I had to rebuild some of the guts of my simulation to prepare it for the big changes that are to come later this week. So I only have two results to report today. The code can now take arbitrary waveforms. I tested it with spherical waves. I injected 12 spherical waves into the field, all originating 50m away from the test mass with arbitrary azimuths. The 12 waves are distributed over 4 frequencies, {10,14,18,22}Hz with equal spectral density (so 3 waves per frequency). The displacement field is far more complex than the plane-wave fields and looks more like a rough endoplasmic reticulum:

The spatial spectra are not so much different from the plane-wave spectra:

The white dots now indicate the back-azimuth of the injected waves, not their propagation direction. And we can finally compare subtraction performance for plane-wave and spherical-wave fields:

Here the plane-wave simulation is done with 12 plane waves at the same 4 frequencies as the spherical waves, and in both cases I chose a 20 seismometer 4*pi spiral array. Note that the subtraction performance is pretty much identical since the NN was generally stronger in the spherical-wave simulation (dots 5 and 20 in the right figure lie somewhere in between the upper right group of dots in the left figure). This makes me wonder if I shouldn't switch to some absolute measure for the subtraction performance, so that the absolute value of NN does not matter anymore. In the end, we don't want to achieve a subtraction factor, but a subtraction level (i.e. the target sensitivity of the GW detectors).

Anyway, the result is very interesting. I always thought that spherical waves (i.e. local sources) would make everything far more complicated. In fact, it does not. And also the fact that the field consists of waves at 4 different frequencies does not do much harm. (subtration performance decreased a little). Remember that I am currently using a single-tap FIR filter if you want. I thought that you need more taps once you have more frequencies. I was wrong. The next step is the wavelet simulation. This will eventually lead to a final verdict about single-tap v. mutli-tap filtering.

Here is the hour of truth (I think). I ran simulations of wavelets. These are not anymore characterized by a specific frequency, but by a corner frequency. The spectra of these wavelets almost look like a pendulum transfer function, where the resonance frequency now has the meaning of a corner frequency. The width of the peak at the corner frequency depends on the width of the wavelets. These wavelets propagate (without dispersion) from somewhere at some time into and out of the grid. There are always 12 wavelets at four different corner frequencies (same as for the other waves in my previous posts). The NN now has the following time series:

You can see that from time to time a stronger wavelet would pass by and lead to a pulse like excitation of the NN. Now, the first news is that the achieved subtraction factor drops significant compared to the stationary cases (plane waves and spherical waves):

And the 4*pi, 10 seismometer spiral dropped below an average factor of 0.88. But I promised to introduce an absolute figure to quantify subtraction performance. What I am now doing is to subtract the filtered array NN estimation from the real NN and take its standard deviation. The standard deviation of the residual NN should not be larger than the standard deviation of the other noise that is part of the TM displacement. In addition to NN, I add a 1e-16 stddev noise to the TM motion. Here is the absolute filter performance:

As you can see, subtraction still works sufficiently well! I am now pretty much puzzled since I did not expect this at all. Ok, subtraction factors decreased a lot, but they are still good enough. REMINDER: I am using a SINGLE-TAP (multi input channel) Wiener filter to do the subtraction. It is amazing. Ideas to make the problem even more complex and to challenge the filter even more are welcome.

I wanted to write down what I learned from our filter discussion yesterday. There seem to be two different approaches, but the subject is sufficiently complex to be wrong about details. Anyway, I currently believe that one can distinguish between real filters that operate during run time, and estimation algorithms that cannot be implemented in this way since they are acausal. For simplicity, let's focus on FIR filter and linear estimation to represent the two cases.

A) FIR filters

A FIR filter has M tap coefficients per channel. If the data is sampled, then you would take the past M samples (including sample at present time t) of each channel, run them through the FIR and subtract the FIR output from the test-mass sample at time t. This can also be implemented in a feed-forward system so that the test-mass data is not sampled. Test-mass data is only used initially to calclulate the FIR coefficients, unless the FIR is part of an adaptive algorithm. For adaptive filters, you would factor out anything from the FIR that you know already (e.g. your best estimates of transfer functions) and only let it do the optimization around this starting value.

The FIR filter can only work if transfer functions do not change much over time. This is not the case though for Newtonian noise. Imagine the following case:

(S1)-----(TM)----------(S2)

where you have two seismometers around a test mass along a line, one of them can be closer to the test mass than the other. We need to monitor the vertical displacement to estimate NN parallel to the line (at least when surface fields are dominant). If a plane wave propagates upwards, perpendicular to the line, then there will be no NN parallel to this line (because of symmetry). The seismic signals at S1 and S2 are identical. Now a plane wave propagating parallel to the line will produce NN. If the distance between the seismometers happens to be the length of the plane wave, then again, the seismometers will show identical seismic signals, but this time there is NN. An FIR filter would give the same NN prediction in these two cases, but NN is actually different (being absent in the first case). So it is pretty obvious that FIR alone cannot handle this situation.

What is the purpose of the FIR anyway? In the case of noise subtraction, it is a clever time-domain representation of transfer functions. Clever means optimal if the FIR is a Wiener filter. So it contains information of the channels between sensors and test mass, but it does not care at all about information content in the sensor data. This information is (intentionally if you want) averaged out when you calculate the FIR filter coefficients.

B) Linear estimation

So how to deal with information content in sensor data from multiple input channels? We will assume that an FIR can be applied to factor out the transfer functions from this problem. In the surface NN case, this would be the 1/f^2 from NN acceleration to test-mass displacement, and the exp(-2*pi*f*h/c) - h being the height of the test mass above ground - which accounts for the frequency-dependent exponential suppression of NN. Since the information content of the seismic field changes continuously, we cannot train a filter that would be able to represent this information for all times. So it is obvious, that this information needs to be updated continuously.

The problem is very similar to GW data analysis. What we are going to do is to construct a NN template that depends on a few template parameters. We estimate these parameters (maximum likelihood) and then we subtract our best-estimate of the NN signal from the data. This cannot be implemented as feed forward and relies on chopping the data into stretches of M samples (not necessarily the same value for M as in the FIR case). Now what are the template parameters? These are the coefficients used to combine the data stretches of the N sensors. This is great since the templates depend linearly on these parameters. And it is trivial to calculate the maximum-liklihood estimates of the template parameters. The formula is in fact analogous to calculating the Wiener-filter coefficients (optimal linear estimates). If we only use one parameter per channel (as discussed yesterday) or if one should rather chop the sensor data into even smaller stretches and introduce additional template coefficients will depend on the sensor data and how nature links them to the test mass. Results of my current simulation suggest that only one parameter per channel is required.

When I realized that the NN subtraction is a linear estimation problem with templates etc, I immediately realized that one could do higher-order noise subtraction so that we will never be limited by other contributions to the test mass displacement (and here I essentially mean GWs since you don't need to subtract NN below other GWD noise, but maybe below the GW spectrum if other instrumental noise is also weaker). Something to look at in the future (if this scenario is likely or not, i.e. NN > GW > other noise).

The simulation is not a good representation of a real detector. The first step to make it a little more realistic is to simulate variables that are actually measured. So for example, instead of using TM acceleration in my simulation, I need to simulate TM displacement. This is not a big change in terms of simulating the problem, but it forces me to program filters that correct the seismometer data for any transfer functions between seismometers and GWD data before the linear estimation is calculated. This has been programmed now. Just to mention, the last more important step to make the simulation more realistic is to simulate seismic and thermal noise as additional TM displacement. Currently, I am only adding white noise to the TM displacement. If the TM displacement noise is not white, then you would have to modify the optimal linear estimator in the usual way (correlations substituted by integrals in frequency domain using freqeuncy-dependent noise weights).

I am now also applying 5Hz high-pass filters here and there to reduce numerical errors accumulating in time-series integrations. The next three plots are just a check that the results still make sense after all these changes. The first plot is shows the subtraction residuals without correcting for any frequency dependence in the transfer functions between TM displacement and seismometer data:

The dashed line indicates the expected minimum of NN subtraction residuals, which is determined by the TM-displacement noise (which in reality would be seismic noise, thermal noise and GW). The next plot is shows the residuals if one applies filters to take the conversion from TM acceleration into displacement into account:

This is already sufficient for the spiral array to perform more or less optimally. In all simulations, I am injecting a merry mix of wavelets and spherical waves at different frequencies. So the displacement field is as complex as it can get. Last but not least, I modified the filters such that they also take the frequency-dependent exponential suppression of NN into account (because of TM being suspended some distance above ground):

The spiral array was already close to optimal, but the performance of the circular array did improve quite a bit (although 10 simulation runs may not be enough to compare this convincingly with the previous case).

So far, the test mass noise was white noise such that SNR = NN/noise was about 10. Now the simulation generates more realistic TM noise with the following spectrum:

The time series look like:

So the TM displacement is completely dominated by the low-frequency noise (which I cut off below 3Hz to avoid divergent noise). None of the TM noise is correlated with NN. Now this should be true for aLIGO since it is suspension-thermal and radiation-pressure noise limited at lowest frequencies, but who knows. If it was really limited by seismic noise, then we would also deal with the problem that NN and TM noise are correlated.

Anyway, changing to this more realistic TM noise means that nothing works anymore. The linear estimator tries to subtract the dominant low-frequency noise instead of NN. You cannot solve this problem simply by high-pass filtering the data. The NN subtraction problem becomes genuinely frequency-dependent. So what I will start to do now is to program a frequency-dependent linear estimator. I am really curious how well this is going to work. I also need to change my figures of merit. A simple plot of standard-deviation subtraction residuals will always look bad. This is because you cannot subtract any of the NN at lowest frequencies (since TM noise is so strong there). So I need to plot spectra of subtraction noise and make sure that the residuals lie below or at least close to the TM noise spectrum.

Just briefly, my first subtracition spectra:

Much better than I expected, but also not good enough. All spectra in this plot (except for the constant noise model) are averages over 10 simulation runs. The NN is the average NN, and the two "res." curves show the residual after subtraction. It seems that the frequency-dependent linear estimator is working since subtraction performance is consistent with the (frequency-dependent) SNR. Remember that the total integrated SNR=NN/noise is much smaller than 1 due to the low-frequency noise, and therefore you don't achieve any subtraction using the simple time-domain linear estimators. Now the final step is to improve the subtraction performance a little more. I don't have clever ideas how to do this, but there will be a way.

Instead of estimating in the frequency domain, I now have a filter that is defined in frequency domain, but transformed into time domain and then applied to the seismometer data. The filtered seismometer data can then be used for the usual time-domain linear estimators. The results is perfect:

So what's left on the list? Although we don't need this, "historically" I had interest in PCA. Although it is not required anymore, analyzing the eigenvalues of the linear estimators may tell us something about the number of seismometers that we need. And it is simply cool to understand estimation of information in seismic noise fields.

This is how Newtonian-noise subtraction works:

I wanted to push the limits and see when NN subtraction performance starts to break by changing the number of seismometers and the size of the array. For aLIGO, 10 seismometers in a doubly-wound spiral around the test mass with outer radius 8m is definitely ok. Only if I simulate a seismic field that is stronger by a factor 20 than the 90 percentile curve observed at LHO does it start to get problematic. The subtraction residuals in this case look like

The 20 seismometer spiral is still good, but the 10 seismometer spiral does not work anymore. It gets even worse when you consider arrays with circular shape (and one seismometer at the center near the test mass):

This result is in agreement with previous results that circular arrays have trouble in general to subtract NN from locally generated seismic waves or seismic transients (wavelets).

I should emphasize that the basic assumption is that I know what the minimum seismic wavelength is. Currently I associate the minimum wavelength with a Rayleigh overtone, but scattering could make a difference. It is possible that there are scattered waves with significantly smaller wavelength.

If the plan is to use feed-forward cancellation instead of noise templates, then the way to optimize the array design is to understand where gravity perturbations are generated. The following plot shows a typical gravity-perturbation field as seen by the test mass. It is a snapshot at a specific moment in time. The gravity-perturbation force is projected onto the line along the arm (Y=0). Green means no gravity perturbation along the arm generated at this point.

The plot shows that the gravity perturbations along the direction of the arm seen by the test mass are generated very close to the test mass (most of it within a radius of 10m), and that it is generated "behind" and "in front of" the mirror. This follows directly from projecting onto the arm direction. As we already know, for feed-forward, we can completely neglect the existence of seismic waves and focus on actual gravity perturbations. In short, for feed-forward, you would place the seismometers inside the blue-red region and don't worry about any locations in the green. The distance between seismometers should be equal to or less than the distance between red and blue extrema. So even though I haven't simulated feed-forward cancellation yet, I already know how to make it work. Obviously, if subtraction goals are more ambitious than what we need for aLIGO, then feed-forward cancellation of NN would completely fail generating more problems than solving problems. Unless someone wants to deploy hundreds to a few thousand seismometers around each test mass.

Today we brought a rack from Drever lab to 050 W Bridge. This rack will be used for crackle experiment.

   We start setting up the experiment, and we need a rack for electronics equipment, so with Steve's help we got one from Drever lab. We cleaned the rack before brought it in the lab, so there should be no dust.

Next, we will find a lock-in amplifer, maybe a function generator to drive the system.

We plan to work on the experiment on

Tuesdays  afternoon

Thursdays afternoon

Fridays afternoon.

Ming Yuan, tara

We setup the basic Michelson interferometer with one arm which can driven by a PZT and another one whose position is adjustable. 

The laser we got didn't work at the beginning. We found that the power supplier was not functional. Tara borrowed another power supplier for the laser.

The basic Michelson interferometer was setup. One of the mirror attached on copper plate was replaced by a regular mirror with position adjustable. One of the PZT is needed to be fixed.

We observed Dark Fringe by adjusting position of the regular mirror.

We got the signal from a basic Michelson setup with one of the arm being driven by a PZT.

 This is the signal from the oscilloscope.

First, we check the signal when there is no voltage applied to the PZT, the signal is plotted in green.

Then, we drove only one of the mirror by PZT. The voltage is 6Vpkpk, with 7V offset. 

 The signal is plotted in blue when the mirror was driven. We can see strong signal on the scope.

 mingyuan, tara

            We setup the Michelson interferometer with two identical x and y arms. We drove both mirrors at 2 Hz and observed signal at 10 Hz using a lockin amplifier. We saw no significant difference whether the mirror were dirven or not.

           (The pzt for the second mirror is fixed. The wire is soldered back to its electrode.)

             We setup the Michelson interferometer, now with similar setups on two arms. The end mirrors on both arms are attached on metal shims. The shims touch the PZTs which are driven by 2Hz, 6Vpkpk sinusoidal signal with 7 V offset.

             We use a voltage divider(we planned to make one, but we found a nice one in EE lab lying on the floor, so we borrowed it) to adjust the voltage on one of the PZTs to make sure that both mirrors are driven by the same distance. We adjusted the divider to minimize the signal at 2Hz.

              fig 1: With a voltage divider, we can adjust the voltage on the PZT so that both mirrors are pushed by the same distance and the 2Hz common mode is minimized. On the plot, Y axis shows the signal output from the lock in amplifier at 2Hz. The higher value means the stronger signal at 2Hz. X axis is time scale. The setup was 5mV sensitivity range, filter in 300 ms, phase -152.3 degree.

 The signal output from the lock in amp has not been calibrated to length yet. We just want to see the qualitative result.

                 Once we made sure that we minimized the common mode, we tried to measure the possible up converted noise at 10Hz. (We used the internal oscillator in the lockin amplifier for reference signal at 10 Hz.)

               First, we did not drive the mirror, so that we could see the signal at 10 Hz due to background. Then, we drove the mirror at 2 Hz, and observed any possible up-converted noise at 10Hz

              There is nothing conclusive yet. The 2Hz signal that drives the PZTs are plotted here for comparison. From a quick glance, there is no obvious correlation between the noise and the driving signal.

fig2:  Signal from the lock in amp at 10Hz. Setup: sensitivity at 500 uV, in filter 300 ms.

Why are we doing this:

We want to measure any possible up-converted noise when the material under stress is driven at low frequency. For example, the system is driven at 2Hz, there might be broadband noise occurs due to the motion. If there is, we can try driving the system with different amplitude to see if the noise changes or not.

We are trying to chopping the signal today.

   The low noise amplifier can be used as bandpass filters for 10-100 HZ.

   We are trying to figure out the signal squaring. The mixers in the lab only work for high frequency (> 500 KHZ).

   Frank recommends us to use AD734 4-Quadrant multiplier. 

   We checked the electronics lab in Downs and 40 m and couldn't find it. We plan to order some AD734.

 I ordered 5 of AD734 and thinking about how to make a circuit for squaring the signal.

    The "chopping" signal readout technique requires that we square the signals.  Basically we need to (as rana suggested):

(1) square the signal from PD, (after 10-100Hz bandpass) to convert it to power, and band pass it again.

(2) square the driving signal (might be varied from 0.1- 1Hz.) This is illustrated in the diagram as doubling the frequency ("2 x freq" box.)  The driving signal for PZT is offset. So the signal is  V drive = A + B xsin (2pi fdrive t) with A > B. This ensures that the voltage on one end of the PZT is always higher than another end. We might need to high pass this signal first, to get a signal with only 2 fdrive frequency after we square it.      

(3) multiply signal from (1) and (2) to demodulate the signal.

Basically, 3 multipliers are needed.

The first one is for (1), so the input frequency is ~ 10 -100Hz, and the output is 20-200 Hz.

The second multiplier is for (2), the signal is ~ 0.1 - 1 Hz, but this one might have large DC term after we square it.

The third one is for (3), this one has to multiply 2 low f signals together which is quite similar to (2), so the design can be the same.

I'll consult Frank and/or Koji again before finalize the multiplier circuit.

In the mean time, we might try this mixer to multiply the signal. I'll order one.

koji, mingyuan, tara: We designed the circuit for multiplying/ squaring signals with AD734. 

    The details for each signal are discussed here. 

    The "general multiplying circuit" box in the diagram shows how each AD734 will be powered/ fed input signal.

     For the signal from the PD, we need to bandpass(10-100Hz) it first. We plan to use a SR560. To split the signal to x and y input, we will use a T connector. Then square the signal and band pass it again at 0.1 - 100Hz bandwidth.

     For the signal from the function generator which drives the PZT. We will high pass it, by either SR560 or a high pass circuit. We might need a buffer here if the output impedance of the function generator is not high. Split the signal with a T again, and square it.

    After both signals are squared, we multiply them together. Send one to X1 input, another signal goes to Y1 input. Then we FFT the output signal from W.

We made a drawing for a structure hat will hold the maraging blade. The details aren't complete yet. The holes for the clamping will be  identified,  but the sketch shows the rough idea.

     We want to clamp the blade to a structure. The drawing for the clamp will be provided by Ryan (he found it in the dcc.) The structure is consisted of the base and the pillar. Although a monolithic structure is better, it might be to expensive to carve out a big piece of Al block, so Koji suggested that we do it like this. The base will be mounted on the table, and the pillar will be mounted on the base by 4 screws. The height of the pillar is not decided yet. It depends on how big the Al mass block we need to pull down the blade by its weight, and how the mirror for reflecting the beam up will be mounted, but it should be around 6 - 8 inches.

    The mass block will be used for mounting the end mirror of the interferometer + a translational stage. This way we can steer the beam with 2 mirrors and adjust the arm length. We will determine the weight, so we can estimate the size of the mass block, assuming we will use Al.

  We switched the current metal shim with the thicker Aluminium shim. Now both mirrors are also the same. We tested and showed that the shim is not too hard to be pushed by pzt.

First, the thicker Al shims have bigger bending stiffness and more difficult to bend under the surrounding perturbation. Therefore, the signal we got has less noise from the surrounding perturbation.

By using the PZT we have, we can still drive the shim well. With the driving, we observed intensity oscillates from ~50 mV to ~200 mV. 

We also observed a low frequency (~80 mHz) oscillation of the signal. I didn't find the source of this oscillation. The sensitivity of response to driving is lower while the intensity is near the minimum and Maximum and higher while intensity is at the middle.

We measured the weight needed for pulling the blades down, and measured Q, f0 of the blades. For Rom blade, the weight is 1.279 kg, f0 = 2.27 Hz, Q = 300. For Rem blade, the weight is 2.005kg, f0 = 2.35Hz, Q = 475.  The test blades are named Romulus(Rom) and Remus(Rem).

     Why do we do this:

    The maraging blades are designed to be flat when they are in used, so we need to know how much weight do we need to pull them down to their operating level. The weight will determine the size of the load mass we want in the drawing as well. We plan to mount mirror mount on the load mass, so we can align the mirror for the interferometer's end mirror.  Plus, resonance frequencies and Qs of the blades and seismic noise will be used to estimated the noise budget of the setup.

    The weight was applied to the blade until the blade horizontally leveled. Then the total weight was recorded.  After that, we used shadow sensing technique to determine their resonance frequencies and Q factors.

The results are summarized here:

Blade     load mass       f0               Q

Rom         1.279 kg       2.27 Hz     300

Rem         2.005           2.35 Hz      475

.

fig1: determining the weight. The blade mounted on the table appears flat with the right weight.

fig2: Q measurement from Rom

fig3: Q measurement from Rem

 We made a sketch for the weight clamp that will carry the mass block on the end of the blades. This will be done in Solidwork tomorrow.

   We plan  to load a block of mass under the tip of the blade by using a pair of knife edge pieces so that the rubbing between the mass block and the blade is minimized.

 The edge of the blade cannot be too large, or it will be noisy when the blade is driven. On the other hands, if the blade angle is too small (sharper blade), the stress on the blade due to the weight will be too large and cause plastic deformation on the blade, which we don't want. We plan to make it flat ~ 1mm wide, with 120degree open angle.

 The yield tensile strength of maraging steel is ~ 1 -2 GPa. With the contact area at the knife edge we can calculate the maximum clamping force.

The width of the edge is ~ 5cm

The thickness of the edge ~ 1mm.

so the maximum force should not exceed ~ 1 GPa x 0.05 m x 0.001 m ~10^4 newton.

We will use spring washers to make sure that we do not tighten the clamps together with too much force and cause plastic deformation on the blade.

We finalized the drawing for blade clamping system. The drawings are posted here and in Crackle ATF Wiki. We will submit the drawings to the machine shop tomorrow.

         For each blade, the clamping system will consist of: 1)Steel base, 2)Steel pillar, 3) Steel top clamp, 4) Al knife edge top piece,5)Al knife edge bottom piece,and 6) Al end piece.

1) Steel base x1:   The steel base is 3"x3"x0.5" . It has 4 counter sunk holes that allow us to mount the steel pillar on it. It has 3" rails on both sides, so we can mount it on the table. Extra clamps can be used to hold the base on the table.

2) Steel pillar x1:   It is 5.5" height with 2"x2" square cross section.  There are 4 tapped 1/4-20 holes , 1" in depth, on the bottom for mounting it on the base. There are 2 tapped 3/8 , 1" in depth, on top for clamping two clamps along with the blade.

3) Steel top clamping piece x1, This will clamp the blade on the pillar.

4) Aluminum knife edge, top piece x1,

5) Aluminum knife edge, bottom piece x1: (4&5) The two knife edge pieces will be used for loading the mass block on the maraging blade tip. The explanation is written in this entry.

6) Aluminum end piece that holds the mirror mount on the blade tip x1: We want to have a steerable mirror for the IFO. So we need a mirror mount. The block will hold the mount and the blade tip together through screws. This piece is uploaded in the above entry.

 The assembly (without the blade and the mirror mount) is shown below.

We measured the tip tilt angle of the blade while the main part of blades was bent flat.  REM: ~9 degree; ROM: ~ 7 degree. This angle should be able to cancel by mirror holder.

One block of Al was designed to mount mirror holder with the blades. The SolidWork drawing is attached below.

Two screws (2-56, A2, 3/16) will be used to mount the block onto blades through the two holes in the head of blades.

One screw (8-32) will be used to mount the mirror holder onto the block. The mirror hold is light, the block should be able to hold it firmly.

The other drawings will be uploaded by Tara

 I tested the mixer, the demodulated signal from input at 10 - 100 Hz might be too small and too distorted to get reliable data.

  As we want to square/demodulate  signal in 10 - 100 Hz BW. a low frequency mixer might be a good tool. I asked Alastair to buy this mixer for me, and it arrived today.

  The lowest acceptable frequency in the design is 500 Hz, but I don't know how well it works at 10 - 100 Hz so I tested it.

   ==Setup and result==

   I used  SR785 to generate sine wave, then split it with a T and connected the output to LO and RF of the mixer. 

   I tested that the mixer works fine at the designed frequency.  The plot below shows the result from  1kHz signal input.

Next, I changed the frequency to 10 Hz, 50Hz, and 100Hz.

  The demodulated signal is then observed in frequency domain (left column of the plot) and in time domain ( right column of the plot)

I think the peaks at driving frequencies (10Hz, 50Hz, 100Hz and their harmonics) appear because of the offset of the sine input signal.

The results for low frequency  seem to be too distorted. We will test the AD734 chips tomorrow. I got the package this afternoon.

We submitted the drawing to the machine shop today. The works should be done before May 23rd.

   The base/ pillar/ blade clamp will be made from stainless steel. The knife edge pieces and mirror mount at the blade tip will be made from aluminum.

 We tested AD734 on the diagnostic bread board, the result is good.

     We want to square/multiply signals between 10 to 100 Hz, so we use AD734 chip to do the work. The circuit is connected as described here

We try to square the signal. the test signals are sine waves at 10 Hz, 50Hz. The output are nice sine waves, but the gain is high (72dB). The chip rails as the input exceeds 0.5 Vpkpk. We will have to check the signal from the PD in the setup to see if it is higher than 0.5 Vpkpk or not. If so we can change the gain of the chip. Otherwise we can go ahead and use it.

The spectrum of the output, for 10Hz input, there's a peak at 20Hz output. For 50Hz input, there's a peak at 100Hz. The response is flat between this bandwidth.

The low frequency oscillation we mentioned in the previous Log could originate from the creep of the rubber between PZT and the Shim. Because the initial stress caused the creep of the rubber, the Shim relaxed slowly and changed the optical path and caused the low frequency oscillation. This mechanism can explain the phase change between the driving and the signal. Rana recommended to use a spring to replace the rubber. To calculate the spring constant of the spring: Spring constant of the Shim, ks = 3EI/L^3; Amplitude of displacement of PZT ~ A; Amplitude of displacement of the Shim ~ B; the spring constant of the spring ~ k;

k = ks*B/(A-B)

From current dimension, ks ~ 10000 N/m. If we don't want to drive PZT too hard, assume A = 2B; k = ks = 10000 N/m.

I was flabbergasted when I saw this. There are many really good seismometers with very simple mechanical design and electronics. This is a nice one with complicated mechanics and electronics.

RA: Awesome.

Some useful things to remember for the AD734:

The transfer function when wired as a multiplying circuit is: W = ((X1-X2)*(Y1-Y2) / 10V) + Z2
For this to be true the Z1 pin should be wired to the output W, to provide feedback, which isn't shown explicitly on Tara's general multiplying circuit diagram. Also for testing the chip inputs were wired as differential, not with one leg grounded as shown on the GMC diagram.

The 10 V comes from the default division voltage when the denominator control inputs (U0, U1, U2) are grounded. If you want some added offset to the output you can send it to the Z2 pin.

The input impedance is listed as 50k for all X, Y, and Z pins.

We measured the noise with 0V X/Y inputs, it was around 1 mV/rtHz at 10 Hz, as you can see in Tara's earlier post, slightly improving at higher frequency.
The input noise is listed as 1 uV/rtHz from 100 Hz to 1 MHz. The amplifier gain is listed as 72 dB which is ~ 4000x, and we were at the default denominator of 10V so this corresponds to a noise of 1e-3 * 10 / 4000 = 2.5 uV/rtHz at the input, seems reasonable compared to spec sheet. The signal to be squared in the creak setup (the output of the Michelson) will have to be bandpassed first, probably by an SR560, so gain can be applied there to get in over the multiplier noise floor.

As Tara noted the output does rail for signal amplitudes well below the listed maximum input, so we need a better understanding of how to control the gain.

We brought the setup back. The interferometer is working and more stable. We will try extracting signal next.

    From this entry, we noticed the 180 degree phase shift in the signal when one arm was driven. The signal from PD followed the driving signal before drifted up and phase shifted by 180 with respect to the driving signal. We believed that this was the effect from the drift of the arm length. Suppose that we operate the IFO at the fringe's maximum slope. The drift in arm length will move the operating point on the fringe, and we might end up on the other side of the fringe which will show up in the 180 degree phase shift of the signal.

    The mirror was pushed by a piece of soft rubber which was glued to a pzt. Another end of the pzt was glued to a piece of plastic. This plastic piece was clamped on a translational stage. We thought that the soft rubber, the plastic and the translational stage caused the drift of the arm length.

So we tried to improved this by

• replacing the rubber and plastic with two pieces of magnets. One was glued on the back of the mirror, another one was glued to the pzt. This did not work, the combination of the force, and the shim stiffness, had to be matched so the mirror position can be adjusted without letting the magnets touch each other. So we tried
• replacing the rubber and plastic with stainless steel nuts, one nut is for clamping, another one is for pushing the mirror.
• 

we haven't got rid of the stage because we still need it for position adjustment purpose. We will use dc voltage offset on pzt to adjust the position later once we can add dc signal to the driving voltage.  Currently, we use a single function generator to drive both pzt simultaneously.

     With new pushing scheme, the drift becomes much less than before. The signal is in phase for more than a minute or two which should be enough for chopping technique later. The picture below shows the signal from driving voltage @ 2Hz(blue), and readout from PD at maximum slope (yellow).

     Once we made sure that the signal was quite stable, (that is, the operating point stays at the maximum slope most of the time), we measured the background noise. This is a readout from PD and maximum slope on the fringe without driving voltage applied on the pzt. Then we measured the signal when one arm was driven at 2 Hz. Finally, we drove two arms at 2Hz and adjust the voltage on the pzt so that the 2Hz common mode cancelled out.

 The plot shows the noise of the setup: 1) the background 2) when one arm was driven at 2 Hz. 3) Both arms are driven, with common mode at 2Hz minimized.

 We will try squaring the signal next. The read out from PD is ~ 200 mV. This value will determine if we need a divider for the signal or not.

We used a big box to cover the optical loop. The interferometer is more stable now.

We build other two AD734 chip circuits for signal square and multiplier.

We already tested that we could square the driving signal and PD signal.

The square of the PD signal has a big offset from the AD 734 circuit. We need figure out how to take the offset out.

We tried read out the signal from chopping technique.We could not see anything yet.

The signal when both IFO arms were driven were similar to the signal when there was no driving.

   After we made the necessary electronics for chopping technique, we tested if we could see the signal or not.

    ==setup==

    We used a 4 mW HeNe laser as a source with a simple Michelson interferometer setup. We tried to operate at the maximum slope of the fringe. Each mirror was attached to a metal shim which could be pushed by a PZT behind it, see here . We drove the mirror with the same distance so that the common mode was canceled and only incoherent noise from crackle in each blade could be detected.

    The diagram omits the IFO part and the blades. The output beam from the IFO was incident on the PD. We operated at the maximum slope of the fringe. The driving voltage Vdrive was send to the PZTs pushing blades (with mirrors attached on them) at the end of both arms.

    The 1/2 and 1/10 dividers are used to reduce the signal down below 0.5 V. This number comes from the square testing. When the input signal to be squared is larger than 0.6, the output starts to rail. So we use 0.5V to be the upper limit for now.

   ==result==

       The PSD of the signal output when two arms are driven are similar to the background signal (arms are not driven). It might be that the gain setting are not optimized, the setup is too noisy, or problems from offset from the AD734 chip. We will figure that out next. We will also make a sturdy box for multiplying chip. Currently we just use temporary test board to operate the chips for the read out.

 1) We removed the squaring circuit from the test board and built it on a board. The box for the circuit was prepared.

 2) We replaced the crappy beam splitter with a Thorlabs 20mm cube 400-700 nm beamsplitter. The beam power is evenly divided and has no multiple reflections. We measured the noise psd at the AS port.

      1) The circuit for squaring, multiplying signals was temporarily built on a plug-and-play test board which was neither sturdy nor compact. So We used a breadboard available in the EE lab to build the circuit.

The cartoon schematic is shown below.

      A) The signal from PD at AS port is band passed before squared (not shown here), then band passed again before.

      B) The driving voltage for PZT will be high pass to get rid of DC component (not shown here), then divided. We want a divider here because we might need to drive the pzts with higher voltage. The second divider might be unnecessary, but we have it just in case.

     C) Then we multiply  A and B and get the signal out for FFT.

     Currently, the chips have offset added to the output, ~ from -1 to -2 V. We tried adding the offset in Z2 let as suggested in the datasheet, but it killed the signal ??!!!. So we are planning to high pass signals that we care only their AC parts. Currently, we are not sure if we care about DC part of the V drive or not. We have to think about it.

     2) The beam splitter used in the original setup is not really for a beam splitter for Michelson IFO. It is not 50/50, and there are multiple reflections from the surfaces.

Thus, we ordered a cube beam splitter suitable the job and replaced it. It is mounted on a beam splitter mounted directly mounted on a 2" post, so we expect it to be more stable.

We measured the noise from AS port when the armed was not driven vs driven at 1 Hz. The result is shown below.

The calibration from V to differential arm length (Lx - Ly) is approximated from

dx ~ dV x  lambda/ 4 / (Vmax - Vmin)

At the maximum slope of the fringe, as we tap the table, the voltage will fluctuate between Vmax (from constructive interference)and Vmin (destructive interference.)  On the fringe, the differential arm length between maximum to minimum V output is lambda/4 (so the accumulated distance from round trip is lambda/2, a condition for changing from maximum Vout to minimum Vout). We can approximate the slope to be (Vmax - Vmin)/ (lambda/4).

Vmax - Vmin ~ 500 mV, lambda = 660 nm. so

dx = dV x 3x 10^ -7

The result is 5 - 6 orders of magnitude above the shot noise level (~ 1e-17 m/rtHz for this setup.) Noise characterization will be considered next, but from

a quick test of tapping, seismic is the dominating source.

I ordered opto mechanical mounts for turning the beam vertically. See the details in psl log.

I also orderedspring lock washers and wave washers. There will be used when we clamp the guillotine things for putting the load on the tip of the blade.

The pressure from the clamp should not exceed the yield strength of the maraging steel blade. So the spring lock washer should give us some limits of pressure on the blade. There is no specification about how much pressure it would be, so I ordered two kinds of washer for testing.

We figure out the offset issue of the chip AD 734. We measured the noise of chip AD 734 with 50 ohm input terminated.

The noise is shown below for two chips we are using and noise from spectrum analyzer is attached for reference.

The noise of AD 734 is about 1 uV/root(Hz) at around 50 Hz. The sensitivity of of the chip should be:

dV*dV/10 = 1 uV/root(Hz)  =>  dV ~ 3 mV/root(Hz)

We are not sure about that we understand the noise propagation through the chip correctly.  

By mingyuan, tara

We figured out the offset problem in AD734 chips, the box for squaring and multiplying signals is finished.

          The problem from the previous circuit was that the ground from the signal was grounded with the load ground. This time the load ground is separated from the signal ground, Z2 is grounded to load ground.  These corrections fix the offset problem and the maximum allowed input ( was 0.6 V.) Now the input can be up to 10V. The output, Z, is (X1-X2)x(Y1-Y2)/10 as described in the datasheet. Now the chip are connected as shown below.

          We are thinking about not using the default denominator (/10) for a multiplying chip (we certainly need it for squaring chips, otherwise the output will rail), because after the signals (from PD and driving voltage) are squared, their dc levels are ~3 V. When the two are multiplied together, the voltage output drops to 3x3/10 = 0.9 V. So if we can have denominator = 1, the signal will be larger. However, we have to understand how the noise in the chip works first. See Mingyuan's entry about input referred noise of the chip ,it is roughly 3 mV/rtHz. If the SNR remains constant regardless of the denominator, we might not need to worry about it.

Today we measured the current system noise by the signal squaring and multiplier system we built.
The interferometer is quite stable now and the phase could be stable for more than half hour.
The plates are droven by 2 Hz 3 Vpp sinusoid signal with 4 V offset in common mode.
The signal From PD is band passed by SR560 with 200 gain and squared by AD 734 chip.
The driving signal is also band passed by S560 and squared by another AD 734 chip.
The two squared signal are multiplied by one AD 734 chip. The signal from multiplier is feed
to spectrum analyzer. We also measured the noise spectrum without driving plates. The results look the same.
The signal is very sensitive to talking and walking nearby the table. We suspect that the seismic noise dominates the noise.
The possible noise source:

• shot noise
• seismic noise
• Thermal noise of the plates
• Thermal noise of the mirror
• PZT noise and rubbing
• Air flowing
• Laser power fluctuation
• Laser frequency noise
• noise from AD 734
• noise from other electronics

Tara will upload the plots later.

By Mingyuan Tara

We measured the FFT of the demodulated signal from chopping technique. We did not see much. The background noise is still too high.

      With everything ready, we used chopping technique to measure crackling noise. We measure the PSD from the demodulated signal between a) the mirrors being driven at 2Hz, and b) background noise, when the system was at rest, no driving force applied to the mirrors. We did this to check if we can see any signal due to crackling noise/ rubbing noise/ pzt noise or any noise originated from the driving mechanism or not. The result is not quite clear, we see a few peaks from the driven system around 40 Hz, but we have yet to confirm and identify them.

     ==setup==

        The setup is shown in the diagram below. For each bandpass through SR560, we added the gain to the signal as much as possible without railing the signal. Note that in this setup we did not bandpass the signal from PD after we square it , as shown in previous entries. Because Mingyuan did not understand why would we need to and I could not answer him properly, so I agreed to let him have it his way.

          When we measured the data from the driven system (red curve in the plot), the setup is as shown in the diagram. However, for background measurement (blue curve in the plot), we want to keep the DC supply provided by the function generator to the pzt so that the sensitivity of the signal remain the same. Hence, we used a second function generator to send in similar driving voltage to the squaring box, while the first function generator was set to the dc output voltage to supply the pzts, no sinusoidal output.   (We made a mistake by just unplugging the Vdrive to the pzt and to the squaring box, and the noise level dropped so much.) 

==result==

     The red and blue curve shows the psd of the demodulated signal when the blades were driven, and the static case respectively. The peak at 4 Hz that presents in both cases are from the square of the driving signal at 2Hz.

     There are a few  peaks around 36 - 40 Hz when the blades were driven. We could not see this in the SR785 monitor because the monitor was so faint. I just saw this after I plotted the data.  The peaks might come from some resonances in the setup. We expect crackling noise to be more broad band. We will confirm and identify the source of the peak to make sure that we can see some signal from the driving (it can be rubbing between metal, pzt noise, crackle.)

We will repeat the same measurement, and try changing driving frequency/ amplitude, to see if the signal changes or not.

To have the ability of controlling the phase, we need adjust DC voltage of one of the PZTs independently.
We use the function generator to generate AC driving with a DC offset for one of the PZTs and use a OP270
chip to add the driving signal with another DC voltage for another PZT. By changing this DC voltage, we can
control the phase of interference signal. We adjust the voltage to put the PD intensity in the middle to have
the best sensitivity.
We did the same measurement as last time to check the peaks we observed. By use the same condition, we do see
a few extra peaks while the plates are being drovn at 2 Hz. We also changed the driving frequency to 0.7 Hz and
did the same measurement. The results looks different.
  Tara will upload the data later.

By mingyuan, tara

We built a simple voltage summing circuit for adding DC level to the pzt. This circuit allows us to fine tune the inteferometer's differential arm length, so that we can operate at the fringe's maximum slope. Then we checked the peaks we observed from last time. It turned out to be harmonics from the common mode from driving.

The circuit schematic is shown below. The result Vout = Vin1 + Vin2. 

The adding circuit is used as shown in the schematic (highlighted in yellow.)

 *Later, we can use this summing circuit in a feedback control loop for locking the interferometer.

Then we used this circuit in the setup and repeat the measurement to check the peaks we observed last time. With the same setup, we observed the peaks again, but they probably are harmonics from 4Hz from common mode motion which was not perfectly cancelled.

     We repeated the measurement again with 0.7 Hz driving, and the peaks disappeared. The signal between driving and not driving the arms are very similar. The shape of the PSD changes slightly because of the lower amplitude of the driving signal, as we low pass the signal at 0.1 Hz.

We do need a seismic isolation and vacuum chamber. Right now, sound from people speaking in the lab can disturb the measurement.

 a few things we have to consider soon, before we use the maraging steel blades pulled down by a mass block in the experiment.

1) how should we push the blades?  capacitor plate? magnetic coil?

2) When can we move and get a better table, so that we can decide on seismic isolation stage.

3) We have to start looking for vacuum bell jar for the experiment.

4) lock the interferometer?

5) will we get an npro laser for the experiment?

We get all the part for blade spring holders, so we determine the setup and get the approximate minimum diameter of bell jar to be 18".

We decided to place the blade in the same direction, so that seismic in horizontal direction coupling into the blade will be common mode.

The masses that will pull the blades down have not been installed yet. We need longer 1/4-20 screws, probably ~2.5".

Note: I fixed the table legs so that 4 of them touch the floor. However, the steel chamber for reference cavity for cryo lab is left on our table. Its  presses on one end of the table and lifts one end completely off the rubber support, see the below picture to see the gap between the rubber and the table.

We want to remove the chamber first, so the table rests evenly on four legs before we measure the seismic noise on the table.

I should begin this ELOG with the warning that this is the first time I've ever used MATLAB...

First item on the agenda is to create a simulation which models the displacement due to the crackling noise on the blade spring. This can be done using MATLAB in just a few lines and generates a plot (see first attachment).

For this simulation, I picked random numbers for the constants, which is why it wil likely look a little funny. 

t =       1:0.1:100;  %time range

fdrive =  0.1   % driving frequency

MaxA =    2    %maximum amplitude

alpha =   1          %some constant

deltax =  MaxA.*sin(2.*pi.*fdrive.*t)

k0 =      1   % spring constant

Force =   k0.*deltax    %driving force

noise =   rand(1,1001); %some random noise function: our crackling noise?

k =       k0+noise.*alpha.*deltax;

dx =      Force./k

crackle = dx-deltax;  % crackling noise measured in terms of displacement

The next step is to attend to chopping. The second attachment is the circuit I drew to do this, but it differs a bit from what I've seen posted on this ELOG here. Which one is right??

From what I understand, the first simulation (above) results must be injected into the beginning of the top part of this circuit (coming from the PD). Then the signal needs to be bandpassed, squared, bandpassed again, the mixed with a source signal and time averaged to isolate the noise. On the subject of bandpassing: I've been reading up on trying to do this in MATLAB. There seem to be a few suggestions on the Internet, but none of them have worked for me...(then again, I'm probably doing it wrong). The crackling plot shows displacement at a range of frequencies, but I imagine that the chopping circuit will have more to do with voltages. How does this translate?

TO DO list:

-- Try to create a bandpass filter in MATLAB

-- Try to create a mixer in MATLAB

The rest of the chopping circuit has been designed (see first attachment). 

***NOTE: adjustments have been made to the crackling circuit, where the AC source signal is NO LONGER BEING SENT THROUGH THE SIGNAL SQUARER)

Because there are four "outputs", there will be four plots generated. There is a number at the "output" of each in the first diagram attached, which I associated with its corresponding plot number. Here is a sample of the MATLAB code I used for Circuit 1:

fs = 500;

ts = 1/fs;

t = (0:ts:100); %time vector

k0 = 2; %ideal spring constant

fdrive = 0.1; %driving frequency

Amp = 1; %max amplitude

dist = Amp*sin(2*pi*fdrive*t); %spring position

vel = 2*pi*fdrive*Amp*cos(2*pi*fdrive*t); %spring velocity

noise_t = rand(1,50001)*2-1; %noise function

Force = k0*dist; %ideal spring force

alpha = 0.5;

k = k0 + noise_t.*alpha.*k0.*(dist./Amp); % k = k0 + dk

dx = (k - k0).*dist/-k0;

crackle = dx/(alpha*Amp); % = noise*sin^2(2*pi*fdrive*tt)

%Bandpass#1 

[B1,A1] = butter(2,[10 100]/(fs/2));

y1 = filter(B1,A1,crackle);

%squared signal

ysq = y1.^2;

%Bandpass#2

[B2,A2] = butter(2,[10 200]/(fs/2));

y2 = filter(B2,A2,ysq);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%Circuit1%%%

%Source

Vsin = sin(2*pi*fdrive*t);

%Doubled source

V1 = 2.*Vsin;

%Mixer

ymix1 = y2.*V1; 

%Low Pass Filter

[B3,A3] = butter(2,0.1); % error message when fc not within (0,1)

ylpf1 = filter(B3,A3,ymix1);

 We went down to the SUS lab and ran some tests to get measurements we could use to calculate the Q and b (dampening factor/constant?) of the blade springs.

The setup was fairly simple (see attachment below): a laser beam was set up such that its path to a photo diode would be interrupted by the movement of a mass (which was attached to the spring blade). The resulting wave function as seen through an oscilloscope hooked up to the PD would give us the necessary data to calculate Q and b.

Given these sets of data, we can reference (this) to find that Q=4.53 f0 T_1/2. Here T_1/2 is the "decay by half life of amplitude", or the time it takes for the amplitude to be half of when it begins, and can be checked by plugging into the equation and seeing if the resulting expression is true: Amplitude( T_1/2)/ Amplitude(t@0) = 1/2 .

TO DO list:

-- Take the oscilloscope data and figure out how to calculate  T_1/2, so that Q can be calculated

-- Think about how to calculate b

Our project is to set up a basic Michelson interferometer to measure and characterize the crackling in blade springs.  That crackling signal will likely be buried under other sources of noise and other parts of the signal, so we will use a chopping technique to extract the crackling signal.  We first set up a Matlab simulation of the chopping technique using a constructed "crackle" signal.  The code and explanation of that simulation are attached.

ETA: changed such that crackle = delta x(t), not delta x(t) - x(t).

I figured out how to plot the graphs given data points gathered by the oscilloscope.Results have been published below.... 

NOTE: there are two blades ("Romulus" and "Remus"). There are two plots per blade: the one with the noticeable sinusoidal shape will be used for Q calculation (see here), whereas the one which looks like a compressed version thereof helps us see how the amplitude of the oscillations decreases over time, exhibiting the "damped" motion, from which we will somehow calculation b.

I had an idea for calculating T_1/2: if Amplitude( T_1/2)/ Amplitude(t@0) = 1/2 . is true, then I just need to find a maximum in the y values (in the voltage data for the graph, since it is not a smooth function), find the closest minimum, then take the difference. This would give me some point near where the amplitude is at "zero". Then all that would have to be done is to find the corresponding x values (time, in seconds) to this maximum and middle "zero" point, and subtract these time values to get the T_1/2 value. It's pretty tricky to implement in MATLAB.

Somehow that doesn't seem right though. If one tried to visualize that, wouldn't it seem like we were just measuring the time interval it takes to get through 1/4 of the wave's period? I don't think I understand what is meant by T_1/2....

Today we started to set up the experiment which will eventually allow us to characterize the noise of the blade spring crackling. The configuration was an analog of the final configuration, where a controlled voltage over a PZT was used as the driving force on the ETMX only.

The first plot, labeled "Spectrum 1" represents the power spectral density plot of the all the noises prevalent in the configuration (i.e., seismic noise, shot noises, fluctuations due to air currents, etc).

"Spectrum 2" is similar, except that the only noise present is 'dark noise', which is the extra signal the PD gets when the laser beams are blocked from hitting it. This 'dark noise' can be thought of as some sort of background noise.

By observation, we can compare the orders of magnitude at which both the sum noise and dark noise curves exist.... Spectrum 1 is around the order of ~10^-3 to 10^-2  whereas Spectrum 2 is around the order of ~10^-5. This confirms that the dark noise occurs within the range of values of the sum noise.

 Measurements taken on June 21.

Data taken June 22

We measured our initial set-up (with mirrors on PZTs, not masses on springs) noise and found a conversion factor from volts on the spectrum analyzer to meters (distance moved by the mirror).

Notes: I'm assuming that peak in the dark noise is from the lights, even though we have a plastic bin around the set-up.  Also, I used the most common wavelength value for HeNe lasers (from wikipedia).  I will confirm this value later.

ETA: Noise curves have been added for the PZT set-up.  Different curves are at different AC amplitudes and AC frequencies.  The curves do not change much in the 10-100Hz range, with varying low AC frequencies).