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\begin{document}

\vspace*{50mm}
\begin{center}
{\bf \Large Study report on }

\vspace*{5mm}
\hbox{\bf \LARGE the LCGT Suspension Point Interferometer}
\end{center}


\vspace*{20mm}
\begin{flushright}
{\large LCGT SPI Special Working Group{\hspace*{1cm}}}

\vspace*{5mm}
{\large September 7, 2009 (Revised November 8, 2009){\hspace*{1cm}}}
\end{flushright}

\vspace*{5mm}

\newpage
\tableofcontents
\newpage
\section{Executive summary}
The LCGT SPI special working group is charged with the
investigations on the necessity and feasibility of the suspension point
interferometer for LCGT. The biggest question to be answered is whether
we need to install an SPI to LCGT or not. For this purpose, we examined
the necessity of the SPI mainly from two aspects: (1) the reduction of
the vibration introduced from heat-link wires and (2) the reduction of
RMS mirror motion for robust lock acquisition. We also investigated
alternative techniques which can replace or supplement the SPI.

This document is a result of the discussions in the working group. In
this section, we present our recommendations and a brief explanation of the
reasoning behind them.  Model calculations are presented in
section\,\ref{SPI Expected Performence} to show the expected performance
of the LCGT SPI. In section\,\ref{Heat link vibration isolation}, we
will examine the necessity of the SPI for heat-link vibration isolation.
The lock acquisition scheme for LCGT is reviewed in
section\,\ref{section: lock acquisition} and the importance of SPI as a
lock acquisition aid is assessed.  Finally the current design of the
LCGT SPI is discussed in section\,\ref{section: SPI}.

\subsection{Recommendations}
The LCGT SPI special working group proposes the following actions be
taken regarding the LCGT SPI.

\begin{center}
\begin{itembox}[l]{\bf Recommendations}
\begin{itemize}
 \item We shall not install an SPI for LCGT
 \item We shall continue to put R\&D efforts on the alternative
       techniques, such as cryogenic active vibration isolation stage
       (section\,\ref{Cryogenic active vibration isolation}),
       hierarchical control (section\,\ref{section: hierarchical
       control}), offset lock (section\,\ref{section: offset lock}) and
       green laser pre-lock (section\,\ref{section: green laser lock}).
 \item In order to keep the possibility of retrofitting an SPI in the
       future, we shall keep the vacuum tube size large enough
       (1\,m). The beam height of the main laser in the vacuum tube
       shall be offset from the center to accommodate the SPI
       beam in the future.
\end{itemize}
%\end{quote}
\end{itembox}
\end{center}

\subsection{Summary of the reasoning behind the recommendations}
The above recommendations were drawn from the following reasons.

\begin{itemize}
 \item Simulations show that the target sensitivity of LCGT is not
       compromised by the heat-link vibration even without an SPI
       (figure\,\ref{fig: target sense}).
 \item The seismic noise is limited by the vertical vibration
       coupling from the heat-links above 6\,Hz (i.e. in the observation
       band). Since vertical vibrations cannot be suppressed by an SPI,
       an improvement of the seismic noise safety factor cannot be
       expected from SPI.
 \item LCGT's requirement for the RMS displacement noise (0.1\,$\mu $m
       above 0.1\,Hz) is satisfied without an SPI in the normal seismic
       period (figure\,\ref{fig: SeismicRMS}). When the micro seismic
       motion is high, the RMS is estimated to be 0.2\,$\mu$m, which is
       only a factor of two above the requirement (figure\,\ref{fig:
       SeismicRMS high}). We do not think it is worth spending a lot of
       resources on SPI to reduce this by a factor of 2. Moreover,
       almost the same effect as the RMS reduction by the SPI can be
       achieved by a carefully designed hierarchical control
       (section\,\ref{section: hierarchical control}).
 \item To avoid a large kick by the radiation pressure, we will use
       the offset lock technique for the lock acquisition regardless of
       the seismic noise (section\,\ref{section: lock acquisition
       radiation pressure}). Since the offset lock has a larger linear
       range than the PDH signal, the lock acquisition can be robust
       even without an SPI.
 \item In order to give more safety margins to both the observation band
       seismic noise and the RMS motion, we should pursue alternative
       techniques for the suppression of the heat-link vibration and
       robust lock acquisition. The alternative techniques proposed in
       this report are not intrusive to the core part of the
       interferometer and easily retrofitted when the technology is
       ready. In contrast, the SPI requires a major modification to the
       penultimate mass design. If it fails, there is a risk of
       degrading the interferometer sensitivity. Considering the current
       maturity of the LCGT SPI design study, we estimate that 
       resources necessary to successfully implement a practical SPI to
       LCGT is much more than the cost required to develop these
       alternative techniques.
 \item Even though SPI is not necessary for the current LCGT, next
       generation upgrades to the LCGT may require an SPI. We may also
       want to use a larger beam size in the future. Since it is
       very costly to replace the vacuum envelope once it is
       built, we have to have a margin for future upgrades. Therefore,
       we propose to keep the beam tube size the same as the original
       design (1\,m diameter).
\end{itemize}

More detailed discussion and calculations to support the above
reasoning will be presented in the following sections.


\section{Expected performance of the LCGT SPI}
\label{SPI Expected Performence}

\subsection{Suspension Model}
\label{section: suspension model} In order to estimate the performance
of the LCGT SPI, we developed a one dimensional model of the LCGT
suspension systems with an SPI.  A conceptual diagram of the LCGT
suspension system is shown in figure\,\ref{fig: LCGT Suspension}. The
overall height of the suspension is about 4\,m. This model includes
heat-links connecting the 4\,K cryo-shield and the suspension platform.
Details of the heat-link design is explained in section\,\ref{Heat link
vibration isolation}.

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=6cm]{drawings/LCGT-Suspension.eps}
\caption{Conceptual diagram of the LCGT suspension system. IP: Inverted
  Pendulum, PM: Penultimate Mass, MB: Magnet Box, TM: Test Mass, RM:
  Recoil Mass.}
\label{fig: LCGT Suspension}
 \end{center}
\end{figure}

The SPI is formed at the PM (penultimate mass) stage.
The feedback filter for the SPI servo is made to set the unity gain
frequency around 50\,Hz. This filter has a low frequency gain boost.
The overall open loop transfer function of the SPI servo is shown in
figure\,\ref{fig: SPI OPLTF}.

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=12cm]{plot/OPLTF.eps}
\caption{SPI servo's open loop transfer function}
\label{fig: SPI OPLTF}
 \end{center}
\end{figure} 

\subsection{Observation band seismic noise}
\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=13cm]{plot/TargetSense.eps}
\caption{Seismic noise and the LCGT target sensitivity.}
\label{fig: target sense}
 \end{center}
\end{figure}

The seismic noise contribution to the LCGT noise is calculated using the
above mentioned model.  Figure\,\ref{fig: target sense} shows the
estimated seismic noise with and without the SPI along with the target
sensitivity of LCGT. The vertical seismic noise was calculated by an
independent one dimensional vertical suspension model and assuming 1\%
vertical-to-horizontal coupling.

The seismic noise does not limit the sensitivity of the interferometer
above 2\,Hz even without the SPI. The safety margin for the seismic
noise without SPI is more than a factor of 6 above 5\,Hz. The horizontal
seismic noise can be suppressed by the SPI by more than 20\,dB. However,
the vertical noise coupling becomes dominant above 6\,Hz. Therefore, the
ability of the SPI to reduce the seismic noise in the observation band
is very limited.

\subsection{RMS motion}

\subsubsection{Displacement RMS}

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=13cm]{plot/SeismicRMS.eps} 
\caption{RMS displacement noise when the seismic noise is normal. Total
  Seismic means the overall motion of the mirrors caused by all kinds of
  seismic vibrations.}  
\label{fig: SeismicRMS}
 \end{center}
\end{figure}


\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=13cm]{plot/SeismicRMS_High.eps}
\caption{RMS displacement noise when the micro seismic motion is high. Total
  Seismic means the overall motion of the mirrors caused by all kinds of
  seismic vibrations.}
\label{fig: SeismicRMS high}
 \end{center}
\end{figure}

Displacement RMSs of the test masses are calculated by integrating the
displacement noise spectra (figure\,\ref{fig: SeismicRMS}).  LCGT's
requirement for the displacement RMS is less than 100\,nm integrated
above 0.1\,Hz. It is an RMS of the open-loop mirror motion (the motion
of the mirror which will occur if there is no feedback from the
interferometer signal). The estimated RMS is 10\,nm for the no-SPI case
and 1\,nm with the SPI. Therefore, the requirement is satisfied even
without the SPI.

When there is a storm outside, the micro seismic motion in the Kamioka
mine becomes high. Roughly speaking, such a condition happens once a
week during the winter (Dec. - Mar.) and once a month in the other
seasons.  Figure\,\ref{fig: SeismicRMS high} shows the seismic noise
spectra calculated with a typical ground motion spectrum of high micro
seismic times. In this case, the RMS displacement is 200\,nm without the
SPI and 20\,nm with the SPI. The no-SPI case slightly exceeds the
requirement.

An important thing to notice is that the seismic noise is still below
the target sensitivity in the high seismic period. This is because the
heightened micro-seismic motion only affects frequencies below 1\,Hz.
Up conversions of large low frequency noise through non-linearity of the
detector response could still be a problem. However, this problem can be
mitigated by the hierarchical control discussed below.

\subsubsection{Hierarchical control}
\label{section: hierarchical control} 

A large RMS displacement noise can make the interferometer
unstable. Resultant frequent lock losses will decrease the duty cycle of
the detector. We also have to increase the control bandwidths of
the auxiliary degrees of freedom, such as the Michelson part, to keep the
interferometer at the operational point against large low frequency
excitations. This may increase the control noise couplings.  Also up
conversion of the low frequency noises to the observation band through
non-linearities in the detector can be serious.

The above problems all arise from insufficiently suppressed residual
motion of the test masses. One way to solve this problem is to reduce
the noise itself, which is what an SPI is supposed to do. Another
solution is to increase the servo gain and strongly suppress the
residual motion.

In general, we cannot make the test mass actuators too strong because
the actuator noise then contaminates the sensitivity. However, if we
feedback the error signal to the upper stages of the suspension system,
the actuator noises are naturally filtered out at high frequencies, thus
not contaminating the observation band sensitivity. A hybrid actuator
formed by strong upper stage actuators for low frequencies and weak test mass
actuators for high frequencies will allow us to strongly suppress the
low frequency vibrations. This scheme is called hierarchical control.

Actually, there is an equivalence principle between SPI and hierarchical
control. The SPI senses the same degree of freedom as the main
interferometer (beam direction) and feed the signal back to the
penultimate stage actuators. Since there is a linear relationship
between the motion of the penultimate masses and the test masses, the
main interferometer can obtain the same information as the SPI unless
the seismic motion is attenuated below other noises by the final stage
pendulum, in which case we don't care the seismic noise anyway. Feeding
back the error signal from the main interferometer to the penultimate
stage is therefore equivalent to an SPI's action in principle.

However, there are two important differences. One is that the
hierarchical control cannot reduce the seismic ``noise'' appearing at
the gravitational wave channel. It only reduces the residual motion of
the mirrors. The other difference is that the SPI can suppress the noise
of the SPI actuators by the servo, whereas the hierarchical control does
not have this active suppression. It has only passive attenuation by the
suspension transfer function.

Even with these differences, still the hierarchical control can provide
most of the benefits of SPI regarding the RMS reduction.

One thing to note is that the hierarchical control only works when the
interferometer is locked. Therefore, it cannot be used to reduce the RMS
speed of the test masses to help lock acquisition.

It is also worth mentioning that hierarchical control schemes have been
successfully implemented by VIRGO and GEO\,[refs]. Advanced LIGO will
also employ this actuation topology. We should make use of the
experiences accumulated by those projects when designing the LCGT
hierarchical control.

\subsubsection{Speed RMS}
The mirror speed spectra are shown in figure\,\ref{fig: Speed RMS} for
a normal seismic period and figure\,\ref{fig: Speed RMS high} for a high
micro seismic period.



\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=13cm]{plot/Speed.eps}
\caption{Mirror RMS speed}
\label{fig: Speed RMS}
 \end{center}
\end{figure}

The RMS mirror speed for the normal seismic period is 20\,nm/sec without
the SPI and 2\,nm/sec with the SPI (integrated above 0.1\,Hz). It
becomes 200\,nm/sec (no SPI) and 20\,nm/sec (with SPI) when the micro
seismic activity is high.


\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=13cm]{plot/Speed_High.eps}
\caption{Mirror RMS speed when the micro seismic motion is high}
\label{fig: Speed RMS high}
 \end{center}
\end{figure}

The mirror speed mainly affects the robustness of the lock
acquisition. Since we do not have to worry about the sensitivity of the
interferometer at this stage, we may be able to reduce the RMS speed by
applying aggressive damping to the suspension system.  Other lock
acquisition tricks are discussed in section\,\ref{section: lock
acquisition}.


\section{Heat link vibration isolation}
\label{Heat link vibration isolation}
In this section, we discuss the necessity of SPI for heat-link
vibration isolation.
\subsection{Heat-link design}
The heat-links used to subtract heat from the mirrors are 99.9999\,\%
pure aluminum wires. 7 heat-links with 1\,mm diameter connect the
suspension platform and the cryo-shield. These are U shaped with the radius
of curvature 25\,cm. The platform and the penultimate mass are connected
by 5 heat-links with the diameter 3\,mm These are U shaped with 20\,cm ROC. 

The spring constants of a U shaped wire can be calculated by the
following formulae.
\begin{equation}
k_\mathrm{x}= \frac{E d^4}{32 R^3}
\end{equation}
\begin{equation}
k_\mathrm{y}= \frac{G d^4}{32 R^3}
\end{equation}
\begin{equation}
k_\mathrm{z}= \frac{E d^4}{96 R^3}
\end{equation}
$E$ is the Young's modulus, $G$ is the shear modulus, $d$ is the
diameter of the wire and $R$ is the bending radius of curvature.
Since $E>G$ for aluminum, $k_\mathrm{x}$ has the largest value among the
three.

Combined with the 120\,kg mass of the suspension platform, the resonant
frequency of a mass-spring system formed by the platform and the
heat-links is about 14\,mHz for $x$-direction and 8\,mHz for $z$-direction. These values are used in the suspension model explained in
section\,\ref{section: suspension model}.

As one can see in the above formulae, the stiffness of the heat-links is
proportional to the fourth power of the wire diameter. On the other
hand, the thermal conductivity of the wire is proportional to the cross
section of the wires ($\propto d^2$). Therefore, we can increase the
number of heat-links keeping the total cross section constant to reduce
the stiffness of the heat links if necessary.

The above mentioned model of the heatlink takes into account only the
stiffness arising from static deformations of a heatlink, which is a
good approximation for low frequency behavior of the heatlink. However,
it does not include the inertia of the heatlink mass nor the internal
vibration modes. For more accurate estimates of the vibration isolation
performance of heatlinks, we should perform detailed finite element
analyses.

 
\subsection{Estimated vibration introduced from the heat-links}
\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=13cm]{plot/SeismicBreakdownNoSPI.eps}
\caption{Breakdown of the seismic noise without SPI}
\label{fig: Seismic Breakdown No SPI}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[width=13cm]{plot/SeismicBreakdownSPI.eps}
\caption{Breakdown of the seismic noise with SPI}
\label{fig: Seismic Breakdown with SPI}
 \end{center}
\end{figure}

Using the stiffness of the heat-links shown in the previous section, the
seismic noise introduced through the heat-links are calculated using the
suspension model explained in section\,\ref{section: suspension model}.
The breakdowns of the seismic noise are shown in figure\,\ref{fig: Seismic
Breakdown No SPI} (without SPI) and figure\,\ref{fig: Seismic Breakdown
with SPI} (with SPI). We used the ground motion spectrum of a normal
seismic period and assumed that the motion of the cryo-shield is the
same level as the ground motion.

In the case of no SPI, the seismic noise is dominated by the horizontal
vibration from the heat-links above 2\,Hz. Then it is superseded by the
horizontal vibration from the heat-links above 6\,Hz.  When the SPI is
used, the vertical heat-link vibration becomes dominant above 1.5\,Hz.
In both cases, the vibration introduced from the heat-links does not
contaminate the target sensitivity.


\subsection{Cryogenic active vibration isolation}
\label{Cryogenic active vibration isolation} The seismic noise in the
observation band (above 10\,Hz) is dominated by the vertical heat-link
vibration. In order to give more safety margin to the seismic noise, we
have to suppress the vertical heat-link vibration. Since SPI does not work for
vertical vibrations, we have to use a scheme which also works
for vertical motion.

It is usually a good idea to do an active vibration isolation close to
the origin of the vibration source because the vibration is still large
and easy to sense.  Therefore it is an interesting option to insert an
active vibration isolation stage in the heat-link chain just after the
4k shield.

\begin{figure}[htbp]
 \begin{center}
 \includegraphics[height=6cm]{drawings/CryogenicActiveVibrationIsolation.eps}
\caption{Cryogenic active vibration isolation}
\label{fig:Cryogenic active vibration isolation}
 \end{center}
\end{figure}

The basic concept of this scheme is shown in figure\,\ref{fig:Cryogenic
active vibration isolation}. An accelerometer is connected to the 4k
shield through a semi-rigid heat conducting structure such as
bellows. This structure should have a good heat conductance so that the
temperature of the accelerometer body is almost the same as the 4k
shield. At the same time, this connection has to be flexible enough that
the accelerometer can be moved relative to the 4k shield to cancel the
vibration introduced from the shield. The accelerometer is supported
from the ground through an actuator, such as a piezo hexapod, so that
the accelerometer can be moved. The vibration sensed by the
accelerometer is fed back to the actuator to cancel it. Thus the body of
the accelerometer is quieter than the 4k shield. Then the heat-link
wires connected to the mirror suspension system is originated from this
isolated accelerometer body.

Since loose heat-links may have cross-couplings of vibrations in all
DOFs to the in-line direction, active vibration isolation probably
has to be performed for all 6 DOFs. Similar vibration isolation systems
are demonstrated at the room temperature.  The biggest concern for us is
whether good cryogenic accelerometers and actuators are
available. Piezos are known to work at cryogenic
temperatures\,\cite{cryo-pzt}. There are commercial cryogenic
accelerometers using PZTs, but the sensitivity may not be enough for our
purpose. We have to investigate on this further more.

\subsection{Conclusion for heat-link vibration isolation}
The model calculation shows that the vibration introduced from
the heat-links does not contaminate the target sensitivity of LCGT.
In the observation band, the seismic noise is dominated by the vertical vibration
of the heat-links. SPI is not effective to reduce this noise. If we need
further reduction of the seismic noise in the observation band, a new
technique such as the cryogenic active vibration isolation is required.
The heat-links do not contribute to the RMS motion of the mirrors at all.
Therefore, SPI is not necessary from the point of view of the heat-link
vibration isolation.

\section{Lock acquisition}
\label{section: lock acquisition} By the reduction of the RMS mirror
motions, an SPI can help the lock acquisition of an interferometer. In
this section, we will evaluate the significance of SPI for LCGT lock
acquisition. For this purpose, we first summarize the general lock
acquisition strategy for LCGT. Then several problems of locking a high
finesse cavity are reviewed. Finally we explain two techniques (offset
lock and green laser pre-lock) to circumvent these problems.

\subsection{LCGT lock acquisition procedure}
Lock acquisition is defined as a process to bring the mirrors of an
interferometer to the desired operational positions. In the case of a
DRFPMI, we have to lock four cavities (two recycling and two arm
cavities) and a Michelson interferometer. There are specific steps to
be taken to robustly lock all the degrees of freedom.

The first step for the lock acquisition is to lock the central part,
i.e. PRC, MICH and SRC. Basically we wait for those three degrees of
freedom to come close enough to the operational points by chance to
produce sensible error signals. Then the servos draw the mirrors to the
operational points. During this process, the arm cavities should not
resonate the carrier and the sidebands. Otherwise the resonance will
destroy the error signals for the central part.

The error signals used for the initial lock acquisition of the central
part have to have wide linear ranges and robustness against the motions
of other degrees of freedom. These signals may not be the same as the
ones used during the observational operation of the interferometer
because the wide range signals generally have higher noise.  It is also
desirable that the error signals are insensitive to the power build up
of the carrier in the PRC and arm cavities. For this purpose, the third
order harmonic demodulation or double demodulation between the sidebands
are suitable. However, these signals often have smaller linear
range. Therefore, we may have to use single demodulation signals for the
initial lock and then switch to those carrier insensitive signals before
the arm cavities are fully locked. To do this, we need to use the offset
lock technique or the green laser pre-lock (explained later in this
section) to temporarily prevent the arms from resonating.

After the central part is locked, the arm cavities are locked. If the
arm cavities resonate sidebands before they are locked to the carrier,
the error signals of the central part will be disturbed and we may loose
the lock. Therefore, it is very important that we lock the arm cavities
at the first attempt.

Because of the radiation pressure problem (section\,\ref{section: lock
acquisition radiation pressure}), we will lock the arm cavities with
some offset from the resonance using the DC transmission power. This is
called offset locking. The offset lock error signal has a wider linear
range than the conventional Pound Drever Hall (PDH) error signal. Offset
lock also allows us to switch the error signals of the central part to
carrier insensitive signals such as the third harmonics demodulation.

Finally we reduce the offsets of the arm cavities slowly to fully lock the
interferometer. The error signals for the arm cavity control have to be
switched to RF or DC readouts during this process. The servo filters
are dynamically changed to compensate for the optical gain change.

The role of the SPI for the lock acquisition is to reduce the RMS speed
of the mirrors so that the actuators can easily stop the mirrors at the
operational point. This is especially important for the lock of the arm
cavities since the finesse of the arm cavities is high (more than 1000).
In the following sections, we will focus on the lock of a single
Fabry-Perot cavity to evaluate the difficulty of arm cavity lock.
\subsection{Obstacles for arm cavity lock}
\subsubsection{Impulse limit velocity for mass lock acquisition}

In order to stop a mirror at a resonance, kinetic energy of the mirror
has to be reduced to zero by the test mass actuators.  For a high
finesse cavity we can sense the mirror position in a very narrow region
around the resonances. The half-width at half-maximum of a cavity
resonance in terms of the cavity length is
\begin{equation}
	\Delta L = \frac{\lambda}{2 {\cal F}} \;.
\end{equation}
In order to stop the mirror, the impulse given to the mirror from the
actuators has to be larger than the initial kinematic energy of the
mirror.  Therefore, the following inequalities have to be satisfied.
\begin{eqnarray}
	\frac{1}{2} m \, v^2  &<&  F_\textrm{max} \; \Delta L \\
        \nonumber \\
        v &<& v_\textrm{impulse} = \sqrt{ \: 2 \, a_\textrm{max} \; \Delta L}
\end{eqnarray}
Here $m$, $v$ and $F_\textrm{max}$ are mass, speed of the mirror
and the maximum force of the actuator,
respectively. $a_\textrm{max}=F_\mathrm{max}/m$.  From the experience at
TAMA, the critical mirror velocity ($v_\textrm{c}$) for a successful
lock acquisition is about a half of the above impulse limit velocity
$v_\textrm{impulse}$.

The test mass actuators are designed to have 
\begin{eqnarray}
        a_\textrm{max} &=& (2 \pi f)\,^2\; \mathrm{[1/s^2]} \times \;
        \frac{ 3 \times 10^{-9} }{ f^2 } \;\; \textrm{[m/V]} \;
        \times 10 \; \textrm{[V]} \nonumber \\
        &=& 1.2 \times 10^{-6}\;\mathrm{[m/s^2]}\,,
\end{eqnarray}
to avoid contamination of the sensitivity from the actuator driver
noise.

Impulse limit velocities and critical velocities are listed for various
values of finesse in table\,\ref{table: ciritical velocity}. For the main laser
carrier (1064\,nm), the critical velocity is very small
(16\,nm/sec) and it is comparable to the RMS speed of the test masses
during a normal seismic period without SPI and during a high seismic
period with SPI. Lock acquisition will be very difficult without the SPI
when the micro seismic motion is high.

\begin{table}[htbp]
   \begin{center}
   \begin{tabular}{rc|ccc||c||}
               && Finesse ${\cal F}$ & $\Delta L$ & $v_\textrm{impulse}$ &
               $v_\textrm{c}$ \\ \hline
   Arm cavity for 1064 nm  &&   1250  & 0.426 nm  & 32 nm/s & 16 nm/s \\
              for  532 nm  & Green Lock &    100  & 2.660 nm  & 80 nm/s & 40 nm/s \\
                & Green Lock &     10  & 26.60 nm  & 253 nm/s & 126 nm/s
   \end{tabular}
   \caption{Critical velocity of mass lock acquisition}
    \label{table: ciritical velocity}
   \end{center}
\end{table}

\subsubsection{Ringing velocity}
A high finesse cavity has a long storage time of photons.  This produces
a ringing in the transient response of the cavity when the mirrors pass
by a resonance. The amount of the ringing depends on the speed of the
mirrors. Typical velocity where the ringing becomes significant can be
estimated by dividing the resonance width by the storage time.  The
resonance width of the cavity is represented as $\lambda/2 \:/ {\cal
F}$.  Here $\lambda$ and ${\cal F}$ are wavelength of the laser and
finesse of the cavity, respectively. The storage time $\tau$ is
represented as $2L/c \times {\cal F}/\pi$. As a result, typical ringing
velocity is represented as follows.

\begin{eqnarray*}
v_\textrm{ringing} &=& \frac{\lambda}{2 \,{\cal F}} \times \frac{1}{\tau} \;\;= \;\;
\frac{ \pi \, c  \, \lambda}{4  \: L \: {\cal F}^2 } \\
% \noalign{In the case of LCGT, arm cavity has following parameters.}
% \nonumber \\
% \lambda &=& 1064 \; \textrm{nm}, \\\nonumber \\
% L &=& 3000, \\
% {\cal F} &=& 1550
% \nonumber \\
% \therefore \;\; v_\textrm{ringing} &=& 35 \; \textrm{nm/s}
\end{eqnarray*}

Table\,\ref{table:ringing velocity} shows the ringing velocities for
several values of arm finesse.

\begin{table}[htbp]
   \begin{center}
   \begin{tabular}{|c|c|}
    Finesse & Ringing velocity \\ \hline
    1550 &  35 nm/s \\
    1250 &  53 nm/s \\
     777 & 138 nm/s \\ \hline
     100 & 4175 nm/s \\
   \end{tabular}
    \caption{Ringing velocity for several arm finesse values}
\label{table:ringing velocity}
   \end{center}
\end{table}

\subsubsection{Radiation pressure}
\label{section: lock acquisition radiation pressure}

\begin{figure}[htbp]
	\begin{center}
		\includegraphics[width=12cm]{power.eps} \caption{Lock
	event with the mirror speed of 60\,nm/s in various
	input powers. 0.2\,\% of the full power corresponds to 1.5\,kW inside
	the arm cavity when fully locked. 0.5\,\% and 1\,\% correspond
	to 3.5\,kW and 7\,kW respectively.}  \label{fig:sus1}
	\end{center}
\end{figure}

If the power inside cavity reaches several \,kW, lock acquisition will
be disturbed by the radiation pressure because the pressure kicks the
mirrors out of the resonance. This effect was simulated using the E2E
simulation engine for a cavity of finesse 1250. In this simulation, a
technique called guide lock was used to increase the chance of lock
acquisition. In this method, we measure the velocity of the mirror using
the first pass of a resonance and then a properly calculated impulse is
given to the mirror to gently send it back to the resonant point.

The time series of the cavity power during lock attempts are shown in
figure\,\ref{fig:sus1} for several input powers. Except for the 0.2\,\%
case, the mirrors are kicked by the radiation pressure many times before
locked. In the case of 1\,\% power, the cavity could not be locked at
all.

The above simulation only took into account the radiation pressure
effect in the longitudinal direction. In reality, the radiation pressure
also causes an instability in the rotational degrees of
freedom. Especially the horizontal push by the radiation pressure
induces a pitch tilt of the mirrors. Then the pitch angular instability
caused by the radiation pressure is excited and it could destroy the
cavity lock.  Figure\,\ref{fig:angle_oscillation} shows the behavior of
the mirror alignment when the arm cavity power is ramped up. The angular
radiation pressure induces an oscillation in pitch and eventually the
lock is lost.

\begin{figure}[htbp]
	\begin{center}
		\includegraphics[width=13cm]{angle_oscillation.eps}
	\caption{Radiation pressure effects on angle motion without
	alignment control. Left graph is power inside the cavity. After
	lock is acquired with very low power, input power is ramped up
	to full power in 15 seconds. The central graph shows the
	horizontal position of the mirrors moved by radiation
	pressure. The right graph shows the pitch angle change. $m0$ to
	$m3$ refer to the four test masses of an advanced LIGO style
	quad-suspension. Lock is lost due to the pitch offset when
	the power inside the cavity reaches about 10\,\% of full power.}
	\label{fig:angle_oscillation}
	\end{center}
\end{figure}

In order to avoid the radiation pressure problems, we have to lock the
cavity with a low input power. However, reducing the laser power to
0.2\,\% of the full power may not be practical as it changes various
parameters of the interferometer drastically, especially the power
falling onto the detection ports. Offset lock is a way to mitigate this
problem and discussed in detail in section\,\ref{section: offset lock}.

\subsubsection{Arm lock simulation}

Here we show simulation results of arm cavity lock using E2E. This
simulation was actually done using the parameters of the old advanced
LIGO arm cavity design (4\,km, finesse 1250) by O. Miyakawa. Since the
characteristics of the cavity is very similar to the LCGT arm cavities,
we show the results here as a reference. The laser power was largely
reduced to eliminate the radiation pressure effect.

In the simulation, the PDH error signal was divided by the transmitted
power of the cavity to gain a wider liner region. According to the
simulation, it is almost impossible to lock the cavity due to the
ringing of the error signal, if the mirror speed exceeds 25\,nm/sec.

The probability of successful lock for a single encounter with a
resonance was calculated for several different mirror speeds
(Table\,\ref{tbl:lockability}). The results indicate that we have to have an
extremely slow mirror motion (less than 1\,nm/sec) in order to lock the arm cavities 
at the first attempt using the PDH signals.

\begin{table}[htbp]
\begin{center}
\begin{tabular}{c|c|c|c|c}

 Mirror speed &25\,nm/s&5\,nm/s&2.5\,nm/s&0.5\,nm/s \\
\hline
Lock probability& 9.5\%&32\% &56\%&98\%  \\

\end{tabular}
\caption{Lock probability with various mirror speeds.}
\label{tbl:lockability}
\end{center}
\end{table}

\subsection{Advanced lock acquisition techniques}
\subsubsection{Offset locking}
\label{section: offset lock}

Offset lock is a way to prevent large build up of the carrier power
inside an arm cavity. The cavity is first locked at a point far from the
resonance using the DC transmitted power as the error signal. This is
useful to avoid a strong kick of the mirrors by the radiation pressure.
It also gives us a time to switch various error signals and servo
topology before building up the carrier power in the cavities.  After
the arm cavities are locked with a CARM (common arm length) offset, it
is slowly reduced to fully lock the interferometer.

Figure\,\ref{fig:offset lock error signal} shows results from E2E
simulations of offset lock signals. An LCGT arm cavity was swept at
various speed and the error signals of offset lock
($1/\sqrt{P_\mathrm{trans}}$) are plotted. $P_{\mathrm{trans}}$ is
normalized to be 1 when the cavity is fully resonant.  When the mirrors
are moving fast, the cavity cannot build up the power enough. This the
reason for the saturation of the error signal around the resonance for
high speed cases. In reality, the servo feedback tries to slow down the
mirrors. Therefore, the saturation of the error signal will be less
significant. 

The error signal has a large linear range.  We will have to switch
between several PDs with different gains to achieve a good error signal
over a large range of the transmitted power.

\begin{figure}[htbp]
   \begin{center}
   \includegraphics[width=12cm]{inv_sqrt.eps}
   \caption{Transmitted light, $1/\sqrt{P_{\mathrm{trans}}}$ + offset vs
    Cavity microscopic length. $P_{\mathrm{trans}}$ is normalized with
    the transmission power at the full lock.}
   \label{fig:offset lock error signal}
   \end{center}
\end{figure}

\subsubsection{Green laser pre-lock}
\label{section: green laser lock} 

Green laser pre-lock is an advanced technique for lock acquisition
proposed by LIGO people\,\cite{GreenLock}. It makes use of frequency
doubled Nd:YAG lasers (emit green light) installed at the end stations
to lock the arm cavities prior to lock the central part of the
interferometer. The arm cavities are kept away from resonances of the
carrier and the sidebands of the main laser.  In this way, we can avoid
disturbances of the error signals from unwanted resonances of the
carrier and the sidebands in the arms while attempting to lock the
central part.

The green lasers at the end stations are phase locked to the main laser
to suppress the relative frequency noise. For this purpose, a small
amount of the main laser light is picked off and sent to the end stations
through optical fibers.  Long fibers may introduce extra phase
noises. There is a technique already demonstrated to cancel this noise
by sending back a part of the light delivered to the end station through
the same fiber\,\cite{FiberNoiseCancel}.

The test mass mirrors are coated with a dichroic coating to have some
reflectivities for the green laser. The finesse of the arm cavities is
set not too high ($\sim$100) so that the pre-lock is easy. Actually the
laser frequency can be used to initially lock the green laser to the
cavities.  Therefore, the lock acquisition of the green lasers should be
almost instantaneous. After the cavities are locked with the frequency
actuators, the feedback points can be handed off to the mirror
actuators.


\paragraph{Benefits of green laser pre-lock}
As was mentioned above, the green pre-lock prevents accidental
resonances of the carrier and sidebands in the arms during the initial
lock acquisition of the central part. In addition, this scheme allows us
to control the microscopic cavity length freely by changing the phase
lock frequency. This is a great benefit because then we can always make
sure the state of the arm cavities after the central part is
locked.

Figure\,\ref{fig:lock point} shows a resonance curve of an arm cavity as
a function of the cavity length. If we use the offset lock scheme, the
cavity length has to be in the non-shaded region after the central part
is locked. Otherwise, the cavity has to pass at least one sideband or
carrier resonance to reach the desired offset lock point. Also the
direction of the mirror motion has to be as indicated by the light blue
arrow in the figure. For the offset lock to be successful, both the arm
cavities must satisfy these conditions by chance at the time the central
part is locked.

If we use the green laser pre-lock, the location of the cavity in the
resonance curve can always be pre-determined before the central part is
locked. An easiest way to ensure this is to mis-align the PRM and scan
the arm cavity length with the green laser lock before attempting to
lock the central part.


\begin{figure}[htbp]
   \begin{center}
   \includegraphics[width=8cm]{drawings/Lock-Point.eps}
   \caption{Cavity length conditions for successful offset lock}
   \label{fig:lock point}
   \end{center}
\end{figure}


\subsection{Conclusion for lock acquisition}
In order to avoid a large kick by radiation pressure, offset lock is
inevitable. Offset lock also provides a pause before going to the full
lock for switching the error signals and preparing the servo topologies
for the full lock. 

Since the linear range of the offset lock error signal is very large, we
should be able to lock the arm cavities even with the 200\,nm/sec RMS
speed.

If we implement the green laser pre-lock, the RMS speed of the free
swinging mirrors becomes totally irrelevant. Since the green laser
pre-lock provides fascinating capabilities for lock acquisition, we
should pursue this technique further more.

From these considerations, we conclude that SPI is not essential for
lock acquisition.

\section{Vacuum tubes}
The default diameter of the vacuum tubes for LCGT (1\,m) was chosen to
accommodate SPI.  The scattered light noise was estimated for this tube
diameter with two laser beams and confirmed to be a factor of 20 smaller than
the requirement\cite{Vacuum}.

If we do not employ the SPI, but still use the same tube diameter, the
safety margin will be much larger. Therefore, there is a room for
reducing the beam tube diameter to cut the construction cost.  According
to our calculations, we can reduce the diameter to 60\,cm at the expense
of an increase in the estimated scattered light noise by a factor of 1.6
(from $5\times10^{-22}\mathrm{m/\sqrt{Hz}}$ to
$8\times10^{-22}\mathrm{m/\sqrt{Hz}}$).  Even with this diameter, We
still have a safety factor of 75 for the scattered light noise from the
design sensitivity. Therefore, we can afford this down scaling if
necessary.  The estimated cost reduction by this is about 2.5 billion
JPY.

However, since the vacuum tubes are hard to replace once they are
installed, we have to be careful when choosing smaller diameter tubes.
Having large diameter vacuum tubes will allow us to retrofit SPI in the
future upgrades or to use larger laser beams when larger mirrors become
available.


\newpage
\appendix
\section{Design of the LCGT Suspension Point Interferometer}
\label{section: SPI} 

In this section, we review and discuss the design of the LCGT SPI in
order to assess the technical feasibility of the SPI.  Although we do
not recommend to install an SPI to LCGT as the conclusion of the working
group, the current design of the LCGT SPI is presented here for the
purpose of documentation of the discussions within the working group.

The first section reviews the working principle of SPI. Then we will
present the current design of the LCGT SPI.

\subsection{Working Principle}
\label{section Working Principle of the SPI}

Suspension point interferometer (SPI) is an active vibration isolation
scheme for interferometric gravitational wave detectors. The SPI makes
use of an auxiliary interferometer to monitor the seismic vibration
transmitted through the suspension wires. The seismic vibration is
actively suppressed by an appropriate feedback control using the signal
from the auxiliary interferometer. The name ``suspension point
interferometer'' is used because the auxiliary interferometer is usually
placed at the suspension points of the mirrors of the main
interferometer.

\begin{figure}[htbp]
\begin{center}
 \includegraphics[height=7cm]{drawings/SPI-FP-Michelson.eps} 
\end{center}
\caption{A conceptual configuration of suspension-point interferometers
 installed to a Fabry-Perot Michelson interferometer}
\label{SPI-FP-MI}
\end{figure}

Figure \ref{SPI-FP-MI} shows a basic configuration of a
Fabry-Perot-Michelson interferometer equipped with a suspension point
interferometer. The lower interferometer, which is composed of blue
optical components in the figure, is called ``main interferometer (MIF)''
and used to detect gravitational waves.  The upper interferometer
(colored green in the figure) is called ``suspension point
interferometer (SPI)''.  The vibration
transmitted through the wires is monitored by the SPI. Then forces to
nullify the vibration are applied to the mirrors of the SPI. Through this,
the mirrors of the MIF are isolated from external vibrations.

In this scheme the reference of motion is not the local inertial frame;
instead, the SPI measures the same degree of freedom as the MIF,
i.e. the distance between the front and end mirrors. Each mirror serves
as a reference for its companion mirror.  Unlike conventional active
vibration isolation schemes using accelerometers, the SPI has a DC
performance because the sensor is essentially a position sensor.

In figure \ref{SPI-FP-MI}, the SPI is illustrated as a recombined
Fabry-Perot-Michelson interferometer. Actual configuration of the 
LCGT SPI does not necessarily have to be recombined. The interferometer
configuration of the LCGT SPI is discussed in section \ref{Interferometer Configuration}.

\subsubsection{Rigid Bar Picture}
\label{Rigid Bar Picture}
\begin{figure}[htbp]
\begin{center}
\begin{minipage}[c]{6cm}
\begin{center}
\includegraphics[height=4cm]{drawings/SPI-rigid-1.eps}

\textbf{(a)}
\end{center}
\end{minipage}
\begin{minipage}[c]{6cm}
\begin{center}
\includegraphics[height=4cm]{drawings/SPI-rigid-2.eps}

\textbf{(b)}
 \end{center}
\end{minipage}
\caption{Rigid bar picture of SPI.}
\label{SPI rigid}
\end{center}
\end{figure}

The working principle of SPI is shown by conceptual drawings in
Fig.\ref{SPI rigid}. In this case, only a Fabry-Perot arm is shown.
Once the SPI is locked, the differential motion of the mirrors of the
SPI is suppressed to the accuracy of the servo loop. Therefore, the SPI
can be considered as a rigid bar hung across the two suspension systems
(Fig.\ref{SPI rigid}\,(b)). If the pendulums hanging from this virtual
rigid bar are identical, no differential motion will be transmitted to
the MIF. Consequently the seismic noise in the MIF is significantly
reduced.  However in reality, the pendulums are not exactly identical. A
fraction of the common motion at the SPI stage is converted to
differential motion at the MIF by asymmetry of the suspensions.

As a measure of  this effect, we define the common mode rejection ratio
(CMRR) of SPI as:
\begin{equation}
 \mathrm{CMRR}\equiv 2\left|\frac{H_1\left(\omega\right)-H_2\left(\omega\right)}{H_1\left(\omega\right)+H_2\left(\omega\right)}\right|
\end{equation} 
Here $H_1\left(\omega\right)$ and $H_2\left(\omega\right)$ are the
transfer functions of the suspensions hanging below the SPI stage.
In the case of single
pendulums as shown in Fig.\ref{SPI rigid}\,(b), the CMRR is
approximately written as,
\begin{equation}
\mr{CMRR} \simeq \left|H_1(\omega)\right|\sqrt{\left(\frac{\Delta
	     l}{g}\right)^2\omega^4 +
\left(\frac{l\gamma}{mg}\right)^2\left(\frac{\Delta
 l}{l}+\frac{\Delta\gamma}{\gamma}-\frac{\Delta m}{m}\right)^2\omega^2}.
\label{CMRR approx} 
\end{equation}
\begin{equation}
 H_1\left(\omega\right) \equiv
  \frac{1}{1+i\frac{l\gamma}{mg}\omega-\frac{l}{g}\omega^2},
\label{H1}
\end{equation}
where $l$, $m$, $\gamma$ are the length, mass and damping factor of
the pendulum 1 and the letters with $\Delta$ represent the difference between
the corresponding parameters of the two pendulums.
The frequency dependence of CMRR is plotted in Fig.\ref{CMRR plot} in the
case of 1\% asymmetry. At frequencies higher than the resonance of the
pendulum, the CMRR approaches to a plateau at $1/(\omega_1-\omega_2)^2$,
where $\omega_1$ and $\omega_2$ are the resonant frequencies of the pendulums.
At lower frequencies the CMRR improves dramatically as the frequency
goes down. The CMRR is poor at the resonant frequency. In the case of
multiple pendulums, the shape of the CMRR around the resonances changes
but the general trend is the same.

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[height=7cm]{plot/CMRR.eps}
\caption{Frequency dependence of CMRR: The blue line shows the CMRR with
  all asymmetries included. The green circles show the CMRR with only
  $\Delta l$. The red line is the CMRR with only $\Delta m$. The pink
  triangles show the CMRR with only $\Delta \gamma$. All asymmetries are
  1\%.}
\label{CMRR plot}
 \end{center}
\end{figure}

CMRR is one of the limiting factors of the performance of an SPI. Other
limiting factors includes coupling from other degrees of freedom, servo
gain of the SPI, noise of the SPI and so on.

\subsection{Design of the LCGT SPI}
\label{Design of LCGT SPI}
In this section, we discuss the design of the LCGT SPI.
\subsubsection{Interferometer Configuration}
\label{Interferometer Configuration} There are several possibilities for
the interferometer configuration of the LCGT SPI. The most basic one is
to use an independent Fabry-Perot cavity for each arm.
Since the arm cavities are most important both in terms of the noise
reduction and lock acquisition, this configuration is the baseline
design for the LCGT SPI.

By using more complex interferometer configurations similar to the main
interferometer, we can suppress the motions in the auxiliary degrees of
freedom, such as MICH, PRCL and SRCL. In general, the coupling of the
motions in those DOFs to the GW channel is much smaller than the arm
cavity contribution. Moreover, the suspensions used for the auxiliary
mirrors (PRM, BS, SRM) are not inferior to the ones used for the main
optics. Therefore, the vibration noises of the aux. DOFs are expected
to be the same order as the arm cavities. This means it is not necessary
to further suppress the vibrations in the aux. DOFs by SPI. 

Heat links are not attached to the PRM, BS and SRM, thus no need to
worry about the extra noise introduced. However, the ITMs will be
connected to the largest number of heat links among the core optics in
order to subtract large heat generated by the laser light passing
through the substrate. As mentioned in section \ref{Rigid Bar Picture},
the common motion of the two mirrors forming an arm cavity is not
suppressed by the baseline SPI. This common motion is seen as the MICH
motion by the main interferometer. In general, the contribution of MICH
to the GW channel is smaller than the arm cavities roughly by a factor
of finesse. The finesse of the LCGT arm cavity is at least
500. Therefore, if the MICH seismic noise from the heat-links is more
than 500 times larger than the target displacement noise of the
interferometer, we need an SPI for MICH, i.e. the SPI has to be a
recombined Fabry-Perot Michelson interferometer. However, as we see in
section \ref{Heat link vibration isolation}, this is unlikely to be the
case.

The other benefit of introducing the SPI for auxiliary DOFs is to
alleviate the coupling of the shot noises in the auxiliary DOFs through
the feedback loops. The story goes like this: Usually, the shot noises
of the error signals for the auxiliary DOFs are not as good as the one
for the GW signal because they do not use the full carrier power
enhanced by the arm cavities. Since we feed back those noisier signals
to control the auxiliary DOFs, these DOFs move more than the main DOF
(the differential arm length). These noises are coupled to the GW
channel through non-diagonal elements in the sensing matrix. This is
called the loop noise. It can be reduced if we can reduce the feedback
bandwidth of the auxiliary DOFs. By using the SPI for those DOFs, we can
reduce the RMS motion of the aux. DOFs, which enables us to use smaller
control bandwidths. However, according to the simulation study in the
interferometer bandwidth working group, the loop noise coupling can be
made small enough by choosing appropriate signal ports and a moderate
feed forward gain without compromising the feedback bandwidth of the
auxiliary DOFs\,\cite{LCGT-BW}. 

The basic operation of the baseline SPI (a simple Fabry-Perot) was
demonstrated\,\cite{aso-phd}. However, stabilization of the auxiliary
DOFs using SPI has not yet demonstrated experimentally. 

\subsubsection{Location of SPI in the Suspension Chain}
\label{Location of SPI in the Suspension Chain}
In the previous sections, the SPI is assumed to be formed at the
penultimate stages of the suspensions. This is the default choice of
location for the LCGT SPI. However, there are several other choices for
the location of the SPI in the suspension chain.

\begin{figure}[htbp]
\begin{center}
 \includegraphics[height=7cm]{drawings/SPI-Location.eps}
\end{center}
\caption{A conceptual diagram of the LCGT cryogenic suspension.}
\label{SPI-Location}
\end{figure}

Figure\,\ref{SPI-Location} shows a conceptual diagram of the cryogenic
suspension of LCGT. Heat-link wires are shown in blue curves. 
There are two possibilities for the first anchor point of the
heat-links, the suspension platform and the penultimate recoil mass.

If we anchor the heat-links on the recoil mass for the penultimate stage,
we could form an SPI using these masses to suppress the heat-link
vibrations. One advantage of this configuration is that we can prevent
or greatly relieve the independent alignment problem, which is
discussed in section \ref{Suspension Design}. Another advantage is that
we can sense the heat-link vibration before it gets attenuated,
therefore, the noise requirement for the SPI will be less stringent.

However, we cannot use the actuator acting between the recoil mass and
the penultimate mass. If we do so, the feedback force to cancel the
heat-link vibration will also actuate the penultimate mass.  This will
just copy the heat-link vibration onto the penultimate mass, thus making
the situation worse. Therefore, we have to prepare another set of
actuators to control the SPI on the recoil masses.

Another problem with this configuration is the inter-degree-of-freedom
couplings caused by the heat-link. An SPI formed by the penultimate
recoil masses will only suppress the heat-link vibration in the in-line
DOF. Therefore, the penultimate recoil mass will still be largely shaken
in, for example, the perpendicular direction to the beam by the
heat-links. Since the heat-links are soft and loose wires, it is
possible that this perpendicular motion is converted to the in-line
motion at the next stage. The amount of this kind of coupling has to be
experimentally evaluated.

One last disadvantage of the penultimate recoil mass SPI is that it
cannot suppress the seismic vibration coming from the above. We could
still copy the quietness of the recoil mass to the penultimate mass by
using a local sensor between the two masses. However, the applicable
frequency will still be severely limited by the noise of the local
sensor.

If we do not anchor the heat-link on the penultimate recoil mass (like
shown by the dashed heat-link in figure\,\ref{SPI-Location} and this is
the default configuration), we could
use the suspension platforms as SPI masses. This configuration can
suppress both the seismic and heat-link vibrations. Although the
orientations of the platform and the main mirror are still coupled, we
may be able to decouple them by counter actuating the penultimate mass
(see the discussions in section\,\ref{Suspension Design}).

The biggest problem of the platform SPI is that the platform is not
exposed to the beam tube in the default design. In order to directly
expose it to the tube, we have to make the tube very large. Steering the
beam down to the vacuum tube height by hanging a periscope like
structure is possible but it will complicate the system unnecessarily.
 
\subsubsection{Suspension Design}
\label{Suspension Design}
There is no detailed design for the suspension of the LCGT SPI mirrors yet.
In this section, we describe a conceptual design and discuss the
problems to be solved. In this section, the location of the SPI mirror is assumed to be the
penultimate stage of the suspension chain.

One big problem of SPI for the practical implementation is that the
orientations of the SPI mirror and the main mirror hung from it are
strongly coupled. Since the laser beams for the SPI and the main
interferometer are not perfectly parallel, we need to adjust the
alignment of the SPI and MIF mirrors independently.  If we simply rotate
the whole SPI stage, the main mirror necessarily follow this
rotation. Counter rotating the main mirror by actuators on the main
mirror is not a good idea because the actuators on the mirror are
designed to be weak enough not to introduce actuator noise.  

If the angular actuation range of the main mass actuator is
$\theta_{\mathrm max}$, the actuator noise spectrum can be naively
calculated by the following formula\,\cite{LCGT Design Doc},
\begin{equation}
x_{\mathrm n}(f)= r \frac{m}{I_{\mathrm m}}\frac{V_\mathrm{max}}{V_{\mathrm n}}\left(\frac{f}{f_0}\right)^2 \theta_\mathrm{max}
\end{equation}
where $r$ is the distance between the actuators (we assumed that four
actuators are attached on the back of the mirror), $m$ and $I_{\mathrm
m}$ are the mass and the moment of inertia of the main mirror,
$V_\mathrm{max}$ and $V_\mathrm{n}$ are the maximum input voltage and
the input equivalent noise of the actuator driver and $f_0$ is the
resonant frequency of the main mirror in the rotational degree of
freedom in question. The estimated actuator noise is plotted using $r=0.1$\,m, $m=30$\,kg, $I_{\mathrm
m}=0.176\,\mathrm{kg\cdot m^2}$, $V_\mathrm{max}=10$\,V,
$V_\mathrm{n}=1\,\mathrm{nV/\sqrt{Hz}}$, $f_0=0.8$\,Hz and
$\theta_\mathrm{max}=5\times 10^{-6}$\,rad, along with the classical
noise budget of LCGT (figure\,\ref{fig:ActuatorNoise}).

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[height=7cm]{plot/ActuatorNoise.eps}
\label{fig:ActuatorNoise}
\caption{Actuator noise of the main mirror and the classical noise of LCGT}
 \end{center}
\end{figure}
 
This result means that we can have only have $5\times 10^{-6}$\,rad
$\approx3\times10^{-4}$\,deg. of angular actuation range. This is
obviously not enough. As suggested in \cite{LCGT Design Doc}, we could allocate dedicated
actuators for rotational degrees of freedom, which has only a small
coupling (like 1/100) to the in-line direction. However, even a 1/100
coupling would allow us to move the mirror only by $3\times10^{-2}$\,deg.
Moreover, we do not want to attach many actuators on the main mirror
because they may increase the thermal noise.

Since the counter actuation does not seem promising, we have to do some
trick to the SPI mirror. The basic idea is to make the SPI mirror
a composite structure and rotate the reflective surface with respect to the
body. A conceptual design is shown in figure\,\ref{fig:SPI-Mirror}.

\begin{figure}[htbp]
 \begin{center}
  \includegraphics[height=8cm]{drawings/SPI-Mirror.eps}
\caption{A conceptual design of the SPI mirror suspension}
\label{fig:SPI-Mirror}
 \end{center}
\end{figure}

The SPI mirror is supported from the outer ring (the body of the SPI
stage) by a gimbal structure thorough torsion bearings. Therefore, the
mirror can be rotated with respect to the outer ring. We will attach
coil magnet actuators on the mirror and apply internal (rotational)
forces between the outer ring and the mirror to adjust the orientation
of the mirror. The main mirror is hung from the outer ring, therefore
it is not moved by the adjustment to the SPI mirror orientation.

Since this configuration supports the mirror through a complicated rigid
structure, the mechanical quality factor will be degraded. Although the
thermal noise requirement for the SPI mirror is not as stringent as the
one for the main mirror, still we have to carefully check this before
employing this idea. 

The other worry is that the torsion bearings may have some mechanical
resonances in the observation band. This may contaminate the sensitivity
of the main interferometer. Also the mechanical resonances may limit the UGF of the SPI
servo hence limiting the SPI's ability to suppress vibrations.

There exists only a conceptual idea for the SPI suspension at this
moment. We need more R\&D on this in order to implement an SPI for LCGT
in practice.

\subsubsection{Noise Requirement}
\label{SPI Noise Requirement} The noise requirement of the LCGT SPI, if
its optical configuration is an independent Fabry-Perot cavity for each
arm, is plotted in Fig.\,\ref{Noise requirememt, SPI noise requirement}
as a gray curve.  The noise of the SPI is fed back to the SPI mirror by
the servo. This vibration is transmitted to the test mass through the
final pendulum of the suspension system.  The noise requirement is
calculated by requiring that the noise introduced from the SPI is less
than the estimated classical noises for the main interferometer by a
safety factor of 10. Since the quantum noise becomes dominant for the
main interferometer at frequencies above 10\,Hz or so, this requirement
is more stringent than necessary. But we adopted this approach as the
quantum noise level of the LCGT's target sensitivity is still not fully
decided.

Assuming that the loop gain of SPI servo has a simple $f^{-1}$
shape and its bandwidth is 50\,Hz, the contributions of various noises
are calculated.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth,keepaspectratio=true]{./plot/NoiseRequirement_SPIRequirement.eps}
\end{center}
\caption{Noise requirement of the LCGT SPI. This requirement is
calculated from the LCGT classical noises, a transfer function of the
final stage suspension ($f_{\mathrm{c}}=0.8\,\mathrm{Hz}, Q=1000$) and a
safety factor of 10.}  \label{Noise requirememt, SPI noise requirement}
\end{figure}

\paragraph{Thermal noise of suspension:}

At frequencies higher than the resonant frequency of the suspension, the
suspension thermal noise has a $f^{-2.5}$ dependence. Since the suspension
thermal noise is a real vibration of the SPI mass, the SPI suppresses
this noise. Therefore the requirement for the suspension thermal noise
above the SPI stage is relieved by the SPI's servo gain. Quantitatively,
the requirement of the suspension thermal noise can be written as:
\begin{align}
&h_{\mathrm{thermal(suspension)}}<1\times 10^{-18}\times f^{-2.5}~~1/\sqrt{\mathrm{Hz}},\\
&x_{\mathrm{thermal(suspension)}}<3\times 10^{-15}\times f^{-2.5}~~\mathrm{m}/\sqrt{\mathrm{Hz}}.
\end{align}

\paragraph{Thermal noise of mirror:}

The thermal noise of mirror has a $f^{-0.5}$ dependence and increases the
noise level of the MIF because it is converted to length motion by the SPI
servo. Therefore, the requirement of the mirror thermal noise is
written as:
\begin{align}
&h_{\mathrm{thermal(mirror)}}<3.9\times 10^{-21}\times f^{-0.5}~~1/\sqrt{\mathrm{Hz}},\\
&x_{\mathrm{thermal(mirror)}}<1.2\times 10^{-17}\times f^{-0.5}~~\mathrm{m}/\sqrt{\mathrm{Hz}}.
\end{align}
In order to realize this requirement, Q-factor of the mirror should be
higher than 165 if the mirror geometry is a simple cylindrical
shape. However, since actual mirror geometry of the SPI mass is
complicated as shown in Fig. \ref{fig:SPI-Mirror}, we will need to
analyze it more in detail.

\paragraph{Coil driver noise:}

Coil driver noise has a $f^{-2}$ dependence at a higher frequency than
the resonant frequency of the suspension. It is suppressed by the SPI
servo within the control bandwidth. The requirement for the raw coil
driver noise (without the servo suppression) is:
\begin{align}
&h_{\mathrm{coil}}<4.8\times 10^{-19}\times f^{-2}~~1/\sqrt{\mathrm{Hz}},\\
&x_{\mathrm{coil}}<1.4\times 10^{-15}\times f^{-2}~~\mathrm{m}/\sqrt{\mathrm{Hz}}.
\end{align}
If the coil driver noise is $1\,\mathrm{nV}/\sqrt{\mathrm{Hz}}$, we have
to choose the actuator efficiency to be less than $1.4 \times
10^{-6}/f^{2}~\mathrm{m}/\mathrm{V}$. Assuming that we use operational
amplifiers with the output voltage range of $\pm 15\mathrm{V}$ and the
suspension's resonant frequency is 0.8\,Hz, the maximum actuation range
is 66\,$\mathrm{\mu m}$.

\paragraph{Shot noise:}
Since the noise requirement in the high frequency is decreased by the
final stage suspension, the shot noise level can be chosen relatively
freely as shown in Fig. \ref{Noise requirememt, Shot noise}. The lower
the finesse of the cavities is, the easier the lock acquisition
is. However, since acceptable laser power on PD is limited, finesse of
50 seems to be a better solution.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth,keepaspectratio=true]{./plot/NoiseRequirement_ShotNoise.eps}
\end{center}
\caption{Shot noise. Laser power and finesse of the arm cavities can be chosen relatively freely.}
\label{Noise requirememt, Shot noise}
\end{figure}

\subsubsection{Input Optics}

A laser for the SPI should be phase-locked to the main laser. We can
select two solutions to obtain a phase-locked laser beam. One is to
picked up a fraction of the main laser as shown in Fig. \ref{Input optics, a
fraction of main laser}, and the other is that another laser is prepared and is
phase-locked to the main laser as shown in Fig. \ref{Input optics, another
laser}. Since each configuration has some advantages and disadvantages,
we describe them in the following.

First, we describe the configuration shown in Fig. \ref{Input optics, a
fraction of main laser}. The biggest advantage of this configuration
is that it does not require any control system for phase locking. In
addition, laser power available is not limited severely. A disadvantage
of this configuration is that the SPI operation is affected by the MIF's
operation directly.

The configuration shown in Fig. \ref{Input optics, another laser} uses a
fiber coupled secondary laser for SPI. The biggest advantage of this
configuration is that we can lead laser beam to MMT4 easily by using an
optical fiber. However, we must suppress the noises induced by the
optical fiber. Another advantage is that SPI operation is relatively
independent of the MIF. Even if the MIF looses lock, the SPI may be able
to stay locked. This configuration has two disadvantages. One is that we
need a tight phase-lock control of the secondary laser. The other one is
that the available laser power is limited to about 2\,W which is the
highest power of commercial laser with reliable performance to be used
in a gravitational wave detector.

Since the length of Fabry-Perot cavities in SPI is stabilized relative
to the frequency of the laser, the phase-locked loop for the secondary
needs a large enough gain to stabilize the laser frequency of a free
running laser to that of the SPI requirement shown in Fig. \ref{Noise
requirememt, SPI noise requirement}. Typically, a frequency noise of an
NPRO is 300\,$\mathrm{Hz}/\sqrt{\mathrm{Hz}}$ at 100\,Hz, which is
equivalent to a strain sensitivity of $1\times
10^{-12}\,/\sqrt{\mathrm{Hz}}$ at 100\,Hz. Accordingly, the loop gain
needed for the phase lock is about 10$^{9}$ at 100\,Hz. In addition, a
photo detector with low noise and high current capacity is needed for
the phase lock.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth,keepaspectratio=true]{./drawings/InputOptics_Laser1.eps}
\end{center}
\caption{Optical configuration (A). The laser light for the SPI is picked
 off from the main laser.}
\label{Input optics, a fraction of main laser}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth,keepaspectratio=true]{./drawings/InputOptics_Laser2.eps}
\end{center}
\caption{Optical configuration (B). A dedicated laser, which is
 phase-locked to the main laser, is used for the SPI.}
\label{Input optics, another laser}
\end{figure}


\subsubsection{Comments on technical feasibility}

The biggest problem for the practical implementation of the SPI for LCGT
is the independent alignment of the test masses and the SPI mirrors. The
proposed suspension design to mitigate this problem is only a conceptual
one and requires a detailed analysis and experimental tests. Since the
structure of the SPI masses is expected to be complex, we may have to
deal with extra mechanical resonances, which might compromise the
sensitivity of the main interferometer.

There is no demonstration of SPI with interferometer configurations
other than a Fabry-Perot. If we need to install the SPI to auxiliary
DOFs, this is a big uncertainty. We will need to do a prototype
experiment for this.


\section{Members of the working group}
Chair: Yoichi Aso (University of Tokyo)\\\noindent 
\hbox{Masaki Ando (Kyoto University)}, \hbox{Koji Arai (Caltech)},
\hbox{Kiwamu Izumi (NAOJ)}\\\noindent
\hbox{Osamu Miyakawa (ICRR)}, \hbox{Shinji Miyoki (ICRR)},
\hbox{Shigenori Moriwaki (University of Tokyo)}\\\noindent
\hbox{Shigeo Nagano (NICT)}, \hbox{Noriaki Ohmae (University of Tokyo)},
\hbox{Kentaro Somiya (Caltech)}\\\noindent
\hbox{Toshikazu Suzuki (KEK)}, \hbox{Ryutaro Takahashi (NAOJ)},
\hbox{Daisuke Tatsumi (NAOJ)}\\\noindent
\hbox{Takashi Uchiyama (ICRR)} and the LCGT SPI special working group.


\section{Acronyms}
\begin{itemize}
 \item PRC: Power Recycling Cavity
 \item PRCL: Power Recycling Cavity Length
 \item SRC: Signal Recycling Cavity
 \item SRCL: Signal Recycling Cavity Length
 \item MICH: Michelson
 \item CARM: Common arm length
 \item DARM: Differential arm length
 \item MIF: Main Interferometer
 \item PDH: Pound Drever Hall
 \item DOF: Degree(s) Of Freedom
 \item ROC: Radius Of Curvature
\end{itemize}



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\end{document}
