Here is a summary of a simulation of the error signal behavior in the PRMI configuration. The main parameters are:
L_PRC = 6.7538 m
Schnup = 0.0342 m
fmod1 = 11.065399e6 Hz
fmod2 = 5 * fmod1

These two plots shows the response of the POP22 and POP110 signals (in almost arbitrary units) to a PRCL sweep around the resonance. The splitting of the 55 sideband peaks is well visible in the second plot. It is due to the fact that the 55MHz sidebands are not perfectly matched to the PRC length

The same thing when sweeping MICH. The peaks are wider and it is not possible to see the splitting.

These are the error signals (REFL11_I/Q and REFL_55_I/Q) as a function of the PRCL (left) and MICH (right) sweep. Here the demodulation phases are not properly tuned. This is just to show that when the phase is wrong, you can get multiple zero crossings (in this case only in the Q signals, but in general also in I) close to resonance.

If the phases are tuned in order to maximize the slope of the I signals with respect to PRCL, one gets these "optimized phase" responses. It is that the phase does not correspond to the one that makes the PRCL peak to peak signal small in Q. The Q signals are indeed flat around resonance for a PRCL motion, but they deviate quite a lot from zero when moving more far from resonance. Moreover, both the REFL_55 error signals (I for PRCL and Q for MICH) are crossing again zero at two additional positions, but those are quite far from the resonance point.

These plots just show the PRCL and MICH error signals together with the POP22 and POP110 signals, to give an idea of the level of triggering that might be needed to be inside the linear range. It seems that if we trigger on POP22 when using the REFL55 signal we loose a bit of linear range, but not that much.

If you reached this point it means you're really interested in this topic, or maybe you have nothing better to do... However, this plot shows the effect of linearization of the error signal, obtained dividing them by the proper POP22/110 signal. The linear range is increased, but unfortunately for the 55 signals, the additional zero crossing I was mentioning before creates two sharp features. Those are however quite outside the triggering region, so they should not be harmful.

I had a look at the POP110 signal, with the PRMI flashing.

1) The LSCoffset script does not zero any more POP22_I_ERR offset. I did it by hand

2) The gain of POP22 is changed a lot, as well as the sign: now sidebands are resonant when POP22_I is negative

3) POP110 seems to deliver good signals. The plot attached shows that when we cross the sideband resonance, there is a clear splitting of the peak. If we rely on the simulations I posted in entry 8401, the full width at half height of the POP_22 peak is of the order of 5 nm. Using this as a calibration, we find a splitting of the order of 7 nm, which is not far from the simulated one (5 nm)

The attached plot shows that also the behaviour of the REFL11 and 55 signals is qualitatively equal to the simulation outcome.

 I simulated how the 3f signal is affected by the resonance condition of the arms.

To keep it simple, I only simulated a double cavity. The attached plot shows the result. In x there is the arm cavity detuning from resonance (in log scale to show what happens close to the 0 value). In the y axis there is the PRC detuning. So every vertical slice of the upper plot gives a PDH signal for a given arm detuning. The bottom plot shows the power build up inside the arm, which is dominated by the carrier.

The 3f signal is not perturbed in any significant way by the arm resonance condition. This is good and what we expected.

However, in this simulation I had to ensure that the 1f sidebands are not perfectly anti-resonant inside the arms. They are indeed quite far away from resonance. If the modulation frequency is chosen in order to make the 1f sidebands exactly ant-resonant, the 2f will be resonant. This screws up the signal: REFL_3f is made of two contributions of equal amplitude, one on the PRC sidebands resonance and the other on the PRC carrier resonance. When the arm tuning goes to zero, these two cancels out and there is no more PDH...

However, this is a limit case, since the frequency show match perfectly. If the modulation frequency is few arm line widths away from perfect anti-resonance, we have no problem.

 From the simulation there is no visible change in the gain.

 The idea of introducing a large MICH offset to reduce the PRC finesse might help us to get rid of the transmitted power signal. We might be able to increase enough the line width of the double cavity to make it larger than the ASL length fluctuations. Then we can switch from ASL to the IR demodulated signal without transitioning through the power signal.

Interesting results. When you compute the effect of ETM motion, you maybe should also consider that moving around the arm cavity axis changes the matching of the input beam with the cavity, and thus the coupling between PRC and arms. But I believe this effect is of the same order of the one you computed, so maybe there is only one or two factors of two to add. This do not change significantly the conclusion.

Instead, the numbers you're giving for PRM motion are interesting. Since I almost never believe computations before I see that an experiment agrees with them, I suggest that you try to prove experimentally your statement. The simplest way is to use a scatter plot as I suggested the past week: you plot the carrier arm power vs PRM optical lever signals in a scatter plot. If there is no correlation between the two motions, you should see a round fuzzy ball in the plot. Otherwise, you will se some non trivial shape. Here is an example: https://tds.ego-gw.it/itf/osl_virgo/index.php?callRep=18918

I'm not as good as a fit, but it seems that there is a loop delay of about 30 microseconds, looking at the high frequency phase. This might explain the limitation on the BW. Eric should be able to get the delay out of the fit with some care.

Measurement 

Looking back at what I did in april (see log #8411) I realized that it is possible to get an estimate of how much the PRC length is wrong looking at the splitting of the sideband resonant peak as visible in the POP_110_I signal. With the help of Jenne the PRMI was aligned and left swinging. The first plot shows a typical example of the peak splitting of 55MHz sidebands. This is much larger than what was observed in April.

When the sidebands resonate inside the PRC they get a differential dephasing given by 

dPhi = 4*pi*f_mod/c * dL

where dL is the cavity length error with respect to the one that makes the sidebands perfectly resonant when the arms are not there. This is not exactly the error we are interested in, since we should take into account the shift from anti-resonance of the SBs in the arm cavities.

Nevertheless, I can measure the splitting of the peak in units of the peaks full width at half maximum (FWHM). I did this fitting few peaks with the sum of two Airy peaks. Here is an example of the result

The average splitting is 1.8 times the FWHM. Knowing the PRC finesse, one can determine the length error:

dL = c / (4 * f_mod * Finesse) * (dPhi / FWHM)

Assuming a finesse of 60, I get a length error of 4 cm.

To get another estimate, we kicked the PRM in order to get some almost linear sweeps of the PRC length. Here is one of the best results:

The distance between consecutive peaks is the free spectral range (FSR) of the PRC cavity. Again, I can measure the peak splitting in units of the FSR and determine the length error:

dL = c / (4 * f_mod) * (dPhi / FSR)

The result is again a length error of 4 cm.

Simulation

An error of 4 cm seems pretty big. Therefore I set up a quick simulation with MIST to check if this makes sense. Indeed, if I simulate a PRMI with the 40m parameters and move the PRC length from the optimal one, I get the following result for POP_110_I, which is consistent with the measurement.

 Therefore, we can quite confidently assume that the PRC is off by 4 cm with respect to the position that would make the 55 MHz sideband resonant without arms. Unfortunately, it is not possible with this technique to infere the direction of the error.

Maybe I'm getting confused, but I still believe there is no way to decide the direction from yesterday's measurement.

Let's say for example that the arm sideband detuning from antiresonance is equivalent to a PRC length change of +1cm away from the position of ideal resonance of the sidebands without arms. Then we can get a measured separation of the sidebands, without arms, corresponding to 5cm both if the PRC is off by +4cm or by -6cm...

I ran a simulation of a double cavity with a PRC length mismatched w.r.t. the modulation frequency. I summarized the results in the attached PDF. I think it would be important to have a cross check of the results.

In brief:

A mismatch between PRC length and modulation frequency do have an effect on error signals

Multiple zeros appear in REFL_3f/PRCL that can be removed by careful tuning of the demodulation phase (however, the shape of the signal makes difficult to understand which phase is good…)

No visible effect on REFL_1f/CARM

But a large PRCL signal appears in REFL_1f_I, which is used to control CARM. This is not good.

A mismatch of the order of 0.5 cm has a small effect.

 [Gabriele, EricQ]

Actually it is difficult to see any laser frequency line in the dark fringe signal, since the Schnupp asymmetry is small. It is much better to use a differential MICH excitation which gives a better signal at the dark port.

We repeated the simulation explained before. We can use both the AS55 or the AS11 signals, bout the first one has a limited linear range and the expected 4cm value is very close to saturation.

 Then there must be something else slowly drifting. It was very clear that the good alignment of the IFO was every time lost after few minutes...

[Gabriele, EricQ]

We wanted to try the PRC length measurement,but we ended up spending all the afternoon to lock the PRMI on sidebands. Here are some results

• Lock of PRM on carrier is easy with MICH on AS55_Q and PRCL on REFL55_I. The optimal gains to avoid correction gain peaking are MICH=-5 and PRCL=0.012. The alignment today was must more stable over time than on Friday
• We wanted to move MICH on REFL55_Q. After a few trials we discovered that REFL55_Q is not seeing any MICH signal at all. This is quite strange and we don't understand this.
• Locking PRMI on carrier using REFL165_I and Q was difficult. We thought that was due to a RF amplifier installed in December. We tried to remove it, but since this did not help, we put it back
• We could lock PRMI on carrier and sidebands using REFL11 signals. The optimal demodulation phase is 155. Lock on carrier was achieved with gains MICH=5, PRCL=0.7. Lock on sidebands with MICH gain=-10 and PRCL gain=-0.3. The MICH correction is very high and exciting a tone at few hundreds Hz. Maybe a violin mode of the ITMs?
• Finally we could tune the REFL165 phase to 118.5 and lock on carrier. The lock was not very stable. Gain are: lock on carrier MICH=-0.5, PRCL=0.05. The error signal has a very big offset: to get a dark fringe we had to add a MICH offset of 500 counts. We also had to engage the CLP400 filter to avoid saturating ITMs corrections with MICH.

[Gabriele, EricQ] 

Finally, we managed to lock PRMI on sidebands:

• MICH locked on AS55_Q with gain -10. Demod phase of AS55=18
• PRCL locked on REFL55_I with gain -0.04. Demod phase of REFL55=88
• Triggering on POP110_I at 50/10
• Filter "1:5" of MICH engaged, this improved a lot the stability

 [EricQ, Gabriele]

We could carry out the measurement of PRC length. The AS110 photodiode was plugged into REFL11. So REFL11 is giving us the AS11 signal. Here is the procedure.

1. Lock MICH.
2. Add a line in MICH (amplitude 20000 counts)
3. Tune AS11 demod phase to have the line in I.
4. Change the demod phase by steps of 1 degree around the rough optimum, taking one minute of data at each point
5. Lock PRMI on sidebands
6. Add a line in MICH (amplitude 500 counts)
7. Tune AS11 demod phase to have the line in I.
8. Change the demod phase by steps of 1 degree around the rough optimum, taking one minute of data at each point

We repeated the same measurement also using AS55, with the same procedure.

Roughly, the phase difference for AS11 was 11 degrees and for AS55 it was 23 degrees. A more detailed analysis and a calibration in terms of PRC length will follow.

 I analyzed the data we took yesterday, both using AS11 and AS55. For each value of the phase I estimated the Q/P ratio using a demodulation code. Then I used a linear regression fit to estimate the zero crossing point.

Here are the plots of the data points with the fits:

The measurements a re more noisy in the PRMI configuration, as expected since we had a lot of angular motion. Also, the AS11 data is more noisy. However, the estimated phase differences between PRMI and MICH configurations are:

• AS11 = -10.9 +- 1.0 degrees
• AS55 = -21.1 +- 0.4 degrees

The simulation already described in slogs 9539 and 9541 provides the calibration in terms of PRC length. Here are the curves

The corresponding length errors are

• AS11 = 1.44 +- 0.13 cm
• AS55 = 0.59 +- 0.01 cm

The two results are not consistent one with the other and they are both not consistent with the previous estimate of 4 cm based on the 55MHz sideband peak splitting.

I don't know the reason for this incongruence. I checked the simulation, repeating it with Optickle and I got the same results. So I'm confident that the simulation is not completely wrong.

I also tried to understand which parameters of the IFO can affect the result. The following ones have no impact

• Beam matching
• ITM curvatures
• Schnupp asymmetry
• Distance PR-BS
• ITM and PRM misalignments

The only parameters that could affect the curves are offsets in MICH and PRCL locking point. We should check if this is happening. A first quick look (with EricQ) seems to indicate that we indeed have an offset in PRCL. However, tonight the PRMI is not locking stably on the sidebands. 

If possibile, we will repeat the measurement later on tonight, checking first the PRCL offset.

 I analyzed the data taken yesterday. 

The AS11 data in PRMI configuration is very bad, while the AS55 seems good enough:

The phase differences are 

AS11 = 21 +- 18 degrees (almost useless due to the large error)

AS55 = 11.0 +- 0.4 degrees

The AS55 phase difference is not the same measured in the last trial, but about half of it. The new length estimates are:

AS11 = 3.2 +- 2.8 cm

AS55 = 0.47 +- 0.01 cm

We can probably forget about the AS11 measurement, but the AS55 result is different from the previous estimate... Maybe this is due to the fact that Eric adjusted the PRCL offset, but then we're going in the wrong direction....

 Yesterday night I plugged back the REFL11 RF cable into the corresponding demodulation board.

Here is how to measure the PRC length with a set of distance measurements in the optical setup. 

We need to take distance measurements between reference points on each mirror suspension. For the large ones (SOS) that are used for BS, PRM and ITMs, the reference points are the corners of the second rectangular base: not the one directly in contact with the optical bench (since the chamfers make difficult to define a clear corner), but the rectangular one just above it. For the small suspensions (TT) the points are directly the corners of the base plates.

From the mechanical drawings of the two kind of suspensions I got the distances between the mirror centers and the reference corners. The mirror is not centered in the base, so it is a good idea to cross check if the numbers are correct with some measurements on the dummy suspensions.

I assumed the dimensions of the mirrors, as well as the beam incidence angles are known and we don't need to measure them again. Small errors in the angles should have small impact on the results.

I wrote a MATLAB script that takes as input the measured distances and produce the optical path lengths. The script also produce a drawing of the setup as reconstructed, showing the measurement points, the mirrors, the reference base plates,  and the beam path. Here is an example output, that can be used to understand which are the five distances to be measured. I used dummy measured distances to produce it.

In red the beam path in vacuum and in magenta the beam path in the substrate. The mirrors are the blue rectangles inside the reference bases which are in black. The thick lines are the HR faces. The green points are the measurement points and the green lines the distances to be measured. The names on the measurement lines are those used in the MATLAB script. 

The MATLAB scripts are attached to this elog. The main file is survey_v2.m, which contains all the parameters and the measured values. Update it with the real numbers and run it to get the results, including the graphic output. The other files are auxiliary functions to create the graphics. I checked many times the code and the computations, but I can't be sure that there are no errors, since there's no way to check if the output is correct... The plot is produced in a way which is somehow independent from the computations, so if it makes sense this gives at least a self consistency test. 

 [Manasa, EricQ, Gabriele]

We managed to measure the PRC length using a procedure close to the one described in slog 9573.

We had to modify a bit the reference points, since some of them were not accessible. The distances between points into the BS chamber were measured using a ruler. The distances between points on different chambers were measured using the Leica measurement tool. In total we measured five distances, shown in green in the attached map.

We also measured three additional distances that we used to cross check the results. These are shown in the map in magenta.

The values of the optical lengths we measured are:

LX    = 6828.96 mm
LY    = 6791.74 mm
LPRC  = 6810.35 mm
LX-LY = 37.2196 mm  

The three reference distances are computed by the script and they match well the measured one, within half centimeter:

M32_MP1  = 117.929 mm   (measured = 119 mm)
MP2_MB3  = 242.221 mm   (measured = 249 mm)
M23_MX1p = 220.442 mm   (measured = 226 mm)

See the attached map to see what the names correspond to.

The nominal PRC length (the one that makes SB resonant without arms) can be computed from the IMC length and it is 6777 mm. So, the power recycling cavity is 33 mm too long w.r.t. the nominal length. This is in good agreement with the estimate we got with the SB splitting method (4cm).

According to the simulation in the wiki page the length we want to have the SB resonate when the arms are there is 6753 mm. So the cavity is 57 mm too long.

Attached the new version of the script used for the computation.

 [Manasa, EricQ, Gabriele]

Today we changed the PRC length translating PR2 by 27 mm in the direction of the corner. After this movement we had to realign the PRC cavity to get the beam centered on PRM, PR2, PR3, BS (with apertures) and ITMY (with aperture). To realign we had to move a bit both PR2 and PR3. We could also see some flashes back from the ETMY . //Edit by Manasa : We could see the ETMY reflection close to the center of the ITMY but the arm is not aligned or flashing as yet//. 

After the realignment we measured again the PRC length with the same method of yesterday. We only had to change one of the length to measure, because it was no more accessible today. The new map is attached as well as the new script (the script contains also the SRC length estimation, with random numbers in it).

The new PRC length is 6753 mm, which is exactly our target!

The consistency checks are within 5 mm, which is not bad.

We also measured some distances to estimate the SRC length, but right now I'm a bit confused looking at the notes and it seems there is one missing distance (number 1 in the notes). We'll have to check it again tomorrow.

 Today we measured the missing distance to reconstruct SRC length.

I also changed the way the mirror positions are reconstructed. In total for PRC and SRC we took 13 measurements between different points. The script runs a global fit to these distances based on eight distances and four incidence angles on PR2, PR2, SR2 and SR3. The optimal values are those that minimize the maximum error of the 13 measurements with respect to the ones reconstructed on the base of the parameters. The new script is attached (sorry, the code is not the cleanest one I ever wrote...)

The reconstructed distances are:

Reconstructed lengths [mm]:
LX    = 6771
LY    = 6734
LPRC  = 6752
LX-LY = 37
LSX   = 5493
LSY   = 5456
LSRC  = 5474

The angles of incidence of the beam on the mirrors are very close to those coming from the CAD drawing (within 0.15 degrees):

Reconstructed angles [deg]:
aoi PR3 = 41.11 (CAD 41)
aoi PR2 = 1.48  (CAD 1.5)
aoi SR3 = 43.90 (CAD 44)
aoi SR2 = 5.64  (CAD 5.5)

The errors in the measured distances w.r.t. the reconstructed one are all smaller than 1.5 mm. This seems a good check of the global consistency of the measurement and of the reconstruction method.

NOTES: in the reconstruction, the BS is assumed to be exactly at 45 degrees; wedges are not considered.

 Very good news! We should have a look at the POP110 sideband peak splitting, to see if we really got the right PRC length... 

I guess that you get an ellipse when the transfer functions to I and Q have a different phase. One mechanism could be that when driving MICH we make some residual PRCL and this couples with a different transfer function to both I and Q. However, I would expect no phase lag in the PRMI configuration, since there is no enough optical delay in the system to give significant dephasing at few hundreds Hz. This effect might come from mechanical resonances.

It is worth measuring the optical transfer functions from both PRCL and MICH to REFL signals at all frequencies, to see if we have strange features in the TFs.

I guess this is normal. DARM has (almost) the same effect of MICH on the corner interferometer signals, just increased in amplitude by the arm cavity amplification. When the arm is out of resonance, DARM is almost completely depressed and it is not affecting MICH at all. On the other hand, when the cavities are exactly at resonance, DARM signal is amplified w.r.t. MICH by the cavity gain (2F/pi).

Since DARM is still controlled with ALS, it is probably noisy. The closer to resonance you move the cavities, the more ALS noise in DARM will affect MICH.

 Koji asked me to perform a simulation of the response of POP QPD DC signal to mirror motions, as a function of the CARM offset. Later than promised, here are the first round of results.

I simulated a double cavity, and the PRC is folded with parameters close to the 40m configuration. POP is extracted in transmission of PR2 (1ppm, forward beam). For the moment I just placed the QPD one meter from PR2, if needed we can adjust the Gouy phase. There are two QPDs in the simulation: one senses all the field coming out in POP, the other one is filtered to sense only the contribution from the carrier field. The difference can be used to compute what a POP_2F_QPD would sense. All mirrors are moved at 1 Hz and the QPD signals are simulated:

This shows the signal on the POP QPD when all fields (carrier and 55 MHz sidebands) are sensed. This is what a real DC QPD will see. As expected at low offset ETM is dominant, while at large offset the PRC mirrors are dominant. It's interesting to note that for any mirror, there is one offset where the signal disappears.

This is the contribution coming only from the carrier. This is what an ideal QPD with an optical low pass will sense. The contribution from the carrier increases with decreasing offset, as expected since there is more power.

Finally, this is what a 2F QPD will sense. The contribution is always dominated by the PRC mirrors, and the ETM is negligible.

The zeros in the real QPD signal is clearly coming from a  cancellation of the contributions from carrier and sidebands.

The code is attached.

 In addition to the simulation described in my previous elog, I simulated the signal on a quadrant photodetector demodulated at 2F. The input laser beam is modulated at 11MHz up to the fifth order. There is no additional 55 MHz modulation.

The QPD demodulated at 2F shows good signals for PRC control for all CARM offsets, as expected from the previous simulation.

 Jenne asked me to simulate the signals on POP QPD when moving different mirrors, as a function of the Gouy phase where the QPD is placed.

I used the opportunity to create a MIST simulation file of the entire 40m interferometer, essentially based on my aLIGO configuration file. I used the recycling cavity lengths obtained from our survey, and other parameters from the wiki page. The configuration file is attached (fortymeters.mist).

Coming back to the main simulation, here is the result, both for the "regular" POP QPD and for a 22MHz demodulated one. The Gouy phase is measured starting from PR2. Cavity mirrors are easily decoupled from PRM in the "regular" QPD. As already demonstrated in a previous simulation, ETMs signals are very small in the 22 MHz QPD. Moreover, it is possible to zero the contribution from ITMs by choosing the right Gouy phase, at the price of a reduction of the PRM signal by a factor of 3-4. Simulation files are attached.

1) Introduction

In brief, I trained a deep neural network (DNN) to recosntuct the cavity length, using as input only the transmitted power and the reflection PDH signals. The training was performed with simulated data, computed along 0.25s long trajectories sampled at 8kHz, with random ending point in the [-lambda/4, lambda/4] unique region and with random velocity.

The goal of thsi work is to validate the whole approach of length reconstruction witn DNN in the Fabry-Perot case, by comparing the DNN reconstruction with the ALS caivity lenght measurement. The final target is to deploy a system to lock PRMI and DRMI. Actually, the Fabry-Perot cavity problem is harder for a DNN: the cavity linewidth is quite narrow, forcing me to use very high sampling frequency (8kHz) to be able to capture a few samples at each resonance crossing. I'm using a recurrent neural network (RNN), in the input layers of the DNN, and this is traine using truncated backpropagation in time (TBPT): during training each layer of RNN is unrolled into as many copies as there are input time samples (8192 * 0.25 = 2048). So in practice I'm training a DNN with >2000 layers! The limit here is computational, mostly the GPU memory. That's why I'm not able to use longer data stretches.

But in brief, the DNN reconstruction is performing well for the first attempt.

2) Training simulation

In the results shown below, I'm using a pre-trained network with parameters that do not match very well the actual data, in particular for the distribution of mirror velocity and the sensing noises. I'm working on improving the training.

I used the following parameters for the Fabry-Perot cavity:

The uncertaint is assumed to be the 90% confidence level of a gaussian distribution. The DNN is trained on 100000 examples, each one a 0.25/8kHz long trajectory with random velocity between 0.1 and 5 um/s, and ending point distributed as follow: 33% uniform on the [-lambda/4, lambda/4] region, plus 33% gaussian distribution peaked at the center with 5 nm width. In addition there are 33% more static examples, distributed near the center. 

For each point along the trajectory, the signals TRA, POX11_I and POX11_Q are computed and used as input to the DNN.

3) Experimental data

Gautam collected about 10 minutes of data with the free swinging cavity, with ALS locked on the arm. Some more data were collected with the cavity driven, to increase the motion. I used the driven dataset in the analysis below.

3.1) ALS calibration

The ALS signal is calibrated in green Hz. After converting it to meters, I checked the calibration by measuring the distance between carrier peaks. It turned out that the ALS signal is undercalibrated by about 26%. After correcting for this, I found that there is a small non-linearity in the ALS response over multiple FSR. So I binned the ALS signal over the entire range and averaged the TRA power in each bin, to get the transmission signals as a function of ALS (in nm) below:

I used a peak detection algorithm to extract the carrier and 11 MHz sideband peaks, and compared them with the nominal positions. The difference between the expected and measured peak positions as a function of the ALS signal is shown below, with a quadratic fit that I used to improve the ALS calibration

The result is

The ALS calibrated z error from the peak position is of the order of 3 nm (one sigma)

3.2) Mirror velocity

Using the calibrated ALS signal, I computed the cavity length velocity. The histogram below shows that this is well described by a gaussian with width of about 3 um/s. In my DNN training I used a different velocity distribution, but this shouldn't have a big impact. I'm retraining with a different distirbution.

4) DNN results

The plot below shows a stretch of time domain DNN reconstruction, compared with the ALS calibrated signal. The DNN output is limited in the [-lambda/4, lambda/4] region, so the ALS signal is also wrapped in the same region. In general the DNN reconstruction follows reasonably well the real motion, mostly failing when the velocity is small and the cavity is simultanously out of resonance. This is a limitation that i see also in simulation, and it is due to the short training time of 0.25s.

I did not hand-pick a good period, this is representative of the average performance. To get a better understanding of the performance, here's a histogram of the error for 100 seconds of data:

The central peak was fitted with a gaussian, just to give a rough idea of its width, although the tails are much wider. A more interesting plot is the hisrogram below of the reconstructed position as a function of the ALS position, Ideally one would expect a perfect diagonal. The result isn't too far from the expectation:

The largest off diagonal peak is at (-27, 125) and marked with the red cross. Its origin is more clear in the plot below, which shows the mean, RMS and maximum error as a function of the cavity length. The second peak corresponds to where the 55 MHz sideband resonate. In my training model, there were no 55 MHz sidebands nor higher order modes. 

5) Conclusions and next steps

The DNN reconstruction performance is already quite good, considering that the DNN couldn't be trained optimally because of computation power limitations. This is a validation of the whole idea of training the DNN offline on a simulation and then deploy the system online.

I'm working to improve the results by

• training on a more realistic distribution of velocity
• adding the 55 MHz sidebands
• maybe adding HOMs
• tune the DNN architecture

However I won't spend too much time on this, since I think the idea has been already validated.

Update

I included the 55 MHz sideband and higher order modes in my training examples. To keep things simple, I just assumed there are higher order modes up to n+m=4 in the input beam. The power in each HOM is randomly chosen from a random gaussian distribution with width determined from experimental cavity scans. I used a value of 0.913+-0.01 rad for the Gouy phase (again estimated from cavity scans, but in reasonable agreement with the nominal radius of curvature of ETMX)

Results are improved. The plot belows show the performance of the neural network on 100s of experimental data

For reference, the plots below show the performance of the same network on simulated data (that includes sensing noise but no higher order modes)

Introduction

I trained a deep neural network (DNN) to reconstruct MICH and PRCL degrees of freedom in the PRMI configuration. For details on the DNN architecture please refer to G1701455 or G1701589. Or if you really want all the details you can look at the code. I used the following signals as input to the DNN: POPDC, POP22_Q, ASDC, REFL11_I/Q, REFL55_I/Q, AS55_I/Q.

Gautam took some PRMI data in free swinging and driven configuration:

• 1187819331 + 10mins: Free swinging PRMI (after first locking PRMI on carrier and dither aligning).
• 1187820070 + 5mins: PRM driven at low freq.
• 1187820446 + 5mins: BS driven at low freq.

In contrast to the Fabry-Perot cavity case, we don't have a direct measurement of the real PRCL/MICH degrees of freedom, so it's more difficult to assess if the DNN is working well.

Results

All MICH and PRCL values are wrapped into the unique region [-lambda/4, lambda/4]^2. It's even a bit more complicated than simpling wrapping. Indeed, MICH is periodic over [-lambda/2, lambda/2]. However, the Michelson interferometer reflectivity (as seen from PRC) in the first half of the segment is  the same as in the second half, except for a change in sign. This change of sign in Michelson reflectivity can be compensated by moving PRCL by lambda/4, thus generating a pi phase shift in the PRC round trip propagation that compensate for the MICH sign change. Therefore, the unit cell of unique values for all signals can be taken as [-lambda/4, lambda/4] x [-lambda/4, lambda/4] for MICH x PRCL. But when we hit the border of the MICH region, PRCL is also affected by addtion of lambda/4. Graphically, the square regions A B C below are all equivalent, as well as more that are not highlighted:

This makes it a bit hard to un-wrap the resonstructed signal, especially when you add in the factor that in the reconstruction the wrapping is "soft".

The plot below shows an example of the time domain reconstruction of MICH/PRCL during the free swinging period.

It's hard to tell if the positions look reasonable, with all the wrapping going on.

Two-dimensional maps of signals

Here's an attempt at validating the DNN reconstruction. Using the reconstructed MICH/PRCL signal, I can create a 2d map of the values of the optical signals. I binned the reconstructed MICH/PRCL in a 51x51 grid, and computed the mean value of all optical signals for each bin. The result is shown in the plot below, directly compared with the expectation from a simulation.

The power signals (POP_DC, AS_DC, PO22_Q) looks reasonably good. REFL11_I/Q also looks good (please note that due to an early mistake in my code, I reversed the convention for I/Q, so PRCL signal is maximized in Q instead than in I). The 55MHz signals look a bit less clear...

Steps forward

• I'm quite confident in the tuning of demodulation phase and signs for REFL11 and POP22, but less so for REFL55 and not sure at all for AS55. So it would be useful to measure a full sensing matrix of PRCL and MICH against those signals, to compare with my simulation
• I'm working on an idea to fine tune the DNN using the real interferometer data, more to follow when the idea crystallizes in a clear form.

This is an update on the results already presented earlier (refer to elog 13274 for more introductory details). I improved significantly the results with the following tricks:

• I retuned the demodulation phase of AS55, this time ensuring that the (alleged) MICH motion is visible mostly in Q when crossing a carrier resonance. Further fine tunings of phases will be possible once we have a measurement of the length optical matrix

• I fine tuned the netwrok by training it again using the real data. The ides is the following. I started with the network trained on the simulated data, and froze the parameters of the input recurrent layers. I fed the real signal to the network, computed the reconstructed PRCL/MICH, and fed them to my PRMI model to compute simulated signals. I allowed some of the parameters of the models to vary (expecially demodulation phases). I then trained again the network by trying to match the model predicted signals with the real input signals. I allowed only the parameters of the fully connected layers to vary (mostly for technical reasons, I'm working on re-training also the recurrent layers)

An example of time domain reconstruction is visible below. It already looks better than the old results:

As before, to better evaluate the performance I plotted averaged values of the real signals as a function of the reconstructed MICH and PRCL positions. The results are compared with simulation below. They match quite well (left real data, right simualtion expectation)

One thing to better understand is that MICH seems to be somewhat compressed: most of the output values are between -100 and +100 nm, instead of the expected -lambda/4, lambda/4. The reason is still unclear to me. It might be a bug that I haven't been able to track down yet.

Introduction

This is an update of my previous reports on applications of deep learning to the reconstruction of PRMI degrees of freedom (MICH/PRCL) from real free swinging data. The results shown here are improved with respect to elog 13274 and 13294. The training is performed in two steps, the first one using simulated data, and the second one fine tuning the parameters on real data.

First step: training with simulation

This step is exactly the same already described in the previous entries and in my talks at the CSWG and LVC. For details on the DNN architecture please refer to G1701455 or G1701589. Or if you really want all the details you can look at the code. I used the following signals as input to the DNN: POPDC, POP22_Q, ASDC, REFL11_I/Q, REFL55_I/Q, AS55_I/Q. The network is trained using linear trajectories in the PRCL/MICH space, and signals obtained from a model that simulates the PRMI behavior in the plane wave approximation. A total of 150000 trajectories are used. The model includes uncertainties in all the optical parameters of the 40m PRMI configuration, so that the optical signals for each trajectory are actually computed using random optical parameteres, drwn from gaussian distributions with proper mean and width. Also, white random gaussian sensing noise is added to all signals with levels comparable to the measured sensing noise.

The typical performance on real data of a network pre-trained in this way was already described in elog 13274, and although being reasoble, it was not too good.

Second step: training with real data

Real free swinging data is used in this step. I fine tuned the demodulation phases of the real signals. Please note that due to an old mistake, my convention for phases is 90 degrees off, so for example REFL11 is tuned such that PRCL is maximized in Q instead of I. Regardless of this convention confusion, here's how I tuned the phases:

• REFL11: PRCL is all in Q when crossing the carrier resonance
• REFL55: PRCL is all in Q when crossing the carrier resonance
• AS55: MICH is all in Q when crossing the PRCL carrier resonance
• POP22: signal peaking in Q when crossing carrier or sideband resonances. Carrier resonance crossing gives positive sign

Then I built the following training architecture. The neural network takes the real signals and produces estimates of PRCL and MICH for each time sample. Those estimates are used as the input for the PRMI model, to produce the corresponding simulated optical signals. My cost function is the squared difference of the simulated versus real signals. The training data is generated from the real signals, by selection 100000 random 0.25s long chunks: the history of real signal over the whole 0.25s is used as input, and only the last sample is used for the cost function computation. The weights and biases of the neural network, as well as the model parameters are allowed to change during the learning process. The model parameters are regularized to suppress large deviations from the nominal values.

One side note here. At first sight it might seems weird that I'm actually fedding as input the last sample and at the same time using it as the reference for the loss function. However, you have to remember that there is no "direct" path from input to output: instead all goes through the estimated MICH/PRCL degrees of freedom, and the optical model. So this actually forces the network to tune the reconstruction to the model. This approach is very similar to the auto-encoder architectures used in unsupervised feature learning in image recognition.

Results

After trainng the network with the two previous steps, I can produce time domain plots like the one below, which show MICH and PRCL signals behaving reasonably well:

To get a feeling of how good the reconstruction is, I produced the 2d maps shown below. I divided the MICH/PRCL plane in 51x51 bins, and averaged the real optical signals with binning determined by the reconstructed MICH and PRCL degrees of freedom. For comparison the expected simulation results are shown. I would say that reconstructed and simulated results match quite well. It looks like MICH reconstruction is still a bit "compressed", but this should not be a big issue, since it should still work for lock acquisition.

Next steps

There a few things that can be done to futher tune the network. Those are mostly details, and I don't expect significant improvements. However, I think the results are good enough to move on to the next step, which is the on-line implementation of the neural network in the real time system.

Here are the demodulation phases and rotation matrices tuned for the network. For the matrices, I am assuming that the input is [I, Q] and the output is [I,Q].

POP22
phi = 153 degrees
[[-0.894, 0.447],
 [-0.447, -0.894]]

REFL11
phi = 93 degrees
[[-0.058, 0.998],
 [-0.998, -0.058]]

REFL55
phi = -90 degrees
[[0.000, -1.000],
 [1.000, 0.000]]

AS55
phi = 7 degrees
[[0.993, 0.122],
 [-0.122, 0.993]]

We discovered that the analog whitening filter of the REFL55_I board is not switching when we operate the button on the user interface. We checked with the Stanford analyzer that the transfer function always correspond to the whitening on.

The digital one is actually switching. We decided to keep the digital de-whitening on to compensate for the analog one. Otherwise we get a very bad shape of the PDH signal. Sorry Rana...

 I forgot to say that the analog gain of the REFL55 channels has been reduced to 9db

 (Manasa)

The ITMx Oplev was misaligned. Switched the ITMx Oplev back on and fixed the alignment.

EDIT, JCD:  This is totally my fault, sorry.  I turned it off the other day when I was working on the POP layout, and forgot to turn the laser back on.  Also, I moved the fork on the lens directly in front of the laser (in order to accommodate one of the G&H mirrors), and I nudged that lens a bit, in both X and Y directions (although very minimally along the beam path).  Anyhow, bad Jenne for forgetting to elog this part of my work.

 There were 4 cables running over the front side of rack 1Y4 such that the front door could not be closed. I re-routed them (one at a time) through the opening on the top of the rack. The concerned channels were

• Green refl mon
• Err mon
• Pzt out (temp) (has been marked "Door Damaged BNC")
• Laser temp ctrl

Before and after pics attached.

Before

.

After

Seems that the GPS is out of sync on donatella. We could not get any data from diaggui...

Summary:

In order to switch on the angular alignment for the IMC mirrors, we needed to center the laser onto the quad-photodiodes at the IMC and the AS Table(WFS1 and WFS2)

I and Gautam went to the IMC table and did the dc centering for the quad-photodiode by varying the beamsplitter angles. After this, we turned the WFS loops off and performed beam centering for the Quad PDs at the AS Table, the WFS1 and WFS2.

Once we had the beam approximately centered for all of the above 3 PDs, we turned on the locking for IMC, and it seems to work just fine. We are waiting for another hour for switching on the angular allignment for the mirrors to make sure the alignment holds with WFS turned off.

[Yehonathan, Gavin]

Measuring POX11_Q_MON and injecting a signal into the ITMX_UL_IN port a signal could not be seen on the function generator. After debugging the source of the issue was two fold:

• By using the quadrant drives for coils (UL, UR etc) a signal has to pass through a switch before reaching the driver. To resolve this the signal input was switched to POS_IN (driving the entire coil at once rather than in quadrants) which has no switch to bypass.
• The averaging on the Stanford SR785 was set too low. By increasing the averages from 10 to 25 the signal became more visible.

Unrelated to these issues the signal input was switched to POY11_Q_MON and ITMY_POS_IN as part of the debugging process. The function generator used was switched from the Stanford to the Siglant SDG 1032X.

An unrelated issue but note worthy was the Lenovo 40m laptop used to measure the IFO state (locked or unlocked) ran out of battery in a very short timespan.

To gauge where the resonance of the test masses are FEA model of a simple 40m test mass was computed to give an esitimate at what frequency the eigenmodes exist. For the first two modes the model gave resonances at 20.366 kHz (butterfly mode) and 28.820 kHz (drumhead mode). Then by measuring with an acquisition time of 1 s at they frequencies on the SR785 and injecting broad band white noise with a mean of 0 V and a stdev of 2 V, small peaks were seen above the noise at 20.260 kHz and 28.846 kHz. By then injecting a sine wave at those frequencies with 9 Vpp, the peak became clearly visible above the noise floor.

The last step is to measure the natural decay of these modes when the excitation is turned off. It is difficult to tell currently if these are indeed eigenmodes or just large cavity injections with an associated stabilisation time (what could appear as a ringdown decay). More investigation is required.

[Yehonathan, Gavin]

As the resonant modes of the 40m TMs are at high frequencies (starting at 28.8 kHz) we started background checks to understand if we would be able to see resonant frequency excitations in the DCPD output. We used the SR785 in the Q_OUT_DEMODULATOR port of the INPUT_MODE_CLEANER to measure around this frequency. Currently we could not see any natural excitation about the noise floor indicating it may not be possible to see such a small excitation. In any case we are conducting additional measurements in the I_MON port of 1Y2_POY11 to understand if this is a certainty.

Summary of This Week's Activities:

6/16: LIGO Orientation; First Weekly Meeting; 40m tour with Jenne; Removed WFS Box Upper Panel, Inserted Cable, Reinstalled panel

6/17: Read Chapter 1 of Control Systems Book; LIGO Safety Meeting; Koji's Talk about PDH Techniques, Fabry-Perot Cavities, and Sensing/Control; Meeting w/ Nancy and Koji

6/18: LIGO Talk Part II; Glossed over "LASERS" book; Read Control Systems Book Chapter 2; Literary Discussion Circle

6/21: Modecleaner Matrix Discussion with Nancy; Suggested Strategy: construct row-by-row with perturbations to each d.f. --> Leads to some questions on how to experimentally do this.

6/22: Learned Simulink; Learned some Terminal from Joe and Jenne; LIGO Meeting; Rana's Talk; Christian's Talk; Simulink Intro Tutorial

6/23 (morning): Simulink Controls Tutorial; Successfully got a preliminary feedback loop working (hooray for small accomplishments!)

Outlook for the Upcoming Week:

Tutorials (in order of priority): Finish Simulink Tutorials, Work through COMSOL Tutorials

Reading (in order of priority): Jenne's SURF Paper, Controls Book, COMSOL documentation, Lasers by Siegman.

Work: Primarily COMSOL-related and pre-discussed with Rana

Summary of this Week's Activities:

6/30: 2nd and 3rd drafts of Progress Report

7/1: 4th draft and final drafts of Progress Report; submitted to SFP

7/5: Began working through busbar COMSOL example

7/6: LIGO meeting and lecture; meeting with Koji and Steve to find drawing of stacks; read through Giaime's thesis, Chapter 2 as well as two other relevant papers.

7/7: Continued working on busbar in COMSOL; should finish this as well as get good headway on stack design by the end of the day.

Single layer of stack successfully modeled in COMSOL. I'm working on trying to add Viton springs now and extend it to a full stack. Having some difficulty with finding consistent parameters to work with.

For the last week, I have been attempting to determine the mirror stack transfer function which relates mechanical mirror response to a given ground-motion drive. The idea is to model the stacks in COMSOL and ultimately apply mechanical tests for manual calculation.

Procuring the correct drawings to base my 3D models off of was no simple task. There are two binders full of a random assortment of drawings, and some of them are associated with the smaller chambers, while others are appropriate for the main chamber, which is what we're interested in right now. For future workers, I suggest checking that your drawings are appropriate for the task at hand with other people before wasting time beginning the painstaking process of CAD design in COMSOL.

The drawings that I ultimately decided to use are attached below. They detail four layers of stacks, each which arrange 15, 12, 8, and 5 (going from bottom to top) Viton damping springs in an orderly fashion. The numbers have been chosen to keep the load per spring as constant as possible, in order for all springs to oscillate with as close a resonant frequency to each other as possible.

After making some minor simplifications, I have finished modeling the top stack:

After triangular meshing, before my many failed attempts at mechanical testing:

Both the Viton and steel portions have been given their material properties, and so I should be (theoretically ) ready to perform some eigenfrequency calculations on this simplified system. If my predictions are correct, these eigenfrequencies shouldn’t be too far of the eigenfrequencies of the 4-layer stacked system, because of the layout of the springs. I’ve tried doing mechanical tests on the top stack alone, but there hasn’t been much progress yet on this end, mostly because of some boundary value exceptions that COMSOL keeps throwing at me.

In the next couple weeks or so, I plan to extend this model to combine all four layers into one completed stack, along with a simple steel base and mirror platform. I also plan to figure out how to make eigenfrequency and transfer function measurements.

Lastly, to anyone who is experienced with COMSOL, I am facing two major roadblocks and could really use your help:

1) How to import one model into another, merge models together, or copy and paste the same model over and over.

2) Understanding and debugging run-time errors during mechanical testing.

If you have any idea how to attack either of these issues, please talk to me! Thanks!

Eigenfrequency analysis has been successfully completed in COMSOL on both a tutorial camshaft, as well as a homemade metal bar.

Upon increasing in complexity to the busbar, I once again began getting into run time errors and increased lag. It seems that this is due to undefined eigenvalues when solving the linear matrices. I tried many boundary values as well as initial conditions in case this was the issue, but it was not. There seems to be some sort of an internal inconsistency. This is no longer a matter of tweaking parameters.

Next steps:

1) Try using the same techniques on the actual mirror stacks to see if we get lucky.

2) In the likely case that this doesn't happen, continue the debugging process. If necessary, a good deal of time may need to be spent learning the COMSOL lower-level jargon.

Viton flouroelastomers have somewhat variable material properties. The following parameters are being used for eigenfrequency analysis.

Young's Modulus: 72,500-87,000 psi (cite: http://www.row-inc.com/pfa.html) *Accurate for PFAs

Poisson's Ratio: 0.48-0.50 (cite: http://www.engineeringtoolbox.com/poissons-ratio-d_1224.html) *Accurate for rubber

Density: 1800 kg/m^3 (cite: http://physics.nist.gov/cgi-bin/Star/compos.pl?matno=275)