[Tara, Evan]

We replaced the Lambertian diffuser with AlGaAs mirror 137B1. We intentionally induced a nonzero AOI of the incident beam, so that the reflected beam could be dumped cleanly. At a distance of 25.7(3) cm back from the mirror, the reflected and incident beams were separated by 1.3(1) cm, giving an AOI of 1.45(11)°.

1. We measured the incident laser power as 9.94(2) mW.
2. We set the exposure time of the camera to 250 ms.
3. We swung the boom to 13°, 16°, 19°, 22°, 25°, 28°, 31°, and 34°. At each angle, we took 5 CCD images with the beam incident, and 1 CCD image with the beam blocked.
4. We measured the incident laser power as 9.95(2) mW.
5. Because the scattered power had fallen off sharply by 30°, we turned up the exposure time to 1.00 s.
6. We swung the boom to 31°, 34°, 37°, 40°, and 43°. At each angle, we took 5 CCD images with the beam incident, and 1 CCD image with the beam blocked.
7. We measured the incident laser power as 10.08(2) mW.
8. We swung the boom to 46°, 49°, 52°, 55°, 58°, 61°, 64°, 67°, and 70°. At each angle, we took 5 CCD images with the beam incident, and 1 CCD image with the beam blocked.
9. We measured the incident laser power as 10.06(2) mW.

For all of these measurements, the two ND filters (OD1.5+OD3.0) were not attached; just the RG1000. With the ThorLabs power meter, we measured the combined transmissivity of these two ND filters to be 1865(14) ppm.

The first attachment shows an example CCD image. The second attachment shows the raw counts, the inferred scattered power, and the BRDF.

Yes.

We added an OD1.5, an OD3.0, and an RG1000 in front of the camera lens (note that these ODs are probably specked for something other than 1064 nm). Then we increased the exposure time to 20 ms. For the AlGaAs measurement, we may need to increase it even further in order to get good statistics.

Then we fixed the boom at 25° and varied the power using the upstream HWP + PBS combo.

For each power level, we took a measurement with the power meter, then 10 CCD images, then another measurement with the power meter. From this we are able to extract nominal values and uncertanties for the power level and the counts. The result is attached. The calibration has about a 4% uncertainty.

Note (Tara): The power measurement includes the solid angle of 3.375 x10^-3 str ( detector diameter = 0.4 inch, distance from the sample = 15.5 cm)

 I put the boom at 15° and took four sets of five exposures. Then I ran my image processing code again to get an uncertainty in the count values. I get the following:

• Beam incident, room lights on: 546(31) × 10³ cts
• Beam blocked, room lights on: 417(9) × 10³ cts
• Beam incident, room lights off: 547(34) × 10³ cts
• Beam blocked, room lights off: 410(2) × 10³ cts

For each set of five, the nominal value is the mean and the uncertainty is the standard deviation of the total counts within the 200×200 pixel region around the beam. Again the exposure time is 100 µs and there was an RG1000 filter in front of the camera lens.

Using a fractional uncertainty of 31/546 = 0.057 for yesterday's background-subtracted total counts, I reran the calibration code. The new plot is attached. The calibration slope (and its uncertainty) doesn't change much, but we can see that the uncertainties in the total counts are quite large. Do we need to improve this before moving on to the AlGaAs BRDF measurement?

[Tara, Evan, Josh, et al.]

Today we did some characterization and calibration of the scattered light apparatus.

To start with, we examined an AlGaAs mirror (s/n 173). We found that there was a great deal of diffuse light transmitted through the mirror (as seen in fig. 3 of ctn:1456). On Josh's suggestion, we put down an iris about 5" in front of the mirror. We stopped it down just enough so that both the incident and reflected beams could clear the aperture. This made the diffuse stuff disappear.

Next, we swapped out the AlGaAs mirror for a Lambertian diffuser (the same one used in Magaña-Sandoval et al.). Tara affixed the power meter to the camera boom in such a way that it could be raised or lowered in front of the camera.

We adjusted the incident power on the diffuser to be 3.00(1) mW. We then swung the boom in 5° increments from 10° to 70° from normal incidence. At each angle, we took the following:

1. Power incident on power meter with beam blocked
2. Power incident on power meter with beam unblocked
3. CCD image with beam blocked
4. CCD image with beam unblocked.

The beam was blocked using a dump located immediately upstream of the steering mirror.

The first attachment is the BRDF of the diffuser based on the power data. The second is the inferred calibration between total CCD counts (with background counts subtracted) and scattered power. The correlation is not great. We may want to retake this data with the room lights off, and also we may want to take multiple exposures per angle setting (that way we can make some estimate of the uncertainty in the CCD counts). The third attachment shows the analyzed CCD region for the 10° images; I've restricted the analysis to a 200×200 pixel region around the diffuser.

The exposure time was 100 µs, and there was a 1 µm long-pass filter (RG1000) affixed to the camera lens.

Data, CCD images, and plot-generating code are on the SVN at CTNLab/measurements/2014_07_24.

I rechecked the CCD response vs exposure time and power. The results are linear.

After some adjustments (strain relief on the camera's cables, clamping down the camera properly), I made sure that the camera is more stable and repeated the measurement. The CCD response is linear with the incident power on the sample (this is under the assumption that the scattered power is directly proportional to the incident power). 

Fig1:  CCD response vs incident power. The camera response is linear.

== AlGaAs Samples==

I prepared the sample for measurements. All the samples are quite dirty, especially on the flat sides. So I wiped all of them. I still cannot get rid off some water marks on the annulus of the mirror. It might cause some problems when I optical contact the mirrors. I'll try to clean them later.

fig2: one of the AlGaAs mirrors before cleaning. 

I put one of the samples in the scattered light setup. The transmitted beam has a lot of diffused light behind the mirror. The amount of the diffused light changes with the beam direction. I'm not sure exactly why. I'll try to investigate it more. But the scattered light from the sample is very small. Most of the light is from debris on the surface, not the micro roughness of the sample. The amount of scattered light significantly changes with the beam position on the mirror.

fig3: diffused light behind the mirror. It might come from the reflection inside the substrate because the incident beam is not normal to the surface.

I'm checking the linearity of power and exposure on the camera. The ccd counts are quite linear with the exposure setup, but I have to check the power again.

==ccd count vs exposure setup==

  The exposure time on the camera can be set to adjust the brightness of the image. Since we might have to adjust it to make sure that the images won't be saturated, it is necessary to check if the ccd count response linearly to the exposure setup or not.

I used a silver mirror as a test sample. The incident power is constant, and the camera position is fixed. Then adjust the exposure from 5k to 30k. I'm not sure if it is in nano second or microsecond unit. [Edit, 20140725: according to page 18 of the manual for the Prosilica GC750, the available exposure options are 30 µs to 60 s, in 1 µs increments. —Evan] But from fig1, the ccd count is quite linearly proportional to the exposure value.

It turns out that when I try to calibrate a sample, the incident power on the sample has to be more (so the power meter can measure some scattered power) and the camera can be saturated. The exposure value has to be around 1000, and I have not checked the response at this level. I might have to remeasure it.

==ccd count vs power== 

 This measurement is similar to the above. But this time the incident power (to the sample) is varied. The result is not linear. I check the images and see that the bright spot moves. The camera might move during the measurement. I'll repeat this again. It will be complicated for the calibration if the ccd count is not linear with the power.

== To do==

• check the ccd count for exposure value down to lowest setting.
• check the ccd count for different power incident.
• check the ccd count with different ND filter in front of the camera.

VCO driver noise measurement

To measure the noise of the PSL VCO driver, we used the same PLL set-up from previous noise measurements.  The PLL consisted of the following: IFR/Marconi 2023A, the SR560, Mini-Circuits Frequency Mixer ZX05-1MHW-S+-0.5-600MHz, Mini-Circuits 15542 BLP-5+ Low Pass Filter 50 Ohm DC-5MHz, and the Stanford Research Systems Model SR560- Low Noise Pre-amplifier with a gain of 200 V/V.  We connected another VCO to the RF port of the mixer.   The Marconi had a carrier frequency of 80 MHz, an RF level of 13 dBm and FM dev set to 1 KHz Ext. DC.  

We connected the VCO to a power supply by hooking up a 9-pin dsub breakout box into the VME interface.  The VCO driver needs 24V from the power supply.  From opening up the box, we found that there are three test points in the VME interface: TP1, TP2 and TP3.  TP1 corresponds to -24V, TP2 corresponds to +24V and TP3 is ground.  Additionally, we needed to figure out what pins to hook up the positive, negative and ground cables onto the breakout box.  +24 V corresponds to pins 9 and 4, -24 V corresponds to pin 5 and ground corresponds to pins 8 and 3.  There are also two switches that need to be connected to the ground in order for the driver to function properly.  The test switch, which corresponds to pin 1 and the wide switch, which corresponds to pin 6 are both connected to pin 3(ground).  We used the TENMA Laboratory DC Power Supply 72-2080 and it was set to 24 V and .5 A.  

After locking the frequencies, we measured the transfer function and ASD with an FFT analyzer (Agilent 35670A Dynamic Signal Analyzer).  The following data was obtained:

Marconi Noise Measurement

 To verify whether or not the noise was from the AOM driver, we measured the noise in the Marconi.   We set up a PLL to do this measurement.  We used the IFR/Marconi 2023A, the SR560, Mini-Circuits Frequency Mixer ZX05-1MHW-S+-0.5-600MHz, Mini-Circuits 15542 BLP-5+ Low Pass Filter 50 Ohm DC-5MHz, and the Stanford Research Systems Model SR560- Low Noise Pre-amplifier with a gain of 500 V/V.  We  connected another Marconi to the RF port of the mixer.   Marconi 1 had a carrier frequency of 80 MHz, an RF level of 13 dBm and FM dev set to 1 KHz Ext. DC.  The second Marconi that we used had a carrier frequency of 80 MHz, an RF level of 2 dBm to match that of the AOM driver and FM dev disabled to prevent noise.  

 Set-up:

Measurement: After locking the frequencies, we looked at the measured the PSD on an FFT analyzer (Agilent 35670A Dynamic Signal Analyzer) and obtained a measurement of the ASD.

It appears that the noise is coming from the AOM driver.  

I'm testing the setup and a code for extracting scattered light from the images. 

I used a red laser pointer to test the scattered light setup. Then took a picture with no light (fig1) and a picture with the incident light (fig2). The scattered light can be extracted by subtract fig1(background) from fig2.

The snapshots saved by SampleViewer are in .bmp file. When it is read by MATLAB, the file will contain 480x752x3 matrix element, Each are varied between 0 and 255. The values are proportional to the brightness (how many photons hit the cell). 480x752 is the resolution of the image, x3 are for R G B color. In our case, the image is greyscale and the values are identical. The code can be found in the attached file.  

fig1: The test mirror without incident beam taken as a background image. The image is enhanced by a factor of 5 (by matlab).

fig2: The test mirror with a red incident beam around the center. The image is enhanced by a factor of 5.

fig3: the image is created by subtracting data of fig1 (background) from fig2 (scattered light) and enhanced by a factor of 100. The scattered light on both surfaces can be seen clearly around the center.

 ==To do next==

• From fig 3, the background can be seen even after subtraction, so some black curtains and beam dumps should be added behind the mirror.
•  A room light filter should be installed in front of the camera.
• I'll see if we can find a sample with known scattering loss, so that we can compare how accurate the measurement is.

We looked at the beat signal of the reflected beam and the beam that is double-passed through the AOM on the oscilloscope.  The first beam has a power of 813 micro-watts and beam that is double-passed through the AOM has a power of 10 micro-watts.  The beat signal fluctuation was between -24 dBm and -13 dBm.  A PLL was used to lock the frequencies.  We used the IFR/Marconi 2023A and the SR560.  The Marconi had a carrier frequency of 160 MHz, an RF level of 13.0 dBm, and FM dev of 10KHz.  The gain on the SR 560 was 200 V/V.  Once the frequencies were locked, we measured the PSD on a spectrum analyzer.  

After obtaining this data, we measured the phase noise of the AOM driver (Crystal Technologies 1080AF-AIF0-2.0 S/N 10351) since it was suspected to be a large source of noise.  We set up a PLL to do this measurement.  We used the IFR/Marconi 2023A, the SR560, Mini-Circuits Frequency Mixer ZX05-1MHW-S+-0.5-600MHz, Mini-Circuits 15542 BLP-5+ Low Pass Filter 50 Ohm DC-5MHz, and the Stanford Research Systems Model SR560- Low Noise Pre-amplifier with a gain of 500 V/V.  The AOM driver was connected to the Stanford Research Systems-Model DS345 30 MHz Synthesized Function Generator with a frequency of 6.00 Hz and Amplitude of .7 Volts and the TENMA Laboratory DC Power Supply 72-2080 with a current of .5 Amps and voltage of 28 Volts.  The AOM driver output had a 10 dB heatsink attenuator followed by a Mini-Circuits 50 ohm-31030 15542 VAT-9+ 9dB attentuator  and another Mini-Circuits 50 ohm-30727 15542 VAT-10+ 10 dB attenuator, which was connected to the RF port of the mixer.  

The settings on the Marconi for locked frequencies: Carrier Frequency: 79.992423 Hz, RF Level: 13 dBm, FM dev: 1.00 KHz, Ext DC

After locking the frequencies, we looked at the measured the PSD on an FFT analyzer (Agilent 35670A Dynamic Signal Analyzer) and obtained a measurement of the PSD.  After plotting the results, we found that the driver noise data almost completely lines up with the measurement of the noise in the set-up.  This is most like the driver noise, but it is possible that there is noise from the Marconi or SR 560, so a similar set-up will be used to analyze noise in the Marconi to determine whether or not the noise is from the AOM driver.  

How hot do you need to heat it? if the thermal expansion of aluminum/steel is much higher than that of fused silica, then just heating the end cap might be a better idea. Thermal conductivity is also better.

I'm setting up a scattered light measurement for AlGaAs samples. The methods are summarized below.

I discussed with Manasa about the setup and how to do the measurement. The goal is to measure  scattered losses from AlGaAs samples from a normal incident beam. The setup is shown below.

==setup==

The setup is in the ATF lab, on the unused optical table. It is too crowded on CTN table. So I will need a to borrow a 1064 laser from somewhere. 

The incident beam will have to be slightly angle from the normal angle in order to dump the beam properly. 

The arm holds the camera, it can rotate to change the angle to cover the measurement from around 10 degrees to ~70 degrees.

==calibration method==

• We can take a picture from a diffuser plate, make sure it is not saturated. 
• Then use a power meter, measure the power fall on the camera.
• compare the output of the camera and the measured power
• I have to think about how to make sure the solid angle of the camera aperture and the power meter are the same.

==measurement and data analysis==

• For each position, take one picture without the beam on the mirror and one with the beam on the mirror. The first one will be used for subtracting the ambient noise from a picture when the beam is on the mirror. 
• Make sure that no pixel is saturated
• For each pair of picture, we will use Matlab to count the output, then use the calibration to convert to power.
• integrate over half the sphere. I have to think about this to make sure I get it right.

The CTE of fused quartz is something like 0.5×10⁻⁶ K⁻¹, and the CTE of steel is more like 15×10⁻⁶ K⁻¹. So I suspect there's not much point in heating the glass spacer if I'm going to leave the steel end cap open to air.

A possible solution is to put a heater on the end cap, but I worry that the differential expansion of steel vs. glass will cause the end cap to pop off the spacer (it looks like it's only held on by epoxy).

A better solution is to improve the insulation on the back end of the PMC. I'll do that next.

Installation of optics for fiber phase noise measurement

After light passes through the AOM, it is reflected back through the AOM and into the fiber.  We installed a 50/50 beamsplitter, quarter wave plate, mirror, lens and photodiode to do the beat measurement.  It is required that the beam spot size is 1/3 the diameter of the photodetector.  We installed a lens at the appropriate distance to obtain a waist that is roughly 50 microns.  We hooked up the photodiode to an oscilloscope and found that the voltage fluctuates between 100-500 mV.  We are not sure why the voltage is fluctuating, but we will continue to investigate the cause.  

I took some hard yellow foam, made it into a U-shape, and wrapped it with a combination of aluminum and duct tape.

This insulation fits snugly over the PMC and its copper shield. In retrospect, the foam is probably a little too thick. I had to temporarily move the beam dump at the input of the Faraday isolator.

Putting 20 V across the 105 Ω heater produces a change of 5 V on the PMC PZT (when locked). So we need better insulation or more heating.

I took a 105 Ω Kapton heater, stuck it to a 15 cm x 25 cm patch of copper foil (thanks Steve), and wrapped the foil around the PMC. This required undoing the top braces on the mount. Currently, the PMC is just sitting on its kinematic contacts. Virtually no tune-up of the pointing was required.

There is currently no insulation, so (perhaps unsurprisingly) the heater doesn't have much of an effect on the PMC's PZT voltage.

The calculation for LIGO astronomical reach with uncertainties are updated. See the details below.

       I got the updated GWINC from Nic. When I run the nomm file, the BNS range is 189.5 Mpc. The nominal value of the refractive index for nH in the code is 2.06 which is the value for the pure Ta2O5. The refractive index for Ta2O5 doped with TiO2 25% (Ta2O5:TiO2) is 2.119 (see Harry et al 2007 paper). So when I changed nH to 2.119, the BNS range became 192 Mpc due to the thinner coating. 

     The ring down measurements from Harry 2007 paper measure parallel loss of the coating from multilayer coating, then extract PhiH from the measurement. Again, this is done by assuming the knowledge of phiL, YL and YH. The loss angle of silica (phiL) used in the paper is 1e-4 (from Crooks 2006 paper) while the value in GWINC is 0.4e-4. In this calculation, I use phiL = 1e-4 because of a couple reasons:

1. The value of 1e-4 is determined in crooks paper. Their samples and the samples used in Harry 2007 paper are for fused silica with the same heat treatment (annealing temp ~ 400-600 C). I think 0.4e-4 is from higher heat treatment (up to 700-800 C) (see Penn 2010 LVC march presentation, here).
2. In Harry 2007 measurements, sample 1 is a multilayer structure of fused silica and Ta2O5: TiO2(6%). Sample6 is a single layer of Ta2O5: TiO2(6%). Sample 6 can determine the parallel loss of phiH directly without the knowledge of fused slilca's material parameters (thin form). When the loss angle of Ta2O5:TiO2(6%) as measured from sample6 is combine with the loss angle of fused silica = 1e-4, the result from the measurement from Sample1 and the calculation agree

So the value phiL = 1e-4 is only used for extracting phiH as a function of YH, while phiL = 0.4e-4 is used as a nominal value in noise budget calculation.

Running the code

The calculation is done in the code name BNS_score.m (see the attached zipped file). The file calls on IFOModel_rnd2.m that generates random material parameters by normrnd command. The plot can be made by running plot_hist.m file.

fig1: histograms and gaussian fits for coating BR noise and total noise at 100Hz. 

fig2: The astronomical range for BNS, in MPC unit.The histogram is from 20k samples. The average is at 189.2 Mpc, while the mode is around 188 MPc.

This is now built, with a few modifications:

• The power supply bypass capacitors are 1 µF tantalum caps.
• The RC low-pass is a  4.8 kΩ resistor and a 3.3 µF capacitor, giving a 10 Hz pole.
• The base of the heating resistor is tied to the negative rail instead of ground, in order to give a greater actuation range.

Both the in-amp stage and the mosfet stage seem to work fine using a 2 kΩ resistor in place of the heater (the actual heater is more like 120 Ω, but the axial resistors in the e-shop are only rated to 0.25 W).

Instead of using a power supply as a current buffer, we can use a mosfet like so:

This is based loosely on the aLIGO PMC heater (D1001618-v1, p. 10).

If we instead want to run off of a unipolar supply, we can replace the AD620 with a noninverting op-amp. We'll lose the common-mode rejection, though.

 Manassa is helping me installing a camera for scattering measurement. The work is in progress.

I'm borrowing a Prosilica gc750 from the 40m. It will be used for scattering measurement on AlGaAs samples. It is a good idea to have a setup that can quantitatively measure scatter loss on mirrors. 

First I tried to install it on the small Acer laptop used with win cam, but it did not work. I'm not sure if the ethernet card of the laptop does not support the camera or not. Now I'm trying to install it on my mac book instead, since Manassa claimed that it worked on her macbook. 

I'll write a step by step installation guide once we succeed.

==note about the AlGaAs samples== 

I used a green laser pointer to check scattering loss on one sample. I couldn't see any green spot of the laser with my eyes. This means that the scattering is probably less than 100 ppm (according to Josh). Once we use the camera to measure it and it turn out to be smaller. We will probably go to Fullerton to have the samples measured there for better accuracy. 

A list of small tasks and some data points:

As of yesterday, the tank is floated. This required minimal realignment of the input pointing into the cavities.

Adjusted powers so that there is 1.04 mW incident on north and 0.96 mW incident on south.

For the south PDH loop, the highest gain we can get seems to be 590 common and 710 fast. Not great. The south error signal has terrible 270 kHz oscillation as well (~100 mVpp).

For the north PDH loop, the highest gain we can get is 807 common and 908 fast. Not perfect, but better than south. No 270 kHz oscillation here.

South path adjustments

RAM on the south PD was terrible: 259(28) mVpp at the PDH frequency; the uncertainty is dominated by slow breathing of the RAM amplitude. I need the PD rf transimpedance to convert this voltage to an actual RAM.

I tried adjusting the alignment of the south EOM, with little effect. The big effect came from slightly rotating the λ/2 plate immediately preceding the EOM: rotating by less than half a degree takes the RAM from >200 mVpp through zero and back to 200 mVpp again. The λ/2 plate was in one of those no-frills rotational mounts where sub-degree precision can be achieved only by nudging, so I instead put the waveplate into a precision mount with a worm drive and a knob. I then tuned the rotation to null the RAM on a scope. There is still some breathing of the amplitude, so that at times the RAM is 40 mVpp. Not good, but better than before.

I measured the south modulation index both before and after this change. I swept the south laser frequency and watched the transmission on the ISS PD. Before, the carrier and single-sideband transmission peaks were 1.91(1) V and 39.6(1.2) mV, respectively, and after they were 1.72(2) V and 38.2(6) mV, respectively. This means the modulation index actually increased from 0.288(4) to 0.298(3) (using the Γ²/4 approximation).

Beat measurement

Installation of optics for fiber phase noise measurement

Following the fiber output, which has a waist of ~50 microns, we calculated the proper lens to use as well as the proper distance to place the objects so that we would have a waist of approximately 150 microns going into the AOM.  Roughly 3.5 inches from the fiber output, we placed a lens: KBX052 with a focal length of 50.2 mm, followed by an AOM: 3080-194, as well as the AOM driver(1080AF-AIF0-2.0) 3 inches away from the AOM to the right. After the light passes through the AOM, we placed another lens: PLCX-24.5-36.1-C-1064, which gives another waist at the mirror placed at the end of this setup.  After the light passes through this lens, we placed a quarter wave plate: Z-17.5-A-.25-B-1064, which is followed by a mirror: PR1-1064-98-1037.

South laser slow at 1.234 V, north laser slow at 5.558 V, beat is 120(1) MHz at +5.5(2) dBm. South and north alignment has not yet been tuned up.

SR785 appears to have broken screen.

I wanted enough power to accommodate both the fiber noise measurement and the south cavity locking. I moved the HWP after the PMC from 338 degrees to 79.5 degrees. Then I moved the HWP after the south EOAM from 249.5 degrees to 280.0 degrees. This gives 1.5 mW transmitted through the PBS toward the south refcav, and a few milliwatts reflecting off the PBS and going toward the fiber.

It looks like we still have good mode-maching into the south cavity; transmission is easily seen on the camera.

 I just saw you comment. I'll find an update version for GWINC.

Anyway, I have a code to plot the result. I will use it on an updated code.

Some material parameters in the calculation are:

•  The concentration of TiO2 is 25 % based on Harry 2010 paper (Class. Quantum Grav. 27 (2010) 084006).
• Young's Modulus of Ta2O5 doped with TiO2 (25%) is ~ 140 +/- 6 GPa (one sigma).  (Abernathy et al. 2014). 
• Loss angle of TiO2 doped Ta2O5 is  (2.3 +/- 0.4) x 10^-4  [Martin 2008, Martin2009] (@300K, heat treated) 
• Loss angle of SiO2 is (6.3 +/- 2.6) x10^-5   [LMA2014, Crooks2006, Crooks2004]

In the IFOModel_rnd.m file which is a copy of IFOModel.m for material params, I use normrnd(mean,sigma) to generate the random value of the material parameters.

Note, for loss angle of SiO2, I have to use abs command to make sure that all the generated values are greater than zero.

This is because the mean is comparable to the standard deviation, and sometime it gives negative values.

Note: I have not taken the coherent between the loss and Young's modulus of Ta2O5 into account yet. I have to read how they measure this more carefully.

Here are prelim results from the above numbers.

above: a histogram of BNS range, due to uncertainties in loss angles/young's moduli of the coatings.

The mean of the histogram is slightly less than the nominal value from GWINC because the mean values of loss angles for fused silica (6e-5) is slightly higher than the original value (4e-5) used in the code.

above: histograms and Gaussian fits for BR noise (blue/green) and total noise and its Gaussian fit (red/cyan) at 100 Hz in the strain unit.

Supplies

• Foam (already have)
• Aluminum tape
• Kapton heater (already have)
• High-current buffer (maybe can use linear power supply with aux inputs on back)

Computing

• Set up or co-opt channel to output to heater
• Modify PMC MEDM screen to accommodate heater

Installation of optics for fiber phase noise measurement

To couple CTN light into the fiber, we decided to pick off using the reflected port of the PBS directly after the south EOAM. In order to mode-match into the fiber, we installed two lenses (and a steering mirror) between the PBS and the fiber.

Mode matching details are as follows: the round trip length of the PMC is 42 cm and the radius of curvature of the concave mirror is 1 m; this gives a waist of 370 microns.  From there, we calculated the proper lenses needed: PLCX-25.4-64.4-UV-1064 (lens 1, focal length 124 mm) and a PLCX-25.4-128.8-UV-1064 (lens 2, focal length 250 mm). Between the two lenses is a mirror, Y1-1037-45-S, which is tilted at a ~45 degree angle to guide the light from lens 1 to lens 2.  Lens 1 and Lens 2 are roughly 2 inches away from each other. There is a fiber coupler placed 4 inches away from the second lens.

Currently, is about 1.4 mW going into the fiber, and about 150 uW coming out.

Edit: We did some more aligning and found that there is 2.2 mW going into the fiber and .7 mW coming out.

PMC alignment tune up, FI power adjustment

Coupling through the PMC was very bad today; we saw 12 mW incident and ~1 mW transmitted. I (Evan) touched the three steering mirrors before the PMC and brought the transmission up to 5 mW.

In order to have more power incident on the fiber, we changed the angle of the HWP immediately after the PMC from 306.5 degrees to 338.0 degrees.

 That GWINC link is more than a year old. You're best off just updating your CVS checkout of the code, or getting a new zip file from someone else if your CVS is broken. When I run gwinc with nomm.m, I get R_BNS = 189.5 Mpc.

I'm estimating the BNS range of aLIGO. Here is a quick note about the calculation.

•  The range can be calculated by GWINC, more details about the calculation can be found in T090049
•  This is for NS-NS (1.4M_solar) CBC.
•  The change in sensitivity due to material parameters results in a different LIGO detection range.
• One sigma here means one standard deviation. +/- one sigma cover 68% of the normal distribution. 

For example, the normal configuration for aLIGO will have BNS in spiral range equal to 178.29 Mpc (based on the current code available on gwinc.

I spent some more time talking to Larry (who is a national treasure) about how to properly estimate the statistical uncertainty on the fit to φ_c. Among other things, he suggested dispensing with the nonlinear fit to S versus f and instead performing a linear fit to log S versus log f.

I implemented this, and I got a fit of 4.43(25)×10⁻⁴ for φ_c, and PSD slope of −1.004(11). Here I've also settled on 144(42) GPa for the Young modulus of tantala.

Then putting this value for φ_c into the Bayesian notebook, I find a MAP of 1.1×10⁻⁴ for the silica loss angle, and 7.8×10⁻⁴ for the tantala loss angle. The percentile values of the PDFs don't change much.

I've incorporated these into the paper, which is now on the DCC at P1400072v3. I've gone through the noise budget code and the Bayesian notebook in order to check that all numerical values are consistent, and that they are reported correctly in the paper.

 Using this new value of φ_||, I reran the Bayesian analysis notebook and also (on Larry's suggestion) generated a plot of the marginalized PDFs with shading to indicate the 16th, 50th, and 84th percentiles. The maximum a posteriori estimates of the silica and tantala loss angles are 0.6×10⁻⁴ and 8.0×10⁻⁴, respectively. I've incorporated these new plots into the paper.

I have finished implementing the comments from the LSC P&P review.

On Steve Penn's suggestion, I went back to Harry et al. (2002) and Penn et al. (2003) and attempted to rederive the parallel loss angle φ_||, along with the experimental uncertainties.

Harry's original number (using a coating thickness that is 5 times too high) was φ_||,wrong = (1.0±0.3)×10⁻⁴. I found φ_||,wrong = (0.98±0.14)×10⁻⁴.

Penn's corrected number is φ_|| = 5.2×10⁻⁴, with no error bar. I found φ_|| = (5.17±0.75)×10⁻⁴.

You can see my working in the attached pdf and ipynb.

 I'm working on estimating aLIGO sensitivity when material uncertainties are taken into account. I have a result for a reference cavity, uncertainty due to Ta2O5's Young's modulus might have smaller effect than we previously expected. All plots and code are attached below.

 ==Intro==

  GWINC does not take any uncertainties in material parameters into account, so its noise budget does not have any error bar. We want to know how the noise budget might change due to imprecise knowledge of the material parameters. One particular issue is coating thermal noise that is dominating around 30 - 200 Hz, so we want to know how its level will change with material parameters. Some import ant parameters are loss angles and Young's moduli of each material. 

  In Hong et al 2013 paper, there is a plot of the calculated coating Brownian noise vs Ta2O5's Young's modulus (YH). The calculated coating BR noise is calculated with the corresponding YH while other parameters are fixed. This would be ok if each parameters were independently measured. In reality, loss angles are measured from ring down measurements, and YH and YL are used to calculated the material loss angles (phiH/phiL), see Penn et al. 2003. So to make the calculation reflects the real situation, we should take the correlation between phiH/phiL and YH/YL into account when we calculate coating BR noise. So the goal is to estimate coating BR noise for aLIGO with some uncertainties from loss angles and Young's moduli of the coatings.

==calculation== 

calculate BR noise vs YL and YH  (see PSL:1408 for the original code) using numbers from our setup (these can be changed later when we want to apply for aLIGO calculation). The code calculates BR noise with phiL = 1e-4, phiH = 8e-4. the numbers are from our measurement and another ring down measurement. This does not take the correlation between loss any YL/YH into account.  I do this to compare my code to Hong's result, and they agree. YL and YH are varied between 80% and 120% of their nominal values (YL = 72 GPa, YH = 140 GPa).

above: Fig1:Thermal noise level of 28 Layer QWL structure, spot size = 180 um as a function of YH and YL

above:Fig2: three slices from the 3-d plot for different values of YL, Y_L min and Y_L max are 80% and 120% of the nominal value. 

 above:Fig3: three slices from the 3-d plot for different values of YH, Y_H min and Y_H max are 80% and 120% of the nominal value. 

Next, let's assume that the values of SiO2 are well measured and the error is much smaller than those of Ta2O5, so we can fix phiL and YL. Then recalculate BR noise when phiH and YH are correlated. I use a calculation from ring down measurement (see PSL:1412 or Harry 2002 or Penn 2003). The equation is

constant = phi_parallel = (YL*dL*phiL + YH*dH*phiH) / (YL*dL + YH*dH)   

from this equation, we can write phiH as a function of YH assuming that other parameters are constant. Currently, I'm using numbers from CTN setup.   

above: Coating BR noise as phi_H is varied along Y_H (green) compared with the previous calculation (Blue) from fig 2. The two traces cross at YH = 140 GPa.  Note that in this plot, YH is varied between 50% and 200% of the nominal value. We see that the uncertainty of coating noise due to YH becomes smaller compared to the previous calculation done in Hong paper. 

==what to do next==

• Apply my calculation to GWINC: Right now it is a stand alone code, but I plan to write a code that uses parameters and function mostly from GWINC because it will be easy if we want to plot it later for aLIGO noise budget. 
• Find out the uncertainties of YH and phiH  for Ta2O5 doped with TiO2, check the amount of TiO2 for aLIGO coating.   
• Plot aLIGO noise budget , and estimate the inspiral range due to the sensitivity.

==Note==

From fig3, uncertainty in YL does not change the BR noise level that much, but this calculation assumes no correlation between YL and phiL. I have not been able to include uncertainty in YL and see the effect on phiL yet, because that will need more constraint equation. But I should check if it will greatly change phiL and affect the total BR noise calculation or not.

I used the 633 nm fiber illuminator and the ThorLabs power meter (set to 633 nm) to test the 60 m polarization-maintaining fiber that we have.

Power right out of the illuminator was 1.25(2) mW, and the power out of the fiber was 0.45(1) mW. Since this fiber is only specked to work above 980 nm, I'm not sure how to interpret this number.

I'd like to compare to the 35 m PX980-XP fiber we have strung from CTN to Crackle.

I performed the same test with the 1060XP fiber (50 m, not polarization maintaining). I got 0.12(1) mW transmission.

I did an optimized structure for ITM and plotted the estimated noise budget of AdvLIGO using optimized AlGaAs coating on ETM and ITM. More details will be added later.

Above: Optimized structure of ITM

Above: AdvLIGO with Optimized AlGaAs coatings on SiO2 substrate, room temp. The plot is generated by GWINC.

Here is a prelim result for AlGaAs TO opt for ETM coating.

The optimization is named opt_ETM5 in .mat file. The structure is in optical length unit ( the physical thickness = (opt length) * 1064e-9 / n). The first layer is the air-coating GaAs layer . For the current optimization (opt_ETM5.mat) the transmission is 5.4 ppm, the reflected phase is off by about 2 degrees. 

ETM parameters used in the optimization

• Spot radius (1/e^2 power) = 6 cm
• Substrate radius =19 cm (GWINC)
• Substrate thickness = 20 cm (GWINC)
• Target transmission = 5 ppm

Note about optimization:

•  The current optimization is just a prelim result for demonstrating that it is possible to optimize coating structure for ETM.
• I have not included any uncertainties in the result.
• The cost function used for this optimization is a only for nH = 3.48, nL = 2.977. So the coating properties might change rapidly with the uncertainties of the refractive indices see PSL:1356. I have not used the cost function with multi-value refractive indicesbecause this optimization is intended for parameter configuration (adjusting the target TO noise level, weight functions for each parameters, numbers of layer). 
• The optimization was done for room temperature (300 K) @1064nm.
• All the codes are attached in the zip file.

 To run the code:

• run optETM.m with the current parameters. The result structure will be called xout in matlab workspace.
• use CoatingThermalNoiseCalcETM.m to plot the noise budget.
• I might miss some functions in the zip file, one is multidiel_rt.m , those are available on svn directory

Add the measurement from AlGaAs coating, and Silicon refcav (see CRYO:1045). The source file, figs, and eps files are attached in the zip file.

I plotted our measurement together with other experiments. The source file and fig files are attached below.

Details about each experiment (cavity length, wavelength), are included in the source file.

Rather than using individual loss angles from Penn as the prior pdf, I've instead reanalyzed the data from Harry et al. (2002).

The ipynb for this is on the SVN in the noise budget folder.

Zach helped me re-setting up the channels for PSL lab.  The two channels are: 

• channel (#29 on the panel): C2:ATF-PSL2_OUT_DAQ
• channel (#30 on the panel): C2:ATF-PSL3_OUT_DAQ

The sampling rate is set to 10kHz (8192 Hz).  

 Anti-alias, provided by foton, cheby2, low pass at 4096 Hz, is induced in both channels.

calibration 2^15 count for 20 V -> 20V/ 2^15  = 6.1035e-4 V/count 

==Note about start configuring MDV==

Run this from the terminal on ws2:
 eval `/cvs/cds/caltech/apps/linux/ligotools/bin/use_ligotools`

 Then, start matlab  and run:
 run /cvs/cds/project/mDV/mdv_config

(there are some error messages about the paths that can't be added (matapps_SDE, matapps_path , frame cache, ligotools_matlab, home_pwd.) ,but they are irrelevant.  

Now gps and get_data commands are working. We checked with the test signal and see both time domain, and frequency domain. The anti-alias filter is working fine.  

I reran the notebook with the following modifications:

1. I changed the prior on φ₂ to 4.4(2) × 10⁻⁴, which is the correct value from Penn. I'm not sure why I had the uncertainty at 4.4(7) × 10⁻⁴ before.
2. I changed the uncertainty on the Young modulus of tantala to 30% (up from 10%), since it sounds like we don't really believe the literature values for the Young modulus all that much.

The posterior estimate for the loss angles is now φ₁ = 1.4(3) × 10⁻⁴ and φ₂ = 4.9(2) × 10⁻⁴, which is much more in line with previously measured values. See the first set of plots.

Comparison with Penn et al.

Since we're using a prior pdf generated from Penn et al., it seems wise to check out what happens if we use a likelihood function that's generated from the same formalism that Penn et al. use. Their eq. 6 gives the relation between φ1, φ2, and φc:

(N1 d1 E1 + N2 d2 E2) φc = N1 d1 E1 φ1 + N2 d2 E2 φ2

where N1 is the number of silica layers, d1 is the thickness of each silica layer, E1 is the Young modulus of silica, and likewise for the tantala parameters. The results are attached in the second set of plots. The posterior estimate is φ₁ = 0.7(3) × 10⁻⁴ and φ₂ = 4.9(2) × 10⁻⁴, in pretty good agreement with what we get with the likelihood from Hong.

Discussion

What I've done above (decreasing the uncertainty on the prior and increasing the uncertainty in the Young modulus) amounts to strengthening the effect prior and weakening the effect of the likelihood. So it's not surprising that the posterior is now closer to the prior.

This does not resolve the issue that both the likelihood functions have slopes that are (we think) too steep. If, for example, we assumed an informative prior for φ₁ [1.0(2) × 10⁻⁴, say] but left the prior for φ₂ flat, our posterior would give a value of φ2 that is very high (9 × 10⁻⁴ in this case).

[Edit, 2014–04–17: On Larry's suggestion, I tried marginalizing instead of just slicing through the MPE. The results are the same. —Evan]

I removed the 90% reflector from the north transmission path on the ISS breadboard and then installed the fiber launcher.

The ThorLabs power meter says 440 uW going into the fiber on the CTN side; the ThorLabs fiber power meter says 260 uW coming out on the ATF side.

Here's a naive attempt at a Bayesian estimate of the loss angles of silica (φ₁) and tantala (φ₂). The attached zip file contains the IPython notebook used to generate these plots.

To construct a joint prior pdf for φ₁ and φ₂, I used the estimates from Penn (2003), which are φ₁ = 0.5(3) × 10⁻⁴ and φ₂ = 4.4(7) × 10⁻⁴ and assumed the uncertainties were 1σ with Gaussian statistics.

For the likelihood I used the relationship between φ₁, φ₂, and Numata's φ_c. This is derived from the Hong paper, and is described in the pdf inside the zip attachment.

Next steps from here:

1. We need to check with someone in the data group or in Cahill to make sure this is the right way to do the analysis.
2. We need to think harder about constructing our prior pdf.
3. We need to think harder about our uncertainties (have I accidentally double-counted some of the uncertainties?)

Heat treatment after coating changes loss angles of both SiO2 and Ta2O5. Our coating might really have higher loss (maybe because of the low temp annealing), regardless of the actual values of coatings Young's moduli.

I went through the paper by LMA2014 that measured loss in SiO2 and Ta2O5 using interferometry on a cantilever blade. They could also extract Young's moduli from thin film SiO2 and Ta2O5, their results are ~ 70GPa and 118 GPa respectively.

One interesting result is that losses are reduced with heat treatment after coating process.

SiO2 loss before heat treatment is ~ 6e-4,  and it goes down to ~ 0.6e-4 after the annealing (from broadband measurement).

From ring down measurement, SiO2 before heat treatment is ~ 4e-4. no result for the measurement after annealing.

For Ta2O5,from broadband measurement, the loss after heat treatment is ~ 4.7e-4, no result from the before heat treatment is reported.

From ring down, the loss is ~ 11.4e-4 before annealing, and down to ~ 4.9e-4 after annealing. 

Their annealing process is described in the paper.  I should find out more how losses of both materials change with different heat treatment i.e. time/ temperature/cooling, then see if any information about our mirrors can be retrieved from REO or not.

Right now, we have only the information about phiH and phiL as phiH = a*phiL + b. I still need another relation to get phiH and phiL individually. My plan is finding information about heat treatment vs loss, like the picture below (I still need to find for Ta2O5). Otherwise, it is hard to say anything about the loss from each material.

Most reports have different annealing temp, (I'm not considering time/ heating rate/ cooling rate right now, but they might be important) So I can compare loss vs annealing temp.

• LMA2014 without annealing and annealing @ 500C, for SiO2 and Ta2O5, thin film.
•  G1000356: Thin SiO2 at 300C, 600C 800C, 900C. This shows that loss decreases as annealing temp increases, see the picture below. Notice that loss at 600C is comparable to what report in LMA (0.6e-4, @500C).

It is hard to extract the similar plot as above for Ta2O5 from Martin2010 paper. I'll try to ask Ian Martin if he can give me the raw data. 

From the loss vs annealing temp I found out below, it seems that the annealing temp for our mirrors will be less than 300C. Since at 300C, silica loss is ~ 1.5e-4, tantala loss is ~ 4e-4. These numbers give the estimated BR noise below our measurement. 

Note:

•  find out if Young's moduli change with annealing process. 
• Coating layer thickness increases after heat treatment effect. This directly increases thermal noise. Find out how much it could be.

ref: silica loss with vs different heat treatment temperature. https://dcc.ligo.org/DocDB/0010/G1000356/001/PennCoatingMarch10.pdf

Martin2010: Class. Quantum Grav. 27 (2010) 225020 (13pp): Tantala loss with different heat treatment temperature

 I used results from ring down measurement in Penn 2003, without assuming the values of YL,YH. If the actual Young's moduli of both materials are about 60% of their nominal values, the calculation of BR noise will match our measurement within 3%. 

I used ring down drumhead mode from sample C2 and F2 since the phi_coating as reported in the paper is about the same as the phi_coating obtained from the analytical result (see previous entry). With these two eqs, I can write

Ysub * D/3 * phitot_1  = phiL*YL*dL_1 + phiH*YH*dH_1-------(1)  (see previous entry, last eq).

Ysub * D/3 * phitot_2  = phiL*YL*dL_2 + phiH*YH*dH_2-------(2)  .

phi tot_1 and _2 are 1/Qtot from the two samples. D is the thickness of the substrate (0.25 cm). dL and dH are the physical thickness of siO2 and Ta2O5 in each sample.

For any fixed values of YH and YL, the two eqs will solve for a pair of phiL and phiH.

First, I checked the validity of these two ring down measurements by using YL = 72 GPa, YH= 140GPa. The results are 

PhiL = 1.29e-4, phiH = 4.13e-4. These numbers agree with the reported values.

Then, I varied YH from 0.5*YH_0 to 2*YH_0 and YL from 0.5*YL_0 to 2*YL_0 ( YH_0 = 140GPa, YL_0 = 72GPa), and solved for the corresponding phiL and phiH. Then with all 4 parameters, BR noise can be calculated.

Below is a plot of ratio of BR calculation and our measurement, vs YH. Each trace represents different value of YL.

Each point on the plot will have information about phiL and phiH. If YL = 43 GPa (0.6*72GPa) and YH = 84 GPa (0.6*140GPa), the loss angles extracted from the ring down measurements are phiL = 2.15e-4 and phiH = 6.9 e-4. All these four parameters give the estimated BR noise comparable to our measurement to 2% (in PSD unit). 

==Conclusion==

 I'm trying to explain why our measurement is larger than the estimated calculation using numbers from literature. But we have good reasons to believe that the measurement is really BR coating since 

• The data has a correct slope, 1/f in PSD
• Scale with 1/w^2 in PSD, (BR noise from substrate/ spacer will have different scaling)
• Agree with Numata2003.

It is possible that loss angles in our coating is lossier than usual. But there are still other possible explanations. The results from ring down measurements rely on the values of Young's moduli of the coating materials. If the actual values divert from the nominal values, the losses will be changed as well. So I used the result from the ring down measurement, without assuming any values of YH and YL, then extracted values of phiH and phiL using different combinations of YH and YL and  calculated the coating noise according to each set of parameters.  If YL and YH have lower Young's moduli than their nominal values, coating BR noise will be higher and agree with our measurement. 

One might argue that 0.6 YL and 0.6 YH are too low.  Ta2O5 was measured with nano indentation to be ~ 140 GPa (Abernathy). Other references measured Ta2O5 ~ 100 GPa (see ref 16, 20 in Crooks2006 paper). So, uncertainty around 40% might be possible.  

In addition, this calculation also assume phi_bulk = phi_shear.  But the different value of phiB/phiS can also change the calculation between 0.5*S_0 to 1.6*S_0, for different values of phi bulk/phi shear ratio is varied by a factor of 5(see Hong2013). These values also change the noise level significantly.

So with the uncertainties in Young's moduli, the loss angles from ring down measurements can be changed significantly.  If the Young's moduli of the coatings are smaller than the nominal values, the loss angles calculated from a ring down result will be higher, and  it resuls in a higher level of coating BR noise calculation.

==Note== 

I'm surprised that for the value of 0.6*YL_0 and 0.6*YH_0 used above, with the loss angles of phiH = 6.89e-4 and phiL = 2.15 e-4, the calculated BR noise is almost the same as when I use the nominal value of YH,YL with the same loss (2.15 and 6.89e-4) see, PSL:1408.  I double checked the result, but I did not see anything wrong in the calculation. It turns out that the BR calculation is not very sensitive to YL, YH, but it is directly proportional to phiH, phiL. However, the values of phiH, phiL obtained from a ring down measurement are very sensitive to YL and YH  as we can see from the plot above.  

This formula is only sensitive to the ratio YL/YH (which I've called E1/E2).

I took the parameters from Penn, chose two fiducial coating thicknesses (a λ/4 + λ/4 coating, and a 3λ/8 + λ/8 coating), and used this (along with Penn's reported values for E₁, E₂, φ₁, and φ₂) to compute two fiducial values for φ_c. Then I solved these two equations for φ₁ and φ₂, and allowed them to vary parametrically with the ratio E₁/E₂.

 I just talked to Matt and learned that:

• The measurement from the above paper (that everybody quotes) is wrong, they just measured the Young's moduli of the substrate. However, the actual value of Y_Ta2O5 is ~ 140 GPa (use +/- 30% for conventional uncertainty) (Abernathy's from 500 nm sample) and the measurement for Y_SiO2  is ~ 70 GPa ( from 2 um sample)
• The term ion beam sputter (IBS) for our coating is actually IAD.
•  Note about coating loss in Penn2003
• T

The measurement from Penn extracts phi1 and phi2 from

 phi_c = (Y1 *d1*phi1 + Y2*d2*phi2) / (d1*Y1 + d2*Y2).

Phi_c is calculated from the total phi of the ring down system.

The dissipated energy comes from two part, the substrate and the coating. With the assumption that phi sub is much smaller than phi coat, we can write 

phi_total (measured from ring down) =  |energy in coating| / |Energy in substrate|  * coating loss, and the ratio Ec/Es can be obtained from FEA.  For drum head mode it is ~ 1500 (From Penn paper), see the picture, top panel.

This Ec/Es also depends on the Young's moduli, so the calculated phi_c also has Y as a parameter.  The calculation I did before takes phi_c from the reported values, so it is not correct.

To get the correct phi_c, the ratio of Ec/Es has to be changed with Y. Crooks PDH thesis has an analytical expression for the drum head mode of a cylindrical substrate. The analytical result is comparable to the FEA result used in Penn2003 within 5%. Note that the young's modulus of the coating is the volume average (Yc tc = y1*t1 +y2*t2) where tc is the total thickness, tx = thickness of material x. See the middle panel in the picture above. 

For the next step, instead of using the report value of phi1,phi2 and Y1,Y2 to reconstruct phi_c (Penn2003). I will use the measured phi_tot (for drum mode) then use that as a constraint on Y1, Y2, phi1, and Phi2 instead, see bottom panel in the picture. This should give a correct dependent among these variables.

 Ta2O5 Young's modulus is quoted to be 140 GPa from this paper Martin1993,  but that is the value of Ta2O5 deposited on Silicon substrate cf fig5, top plot. The deposition technique is IAD. I'm not sure if it is the same as ion beam sputtering or not. I'm looking into it.

Anyway, the Young's modlus of Ta2O5 can be down to 70 GPa for IAD technique on glass substrate, as the paper says in the conclusion section. 

 Note that Crooks2006 mentions other papers measure YTa2O5 to be around 100-110 GPa as well. I'm looking into it.

I'm checking how loss angle of Ta2O5 is related to its Young's modulus (as used in ring down measurements), then I use that relation in error calculation for BR noise in coating. The uncertainties in Young's moduli of SiO2/Ta2O5 might lead to errors in loss angles and BR noise in coating.

==Background==

Many ring down measurements (see Penn2003, Crooks2004, Crooks2006), observed loss from a disc substrate with multilayer coatings of Ta2O5/SiO2. The loss in the coating (ring down mode) is written as  

phi_c = (phi1*Y1*d1 + phi2*Y2*d2 )/ (Y1*d1 + Y2*d2)    --------(1)

Where phi_c is determined from the measurement. Y is the young's modulus, phi is loss angle of material, d is physical thickness of the material.

Then phi1 and phi2 is determined with the assumption that Y1 and Y2 are known.  

So, the reported value of phi Ta2O5 is directly related to its Young's modulus. The uncertainty calculation of BR noise where Y, phi are varied independently might not reflect the real situation.

For example, I recalculated phi_c (of QWL structure) using phiH phiL of  4e-4, 1e-4. YH = 140 Gpa, YL = 72GPa. Then I rearranged eq(1) so that phiH can be written as a function of YH to see how the loss angle of Ta2O5 (H) will change with its Young's modulus assuming that YL and phiL are fixed.

fig1: How phi Ta2O5 changes with Young's modulus.

==BR calculation with loss parametrized by Young's modulus==

With the loss angles parametrized by the Young's modulus, I calculate the estimated thermal noise compared to our measurement (using Hong2013) 

fig2: ratio of BR calculated and our result. 

It is interesting that, even with the lower phiH as YH increases, the total BR noise increases. And the nominal value that we have been using (YTa2O5 = 140 GPa) yields almost the minimum value of BR noise calculation. 

==next step== 

  So far, the calculation is done assuming phiL = 1e-4, YL = 72e9.  The next step is to varied phiL, YL, phiH, YH all together ( with the constraint given by eq1) and see how BR noise changes.

  I'm also checking how large the errors are in the measurements for Young's modulus (both SiO2/Ta2O5). Crooks2006 reports the value of Young's modulus of Ta2O5 with the assumption that Y_SiO2 is 72e9. This might give another constraint. 

 I checked Brownian coating noise level with uncertainties in coating parameters. The measured result is barely at the edge of the confident interval. 

Hong2013 look into coating noise level when materials' parameters are changed. One example is the Young's modulus of Ta2O5. With the assumption phi bulk = phi shear, if Y_Ta2O5 is varied between 70e9 to 280e9 (nominal value = 140e9), coating thermal noise can be changed by a factor of 0.9 to 1.5 from the nominal value (in PSD m^2/Hz unit). It seems that the range is quite large compared to the numbers measured by various groups, (see PSL895 for error in material parameters). I used a smaller range, but I varied other parameters as well.

==Note about uncertainties in calculation==

I used rand command in Matlab to generate random values. The reasons are 1) for reported loss angles, say 4+/- 2e-4, if I use Gaussian dist, with sigma =2, mean = 4, sometimes the generated value will be negative, and 2) since we are only trying to see the possible range of the estimated noise level, not the real statistic value, rand should be ok at this point.

==1:fixed loss angles==

  First I checked how much the parameters effect the calculation if the loss angles are fixed (phi silica = 1e-4, phi tantala = 4e-4). Y tantala is chosen between (70-280 GPa), Y silica is varied between72e9 +/- 10%,  Poisson's ratio are varied between 10percent for coating materials. All substrate parameters are fixed, since they should be relatively well measured compared to that of the coatings.  The result is around 0.5-0.85 of the measurement (in PSD m^2/Hz).

  For a more conservative value of Ta2O5 ( 140+/-40 GPa), the result is a factor of 0.5-0.64 of the measurement.

 ==2:varied loss angles==

In this study, I varied loss angles of phi_silica = [0.8-1.2] x10^-4, phi_ta2O5 = [3,5]x10^-4, these numbers are reported from several measurement. Then I change the uncertainties range of Y_Ta2O5 in my calculation

•  Y_ta2o5 = [100,180] GPa, the result is around [0.4,0.88] of the measurement.
•  Y_ta2o5 = [70,280] GPa, the result is around [0.4,1.09] of the measurement. A histogram of the ratio between noise level and the measurement is shown below (from 1e4 runs). The measurement value (Sx/Smeasured = 1) is barely at the edge of the confident interval (and not from Gaussian distribution either).
• 

==note and comment==

 Both Hong's and Harry's calculation provide quite the same value (within 3%). So I show only histogram obtained from Hong's calculation. I don't know why the study shown in Hong paper choose the value of Y tantala between 70-280 GPa, most of the measurements report smaller uncertainty. But with that higher value of Y_Ta2O5, it can explain the measured noise level from our measurement. However, I doubt that this argument is valid, since most of the ring down measurements to evaluate phi_Ta2O5 assume Y_tantala ~ 140GPa. Then the loss angle of Ta2O5 should carry some information about Y_Ta2O5 in it and cannot be treated as an independent parameter like this calculation. I'll look into the ring down papers to see how much Y_Ta2O5 affects the extraction of its loss angle.