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\title{Parameters of the OPO for squeezed vacuum injection (SQZ) in O3}
\author{L. McCuller, D. Sigg, L. Barsotti}
\ligodccnumber{T}{20}{00273}{v1} \ligodistribution{LIGO Laboratory}
%\ligodraft1500019


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\section{Overview}
This document describes the parameters of the SQZ OPO as implemented during O3.

The main difference with respect to the OPO design described in \dcc{T1700104-v2} is the reflectivity of the M1 mirror reflectivity @ 532nm (input coupler for green), which was changed from the original $R = 87.5\%$ to  $R = 98\%$ (see \dcc{E1700299}).

ECR \dcc{E1800193} describes the motivation for the change, mainly to increase the green finesse of the OPO so as to require less input green power to pump the OPO, therefore reducing the amount of green light delivered by the fiber system. 

\section{O3 OPO Parameters}

Table~\ref{tab:mirrors} shows the mirrors coatings of the OPO M1 and M2 mirrors during O3.

\begin{table}[h!]
\centering
 \begin{tabular}{|r | c|| r | c |} 
 \hline
M1 1064nm R &amp; 87.5\% &amp; T &amp; 0.125\% \\
M2 1064nm R &amp; 99.85\% &amp; T &amp; 0.15\% \\
M1 532nm R &amp; 98.0\% &amp; T &amp; 2\% \\
M2 532nm R &amp; ~1.0 &amp; T &amp; $&lt;0.10$ppm \\
                      \hline
 \end{tabular}\label{tab:mirrors}
\end{table}

The O3 OPO 1064nm finesse is 49.6 and the FSR is 851MHz, so the unsqueezed
cavity pole is 8.58MHz.

The theoretical escape loss at 1064nm is $T_{\text{M2}} / T_{\text{M1}} =1.2\% $. This
number is consistent with the measured escape efficiency of 98\%-99\%, indicating that as far as we know, the dominant loss in the OPO is the M2 CLF coupling mirror, not the crystal itself.

The power transmissivity of the on resonance CLF upper-side-band at 3.125MHz is
3.94\%. With the OPO at NLG=14.8, generating 14.5db of squeezing, the
transmission is enhanced by parametric gain to be 8.6\%.

The theoretical 532nm OPO finesse in the ideal lossless case would be $\sim 315$. 
Based on a study on O3 data (\dcc{T2000150}), the OPO round-trip 532nm loss ($L_{\text{rt532}}$) has been increasing from $\sim 2.1\%$ to $\sim 3.75\%$, so the effective finesse of the OPO during O3 has been between 150 and 110.

The corresponding power buildup factor G, defined as:
\begin{align}
     G &amp;= \frac{T_{\text{M1}}}{(1 - r_{\text{M1}}\sqrt{1 - L_{\text{rt532}}})^2} 
 \end{align}
 
has been therefore between $G = 50$ and $G = 25$.

Given the 1064nm finesse of $\sim50$, the O3 OPO cavity hits threshold at around 900mW of
internal circulating pump power. %See fig.\ref{fig:O4circulating}. This is where
%the 532nm losses become critically important, because if the losses are $&lt;0.8\%$,
%then the power buildup is $&gt;100$, but if they are larger than the input coupler,
%than the power buildup drops very quickly as the cavity becomes critically
%coupled or under coupled.
This entails that the threshold in units of input pump power (in the 00 mode) can be between $20 - 35$mW. This is consistent with the variations of threshold measurements taken at different times after a crystal position move (for example, here is the latest threshold measurement performed at LLO, \href{https://alog.ligo-la.caltech.edu/aLOG/index.php?callRep=49428}{log entry 49428}, showing a threshold of 35 mW).
\begin{comment}
\section{Threshold Modification}

Assuming we cannot improve the power buildup in the 532nm pump, the question is
can we increase the 1064nm finesse to lower the threshold in units of
circulating pump power. If we do that, what are the implications for

\begin{enumerate}
  \item Escape Efficiency or Escape Loss, which is finesse-enhanced.
  \item CLF and RLF upper-side-band power needed for length sensing the filter
    cavity, and its transmission through the OPO.
  \item Sensitivity to thermal detuning noise for both the phase noise in
    squeezing and potentially the filter cavity length signal.
\end{enumerate}

I have not yet done the analysis on the filter cavity signals, but following is the
full analysis on the OPO/squeezer closed-loop system to analyze the changes to
squeezer sensitivities.

You can see from the parameters calculated that the first two can be compensated
with the M2 1064nm transmissivity. After the parameters I analyze the thermal
issue which a bit more technical.

\section{O4 OPO Parameters}

\begin{table}[h!]
\centering
 \begin{tabular}{|r | c|| r | c |} 
 \hline
M1 1064nm R &amp; 93.0\% &amp; T &amp; 07.0\% \\
M2 1064nm R &amp; 99.9\% &amp; T &amp; 0.1\% \\
M1 532nm R &amp; 98.0\% &amp; T &amp; 2.0\% \\
M2 532nm R &amp; ~1.0 &amp; T &amp; 0.10\% \\
                      \hline
 \end{tabular}
\end{table}

The new OPO 1064nm finesse is 88.5 and the FSR is 851Mhz, so the unsqueezed
cavity pole becomes 4.81MHz. Note that now the CLF will be more out of the OPO resonance.
This will affect how changing the NLG changes the CLF1F error signals. Since the
cavity pole lowers in frequency as the NLG is increased, this should make NLG
increases have a smaller effect on the phasing CLF1F error signals.

Escape Loss: $1.4\% = T_{\text{M2}} / T_{\text{M1}}$ at 1064. This is because
the transmissivity of the M2 1064nm side is slightly reduced to compensate for the
increased finesse. Escape efficiency is worse with the new mirrors, but only by
a small amount. If the crystal is also lossy at 1064, those contributions will
be magnified by a approximately factor of 2. Assuming the worst escape
efficiency we have measured of 98\%, this means that the crystal loss
contribution is $\sim 1\%$ with the old finesse and at most $2\%$ at the new
higher finesse.

The CLF on resonance power tranmsissivity of the upper-side-band at 3.125MHz becomes
3.85\%. With the OPO at NLG=15.3, generating 14.5db of squeezing, the
transmission is enhanced by parametric gain to be 4.8\%. It is enhanced less for
this OPO because the NLG gain on the CLF is canceled from the OPO bandwidth
reducing due to the parametric gain. This happens now since the CLF is not well
within the OPO cavity-pole.

The 532nm finesse is 300 assuming no 532nm losses, and 152 assuming the same
critical-coupling losses as for the O3 case. The slight decrease in Finesse is
from the increased M2 532nm transmission.

This cavity with the new 1064nm finesse hits threshold at around 300mW of
internal circulating pump power - 3x reduction in the required pump power. See fig.\ref{fig:O4circulating}.
This is where the 532nm losses become critically

This entails that the threshold in units of input pump power can be between 3mW
if the loss is low, all the way to 8mW or 10mW if the cavity is critically or
under-coupled.

\section{Thermal Phase Detuning}


You can see from Fig.\ref{fig:temp_spec} that the OPO temperature loop has
between 20 - 50 uK RMS noise, depending on if you believe the noise is real over
200mHz, or if it is sensing noise injected by the 50mHz loop. Using the dndT
numbers from the VOPO final optical design, as well as the dLdT and indexes, one
can calculate the the PPKTP crystal converts temperature noise into differential phase
(1064nm vs 532nm) with the rate $1.16{\cdot}10^{-5}\frac{\text{rad}}{\text{K}}$ for
dndT, and the rate  $4.9{\cdot}10^{-7}\frac{\text{rad}}{\text{K}}$ for dLdT.

Together, this entails about $10^{-5}$rad of noise on the 1064nm beam in the OPO.
This gets enhanced by the 1064nm cavity finesse, so I have analyzed the sensing
matrix elements for the open loop tables \ref{tab:O3opoOL} and \ref{tab:O4opoOL}
``OPO 1064 [rad]'' and its conversion to ``SQZ\_phase'' (also in units of
radians of squeezing phase modulation). This shows that in the open loop, the
noise creates squeezing noise 10-20x as large, which is still under 1mrad and
should not be a major concern.

The closed-loop case is also analyzed in tables \ref{tab:O3opoCL} and
\ref{tab:O4opoCL}. For these tables, the standard OPO 532nm PDH, pump and CLF
ISS's, CLF2F loop and CLF1F/LO loop are all closed with infinite loop gains. The
matrix is modified to reflect that closed-loop response. You see that there is
still some coupling of 1064nm OPO dephasing to squeezer dephasing. This is due to
the CLF being an imperfect witness since it is injected in a different port than
the squeezed vacuum, and because it is at a different frequency and experiences
the cavity differently than the squeezed vacuum.

The closed loop sensing tables show that the thermal noise coupling will roughly
triple at the higher finesse, but still represent a rather small phase noise
contribution. I will augment this analysis to include the effective length noise
these signals will imprint on the filter cavity when it is controlled, but I
have not yet done so, since the increase in sensitivity is less than an order of
magnitude.

How these tables are generated and how this model works will be written up fully
in the filter cavity controls document.

\section{Figures (O3 Parameters)}
\autofigure{
  folder=./tresults/Apl_OPO/, 
  file=OPO_SQZ_circulating,
  width=0.8\textwidth,
  label={O3circulating},
  caption={
    Output squeezing level as a function of circulating power, using a cavity
    model established with 900mW causing threshold. Although this is immediately
    out of the cavity, the squeezing is degraded here by the imperfect escape
    efficiency.
  },
}

\begin{table}[h!]
\resizebox{1\linewidth}{!}{
\begin{tabular}{lrrrrrrrr}
\hline
 14.53 dB Gen.   &amp;     LO [rad] &amp;   OPO 1064 [rad] &amp;   CLFVCO [rad] &amp;   CLFVCO [rin] &amp;   pump [rin] &amp;   pump [rad] &amp;   OPO 532 [rad] &amp;   OPO len [nm] \\
\hline
 SQZ\_phase       &amp; -1           &amp;     -9.17075     &amp;    0           &amp;    0           &amp;  0           &amp;  0.5         &amp;     24.4998     &amp;    0.470403    \\
 CLF2F\_I         &amp;  0           &amp;     -0.016373    &amp;    0.0416985   &amp;   -4.28561e-14 &amp; -0.0126618   &amp; -0.0208493   &amp;     -1.01848    &amp;   -0.0243131   \\
 CLF2F\_Q         &amp; -2.31474e-15 &amp;      0.00800397  &amp;   -4.16986e-05 &amp;    0.0208545   &amp;  0.017191    &amp; -1.04246e-05 &amp;     -0.00670054 &amp;    5.973e-05   \\
 CLF2F\_RIN       &amp;  0           &amp;      0.793373    &amp;   -4.44089e-13 &amp;    1.00025     &amp;  0.0149817   &amp;  0           &amp;     -0.00366253 &amp;    0.00935566  \\
 CLF1F\_I         &amp; -0.262833    &amp;     -1.74384     &amp;    0.0810337   &amp;    3.39597e-15 &amp;  0.003244    &amp;  0.0908428   &amp;      4.43958    &amp;    0.0845094   \\
 CLF1F\_Q         &amp; -0.227502    &amp;     -2.18569     &amp;   -4.05168e-05 &amp;    0.0405168   &amp; -0.0175195   &amp;  0.113776    &amp;      5.58776    &amp;    0.105919    \\
 CLF1F\_RIN       &amp;  0           &amp;     -5.86119     &amp;    2.22045e-13 &amp;    1.00025     &amp;  0.989941    &amp; -2.55351e-12 &amp;     -0.242496   &amp;   -0.0704392   \\
 PUMP\_INT\_RIN    &amp;  0           &amp;      0           &amp;    0           &amp;    0           &amp;  1           &amp;  2.22045e-13 &amp;     -0.244994   &amp;   -0.00136698  \\
 PUMP\_REFL\_I     &amp;  0           &amp;      0           &amp;    0           &amp;    0           &amp;  0           &amp;  5.47273e-16 &amp;      5.19121    &amp;    0.122624    \\
 PUMP\_REFL\_Q     &amp; -6.07596e-29 &amp;     -6.07596e-28 &amp;   -6.07596e-29 &amp;   -6.07596e-29 &amp;  2.73568e-16 &amp; -2.73637e-19 &amp;     -0.0256965  &amp;   -0.000143377 \\
\hline
\end{tabular}
}
\caption{
  Open-Loop Sensing Matrix for OPO with O3 Mirror Configuration. The CLF
  I and Q phases are in units of CLF demodulation field normalized by the CLF
  single-sideband field incident on M2. This sensing matrix represent the OPO
  with sufficient pump power to generate 14.5db of (anti)squeezing at its output.
}
\label{tab:O3opoOL}
\end{table}

\begin{table}[h!]
\centering
\resizebox{1\linewidth}{!}{
\begin{tabular}{lrrrrrrrr}
\hline
 14.53 dB Gen.    &amp;   OPO 1064 [rad] &amp;   pump [rad] &amp;   OPO len [nm] &amp;   PUMP\_REFL\_I SG &amp;   CLF2F\_RIN SG &amp;   PUMP\_INT\_RIN SG &amp;    CLF2F\_I SG &amp;   CLF1F\_I SG \\
\hline
 SQZ\_phase        &amp;      -2.65704    &amp;  0.000216278 &amp;   -0.0322718   &amp;     -0.0100507   &amp;    3.29706e-13 &amp;       0.105961    &amp;   7.39375     &amp; -3.8047      \\
 CLF2F\_Q          &amp;      -0.00855367 &amp; -3.12739e-05 &amp;   -5.36606e-05 &amp;      0.00067626  &amp;   -0.0208493   &amp;      -0.0168659   &amp;   0.001       &amp; -8.80687e-15 \\
 CLF1F\_Q          &amp;      -0.735959   &amp;  5.41366e-05 &amp;   -0.00907431  &amp;     -0.0039677   &amp;   -0.0405067   &amp;       0.0422449   &amp;   1.68306     &amp; -0.865575    \\
 CLF1F\_RIN        &amp;      -6.65456    &amp; -2.43693e-12 &amp;   -0.0784627   &amp;     -4.94956e-06 &amp;   -1           &amp;      -0.974959    &amp;  -1.5975e-11  &amp;  0           \\
 PUMP\_REFL\_Q      &amp;      -1.8749e-28 &amp;  2.43537e-18 &amp;    0.000463614 &amp;      0.00495     &amp;    6.07444e-29 &amp;      -2.73568e-16 &amp;   1.90636e-27 &amp; -2.31172e-28 \\
 OPO 532 [rad] CL &amp;       0          &amp; -1.05423e-16 &amp;   -0.0236215   &amp;     -0.192633    &amp;    0           &amp;       0           &amp;   0           &amp;  0           \\
 CLFVCO [rin] CL  &amp;      -0.793175   &amp;  2.25315e-13 &amp;   -0.00937361  &amp;      1.52162e-06 &amp;   -0.99975     &amp;       0.014978    &amp;  -1.06473e-11 &amp;  0           \\
 pump [rin] CL    &amp;      -0          &amp; -2.2207e-13  &amp;   -0.00442016  &amp;     -0.047194    &amp;    0           &amp;      -1           &amp;   0           &amp;  0           \\
 CLFVCO [rad] CL  &amp;       0.392651   &amp;  0.5         &amp;    0.00477155  &amp;     -4.71938     &amp;   -1.0275e-12  &amp;      -0.303652    &amp; -23.9817      &amp; -0           \\
 LO [rad] CL      &amp;      -6.51371    &amp;  0.499784    &amp;   -0.0760483   &amp;     -4.70943     &amp;   -3.29706e-13 &amp;      -0.105961    &amp;  -7.39375     &amp;  3.8047      \\
\hline
\end{tabular}
}
\caption{
  Closed-Loop Sensing Matrix for OPO with O3 Mirror Configuration.
  Normalizations are the same as for the open loop table. The CL flags represent
  the gains to the actuators while the loop is closed. The SG flags represent
  the coupling of sensing noise.
}
\label{tab:O3opoCL}
\end{table}

\section{Figures (O4 Parameters)}

\autofigure{
  folder=./tresults/Apl_OPO_O4/, 
  file=OPO_SQZ_circulating,
  width=0.8\textwidth,
  label={O4circulating},
  caption={
    Output squeezing level as a function of circulating power, using a cavity
    model established with 900mW causing threshold. Although this is immediately
    out of the cavity, the squeezing is degraded here by the imperfect escape
    efficiency.
  },
}


\begin{table}[h!]
\resizebox{1\linewidth}{!}{
\begin{tabular}{lrrrrrrrr}
\hline
 14.45 dB Gen.   &amp;    LO [rad] &amp;   OPO 1064 [rad] &amp;   CLFVCO [rad] &amp;   CLFVCO [rin] &amp;   pump [rin] &amp;   pump [rad] &amp;   OPO 532 [rad] &amp;   OPO len [nm] \\
\hline
 SQZ\_phase       &amp; -1          &amp;     -17.122      &amp;    0           &amp;    0           &amp;  0           &amp;  0.5         &amp;     24.4998     &amp;    0.376495    \\
 CLF2F\_I         &amp;  0          &amp;      -0.014685   &amp;    0.0316763   &amp;    1.25697e-13 &amp; -0.0068251   &amp; -0.0158382   &amp;     -0.774379   &amp;   -0.0185053   \\
 CLF2F\_Q         &amp;  0          &amp;       0.00940703 &amp;   -3.16763e-05 &amp;    0.0158421   &amp;  0.00838236  &amp; -7.91908e-06 &amp;     -0.00394792 &amp;    8.94873e-05 \\
 CLF2F\_RIN       &amp;  0          &amp;       1.19315    &amp;    0           &amp;    1.00025     &amp;  0.018631    &amp; -2.22045e-13 &amp;     -0.00456289 &amp;    0.0140916   \\
 CLF1F\_I         &amp; -0.163038   &amp;      -1.36167    &amp;    0.0864372   &amp;   -6.50123e-15 &amp;  0.00585498  &amp;  0.0382553   &amp;      1.8645     &amp;    0.028124    \\
 CLF1F\_Q         &amp; -0.179556   &amp;      -3.20362    &amp;   -4.32186e-05 &amp;    0.0432186   &amp; -0.0168758   &amp;  0.0897755   &amp;      4.40667    &amp;    0.0661024   \\
 CLF1F\_RIN       &amp;  0          &amp;     -17.1471     &amp;    0           &amp;    1.00025     &amp;  0.338314    &amp;  1.33227e-12 &amp;     -0.0829295  &amp;   -0.20331     \\
 PUMP\_INT\_RIN    &amp;  0          &amp;       0          &amp;    0           &amp;    0           &amp;  1           &amp;  0           &amp;     -0.244994   &amp;   -0.00136698  \\
 PUMP\_REFL\_I     &amp;  0          &amp;       0          &amp;    0           &amp;    0           &amp;  0           &amp;  3.01264e-16 &amp;      2.85767    &amp;    0.0675025   \\
 PUMP\_REFL\_Q     &amp; -3.3447e-29 &amp;      -3.3447e-28 &amp;   -3.3447e-29  &amp;   -3.3447e-29  &amp;  1.50594e-16 &amp; -1.50632e-19 &amp;     -0.0141455  &amp;   -7.89266e-05 \\
\hline
\end{tabular}

}
\caption{
  Open-Loop Sensing Matrix for OPO with O4 Mirror Configuration. The CLF
  I and Q phases are in units of CLF demodulation field normalized by the CLF
  single-sideband field incident on M2. This sensing matrix represent the OPO
  with sufficient pump power to generate 14.5db of (anti)squeezing at its output.
}
\label{tab:O4opoOL}
\end{table}

\begin{table}[h!]
\centering
\resizebox{1\linewidth}{!}{
\begin{tabular}{lrrrrrrrr}
\hline
 14.45 dB Gen.    &amp;   OPO 1064 [rad] &amp;   pump [rad] &amp;   OPO len [nm] &amp;   PUMP\_REFL\_I SG &amp;   CLF2F\_RIN SG &amp;   PUMP\_INT\_RIN SG &amp;    CLF2F\_I SG &amp;   CLF1F\_I SG \\
\hline
 SQZ\_phase        &amp;     -9.01587     &amp;  0.00027513  &amp;   -0.107497    &amp;     -0.0231627   &amp;   -2.14313e-12 &amp;       0.150144    &amp;  16.737       &amp; -6.13355     \\
 CLF2F\_Q          &amp;     -0.00950491  &amp; -2.37572e-05 &amp;   -7.80784e-05 &amp;      0.000934459 &amp;   -0.0158382   &amp;      -0.00808046  &amp;   0.001       &amp;  0           \\
 CLF1F\_Q          &amp;     -1.79969     &amp;  2.53327e-05 &amp;   -0.021516    &amp;     -0.00499495  &amp;   -0.0432078   &amp;       0.0446493   &amp;   3.0066      &amp; -1.10132     \\
 CLF1F\_RIN        &amp;    -18.3403      &amp;  1.55431e-12 &amp;   -0.216964    &amp;      1.61862e-05 &amp;   -1           &amp;      -0.319683    &amp;  -0           &amp;  0           \\
 PUMP\_REFL\_Q      &amp;     -3.89543e-29 &amp;  1.34063e-18 &amp;    0.000255211 &amp;      0.00495     &amp;    3.34387e-29 &amp;      -1.50594e-16 &amp;   1.61571e-27 &amp; -2.05149e-28 \\
 OPO 532 [rad] CL &amp;      0           &amp; -1.05423e-16 &amp;   -0.0236215   &amp;     -0.349936    &amp;    0           &amp;      -0           &amp;  -0           &amp;  0           \\
 CLFVCO [rin] CL  &amp;     -1.19285     &amp;  2.21989e-13 &amp;   -0.0141135   &amp;      5.5536e-07  &amp;   -0.99975     &amp;       0.0186263   &amp;  -0           &amp;  0           \\
 pump [rin] CL    &amp;     -0           &amp; -2.5828e-17  &amp;   -0.00442016  &amp;     -0.0857321   &amp;    0           &amp;      -1           &amp;   0           &amp;  0           \\
 CLFVCO [rad] CL  &amp;      0.463595    &amp;  0.5         &amp;    0.00578076  &amp;     -8.57322     &amp;    3.96717e-12 &amp;      -0.215464    &amp; -31.5693      &amp; -0           \\
 LO [rad] CL      &amp;     -8.10609     &amp;  0.499725    &amp;   -0.0947305   &amp;     -8.55019     &amp;    2.14313e-12 &amp;      -0.150144    &amp; -16.737       &amp;  6.13355     \\
\hline
\end{tabular}
}
\caption{
  Closed-Loop Sensing Matrix for OPO with O4 Mirror Configuration.
  Normalizations are the same as for the open loop table. The CL flags represent
  the gains to the actuators while the loop is closed. The SG flags represent
  the coupling of sensing noise.
}
\label{tab:O4opoCL}
\end{table}

\section{Figures (other)}

\autofigure{
  folder=./figures/, 
  file=H1_OPO_TEMP_spec,
  width=0.8\textwidth,
  label={temp_spec},
  caption={
    Spectrum of the temperature controller loop spectrum. It is in units of Celsius/rtHz
  },
}
\end{comment}
\bibliographystyle{ieeetr}%plain}

\end{document}
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