Rueben from Technical Safety Services installed and calibrated an Apex 1000 airflow monitor for the fume hood in CTN. Its power supply should remain plugged in to the outlet directly below the unit. Attachment 1 is the testing certificate. We can contact Shelley in H&S tomorrow to receive their report on the work.

Martin Fejer recently gave two talks in a coatings workshop where he showed calculations regarding the thermal photoelastic channel. I have not been able to under the logic behind some of the calculations yet, nevertheless, I used his formulas for our coatings to get an alternative idea of this noise coupling.

Major difference

• Fejer argues that free body thermal expansion does not generate any strain, and it is only when the substrate is present to counteract with it, that such strain is generated.
• Hence, the calculation goes as: thermal expansion -> stress in presence of substrate -> strain -> photoelastic effect.
• So instead of the simple contribution for photoelastic tensor and thermal expansion that I take, the term is:
  
• This gives an effective (averaged with layer thickness weighting) coefficient of thermal photoelasticity of 1.45e-5 K^-1 instead of 4.30e-5 K^-1 from my calculations. That's a reduction by a factor of roughly 3.

Updates

• Attached is the photothermal transfer function calculated with TPE contribution as calculated by Fejer. This makes the situation bit more messy on what to trust.
• I updated the noise budget with two new noise traces, the thermo-photoelastic contribution as calculated by Fejer and the total thermo-optic noise as calculated by Fejer.

I just received more calculation notes of Fejer (through Yuta) which I'll study and try to make more sense of this calculation. It also contains the calculations of sough-after birefringence noise.. But in his presentation as well, he stated that birefringence noise is not sourced through termperature fluctuations and is not part of thermo-optic noise (something I didn't understand again).

Not easily, but it is doable if we resurrect the south path only. I estimate ~1 month of work for that if things go fine.

With circularly polarized light, Tara could be addressing any of the two possible resonances, with only effect of suffering in modematching with the cavity. So it should be a 50/50 chance that they measured it in a different polarization. However, the nature of thermal photoelastic measurement is same in both polarizations. The photoelastic tensor for GaAs (cubic symmetry), in theroy, does not create birefringence, or afect different polarizations differently. The source of birefringence in these coatings is not known.

The photothermal transfer function measurement made back in 2014 showed some cancellation of thermo-optic noise, but there were some irregularities with the modelled transfer function even back then. Here in attachment 1, I have plotted the measured photothermal transfer function, along with the estimated transfer function with and without adding a term for thermal photoelastic (TPE) channel.

Notes:

• The estimated transfer function without TPE (as was estimated back then) does match well with the measured transfer function on the south cavity below 200 Hz.
• However, the north cavity measurement did not match well.
• The estimated transfer function with TPE (green) is in between south and north measurements at least in magnitude above 200 Hz.
• However, the phase of estimated transfer functions (with or without TPE) do not match well with any of the measurements. This phase discrepancy is worrisome.
• Looking at these estimated transfer functions and measured transfer functions, which model do you think explains the measured data better?

Updated noise budget:

I was wondering if photothermal noise would get amplified due to the TPE effect. We were not using a measured photothermal transfer function in our noise budget for this noise contribution and relied on a theoretical model instead. For comparison, I added noise traces for three cases, Estimated photothermal noise with and without PTE, and photothermal noise using measured TF. In all these cases though, the ISS in the experiment suppressed RIN enough that photothermal noise did not matter to beatnote frequency noise.

Interesting.

The obvious go to measurment here would be two-lasers-one-cavity to measure the residual between the two polarisaiton modes of one of the cavities. Is the experiment in a state where this could be done easily?

If I recall correctly Tara had this set up with an optical circulator on the input side which Antonio and I switched to linear polarisasion with Faraday isolator.  The mode splitting of the AlGaAs coatings would take care of only selecting one polarisation mode, but is it posisble that the latter measurments sampled a different polarisation to the original thermo-optic measurment? Just a thought.

I made a few changes in my calculations today, which changed the noise contribution of this photoelastic noise (coatTPE) to roughly half of the individual contribution from coating thermo-refractive (coatTR). If this was true, it would significantly affect thermo-optic optimization, although not totally destroying it. I admit there is an outcome bias in this statement, but this noise estimate fits very well with the noise floor measured by CTN lab.

Changes in the calculation:

I made two changes in total:

• I'm using original coefficients of thermal expansion for each layer instead of the "effective" coefficients used in calculations of thermo-optic noise as per Evans et al. PRD 78, 102003 (2008)
• I removed the use of young's modulus and the crystal's elasticity tensor.

So now, the noise calculation is as follows:

• The temperature fluctuations cause isotropic strain fluctuations in the coating layers related through coefficient of thermal expansion
  
• The strain fluctuations cause changes in the refractive index of the layers through photoelastic tensor
  
  In the last step above, I assumed isotropic bulk strain in the layers (which is expected for this cubic lattice), thus
  
• The product of the above two numbers give the coefficient of thermo-photoelastic effect as:
  
  I averaged this coefficient over all coating layers weighted by their thicknesses.
• The noise contribution comes same as coatTR term as they both are channels causing dn/dT.

Notes:

• The above calculation does not take into account any birefringence in the layers that could be caused by this effect. In fact, the cubic crystal symmetry of GaAs does not allow for birefringence to occur in usual formalism and the only way it could happen is due to a large strain in one direction breaking the symmetries. Thus, I would not call this noise "birefringence noise", but it is a credible noise source in it's own right.
• Note that the themo-optic cancellation is only partially happening now, but the thermo-optic noise is still much less than the simple quadrature sum of the noises. We can maybe check back our measurements in our previous paper if the measured photothermal transfer function allows this.
• Maybe this noise source is not perfectly coherent with coatTE and coatTR and needs to be added a bit differently.

About the plot:

• The trace marked "Coating Thermo-Optic" is a coherently summed noise of coatTR, coatTE, and coatTPE.
• The trace marked "Coating Thermo-elastic + Thermo-refractive" is what we previously used to calculate as thermo-optic noise.
• "Measured Beat" is the best measurement we made and is a median over 50 lowest noise measurements made in June of 2020.
• "Coating Brownian" trace is calculated using bulk loss angle value of 4.878e-5 which was measured by Penn et al. in indirect measurement.

I think we need to regroup and discuss this further.

I followed the analysis of this recently published paper Jan Meyer et al 2022 Class. Quantum Grav. 39 135001 to calculate the birefringence noise in the CTN experiment. Interestingly, the contribution from birefringence noise after my first attempt at this calculation looks very close to what we were calculating as coating thermo-refractive noise before. If this were true, our experiment would have seen it much before. In fact, we wouldn't have seen thermo-optic cancellation as Tara experimentally verified here. So something is missing

What is birefringence noise?

After going through some literature and reading properly Meyer et al, I have the following understanding of the birefringence noise (and why it is called so).

• The temperature fluctuations cause length fluctuations in the coating layers (through the coefficient of thermal expansion)
• The length fluctuations cause stress fluctuations in the coating layers (through Young's modulus Y).
• The stress fluctuations get converted into refractive index fluctuations through the photoelastic effect (through photoelastic tensor)
• The photoelastic tensor for cubic crystals like GaAs has 3 independent values, P11, P12 and P44. (Section 12: Table 1. Elasto-Optic Coefficients: Cubic Crystals (43m, 432, m3m))
• This induces refractive index fluctuations that are different for the fast and slow axes. The difference between the two fluctuations causes a phase shift in reflected light from each layer. That's why this can be birefringence noise. In an unstressed isotropic material, this pathway should not exist.

Is this different from thermo-refractive noise?

This is a question I am still not sure how to answer. My understanding is that the common mode change in refractive indices of both axes drives the thermo-refractive noise. This means I should be able to derive the coefficient of thermo-refraction using the same formalism.

Calculation:

Both thermo-refractive noise and thermo-photoelastic noise show up as dn/dT terms in the thermo-optic noise summation, just through different physical processes. This could mean that experimentally measured coefficients of thermo-refraction already include birefringent contribution if any. In my calculations for the plots presented here, I got the following values of the two coefficients:

Coefficient of thermo-refraction (Effective for coating): 8.289e-05

Coefficient of thermo-photoelastic effect (Effective for coating, using Eq.11 of Meyer et al.): 8.290e-05

It was very surprising to me to see that both these coefficients came out to be within 1% of each other.

Because of this, when we add the noise sources coherently (since they are all driven by the same thermal fluctuations), the thermo-optic cancellation that we have experimentally proved does not work anymore. So something must be wrong with my calculation.

Possible explanations:

• Calculaiton error in my code. I'll double check tomorrow.
• Somehow the thermorefractive noise already takes into account the birefringent noise, through the coefficient of thermo-refraciton that we use as seed in our thermo-refractive noise calculation. This would explain how the witnessed themo-optic cancellation was achieved.
• Meyer et al. is calculating birefringent noise for the substrate. Maybe the tensorial calculations are different for coatings.

[Anchal, Paco]

CTN was raided today afternoon between 2 pm and 3 pm by 40m tribes. They have taken away precious Acromag units which are a very scarce resource these days. Following units were taken (Attachment 1):

• 1 XT1111 from blue cabinet.
• 4 XT1221 which were used at following places (see network):
  • |10.0.1.42  | 00:01:C3:00:98:59  | CTN Lab  | Acromag XT1221: 8-Channel Differential Analog Voltage Input Module  (CTN Slow controls)|
  • |10.0.1.50  | ?  | CTN Lab  | Acromag XT1221 8-Channel Differential Analog Voltage Input Module (Temperature Sensor monitor channels)  |
  • | 10.0.1.46 | 00:01:C3:00:98:55  | CTN Lab  | Acromag XT1221: 8-Channel Differential Analog Voltage Input Module  (PMC North remote controls)|
  • | 10.0.1.48  | 00:01:C3:01:12:FF  | CTN Lab  | Acromag XT1221: 8-Channel Differential Analog Voltage Input Module  (PMC South remote controls)|

3 rack mount units were affected:

CTN Slow Controls chassis:

• 2 XT1221 were removed from here. These can be installed back if Acromags become available and we need this experiment again
• 1 XT1541 is still mounted on this chassis.

PMC Servo Card Chassis:

• 1 XT1221 was removed from each servo card chassis (see attachment 2(North) -3 (South)).
• These chassis are otherwise functional and one can repopulate the XT1221 to make them function again (see CTN/2248 for instructions).

All these units are stored in the flowbench side wire rack (see attachment 4).

I borrowed one two unused SR560 from CTN to Cryo. The first one had periodic noise at 100 Hz (operated in low noise AC coupled mode, G=100, with a 30 kHz lowpass).

Returned all remaining stuff to CTN:

1. Wenzel 5001-13905 24.483 MHz Crystal Oscillator
2. One small 5/16 spanner
3. 4 SMA-F to BNC-F adaptors
4. 4 SMA-M to BNC-F adaptors
5. 2 10 dB BNC Attenuators
6. 4 BNC cables

Also returned the Red Pitaya accessory box to CTN. I've kept Red Pitaya at home for more playing.

I added script SRIMD.py in 40m/labutils/netgpibdata which allows one to measure second order intermodulation product while sweeping modulation strength, modulation frequency or the intermodulation frequency. I used this to measure the non-linearity of SR560 in DC coupling mode with gain of 1 (so just a buffer).

IP2 Characterization

• Generally the second order intercept product increases in strength proportional to the strength of modulation frequency with some power between 1 and 2.
• The modulation frequency strength where the intermodulation product is as strong as the original modulation frequency signal is known as intercept point 2 or IP2.
• For SR560 characterization, I sent modulation signal at 50 kHz and set intermodulation frequency to 96 Hz.
• The script sends two tones at 50 kHz and 50khz -96 Hz at increasing amplitudes and measured the FFT bin around 96  Hz with dinwidth set by user. I used 32 Hz bin width.
• In attachment 1, you could see that beyond 0.1 V amplitude of modulation signal, the intermodulation product rises above the instrument noise floor.
• But it weirdly dips near 0.8 V value, which I'm not sure why?
• Maybe the modulation signal itself is too fast at this amplitude and causes some slew rate limitation at the input stage of SR560, reducing the non-linear effect downstream.
• Usually one sees a straight curve otherwise and use that to calculate the IP2 which I have not done here.

IMD2TF Characterization

• First of all, this is a made up name as I couldn't think of what else to call it.
• Here, we keep the amplitude constant to some known value for which intermodulation signal is observable above the noise floor.
• Then we sweep the modulation frequency and intermodulation frequency both, to get a 2-dimensional "transfer function" of signal/noise from higher frequencies to lower frequencies.
• Here I kept the source amplitude to 0.4V and swept the modulation frequency from 10kHz to 100kHz and swept the intermodulation frequency from 96 Hz to 1408 Hz, with integration bandwidth set to 32 Hz.
• I'm not completely sure how to utilize this information right now, but it gives us an idea of how much noise from a higher frequency band can jump to a lower frequency band due to the 2nd order intermodulation effect.

Edit Wed Feb 17 15:34:40 2021:

Adding self-measurement of SR785 for self-induced intermodulation in Attachment 3 and Attachment 4. From these measurements at least, it doesn't seem like SR785 overloaded the intermodulation presented by SR560 anywhere.

I noticed small amount of water on the floor (Attachment 1) on the west end of the lab. Immediately above it is a pipe which I don't know what it does. One can see another drop forming at the edge of this pipe (Attachment 2). This water is slowly dripping on the side of the pipe (Attachment 3). I could trace it out to coming from somewhere on the top (Attachment 4 and 5).

Maybe this is just some condensation because of increased humidity in the air. But maybe this is some troubling sign. What should I do?

I have brought back HP E3630A triple output DC power supply.

Moved the rack-mounted Marconi 2023A (#539) and SRS FS725 Rb Clock to crackle lab (See SUS_Lab/1876).

Borrowed two broadband PDs (new focus 1811) and one power supply unit (new focus 0901) from CTN into Crackle.

Borrowed Universal PDH box from CTN lab (D0901351) and a mounted Faraday isolator (Thorlabs) for use in Crackle.

I transferred two Gold Box RFPDs labeled SN002 and SN004 (both resonant at 14.75 MHz) to 40m. I handed them to Gautam on Oct 22, 2020. This elog post is late. The last measurement of their transimpedance was at CTN/2232.

I have brought home the following items (provided by Anchal):

1. Low-noise Preamplifier (Model SR560)

2. Two serial-to-USB adapters

I can't answer the last question, but I just tell you my experience on the AC power supply. The OMC Lab used to have a LWE NPRO and every time I plugged low quality AC adapters (like one for CCD adapter), NPRO came back to StandBy. So I suppose that you have a large electrical spike on the AC line when the North laser shuts down for whatever reason.

I shorted the interlock terminals on the North laser power supply and still as soon as I turn the key to 'ON' position, the south laser drops to standby mode and the north laser power supply display does not switch on with the status yellow led blinking asynchronously. I still do not understand why the two laser operations are coupled. South laser power supply does not share anything other than the power distribution board with the North power supply. Could it be that something in the north power supply has created a short circuit in the power drawing portion?

Is it worth it to fix this path?

As Koji suggested, I can use a spare LWE NPRO controller but do we want to put more resources and time into this experiment? We have acquired loads of measurements over 4 months in the quietest environment already. So I'm not sure if it is worth it.

Measurement and estimation method:

• 125N-1064 Thermal continuous tuning coefficient is 5 GHz/V
• The south cavity is locked with nominal settings with slow PID switched on.
• The cavity was first allowed to equilibrate at the nominal setpoint of cavity heater currents of 0.5 W common power and -0.09358 W Differential power.
• After more than 9 hours, the cavity heaters are switched off and the cavities are allowed to reach equilibrium with the vacuum can.
• The slow voltage control of the south cavity moves slowly in response to the temperature change of the cavity.
• Cavity frequency change to length change factor is  m/Hz.
• Cavity spacer is made out of fused silica whose coefficient of expansion is  [from Accuratus SiO2 datasheet]
• Therefore, NPRO temperature tuning slow voltage to cavity temperature conversion factor is:

• After waiting for 60 hours, the cavity finally cooled down to equilibrium with the vacuum can. The out-of-loop temperature of the vacuum can at this point is used as the equilibrium temperature of the cold cavity.
• The voltage change in slow voltage control of south laser from this point to setpoint is used to estimate the cavity temperature at beatnote measurement which came out to be around 37 degrees Celcius. I'm taking a generous uncertainty of  during noise budget calculations to account for miscalibration of the vacuum can temperature sensor and other errors in this method.
• The noise budget calculation was updated, Bayesian inference updated and the results in the paper draft have been updated.

Code

Borrowed 1 (new focus) broadband EOM from CTN for temporary use in Crackle (2 um OPO experiment)

I transferred the following form my home to Cryo lab (Cryo_Lab/2587) today:

1. Moku with SD Card inserted and charger.
2. Ipad pro with USB-C to USB-A charging cord

I suspected the interlock failure. Can you replace the interlocking wire with a piece of wire for the troubleshooting?

There probably is a spare LWE NPRO controller at the 40m.

The north laser power supply display is not working and when the key is turned to ON (1) position, the status yellow light is blinking. I'm able to switch on the south laser though with normal operation. But as soon as the key of the north laser power supply is turned on, the south laser goes back to standby mode. This is happening even when the interlock wiring for the north laser is disconnected which is the only connection between the two laser power supplies (other than them drawing power from the same distribution box). This is weird. if anyone has seen this behavior before, please let me know. I couldn't find any reference to this behavior in the manual.

I have brought home the following item, provided by Paco:

1. Black Dell Laptop and charger.

I set up Red Pitaya, Wenzel Crystal, and Moku at my apartment and took frequency noise measurements of Red Pitaya and Wenzel Crystal with Moku.

Method:

• Wenzel Crystal was powered on for more than 5 hours when the data was taken and has an output of roughly 24483493 Hz. This was fed to Input 2 of Moku with a 10dB attenuator at front.
• Red pitaya was on signal generator mode set to 244833644 Hz with 410 mV amplitude. This was fed to Input 1 of Moku.
• Measurement files RedPitayaAndWenzelCrystalFreqTS_20201002_163902* were taken with 10 kHz PLL bandwidth for 40 seconds at 125 kHz sampling rate. So the noise values are trustworthy upto 10 kHz only.
• Measurement files RedPitayaAndWenzelCrystalFreqTS_20201002_165417* were taken with 2.5 kHz PLL bandwidth for 400 seconds at 15.625 kHz sampling rate. So the noise values are trustworthy upto 2.5 kHz only.
• Measurement MokuSelfFreqNoiseLongCablePhasemeterData_20190617_180030_ASD.txt was taken by feeding the output of Moku signal generator to its own phase meter through a long cable. Measurement details can be found at CTN:2357.
• All measurement files have headers to indicate any other parameter about the measurement.

Plots:

The plots in RedPitayaAndWenzelCrystalNoiseComp.pdf show the comparison of frequency noise of Red Pitaya and Wenzel Crystal measured with different bandwidths. Last two plots show all measurements at once where the last plot is shown for phase noise with integrated rms noise also plotted as dashed curves.

Large time-series data files are stored here: https://drive.google.com/drive/folders/1Y1JndCos8-cW4TQETRNVNybFcZhrVUCz?usp=sharing

Attachment 2 contains the calculated ASD data.

Update Thu Oct 22 21:47:21 2020

Attachment 3: Moku phasemeter block diagram sent to me by Liquid Instruments folks.

I have brought home the following items from CTN Lab today:

1. Moku with SD Card inserted and charger.
2. Ipad pro with USB-C to USB-A charging cord
3. Wenzel 5001-13905 24.483 MHz Crystal Oscillator
4. HP E3630A triple output DC power supply
5. One small 5/16 spanner
6. 4 SMA-F to BNC-F adaptors
7. 4 SMA-M to BNC-F adaptors
8. 2 10 dB BNC Attenuators
9. 4 BNC cables

Additionally, I got 4 BNC-Tee and a few plastic boxes from EE shop. Apart from this, I got a box full of stuff with red pitaya and related accessories from Rana.

I entered CTN just before (Wed Sep 23 00:22:11 2020 ) to borrow a spectrum analyzer, which I took to Cryo. Wore shoe covers, goggles. Sanitized goggles and door after.

I have switched on HEPA filters to high, both on top of the main table and on top of the flow bench.

Continuous measurement is stopped hereby. This experiment is finished.

I added the possibility of having a power-law dependence of bulk loss angle on frequency. This model of course matches better with our experimental results but I am honestly not sure if this much slope makes any sense.

Auto-updating Best Measurement analyzed with allowing a power-law slope on Bulk Loss Angle:

RXA: I deleted this inline image since it seemed to be slowing down ELOG (2020-July-02)

Major Questions:

• What are the known reasons for the frequency dependence of the loss angle?
• Do we have any prior knowledge about such frequency dependence which we can put in the analysis as prior distribution?
• Is this method just overfitting our measurement data?

Analysis Code

Final result of CTN experiment as of June 24  9 pm:

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval.

The analysis is attached.

Adding Effecting Coating Loss Angle (Edit Fri Jun 5 18:23:32 2020 ):

If all layers have an effective coating loss angle, then using gwinc's calculation (Yam et al. Eq.1), we would have an effective coating loss angle of:

This is worse than both Tantala (3.6e-4) and Silica (0.4e-4) currently in use at AdvLIGO.

Also, I'm unsure now if our definition of Bulk and Shear loss angle is truly the same as the definitions of Penn et al. because they seem to get an order of magnitude lower coating loss angle from their bulk loss angle.

Analysis Code

Automatically updating results from now on:

Final result of CTN experiment as of June 23  5 pm:

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval.

The analysis is attached.

Adding Effecting Coating Loss Angle (Edit Fri Jun 5 18:23:32 2020 ):

If all layers have an effective coating loss angle, then using gwinc's calculation (Yam et al. Eq.1), we would have an effective coating loss angle of:

This is worse than both Tantala (3.6e-4) and Silica (0.4e-4) currently in use at AdvLIGO.

Also, I'm unsure now if our definition of Bulk and Shear loss angle is truly the same as the definitions of Penn et al. because they seem to get an order of magnitude lower coating loss angle from their bulk loss angle.

Analysis Code

I've just finished a preliminary draft of CTN paper. This is of course far from final and most figures are placeholders. This is my first time writing a paper alone, so expect a lot of naive mistakes. As of now, I have tried to put in as much info as I could think of about the experiment, calculations, and analysis.
I would like organized feedback through issue tracker in this repo:
https://git.ligo.org/cit-ctnlab/ctn_paper
Please feel free to contribute in writing as well. Some contribution guidelines are mentioned in the repo readme.

Final result of CTN experiment as of June 15th 5 pm:

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval.

The analysis is attached.

Adding Effecting Coating Loss Angle (Edit Fri Jun 5 18:23:32 2020 ):

If all layers have an effective coating loss angle, then using gwinc's calculation (Yam et al. Eq.1), we would have an effective coating loss angle of:

This is worse than both Tantala (3.6e-4) and Silica (0.4e-4) currently in use at AdvLIGO.

Also, I'm unsure now if our definition of Bulk and Shear loss angle is truly the same as the definitions of Penn et al. because they seem to get an order of magnitude lower coating loss angle from their bulk loss angle.

Analysis Code

I figured out why folks befor eme had to use a different definition of effective coating coefficent of thermal expansion (CTE) as a simple weighted average of individual CTE of each layer instead of weighted average of modified CTE due to presence of substrate. The reason is that the modification factor is incorporated in another parameter gamma_1 and gama_2 in Farsi et al. Eq, A43. So they had to use a different definition of effective coating CTE since Farsi et al. treat it differently. That's my guess anyway since thermo-optic cancellation was demonstrated experimentally.

I realized that in my noise budget I was using higher incident power on the cavities which was the case earlier. I have made the code such that now it will update photothermal noise and pdhShot noise according to DC power measured during the experiment. The updated result for the best measurement yet brings down our estimate of the bulk loss angle a little bit.

Final result of CTN experiment as of June 11th 2 pm:

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval.

The analysis is attached.

Adding Effecting Coating Loss Angle (Edit Fri Jun 5 18:23:32 2020 ):

If all layers have an effective coating loss angle, then using gwinc's calculation (Yam et al. Eq.1), we would have an effective coating loss angle of:

This is worse than both Tantala (3.6e-4) and Silica (0.4e-4) currently in use at AdvLIGO.

Also, I'm unsure now if our definition of Bulk and Shear loss angle is truly the same as the definitions of Penn et al. because they seem to get an order of magnitude lower coating loss angle from their bulk loss angle.

Analysis Code

I've made a brief summary of the talks and topics I saw in DAMOP 2020 conference which happened virtually last week. Here is the link:

https://docs.google.com/document/d/1qsOVHgfq-Fsxd72qOwUWAq6JO2XRvewyd0fMzVfbHtk/edit?usp=sharing

You would need to sign in using your Caltech email address (jdoe@caltech.edu) to access the file.

Better measurement captured today.

Final result of CTN experiment as of June 4th 9 am:

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval.

The analysis is attached. This result will be displayed in upcoming DAMOP conference and would be updated in paper if any lower measurement is made.

Adding Effecting Coating Loss Angle (Edit Fri Jun 5 18:23:32 2020 ):

If all layers have an effective coating loss angle, then using gwinc's calculation (Yam et al. Eq.1), we would have effective coating loss angle of:

This is worse than both Tantala (3.6e-4) and Silica (0.4e-4) currently in use at AdvLIGO.

Also, I'm unsure now if our definition of Bulk and Shear loss angle is truly same as the definitions of Penn et al. because they seem to get an order of magnitude lower coating loss angle from their bulk loss angle.

Analysis Code

what is the effective phi_coating ? I think usually people present bulk/shear + phi_coating.

I've implemented all the proper analysis norms that Jon suggested and are mentioned in the previous post. Following is the gist of the analysis:

• All measurements taken to date are sifted through and the sum of PSD bins between 70 Hz to 600 Hz (excluding 60 Hz harmonics and region between 260 Hz to 290 Hz (Known bad region)) is summed. The least noise measurement is chosen then.
• If time-series data is available (which at the moment is available for lowest noise measurement of May 29^th taken at 1 am), following is done:
  • Following steps are repeated for the frequency range 70 Hz to 100 Hz and 100 Hz to 600 Hz with timeSegement values 5s and 0.5s respectively.
  • The time series data is divided into pieces of length timeSegment with half overlap.
  • For each timeSegment welch function is run with npersegment equal to length of time series data. So each welch function returns PSD for corresponding timeSegement.
  • In each array of such PSD, rebining is done by taking median of 5 consecutive frequency bins. This makes the PSD data with bin widths of 1 Hz and 10 Hz respectively.
  • The PSD data for each segement is then reduced by using only the bins in the frequency range and removing 60 Hz harmonics and the above mentioned bad region.
  • Logarithm of this welch data is taken.
  • It was found that this logarithm of PSD data is close to Gaussian distributed with a skewness towards lower values. Since this is logarithm of PSD, it can take both positive and negative values and is a known practice to do to reach to normally distributed data.
  • A skew-normal distribution is fitted to each frequency bin across different timeSegments.
  • The fitted parameters of the skew-normal distribution are stored for each frequency bin in a list and passed for further analysis.
• Prior distribution of Bulk Loss Angle is taken to be uniform. Shear loss angle is fixed to 5.2 x 10^-7 from Penn et al..
• The Log Likelihood function is calculated in the following manner:
  • For each frequency bin in the PSD distribution list, the estimated total noise is calculated for the given value of bulk loss angle.
  • Probability of this total estimated noise is calculated with the skew-normal function fitted for each frequency bin and logarithm is taken.
  • Each frequency bin is supposed to be independent now since we have rebinned, so the log-likelihood of each frequency bin is added to get total log-likelihood value for that bulk loss angle.
• Bayesian probability distribution is calculated from sum of log-likelihood and log-prior distribution.
• Maximum of the Bayesian probability distribution is taken as the most likely estimate.
• The upper and lower limits are calculated by going away from most likely estimate in equal amounts on both sides until 90% of the Bayesian probability is covered.

Final result of CTN experiment as of now:

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval.

The analysis is attached. This result will be displayed in upcoming DAMOP conference and would be updated in paper if any lower measurement is made.

Analysis Code

Thu Jun 4 09:17:12 2020 Result updated. Check CTN:2580.

I'm listing first few comments from Jon that I implemented:

• Data cleaning can not be done by looking at the data itself. Some outside knowledge can be used to clean data. So, I removed all the empirical cleaning procedures and instead just removed frequency bins of 60 Hz harmonics and their neighboring bins. With HEPA filters off, the latest data is much cleaner and the peaks are mostly around these harmonics only.
• I removed the neighboring bins of 60 Hz harmonics as Jon pointed out that PSD data points are not independent variables and their correlation depends on the windowing used. For Hann window, immediate neighbors are 50% correlated and the next neighbors are 5%.
• The Hard ceiling approach is not correct because the likelihood of a frequency bin data point gets changed due to some other far away frequency bin. Here I've plotted probability distributions with and without hard ceiling to see how it affects our results.

Bayesian Analysis (Normal):

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval.

Note that this allows estimated noise to be more than measured noise in some frequency bins.

Bayesian Analysis (If Hard Ceiling is used):

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval.

Remaining steps to be implemented:

There are more things that Jon suggested which I'm listing here:

• I'm trying to catch next stable measurement with saving the time series data.
• The PSD data points are not normal distributed since "PSD = ASD^2 = y1^2 + y2^2. So the PSD is the sum of squared Gaussian variables, which is also not Gaussian (i.e., if a random variable can only assume positive values, it's not Gaussian-distributed)."
• So I'm going to take PSD for 1s segements of data from the measurement and create a distribution for PSD at each frequency bin of interest (50Hz to 600 Hz) at a resolution of 1 Hz.
• This distribution would give a better measure of likelihood function than assuming them to be normal distributed.
• As mentioned above, neighboring frequency bins are always correlated in PSD data. To get rid of this, Jon suggested following
  "the easiest way to handle this is to average every 5 consecutive frequency bins.
  This "rebins" the PSD to a slightly lower frequency resolution at which every data point is now independent. You can do this bin-averaging inside the Welch routine that is generating the sample distributions: For each individual PSD, take the average of every 5 bins across the band of interest, then save those bin-averages (instead of the full-resolution values) into the persistent array of PSD values. Doing this will allow the likelihoods to decouple as before, and will also reduce the computational burden of computing the sample distributions by a factor of 5."
• I'll update the results once I do this analysis with some new measurements with time-series data.

Analysis Code

Today we measured further low noise beatnote frequency noise. I reran the two notebooks and I'm attaching the results here:

Bayesian Analysis with frequency cleaning:

This method only selects a few frequency bins where the spectrum is relatively flat and estimates loss angle based on these bins only. This method rejects any loss angle vaue that results in estimated noise more than measured noise in the selected frequency bins.

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval.

Bayesian Analysis with Hard Ceiling:

This method uses all frequency bins between 50 H and 600 Hz and uses them to estimate loss angle value This method rejects any loss angle value that results in estimated noise more than measured noise in the selected frequency bins.

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval.

Analysis Code

I realized that using only the cleaned out frequencies and a condition that estimated power never goes above them at those places is double conditioning. In fact, we can just look at a wide frequency band, between 50 Hz to 600 Hz and use all data points with a hard ceiling condition that estimated noise never goes above the measured noise in any frequency bin in this region. Surprisingly, this method estimates a lower loss angle with more certainty. This happened because, 1) More data points are being used and 2) As Aaron pointed out, there were many useful data bins between 50 Hz and 100 Hz. I'm putting this result separately to understand the contrast in the results. Note that still we are using a uniform prior for Bulk Loss Angle and shear loss angle value from Penn et al.

The estimate of the bulk loss angle with this method is:

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval. This result has an entire uncertain region from Penn et al.  within it.

Which is a more fair technique: this post or  CTN:2574 ?

Analysis Code

I talked to Kevin and he suggested a simpler straight forward Bayesian Analysis for the result. Following is the gist:

• Since Shear Loss Angle's contribution is so little to the coatings' brownian noise, there is no point in trying to estimate it from our experiment. It will be unconstrained in the search always and would simply result in the whatever prior distribution we will take.
• So, I accepted defeat there and simply used Shear Loss Angle value estimated by Penn et al. which is 5.2 x 10^-7.
• So now the Bayesian Analysis is just one dimensional for Bulk Loss Angle.
• Kevin helped me inrealizing that error bars in the estimated noise are useless in bayesian analysis. The model is always supposed to be accurate.
• So the log likelihood function would be -0.5*((data - model)/data_std)**2) for each frequency bin considered and we can add them all up.
• Going to log space helped a lot as earlier probablitis were becoming zero on multiplication but addition of log likelihood is better between different frequencies.
• I'm still using the hard condition that measured noise should never be lower than estimated noise at any frequency bin.
• Finally, the estimated value is quoted as the most likely value with limits defined by the region covering 90% of the posterior probability distribution.

This gives us:

with shear loss angle taken from Penn et al. which is 5.2 x 10^-7. The limits are 90% confidence interval.

Now this isn't a very good result as we would want, but this is the best we can report properly without garbage assumptions or tricks. I'm trying to see if we can get a lower noise readout in next few weeks, but otherwise, this is it, CTN lab will rest afterward.

Analysis Code

It's typically much easier to overestimate than underestimate the loss angle with a ringdown measurement (eg, you underestimated clamping loss and thus are not dominated by material dissipation). So, it's a little surprising that you would find a higher loss angle than Penn et all. That said, I don't see a model uncertainty for their dilution factors, which can be tricky to model for thin films.

Yeah but this is the noise that we are seeing. I would have liked to see lower noise than this.

If you're assuming a flat prior for bulk loss, you might do the same for shear loss. Since you're measuring shear losses consistent with zero, I'd be interested to see how much if at all this changes your estimate.

Since I have only one number (the noise ASD) and two parameters (Bulk and Shear loss angle), I can't faithfully estimate both. The dependence of noise due to the two-loss angles is also similar to show any change in frequency dependence. I tried giving a uniform prior to Shear Loss Angle and the most likely outcome always hit the upper bound (decreasing the estimate of Bulk Loss Angle). For example, when uniform prior to shear was up to 1x 10^-5, the most likely result became = 8.8x10^-4, = 1 x 10^-5. So it doesn't make sense to have orders of magnitude disagreement with Penn et al. results on shear loss angle to have slightly more agreement on bulk loss angle. Hence I took their result for the shear loss angle as a prior distribution. I'll be interested in knowing if their are alternate ways to do this.

I'm also surprised that you aren't using the measurements just below 100Hz. These seem to have a spectrum consistent with brownian noise in the bucket between two broad peaks. Were these rejected in your cleaning procedure?

Yeah, they got rejected in the cleaning procedure because of too much fluctuations between neighboring points. But I wonder if that's because my empirically found threshold is good only for 100 Hz to 1kHz range because number of averaging is lesser in lower frequency bins. I'm using a modified version of welch to calculate the PSD (see the code here), which runs welch function with different npersegment for the different range of frequencies to get the maximum averaging possible with given data for each frequency bin.

Is your procedure for deriving a measured noise Gaussian well justified? Why assume Gaussian measurement noise at all, rather than a probability distribution given by the measured distribution of ASD?

The time-series data of the 60s for each measurement is about 1 Gb in size. Hence, we delete it after running the PSD estimation which gives out the median and the 15.865 percentile and 84.135 percentile points. I can try preserving the time series data for few measurements to see how the distribution is but I assumed it to be gaussian since they are 600 samples in the range 100 Hz to 1 kHz, so I expected the central limit theorem to kick-in by this point. Taking out the median is important as the median is agnostic to outliers and gives a better estimate of true mean in presence of glitches.

It's not clear to me where your estimated Gaussian is coming from. Are you making a statement like "given a choice of model parameters \phi_bulk and \phi_shear, the model predicts a measured ASD at frequency f_m will have mean \mu_m and standard deviation \sigma_m"? 

Estimated Gaussian is coming out of a complex noise budget calculation code that uses the uncertainties package to propagate uncertainties in the known variables of the experiment and measurement uncertainties of some of the estimate curves to the final total noise estimate. I explained in the "other methods tried" section of the original post, the technically correct method of estimation of the observed sample mean and sample standard deviation would be using gaussian and distributions for them respectively. I tried doing this but my data is too noisy for the different frequency bins to agree with each other on an estimate resulting in zero likelihood in all of the parameter space I'm spanning. This suggests that the data is not well-shaped either according to the required frequency dependence for this method to work. So I'm not making this statement. The statement I'm making is, "given a choice of model parameters and , the model predicts a Gaussian distribution of total noise and the likelihood function calculates what is the overlap of this estimated probability distribution with the observed probability distribution.

I found taking a deep dive into Feldman Cousins method for constructing frequentist confidence intervals highly instructive for constructing an unbiased likelihood function when you want to exclude a nonphysical region of parameter space. I'll admit both a historical and philosophical bias here though :)

Thanks for the suggestion. I'll look into it.

Can this method ever reject the hypothesis that you're seeing Brownian noise? I don't see how you could get any distribution other than a half-gaussian peaked at the bulk loss required to explain your noise floor. I think you instead want to construct a likelihood function that tells you whether your noise floor has the frequency dependence of Brownian noise.

Yes, you are right. I don't think this method can ever reject the hypothesis that I'm seeing Brownian noise. I do not see any other alternative though as such as I could think of. The technically correct method, as I mentioned above, would favor the same frequency dependence which we are not seeing in the data :(. Hence, that likelihood estimation method rejected the hypothesis that we are seeing Brownian noise and gave zero likelihood for all of the parameter space. Follow up questions:

• Does this mean that the measured noise is simply something else and the experiment is far from being finished?
• Is there another method for calculating likelihood function which is somewhat in the midway between the two I have tried?
• Is the strong condition in likelihood function that "if estimated noise is more than measured noise, return zero" not a good assumption?

Wow, very suggestive ASD. A couple questions/thoughts/concerns:

• It's typically much easier to overestimate than underestimate the loss angle with a ringdown measurement (eg, you underestimated clamping loss and thus are not dominated by material dissipation). So, it's a little surprising that you would find a higher loss angle than Penn et all. That said, I don't see a model uncertainty for their dilution factors, which can be tricky to model for thin films.
• If you're assuming a flat prior for bulk loss, you might do the same for shear loss. Since you're measuring shear losses consistent with zero, I'd be interested to see how much if at all this changes your estimate.
• I'm also surprised that you aren't using the measurements just below 100Hz. These seem to have a spectrum consistent with brownian noise in the bucket between two broad peaks. Were these rejected in your cleaning procedure?
• Is your procedure for deriving a measured noise Gaussian well justified? Why assume Gaussian measurement noise at all, rather than a probability distribution given by the measured distribution of ASD?
• It's not clear to me where your estimated Gaussian is coming from. Are you making a statement like "given a choice of model parameters \phi_bulk and \phi_shear, the model predicts a measured ASD at frequency f_m will have mean \mu_m and standard deviation \sigma_m"? 
• I found taking a deep dive into Feldman Cousins method for constructing frequentist confidence intervals highly instructive for constructing an unbiased likelihood function when you want to exclude a nonphysical region of parameter space. I'll admit both a historical and philosophical bias here though :)
• Can this method ever reject the hypothesis that you're seeing Brownian noise? I don't see how you could get any distribution other than a half-gaussian peaked at the bulk loss required to explain your noise floor.
  • I think you instead want to construct a likelihood function that tells you whether your noise floor has the frequency dependence of Brownian noise.

I did this analysis last with bare-bones method in CTN:2439. Now I've improved this much more. Following are some salient features:

• Assuming Uniform prior distribution of Bulk Loss Angle since the overlap with Penn et al. is so low that our measurement is inconsistent with theirs ((5.33 +- 0.03) x 10^-4 )if we take into account their extremely low standard deviation associated to bulk loss angle.
• Assuming Normal Distributed prior distribution for Shear Loss Angle matching Penn et al. reported value of (2.6 +- 2.6) x 10^-7. This is done because we can faithfully infere only one of the two loss angles.
• The likelihood function is estimated in the following manner:
  • Data cleaning:
    • Frequency points are identified between 50 Hz to 700 Hz where the derivative of Beat Note Frequency noise PSD with respect to frequency is less than 2.5 x 10^-5 Hz²/Hz²..
    • This was just found empirically. This retains all low points in the data away from the noise peaks.
  • Measured noise Gaussian:
    • At each "clean" frequency point, a gaussian distribution of measured beat note frequency noise ASD is assumed.
    • This gaussian is assumed to have a mean of the corresponding measured 'median' value.
    • The standard deviation is equal to half of the difference between 15.865 percentile and 84.135 percentile points. These points correspond to mean +- standard deviation for a normal distribution
  • Estimated Gaussian and overlap:
    • For an iterable value of Bulk and Shear Loss Angle, total noise is estimated with estimated uncertainty. This gives a gaussian for the estimated noise.
    • The overlap of two Gaussians is calculated as the overlap area. This area which is 0 for no overlap and 1 for complete overlap is taken as the likelihood function.
    • However, any estimate of noise that goes above the measured nosie is given a likelihood of zero. Hence the likelihood function in the end looks like half gaussian.
    • The likelihood for different clean data points is multiplied together to get the final likelihood value.
• The Product of prior distribution and likelihood function is taken as the Bayesian Inferred Probability (unnormalized).
• The maximum of this distribution is taken as the most likely inferred values of the loss angles.
• The standard deviation for the loss angles is calculated from the half-maximum points of this distribution.

Final results are calculated for data taken at 3 am on March 11th, 2020 as it was found to be the least noise measurement so far:

Bulk Loss Angle: (8.8 +- 0.5) x 10^-4.

Shear Loss Angle: (2.6 +- 2.85) x 10 ^-7.

Figures of the analysis are attached. I would like to know if I am doing something wrong in this analysis or if people have any suggestions to improve it.

The measurement instance used was taken with HEP filter on but at low. I expect to measure even lower noise with the filters completely off and optimizing the ISS as soon as I can go back to lab.

Other methods tried:

Mentioning these for the sake of completeness.

• Tried using a prior distribution for Bulk Loss Angle as a gaussian from Penn et al. measured value. The likelihood function just became zero everywhere. So our measurements are not consistent at all. This is also because the error bars in their reported Bulk Loss Angle are extremely
• Technically, the correct method for likelihood estimation would be following:
  • Using the mean () and standard deviation () of estimated total noise, the mean of the measured noise would be a gaussian distribution with mean and variance where N is the number of averaging in PSD calculation (600 in our case).
  • If standard deviation of the measured noise is , then would be a distribution with N-1 degrees of freedom.
  • These functions can be used to get the probability of observed mean and standard deviation in the measured noise with a prior distribution of the total estimated noise distribution.
  • I tried using this method for likelihood estimation and while it works for a single frequency point, it gives zero likelihood for multiple frequency points.
  • This indicated that the shape of the measured noise doesn't match well enough with the estimated noise to use this method. Hence, I went to the overlap method instead.

Analysis Code

I will be presenting a poster on the latest results from CTN Lab in the upcoming virtual DAMOP 2020 conference. The following link is poster as of now:

CTN_DAMOP2020_Poster.pdf

Comments/suggestions/mockery is most welcome.

I will need to go to the lab for the following things:

• Take a transfer function measurement of RIN to beatnote frequency noise.
• Take a final measurement with HEPA filters off.
• Adjust gain levels of the ISS (North ISS is oscillating at this time)
• If possible, take Noise Injection measurement with RIN.

The first two are important to get a good result to show in this poster. I hope the lab access opens up before June 1st, or I get some special access for a day or two.

I did this analysis to calculate how much of Seismic noise couples to the cavity resonance frequency due to the birefringence of the mirror.

Short version:

The seismic noise can twist the cavity if the support points are not exactly symmetric which is highly possible. The twist in the cavity will change the relative angle between the fast axes of the mirrors (which should be close to 90 degrees normally). This twist changes the resonant frequency of the cavity as the phase shift due to the mirrors fluctuate.

Edit Tue May 5 11:09:53 2020 :

I added an estimate of this coupling by using some numbers from Cole et al . "Tenfold reduction of Brownian noise in high-reflectivity optical coatings" Supplementary Information. The worst worst-case scenario gives a vertical seismic acceleration coupling to cavity strain of 5x10^-13 s²/m (when the end mirros are at near 90 degrees to each other and the supports are misaligned differentially to cause normal force misalignment of 5 degrees). For comparison, the seismic coupling to cavity longitudinal strain is 6x10^-12 (from Tara's thesis). Note, that Tara took into account common-mode rejection of this coupling between the two cavities while in my estimate, I didn't assume that. So it is truly the worst worst worst-case scenario and even then an order of magnitude less than the usual seismic coupling we use in our noise budget calculations where Seismic noise is not dominating the experiment anywhere.

So the conclusion is that this effect is negligible in comparison to the seismic coupling through bending of the cavity.