I rechecked the TF between power fluctuation and frequency noise in beat measurement that I did last year. The estimated result agrees more with the measured result. This can be used to estimate the requirement for ISS for SiO2/Ta2O5 and AlGaAs coatings.
The calculation is taken from Farsi etal 2012 (J. Appl. Phys. 111, 043101), and compared with the measurement from 8" cavities, SiO2/Ta2O5 QWL with SiO2 1/2 wave cap. The code I wrote before has several mistakes, so I fixed them.
Mistakes in the original code:
Above: Measurement(purple) from SiO2/Ta2O5 coatings and analytical result (cyan) in comparison. Finesse = 7500 (old ACAV), absorbtion = 5ppm. The slope at high frequency seems to be real TO noise. Notice that phases from TE and TR have different sign and cancel one another.
==for TO optimized AlGaAs coatings==
Above: Calculation for RIN induced thermo noise for optimized AlGaAs coatings in Hz/Watt unit. The calculation is for 200 ppm transmission,-> Finesse ~14 000. 1.45" cavity. The cancellation in coatings will reduce the noise. The estimated effect is plot against the measurement from 8" cavity, T=300ppm, SiO2,Ta2O5 cavity.
We might have to make sure that RIN is small enough, since this time we will have no common mode rejection like what we had with just a single laser. I'll add the estimated requirement later.
I estimated the requirement for laser RIN for AlGaAs coatings. The result is a factor of 5 more stringent from what we need for SiO2/Ta2O5 cavity.
See some calculation about RIN requirement PSL:1270.
I estimated the RIN induced TO noise in AlGaAs cavities. Due to the TO optimization, the effect will be small and we will see only the effect from the substrate, see RIN induced noise estimate.
This will be quite serious, if we do not have a good ISS, since we will not have common mode rejection like what we had with the single laser setup anymore. I'll look up what was the RIN performance we had before.
Noise hunting is in progress, I checked the error noise from ACAV and RCAV loops and compared them to the beat. The beat is about an order of magnitude higher than the sum of error noise.
NOte: slope of error signal RCAV = 1.57 MHz/V (13 dBm from Marconi, throug 4-way splitter, to BB EOM, 1mW input power).
ABOVE: beat signal in comparison with noise at error points from ACAV and RCAV loops. The beat signal is about an order of magnitude higher than the error noise.
I'm working on optimization and noise characterization of the setup. Before measuring the beat I have to make sure that:
I think the gain in the TTFSS is the problem. For ACAV, the scattered light from the window interferes with the main beam and causes the loop to oscillate when the gain is up. For RCAV, the EOM is a broadband one and does not have enough gain. The bump in the frquency lower than 100Hz is probably the contribution from scattered light. I have not properly dumped all beams yet.
Also I noticed that the beat signal has weird sidebands at +/- 100kHz, 200kHz, and 300kHz, see the figure below. I'm not sure why, I have not seen it before. I might saturate the PD making it distorted from a perfect sine wave. I'll investigate this.
Noise hunting is in progress. Today I identified that scattered light from the window is one of the problem.
I spent sometime making sure that all the beams in the input optic and the beat areas were dumped properly. I also tightened all the screws on the optics and the mounts on the table.
I mentioned in the previous entry that for RCAV, the reflected beams from the cavity and the vacuum window overlapped a little bit. The window beam was much smaller and actually closer to the edge of the main beam, so I used an iris to remove the outer path, and let only the beam in the center area go through to the RFPD. With that I could increase the gain in RCAV loop to Common/Fast = 624/750, where they used to be ~ 600/600 before. The iris might introduce some extra scattered lights, since it clips a part of the beam.
The scattered noise around DC to 100 Hz is reduced a bit, see the below figure. However, not much improvement in the flat region (100Hz and above). Plus, some mechanical peaks around 1kHz appear with higher level than before.
I expected the scattered noise will be even lower if the cavities are tilted a bit to avoid the beams overlapping. At higher frequency, it might be the gain limit from RCAV loop where the modulation depth is very small.
Next thing to do is to increase more power in the modulation depth for RCAV.
I found out that the sidebands in the beat signal mentioned in the previous entry changed with the gain of the TTFSS (both ACAV and RCAV). With higher gain, the sidebands are suppressed more. It might have to do with the PZT resonant of the NPRO.
I installed an electro-optic amplitude modulator (EOAM) in RCAV path. Better optimization will be needed to reduce extra noise.
above, the setup for ISS actuator, the first 1/2 wave plate rotates p-beam to s-beam, EOAM, 1/4 Wave plate that tuned so that the output beam is 45 degree so the power transmitted through the final PBS is reduced in half.
After the EOAM was added, I checked the beat noise and saw a bump at ~ 2 kHz, see the figure below(blue plot). This was from the EOAM even though there was no input drive. It disappeared after I changed the EOAM position by rotating it a bit( yellow plot). I have not finished with optimizing it yet. I'm thinking about what kind of mechanism that causes the noise here.
I went through all the code with Evan and found another mistake. This time the code should be correct, and the result is close to what we measured a year ago.
The calculation in PSL:1014 is wrong. There should be no square root for the absorption power (Finesse/pi * absorption). With that correction, and an assumption of absorption of 18ppm in the mirrors (9ppm on each) with Finesse of 7000, see PSL:425. The result matches with the calculation quite well.
The validity of this result depends on the absorption factor and cavity finesse. The finesse was measured, but the absorption measurement has never been done. So it might be good to think about how to measure that.
We did the same measurement with the current ACAV 1.45" cavity. Evan will post the result later.
The bump at 2kHz in the beat signal that I saw before was also from RFAM. By adjusting the 1/2 waveplate in front of the sideband EOM, the bump disappears. I still don't understand why adjusting the EOAM can reduce the bump from RFAM.
As I planned to add the eom driver to the BB EOM for sideband in RCAV path, I wanted to see the improvement without worrying about the EOAM optimzation. So I removed the EOAM, but I still saw the bump I observed before. This time it came from the RFAM. By adjusting the wave plate to match the polarization of the input beam to the EOM axis, the bump is gone.
above: From right to left, 1) laser for RCAV, 2)&3) 1/2 and 1/4 wave plates, 4) lense, 5) Faraday isolator, 6) 1/2 wave plate, 7)BB EOM for frequency locking, 8) BB EOM for side band, the EOM driver is attached to the side (in aluminum foil wrapped box). RFAM is minimized by adjusting (6) 1/2 wave plate.
I added the EOM driver, however it was not yet modified for 14.75 MHz, so the amplification is small, see PSL:1234 . After adjusting the phase of the demodulated sigmal, the error signal slope is increased by a factor of 2. Then I remeasured the beat signal, and the beat was better by ~ a factor of 2 at high frequency. So I think now the signal is gain limited (in RCAV loop) at high frequency. This makes me confused why the error noise from RCAV loop does not match the beat signal in PSL:1307. I have to re check my work.
The next few things to do are:
I closed the chamber. The turbo pump is on and pumping down.
I realigned the beams so the visibilities for both cavities were 80% or more. This made sure that the beams' path would be close to the optimized path.
Now, the window reflection won't overlap with the cavity reflection, and can be dumped properly.
Note about a few things to do:
FYI for torque wrench setting for CTN cavity:
The CTN cavity is 10" OD, the Torque required is 190 InchPound.
Since the optimized layer structure is designed, I'm checking how the coatings properties change with error in layer thickness.
G.Cole said that they can control each layer thickness within 0.3%. So I tested the optimized coatings properties by adding some random number within +/- 0.5% on each layer thickness. The results are shown below for 10 000 test.
The error check does the following:
The figure below is an example of the varying layer thickness added by rand command. They are confined within 0.5%.
1) result from the error in thickness control
Above: histograms of the important values. top left, reflected phase. top right, ratio between PSD of Brownian noise and Thermo optic noise at 100 Hz. Bottom left, transmission. Bottom right, total coating thickness error.
comments: this test is chosen for 0.5% error which is almost a factor of 2 worse than what they claimed (0.3%), so the actual result should be better. I assumed 0.5% errof because of the irregular layer structure of the optimized coatings, there might be some more error in the manufacturing process.
2) result from different calculated Beta values:
Here I checked what happen if the beta calculation was wrong, and the error is still within 0.5% in each layer.
In Evans paper, the effect from "Thermo-refractive" comes from the phase changes of the wave travels in each layer. So it includes the effect from dn/dT and dz. The effective beta for each layer is given as
where alpha bar is
Where s denotes substrate, k denotes the material in each layer (high or low indices).
So my, calculation & optimization have been using these equations.
However, in the original GWINC code for TO calculation, the calculation [B8], alphabark( used in dTR) is not the same as A1, but rather.
alphaH * (1 + sigH) / (1 - sigH)
see getCoatLayerAGS.m. Line 16-17.
This is used in the calculation for beta effective in getCoatTOphase. Line73-74. Notice that for dTE, the alpha_bar_k is the same as used in Evans. (line 77).
the comment says "Yamamoto thermo-refractive correction". I emailed kazuhiro yamamoto, but never got a response back. So I keep using the same formula as in Evans because I don't see the reason why the expansion contribution should be different between TE and TR.
So this is the nb plot for TO noise from the optimized coating, if using yamamoto TR correction.
Above: nb from the optimized coatings, using Yamamoto TR correction. The cancellation becomes worse, but TO is still lower than other noise.
Finally, I repeat the same error analysis for random noise in the thickness (+/- 0.5%).
Most of the parameters behave similarly, except the cancellation (upper right plot). Now BR is only ~ x12 larger than TO noise because of the worse cancellation. Good news is, it still below Brownian noise, the cancellation still somehow works.
We modified the EOM driver, so that the resonant frequency is now~ 14.75MHz. The full test will be done later.
As mentioned in PSL:1311, the resonant frequency on the EOM driver was not at 14.75MHz. Evan and I discussed about how to modify it and decided tof change L4 from 1.4uH to 3 uH, see the schematic here.
above, the driver after the inductor was replaced. The new one has a shield to reduce any magnetic field leakage. The legs are not fit with the footprint on the PCB, so I had to solder it to another wire to reach the footprint.
above: the TF of the driver measured between the drive and the mon output. Red trace shows the TF before the modification. Yellow trace shows the TF after the modification, notice the peak is at 14.75MHz, the Q is about the same.
Here is a summary for how I verify the codes for TO calculation.
So far, we have been using a set of modified GWINC codes to calculate TO noise, but I have not mentioned how did I make sure that the codes were reliable. So I'll try to explain how I check the codes here.
==What do we compute?==
For the TO nosie calculation and the optimization, we are interested in:
==Beta calculation check==
For TR coefficient we can compare GWINC with an analytical result (see Gorodetsky,2008, and Evans 2008) (when # of layers ~ 50 or more), see psl:1181. I tried the solution with nH, 1/4 cap and nL, 1/4 and 1/2 cap. All results agree.
==Alpha calculation check==
There is no complication in this calculation. The effective alpha is just the sum of all layers. This calculation is quite straight forward.
This was done by reducing the coating layers to one or two layers and comparing with an analytical solution by hand. I checked this and the results agreed.
So I think the calculations for TO noise is valid, the noise estimated from the optimized coatings is done with some error check (previous entry). I think we should be ready to order.
The turbo pump is removed, and the ion pump is on. The initial value is ~7mA.
I removed the turbo pump and turn on the ion pump, see the procedure on wiki page. The initial value on the ion pump is ~ 7mA, similar to the last time we opened the chamber although this time I left the turbo pump on 4 days instead of 2 days. So I think this is the limit of this turbo pump.
I updated the optimization and error analysis. The error in optimized structure is comparable to that of a standard quarter wave length structure.
After a discussion with Rana, Garrett, and Matt, I fixed the thermo-optic calculation, and the error analysis done in PSL:PSL:1315. The modifications are
1) fix the TO calculation (Yamamoto TR correction): There is a modification for TR correction that is not in Evans etal 2008, paper. I contacted M. Evans to ask about the details of this correction which is done in GWINC.
2) Try another optimized coatings with the correct TO calculation: After the correction, I ran doAlGaAs.m code, cf PSL:1269 using fmincon function , to find another optimized structure. The result is shown below.
above) layer structure in optical thickness, the .fig and .mat file are attached below. Note .mat file contains 54 layers, you need to add 1/4 cap to the first entry to calculate the noise budget.
above) noise budget of the optimized coating.
3) Repeat the error analysis : This time I used the following assumptions (from G Cole)
Fig1: Above, percentage of error distribution between the two materials used in the calculation. nH(red) has 2 sigma = 1% and nL(blue) has 2sigma=1%.The same error distributions are used for both optimized layers and QWL layers in comparion, see fig2.
The section below is the algorithm used to distribute the error, this one makes the error between the two materials to be the same sign. The whole code can be found on svn.
mu1 = 0;
sigma1 = 0.5; % 2sigma is 1percent;
mu2 = 0;
sigma2 = 1;
run_num = 5e4; % how many test we want
errH = normrnd(mu1,sigma1,[run_num,1]); %errH in percent unit
errL = normrnd(mu2,sigma2,[run_num,1]); %errL in percent unit
errL = abs(errL).*sign(errH); %make sure that errH and errL have the same sign
dOpt = xout; % xout from doAlGaAs (optimized layer)
dOpt = [ 1/4 ; dOpt]; % got 54 layer no cap from doALGaAs, need to add the cap back
dOpt_e = zeros(length(dOpt),1);
for ii = 1:run_num;
dOpt_e(1:2:end)= dOpt(1:2:end)*(1+ errH(ii)/100 );
dOpt_e(2:2:end)= dOpt(2:2:end)*(1+ errL(ii)/100 );
This time I calculated the change in reflection phase (TOP left), the ratio between TO noise from the coatings with error and the coatings with no error(top right), transmission (bottom left), and ratio of BR noise ( bottom right). The result from the optimized coating(blue) is compared with the QWL coating (black).
Fig2: Error analysis, in 5e4 run. Blue: from optimized coatings Black:from 55 QWL coatings, from 5x10^4 runs.
Reflection phase: The reflection phase can be away up to ~6 degree. The power at the surface will be ~Finesse/pi * Power input * sin^2 (6degree) ~ 50mW. Seems high, but this is about a regular power used in the lab.
Ratio of PSD TO/TO_0 : At worse, it seems the PSD TO noise will be ~ a factor of 10 higher than the design. However, this will be still lower than BR noise. I plotted only the error distribution for optimized coatings because for QWL coatings, the ratio will be about the same, since TO is dominated by TE.
Transmission: Most of the results are within 197-200 ppm. The optimized coating has transmission ~ 197ppm. The QWL with 55 layers has transmission ~100ppm.
Ratio of BR: not much change here.
It's a quiet night, so I went down the lab to measure the beat signal. We are getting close. I think I have to review my noise budget calculation and estimate the error in the measurement carefully.
So after a few things Evan and I did a few days ago:
Then I measured the beat signal.
We reduce some noise from scattered light at frequency below 100 Hz, we are limited by some white noise at high frequency ~ above 1 kHz.
fig1: measurement vs noise budget
fig2: zoom in. The slope of the measured signal agrees well with the slope of thermal noise.
Coating optimization and error analysis are updated, see PSL:1320.
Short note from tonight measurement:
1) scattered bump from dc to 100Hz is mostly from seismic. It is worse during the day. It gets smaller at around 3-4 am. Unless we have a better seismic isolation, we might not be able to see anything below 100Hz.
2) RIN shape from RCAV changes, reasons still unknown. (DC level 0.7 V)
3) I might see the effect from RIN induced TO noise at frequency ~ 1-3 kHz. (compare RIN and beat).
I'll get into details tomorrow.
The measured RIN is measured and converted to frequency noise via photo thermal effect then compared to beat. The effect seems to be significant now since we lost the common mode rejection.
I measured RIN after ACAV (there is only one PD behind ACAV right now. we will add another one for RCAV soon). The magnitude is comparable from what we measured before but the peaks seem to change, see PSLPSL:1326, :PSL:1308, (8"cavity) PSL:742 .
The peaks around kHz are more clear. I'm not sure where they are from, but I think it is associated with vibration on mirror mounts that causes beam jitter. Because the peaks look like mechanical peaks, and this time the cavities are shorter, the beamsize is smaller from 8" cavities, the same beam misalignment will cause the power coupled into the cavities to change more compared to that of 8" cavity. We can check that by mis-aligning the input beam a bit, and see if RIN becomes larger or not.
The coupling from RIN to frequency noise is discussed in PSL:1328
I applied that to the measurement and here is the result. Note, only the effect from one cavity (ACAV) is taken into account.
The peaks seems to match up, especially around 20-30Hz and around 1kHz, see the zoomed in picture below. This makes me think that we might be limited by RIN noise now.
To Do next:
I'm trying to understand the measured RIN in the setup. The evidence suggests that the measured RIN in 100Hz- 6kHz, is real intensity noise and not associated with alignment + jitter.
As seen in PSL:1329 that we might be limited by RIN at high frequency, I tried to figure out what cause the shape of the RIN around kHz to be mechanical -like peaks. So the problem can be minimized, and does not have to rely on ISS that much.
My assumption was that they were from mirror mounts, because
So to test this, I measured RIN before and after ACAV (NOTE:ACAV path has PMC in it), when
above, beam path in front of ACAV, before the beam enters ACAV. The PD for RIN measurement is circled in blue.
above, beam path behind ACAV.
If the measured RIN was from the jitter, RIN after the cavity should change with the alignment, and RIN before the cavity should not change much. I made sure that the spotsize on both PDs are significantly smaller than the PD to make sure that any jitter in front of the cavity should not change the power level that much.
==comments about the result==
==To do next==
The cause of the peaks around 1kHz in RIN is solved, PMC is the reason. After damping it, the peaks disappears.
Short notes from tonight measurement:
need to buy:
I checked the calculation. I think most of the discrepancies are from the thick coating correction calculation (from Evans etal paper). The error is frequency dependent, and the calculations that involve frequency dependence are temperature fluctuation and thick coating correction. The temperature fluctuations are the same from our results. So it is most likely the thick coating correction. I checked and the corrections did differ at high frequency.
I need to take a closer look to tell exactly where the errors are. Since the error is small and only at high frequency (around the shot noise limit, 10kHz), I don't think it will be a problem for us.
Optimized coatings structure.
Tara noticed an accidental re-definition in my old code. I fixed it, and updated the svn. This fixes most of the discrepancies, but shifts the difference in thermo-optic to the low-frequency region.
Attachment 1 is the comparison from case 3 between mine and Tara's calculations of his optimized coating structure.
Attachment 2 is the comparison from case 2 between mine and Tara's calculations of a 55-layer 1/4-wavelength stack.
I discussed the calculation with Matt. The error in TO noise is large because it is a fraction of something small. Mostly it comes from TE part. The error in TO noise appears large (10%-20%) because the TO level is small. Otherwise, the rests are in good agreement, and I think we should be able to order soon.
Below, summary of the calculation, dTE is alpha_effective * coating thickness, dTO is beta effective * lambda. 0.2% difference in dTE and 0% difference in dTR can cause error upto 40% in dTO when dTE and dTR cancel each other really well. But this will be insignificant, since the final TO levels are still in the same magnitude.
The summary of the TO cancellation is in wiki page AlGaAs
Details for AlGaAs coatings order
Above, plot of ratio of power due to finite size mirror P(r) / P0, P(r) is the power of the beam at radius r from the center. G Cole said that the wafer can be made to 8mm diameter. diameter between 5-8 mm should be good for us.
I'm using Matt's code to do error analysis for AlGaAs coatings. This time I vary materials' parameters and compare the thermo optic noise, reflected phase and transmission. There is no alarming parameter that will be critical in TO optimization, but the values of refractive indices will change the transmission considerably.
Eric, Matt and I discussed about this to make sure that even with the errors in some parameters, the optimization will still work.
Parameters in calculation and one sigma estimated from Matt
% Coating stuff
betaL = 1.7924e-4 +/- 0.07e-4; %dn/dT
betaH = 3.66e-4 +/-0.07e-4 ;
CL = 1.6982e6 +/- 5% ; % Heat Capacity per volume
CH = 1.754445e6 +/- 5%;
kL = 69.8672 +/- 5% ; % Thermal Conductivity
kH = 55 +/- 5%;
alphaL = 5.2424e-6 +/- 5%; % Thermal expansion
alphaH = (5.73e-6 ) +/- 5%;
sigmaL = 0.32 +/- 10%; % Poisson Ratio
sigmaH = 0.32 +/- 10% ;
EL = 100e9 +/-20e9; % Young's modulus
EH = 100e9 +/-20e9;
nH = 3.51 +/-0.03 ; % Index of refraction
nL = 3.0 +/-0.03 ;
* Note: when I change nH and nL value, I keep the physical thickness of the layers constant. This is done under the assumption that the manufacturing process controls the physical thickness. The optical thickness in the calculation will be changed, as the actual dOpt = physical thickness * actual n / lambda. And averaged values of coatings will depend on physical thickness.
This is fixed in Line 120-180
== Effect on TO cancellation from each parameters==
First, I calculate the TO cancellation when one of the parameter changes. Some parameters, for examples, Poisson ratios, Young's moduli, are chosen to be the same for both AlAs and GaAs. In this test, I vary only one of them individually, to see which parameters are important. The numbers indicate the ratio between the PSD of TO noise with change in the parameter and the optimized TO noise . Not the standard deviation of the parameters.
Turns out that the change in Young's moduli and Poisson's ratios are quite important.
==Effect on TO cancellation, from all paramerters==
Then, I calculate the TO noise when all parameters vary in Gaussian distribution, similar to what I did before,all parameters are uncorrelated. The histograms from 1000 runs are shown below.
I'll try more run overnight. Each run takes about 1 second.
== combined effect from errors in layer thickness and material parameters==
Since the comparison does not need to calculate the thermal fluctuations and finite size correction all the time, I cut that calculation out and save some computation time. Now I compare errors from
Each simulation contains 5e4 runs. The Transmission varies a lot depending on the material parameters ( mostly refractive indices, see the cyan plot).
The cancellation seems still ok. Most of the time it will not be 10 times larger than the optimized one. Only the transmission that seems to be a problem, but this is highly depends on refractive indices. It's weird that the error makes the mean of the transmission smaller.
If that's true, then it means that a 1% deviation in the index of refraction of the low index material can by a 10x increase in the TO noise. Is this really true?
That surprises me too, but, that's what the calculation gives me. It is also strange that deviation in nH has smaller effect on to TO noise than nL does. I'm checking it. I ran the code one more time, and still got the same result.
Note: when I calculate the error in refractive indices, I assume that the physical thickness is constant = x * lambda/ n_0. Where x is the optical thicknesss. But if the the actual refractive index is not n_0, it means the optical length is not x, but x*n/n_0. I think this is a valid assumption, if they control the physical thickness during the manufacturing process.
update:Tue Sep 24 02:09:28 2013
The TO noise level does really change a lot when nL is nL + sigma (nL=3.0+ 0.03), dark green trace. Most of the change comes from TR noise level (not shown in the plot). TE noise remains about the same level. It might be worth a try to find another optimization that is less sensitive to the change in value of n. I'll spend sometime working on it.
I'm trying to find another optimization that is less sensitive to change in nH and nL. Here is a few thought and a few examples.
We have seen that uncertainties (withing +/- 1%)in nH and nL result in higher TO noise (up to 10 time as much) in the coating. So we are trying to see if there is another possible optimized structure that is less sensitive to the values of n. We estimate the value of nH to be 3.51 +/- 0.03, and nL to be 3.0 +/-0.03. (The numbers we have used so far are nH/nL = 3.51/3.0, while G.Cole etal use nH/nL = 3.48/2.977.
The algorithm is similar to what I did before[PSL]. But this time the cost function is taken from different values of refractive indices. The values of nH and nL used in this optimization are
The cost function is the sum of the TO noise level at 100Hz, Transmission, and reflected phase, calculated from 9 possible pairs of nH and nL values. The weight number from each parameters (which parameter is more important) are chosen to be 1, as a test run. I have not had time to try other values yet, but the prelim result seems to be ok.
[Details about the codes, attached codes]
Note about the calculation,
The calculation follows these facts:
==results from QWL (55layers) and 4 other optimized coatings.==
Each plot has three traces (blue, black, red) for different values of nH (3.48, 3.51, 3.54). nL is varied on x-axis from 2.97 to 3.03. The first result is from QWL coating, with 55 layers. This serves as a reference, to see how much each property changes with the uncertainty in nH and nL.
I tried to change the cost function in the optimization code and numbers of layer to see if better optimized structure can be done. The optimized structure (V3,4,5) seems to be less sensitive to the values of n, see below.
Above: from QWL coatings, 55 layers. nominal transmission = 100ppm. We can see that the transmission of QWL coatings is still quite sensitive to uncertainties in nH and nL.
Above: First optimization reported before, TO noise is larger by a factor of 10 in certain case, and transmission can be up to 500 ppm. This coating is very sensitive to the change in refractive indices.
Above: opt3, obtained from the code using the new cost function discussed above. 55 layers, nominal transmission = 150ppm. The TO noise is less dependent on nH and nL, but the transmission is still quite high.
Above: opt4, the weight parameter for transmission is changed to 3, 57 layers.
above opt5,the weight parameter for transmission is changed to 50, Lower/Upper thickness bound = 0.1/0.5 lambda, 59 layers
Above: Opt6, the weight parameter for transmission is changed to 500, Lower/Upper thickness bound = 0.1/1.2 lambda, 59 layers
From the results, optimized structure # 3,4,5 seem to be good candidates. So I ran another monte carlo error analysis on opt1 (as a reference), opt3, opt4, and opt5, assuming errors in both material properties and coating thickness. Each one has 5e4 runs. Surprisingly, the results from all designs are very similar (see the plot below). It is possible that, by making the coatings less sensitive to changes in nH/nL, it is more sensitive to other parameters (which I have to check like I did before). Or the properties are more dependent on coating thickness, not material parameters (this is not likely, see psl:1345). Or perhaps, there might be a mistake in the monte carlo run. I'll check this too.
I'll update the coating structure and forward it in google doc soon.
The new optimization is less sensitive to the values of refractive indices, but the overall error will not change much if other material parameters have the uncertainties as we estimate.
Summary: see update of error analysis in PSL:1356. The issues from the previous entry are cleared
1) show error analysis
New legs were installed. The table is floated. The cables for signals/ power supplies will be reconnected later
Evan and I replaced the old legs. I made sure that the leak was not in the connections and the tube. After the legs replacement, the air pump can reach 25 psi within ~25 minutes and the table can be floated.
The regulating valves are adjusted and the table is leveled.
I recalculated the coatings properties, with the values of nH and nL to be 3.48 and 2.977. Note about each optimization is included here. Transmission plots are added in google spread sheet. I'll finish the calculation for E field in each layer soon.
Note about each optimized coating version: different versions were obtained from different cost functions, and different number of layers.
Judging from TO noise level, Transmission and reflected phase, I think opt4 is the best choice for us. The structure consist of thick nH layers and thin nL layers. This is good for us in terms of thickness control.
Electric field in coating layer is calculated. This will be used in loss calculation in AlGaAs coatings.
1) average E field in layer is the transmitted E field in the layer.
I attached a short matlab file for a simulation of the combined field. Ein in each layer will be the transmitted beam through the layers. For a value of r close to 1, we get a standing wave. Try changing the value of r in test_refl.m to see the effect
2) Calculation for the transmitted field in each layer
I borrow the notation from Evns etal paper (rbar), the calculation code multidiel_rt.m is attached below. Note: the final transmission calculated in the code is the transmission from the coating to the substrate. To calculate the transmission to the air, multiply the last transmission by 2*n_sub/(n_sub + n_air) which is the transmission from sub to air. Since the thickness of the substrate is not known with the exact number, it will not be exact to the transmision calculated in GWINC or Matt A's code (which do not take the sub-air surface into account), but they will be close, because the reflected beam in the last interface will be small compare to those in the coatings.
The penetration of E field for QWL and different optimized coatings are shown here. The transmissions in the legend are calculated from MattA./GWINC and the values in the parenthesis are calculated from multidiel_rt.m which include the effect from the substrate-air surface. This makes the values in the parenthesis smaller (as more is reflected back and less is transmitted).
I checked the dependent of coatings properties with the uncertainty in x (amount of Al in Al_x Ga_(1-x) As). The effect is already within the uncertainties in materials parameters we did before and will not be a problem.
G. Cole told us about the variations in Al contents in the coatings. Right now the values are 92% +/- 0.6%.
(92.10, 91.43, 91.34, 91.57, 92.73, 92.67). Although the deviation is small, the Al content does not always hit 92%, but 92+/- sigma%. So I decided to check the effect of x on the optimization.
The materials properties that change with x are heat capacity, alpha, beta, heat conductivity and n. The values of those as functions of x can be found on ioffee except n. So I looked through a couple of sources ( rpi, sadao) to get n as a function of x, (Note: E0 and D0 are in eV, they have to be converted to Joules when you calculate chi and chi_so). GaAs (nH) has a well defined value ~ 3.48+-0.001, nL has a bit more uncertainty, but it is within the approximated standard deviation of 0.03 . The table below has numbers from the sources. For RPI, I use linear approximation to get nL for x = 0.92 @ 1064nm.
The dependent of n on x is about -0.578 *dx. The numbers from RPI and Sadao are about the same. This means that for the error of 0.6% in Al. nL can change by 0.578*0.006 = 0.0035. The number is almost a factor of ten smaller than the standard deviation of nL and nH I used in previous calculation (0.03). For examples,
This means that the uncertainty in nL/nH (+/- 0.03) we used are much larger than the effect coming from uncertainty in x. This is true for other parameters as well.
After installing the table legs, I have been trying to measure the beat. However, there is an unknown scattered light noise up to 400 Hz. I'm still trying to fix that.
Here are some bullets about what happened, I'll add the details later.
Note: check if the beams in the tank is blocked by wires or not.
I revised the calculation for photo-thermal noise in AlGaAs coatings, the photo thermal noise should not be a limiting source.
photothermal noise arises from the fluctuation in the absorbed laser power (RIN + shot noise, mostly from RIN) on the mirror. The absorbed power heats up the coatings and the mirror. The expansion coefficient and refractive coefficients convert thermal change into phase change in the reflected beam which is the same effect as the change of the position of the mirror surface.
Farsi etal 2012, calculate the displacement noise from the effect. The methods are
When they solve the heat equation, the assume that all the heat is absorbed on the surface of the mirror. This assumption is ok for their case ( SiO2/Ta2O5) with Ta2O5 at the top surface, all QWL, as 74% of the power is absorbed in the first four layers (with the assumption that the absorbed power is proportional to the intensity of the beam, and all absorption in both materials are similar).
However, for AlGaAs coatings with (nH/nL) = (3.48/2.977) The E field goes in the coatings more that it does in SiO2/Ta2O5, see the previous entry. So we might want to look deeper in the calculation and make sure that photo thermal noise will not be a dominating noise source.
==calculation and a hand waving argument==
The plot below shows the intensity of the beam in AlGaAs Coatings, opt4, and the estimated intensity that decreases with exponential square A exp(-z^2/z0^2). X axis is plotted in nm (distance from surface into coatings). The thickness of opt4 is about 4500 nm. To simplify the problem, I use the exponential decay function as the heat source in the diff equation. But I have not been able to solve this differential equation yet. Finding particular solution is impossible. So I tried to solve it numerically with newton's method, see PSL:284. But the solution does not converge. I'm trying green function approach, but i'm still in the middle of it.
However, the coatings optimized for TO noise should still be working. Evans etal 2008 discuss about how the cancellation works because the thermal length is longer than the coating thickness. The calculation (TE and TR) treat that the temperature is coherent in all the coatings ( they also do the thick coatings correction where the heat is not all coherent, and the cancellation starts to fail at several kHz). So the clue here is that the cancellation works if the heat (temperature) in the coatings change coherently.
For photothermal calculation, if we follow the assumption that all heat is absorbed at the surface (as in Farsi etal), we get the result as shown in psl:1298, where most of the effect comes from substrate TE . In reality, where heat is absorbed inside the coatings as shown in the above plot, heat distribution in the coatings will be even more coherent, and the effect from TE and TR should be able to cancel each other better. Plus, higher thermal conductivity of AlGaAs will help distribute the heat through the coatings better.
This means that the whole coatings should see the temperature change more coherently, thus allowing the TO cancellation in the coatings to work. The assumption that heat is absorbed on the surface should put us on an upper limit of the photothermal noise.
This means that photothermal noise in the optimized coatings should be small and will not be a dominating source for the measurement.
I'm optimizing the setup, and clearing the table a little bit.
To do lists
above: old PBS, bad inter surface can be seen.
above: new PBS: all surfaces are clear
Evan found that when common gain is changed, DC offset also changes as well. I'm still looking into the problem.
a part of schematic, the driving signal was sent in through test port (the switch was flipped from off to test), so the signal came through PD line in this page.
We still cannot lock RCAV with TTFSS, so I'm checking the box 2009007 (#7).
Common Gain - DC offset problem
DC offset vs input drive. DC offset is calculated from (Vmax + Vmin) /2 from a sinusoidal signal input. The signal was taken from TP4. The behavior is very non linear and it is impossible to make a table for an appropriate offset level vs common gain setting.
What to do next?
I'm putting EOAM back on ACAV path. The setup is roughly optimized.
(14.75 MHz) EOM , EOAM, quarter waveplate and PBS in ACAV path are put back together. I used a half waveplate in front of the EOM to adjust the beam to S- polarization. Right now all the polarizations optimization (to all EOMs, both ACAV/RCAV path) are adjusted to S-polarization with respect to the table. We may have to fine tune it later to match the E field in the EOMs. The EOAM setup is optimized. With +/-4 V, the output power can be adjusted to 1mW +/- 0.09 mW (+/- 9%). The performance is comparable to RCAV EOAM. (10%) . I have not add another half waveplate before the EOAM yet. We can add it back later if we need to adjust the input polariztion to the EOAM.
I checked scattered light in the area between PMC and ACAV. There is a reflection from EOAM back to EOM, but I cannot really block it with an iris. It probably bounces of the case of the EOM or going back to the crystal. Anyway I'll block the beam around this path later.
I have not aligned the beam to the cavity yet, since the temperature was changing because I removed the insulation caps to patch them with black out material.
I put black out material (R @1064 ~0.4-0.6%)on the vac tank insulation caps to minimize any possible scattered light source inside the tank that might leak out. It also keep the surface cleaner from all the foam dust.
Do you guys have a plot that shows the required loop gain and the achieveable loop gain with this TTFSS on the same plot?
Not yet, we will add this later. but we measured the noise at error point before and it is well below the estimated coating noise.
Plan for this week
Mon: (See Evan's entry for more detail)
We made a mistake by choosing the input power to the cavities to be 0.25 mW, so today I turned them back to 1mW and measure the beat.
Note about the measurement:
To do next:
I add the photo thermal noise effect in the noise budget. With ISS, photothermal noise should be sufficiently small.
What I did
Comment about the beat
Note about RIN measurement
Note about loss angles: For SiO2 and Ta2O5 loss angles = 1e-4 and 7.5e-4 (a factor of 3 above the regular number), the noise budget matches the measurement well. I'll see if it is the same for the data from 8" cavities or not.
I'm re-arranging the optics in PMC path a bit. The work is in progress, so ACAV path is still down.
I'm investigating why ACAV TTFSS performance is worse than that of RCAV. One thing is that ACAV has the PMC. This area has not been optimized for awhile, so I'm checking everything.
PMC path is back, I aligned the polarization of the input beam to the BB EOM for TTFSS. The visibility of PMC is now ~ 80%.
I created an svn folder for my thesis on CTN measurement.
It can be found here
I compared our beat measurement with results from Numata2003 and TNI. They agree well. I'm quite certain that we reach Brownian thermal noise from coatings.
To make sure that what we measure is real Coating Brownian noise (It could be something else, i.e thermal noise in the support, spacer , or optical bond), we should compare our result to previous measurements to make sure that the numbers agree.
Numata etal and TNI reported coating thermal noise measurement from suspended cavities (no spacer). They adjusted loss in the coatings to fit the measurement. Phi coatings as reported in Numata is 4e-4 while TNI gives phi perp = phi_para = 2.7e-4. Both agree with our result, see the plot below. This means that our result is comparable with what they measured. It should be an evidence to support that we see real coating thermal noise, not contribution from something else (spacer, optical bond between the mirrors and the spacer).
Another evidence is from our previous measurement from 8" cavity.
So It is clear that our beat measurements from both 8" and 1.45" cavities are coating Brownian noise limited (around 50Hz-1kHz).
I realized that we have not checked the eigenmodes of 1.45" cavity yet, so I used comsol to find out several modes. The lowest mode is ~ 46kHz, and the first longitudinal mode is about 60kHz. The frequencies are high enough so that the thermal noise calculation in dc- 10kHz frequency band can be done with quasi-static assumption.
1) I tried a simple cylindrical shape, with the dimension of the spacer. The result for the first longitudinal mode is 74KHz, the analytical result is ~ 77kHz, see PSL:1135. It seems that COMSOL's result and the analytical results are comparable.
2) Then I simulated the whole reference cavity. The lowest body mode is ~ 47kHz. The body expand-contract radially, and should not change the cavity beamline length that much. The first longitudinal mode is ~ 60kHz. The color on the surface shows the rms displacement from all direction.
open loop gain transfer function of RCAV is measured.
1) how to measure OLG TF
The requirement assumes that the residual frequency noise is 5% or less in the total noise. The servo performance is definitely ok for 1.45 inch cavity.
We heard back from G. Cole about the thickness resolution in the AlGaAs coating manufacturing process will be around 0.5 A. So I'm checking how the noise budget will change by rounding up the physical thickness in opt V4 to the next 0.5A. The design will still work. The round up thickness is added in the google document (for opt v4 only).
The estimated growth rate of the crystal is 4.8A/s and shutter speed is assumed to have 0.1 sec time step. This means the smallest step of the thickness control is ~0.5A. So I round up the physical thickness to the next 0.5 A and calculate the coating properties.
1) Rounding up to the next 0.5 Angstrom. The truncating process acts like a random thickness variation in the optimized coatings with maximum error ~ 0.25 Angstrom. The averaged layer thickness is ~ 800 Angstrom.
2)Results when the layers physical thickness are round up to the closest 0.5 A. The noise budget does not change much.
The coatings properties still hold, even with random error in parameters, thickness.
Note: For the error calculation I did before I used 1 sigma to be 1% for AlGaAs, and 0.5% for GaAs. The thinnest layer is AlGaAs at 35 A, so its sigma is about 0.35 A. The average thickness is 90 Angstrom, so the average error is about 0.9 A. The estimated error in the calibration process is already larger than the error from the truncation(0.25A). That's why the error analysis results are still valid.
I got a chance to measure beat measurement. The noise budget is updated and contains all dominant noise traces.
== Beat measurement ==
1) at DC to 10Hz, the contribution is mostly from RIN driven Photothermal noise and a bit of seismic noise, a small peaks around 10Hz is probably from the stack, not the cavity sagging. The hump from DC to ~ 50Hz disappear when it is quiet. I think it is mostly scattered light associated with the seismic noise, not displacement noise due to the vibration.
2) 10Hz to 1kHz is pretty much Coating Brownian noise.
3) At 1kHz and above, it is PLL readout noise and residual frequency noise from the laser, where the gain cannot suppress enough noise. This is mostly from ACAV. The residual frequency noise = free running noise / (1+ OLGTF). The measurement of the open loop gain is explained below.
==TTFSS Loop characterization==
The OLG TF of TTFSS is measured up to 10MHz and compared with the calculation. The schematic explaning how TTFSS actuates on the laser is shown below.
The freqeuncy discriminator can be measured from the slope of the error signal (from Common out1) while scanning the laser. For RCAV Dv = 1/ (194 kHz/V) and 1/(164kHz/V) for ACAV. with 1mW input power.
The adjustable gain stage can be tuned by turning the dial knob. At 400, gain=1, and the gain changes by 10dB with every 250click.
The PZT actuator has a gain of 4.5MHz/V (measured), and the EOM actuator is 15mRad/V (or 15mHz/f Hz/V) (taken from the spec sheet).
OLG measurement is taken: RCAV OLG is measured and plotted against the theoretical approximation, see the below figure.
above: RCAV OLG TF. Note: The calculation and the measurement do not include the integrator with corner frequency at 4.6kHz.
There are some problems with ACAV loop and I could not increase the gain up as much as it used to be and the UGF is around only 200kHz , but the measurement matches the calculation. Right now RCAV servo has a better loop performance.
The calculated OLG TF trace(green) should go down at 1MHz or above because of the opamps' bandwidth. I used ideal Op Amps in the simulation because I don't have some op amps in my liso library. I'll see if I can fix it.
I'm trying to record beat measurement for a few days. The data will be taken from ATF using mDV. There are a few issues about mDV right now, I'm looking into it and asking around.
There is a problem with gps.m that converts the string to gps second. It is used in get_data where we specify the start time. I tried enter the gps second manually but it returns an empty time struct, and the get_data cannot be used.
A reminder entry: psl:978
I looked into the uncertainty in coating thickness of the QWL SiO2/Ta2O5 coating The thickness of 4.53 +/- 0.07 um (~1.5%)seems to be appropriate.
The thermal noise level is directly proportional to the coating thickness, so we want to estimate its uncertainty. The error in the thickness is from
The errors in nL and nH are quite small, nL ~ 1.45 +/ 0.01, nH ~ 2.06+/- 0.01. (From the literature). I also looked around the error in IBS thickness control, they are usually better than 0.1 nm, IBS, but that is the current technology. In literature around 2000s, 2% error seems to be the number estimated for the thickness control (Sullivan 2000, Badoil 2007). As a quick check, I used the same assumption for error propagation similar to that of AlGaAs coating. The result gives ~ 4.53 +/- 0.07 um for coating thickness.
Note that the error here is smaller than the difference in coating thickness for the coatings with or without half wave cap.
For 28 Layer (with cap), the coating thickness is 4.53 um, for 28 layer QWL, the coating thickness is 4.35 um. But after digging up all the information from REO, and peter king they agree that it is 28 QWL with half wave cap. I tried to compare the calculation and the photothermal TF measurement, but the effect is too small to be conclusive about the structure. So the biggest error might come from the fact that the coating has cap or not. The error is about 4%.