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 FabryPerot 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 FabryPerot 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 pretrained 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 FabryPerot 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 nonlinearity 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
z_initial = 1e9 * L*lamba/c *1.26. * ALS
z_corrected = 2.1e06 z^2 1.9e02 z 6.91e+02
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 handpick 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.
