
sensitivities of bKAGRA shown in Figure\,\ref{bKAGRA Target Sensitivity
       BRSE} and \ref{bKAGRA Target Sensitivity DRSE}. These target
       sensitivities are determined as a result of the optimization of
       the optical parameters given fundamental noise sources other than
       quantum noises. Details of the optimization work are described in
       \cite{BW Report}.

 \item bKAGRA 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 KAGRA has to be robust enough to ensure
       stable operation of the interferometer in the environmental
       disturbances of Kamioka mine. The target duty cycle during the
       observation is more than 90\%.

\end{itemize}

\begin{figure}[tbp]
\begin{center}
 \begin{minipage}[c]{7cm}
 \begin{center}
\includegraphics[width=7cm]{plots/NoiseBudget-BRSE.pdf}
\caption{bKAGRA Target Sensitivity in the BRSE mode}
\label{bKAGRA Target Sensitivity BRSE}
 \end{center}
 \end{minipage}
\hspace{1cm}
 \begin{minipage}[c]{7cm}
 \begin{center}
\includegraphics[width=7cm]{plots/NoiseBudget-DRSE.pdf}
\caption{bKAGRA Target Sensitivity in the DRSE mode}
\label{bKAGRA Target Sensitivity DRSE}
 \end{center}
 \end{minipage}
\end{center}
\end{figure}


%}}}

%}}}

%{{{ \chapter{Optical Configuration}

\chapter{Optical Configuration}

%{{{ \section{Overview}

\section{Overview}
The main interferometer part of bKAGRA is a dual recycled Fabry-Perot
Michelson interferometer operating in a resonant-sideband extraction
(RSE) mode. This interferometer is designed to be operated in two modes,
Broadband RSE (BRSE) and Detuned RSE (DRSE). 
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}


%}}}

%{{{ \section{Arm Cavity Parameters}

\section{Arm Cavity Parameters}

The arm cavity (AC) parameters are summarized in Table\,\ref{Arm cavity
parameters}.  The arm cavity length is constrained to less than 3\,km by
the size of the mountain. So we chose the largest one. The finesse of
the arm cavities is about 1500. This rather high finesse is chosen to
avoid high optical power from transmitting through ITMs, thus reducing
the heat generation in the mirror\,\cite{BW Report}. This is critical to
meet the cooling capacity of the cryocoolers. From the finesse, the
reflectivities of the mirrors are determined. The round trip loss of the
cavity including the ETM transmission should be less than 100\,ppm. As
more concrete requirements, we assign 10\,ppm to the transmission and
45\,ppm each to the reflection loss of each mirror.

\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& 1900($\pm$10)\,m& \\ 
   ITM Beam Size&3.5\,cm & $1/e^2$ radius\\ \hline
   ETM Reflectivity&$>$99.9945\%& \\ 
   ETM Radius of Curvature&1900($\pm$10)\,m& \\ 
   ETM Beam Size&3.53\,cm & $1/e^2$ radius\\ \hline
   g-factor&0.335 &$g_1\cdot g_2$ \\ 
   Round Trip Loss&$<$100\,ppm& \\ 
   Finesse&1530 & \\ \hline
  \end{tabular}
\caption{Arm cavity parameters}
\label{Arm cavity parameters}
 \end{center}
\end{table}

\subsubsection{g-factor}
The radius of curvature of the mirrors are selected to realize the
desirable beam spot sizes on the mirrors. From the point of view of the
thermal noise, we want to make the spot size as large as
possible. Considering the size of the mirror (22\,cm diameter) and
requiring the diffraction loss per reflection to be less than 1\,ppm,
the largest possible beam radius is 4.0\,cm. We employ this number as
the ETM spot size. For the ITMs, the dielectric coating is thinner
because of the smaller reflectivity. Thus, the impact on the sensitivity
is minimal, even if we reduce the beam size on the ITMs to 3.5\,cm (the
reduction of the sensitivity in terms of the inspiral range by this
change is about 2Mpc\,\cite{JGW-G1100599}). A smaller beam in the vertex
makes it easier to handle the stray beams in the congested BS chamber.
Therefore, we decided to set the beam size on the ITMs to be 3.5\,cm.

There are two possible choices of mirror curvatures to realize the same
spot sizes. The first set of the ROCs is ITM=14km and ETM=7.5km, which gives
positive g-factors, g1=0.786 and g2=0.602. Another possibility is
ITM=1.68km and ETM=1.87km, corresponding to negative g-factors,
g1=-0.786 and g2=-0.602. In order to decide the polarity of the
g-factors, we considered the higher order mode (HOM) resonances,
parametric instability and the angular optical spring instability.

%}}}

%{{{ \subsection{Arm Cavity Higher Order Mode Resonances}

\subsection{Arm Cavity Higher Order Mode Resonances}
Ideally, the arm cavities should resonate only the TEM00 mode. However,
optical higher modes are not completely anti-resonant to the AC in general.
Therefore, if there is mis-alignment or mode mis-matching, HOMs could
resonate in the AC, potentially increasing the shot noise. If the selected
arm g-factor is a particularly bad one, this HOM coupling could be large. 
In this section, we will check if our arm cavity design is robust
against this problem. We will compare the two cases of g-factors,
the negative one and the positive one.

Figure\,\ref{HOM Power in AC} shows the HOM power ratio to the TEM00
power in the AC. This is the ratio of the intra-cavity optical power, if
TEM00 and TEMnm modes are injected to the AC with the same power. Of
course, it is very unlikely, thus in reality, the ratio is much smaller.

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/negative-g/HOM-Power-in-AC.pdf}
\caption{HOM power in the arm cavity relative to the TEM00 power. The
  mode number is defined as $n+m$ for TEMnm modes.}
\label{HOM Power in AC}
 \end{center}
\end{figure}

When calculating the HOM power, we took into account the fact that for
HOMs, the diffraction loss is higher than TEM00. This is because HOMs
are spatially spread more widely. Figure\,\ref{HOM Finesse} shows the
resonance curves of various HOMs. The power build up is suppressed
quickly as the mode number gets higher. These curves are calculated
using SIS[ref].

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/negative-g/HOM-Finesse.pdf}
\caption{Resonance curves of HOMs taking into account the diffraction
  loss. The HOMs are expanded by the Laguerre-Gaussian
  basis. $\mr{LG}(p,l)$ corresponds to the mode number $2p+l$.}
\label{HOM Finesse}
 \end{center}
\end{figure}


Figure\,\ref{HOM Power in AC} assumed that the g-factor of the cavity is
exactly as designed. In reality, there is always some error in the ROC
of the real mirrors. We set the error tolerance to be $\pm$0.5\% mainly
from the technical feasibility of the mirror polishing.
Figure\,\ref{HOM Power in AC ITM ROC error negative-g} and
Figure\,\ref{HOM Power in AC ETM ROC error negative-g} show the maximum
HOM power ratio (the value of the highest peak in Figure\,\ref{HOM Power
in AC} except for the mode number = 0) as a function of ITM ROC error
and ETM ROC error. The ROCs are swept by $\pm$1\% of the nominal
values. These figures are for the negative g-factor
case. Figure\,\ref{HOM Power in AC ITM ROC error positive-g} and
Figure\,\ref{HOM Power in AC ETM ROC error positive-g} show the same
plots for the positive g-factor case. In both the cases, the HOM power
does not go up so much within the tolerated ROC errors.

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/negative-g/HOM-Power-ITMROC.pdf}
\caption{The maximum HOM power ratio in the AC as a function of ITM ROC
  error for the negative g-factors. The ROC is swept by $\pm$1\% around
  the nominal value.}
\label{HOM Power in AC ITM ROC error negative-g}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/negative-g/HOM-Power-ETMROC.pdf}
\caption{The maximum HOM power ratio in the AC as a function of ETM ROC
  error for the positive g-factors. The ROC is swept by $\pm$1\% around
  the nominal value.}
\label{HOM Power in AC ETM ROC error negative-g}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/positive-g/HOM-Power-ITMROC.pdf}
\caption{The maximum HOM power ratio in the AC as a function of ITM ROC
  error for the positive g-factors. The ROC is swept by $\pm$1\% around
  the nominal value.}
\label{HOM Power in AC ITM ROC error positive-g}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/positive-g/HOM-Power-ETMROC.pdf}
\caption{The maximum HOM power ratio in the AC as a function of ETM ROC
  error for the positive g-factors. The ROC is swept by $\pm$1\% around
  the nominal value.}
\label{HOM Power in AC ETM ROC error positive-g}
 \end{center}
\end{figure}

%{{{ \subsubsection{g-factor and the HOM resonances}
\subsubsection{g-factor}
By comparing the dependence of the HOM resonance in the AC on the
mirror ROC errors between the negative and positive g-factor cases,
there is no significant difference between them. Therefore, both g-factors
are acceptable from the point of view of this problem.
%}}}

%}}}

%{{{ \subsection{RF Sidebands Resonances in the Arm Cavities}
\subsection{RF Sidebands Resonances in the Arm Cavities}

%{{{ \subsubsection{Fine Adjustment of the RF Sideband Frequencies}

\label{RFSB frequency fine adjustment}
\subsubsection{Fine Adjustment of the RF Sideband Frequencies}
As explained in section\,\ref{bKAGRA LSC}, we will use three RF
sidebands (RF SBs), namely f1, f2 and f3, to extract the error signals for the
interferometer control. We also want to prevent the RF sidebands including
their HOMs from accidentally resonating in the arm cavities. The f3 sideband
is not resonant in the PRC, so it does not see the arm cavities at
all. Therefore, we only consider f1 and f2 sidebands in this section.

Before proceeding to check the HOM resonances of the RF sidebands, there
is a subtle but important point to note about the fine adjustment of the
sideband frequencies.  The RF sidebands f1 and f2 are almost
anti-resonant to the arm cavities but not perfectly so. A consequence of
this is that they get finite phase shifts when reflected by the
ACs. Those two sidebands are supposed to resonate in the PRC at the same
time. However, if the phase shifts they get from the AC are arbitrary,
the resonant conditions for them is different, thus not being able to
resonate both at the same time.  A solution to this problem is the
following: The effective length change caused by a phase shift $\phi$
for a modulation sideband with the modulation frequency $\omega_\mr{m}$
is $\Delta L = \phi c/\omega_\mr{m}$. Therefore, if the phase shifts for
the f1 and f2 SBs are proportional to their frequencies, the effective
length change is the same for the two SBs. Then we can just pre-shorten
the PRC length by this amount to fulfill the resonant conditions for
both the SBs at the same time.

In order to adjust the reflection phases for f1 and f2, we need to
change their frequencies. However, as explained in section\,\ref{bKAGRA
LSC}, the ratio of f1 and f2 frequencies has to be 3:8. Therefore, we
can only change the frequencies under the constraint of keeping the
ratio unchanged. This condition is automatically satisfied by requiring
the two sidebands to transmit the MC. That is, the f1 frequency is 3
times the FSR of MC and f2 is 8 times the MC FSR. Therefore, we will
slightly change the MC length from its nominal value to find the optimal
RF SB frequencies which give the AC reflection phases proportional to
them. The precise amount of phase shifts induced on nearly-anti-resonant
fields depends on the finesse of the cavity.  Therefore, the RFSB
frequencies must be adjusted according to the measurement of the real
arm cavities. In the design phase, we assume 100\,ppm of loss in the
arm, resulting in the finesse of 1530.  Figure\,\ref{RFSB Refl phase
ratio} shows the ratio of the reflection phases ($\phi_2/\phi_1$) as a
function of the MC length. The desired value of 8/3 is indicated by the
green horizontal line.  By finding a intersection of the blue curve with
the green line, the tentative numbers for the RFSB frequencies are,
f1=16.881\,MHz and f2=45.016\,MHz. However, again, these numbers should
be corrected based on the finesse measurements of the actual arm
cavities. Figure\,\ref{RFSB Refl phase FSR} shows the relative positions
of the RFSBs in the FSR of the arm cavities.

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/RFSB-Reflection-Phase-Ratio.pdf}
\caption{Ratio of the SB reflection phases by the arm cavities. We want
  to set it to 8/3, which is indicated by the green line.}
\label{RFSB Refl phase ratio}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SB-ReflectionPhase.pdf}
\caption{Relative position of the RFSBs in the FSR of the arm cavities.}
\label{RFSB Refl phase FSR}
 \end{center}
\end{figure}

%}}}

%{{{ \subsubsection{HOM Resonances of the RF Sidebands}
\subsubsection{HOM Resonances of the RF Sidebands}

Once the exact frequencies of the RF SBs are determined, we can check
the resonant conditions of the RF SBs and their HOMs in the arm
cavities. Figure\,\ref{RFSB Refl phase FSR} shows the positions of the
RF SBs and the HOMs in the FSR of the arm cavity. In the figure, both
the HOM resonant curves (Lorentzian-shaped curves with mode numbers) and
the harmonics of the RF SBs (vertical lines: black for f1, red for
f2). If a vertical line overlaps well with one of the HOM resonances,
then this RF SB harmonics may resonate in the ACs when the cavity is 
mis-aligned and the HOM is excited. If it happens to the first harmonics
of the RF SBs, which are used for the signal extraction, the error
signals will be disturbed and in the worst case we will lose the
interferometer lock. Higher order harmonics are not so important, but
still some of them contribute to the error signal through inter-modulations
with other higher order harmonics. Therefore it is better to avoid
overlaps as much as possible.

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/negative-g/RFSB-Resonance-FSR.pdf}
\caption{Positions of the RF SBs and the HOMs in the FSR of the arm
  cavities. The colorful sharp peaks represent the resonant curves of
  the HOMs. The mode number is printed at the top of each resonance. The
  vertical lines are the positions of the RF SBs and their
  harmonics. The numbers associated with the lines indicate the harmonic
  order. The black lines are the f1 harmonics, whereas the red lines are
  the f2 harmonics.}
\label{RFSB HOM FSR}
 \end{center}
\end{figure}

In order to see how the overlap between the RF SB harmonics and the HOM
resonances changes as the ROC of the mirrors, we use a figure of merit
for the overlap defined as follows: For all combinations of an RF SB
harmonics and a HOM, the intra-cavity power is calculated assuming that
the SB is 100\% in this HOM. The diffraction loss of Figure\,\ref{HOM
Finesse} is taken into account. Then we take the power ratio between
this calculated power to the TEM00 power in the cavity. The FOM is the
maximum of this ratio from all the combinations of the RF SB harmonics
and the HOMs. Figure\,\ref{RFSB HOM FOM ITM negative-g}, \ref{RFSB HOM
FOM ETM negative-g}, \ref{RFSB HOM FOM ITM positive-g} and \ref{RFSB HOM
FOM ETM positive-g} show the FOM as a function of ROC errors,
ranging, again, $\pm$1\% of the nominal values.
 
Note that the meaning of the FOM above is not so clear, in a sense that
there is no definite threshold below which the overlap problem is
considered safe. A thorough simulation study may give some quantitative
interpretation of the FOM, but we haven't done it yet. So what we can
say from the above mentioned plots is just that the severity of the
overlap does not change much within the specified error range of the
ROCs.


\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/negative-g/RFSB-HOM-FOM-ITM.pdf}
\caption{Figure of merit of the overlap between the RF SB harmonics and
  the HOMs when the ITM ROC is swept. g-factor is negative.}
\label{RFSB HOM FOM ITM negative-g}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/negative-g/RFSB-HOM-FOM-ETM.pdf}
\caption{Figure of merit of the overlap between the RF SB harmonics and
  the HOMs when the ETM ROC is swept. g-factor is negative.}
\label{RFSB HOM FOM ETM negative-g}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/positive-g/RFSB-HOM-FOM-ITM.pdf}
\caption{Figure of merit of the overlap between the RF SB harmonics and
  the HOMs when the ITM ROC is swept. g-factor is positive.}
\label{RFSB HOM FOM ITM positive-g}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/positive-g/RFSB-HOM-FOM-ETM.pdf}
\caption{Figure of merit of the overlap between the RF SB harmonics and
  the HOMs when the ETM ROC is swept. g-factor is positive.}
\label{RFSB HOM FOM ETM positive-g}
 \end{center}
\end{figure}

%}}}

%{{{ \subsubsection{g-factor and the RF SB resonances}
\subsubsection{g-factor and the RF SB resonances}
By comparing the dependence of the RF SB resonance in the AC on the
mirror ROC errors between the negative and positive g-factor cases,
there is no significant difference between them. Therefore, both g-factors
are acceptable from the point of view of this problem.
%}}}

%}}}

%{{{ \subsection{Parametric Instability}
\subsection{Parametric Instability}
The parametric instability (PI) happens when the shape of an optical higher
order mode in the arm cavity is similar to an elastic mode of the mirror
substrate. In addition, the HOM's resonant frequency offset from the
TEM00 must be very close to the elastic mode's eigen-frequency. The
offset frequencies of the HOMs depend on the g-factor. Therefore, there
are hot spots of g-factor, where PI is large.

The magnitude of the PI is characterized by the parametric gain
$R$\,\cite{PI Braginsky}. $R$ is associated with each elastic mode of
the mirrors. If there is a mode with $R$ of greater than 1, this mode is
unstable. We want to select a g-factor, with which no unstable mode
appears.

Figure\,\ref{PI Plots} shows the parametric gain R as a function of the ROCs of ITM
and ETM. The ROCs are swept by $\pm$2\% of the nominal values. The
colored areas are where R is greater than 1. Details on how to generate
these plots are explained in .

By comparing the plots for the negative and positive g-factors, there is
no significant difference. The most important conclusion drawn from the
plots is that there is no continuous white area which is large enough to
accommodate the $\pm$0.5\% error of the ROCs. Moreover, the accuracy of
the plots is not so good, because of the limited accuracy of the finite
element analysis (using COMSOL). This means we cannot target a
particular g-factor, even if we ignore the manufacturing error, to put
our interferometer at a sweet spot (white area). In the actual
interferometer, what seemed to be a sweet spot in calculation may not
be a sweet spot. Therefore, regardless of the polarity of the g-factors,
we have to be prepared for the PI to happen.

Once PI happens, we have to damp the oscillation somehow. Mainly two
types of schemes are proposed to mitigate the PI. One is to put some
lossy stuff on the side of the mirrors to damp the unstable
modes. Another scheme is basically an active damping of the unstable
mode using the interferometer output as an error signal.  Detailed
design of KAGRA PI damper is yet to be discussed.

\begin{figure}[tbp]
 \begin{center}
\begin{minipage}[c]{7.5cm}
 \begin{center}
  \includegraphics[width=7.5cm]{plots/PI-Positive.pdf}
Positive g-factors
 \end{center}
\end{minipage}
\begin{minipage}[c]{7.5cm}
 \begin{center}
  \includegraphics[width=7.5cm]{plots/PI-Negative.pdf}
Negative g-factors
 \end{center}
\end{minipage}
\caption{Maps of the maximum parametric gain as functions of the ROCs of the test
  masses. Left: positive g-factor, Right: negative g-factor. The ROCs
  are swept by $\pm$2\% of their nominal values. White areas correspond
  to $R_\mr{max}<1$.}
\label{PI Plots}
 \end{center}
\end{figure}

%}}}

%{{{ \subsection{Angular Instability by the Radiation Pressure}
\subsection{Angular Instability by the Radiation Pressure}
The opto-mechanical coupling between the high power optical fields
circulating in the arm cavities and the mirrors create an angular
optical spring effect\,\cite{Sidles Sigg}. For single cavity, there are
always two eigen modes of the angular motions of the mirrors. Out of the
two, one mode has a positive spring constant, meaning that the optical
spring generates a restoring force. The other has a negative spring
constant, potentially leading to an instability. 

In general, the absolute values of the two spring constants are
different. If the g-factors are negative, the negative spring constant is
always smaller than the positive one. For positive g-factors, the order
is the other way around. Therefore, in order to minimize this
instability, negative g-factors is preferable.
More quantitative comparison of the positive and negative g-factors is
presented in section\,\ref{ASC}.

%}}}

%{{{ \subsection{Conclusion on g-factors}
\subsection{Conclusion on g-factors}
Based on the discussions in the above few sections, we decided to select
the negative g-factors, g1=-0.786 and g2=-0.602, as the design of the
KAGRA arm cavities. The main reason for this selection is the radiation
pressure induced angular instability.

The required ROCs to realize the positive and negative g-factors are
different (R1=14km and R2=7.5km for positive, R1=1.68km and R2=1.87km
for negative). Which set of ROCs is easier to manufacture is an
important question to be answered. So far, we received contradictory
answers from different companies: one said larger ROCs are easier to do,
while another say the opposite. Therefore, the manufacturability is put
out of scope of this document. It shall be addressed by the mirror group
and feedback shall be given to us, the MIF group.

%}}}

%{{{ \section{Recycling Cavities}
\section{Recycling Cavities}

%{{{ \subsubsection{Overview}

\subsubsection{Overview}
There are two recycling cavities (RCs) in the KAGRA interferometer: Power
Recycling Cavity (PRC) and Signal Recycling Cavity (SRC). These cavities
are folded in a Z-shape by two telescope mirrors to accumulate extra
Gouy phase. This is necessary to isolate the HOMs in the cavities,
i.e. to stabilize the cavities in terms of the spatial modes.

The parameters of the recycling cavities are listed in Table\,\ref{Recycling cavity parameters}.

\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&458.1285\,m & \\ 
   PRM Beam Size&4.457\,mm &$1/e^2$ radius \\ \hline
   PR2 ROC&-3.0764\,m & \\ 
   PR2 Beam Size&4.457\,mm &$1/e^2$ radius \\ \hline
   PR3 ROC&24.9165\,m & \\ 
   PR3 Beam Size&36.639\,mm &$1/e^2$ radius \\ \hline
   SRM Reflectivity&85\% & \\ 
   SRM ROC&458.1285\,m & \\ 
   SRM Beam Size&4.308\,mm &$1/e^2$ radius \\ \hline
   SR2 ROC&-2.9872\,m & \\ 
   SR2 Beam Size&4.328\,mm &$1/e^2$ radius \\ \hline
   SR3 ROC&24.9165\,m & \\ 
   SR3 Beam Size&36.666\,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
%    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&303.96\,m & \\ 
%    PRM Beam Size&4.03\,mm &$1/e^2$ radius \\ \hline
%    PR2 ROC&-2.7628\,m & \\ 
%    PR2 Beam Size&4.03\,mm &$1/e^2$ radius \\ \hline
%    PR3 ROC&24.574\,m & \\ 
%    PR3 Beam Size&36.479\,mm &$1/e^2$ radius \\ \hline
%    SRM Reflectivity&85\% & \\ 
%    SRM ROC&303.96\,m & \\ 
%    SRM Beam Size&4.03\,mm &$1/e^2$ radius \\ \hline
%    SR2 ROC&-2.7764\,m & \\ 
%    SR2 Beam Size&4.03\,mm &$1/e^2$ radius \\ \hline
%    SR3 ROC&24.584\,m & \\ 
%    SR3 Beam Size&36.327\,mm &$1/e^2$ radius \\ \hline
%   \end{tabular}
% \caption{Old recycling cavity parameters}
% \label{Old 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&14.762\,m & Distance between PRM and PR2\\ 
   Lp2&11.0661\,m & Distance between PR2 and PR3\\ 
   Lp3&15.7638\,m & Distance between PR3 and BS\\ \hline
   Ls1&14.7412\,m & Distance between SRM and SR2\\ 
   Ls2&11.1115\,m & Distance between SR2 and SR3\\
   Ls3&15.7386\,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}

%}}}

%{{{ \subsection{Reflectivities}

\subsection{Reflectivities}
The reflectivities of the PRM and SRM are determined as a part to
optimize the quantum noise shape of the interferometer\,\cite{BW
Report}.

%}}}

%{{{ \subsection{Length and RF SB frequencies}

\subsection{Length and RF SB frequencies}

The length parameters of the recycling cavities (RCs) are selected to resonate the
RF sidebands used for the control signal extraction, which is explained
in section\,\ref{bKAGRA 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. Another sideband, called f3, will be used during
the lock acquisition. It is not resonant in any part of the
interferometer, thus reflected directly back by the PRM. The RF SB
resonant conditions are depicted in Figure\,\ref{Signal Ports}.

%{{{ \subsubsection{Constraints}
\subsubsection{Constraints}

While there are many possible combinations of the RC lengths and the RF
SB frequencies to satisfy the above mentioned resonant conditions, we
have to meet several practical constraints. First of all, the RCs cannot
be too long for the obvious reason of limited space in the mine. From
this point of view, the shorter the better. At the same time, the RCs
cannot be too short. One reason for this is that we have to accommodate
20\,m long cold segments of vacuum pipes between the BS and the
ITMs. This is necessary to reduce the thermal radiation impinging on the
cryogenic ITMs. Another reason is that we need some length to fold the
RCs. If the RCs are short, the folding angles have to be wide. This will
increase the astigmatism because the folding mirrors have curvature.
For those reasons, we want the RC lengths to be about 70\,m.

The next constraint is the range of the RF SB frequencies. We want them
to be moderate, meaning roughly in the range of 10\,MHz to 50\.MHz. In
this frequency range, we can find PDs with reasonable aperture size
($\sim$1\,mm). If the frequency is much higher than 50\,MHz, we have to
use smaller PDs, which is more susceptible to beam jitter, and may take
less power. If the frequency is too low, the laser noise may be larger
at the demodulation frequencies. Especially, we doubt that the intensity
noise is at the shot noise level in less than 10\,MHz.

The two RF SBs have to transmit the mode cleaner (MC). Therefore, the
FSR of the MC has to be a common measure of $f_1$ and $f_2$. The FSR, in
turn determines the length of MC, as $L_\mr{MC} = c/(2 f_\mr{FSR})$. In
order to keep the MC length in the reasonable range (order of 30\,m),
the FSR cannot be too small. This sets a severe constraint on the choice
of the RF SB frequencies.

%}}}

%{{{ \subsubsection{SRCL linear range}
\subsubsection{SRCL linear range}
We explored all possible combinations of the RC lengths and RF SB
frequencies by the algorithm explained in \ref{RCL Algorithm}.  After
the extensive search, there are still many sets of parameters, which
satisfy the above mentioned constraints. In order to determine the final
parameter set, we look at the linear range of the SRCL error
signal. Since we will detune the SRC by adding an offset to the error
signal, the error signal has to have a reasonable slope at the detuned
operation point. This situation is shown in Figure\,\ref{SRCL Sweep}. In
the plot, the SRCL error signals are plotted as functions of SRCL
detuning in terms of the one-way phase change. There are three curves
corresponding to different finesse of the PRC-SRC coupled cavity.  The
blue curve is the one with the final parameter set we selected. At the
center, corresponding to the BRSE operation point, the blue curve has
some slope. At the SRC detuning = 3.5\,degrees, corresponding to the
DRSE operation point and indicated by the black vertical line, the curve
still has some slope, though it is more gradual. On the other hand, the
red curve has almost zero slope at the DRSE operation point. Therefore,
we cannot use this signal for controlling the SRC in DRSE. The green
curve has almost constant slope, but it is less steep than the blue
curve, at least at the BRSE operation point, meaning that the signal is
weaker.  Therefore, we prefer the blue curve.

The linear range of the SRCL error signal is roughly determined by the
finesse of the PRC-SRC coupled cavity for the f1 sideband.  Since the
reflectivities of the PRM and the SRM are already determined by the
optimization of the quantum noise shape, we are left with the Michelson
reflectivity for the f1 sideband to change the finesse of the coupled
cavity. 

The Michelson reflectivity $R_\mr{m}$ depends on the f1 frequency
($f_1$) and the Michelson asymmetry ($l_\mr{m}$) as
$R_\mr{m}\propto\cos(2\pi f_1\cdot l_\mr{m} /c) $, where $c$ is the
speed of light.  When $R_\mr{m}$ is closer to the PRM reflectivity
(0.9), the effective reflectivity of the power-recycled Michelson seen
from the SRC becomes lower. Therefore, the finesse of the SRC gets
smaller, resulting in a wider linear range. 

Even though the blue curve in Figure\,\ref{SRCL Sweep} has a finite
slope at the DRSE operation point, the second derivative is non-zero
there, i.e. the error signal is non-linear. This may non-linearity could
produce up- and down-conversions of error signals. This problem is
examined in the appendix\,\ref{SRCL non-linearity}. The conclusion is
that this effect is small enough to worry about.


\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=11cm]{plots/SRCLSweep.pdf}
\caption{SRCL error signals for three different values of
$R_\mr{m}$. The horizontal axis is the detuning of the SRC in terms of
the one-way phase shift. The vertical axis is the signal from the
POP port demodulated at the f1 frequency. The vertical line shows the
operation point of DRSE (3.5$^\circ$). }
\label{SRCL Sweep}
 \end{center}
\end{figure}

%}}}

%{{{ \subsubsection{Selected Length and Frequencies}
\subsubsection{Selected Length and Frequencies}

The PRC length ($L_\mr{prc}$) is set so that $f_1 = 7.5\times F_\mr{prc}$, where
$F_\mr{prc}\equiv c/(2L_\mr{prc})$ is the FSR of the PRC. Since the carrier is
anti-resonant to the PRC cavity itself\footnote{Because the carrier gets
extra phase shift of $\pi$ from the arm cavities.}, this condition
makes the round trip phase change of the f1 sideband in the PRC an
integral multiple of $2\pi$. The reflectivity of the Michelson part
 for f1 is 0.383 (amplitude reflectivity). Therefore, f1 transmits
through MICH and reaches SRM. The SRC length ($L_\mr{src}$) is the same
as $L_\mr{prc}$, thus satisfying $f_1 = 7.5\times F_\mr{src}$. In this
case, the SRC length is controlled to resonate the carrier. Therefore,
SRC is anti-resonant for f1 by itself. However, there is always a $\pi/2$
phase shift associated with the transmission of MICH. During the round trip
of the PRC-SRC coupled cavity, f1 experiences this phase shift twice
amounting to a sign flip. Therefore, in total, the round trip phase
shift of the coupled cavity is an integral multiple of $2\pi$.

For the f2 sideband, the MICH reflectivity is -1. The FSR of PRC satisfies
$f_2 = 20\times F_\mr{prc}$. Combined with the fact that the carrier is
anti-resonant and the MICH reflection induces a sign flip, f2 is
resonant in the PRC. Since it is perfectly reflected by MICH, the f2
sideband does not resonate in SRC.
 
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. The remaining 5\,m is used to absorb the
Schnupp asymmetry.

As explained in section\,\ref{RFSB frequency fine adjustment}, the two
RF SBs get finite phase shifts as reflected by the ACs. Since we fine
adjusted the frequencies of f1 and f2, the effective length changes by
these phase shifts are the same for both of them. Thus, what we have to
do is to change the PRC length (Lprc) and SRC length (Lsrc) by $dL = c
\phi_1/(4\pi f_1)=c
\phi_2/(4\pi f_2)$, where $\phi_\mr{x}$ and $\omega_\mr{x}$ are the
phase shift and the angular frequency of the f1 or f2 sidebands.
%}}}

%}}}

%{{{\subsection{Gouy phase change in the Recycling Cavities}
\subsection{Gouy phase change in the Recycling Cavities}
The recycling cavities (RCs) are both folded by two additional mirrors
to allow focusing of the beam inside the cavities. The main purpose of
the folding is to add an extra Gouy phase change to the beam traveling
inside the RCs. This additional Gouy phase prevents the HOMs from
resonating in the RCs, thus stabilizing the RCs in terms of the spatial
modes.

Although HOMs are not welcome in most aspects of the interferometer
design, the TEM10 and TEM01 modes are utilized to generate the wavefront
sensing (WFS) signals. When TEM10/TEM01 modes are excited by the
mis-alignment of the arm cavities, they have to travel to the detection
ports, such as REFL. If the RCs are completely anti-resonant to those
modes, these modes are greatly suppressed by the time they reach the
detection ports.  Therefore, we do not want to suppress them too much.
As a compromise, we will set the one-way Gouy phase shift of the RCs to
be less than 20 degrees.

In order to quantitatively see how much HOMs may be built up in the RCs,
we used the simple interferometer model explained in
Appendix\,\ref{Static Model of the Interferometer}. Using the model, we
first calculate the field amplitudes when the carrier in the TEM00 mode
is injected from the back of the PRM. This calculation is repeated for
the f1 and f2 RF sidebands. We record those amplitudes to be used as the
normalization factors later. Now, we repeat the same calculations for
higher order modes. We are particularly interested in the optical power
in the PRC, since it is where we want to suppress the circulation of the 
HOMs. 

After calculating the HOM optical power for the carrier and the RF SBs
in the PRC, we take the ratio of the HOM power to the TEM00 power. The
results are plotted in Figure\,\ref{HOM Power Ratio PRC}. The horizontal
axis is the HOM order $s$, which is defined as $s = n+m$ for
$\mr{HG}_{nm}$ modes and $s = 2p+l$ for $\mr{LG}_{pl}$ modes. The
vertical axis is the power ratio of the HOMs and the fundamental mode
(TEM00). The graph shows, for each mode order, how much power can be
built up given the input beam is purely in the corresponding HOM. Since
the HOM components are much smaller than the fundamental mode in
reality, the actual power build up of HOMs will be much smaller than the
values shown in the graph.

\subsubsection{Diffraction Loss}
When calculating the power in the PRC, we incorporated the fact that the
diffraction loss of the mirrors is higher for HOMs because of their larger
spatial extent. It was done by first calculating the diffraction loss of
the mirrors to each higher order mode. Actually, we only considered the
diffraction losses in the ITMs and ETMs, because on other mirrors, the
beam spot sizes are small enough compared to the mirror size, thus the
diffraction loss is negligible. 

The diffraction loss was calculated by integrating the power profile of
$\mr{HG}_{nm}$ or $\mr{LG}_{pl}$ modes on the mirror surface. Then we
get a value $A$ which is the fraction of the optical power captured by
the aperture of the mirror.  The diffraction loss $L_\mr{d}$ is
$L_\mr{d}=1 - A$. For the same mode order $s$, there are different basis
modes, e.g. $\mr{HG}_{30}$, $\mr{HG}_{21}$ and $\mr{HG}_{12}$ all belong
to $s=3$. We used the smallest value of A in the family of HOMs to
evaluate the worst case scenario. Assuming that the mirror's intrinsic
reflectivity $R$ and the transmissivity $T$ satisfy $R+T=1$, i.e. loss
less, we get $A\cdot (R+T) + L_\mr{d}=1$.  Thus, the effective
reflectivity and transmissivity of the mirror including the diffraction
loss are $A\cdot R$ and $A\cdot T$ respectively. We replace the TM
reflectivities and transmissivities with those numbers when calculating
the field amplitudes for HOMs.

\subsubsection{Gouy Phase Scan}
Figure\,\ref{HOM Power Ratio PRC} was calculated using the one-way Gouy
phase shift of 16.4 degrees for PRC ($\eta_\mr{PRC}$) and 13 degrees for
SRC ($\eta_\mr{SRC}$). Now we will scan the values of the Gouy phase
shift to find the optimal ones. To do so, for each set of
($\eta_\mr{PRC}, \eta_\mr{SRC}$), we compute the PRC power for HOMs, as
shown in Figure\,\ref{HOM Power Ratio PRC}. Then take the maximum of
those values. The procedure is repeated by scanning the Gouy phase shift
values in the two-dimensional plane of ($\eta_\mr{PRC}, \eta_\mr{SRC}$).
The results are shown in
Figures\,\ref{PRC-2DHOMScan-Carrier}\,-\,\ref{PRC-2DHOMScan-Total}.
By looking at Figure\,\ref{PRC-2DHOMScan-Total}, which shows basically
the maximum of
Figures\ref{PRC-2DHOMScan-Carrier}\,-\,\ref{PRC-2DHOMScan-f2}, the area
around ($\eta_\mr{PRC}, \eta_\mr{SRC}$) = (16.5, 13.0) looks like a good
candidate for the Gouy phase shifts. Zoom ups of the region are shown in
Figures\,\ref{PRC-2DHOMScan-Carrier-Zoom}\,-\,\ref{PRC-2DHOMScan-Total-Zoom}. From
those plots, we decided that the one-way Gouy phase shifts of the RCs to
be 16.4\,degrees for PRC and 13\,degrees for SRC.

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/HOM-in-PRC-BRSE.pdf}
\caption{HOM power ratio in the PRC: BRSE.}
\label{HOM Power Ratio PRC}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PRC-2DHOMScan-Carrier-BRSE.pdf}
\caption{The maximum HOM power ratio of the carrier in the PRC: BRSE.}
\label{PRC-2DHOMScan-Carrier}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PRC-2DHOMScan-f1-BRSE.pdf}
\caption{The maximum HOM power ratio of the f1 SB in the PRC: BRSE.}
\label{PRC-2DHOMScan-f1}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PRC-2DHOMScan-f2-BRSE.pdf}
\caption{The maximum HOM power ratio of the f2 SB in the PRC: BRSE.}
\label{PRC-2DHOMScan-f2}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PRC-2DHOMScan-Total-BRSE.pdf}
\caption{The maximum HOM power ratio of the total field in the PRC: BRSE.}
\label{PRC-2DHOMScan-Total}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PRC-2DHOMScan-Carrier-Zoom-BRSE.pdf}
\caption{The maximum HOM power ratio of the carrier in the PRC, zoomed: BRSE.}
\label{PRC-2DHOMScan-Carrier-Zoom}
 \end{center}
\end{figure} 

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PRC-2DHOMScan-f1-Zoom-BRSE.pdf}
\caption{The maximum HOM power ratio of the f1 SB in the PRC, zoomed: BRSE.}
\label{PRC-2DHOMScan-f1-Zoom}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PRC-2DHOMScan-f2-Zoom-BRSE.pdf}
\caption{The maximum HOM power ratio of the f2 SB in the PRC, zoomed: BRSE.}
\label{PRC-2DHOMScan-f2-Zoom}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PRC-2DHOMScan-Total-Zoom-BRSE.pdf}
\caption{The maximum HOM power ratio of the total field in the PRC,
  zoomed: BRSE.}
\label{PRC-2DHOMScan-Total-Zoom}
 \end{center}
\end{figure}

\subsubsection{DRSE}
The above calculations are all done assuming the BRSE
configuration. When the SRC is detuned, the resonant conditions for the
HOMs in the SRC are also changed. Mostly the f1 SB is effected by the
detuning. Figure\,\ref{PRC-2DHOMScan-Total DRSE} and
Figure\,\ref{PRC-2DHOMScan-Total-Zoom DRSE} show the results of the same
Gouy phase scanning as Figure\,\ref{PRC-2DHOMScan-Total} and
Figure\,\ref{PRC-2DHOMScan-Total-Zoom}, but the SRC is detuned.  It is
evident that the difference is very small.

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PRC-2DHOMScan-Total-DRSE.pdf}
\caption{The maximum HOM power ratio of the total field in the PRC: DRSE.}
\label{PRC-2DHOMScan-Total DRSE}
 \end{center}
\end{figure}


\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PRC-2DHOMScan-Total-Zoom-DRSE.pdf}
\caption{The maximum HOM power ratio of the total field in the PRC,
  zoomed: DRSE.}
\label{PRC-2DHOMScan-Total-Zoom DRSE}
 \end{center}
\end{figure}

\subsubsection{Higher Order Modes in SRC}
One more thing to check is the HOM resonances in the SRC. Since the f2
SB is perfectly reflected by MICH, the SRC Gouy phase shift is not a
concern for f2. The f1 SB resonates in the compound cavity formed by the
PRC and SRC, resulting in the strong correlation of the power in the PRC
and SRC. Therefore, it is sufficient to check the PRC power for this
RFSB. The situation for the carrier is a bit complicated. In the normal
operation state, the fundamental mode of the carrier is not resonant in
the SRC. More specifically, the carrier is resonant in the SRC by
itself. However, because the carrier is also resonant in the arm
cavities, the reflectivities of the ITMs flip the sign and the carrier
becomes anti-resonant to the SRC. Therefore, it does not make sense to
the TEM00 power as a normalization factor. Instead, we decided to take,
for each HOM, the ratio of the power in the SRC with the designed
$\eta_\mr{SRC}$ and the power of the same HOM when the SRC is completely
degenerated, i.e. $\eta_\mr{SRC}=0$. The results are shown in
Figure\,\ref{SRC-Carrier HOM resonances BRSE} and
Figure\,\ref{SRC-Carrier HOM resonances DRSE} for the BRSE and DRSE
cases respectively. To calculate the ratio, we injected a beam from the
back of the SRM and for each HOM, calculated the power in the SRC with
$\eta_\mr{SRC}=13.0^\circ$ and $\eta_\mr{SRC}=0^\circ$.

According to Figure\,\ref{SRC-Carrier HOM resonances BRSE}, lower order
modes are well suppressed by making the SRC non-degenerated. There is a
peak at order 14. However, this is not a serious concern, because at
this high order number, even the completely degenerated SRC has a low
power build up gain, due to the diffraction loss. Therefore, the actual
power gain for this mode is not high. Figures\,\ref{SRC-Carrier HOM
resonances RawGain BRSE} and \,\ref{SRC-Carrier HOM resonances RawGain
DRSE} show the raw power gain of the SRC when a beam is injected from
the back of SRM. Even at the peak of mode order = 14, the power gain is
about 1, i.e. not amplified. Note that the SRC has a power gain of 25
for lower order modes if it is completely degenerated.

Overall, the DRSE configuration tends to allow more HOMs to resonate in
the SRC in the lower mode order. This is because the detuning of the SRC
by $3.5^\circ$ cancels out with the Gouy phase change of
$13.0^\circ$. Thus, the net detuning of the 1st order mode, for example,
from the complete resonance is $9.5^\circ$ instead of $13.0^\circ$.

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/Carrier-HOM-in-SRC-BRSE.pdf}
\caption{The SRC HOM power build up compared to the completely
  degenerated SRC: BRSE.}
\label{SRC-Carrier HOM resonances BRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/Carrier-HOM-in-SRC-DRSE.pdf}
\caption{The SRC HOM power build up compared to the completely
  degenerated SRC: DRSE.}
\label{SRC-Carrier HOM resonances DRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/Carrier-HOM-in-SRC-RawGain-BRSE.pdf}
\caption{The SRC HOM power build up gain: BRSE.}
\label{SRC-Carrier HOM resonances RawGain BRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/Carrier-HOM-in-SRC-RawGain-DRSE.pdf}
\caption{The SRC HOM power build up gain: DRSE.}
\label{SRC-Carrier HOM resonances RawGain DRSE}
 \end{center}
\end{figure}

%}}}

%{{{ \subsection{Radius of Curvature of the RC mirrors}
\subsection{Radius of Curvature of the RC mirrors}


The ROCs of the folding mirrors are set to realize the desired Gouy
phase shift.  There are many combinations of the ROCs to achieve
this. We selected a design which works as a beam reducing telescope. In
this design, the beam coming back from the arm cavities, having the beam
radius of 3.5\,cm, is focused by the concave PR3 mirror down to 4.5\,mm on
the PR2.  Then the convex PR2 collimates the beam, to have the same beam
size on the PRM. This design is very convenient because we do not have
to handle large beams at the reflection (REFL) port and the
POP\footnote{Pick Off in the Power recycling cavity} port, which is the
beam transmitted through the PR2. 4.5\,mm beams can directly taken in and
out of the vacuum chambers without using large optical windows, which
are expensive and tend to be fragile.

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 (3.6\,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.

%}}}

%}}}

%{{{ \section{Output Mode Cleaner}


\section{Output mode-cleaner system}

An output mode-cleaner (OMC) filters out junk light coming out to the dark port of the interferometer. The junk light here means any light other than the fundamental mode of the probe beam, namely the average mode of the two arm cavities, at the carrier light frequency. In fact, it is rather reasonable to say that any light other than the fundamental mode of the OMC will be filtered out. Thus, the OMC should be designed first of all not to lose the signal fields at the transmission. It is then also required for the OMC not to let higher-order modes come close to its resonance so that the unused light contributes to increase shot noise. In addition to the higher-order spatial modes of the carrier light, the RF sideband fields should be filtered out by the OMC, otherwise the sideband fields just contribute to increase shot noise in the DC readout scheme. The g-factor of the OMC should be carefully chosen to satisfy all of these conditions.

\begin{figure}[htbp]
	\begin{center}
	 \includegraphics[width=5cm]{figures/OMC.eps}
	\caption{Output mode-cleaner system.}
	\label{fig:OMCsystem}
	\end{center}
\end{figure}

The output mode-cleaner system consists of the OMC and the output mode-matching telescope (OMMT). There will be also a suspended output Faraday Isolator between the first OMMT mirror (OMMT1) and the SRM to avoid the back scattering into the interferometer. Figure~\ref{fig:OMCsystem} shows a schematic view of the system.

For the OMC design, we should first simulate how much higher order modes will come out to the dark port with reasonable values for the test mass RoCs. A simulation software FINESSE with the modal decomposition method is used for the calculation. The mirror size is infinite in the simulation. The maximum mode in the calculation is 5, which is just barely fine. It would be better to cross-check the result with other codes preferably with finite-size mirrors and further higher order modes.

The RoCs of the test masses are 1680~m for the ITMs and 1870~m for the ETMs (concave-concave cavities). A 1~\% error of the RoC is imposed in various ways: commonly in the ITMs (X and Y), commonly in the ETMs, differentially in the ITMs, and differentially in the ETMs. The 1~\% common error of the RoCs can be compensated by tuning the distance of the SR2 and SR3 by about 14~cm. This corresponds to the calculation result for power recycling.~\cite{ROC Error}. Hereafter we shall focus on the differential error of the RoCs. The differential error causes a shot noise increase in two different ways. One is for a loss of the signal field and the other is for an increase of higher-order modes. Figure~\ref{fig:HOM} shows the simulation results. A remarkable difference between ITM errors and ETM errors can be seen in each plot. The reason of the difference is still unknown and is to be investigated.

\begin{figure}[htbp]
	\begin{center}
		\includegraphics[width=7.5cm]{plots/100Hz.eps}
		\includegraphics[width=7.5cm]{plots/junklight.eps}
	\caption{{\it Left}: Signal loss due to the differential RoC errors of the test masses, and {\it Right}: Shot noise increase due to the junk light caused by the differential RoC errors of the test masses.}
	\label{fig:HOM}
	\end{center}
\end{figure}

Table~\ref{tab:HOM} shows the amount of the light in Hermite Gaussian modes at the dark port before the OMC. Here the optical loss of the test masses are 45~ppm in average and $\pm4$~ppm imbalance is assumed between the arm cavities. The finesse imbalance of 0.5~\% and the differential RoC error of 1~\% are also assumed. The higher order modes and the RF sideband fields should be less than a few percent of the reference light (TEM00 carrier light) after the OMC. Required suppression rates for RF sideband TEM00, carrier light TEM20/02, and other modes are 80~dB, 70~dB, and 45~dB, respectively, which corresponds to the shot noise increase of 2~\%, 0.5~\%, and $<0.5$~\%, respectively. With the optical loss of 30~ppm being assumed for each OMC mirror, the finesse should be less than 1000 so that the signal loss can be 2~\% or less. The shot noise increment at the OMC is assumed to be 5~\% or less in total. Should the mirror RoC errors be 2~\% instead, the amount of the second order modes would increase by about a factor of 4.

\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
&TEM00&TEM20&TEM02&TEM40&TEM04&TEM22\\ \hline
RF&85mW&0.1mW&0.1mW&4$\mu$W&4$\mu$W&3$\mu$W\\ \hline
\hline
&TEM00&TEM20&TEM02&TEM40&TEM04&TEM22\\ \hline
DC&1.0mW&8.9mW&8.9mW&30$\mu$W&30$\mu$W&20$\mu$W\\ \hline
\end{tabular}
\caption{Amount of the light in each mode at the dark port before the OMC.}
\label{tab:HOM}
\end{center}
\end{table}

The length and the g-factor of the OMC should be carefully chosen not to let higher order modes come close to the resonance. The longer the OMC, the harder to find a safe region where any mode is far from the resonance. On the other hand, a sufficient length is necessary to reduce the RF sideband TEM00 field. With the finesse of 800, the OMC should better be longer than 90~cm to suppress the RF sideband by 80~dB. Figure~\ref{fig:modes} shows the frequency margin to the closest resonance of the higher order modes up to the 8th mode. The 16.875~MHz RF sideband and its higher order modes are also included. Scanning through the OMC Gouy phase, we have found good regions at around 19~deg, 38~deg, 80~deg, and 99~deg, and the 38~deg is the best in terms of the suppression rate of the 2nd order modes. The second harmonics of the RF sideband, which is not included in the plot, turns out to come close to a resonance, which can be well avoided by tuning the OMC length to be 87~cm.

\begin{figure}[htbp]
	\begin{center}
		\includegraphics[width=10cm]{plots/modes3.eps}
	\caption{Frequency margin to the closest resonance of any higher order modes up to the 8th mode. The black, red, orange, green and blue curves correspond to the OMC length of 90, 80, 70, 60, and 50~cm, respectively. The horizontal line at the center indicates the suppression rate of 45~dB.}
	\label{fig:modes}
	\end{center}
\end{figure}

Since the incident angle to the curved OMMTs is non-zero, the beam is elliptical and the mode matching cannot be perfect with the spherical OMC mirrors. The Gouy phases for the x and y axes are different by about 0.5~deg. With the chosen setup parameters, the Gouy phases for the x and y axes are 37.7~deg and 38.3~deg, respectively. Table~\ref{tab:HOM2} shows the amount of the light fields after the OMC and Table~\ref{tab:OMCsetup} shows the setup parameters for the output mode-cleaner system.

\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
&TEM00&TEM20&TEM02&TEM40&TEM04&TEM22\\ \hline
RF&4$\mu$W&4nW&4nW&$<1$nW&$<1$nW&$<1$nW\\ \hline
\hline
&TEM00&TEM20&TEM02&TEM40&TEM04&TEM22\\ \hline
DC&980$\mu$W&100nW&100nW&$<1$nW&$<1$nW&$<1$nW\\ \hline
\end{tabular}
\caption{Amount of the light in each mode at the dark port after the OMC.}
\label{tab:HOM2}
\end{center}
\end{table}


\begin{table}[htbp]
\begin{center}
\begin{tabular}{c|c}
item&value\\ \hline
SRM-OMMT1 length&6.5~m\\
OMMT1-OMMT2 length&6.2~m\\
OMMT2-OMC1 length&6.266~m\\
OMC round-trip length&1.74~m\\
OMMT1 RoC&34.7~m\\
OMMT2 RoC&34.7~m\\
OMMT1 incident angle&1.9~deg\\
OMMT2 incident angle&1.9~deg\\
OMC mirrors incident angle&6.7~deg\\
OMC1 RoC&7.2~m\\
OMC2-4 RoC&flat\\
beam radius on OMMT1&4.1~mm\\
beam radius on OMMT2&4.3~mm\\
beam radius on OMC1&0.95~mm\\
power reflectivity of OMC1&99.6~\%\\
power reflectivity of OMC3&99.6~\%\\
OMC suspension&Type-C\\
OMC material&Aluminum\\
\end{tabular}
\caption{Setup parameters of the output mode-cleaner system.}
\label{tab:OMCsetup}
\end{center}
\end{table}

%}}}

%}}}

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

\chapter{Length Sensing and Control Scheme}
\label{bKAGRA 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}

%{{{ \section{Overview}

\section{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). These degrees of freedom are collectively
called canonical degrees of freedom in this document. 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 a 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{SRCL non-linearity}.

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}


%}}}

%{{{ \section{Simulation Conditions}

\section{Simulation Conditions}
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 KAGRA svn at \cite{LSC
Code}.

%{{{ \subsubsection{Arm Cavity Asymmetry}

\subsubsection{Arm Cavity Asymmetry}
\label{Arm Cavity Asymmetry}
When performing simulations, we assumed the asymmetry of the arm cavity
loss (round trip) to be $\pm$30\,ppm, while the average loss is 100\,ppm
per round trip. This rather high asymmetry corresponds to about
$\pm$1.5\% asymmetry of the finesse. As a consequence, the DC carrier
power at the AS port without the differential arm cavity length offset
is about 4\,mW. We call this mis-match light or field, because it is
produced by the loss mis-match between the two arm cavities.

The mis-match light itself is useless as a local oscillator for the DC
readout, because it is orthogonal to the GW SBs. However, when combined
with the DC field produced by the differential offset of the arm cavity
lengths (we call it offset light), the homodyne angle can be set to be
other than 90 degrees. By properly choosing the homodyne angle, we can
perform the back action evasion (BAE) measurement, which can beat the
standard quantum limit (SQL). The default homodyne angle of KAGRA for
BRSE is 58\,degrees, which requires about 10\,mW of the offset light,
corresponding to $\pm$2\,pm of DARM offset. For BRSE, the homodyne angle
is 45\,degrees. The offset light is 4\,mW corresponding to $\pm$1.3\,pm
of DARM offset.

The amount of the loss asymmetry is not something we can precisely
control. Therefore, the intensity of the mis-match light may be
significantly larger or smaller than 4\,mW. If the asymmetry is very
large and the mis-match field is too strong, we consider it a failure of
the mirror manufacturing. We have to avoid it by careful quality control
of the mirror fabrication\footnote{which is extreme difficult. We know
it.}.  If the loss imbalance is too small, the required offset light
power to realize the BAE becomes very small. This makes the requirements for
the OMC very stringent, because the OMC has to reduce the RF SB and
other junk fields well below the small local oscillator field.  One
solution to this problem is to increase the loss of one of the mirrors
by, for example, slightly staining the surface. However, this is
probably not a good idea and very difficult to do in a controlled
manner. It is very likely that we damage the mirror. Thus not
recommended. Another option is just to abandon the BAE. In this case, we
lose 10 to 20\,Mpc of the inspiral range\footnote{Need to check the numbers.}.

%}}}

%{{{ \subsubsection{PD}
\subsubsection{PD}

We all know that there is no good RF PD which can receive 1\,W of light
power. In the simulation, the DC light power falling on to any RF PD is
limited to be equal to or smaller than 50\,mW by, if necessary,
inserting an attenuator, so that the shot noise estimates for the
auxiliary DOFs are realistic.

%}}}


%}}}

%{{{ \section{Signal Name Convention}

\section{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.

%}}}

%{{{ \section{Signal Extraction Ports}

\section{Signal Extraction Ports}

The default length sensing scheme for bKAGRA uses two phase modulated RF
sidebands, f1 and f2. The beat notes of these sidebands with the carrier
are detected at the REFL and the POP ports to extract necessary error
signals. The signal sensing matrices are shown in Table\,\ref{Sensing
Matrix LSC SDM BRSE} and Table\,\ref{Sensing Matrix LSC SDM DRSE}. The
diagonal elements are the signals we plan to use as error signals. The
shot noise of each signal is shown in Table\,\ref{Shot noise matrix LSC
SDM BRSE} and Table\,\ref{Shot noise matrix LSC SDM DRSE}. The shot
noise matrices are calculated by first dividing the signal strength (in
[W/m]) of each DOF at each signal port by the shot noise level (in
[$\mr{W/\sqrt{Hz}}$]) of the corresponding signal port, then inverting
the result, to get the displacement equivalent noise in the unit of
[$\mr{m/\sqrt{Hz}}$].

The 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, the
CARM feedback 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 (the gain
hierarchy approach).

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. The
$\pm$30\,ppm loss asymmetry, as described in section\,\ref{Arm Cavity
Asymmetry}, makes the CARM, PRCL and SRCL signals no longer be at
exactly the orthogonal quadrature to the MICH signal. 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 $3\times 3$ matrices of Table\,\ref{Sensing Matrix
LSC SDM BRSE} and Table\,\ref{Sensing Matrix LSC SDM DRSE} are 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}&1.0&$3.3\times 10^{-6}$&$7.2\times 10^{-4}$&$1.8\times 10^{-7}$&$5.0\times 10^{-5}$\\\hline
{\bf REFL\_1I}&$9.6\times 10^{-3}$&1.0&$5.0\times 10^{-3}$&$6.2\times 10^{-2}$&$3.0\times 10^{-2}$\\\hline
{\bf REFL\_1Q}&$7.1\times 10^{-3}$&$2.6\times 10^{-4}$&1.0&$8.5\times 10^{-2}$&$2.5\times 10^{-2}$\\\hline
{\bf POP\_2I}&$5.4\times 10^{-2}$&5.7&$1.8\times 10^{-2}$&1.0&$2.7\times 10^{-4}$\\\hline
{\bf POP\_1I}&$1.8\times 10^{-1}$&19.0&$1.1\times
	       10^{-1}$&2.1&1.0\\\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 BRSE}
 \end{center}
\end{table}

\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}&1.0&$6.6\times 10^{-6}$&$7.1\times 10^{-4}$&$5.8\times 10^{-7}$&$5.8\times 10^{-5}$\\\hline
{\bf REFL\_2I}&$1.4\times 10^{-2}$&1.0&$4.0\times 10^{-3}$&$4.3\times 10^{-2}$&$1.5\times 10^{-4}$\\\hline
{\bf REFL\_1Q}&$2.5\times 10^{-1}$&1.3&1.0&$6.5\times 10^{-2}$&$3.4\times 10^{-2}$\\\hline
{\bf POP\_2I}&$7.7\times 10^{-2}$&5.7&$1.8\times 10^{-2}$&1.0&$1.5\times 10^{-4}$\\\hline
{\bf POP\_1I}&$4.7\times 10^{-1}$&32.8&$1.7\times
	       10^{-1}$&2.0&1.0\\\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}

\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}&$6.7\times 10^{-21}$&$2.0\times 10^{-15}$&$9.3\times 10^{-18}$&$3.8\times 10^{-14}$&$1.3\times 10^{-16}$\\\hline
{\bf REFL\_1I}&$1.0\times 10^{-16}$&$1.0\times 10^{-18}$&$2.0\times 10^{-16}$&$1.6\times 10^{-17}$&$3.3\times 10^{-17}$\\\hline
{\bf REFL\_1Q}&$3.7\times 10^{-14}$&$1.0\times 10^{-12}$&$2.6\times 10^{-16}$&$3.1\times 10^{-15}$&$1.1\times 10^{-14}$\\\hline
{\bf POP\_2I}&$2.8\times 10^{-15}$&$2.6\times 10^{-17}$&$8.5\times 10^{-15}$&$1.5\times 10^{-16}$&$5.5\times 10^{-13}$\\\hline
{\bf POP\_1I}&$1.4\times 10^{-15}$&$1.3\times 10^{-17}$&$2.3\times
	       10^{-15}$&$1.2\times 10^{-16}$&$2.5\times
		       10^{-16}$\\\hline

  \end{tabular}
\caption{Shot noise matrix of LSC in the case of BRSE. The
  numbers represent the displacement equivalent shot noise
  [$\mr{m/\sqrt{Hz}}$].  The interferometer response was evaluated at
  100\,Hz to create this matrix.}
\label{Shot noise matrix LSC SDM BRSE}
 \end{center}
\end{table}

\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}&$4.6\times 10^{-21}$&$6.9\times 10^{-16}$&$6.4\times 10^{-18}$&$7.9\times 10^{-15}$&$7.9\times 10^{-17}$\\\hline
{\bf REFL\_2I}&$4.3\times 10^{-16}$&$5.8\times 10^{-18}$&$1.4\times 10^{-15}$&$1.4\times 10^{-16}$&$3.8\times 10^{-14}$\\\hline
{\bf REFL\_1Q}&$7.2\times 10^{-16}$&$1.3\times 10^{-16}$&$1.8\times 10^{-16}$&$2.8\times 10^{-15}$&$5.3\times 10^{-15}$\\\hline
{\bf POP\_2I}&$1.9\times 10^{-15}$&$2.6\times 10^{-17}$&$8.5\times 10^{-15}$&$1.5\times 10^{-16}$&$1.0\times 10^{-12}$\\\hline
{\bf POP\_1I}&$1.1\times 10^{-15}$&$1.6\times 10^{-17}$&$3.2\times 10^{-15}$&$2.6\times 10^{-16}$&$5.3\times 10^{-16}$\\\hline
  \end{tabular}
\caption{Shot noise matrix of LSC in the case of DRSE. The
  numbers represent the displacement equivalent shot noise
  [$\mr{m/\sqrt{Hz}}$].  The interferometer response was evaluated at
  100\,Hz to create this matrix.}
\label{Shot noise 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). Because the carrier is not involved in the signal generation of
MICH, PRCL and SRCL, 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. The addition of
Mach-Zehnder may introduce additional noise. Moreover, the generation of
AM sidebands is not as easy as one might think. A simple AM generator
wastes a lot of laser power\,\cite{Ohmae Thesis}. A clever idea is
proposed to avoid this problem\,\cite{Ohmae Thesis}. However, it is
still at a proof-of-concept stage.

There is another potential problem with the f3 scheme. In this scheme,
the carrier does not contribute to the the signal generation of the
central part. Therefore, it just increase the shot noise if present at
detection ports. At REFL, the amount of the carrier power returning from
the interferometer depends on the reflectivity matching of the PRM and
the Fabry-Perot Michelson part. The reflectivity of the FPMI depends
heavily on the loss of the arm cavities, hence very difficult to
precisely control. If the matching is poor, a lot of carrier power will
come back to the REFL port. This has a potential of increasing the shot
noise of the signals detected at REFL by a large factor. 

For the above mentioned reasons, we do not employ the f3 scheme during
the observation mode of KAGRA. However, we can still use this scheme
during the lock acquisition, where the noise is not so important but the
stability and robustness of the error signals is critical.

\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}


%}}}

%{{{ \section{Loop Noise}

\section{Loop Noise}

%{{{ \subsection{Servo loop model}
\subsection{Servo loop model}


%{{{ Control Loop Diagram

\begin{figure}[tbp]
\begin{center}
\begin{minipage}{8cm}
\includegraphics[width=8cm]{figures/ControlLoopDiagram.pdf}
\end{minipage}
\begin{minipage}{6cm}
$
 \left(
\begin{array}{c}
 e_\mr{DARM}\\
 e_\mr{CARM}\\
 e_\mr{MICH}\\
 e_\mr{PRCL}\\
 e_\mr{SRCL}
\end{array}
\right)
=D\cdot
\left(
\begin{array}{c}
 x_\mr{ITMX}\\
 x_\mr{ITMY}\\
 x_\mr{ETMX}\\
\vdots\\
 x_\mr{SRM}\\
 x_\mr{freq}\\
 x_\mr{intensity}\\
\vdots
\end{array}
\right)
$
\end{minipage}
\caption{Block diagram of the feedback loops. The real DOF vector
 $\vec{x}$ contains the displacement of each mirror and other dynamic
 degrees of freedom in the interferometer, such as laser frequency and
 intensity. It is converted to the vector $\vec{e}$ of the error signals
 in the canonical DOFs by the detector matrix $D$. All the matrices in
 the figure are frequency dependent.}  \label{Fig:Feedback Diagram}
\end{center}
\end{figure}
%}}}

In general, the auxiliary degrees of freedom have larger shot noise than
DARM. By using those signals for mirror control, we are effectively
injecting extra 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 this
factor. This noise coupling mechanism is called loop noise and one needs
to pay close attention to this when designing an interferometer control scheme.

Figure\,\ref{Fig:Feedback Diagram} shows a block diagram of the servo
loops for the interferometer control. The detector matrix $D$ converts
the real DOF vector $\vec{x}$, which represents the mirror
displacements and other dynamic elements in the interferometer, such as
laser frequency, into a vector of error signals $\vec{e}$ in the
canonical DOFs. Then the sensing noise vector, $\vec{n_\mr{s}}$, is
added to the error signal vector. $\vec{n_\mr{s}}$ represents shot noise
and any other noise added at the sensing stage, such as PD noise.

$D$ and $\vec{n_\mr{s}}$ are calculated by Optickle. The error signals
are filtered by a feedback filter $F$ and fed back to the mirrors
through the actuator matrix $A$, which converts feedback signals in the
canonical DOFs to the real DOFs of the interferometer. Then the disturbance
vector $\vec{n_\mr{d}}$ is added before the sensing matrix
$D$. $\vec{n_\mr{d}}$ represents disturbances to the interferometer,
such as the mirror displacement noises and the laser frequency noise.

The DARM error signal is the first element of the error signal vector
$\vec{e}$.  In the absence of gravitational waves, $\vec{e}$ is written
as,
\begin{equation}
\label{Loop Noise Formula}
 \vec{e} = (I+G)^{-1}\cdot \vec{n_\mr{s}} + (I+G)^{-1}\cdot
  D\cdot \vec{n_\mr{d}},
\end{equation}
\begin{equation}
\label{Define G}
 G\equiv D\cdot A\cdot (I+F')\cdot F,
\end{equation}
where $I$ is the identity matrix.  The off-diagonal elements of
$(I+G)^{-1}$ are responsible for the loop noise couplings.

The shot noise coupling by the control loops was calculated using the
above formula. We assumed the unity gain frequencies of the servo loops shown in
Table\,\ref{Control Loop UGFs}. 

\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& 50\,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}
and Figure\,\ref{Loop Noise Coupling: DRSE, SDM}. The curves labeled
Target are the target sensitivity. It is clear that loop noise couplings
from other degrees of freedom are larger than the target level.

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/ShotNoiseCoupling-BRSE.pdf}
\caption{Loop Noise Coupling: BRSE}
\label{Loop Noise Coupling: BRSE, SDM}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/ShotNoiseCoupling-DRSE.pdf}
\caption{Loop Noise Coupling: DRSE}
\label{Loop Noise Coupling: DRSE, SDM}
 \end{center}
\end{figure}

%}}}

%{{{ \subsection{Feed forward}
\subsection{Feed forward}

The loop noise coupling can be reduced by a technique called feed
forward\,\cite{LoopNoise}.  Its working principle is the
following. Taking MICH as an example, we can measure the transfer
function from the motion of BS\footnote{The feedback point for
MICH. Actually, we also move PRM and SRM by $1/\sqrt{2}$ to compensate
for the changes in PRCL and SRCL. } to the DARM signal. Then we assume
that the error signal of MICH is dominated by the shot noise (or any
sensing noise). This means that BS is moved by the shot noise through
the feedback. From the feedback signal, we know exactly how much the BS
is erroneously moved. Therefore, we can predict how much noise is
injected from this BS 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 feed forward path is
indicated in Figure\,\ref{Fig:Feedback Diagram} by $F'$.

The performance 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, which
means that the loop noise couplings can be reduced by a factor of 100.

In Figure\,\ref{Loop Noise Coupling FF: BRSE, SDM}and Figure\,\ref{Loop
Noise Coupling FF: DRSE, 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 can be reduced well below the
DARM quantum noise. 

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=11cm]{plots/ShotNoiseCouplingFF-BRSE.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=11cm]{plots/ShotNoiseCouplingFF-DRSE.pdf}
\caption{Loop Noise Coupling with Feed Forward: DRSE}
\label{Loop Noise Coupling FF: DRSE, SDM}
 \end{center}
\end{figure}

%}}}

%}}}

%{{{ \section{PD Dynamic Range}
\section{PD Dynamic Range}
\label{PD Dynamic Range}

The intrinsic noises of photo detectors is another important class of
sensing noise.  A PD always receives some offset light, either in RF or DC
depending on the type of PD. At some ports, these offset signals can be
very large. In this case, the dynamic ranges of the PD becomes an issue.

Typically, a low-noise fast operational amplifier (op-amp) used for the
current to voltage conversion of a PD has a dynamic range of about
200\,dB according to the catalog
specifications\,\cite{AD829_Data_Sheet}. However, because of the slew
rate limit, the actual dynamic range at RF is much smaller. Moreover, in
order to minimize the non-linearity of the detector response, we want to
use the op-amps at a much smaller signal level than the slew rate
limit. Therefore, for the following analysis, we assume the dynamic
range to be 160\,dB for RF PDs and 190\,dB for a DC PD.

Once the dynamic range $D$ is specified, the sensing noise, $n_\mr{pd}$,
of a PD, in terms of the equivalent signal light power on the PD, can be
expressed as $n_\mr{pd} = P_\mr{ofs}/D$, where $P_\mr{ofs}$ is the
offset signal power for the PD. Then we can simply replace
$n_\mr{s}$ in (\ref{Loop Noise Formula}) with $n_\mr{pd}$ to calculate
the loop noise couplings for the PD noise.

Figure~\ref{PD Noise BRSE} and Figure~\ref{PD Noise BRSE} show the
calculated PD noise couplings for BRSE and DRSE. The PD noises are large
in the DRSE mode, especially for MICH. It is because the SRC detuning
changes the relative phase of the f1 sidebands with the carrier so that
they no longer form a pure phase modulation.  The result is constant
large RF signals on the PDs for the signals using the f1
sidebands. Since the PD noise is a kind of sensing noise, with the use
of feed forward, it can be reduced. Figure\,\ref{PD Noise BRSE FF} and
Figure\,\ref{PD Noise DRSE FF} show the PD noise contributions after the
feed forward is applied. Even for the DRSE case, the noises are below the
target sensitivity.

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PDNoiseCoupling-BRSE.pdf}
\caption{PD noise coupling: BRSE.}
\label{PD Noise BRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PDNoiseCoupling-DRSE.pdf}
\caption{PD noise coupling: DRSE.}
\label{PD Noise BRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PDNoiseCouplingFF-BRSE.pdf}
\caption{PD noise coupling with feed forward: BRSE.}
\label{PD Noise BRSE FF}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/PDNoiseCouplingFF-DRSE.pdf}
\caption{PD noise coupling with feed forward: DRSE.}
\label{PD Noise DRSE FF}
 \end{center}
\end{figure}

%}}}

%}}}

%{{{ \chapter{Noise Requirements}
\chapter{Noise Requirements}
\label{Chapter: Noise Requirements}
In this chapter, contributions of various noise sources, such as mirror
displacement noise and laser noises, to the DARM error signal are
calculated to set the requirements to those noises. The formalism of the
calculation is basically the same for all the noise sources: Solving the
equation (\ref{Loop Noise Formula}) for $\vec{n_\mr{d}}$ and plugging in
the target sensitivity into the first element of $\vec{e}$, which is the
DARM error signal, we get the critical noise levels for all the
disturbances in $\vec{n_\mr{d}}$. The critical noise level is defined as
the amount of noise which produces the same noise level as the target
sensitivity at the DARM. Since there are many noise sources, we require
that contribution of each noise is 10 times smaller than the target
sensitivity. Assuming all the noises are uncorrelated, we can allow 100
different kinds of noise sources to present before compromising the target
sensitivity. Please note that {\em all the noise requirements below include
the safety factor of 10}.


%{{{ \section{Mirror Displacement Noise}

\section{Mirror Displacement Noise}
\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 degree 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 current estimates of the auxiliary suspension (Type-B SAS) seismic noises are
plotted alongside. Except for at several mechanical resonances, the
displacement noise requirements are met.

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=11cm]{plots/DispNoiseReq-BRSE.pdf}
\caption{Displacement noise requirements for auxiliary mirrors: BRSE}
\label{Displacement Noise Requirent for Each Mirror: BRSE, SDM}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=11cm]{plots/DispNoiseReq-DRSE.pdf}
\caption{Displacement noise requirements for auxiliary mirrors: DRSE}
\label{Displacement Noise Requirent for Each Mirror: DRSE, SDM}
 \end{center}
\end{figure}


%}}}

%{{{\section{Laser Noises}
\section{Laser Noises}
Contributions of the laser noises (frequency and intensity) are
estimated by adding phase and amplitude modulators before the PRM in the
Optickle model. The frequency noise requirement is calculated from the
phase noise requirement by multiplying it with $2\pi f$, where $f$ is
the frequency of interest. The amplitude noise can be converted to the
relative intensity noise (RIN) just by multiplying with 2\footnote{$dP/P
= 2\cdot dE/E$}.

\subsection{Frequency stabilization servo}
The frequency noise stabilization servo will be a complex multiple loop
system. A conceptual diagram of the frequency stabilization system (FSS)
is shown in Figure\,\ref{FSS Concept}.

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{figures/FSS-Topology.pdf}
\caption{Conceptual diagram of the frequency stabilization system}
\label{FSS Concept}
 \end{center}
\end{figure}

At the beginning, the laser is locked to a rigid cavity called pre mode
cleaner (PMC). The PMC serves as the absolute reference at low
frequencies, typically below a few Hz. Then the laser is locked to a
suspended MC. The error signal of the MC is fed back to the MC mirrors
at low frequencies, where the mirror displacement noise is large, and to
the error point of the PMC at high frequencies. Therefore, at high
frequencies the laser frequency is locked to the MC length. Finally, the
CARM used as the absolute reference of laser frequency in the
observation frequency band. The CARM error signal is, again, fed back to
the ETMs at low frequencies, and to the MC end mirror in the middle
frequency range. At very high frequencies (above 50kHz or so), the error
signal will be added  to the MC error point, which is directly passed to
the PMC error point.

Details of the FSS servo topology is yet to be designed. Therefore, in
the next section, we treat whole the input laser system, from the laser source
to the output of the MC, as a black box. Then we assume that the CARM
error signal is fed back to the frequency actuator of this black
box. The feedback UGF is assumed to be 10\,kHz. The frequency noise
requirement calculated with this model is equivalent to the frequency
noise requirement to the MC output.


\subsection{Frequency noise requirement}
The frequency noise requirements with the safety factor of 10 are shown
in Figure\,\ref{Laser freq noise requirement BRSE} and
Figure\,\ref{Laser freq noise requirement DRSE}. As mentioned above,
these are the frequency noise requirements to the MC output. The
frequency noise of the MC output is ultimately determined by the
displacement noise of the MC mirrors (seismic and thermal). Therefore,
it is interesting to convert the frequency noise requirements to the
displacement noise requirements. Figure\,\ref{MC disp requirement BRSE}
and Figure\,\ref{MC disp requirement DRSE} show these requirements
alongside the estimated MC suspension seismic noise.

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/MCFreqReq-BRSE.pdf}
\caption{Laser frequency noise requirement at the output of the MC: BRSE}
\label{Laser freq noise requirement BRSE}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/MCFreqReq-DRSE.pdf}
\caption{Laser frequency noise requirement at the output of the MC: DRSE}
\label{Laser freq noise requirement DRSE}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/MCDispReq-BRSE.pdf}
\caption{Displacement noise requirement for the MC suspension: BRSE}
\label{MC disp requirement BRSE}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/MCDispReq-DRSE.pdf}
\caption{Displacement noise requirement for the MC suspension: DRSE}
\label{MC disp requirement DRSE}
 \end{center}
\end{figure}


\subsection{Intensity noise}
The requirements on the relative intensity noise (RIN) of the laser are
shown in Figure\,\ref{RIN Req BRSE} and Figure\,\ref{RIN Req
DRSE}. Since the Optickle can simulate the radiation pressure properly,
the calculated results include the noise generation by the radiation
pressure induced mirror motion.

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/RINReq-BRSE.pdf}
\caption{Relative Intensity Noise (RIN) requirement: BRSE}
\label{RIN Req BRSE}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/RINReq-DRSE.pdf}
\caption{Relative Intensity Noise (RIN) requirement: DRSE}
\label{RIN Req DRSE}
 \end{center}
\end{figure}

%}}}

%{{{ \section{RF Oscillator Noises}
\section{RF Oscillator Noises}
Phase and amplitude noises of the RF oscillators driving the phase
modulators to generate the RF SBs produce noises in the error signals of
the auxiliary DOFs, which use the RF SBs for error signal extraction.
The feedback of those noises into the mirrors inject noises into DARM
error signal, even though it uses DC readout.

\subsection{Phase Noise}
It is a common practice to express the phase noise of an oscillator in terms of the
single sideband (SSB) phase noise. It is the power ratio of the carrier
and the sideband which is generated by the phase noise at a particular
frequency offset from the carrier frequency. The unit is
dBc. Figure\,\ref{SSB Req BRSE} and Figure\,\ref{SSB Req DRSE} show the
phase noise requirements of the RF oscillators in terms of SSB phase
noise.

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SSB-Req-BRSE.pdf}
\caption{SSB Phase noise requirements: BRSE}
\label{SSB Req BRSE}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SSB-Req-DRSE.pdf}
\caption{SSB Phase noise requirements: DRSE}
\label{SSB Req DRSE}
 \end{center}
\end{figure}

Good commercially available oven controlled crystal oscillators
(OCXOs) have SSB phase noise of better than -150dBc. However, the
requirement of DRSE (about -180dBc at a few tens of Hz) is extremely
difficult to meet. One reason for the requirement being so stringent
for DRSE is that, as explained in section\,\ref{PD Dynamic Range}, the
SRC detuning causes the conversion of the purely phase modulated f1 SB
into a mixture of amplitude and phase modulations. This AM creates
offset in the error signals using the f1 SB. Then the oscillator phase
noise directly changes the amount of the offset. A possible solution to
this problem is to prepare the f1 SB as a mixture of PM and AM in the
beginning. This idea is discussed in appendix\,\ref{PM + AM for f1}.


\subsection{Amplitude Noise}
The requirements on the relative amplitude noise of the RF oscillators
are shown in Figure\,\ref{RF AM Req BRSE} and Figure\,\ref{RF AM Req
DRSE}. Please note that these are {\em amplitude} noise, not {\em intensity}.

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/RFAM-Req-BRSE.pdf}
\caption{RF oscillator amplitude noise requirements: BRSE}
\label{RF AM Req BRSE}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/RFAM-Req-DRSE.pdf}
\caption{RF oscillator amplitude noise requirements: DRSE}
\label{RF AM Req DRSE}
 \end{center}
\end{figure}

%}}}

%{{{ \section{Scattered Light Noise}
\section{Scattered Light Noise}
A generic model of scattered light noise can be constructed as follows: A stray
beam from the interferometer hits something, like a wall of a vacuum
chamber. The surface of this object is vibrating so that the reflected
light from the surface gets some phase fluctuations. A small fraction of the
light reflected or scattered by this surface comes back to the
interferometer. This scattered light field returning to the
interferometer is denoted by $E_\mr{SCL}$. Assuming the power spectrum
density of the surface vibration is $A(\omega)\mr{[m/\sqrt{Hz}]}$, the
phase fluctuation of $E_\mr{SCL}$ is $2 k A(\omega)$, where $k$ is
the wave-number of light. Once entered into the interferometer,
$E_\mr{SCL}$ propagates through the interferometer and interferes with
the other fields to create noises in the error signals. 

The contributions of scattered light to the interferometer outputs are
calculated with the Optickle model by adding injection ports for
scattered light at several locations. For example, in order to simulate
scattered light injection to the back of PRM, a low reflectivity
pick-off mirror of 50\,ppm is inserted just before the main beam from
the MC hits the back of the PRM. Then a light source for simulating a
scattered light field is added to the model. The beam from this source
is passed through a phase modulator and injected to the interferometer
through the pick-off mirror. The transfer functions from this phase
modulator to various signal ports of the interferometer are calculated
to estimate the magnitudes of scattered light noise couplings.

The relative phase between the additional light source and the carrier
field is important. In the real interferometer, this is a random
number. In the simulation, we injected the simulated scattered light in
two orthogonal phases relative to the carrier. Then we took the squared
sum of the transfer functions from the two injection phases to estimate
the worst noise coupling.

The phase modulator explained above is a kind of disturbance to the
interferometer.  So it can be included as an element of $\vec{n_\mr{d}}$. Then
the requirements on this disturbance can be calculated as described in
\ref{Chapter: Noise Requirements}.  Since our model includes the
feedback and feed forward loops, the scattered light noise couplings
through the control loops of the auxiliary DOFs are automatically
included in the results.

The scattered light can be at the carrier frequency or one of the RF SB
frequencies. So we repeated the simulation changing the frequency of the
scattered light source. If the scattered light field with an RF SB
frequency enters the interferometer, it will not directly interfere with
the GW SBs at the AS port. Rather, it will disturb the error signals for
the auxiliary DOFs. Through the feed back loops, the scattered light
induced disturbances appear in the DARM signal.

Figure\,\ref{SCL Req Carrier BRSE} and Figure\,\ref{SCL Req Carrier
DRSE} show the requirements on the scattered light fields at the carrier
frequency entering from various parts of the interferometer.  The
meaning of the curves is the following: The product of the field
amplitude $E_\mr{SCL} \mr{[\sqrt{W}]}$ and the vibration amplitude
$A(\omega)\mr{[m/\sqrt{Hz}]}$ must be smaller than the curves.
Similar requirements can be calculated for scattered light fields with RF SB
frequencies and shown in Figure\,\ref{SCL Req f1 BRSE} to \ref{SCL Req f2 DRSE}.


\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SCL-Req-Carrier-BRSE.pdf}
\caption{Scattered light requirements for the carrier: BRSE}
\label{SCL Req Carrier BRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SCL-Req-f1-BRSE.pdf}
\caption{Scattered light requirements for the f1 RFSB: BRSE}
\label{SCL Req f1 BRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SCL-Req-f2-BRSE.pdf}
\caption{Scattered light requirements for the f2 RFSB: BRSE}
\label{SCL Req f2 BRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SCL-Req-Carrier-DRSE.pdf}
\caption{Scattered light requirements for the carrier: DRSE}
\label{SCL Req Carrier DRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SCL-Req-f1-DRSE.pdf}
\caption{Scattered light requirements for the f1 RFSB: DRSE}
\label{SCL Req f1 DRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SCL-Req-f2-DRSE.pdf}
\caption{Scattered light requirements for the f2 RFSB: DRSE}
\label{SCL Req f2 DRSE}
 \end{center}
\end{figure}


Figure\,\ref{SCL Coupling Carrier BRSE} to Figure\,\ref{SCL Coupling
f2 DRSE} show
the coupling coefficients of the scattered light noise from various
entry points of the interferometer to the DARM error signal. The unit
may look strange, but it is identical to SNXXX/DARM in \cite{Yamamoto
SCL}. Basically, it is the ratio of the scattered light transfer
function and the DARM transfer functions. These numbers are used by the
auxiliary optics group to estimate the scattered light noise.

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SCL-Couplings-Carrier-BRSE.pdf}
\caption{Coupling coefficients of scattered light for the carrier: BRSE}
\label{SCL Coupling Carrier BRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SCL-Couplings-f1-BRSE.pdf}
\caption{Coupling coefficients of scattered light for the f1 RFSB: BRSE}
\label{SCL Coupling f1 BRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SCL-Couplings-f2-BRSE.pdf}
\caption{Coupling coefficients of scattered light for the f2 RFSB: BRSE}
\label{SCL Coupling f2 BRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SCL-Couplings-Carrier-DRSE.pdf}
\caption{Coupling coefficients of scattered light for the carrier: DRSE}
\label{SCL Coupling Carrier DRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SCL-Couplings-f1-DRSE.pdf}
\caption{Coupling coefficients of scattered light for the f1 RFSB: DRSE}
\label{SCL Coupling f1 DRSE}
 \end{center}
\end{figure}

\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=11cm]{plots/SCL-Couplings-f2-DRSE.pdf}
\caption{Coupling coefficients of scattered light for the f2 RFSB: DRSE}
\label{SCL Coupling f2 DRSE}
 \end{center}
\end{figure}

%{{{ \subsection{Light power in the interferometer}
\subsection{Light power in the interferometer}

It is necessary for the estimation of scattered light noises to know the
power of the light fields in the interferometer, because those numbers
are the starting point of the scattered light noise
calculation. Table\,\ref{Power in the IFO BRSE} and Table\,\ref{Power in
the IFO DRSE} show the light power in the various parts of the
interferometer. Most of the values were calculated by the Optickle
model, while the higher order mode power was calculated using Finesse,
assuming $\pm$1\,\% differential ROC error of the arm cavity mirrors.

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c|c|c|c|c|}
\hline
 & Carrier &f1&f2&Carrier HOM\\\hline
Arm Cavity&250\,kW&20\,mW&27\,mW&\\\hline
PRC&515\,W&7.3\,W&1.8\,W&\\\hline
SRC&107\,mW&3.4\,W&1.6\,nW&230\,mW\\\hline
REFL&1.8\,W&78\,mW&91\,mW&\\\hline
AS&24\,mW&770\,mW&0.37\,nW&\\\hline
  \end{tabular}
\caption{Light power in the various parts of the interferometer: BRSE}
\label{Power in the IFO BRSE}
 \end{center}
\end{table}

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c|c|c|c|c|}
\hline
 & Carrier &f1&f2&Carrier HOM\\\hline
Arm Cavity&250\,kW&20\,mW&27\,mW&\\\hline
PRC&515\,W&5.8\,W&1.8\,W&\\\hline
SRC&60\,mW&2.7\,W&1.6\,nW&230\,mW\\\hline
REFL&1.8\,W&240\,mW&91\,mW&\\\hline
AS&13\,mW&610\,mW&0.37\,nW&\\\hline
  \end{tabular}
\caption{Light power in the various parts of the interferometer: DRSE}
\label{Power in the IFO DRSE}
 \end{center}
\end{table}

% \begin{table}[tbp]
%  \begin{center}
%   \begin{tabular}{|c|c|c|c|c|c|}
% \hline
%    Arm cavity (carrier)&250\,kW&Arm cavity (f1) &25\,mW&Arm cavity (f2)&27\,mW\\\hline
%    PRC (carrier) &515\,W&PRC (f1) &7.3\,W&PRC (f2)&1.8\,W\\\hline
%    SRC (carrier)&107\,mW&SRC (f1) &3.4\,W &SRC (f2) &1.6\,nW\\ \hline
%    SRC (carrier HOM)&230\,mW & & & & \\ \hline
%   \end{tabular}
% \caption{Light power in the various parts of the interferometer: BRSE}
% \label{Power in the IFO BRSE}
%  \end{center}
% \end{table}

% \begin{table}[tbp]
%  \begin{center}
%   \begin{tabular}{|c|c|c|c|c|c|}
% \hline
%    Arm cavity (carrier)&250\,kW&Arm cavity (f1) &20\,mW&Arm cavity (f2)&27\,mW\\\hline
%    PRC (carrier) &515\,W&PRC (f1) &5.8\,W&PRC (f2)&1.8\,W\\\hline
%    SRC (carrier)&60\,mW&SRC (f1) &2.7\,W &SRC (f2) &1.6\,nW\\ \hline
%    SRC (carrier HOM)&230\,mW & & & & \\ \hline
%   \end{tabular}
% \caption{Light power in the various parts of the interferometer: DRSE}
% \label{Power in the IFO DRSE}
%  \end{center}
% \end{table}

%}}}

%}}}

%}}}

%{{{ \chapter{Alignment Sensing and Control Scheme}
\chapter{Alignment Sensing and Control Scheme}
\label{ASC}

\section{Overview}
The alignment sensing and control (ASC) scheme of bKAGRA has not yet fully determined. We plan to use a combination of the wave front sensing (WFS) scheme and optical levers. Currently, we have almost finished studying the WFS part with an Optickle model.

\section{Soft and Hard Modes}
The most important thing we have to consider about when designing ASC scheme is the radiation pressure effect. The angular motion of one mirror makes other mirrors to move due to opto-mechanical coupling. Especially, the high circulating power inside the arm cavity ($\sim$380\,kW) introduces strong coupling between angular motions of ITM and ETM. So, in order to make it easier to diagonalize the sensing matrix, the WFS signals are sensed in the soft-hard mode basis\,\cite{Sidles Sigg}.

The soft mode is the mode which two cavity mirrors tilt anti-symmetrically and the radiation pressure torque makes the pendulum mode of the test masses less stiff. The hard mode is symmetric mode and the radiation pressure torque makes the pendulum mode stiffer. The opto-mechanical transfer functions of the test masses are shown in Figure\,\ref{ASC_optomech_negative}. Note that soft mode of yaw motion is unstable(the phase at DC is -180\,deg).

In order to estimate how radiation pressure shifts the resonant frequencies, we first fit the transfer function with a simple pendulum transfer function. From the fitting, an equivalent momentum of inertia ($I_{\rm eq}$) and a mechanical restoring torque ($k_{\rm mech}$) is calculated. The soft and hard mode resonant frequencies can be computed by
\begin{equation}
  f = \frac{1}{2 \pi} \sqrt{\frac{k_{\rm mech}+k_{\rm opt}}{I_{\rm eq}}},
\end{equation}
where $k_{\rm opt}$ is the radiation pressure torque. $k_{\rm opt}$ of the soft(hard) mode is negative(positive) and the combination of $k_{\rm opt}$ differ by the g-factors and the intra-cavity power. When $k_{\rm mech}+k_{\rm opt} < 0$, resonant frequency is imaginary and angular motion is unstable. Resonant frequencies of the soft and hard modes for negative and positive g-factor are listed in Table\,\ref{ASC_resonantfrequencies}. As you can see from the table, the resonant frequency of the yaw soft mode will be too unstable for the positive g-factors. The pitch soft mode will also be unstable if mechanical restoring torque differed $\sim$ 30\% from the suspension model. 

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=10cm]{figures/ASC/optomech_negative_pitch.eps}
\includegraphics[width=10cm]{figures/ASC/optomech_negative_yaw.eps}
\caption{The transfer functions from the torque on the test mass to angle of the test mass(top: pitch, lower: yaw). The blue curve shows the mechanical transfer function in the absence of radiation pressure. The green and red curve show the opto-mechanical transfer functions of the soft mode and hard mode.}
\label{ASC_optomech_negative}
 \end{center}
\end{figure}

\begin{table}[htbp]
 \begin{center}
  \begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{g-factor} & \multirow{2}{*}{$P_{\rm incav}$ [kW]} & \multicolumn{3}{|c|}{pitch [Hz]} & \multicolumn{3}{|c|}{yaw [Hz]} \\
\cline{3-8}
& & $f_{\rm mech}$ & $f_{\rm SOFT}$ & $f_{\rm HARD}$ & $f_{\rm mech}$ & $f_{\rm SOFT}$ & $f_{\rm HARD}$ \\
\hline
\multirow{2}{*}{negative} &250	& \multirow{4}{*}{2.62} & 2.49 & 3.26 & \multirow{4}{*}{0.28} & 0.78$i$ & 1.95 \\
\cline{2-2}\cline{4-5}\cline{7-8}
 			  &380	&  & 2.41 & 3.55 &  & 0.98$i$ & 2.40 \\
\cline{1-2}\cline{4-5}\cline{7-8}
\multirow{2}{*}{positive} &250	&  & 1.77 & 2.75 &  & 1.90$i$ & 0.87 \\
\cline{2-2}\cline{4-5}\cline{7-8}
			  &380	&  & 1.08 & 2.81 &  & 2.36$i$ & 1.05 \\
\hline
  \end{tabular}
\caption{Resonant frequencies under radiation pressure. $i$ represents the instability.}\label{ASC_resonantfrequencies}
 \end{center}
\end{table}

The strategy for the ASC scheme is to make the UGF of the control loop as low as possible ($< 10$\,Hz) in order not to introduce the WFS shot noise. However, if the unstable frequency is high, we have to make the UGF higher than that frequency to make it stable. This is the main reason why we didn't selected the positive g-factors.

\section{Simulation Conditions}
The simulations of the interferometer response including radiation
pressure effects are done using Optickle. This simulation is done for
the negative g-factors of the arm cavities.

\subsubsection*{Arm Cavity Asymmetry}
The Optickle model used for ASC simulation doesn't include the arm cavity asymmetry for the simplicity. The two arm cavities are completely identical and the DC power at AS port is completely dark. So, the shot noise estimation for AS port is optimistic.

\subsubsection*{QPD}
By inserting attenuators, the power impinging upon each QPD is adjusted to $\sim$ 50\,mW, except for the AS port.

\section{Signal Extraction Ports}
The signal extraction ports are basically the same as the LSC scheme, but we also use the arm transmitted ports (TRX and TRY). For each port, two QPDs are placed at different Gouy phases. The suffixes used to identify the two are ``A" and ``B".

The sensing matrix is shown in Figure\,\ref{ASC_WFSSensingMatrix}. CS, CH, DS and DH in the sensing matrix mean, ``Common Soft'', ``Common Hard'', ``Differential Soft'' and ``Differential Hard'' respectively. The Gouy phases are optimized to minimize the degeneracy of the signals.

There is no good WFS sensing port for SR2, so the angular motions of SR2 should be controlled by using an optical lever. Since the local control is not included in our model right now, SR2 isn't controlled at all in the following results.

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=10cm]{figures/ASC/WFSSensingMatrix.eps}
\caption{Normalized WFS sensing matrix. Each row is normalized by the diagonal element. SR2 is not controlled by the WFS.}
\label{ASC_WFSSensingMatrix}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
\vspace{15mm}
\includegraphics[width=8cm,angle=-90]{figures/ASC/ASCstructure.eps}
\caption{Structure of the ASC model.}
\label{ASC_ASCstructure}
 \end{center}
\end{figure}

\section{Angular noise coupling to DARM}
\subsection{Structure of the ASC model}
The structure of the ASC model is summarized in Figure\,\ref{ASC_ASCstructure}. Notes on the each matrix are the following;
\begin{itemize}
\item IFO optical response $D$ [W/rad] represents the amount of the signal from each mirror at each probe. $D$ is calculated by Optickle simulation (sigAC in Optickle) and it includes the radiation pressure effects.

\item Input matrix $I$ is for reconstructing the error signals of the control DOFs. $I$ is ideally computed by taking an inverse of the sensing matrix, but the complete sensing matrix cannot be obtained during the lock acquisition phase. So, in the simulation, we only used large elements in the sensing matrix to compute $I$.

\item Filter $F$ is diagonal filter matrix for each DOF.

\item DOF to MIRROR $M$ is the base transformation matrix from DOFs to mirrors.

\item Actuator $A$ [rad/Nm] is diagonal actuator matrix for each mirrors. Actuator transfer functions are calculated by VIS group and currently we are considering only of actuation from recoil masses.

\item Radiation pressure effect $M$ [rad/rad] is calculated by Optickle simulation (mMech in Optickle). $M$ is the transfer function matrix from each mirror motion to each mirror motion and it represents the opto-mechanical coupling of the mirrors.

\item Beam spot motion (BSM) matrix $B$ [m/rad] is also calculated by Optickle simulation. $B$ is the transfer function matrix from each mirror motion to beam spot motion on each mirror.

\item Seismic noise $\vec{n}_{\rm seis}$ [rad] is the angular motion of each mirror caused by the seismic noise and is calculated by VIS group.

\item Shot noise $\vec{n}_{\rm shot}$ [W] is calculated by DC power on each QPD. We are assuming the limiting noise of each sensor is shot noise.
\end{itemize}

By using these matrices, residual angular motion and beam spot motion can be written as,
\begin{eqnarray}
 \vec{\theta}_{\rm res} &=& R (1+G_{\rm mirror})^{-1} (\vec{n}_{\rm seis}+G_{\rm mirror} D^{-1} \vec{n}_{\rm shot}), \\
 \vec{d}_{\rm spot} &=& B (1+G_{\rm mirror})^{-1} (\vec{n}_{\rm seis}+G_{\rm mirror} D^{-1} \vec{n}_{\rm shot}),
\end{eqnarray}
where
\begin{equation}
 G_{\rm mirrors} = A  M  F I D.
\end{equation}

The cavity length change caused by angular mirror motion is the product of the beam spot displacement and the mirror angle. In the frequency domain, this will be a convolution of the two spectra,
\begin{eqnarray}
  \delta L(f) &=& d_{\rm spot}(f) * \theta_{\rm res} (f) \\
              &\simeq& d^{\rm RMS}_{\rm spot} \theta_{\rm res} (f) + \theta^{\rm RMS}_{\rm res} d_{\rm spot}(f).
\end{eqnarray}

For the test masses, the length change produced by the angular motion directly couples to DARM. For BS and recycling cavity mirrors, we assumed the coupling factor of $\pi/(2F)$ and $1/100$ respectively.

\subsection{Simulation results}
The angular noise coupling to DARM is calculated using the formula presented above. We designed the servo filters to meet the sensitivity. Basically, lower UGF gives lower angular noise coupling at frequencies $>$ 10\,Hz. But for the test mass yaw motion, we have to make the UGF at around 3\,Hz in order to overcome the radiation pressure anti-spring. So, we put some extra gain at 0.04-0.05\,Hz where there is a seismic noise peak to reduce the RMS of the beam spot motion. The UGFs of the servo loops are listed in Table\,\ref{ASC_controlUGFs}.

Figure\,\ref{ASC_A2DARM} shows the angular noise coupling of each mirror to DARM. They are below the DARM noise (bKAGRA design sensitivity) at frequencies $>$ 10\,Hz.

\begin{table}[htbp]
 \begin{center}
  \begin{tabular}{|c|c|c|}
\hline
   & pitch & yaw \\\hline
CS, CH, DS, DH  &0.06\,Hz&3\,Hz\\\hline
other DOFs  &0.1\,Hz& 0.1\,Hz\\\hline
  \end{tabular}
\caption{ASC control loop UGFs}
\label{ASC_controlUGFs}
 \end{center}
\end{table}


\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=10cm]{figures/ASC/A2DARM_pitch.eps}
\includegraphics[width=10cm]{figures/ASC/A2DARM_yaw.eps}
\caption{Angular noise coupling to DARM (top: pitch, lower: yaw). The dotted line shows bKAGRA design sensitivity.}
\label{ASC_A2DARM}
 \end{center}
\end{figure}

%}}}

%{{{ \chapter{Lock Acquisition Scheme}
\chapter{Lock Acquisition Scheme}

%{{{ \section{Overview}
\section{Overview}
Quick and robust lock acquisition is important for maintaining high duty
factor. The lock acquisition procedure of bKAGRA 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 of the arm cavities move around and
randomly pass by the resonances of the RF sidebands. This makes the lock
acquisition very difficult and a 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}
\section{Green Laser Pre-Lock}
\label{Green Laser Pre-Lock}

%{{{ \subsection{Overview}
\subsection{Overview}

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{Noise Analysis}
\subsection{Noise Analysis}
In order to keep the arm cavities quiet enough so that they can be
brought to the full resonances, the relative fluctuations of the main
laser frequency and the arm cavities have to be much smaller than the
line width of the arm cavities. Since CARM has two cavity poles, by the
PRC and the ACs, the line width is very narrow (about 1\,Hz). If we
require the ACs effective length fluctuation, seen from the main laser,
to be 1/100 of the resonance width, the RMS length fluctuations of ACs
have to be kept smaller than $0.33$\,pm.  The green lock system has to be
able to pre-lock the ACs with this stability.  In order to design such a
system, a noise analysis of the green laser lock system was
performed\,\cite{Tatsumi Green}.  It was found that the most stringent
requirement is imposed on the main laser frequency noise out of the MC.

%}}}
%}}}

%{{{ \section{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 the 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.

Because of the susceptibility of the MICH signal to the CARM offset, THD
is not the default lock acquisition scheme of KAGRA. However, 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=11cm]{plots/THD-MICH.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=11cm]{plots/THD-PRCL.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=11cm]{plots/THD-SRCL.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}

%}}}

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

\section{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=11cm]{plots/NRS-MICH.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=11cm]{plots/NRS-PRCL.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=11cm]{plots/NRS-SRCL.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}

%}}}

%}}}

%{{{ \chapter{Optical Layout}

\chapter{Optical Layout}
\label{Optical Layout}

\section{Basic design}

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 KAGRA 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 KAGRA svn\,\cite{KAGRA svn layout}.

%{{{ \section{Wedge angle error tolerance}
\section{Wedge angle error tolerance}
If the wedge angles of transmissive optics, i.e. BS and ITMs, are
different from their designed values, the beams do not propagate
properly in the interferometer. We have to compensate for the error by
tweaking the position and angle of the BS and ITMs. With the SASs, we
can move these mirrors by a few mm without opening the vacuum. From
this, we can set requirements to the wedge angle error\,\cite{Wedge
Error Aso}. The conclusion is that in order to make the required amount
of adjustment to the positions of the BS and ITMs less than a few mm,
the error tolerance has to be about $\pm$1\%.

%}}}

\section{Tunnel Slope}
One thing to be kept in mind here is the tunnel slope. Since the 3\,km
tunnels of KAGRA 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.

%}}}

%{{{ \chapter{Installation/Adjustment Procedure}
\chapter{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\,\cite{ROC Error}.

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

%}}}

%{{{ \chapter{iKAGRA}

\chapter{iKAGRA}
\section{Overview}
iKAGRA is the first milestone of the two phase development of the KAGRA
interferometer. The optical configuration of iKAGRA is a Fabry-Perot
Michelson interferometer. It will be operated at room temperature and
the test mass substrate will be fused silica. 

The main purpose of iKAGRA is to {\em operate
a large interferometer as soon as possible to identify potential
problems associated with the facilities and other components in common
with bKAGRA}, so that we can take earlier actions to address the
discovered issues before starting the bKAGRA commissioning. For this
reason, there is no target sensitivity for iKAGRA. The focus is on
locking and stably operating the interferometer.

\section{Changes from bKAGRA}
\subsection{Mirrors}
The test masses of iKAGRA will be made of fused silica. The size of the
mirrors will be 25\,cm diameter and 10\,cm thick. The radius of
curvatures of the mirrors will be the same as the bKAGRA TMs.
Because the index of refraction is different from sapphire, the wedge
angle of the iKAGRA ITMs is different from the bKAGRA TMs.

We will continue to use the other mirrors used in iKAGRA through bKAGRA.

\subsection{Optical Layout}
\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{figures/iKAGRA-bKAGRA-Layout-Change.pdf}
\caption{Differences between the iKAGRA and bKAGRA optical configurations/layouts.}
\label{iKAGRA OptLayout Change}
 \end{center}
\end{figure}

The differences in the optical layout of iKAGRA and bKAGRA are shown in
Figure\,\ref{iKAGRA OptLayout Change}. The ITMs are moved by 25\,m
towards ETMs. The ETMs are moved by 35\,m towards ITMs. These iKAGRA TMs
will be installed in vacuum chambers different from the bKAGRA TMs so
that the installation and commissioning of Type-A SAS can be done in
parallel with the iKAGRA commissioning. The chambers for iKAGRA TMs will
be used as auxiliary optics chamber, for example housing optical lever
optics, in bKAGRA.

Because ITM AR surfaces have a finite wedge angle, the beams traveling
from the BS to the ITMs are not perfectly orthogonal to each
other\footnote{Otherwise, the deflected beam by the AR wedge will not
make a normal incidence to the HR surface}. Because the ITMs are moved
farther away from the BS, the beam spot position on the ITMs will be
laterally shifted by a few cm. Therefore, the positions of the ITMs will
be shifted by the same amount\footnote{Currently, the wedge angle of the
ITMs are still being adjusted to ease the handling of stray light. The
value of this lateral shift depends on the wedge angle, but a concrete
number is not available yet. Still, we know that this will be a few
cm.}. 

There are gate valves (GVs) separating the ITM chambers from the
3\,km-long vacuum pipes. The initial alignment of the arm cavities will be
done while the ITMs are exposed to the air and closing the GVs to keep
the vacuum of the long segment. For this purpose, there is an optical window
at the center of each GV to allow the beam to pass through. If the
lateral shift of the ITMs is too large, the beam will not go through the
optical window. In this case, we will move the pipe segments containing
the GVs by the same amount to center the beams on the windows. This will
be done by preparing two sets of anchor points for the necessary segments of the
vacuum pipes.

The PRM will not be installed in iKAGRA. The folding mirrors, PR2 and
PR3 will be installed. The whole SRC will not be used in iKAGRA. The SRC
part of the vacuum system will be separated from the rest of the
interferometer during iKAGRA. Then the SRC mirrors will be installed
during the iKAGRA commissioning. In order to get the AS beam of iKAGRA,
a pick off mirror will be installed either in the BS chamber or in a
temporary chamber on the AS side of the BS chamber.


\subsection{Mode Matching}
We will use the same input mode-matching telescope (IMMT) both in iKAGRA
and bKAGRA\footnote{Because we are lazy :-)}.  The IMMT will be
optimized for bKAGRA.  Because the PRM will not be installed in iKAGRA,
and the positions of the arm cavities are different from bKAGRA, the
mode matching of iKAGRA will be degraded. Figure\,\ref{iKAGRA IMMT}
shows the input mode matching of iKAGRA mapped by changing the positions
of the IMMT mirrors centered around the optimized positions for
bKAGRA. It is impossible to recover the mode matching to a very good
value, such as 99\%, even by moving the mirrors by 40\,cm. Therefore, we
will not attempt to re-optimize the IMMT. We will use the IMMT as is
i.e. optimized for bKAGRA. The mode matching for iKAGRA will be then
about 87\%. We found it acceptable from the experience of
TAMA\footnote{The IMMT of TAMA, which used infamous off-axis parabolic
mirrors, had poor quality. Thus the mode matching we got was something
like 95\%. 87\% is worse than this, but not terribly different.}.

\begin{figure}[htbp]
 \begin{center}
\includegraphics[width=11cm]{plots/IMMT-iKAGRA.pdf}
\caption{Input mode matching of iKAGRA, mapped  by changing the positions of the
  IMMT mirrors from the optimized ones for bKAGRA.}
\label{iKAGRA IMMT}
 \end{center}
\end{figure}

\subsection{Interferometer Control}
We will use only the f1 sideband for the length control of iKAGRA. WFS
will not be used, although the installation of hardware and some tests
may be performed.

There are three DOFs to be controlled in length. Sensing ports are
listed in Table\,\ref{iKAGRA sensing ports}

\begin{table}[tbp]
 \begin{center}
  \begin{tabular}{|c|c|}
\hline
   DARM&AS\_1Q \\\hline
   CARM&REFL\_1I \\ \hline
   MICH&REFL\_1Q \\ \hline
  \end{tabular}
\caption{Length sensing ports of iKAGRA}
\label{iKAGRA sensing ports}
 \end{center}
\end{table}



There is no plan to use green lock system for iKAGRA. However, we
will try to install it as soon as possible. Therefore, the hardware may
be installed during iKAGRA phase.

%}}}

%{{{ Appendix
\appendix

%{{{ \chapter{Recycling Cavity Length Determination Algorithm}
\chapter{Recycling Cavity Length Determination Algorithm}
\label{RCL Algorithm}
To be written.
%}}}

%{{{ \chapter{SRCL non-linearity}
\chapter{SRCL non-linearity}
\label{SRCL non-linearity}

To be written.

%}}}

%{{{ \chapter{Static Model of the Interferometer}
\chapter{Static Model of the Interferometer}
\label{Static Model of the Interferometer}
\begin{figure}[tbp]
 \begin{center}
\includegraphics[width=14cm]{figures/RSE-fields-diagram.pdf}j
\caption{Field definitions}
\label{RSE Static Model Fields}
 \end{center}
\end{figure}
Most of the simulation works in this document are done using
Optickle. However, our Optickle model is highly complicated and the
computation is not that fast. For some optimization works, which require many
iterations of simulation with slightly different parameters, we need a
quicker simulation tool even with some omission of details.
The static model of the KAGRA interferometer, presented in this chapter,
is constructed for this purpose.

\paragraph{What does it do ?}
It performs basically the same task as Optickle's sweep() function. This
model solves a set of field equations to get the DC fields inside the
interferometer given a set of phase parameters. A phase parameter is
one-way phase change a beam experiences while traveling a particular
part of the interferometer. A set of the phase parameters defines the
operation state of the interferometer, i.e. where the mirrors are with
respect to the resonances. The output of the model is a set of DC
optical field amplitudes (complex numbers, thus containing the phase
information too).

\paragraph{How is it implemented ?}
The model uses a simplified picture of the KAGRA interferometer as shown
in Figure\,\ref{RSE Static Model Fields}. The folding part is
omitted. $\Phi_\mr{p}$, $\Phi_\mr{s}$, $\Phi_\mr{x}$, $\Phi_\mr{y}$,
$\Phi_\mr{X}$ and $\Phi_\mr{Y}$ represent the one-way phase changes in
particular part of the interferometer. For example, $\Phi_\mr{p}$ is the
phase change between the PRM and the BS. By requiring the fields are in
a steady state, the fields have to satisfy the following equations:

\begin{eqnarray}
E_\mr{pb}& =& e^{-i\Phi_\mr{p}}\left(t_\mr{p} E_\mr{in} - r_\mr{p}
				E_\mr{bp}\right)\\
E_\mr{bp}& =& e^{-i\Phi_\mr{p}}\left(r_\mr{bs} E_\mr{yb} + t_\mr{bs}
				E_\mr{xb}\right)\\
E_\mr{bx}& =& e^{-i\Phi_\mr{x}}\left(t_\mr{bs} E_\mr{pb} - r_\mr{bs}
				E_\mr{sb}\right)\\
E_\mr{xb}& =& e^{-i\Phi_\mr{x}}\left(r_\mr{ix} E_\mr{bx} + t_\mr{ix}
				E_\mr{2x}\right)\\
E_\mr{by}& =& e^{-i\Phi_\mr{y}}\left(r_\mr{bs} E_\mr{pb} + t_\mr{ibs}
				E_\mr{sb}\right)\\
E_\mr{yb}& =& e^{-i\Phi_\mr{y}}\left(r_\mr{iy} E_\mr{by} + t_\mr{iy}
				E_\mr{2y}\right)\\
E_\mr{1x}& =& e^{-i\Phi_\mr{X}}\left(-r_\mr{ix} E_\mr{2x} + t_\mr{ix}
				E_\mr{bx}\right)\\
E_\mr{2x}& =& e^{-i\Phi_\mr{X}}\left(-r_\mr{ex} E_\mr{1x} \right)\\
E_\mr{1y}& =& e^{-i\Phi_\mr{Y}}\left(-r_\mr{iy} E_\mr{2y} + t_\mr{iy}
				E_\mr{by}\right)\\
E_\mr{2y}& =& e^{-i\Phi_\mr{Y}}\left(-r_\mr{ey} E_\mr{1y} \right)\\
E_\mr{bs}& =& e^{-i\Phi_\mr{s}}\left(t_\mr{bs} E_\mr{yb} - r_\mr{bs}
				E_\mr{xb}\right)\\
E_\mr{sb}& =& e^{-i\Phi_\mr{s}}\left( - r_\mr{s}
				E_\mr{bs}\right)
\end{eqnarray}

The solution to the above set of linear equations is a set of field
amplitudes. For example, in order to compute the carrier fields at the
optimal operation point of the interferometer, you put ($\Phi_\mr{p}$,
$\Phi_\mr{s}$, $\Phi_\mr{x}$, $\Phi_\mr{y}$,
$\Phi_\mr{X}$,$\Phi_\mr{Y}$) = (0, $\pi/2$, 0,0,0,0) into the above
equations and solve them.  If you want to simulate an RF SB, the one-way
phase change between the PRM and the BS will be $\Phi_\mr{p} =
\Phi_\mr{p0}+2\pi f_1 L_\mr{p}/c$ , where $\Phi_\mr{p0}$ is the phase
change for the carrier, $f_1$ is the modulation frequency, $c$ is the
speed of light and $L_\mr{p}$ is the distance between the mirrors. All
the other phase changes can be treated similarly. For higher order
modes, you add extra phase change $n\cdot \eta$ to each part of the
interferometer, where $n$ is the order of the mode, $\eta$ is the
one-way Gouy phase change of the corresponding part of the interferometer.

%}}}

%{{{ \chapter{Mixed PM and AM for f1}
\chapter{Mixed PM and AM for f1}
\label{PM + AM for f1}
To be written.

%}}}

%{{{ \chapter{Terminology}

\chapter{Terminology}
\label{Appendix Terminology}
\begin{table}[hp]
 \begin{center}
\caption{Terminology}
\label{Terminology Table}
  \begin{tabular}{|p{7em}|c|p{10em}|}
\hline
   AC&Arm Cavity & \\ \hline
   AM&Amplitude Modulation & \\ \hline
   AS&Anti-symmetric port& \\ \hline
   Auxiliary DOF& &Length degrees of freedom other than DARM \\ \hline
   Canonical DOF&Collective name of DARM, CARM, MICH, PRCL and SRCL & \\ \hline
   CARM&Common Arm Length&\\\hline
   DARM&Differential Arm Length& \\\hline
   DOF&Degrees Of Freedom & \\ \hline
   MC&Mode Cleaner & \\ \hline
   MICH&Michelson Part&L shaped part of the interferometer formed by BS
	   and two ITMs\\\hline
   MZ&Mach-Zehnder & \\ \hline
   PM&Phase Modulation & \\ \hline
   PD&Photo Diode/Detector & \\ \hline
   POP&Pick-off in the Power Recycling Cavity & \\ \hline
   POX&Pick-off at the ITMX & \\ \hline
   POY&Pick-off at the ITMY & \\ \hline
   QPD&Quadrant Photo Diode/Detector & \\ \hline
   REFL&Reflection port& \\ \hline
   RF SB&RF Sideband & \\ \hline
   PRC&Power Recycling Cavity&\\\hline
   PRCL&Power Recycling Cavity Length&\\\hline
   SRC&Signal Recycling Cavity&\\\hline
   SRCL&Signal Recycling Cavity Length&\\\hline
   
  \end{tabular}
 \end{center}
\end{table}

%}}}

%{{{ \chapter{Contributors}

\chapter{Contributors}

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

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

Y. Michimura wrote section\,\ref{ASC}. All the other sections were
written by Y. Aso. K. Somiya and O. Miyakawa were deeply involved in the
development of the LSC scheme. The PI calculation was done by K. Shibata
with the help of K. Yamamoto. The noise analysis of the green lock
system was done by D. Tatsumi and K. Arai. Chen Dan validated the code
for the optical layout.



%}}}

%}}}

%{{{ References

\begin{thebibliography}{00}

 \bibitem{BW Report} K. Somiya, Study report on the new LCGT setup with
	 22cm mirrors, JGW-T1100644 
 \bibitem{JGW-G1100599}K. Somiya, JGW-G1100599
 \bibitem{PI Braginsky} V.B. Braginsky et al., Phys. Lett. A 287, 331
	 (2001).
 \bibitem{Shibata PI} K. Shibata, Parametric instability in the bKAGRA, JGW-G1200844

 \bibitem{Sidles Sigg}  J. A. Sidles and D. Sigg, {\it Physics Letters
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 \bibitem{LSC Code}
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	 http://www.analog.com/
 \bibitem{Yamamoto SCL} H. Yamamoto, LIGO Document T060073
 \bibitem{Tatsumi Green}D. Tatsumi, JGW-T1200788
 \bibitem{Wedge Error Aso}Y. Aso, JGW-T1100489
 \bibitem{ROC Error} K. Agatsuma, JGW-G1100553

\end{thebibliography}
\end{document}


%}}}
