Tonight I succeeded in using the M2 ISS readout board and the 3-mm diodes to do some real intensity stabilization using the SiFi test setup.

First, I built a foam box to use as a temporary enclosure for the PMC and diodes until we get our real box finished:

There are holes for the input and REFL beams, and the diodes are held with makeshift mounts that clamp down on the sockets. Clearly, these aren't as stiff or stable as what we're having built, but they do the job for now. There is a steering mirror before the 50/50 BS so that the position on each diode can be adjusted separately with ease.

I didn't want to open the Chachi ISS box can of worms yet, so I just built my own temporary breadboard circuit. I had done some preliminary SR560 locking, so I knew roughly what I wanted, and I measured the modulator -> PD transfer function again today and verified that it was flat well above 100 kHz. I made a 2-stage pole/zero-style circuit, with a double (removable) pole at 300 Hz and a zero at 10 kHz to bring phase back to -90° around the target UGF of 100 kHz. It looks like this:

I wanted to DC couple, so I came up with an idea to pick off the stable 5-V bias supply from the M2 board and sum it with the (negative) output of the in-loop PD in the first stage of the servo. I had some current-related issues with the summer at first, but these went away when I increased the input resistors a bit (n.b., to fix the gain I had to change some other components, and as a result the controller TF is actually slightly different than shown above, but not much).

Hooking it up, it locked right away with the expected UGF of near 100 kHz (not yet measured, but inferred from the transfer functions and spectra). Here is the stabilization result:

As you can see, the out-of-loop signal is stabilized to the shot noise level (which is  higher than the bare shot noise for half the beam due to the well-understood correlated noise imprinted by the loop) from about 5 kHz down to just below 100 Hz. Below this, there is clearly some differential environmental noise between the PDs. I did some beam scanning to try and minimize with some success, but not much. I'm not sure what the coherence below 20 Hz indicates---the in-loop signal is suppressed to below the measurement noise level, while the OOL signal exhibits excess differential noise, so I don't see why there should be any coherence.

In any case, this is a nice verification that:

• The M2 readout board works with real optical signals
• The intensity feedback system for the SiFi experiment works
• The 3-mm diodes (save for the one bad one---see CRYO:1207) behave nicely, at least for these relatively low powers
• The PMC -> PD scheme shows promise for our future tests with nicer hardware

[Nic, Zach]

Nic got a Glasgow-style cantilever from a group in Taiwan, and a quick test in the rapid cycle chamber showed that it had pretty low loss, so we are running it in the cryostat now. As a reminder, these are the rough dimensions of this style cantilever:

Below is a photo of the box it came in, showing the actual 92-um thickness of this sample, as well as a shot of it in the vacuum chamber. For some reason, this particular sample's clamping tab did not fit in the groove that Nic had built into the clamping block for the other Glasgow cantilevers, so I had to mount it to the side against the flat faces of the clamp (as I've been doing with our larger samples).

This evening, we transferred it over to the cryostat and restored all the electrical connections for what will hopefully be a fruitful cryo run. Here is a ringdown of the fundamental mode (~106 Hz) at room temperature:

The measured decay time of 41 seconds corresponds to a Q of around 14,000, which is about as good as we expect at room temperature. This sample is probably better than our other ones for at least 2 reasons:

1. It is made from a better-quality (FZ) wafer, and
2. It has been manufactured monolithically with a thicker clamping tab, which our modeling suggests is a very effective way to evade clamping losses by keeping strain energy within the silicon.

Given that we didn't see much improvement at all with our other samples when going to low temperature, I believe (2) is by far the biggest effect. The Glasgow wafers only have the clamp-region thickness extended to one side, which is modelled to be worse than if you go both ways, but it is still much better than we can do with our discrete sandwiching.

I filled the LN2 reservoir and the volume is cooling overnight. I did some rough ringdowns at a point when the steel block was registering around 160 K and found greatly improved Qs already (approaching and perhaps exceeding 10⁵). We will continue to make measurements tomorrow.

[Nic, Zach]

We measured the Q of the fundamental (~106 Hz) mode of the Taiwan cantilever in two ways. First, we used Nic's active steady-state method, and then we did a traditional ringdown. The results seem to agree, but the precision of the first method is much better due to the dynamic range of the readout for this mode: the motion becomes nonlinear at an amplitude only a few times greater than the background excitation level. Over a ~4-hr average, the loss is measured to be 1.45 x 10^-6 ± 2.9 x 10^-7, giving a Q of ~6.9 x 10⁵.

Here is a plot of the instantaneous phi from the calibrated control signal. This data has already been fed through a ~1-hr lowpass, and then the data from the initial settling time has been truncated away. The mean and standard deviation of the rest of the points are what is reported.

After this measurement was made, we shut off the servo and allowed the mode to ring down. Here is that ringdown, along with a predicted range of theoretical curves using the result from above. As you can see, they are fairly consistent with what is measured, considering that the system quickly reaches a regime where it is excited by the environment (that is, only the initial part of the ringdown, where the agreement is good, is very trustworthy).

This Q is a couple orders of magnitude lower than what is expected for this mode at this temperature, but it is also only a factor of 2-3 worse than the best measurements using a similar apparatus at Glasgow (to my knowledge).

It bugs me that we don't seem to have any information about what steel looks like at low temperatures. Given my COMSOL strain energy modeling, the energy ratio for this mode is about 3 x 10-^4, so this could be explained by clamp loss if the steel Q is as low as a few hundred. I'm looking into other modes to try and support or refute this hypothesis; since different modes have different energy ratios, we may be able to see what's going on. In parallel, I'm asking Matt and others to find out what is really known about cryogenic steel.

The most recent measurements on the Taiwan-sourced Glasgow-style cantilver (see CRYO:1213) are encouraging, but the best Q measurement at low temperature is still a couple orders of magnitude worse than what is theoretically achievable, and about one order of magnitude worse than our conservative clamp loss estimates. Also, I've done some measurements on other modes (that have different expected clamp loss contributions due to the relative strain energy ratios) to try and sort out what is going on, with little success. Finally, some modes---including the 2nd bending mode at ~650 Hz---exhibited very low Q for no known reason.

One thing I thought about is that, since the Taiwan cantilever did not fit in the groove that was built into the block for the Glasgow-style cantilevers and therefore is just sandwiched between the two large pieces making up the clamp (see CRYO:1211), the clamp is likely pushing down at somewhat of an angle, which could lead to all sorts of non-idealities. Since the other Si samples we have lying around are roughly the size of the clamping region of this cantilever (~300-500 um), I opened up the cryostat today and reclamped the cantilever using a spare broken-off 300-um-thick cantilever piece as a spacer on the other side:

Pumping it all back down, I immediately measured Qs a bit higher than what we saw last time around at room temperature. The last measurement I made before leaving was tau ~ 135 s ==> Q ~ 46000, though it had been increasing up to that point, likely from the residual pressure, which was at ~10^-3 Torr when I left. Compare this with the Q of ~14000 from the last time around, though admittedly I did not record the pressure at which this was measured.

On Tuesday night, when I added the Si spacer to the clamp, I measured a Q of ~46000, but I noted that it had been increasing up to that point, likely due to the residual gas damping (see CRYO:1214). Last night, I made another measurement and found it to be much higher, at ~1.2 x 10⁵ (tau ~ 350 s). This is much better than we have seen at room temperature thus far, so it looks like my spacer addition has helped.

I remeasured this an hour or so later and saw no appreciable increase. I checked again today and it appears as though it may have increased slightly, but it was hard to say for sure due to higher environmental noise. Really, we need the steady-state ringdown to make a good measurement at this level.

The LN2 dewar was refilled today, so I filled the cryostat and we'll see how it looks at low temperature tomorrow.

The cantilever had cooled to around 100 K by this morning, so I set up the mode ringer and began an active measurement on the fundamental mode. The online loss angle measurement for a 3-hr period beginning around an hour after lock is shown below (this is the control signal filtered by a 2nd-order low pass at 0.2 mHz.

As you can see, the loss is hovering around 3 x 10^-6, giving a Q around 3 x 10⁵, which is slightly but significantly lower than what we measured before I added the Si spacer to avoid skewing the clamp (CRYO:1213). I would chalk this up to the spacer actually making the clamp worse, but we did in fact see a huge improvement at room temperature (CRYO:1216). So, like, what the hell man?

I've left it running to collect more data over the weekend. I haven't gone over the temperature readout/control system with Nic, so I set up a simple temperature readout in the meantime so that we can have at least a coarse Q(T) measurement as it warms. To do this, I simply put a 1k resistor inline with the RTD and put 5V across with a lab supply. The second set of RTD leads goes to the temperature readout input in the digital system, so this is now just a DC readout of the voltage across the RTD. The lockin input channel X1:SCQ-TEMPERATURE_LOCKIN_DEMOD_SIG_OUT is calibrated to volts, and is equal to 5 V * R_RTD/(1k + R_RTD).

I reported in the replied-to entry that the Q of the Taiwanese cantilever at low temperatures actually appeared to have gotten lower at low temperatures, relative to the case before the Si spacer was added to the clamp to avoid skewness. However, the data from the longer run over this past weekend (see the ~20-hr stretch below) seem to suggest a Q not significantly different from that measured in CRYO:1213.

Interestingly, the online phi measurement starts out at the higher level I indicated in the previous post, but then slowly approaches a level not inconsistent with the ~1.5 x 10^-6 number from before the spacer addition. The title is misleading, as the system actually approached a minimum temperature of ~90 K on Saturday, but the thermoelastic noise prediction is roughly flat over this temperature band, so that shouldn't be a factor, and the associated deflection from this temperature shift should not be enough to account for this drift via calibration error.

As I discuss in the quote, I had hoped to make a continuous phi measurement as the system warmed leading up to today, but at the time I neglected to consider the thermal deflection, which over such a large temperature swing completely rasters the beam off the QPD. In retrospect, I'm lucky that this effect didn't break the cantilever as the sensing gain was reduced from the misalignment---thankfully, the loop destabilized quickly enough that the watchdog script killed the feedback before anything happened.

So, it looks like we'll have to make this measurement the old fashioned way, point-by-point, which is why I spent time reconfiguring the temperature control today. I'm running an active measurement overnight at 120 K to see if we see a Q bump there.

These things happened over the last few days but haven't been elogged.

Windows installed

I installed the AccuGlass windows into the flanges, but ran into some trouble as one of them cracked. It happened on the 4th one, and I did not use any more force than seemed reasonable given that the glass-on-o-ring contact has to hold the vacuum. I think we might need to get some teflon gaskets to soften the metal-glass contact.

Cavities installed into chamber body, realigned, locked

Despite the above, since we had 5 windows, I finished putting them on the cryostat and went about installing the cavities (mounted to the coldplate and cryostat top) into the main body. I transferred the top from the open-air test rig (see CRYO:1244) into the chamber body, then positioned the whole assembly according to some reference marks I made on the table. Using a fast lens in transmission of one of the cavities, I was able to align the cryostat well enough by hand to see some high-order flashing (the test rig is ever-so-slightly taller than the assembled chamber, so what I saw was mostly TEM_0n modes). With some tweaking of the input alignment, I aligned each beam to its cavity's TEM₀₀. Here is one cavity locked while in the chamber:

It was all very deterministic, so that inspires some confidence in the procedure.

Other stuff

I also tuned up the W RFPD to match the temporary testing TF I set on the E one (readout 30 MHz, notch 60 MHz---see CRYO:1242), and used it to lock one of the cavities.

I attempted to modify the PDH2 from gyro configuration into a useful one for testing here, but I may have changed too many variables at once; I knew I wanted some more gain in the ~1 kHz region, so I used an extra P/Z than what is available with the uPDH box to do so. Somehow, I'm not getting any good locking action even though the TFs are identical at DC and near the UGF of the known working loop at ~30 kHz.

Also, I remembered that there's no hope to do any good locking with the other (standalone, LDC201C) laser driver that I have, since its bandwidth is about a kHz (CRYO:1205). I'm asking Dmass if it's OK if I trade him my standalone temperature controller (TED200C) for one of the integrated units (like the one I have one of), since he's only using the temperature controller half.

[Matthew, Nic, Zach]

Over the last few days, we've done 2 significant measurements:

Flower Mound propeller resonator sandwiched between sapphire washers at room temperature

We clamped one of the propeller resonators between two of the new, larger sapphire washers, using the cylindrical mount in the room-temperature chamber. Unfortunately, the inner diameter of these washers is also too small to fit around the lip in the bottom part of the clamp (as it was with the old washers---see CRYO:1197), so I needed to reuse the steel washer that I fashioned before.

We measured the Q to be a rather low ~2,000.

It's unclear exactly why it should be so bad, but we have never measured a good Q with these thick, uniform resonators. It may be caused by the effect I noted in CRYO:1198, where the mode is highly coupled to the relatively soft and high-loss PEEK. It could also be from clamping non-idealities from the fact that the bottom sapphire washer does not sit flush on the clamp over its whole surface.

Matthew has designed a revised version of the cylindrical clamp. The main improvements are 1) it is bigger and therefore should not allow for as much coupling of the silicon modes to the PEEK base, and 2) it will be made to accomodate the sapphire washers we have. Nic put the order in and it should be in late next week.

Painter fab prototype 2, after surface treatment, in new polished block clamp at ~100 K

I finally got a chance to meet with Justin again to do a surface treatment of the second prototype resonator (the one in CRYO:1264). This consisted of 10 minutes in "piranha" etch (this is 3:1 hot H₂SO₄ and H₂O₂), followed by 10 minutes in room temperature HF. According to Justin, this is a passivation treatment that only affects the surface chemistry of the resonator; it does not repair the surface in terms of the roughness. Since our leading hypothesis was that our low Q for this thin resonator is related to the rough surface, I didn't have high hopes for a big improvement.

Matthew and I put it in the cryostat yesterday and cooled it overnight. This morning, we measured a Q of around 30,000 at 100 K. This is about a factor of 2 lower than before, which is certainly surprising...

Matthew did some real data analysis on this ringdown, and I'll have him attach it to this post.

Update (Matt):

I just attached a couple data analysis plots that I used for determining Q. I calculated a Q of ~28,000 at f₀ = 145Hz. I'm taking an FFT of the ringdown measurement, filtering around the resonant frequency (in the time domian), and then fitting an exponential to the filered signal to measure tau. 

I put together a quick COMSOL model of the Painter 2 resonator that Zack, Nic, and I were testing earlier. The first mode was calculated to be at 150Hz, which matches our data from 7/14 very closely. See below:

Next Dmitry and I began working on simulating the full pinwheel setup, including the SS clamp, PEEK base, and sapphire washers. We calculated the energy stored in each domain using the solid.ws variable in COMSOL as per Nic's suggestion. This involved integrating the energy density over the volume of each mount component. The results are shown in the table below for the first four modes of the resonator:

COMSOL does not perform any displacement calibration by default when operating in frequency domain, so only the ratio of energy values should be considered, not the values themselves.  The important thing to note here is that all of the other materials store much less energy than the si pinwheel. This means that the loss in the PEEK base would need to be 10^5 or 10^6 times larger than the cantilever loss in order to be comperable to cantilever losses. Same thing goes with the SS and sapphire, which are less lossy to begin with. Does this indicate that loss in the PEEK base isn't a problem? If so, we wouldn't benefit from making the switch to macor.

We also did a rough calculation of the Q of the 2.4in pinwheel resonator. Using the formula for thermoelastic loss in the Glasgow paper, we estimated a Q~50,000. We are clearly not there yet, so there must be other important loss sources present in the sysem. Clamping?

Dmitry did some more involved analytical work in COMSOL modeling thermoelastic loss in the pinwheel system. His results show the estimated Q at different modes if we are only limited to thermoelastic losses:

As expected, these results are all less than the 50,000 estimate since the calculations take the entire system (mount, clamp, washers, etc.) into account, but they're still much greater than the things we've been measuring. The Q decreases as f increase since in this regime the loss angle phi goes nearly linearly with drive frequency omega. What does this tell us? We probably aren't limited by thermoelastic losses.

Today we also reclamped the pinwheel resonator and started measuring the 2in cantilever. We did some quick ringdown estimates at room temp and 10^-3 torr and didn't see any significant changes in the Q. We'll take a couple more careful measurements at lower pressure tomorrow but I'm skeptical that we'll see any improvements from the other pinwheel ringdowns.

I took ringdown measurements of 1.8in cantilever at 4.3*10^-4 torr:

Also I tried to further tweak the dimensions of the pinwheel in COMSOL in order to make it predict the frequencies closer to the measured ones. While being able to match the frequencies of the first order and the second order modes, the torsional mode was always far off. So I switched to the anisotropic model – (100) wafer, cantilevers along [110] and [-110] axes.

Using this model, COMSOL predicts the following frequencies for the pinwheel of thickness 0.0112in with cantilevers of length 2.44in, 2.2in (*no experimental data for this one), 2.04in, 1.86in an width 0.385in:

For the longest cantilever, the frequencies are very close to the measured ones.  The ratios of total strain energy (E_clamp), stored outside of the pinwheel, to the strain energy stored in the pinwheel (E_pinwheel) are the following (for the longest cantilever):

*Actually should be regarded as a.u.

Seems that the low Q cannot be explained simply by the energy leakage into the clamping (however any kind of friction is not accounted for in the model) and this source of damping also does not explain different Q's for the three modes (??? - the leakage is the biggest for the second mode and the smallest for the torsional one - that is conistent with what we see in the experiment, but the relative change is too small).

Then I checked whether we get measured Q's if, suppose, we have a “very bad” surface. I used the first model (only pinwheel and sapphire washers) and added thin (19µm) layers to each surface of the pinwheel. Assuming the bulk loss factor is equal to the thermoelastic loss, COMSOL predicts the following quality factors (for the longest cantilever):

Low Q could be caused by a very lossy surface, but still this does not explain lower Q for the second mode and higher Q for the torsional one (??? same as for the clamping loss - the change in Q is consistent, but too small compared to the measured one)

[Nic, Zach, Matthew, Dmitry]

On Tuesday we opened the cryostat and switched the Painter 2 cantilever for the Taiwanese one (CRYO:1211). ESD seemed to be a bit too high above the cantilever and was turned upside down (the cantilever was facing the wrong side of the ESD). Nevertheless it provided enough force to excite both the first mode (106Hz) and the second one (663Hz).

(*) it is an estimate, the gauge actually showed 10^-7 torr

On Wednesday we opened the cryostat again, cleaned the surface of the cantilever (there were several smears on both surfaces), lowered the ESD (so that it is now closer to the cantilever and the side with electrodes is facing the cantilever – see the photo below) and Zach installed the guiding rods for the clamp.

(*) it is an estimate; evacuation time - ca 1.5 hours, the gauge already showed 10^-7 torr.

(**) also an estimage - evacuation time - about 12 hours.

Either the surface became actually dirtier or more dusty, or the previous clamping was better.

Today with Nic's help Matthew and I were getting ourselves familiar with continuous Q measurements technique. We were able to repeat continuous Q measurement for the first mode and got close to setting up one for the second mode. The ESD is close to the node of the second mode, so the excitation is not as effective as for the first one, and the frequency is higher so it is difficult to get UGF high enough.

Nevertheless we were able to perform “proof of concept” experiment where we were driving both frequencies simultaneously and were able to change amplitudes of both modes independently.

After that we once more estimated the current quality factor values:

160Hz  - 3.2*10⁴ 

663Hz -  7*10³

I made a COMSOL model of the cantilever and the clamping. For some reason (most probably the big aspect ratio of cantilever's length to it's thickness) COMSOL's results are mesh dependent, no matter how coarse/fine the meash is. 3D model failed, but 2D one gives consistent results using specially designed mesh (it predicts correct 106 and 663 Hz frequencies and correct quality factors given some bulk loss factor value). Using this model I calculated the dependence of the clamping loss (through energy leakage) on the length of the part of the cantilever that is clamped (L=0..10mm). COMSOL predicts minimum loss if almost all of the cantilever's thick part is clamped  L(min)~9.7mm.

Yesterday Matthew and I opened the cryostat in order to reclamp the cantilever (hoping to get higher Q) and move the ESD to make it face the middle of the cantilever so that the second mode would be excited more effectively.

In attempt to compensate the angle between the upper and the lower parts of the clamp and make the pressure distribution more even, we removed guiding rods and replaced them with screws:

We tried ti clamp it with 4 screws using the central holes, but the holes in the upper and lower parts of the clamping did not match so we used two guiding rod holes instead. We also left about 1/10 of the thick part of the cantilever outside the clamping. This resulted in frequencies shift (105.5Hz for the first mode and 655Hz for the second) and bad quality factor:

We failed to make continuous Q measurements for the 2nd mode, as we got almost no excitation, and the effectiveness of the ESD at the 1st mode also was poor.

Today we reclamped the cantilever (holding it now by almost the full length of the thick part - minimum clamping loss according to COMSOL, two front screws only. Using 500um Si shim we were able to compensate the angle between the upper and the lower part of the clamp). We also removed part of the ESD support to make it closer to the cantilever:

The quality factor of the first mode is worse than the one achieved with the first clamping (CRYO:1276), the one of the second mode - better. We were able to repeat continuous Q measurements for the second mode, got excitation for the second mode but the UGF is still too low.

I have spent the last couple days working on measuring several of the Taiwan cantilever modes simultaneously using the continuous measurement system. This has been more difficult than just opening two instances of the moderinger program and filtering each instance for a separate mode. The first two modes that I'm measuring are clearly coupled: I can hold both modes at two set amplitudes, but as soon as I disengage the 1st mode's lock, the 1st mode begins to increase in amplitude as the 2nd mode feeds energy into it. This results in clearly incorrect calculations such as negative losses and Q's since energy is being pushed into the mode from a source other than the ESD drive signal. I have been playing with bandpass filtering around each drive output to successfully lessen this effect, however the coupling is still significant enough to prevent accurate Q measurements. This is an ongoing problem that I haven't been able to solve yet. ***Edit: I think I fixed it, see CRYO:1285 -Matt

I took some more ringdown and continuous measurements of the first two Taiwan cantielver modes independently (300K and gauge reads 2e-7 torr):

Both modes have a higher Q than when Dmitry and I took measurements on Saturday, and the fundamental mode has improved significantly. I think that the most likely cause is lower pressure. We left the pump on all week and the gauge isn't accurate at very low pressures so we are likely at a lower pressure than we are reading.

On a different note, we got a fresh dewar of LN2 today and Nic set up the temperature measurement system. We were able to cool the cryostat down to 160K. I set up the continuous measurement for the first mode and will be recording Q as the cryostat temperature equilibrates overnight.

I took the first cryo measurements using the Taiwan cantilever on Friday (7/31/15). The cryostat had reached a stable temperature of 97K. I performed two continuous measurements of the first mode--- the Q is large enough in this regime that ringdowns start to become impractical.

These are fairly close results to previous measurements (CRYO:1267) and I think that any improvements are due to improved clamping. We noticed in previous setups that the the upper SS plate wasn't flush with the bottom, so we inserted a si shim. The clamping looks much more even now:

I also noticed that the measurement looked much better at the higher amplitude set point. After playing around with the set point for a while, I think that the best set point is around 50-70% of the amplitude when the displacement signal begins to look nonlinear.

The most interesting result from this experiment was that the second (663Hz) mode's Q decreased dramatically at the lower temperature. I couldn't make a continuous measurement on the mode but I estimated the time constant tau to be ~2s with a ringdown, so the Q is ~4,200. This is almost an order of magnitude worse than previous measurements (CRYO:1278). I talked to Nic about the disparity on Friday and he suggested that the clamping may have changed due to clamping materials' different thermal expansion coefficients. It seems strange to me that second mode would suffer from cryogenic clamping changes so much more than the fundamental mode, so there is clearly some work to be done thinking about these effects. 

Today I was able to successfully perfom continuous Q measurements on the first and second modes of the Taiwan cantilever simultaneously for the first time. The results and system parameters are shown below:

All measurements were taken with a 150V bias voltage on the ESD. These results should be compared to the ringdown and individual continuous measurements I took earlier today:

These simultaneous measurements look very consistent with what I have measured in the past.

I think that there were 3 important parameters that I needed to change in order for this measurement system to work correctly. The first change was doing more aggressive low pass filtering at the RMS stage. Previously this stage was being low pass filtered at 500 or 1000 Hz, which resulted in rapid amplitude oscillations appearing on the drive signal. This was especially troublesome when I was trying to take continuous measurements on the second mode. I moved the low pass filter down to 50Hz and the results were much better. The second change was tuning the phase offset at the sinphase/differentiation stage. Rather than using the 90 degree phase shift, I played with other phase offsets. I settled on a 120 degree shift for the first mode and a 55 degree shift for the second mode by implementing other filters at this stage. I suspect that the original 90 degree shift wasn't optimal due to other accumulated phase shifts throughout the system. Regardless, the phase margin for each mode appears to be better now. The third change was bandpass filtering the output of each drive signal around the resonant mode frequency. Previous PSDs of each drive are shown below:

The MODERINGER and MR1 channels are used for driving the fundamental mode and second mode respectively. Although the peaks are where they should be at 106Hz and 663Hz, I was worried that a significant amount of energy might transferred to the wrong mode by each drive. After bandpass filtering, the outputs look like this:

Once I made these changes I saw much less coupling between modes than before (CRYO:1283). For example, I can now drive one mode and see that the other mode remained unexcited.

In order to check for coupling in the new system, I would place both modes in lock then let one mode ring down and look at the loss of the still-locked mode. For the traces below:

Blue = 1st mode amplitude

Cyan = 1st mode loss (Q^-1)

Green  = 2nd mode amplitude

Red = 2nd mode loss

Yellow = cryostat temp. (not important for these experiments)

In the plot below I had previously locked both modes. I let the 1st mode ring down at t = -2 minutes and look at the effect on the 2nd mode. The 2nd mode loss changes slightly (~20%) but returns to the correct value when the 1st mode decays completely. We can also see that the 1st mode decays naturally, even though the 2nd mode is still being driven.

I repeated the experiment, this time ringing down the 2nd mode and looking at the loss of the 1st mode. I let the 2nd mode ring down at t = -1 minute We can also see that the 2nd mode has a much lower Q since it decays much faster. Again the loss only changes slightly and returns to the correct value when the 2nd mode has decayed away.

I also checked the robustness/speed of the system by putting both modes in lock and then exciting the cantilever by hitting the table and cryostat with a screwdriver. At t = -2 minutes I hit the table. We can see that both modes are temporarily excited, but are quickly driven back to the amplitude set point. The calculated loss of each mode take much longer to settle, but do eventually return to their steady state values. This is a good sign, in previous systems I had occasionally been in a semi-stable regime where an external input (like a screwdriver) could rail the drive output.

Today I was able to get the NDS client working in MATLAB with Chris' help. This allows us to fetch online and offline data from the server and perform separate analysis afterwards. The motivation behind this is to eventually develop a fully automated measurement system where we load the cantilever sample into the cryostat, add LN2, and run the MATLAB scripts to measure the Q of several modes as a function of temperature.

I have been consulting this guide: https://www.lsc-group.phys.uwm.edu/daswg/projects/nds-client/doc/manual/index.html

I made a couple plots using data from when I cooled the cryostat down last week to test out grabbing data from different channels. These are plots of the temperature, RMS fundamental mode amplitude, and Q for 6 hours. Clearly I need to do more low pass filtering on the Q data, since the quality factor should never be less than zero. The amplitude set point was c = 500.

No data was being written to frames / test points were broken for the X1CRY model. We restarded the daqd and ran the startallmodels script on cymac1.

There are still many red lights and error indications on the medm master screen which we could not resolve.,

The timing error (x4000) is one of the problems.

The turbo was touching the chair by the workstation and very easy to kick the power button. I closed the sifi vacuum,. turned off the pump, and moved the pump back towards the table.

It seemed to be in the bad location because of ground flatness (lack thereof), so I shimmed it up with some aluminum foil.

[Brian, DMass]

We've been trying to use the Wilcoxon accelerometers to measure the seismic noise in the cryo lab. We were able to see a signal from a shaker next to the Wilcoxon, however we could not see any seismic noise at low frequencies.

We went through and checked all the channels on the Wilcoxon amplifier. Of the 10 channels, we found 4 of them to be completely bad. Channels 4 and 8 looked ok. 9 and 10 had similar sensitivity to 4 and 8 but had a higher noise floor. 2 had the same noise floor as 4 and 8 but lower sensitivity.

We also found that the gain switch on the amplifier is bad. Flipping it to gain 10 increased the signal by ~5x but the noise floor by ~20x.

We tested a different cable and found similar results between the two. We also tried 5 different Wilcoxon accelerometers and saw fairly similar results across the board. There was a difference of ~40% between the least and most sensitive aceelerometer. The two most sensitive accelerometers are the ones labeled A and D. However they both still could not see any seismic noise.

We then tried using some of the tiny Kistler accelerometers. The Kistlers saw what looked to be seismic noise, even though the Wilcoxons are rated 10x more sensitive.

Tomorrow we will be going to the 40m to try out our Wilcoxons on their amplifier to try and determine where the fault is.

[Brian, DMass]

Today we went to the 40m and checked our Wilcoxons on their setup with Eric Q. All of our accelerometers appeared to be working normally.

We then went back to the cryo lab and tested some of the parts from the 40m with our setup. We found that our power supply was too low voltage. With the correct power supply, our amplifier and accelerometer worked normally. We found a power supply in the cryo lab that works and returned the parts we borrowed to the 40m.

I have started taking measurements with one of the new Glasgow cantilevers (CRYO:1292). First I took some pictures out of curiosity and to see what the cantilever surface looks like before we handle them:

I think that the surface quality looks pretty comparable to the Taiwan cantilever, but I'm not being at all quantitative.

I also replaced the pressure gauge, since the old gauge wasn't giving accurate readings at low pressures (gauge would get stuck and display 2e-7 torr).

I began the measurement process by looking at the impulse response of the oscillator to find the first couple modes:

It looks like the first three modes are at 66, 417, and 1198Hz. I can easily excite the first two modes, so I'm focused on measuring the quality factor at these resonances. The first mode has a very long decay time (~450s) and starts to get clipped at medium to large amplitudes, so I wasn't able to measure the Q with a ringdown very accurately. I suspect the clipping is due to a bad clamping position, but I'll play with the optics some more to see if I can get better alignment. I measured the decay time of the 2nd mode to be ~80s, so Q=1e5. Pretty comparable to the Taiwan cantilever. This is at room temperature and 2.6e-4 torr. The filtered 2nd mode ringdown is shown below:

I tried to set up continuous measurements on the first two modes to look at Q as the pressure continues to decrease, but got the error message:

Cannot open related display:

/opt/rtcds/userapps/CyMAC/medm_screens/x1scq/UGS.adl

Check EPICS_DISPLAY_PATH

when I try to set up the UGF block on the amplitude locked loop stage. It looks like the UGF.adl file in this directory has gone missing! I spent some time trying to fix this but was unsuccessful, so I'll ask around. Hopefully I will be able to take some continuous and cryogenic measurements tomorrow.
 

I was able to get the continuous measurements working today for the first two modes of the new Glasgow cantilever:

I actually had a lot more trouble setting up the system for the first mode this time (whereas the 2nd mode was significantly more difficult using the Taiwan cantilever). I'm guessing it's because the mode frequency was lower than I'm used to tuning, but I'm not totally sure. As described earlier, I also can't ring the mode up to a very large amplitude without running into nonlinearities.

Tonight I'm cooling the cryostat with LN2 so that I can make cryo measurements tomorrow. Dmass requested that I turn off the pump tonight so that he can do some seismic noise measurements, so I'm also going to be checking out how well the pressure is maintained overnight with the crystat sealed.

Update: Clearly we don't have a good seal. The gauge read ~2e-5 torr while the pump was running, but has now settled to 1.1e-2 torr since I turned off the pump and closed the valve. I'll see how well I can cryopump when I add LN2.

After cooling down the cryostat to ~90K, I started trying to take continuous measurements for the first two modes of the Glasgow cantilever. However, this time I noticed some coupling between the modes.

In the plot below, the blue trace is the first mode amplitude and the teal trace is the second mode amplitude. At the start I have the first mode locked at an amplitude of 2000, and the second mode is free-running (excited from earlier measurements). At -30 seconds I lower the set point of the first mode to 1000, and we can see that the amplitude of the second mode increases even though it isn't locked. I decrease the first mode amplitude even further at -10s and the same effect happens again.

This continues to happen even after I bandpass filter the drive output (this resolved the issue for the Taiwan cantilever). The effect is much less significant when the second mode is at lower amplitudes, so I usually take Q measurements on the first mode after I reduce the second mode to as low an amplitude as possible. 

I tried to realign the optics in order to reduce the clipping/nonlinearity I see at higher amplitudes, but didn't see any significant improvements.

Tonight I am going to do a long Q measurement for the first mode. I ran into some problems trying to do this measurement earlier today, but Zach suggested that I average out the measurement longer by using a slower low pass filer (longer cantilever ringdown tau ---> longer averaging time needed).

I've been trying to model the cavities in the second paper Dmass linked (http://journals.aps.org/pra/abstract/10.1103/PhysRevA.75.011801). I used the same parameters as the open circle trace in their figure 3. There was some ambiguity in how the cut depth is defined, especially in the vertical direction. I assumed a square cross section with sidelength c and one corner at (-R,-R) was removed, with R being the spacer OD. Maybe that's not exactly what they did but it should be fairly close. Their cavity is made out of ULE glass, but I kept mine made out of silicon.

The paper stated that their mesh had ~6000 elements which is about the number that COMSOL gives with an automesh setting of 5 ('Normal'). I looped over the support separation z just as they did to try and replicate the results of figure 3. The result is seen in the first attached figure. There is some qualitative agreement but mine is much less smooth than theirs. The numbers are on the same order as well.

I tried refining the mesh to 3 ('Finer'). The result is shown in the second attachment. The trace is much smoother except the large jump at 10mm.

For these graphs I was using a single point as the support, just as in the paper. I tried changing to a 2.5mm x 2.5mm support. Note that the coordinates are to the center of the square so at z=5mm there is still 2.5mm between the edge of the support and the edge of the spacer. The results are shown in the third and fourth attachments at mesh 5 and 3 respectively. Once again, the finer mesh smooths out the jumps in the trace. However in both of these we see a large change between 5mm and 10mm, just as in the second plot.

Even though the results aren't exactly the same as the paper, I can verify that there is a zero crossing with the slotted supports somewhere in the middle of the range. I'm going to look into what's going weird at 10mm and 5mm. I'll also try changing to ULE glass to see if I can match their results more exactly. Another idea I had is to try to loop over the cut depth to see if I can get similar results to our original support geometry near c=0 and try to see why we only see a zero crossing very close to the edge of our cavity.

After talking to Zach on Friday I learned that I need to do much longer averaging on phi/Q measurements. Assuming that the Q of the cantilever is around 1e6, this gives a timescale of around 8 hours. With this in mind I turned on the continuous measurement system for the first mode of the Glasgow cantilever on Friday afternoon around 5 and planned to collect data over the weekend. The mode stayed in lock for around 20 hours. I'm not surprised that I lost the lock as the clamp began to warm up, I usually need to realign optics when the cyrostat makes large changes in temperature. When I came in Monday morning I had completely lost the amplitude signal.

The start time of the plots is at 5pm on Friday. The amplitude set point is 1500.

The average phi was -2.5e-7, and this was after a lot of slow low pass filtering so clearly something is wrong. I also don't know what's going on with the jumps in temperature.

I made a COMSOL model to check our assumption that the coupling of vertical motion to the cavity length can be regarded as DC.

I used a prestressed frequency analysis study in COMSOL to evaluate this.

• First a static gravitational force is applied to the cavity.
• The solution to this is used as the initial condition for a frequency analysis where a a sinusoidal vertical force of g/10 is applied at various different frequencies.
• The cavity supports are fixed constraints in both simulations.
• I used the maximum amplitude of displacement in the direction of the cavity axis at several points on the mirror to calculate the effect of displacement and tilt on the cavity frequency just as Evan did in his model here: http://nodus.ligo.caltech.edu:8080/Cryo_Lab/1031

The results are plotted below. The cavity parameters were L=4" and R=1", the supports are at 60º from the horizontal and seperated by 2".  Up to about 1 kHz, the response is flat at 308 kHz/g which matches nicely with the number Evan's DC model got. According to a COMSOL Eigenmode analysis of the cavity, the sagging mode is at 6.5 kHz which also agrees with this result.

This is half of a very nice transfer function plot - See if you can get COMSOL to give you the phase information too and plot that in tandem (e.g. https://en.wikipedia.org/wiki/Bode_plot). 

Also, I'm mildly curious how the frequency shifts with support distance. Are you using material parameters for the supports, or treating them as an area of constrained geometry?

After playing around a bit more with Comsol, I realized and fixed some slight mistakes. I added in the phase information. The transitions in phase occur very rapidly and with resolution I was running at I couldn't get any points during the transitions. I used a loss factor of 1e-8 for the silicon (as measured here).

This simulation runs by considering the support points of finite size as fixed constraints.

I'm going to try and check effects of the support positions.

I also uploaded my model to the svn under: /trunk/CryoLab/comsol/cryo_cavity_sagging_transfer_functions/

I added the ability to loop over the support separation. I checked values from .5in to 3.5in. All of these were with L=4", R=1", and at 60 degrees.

The first plot shows each different support separation plotted together. The remaining plots are the bode plots for each individual support separation. Up until aout 10kHz, the transfer functions are pretty much the same besides the slightly different gains at low frequency.

I also committed these changes to the SVN.

Date: 8/27/15

Cavity Temperatures:

Modematching:

PM Depth:

RAM Depth:

Notes:

My PM depth is much larger than what David thinks it should be (~0.15). I found an old elog post that measured a gamma of 0.17. The carrier voltage was relatively similar to my measurement (1.42 V). However the side band measurement was much smaller (16.2 mV). I'm not entirely sure what's happening here. I made sure that what I was measuring was in fact the sidebands (the signals went away if I turned off the modulation). It doesn't seem like I'm saturating the PDs. I tried playing with the total laser power in the system to see if that changed things but was still getting gammas >0.4.

Date: 8/28/15

Cavity Temperatures:

Modematching:

PM Depth:

RAM Depth:

Notes:

The numbers remained pretty consistent from yesterday. I lowered the power in each beam to see if that would make much effect. My value for RAM depth went up quite a bit for the East cavity, which had the most power in it yesterday. I think this is because with such low power, the dark noise is at a comparable level to the RAM. When blocking the beam, I noticed the AC voltage remained fairly similar.

I also still am measuring anomalously high levels for the PM depth. I'm still not sure why this is.

Date: 8/31/15

Cavity Temperatures:

Modematching:

PM Depth:

RAM Depth:

Notes:

These measurements were taken with the cavities at a slightly higher temperature than before. I didn't actually calculate the modematching of the East cavity before taking all the measurements and didn't realize it was so low. It definitely could have been aligned better.

My PM depth still seems abnormally high.

In order to make sure my measurements of gamma in elog:1310 are reasonable, I went through and calculated what we should expect for gamma. The path to the EOM is as follows:

Marconi: +10 dBm

ZFSC 2-4-S+ splitter (port 2): -3.3 dB

VAT-6+ attenuator: -6.0 dB

VAT-10+ attenuator: -10 dB

ZX60-100CH+ amplifier: +36.5 dB

The total output is 27.2 dBm or 7.24 Vpp assuming 50 Ohm resistance.

This then goes into a resonant circuit built by DMass with a voltage gain of 12x (elog:896), leaving us at 86.88 Vpp.

This then goes into a Thorlabs EO-PM-NR-C3 EOM with a half-wave voltage of 350 V for 1550 nm light. So 86.88 V * pi rads / 350 V = 0.78 rads.

This is higher than my measured values of ~0.5. This calculation doesn't acccount for all other causes of attenuation and assumes every device is working perfectly as stated by the manufacturer so the calculated value should be seen as an upper bound.

EDIT: According to DMass' reply, the resonant circuit voltage gain is probably closer to 8x than 12x. This gives 57.92 Vpp going into the EOM which should produce a gamma 57.92 V * pi rads / 350 V = 0.52 rads. This is consistent with the measured values.

Date: 9/9/15

Cavity Temperatures:

Modematching:

PM Depth:

RAM Depth:

Notes:

These measuremets were taken with the cavity close to room temperature.

I went through and calculated what to expect for the PM depth in elog:1319. I calculated a value of 0.78, which roughly agrees with what I have been measuring. However, just now when I measured it I found the west laser gamma had fallen to 0.13 for some unexplained reason. The east laser gamma is still 0.50.

The x12 on the resonant EOM circuit is assuredly less than x12 because of lossy RF elements. I would assume x8 is around the true value, which probably makes your measurements of gamma consistent with what you expect.

Try fitting a function of the form:

I used the matlab function lsqnonlin to do this fitting. This function fits real valued functions so I had it fit both the real and imaginary parts. The first plot is done with tau=0 while in the second I allowed it to solve for tau. I used the DC gain as the initial guess for G, 1e8 for Q, the location of the maximum for f_res and 0 for tau. The version with non-zero tau would not converge on the default settings so I played with the parameters until it converged. Both solvers also ended up giving a warning that the value they converged to might not be a true solution. I tried playing around with the settings of the solver and it ended up giving results that were very clearly off. I might try to play around with it a little bit more later.

The fit with no tau agrees pretty well up until the higher order resonances kick in.

The fit with tau also agrees pretty well in magnitude for the first resonance however the phase is all over the place. As I said, the solver had issues with this so I might try a few things to make it work better.

Between the two fits, G and f_res stayed fairly constant between the 2 as expected. However Q decreased by an order of magnitude in the solution with tau.

Wrong: G / (1 - i f / Q - f^2 / f_res^2)
Correct: G / (1 - i f / f_res /Q - f^2 / f_res^2)

Relationship between Q and decay time

There is a way to do the complexfitting by converting C into 2xR

i.e. Conver a data set from
a1 + i b1
a2 + i b2
a3 + i b3
...
an + i bn

to

1 a1
1 a2
1 a3
...
1 an
2 b1
2 b2
...
2 bn

and instead of the complex function f(x)+i g(x), use the 2D fitting function
z(x) = kroneckerDelta(x,1) f(y) +kroneckerDelta(x,2) g(y)

Then the 1D complex fitting is mapped to 2D real fitting.

In any case, your transfer functions might not have sufficient resolution to allow us to do the Q fitting.
 

I tried again with fitting the correct transfer function. The fit is pretty much the same except with a slightly higher gain and resonant frequency. However, I noticed that the solver wasn't changing the value of Q. In the plot below, I initially guessed 5e7. I tried decreasing the initial guess of Q, but the solver wouldn't change it at all until I got to around 1e5. So as Koji said, there's probably not enough resolution to get a good fit for Q.

With the correct transfer function, I couldn't get the solver to converge anymore with a tau parameter I guess because the function it is trying to fit is slightly more complicated now.

The approach with changing a complex valued function to a 2-dimensional real valued function is what I have been doing for the fitting. I guess I didn't make that too clear in my original post.

EDIT: Evan pointed out to me that the complex pole pair should create a phase lag of 180, not a phase lead. I had used a Hermitian transpose (') instead of a standard transpose (.') on the COMSOL data which flipped the sign of the phase. For the fitted data, there was a flipped sign in the transfer function. It should be: G / (1 + i f / f_res /Q - f^2 / f_res^2). I updated the below plot with this in mind. Changing this didn't affect any of the parameters as both the data and the fitting function had reversed phase.

Last time I measured the PM depth I found the west laser to be at gamma = 0.13, which is much lower than the expected value of ~0.5. I looked into this today and was able to get gamma back up to 0.5 for the west laser.

The issue was with the waveplates near the EOM. While playing with them to try and reduce the AM, I accidentally reduced the PM depth as well. Lesson learned: when trying to reduce AM, don't adjust the waveplates too far or you'll also lose PM.

You want the waveplates to be such that:

• PM is big.
• AM is small.
• All the power is in P polarization after hte 2nd waveplate (and will thus pass through the PDH PBS and go to teh cavity). You can toss a power meter on the S port (reflected) of the PBS and watch it while you look at the AM and PM. 

Remember that your cavity sweep has significant modulation depth in amplitude, so you will have a hard time minimizing RFAM on the input monitor PD as it will have a ~10-20% amplitude modulation envelope. While you are getting close to correct waveplate angles, it's fine to look at the light reflected from the PBS and the RFAM on the monitor PD during the sweeps. To really fine tune it you may need to turn off the sweep and either detune the cavity from resonance (ok) or block the input beam (better).

Attached updated PDH phasebudget for the LB1005 + pomona box setup currently in use. 

Also changed the custom low pass shown in the initial PDH phasebudget with an SLP5. Can swapa freely.

The most gain to be had is in the plant (can either deal with using chris W's AM RF injection idea or by using a fast mod path + EOM).

The phase delay is not the driver (confirmed with conversations with Mike Martin and published JILA paper on the PLANEX lasers)

Over the past couple weeks, I have been iterating on the production process for our cantilevers, trying to get as close to Shiuh Chao's recipe as possible, but with our different design. I will make an entry dedicated to explaining the evolution of the process, but for now I want to happily report that I just measured the most recent sample that I brought over from the KNI today, and its room temperature Q appears to be in excess of 10⁵, which is as good as we have ever seen even for the Glasgow/Taiwan cantilevers.

Here are a couple shots of the cantilever itself. I should note that there are *many* nonidealities with this sample (e.g., it is single-side polished, so the backside is very rough; it was masked using a lo-fi technique that has left the etch pattern slightly wiggly; finally, I feel that the mask can stand to be a bit thicker, as I think we had some etch-through in a few regions that should have been protected), so there is still room for improvement. The second photo here is focused on the reflection of the ceiling, so you can see the surface finish of the protected end regions and the central etched region.

Here is the ringdown. The mode frequency is about 100 Hz, so this tau of 400 s corresponds to Q ~ 1.25 x 10⁵.

After having the LN2 dewar refilled, I transferred the newest cantilever (which performed well at room temperature---see CRYO:1332) into the cryostat to make a cryogenic measurement. I had to adjust the clamping block position on the cold plate to put the smooth reflective surface of the cantilever within line of sight of the laser beam, and I think the RTD may have come loose from the block during closeout because it is registering ~100 K after 2 days of cooling.

In any case, I made several-hour measurements at two different amplitudes using the steady state technique, and found a consistent Q of about 5 x 10⁵. This is a bit lower than what is expected given the room temperature performance---when the Glasgow/Taiwain cantilevers showed Q > 10⁵ at room temperature, they almost always had Q  10⁶ at low temperature.

I can think of at least 2 reasons we may be seeing this effect:

1. The clamp state is not good. It is well known that "bad" clamping can happen, and this can only really be evaulated empirically. I will probably try a re-clamp on this once it heats up.
2. Some kind of lossy oxide/gunk has grown on the cantilever since the nicer measurement. When the room temperature measurement was made, the cantilever had just been removed from a hydrofluoric acid clean (to remove the hard nitride etch mask). I cleaned it, carried it across campus, and installed it into the chamber. It sat in that chamber for a few days, then in the lab air for on the order of an hour while the cryostat was cycled. So, conceivably, it could have gotten dirtier in the meantime.

I will also re-measure the cantilever at room temperature in the simple chamber, to see if I can recover the result from the other day.

(P.S.: Why do images automatically get reduced now and even when you try to set their size larger upon upload it just magnifies the low-resolution copy? Pooey.)

I re-measured the same cantilever in the room-temperature chamber again, and again its Q was above 100,000.

I re-installed it into the cryostat chamber, under the hypothesis that the clamping condition was bad in the previous cryogenic measurement. Instead of cooling, I left it pumping overnight at room temperature (the small pump aperture makes it take several hours to reach a good vacuum without cryopumping). When I came back the next day, at 10 uTorr, the Q was measured to be only around 30,000. So, it seems that---as I have noticed in the past---the Q is consistently lower in the cryostat.

I then took the clamp and PEEK base from the cryostat coldplate and put it in the simple chamber. I wanted to repeat the measurement in this chamber without reclamping the cantilever, but unfortunately I had to becasue I needed the reflective side up instead of down (the 45 deg mirror is below the cantilever on the coldplate but above it in the room temperature chamber). So, I reclamped it, and then measured the Q to be roughly as low as it was just before in the cryostat. So, perhaps it is the PEEK and/or newer clamp, and not the cryostat itself, which is adding the loss.

Finally, I went to take off the PEEK base and just measure with the clamp alone, but within the simple chamber and not the cryostat. Sadly, in this process, I broke the cantilever. I installed another one from the same batch and recorded a Q near 100,000 again.

So, this needs to be repeated with a better control now that I broke the original cantilever, but it seems that the PEEK spacer has been causing extra loss in the measurement. We will likely need to get a glass spacer.

I measured the transfer function through the LB1005 and the custom Pomona box at the output of the LB1005. Examining the Pomona boxes, I found they were not the same as in elog:1152. The architecture is the same with a resistor in parallel with a capacitor and another resistor to form a low pass filter. The values for the resistors and capacitors are not the same as in the elog post and are not even the same between the two paths. In the West path, the first resistor is 1 kOhm and the second is 10 Ohm. In the East path, the first resistor is 2.69 kOhm (it says 2.72 kOhm on the resistor but I measured 2.69 kOhm with a multimeter) and the second is 27 Ohms. The capacitors were unlabeled so I found their capacitances by the transfer function measurement. Fitting by hand, I found 1 uF fit the data. I confirmed this with a least squares regression and got 1.04 uF and 1.06 uF for the West and East cavities.

The measurement was made by using a splitter to input the HP's signal generator into the HP's R input and the A input of the LB1005. The output of the Pomona box was connected to the A input of the HP. The result was averaged over 10 measurements. Note that the input impedance of the HP is 50 Ohms. The settings on the LB1005 were PI Corner = 30 kHz, proportional gain = 20 dB, low frequency gain limit (LFGL) = 30 dB. The LB1005 was run with the LFGL to avoid overloading the HP at low frequencies.

At frequencies less than 60 Hz the measured transfer function drops off completely. This could be just from the HP as noted in elog:1331. At frequencies greater than 5 MHz the transfer function drops off again. At this point the LB1005 overload indicator began flashing red as well.

The calculated line in the graphs was made using the measured values of the resistors, a value of 1 uF for the capacitors, the 50 Ohm input impedance of the HP, and the LFGL transfer function of the LB1005 given in it's manual. The magnitude of the transfer function agrees very well for both. The phase does not fit as well with the measured data actually having a better phase margin in the range of 5-100 kHz by up to 15 degrees. Above 1 MHz the measured phase begins dropping rapidly, possibly from delays due to the wiring.

I measured the drop in voltage at the PDH RFPDs from a misalignment of the quarter waveplate right before the cryostat. These measurements were made on the east cavity as the degree markings on the west cavity waveplate are facing the cryostat and are very hard to read. I did the measurement twice for each angle I measured. The DC voltage of the RFPD was measured using the mean measurement on an oscilloscope.

As seen in the plot below, the data is quadratic. The fitted equation is y=-0.57982x^2+13.807x+169.32.

If we assume we are at the maximum, all we care about is the quadratic term. We can normalize this based on the max voltage (252 mV). Since the incident power is directly proportional to the voltage of the PD, this normalized value also applies to the power. So the relative power loss is 0.23%/deg^2. For a 1 degree offset, we only lose 0.23% of the power.

David asked me to look at the radius of curvature change in our cavities due to thermal expansion. Since alpha is a function of temperature, we need to use an effective alpha given by the integral of alpha(T)*dT. Using the function in elog:39 for alpha(T) and integrating from 300K to 125K, I calculated an effective alpha of -2.614e-4. The negative sign indicates that the silicon will contract when cooled. I also calculated this by integrating the NIST data here with the trapezoidal method and got a similar result of -2.620e-4.

Since all dimensions of our mirrors will shrink equally by alpha, the radius of curvature R should also shrink by alpha. I worked this out geometrically just to ensure that it makes sense. So our new R is R(125K)=(1+alpha)*R(300K).

Additionally, our cavity length L will shrink by alpha: L(125K)=(1+alpha)*L(300K)

From David's thesis, R and L have been measured to a relative precision of about 0.04. This is 2 orders of magnitude bigger than the thermal effects so the thermal effects are well within our current error bars.

I added/some updated some wiki pages over the past couple days.

At https://nodus.ligo.caltech.edu:30889/CryoWiki/doku.php?id=documents:frequency_readout I summarized some options we have to readout the beat frequencies between the different cavities. For now a lot of the information is trivial, but I wanted to have it all in one place. I plan to expand it as I obtain data on performance of the different methods.

I also added a table to https://nodus.ligo.caltech.edu:30889/CryoWiki/doku.php?id=documents:2nd_gen_cavities which I will keep up-to-date as I hear back from coaters and polishers.