The codes for optimizing Thermo-optic noise in coatings are up on svn.

I adopt some codes that have been on svn for awhile and modified them for AlGaAs coatings. There are two main codes

      1) DoAlGaAs.m

         This file is modified from DoETM.m found in .../iscmodeling/coating/AlGaAs/doETM.m . The optimization method is using Matlab's fmincon function to search for coatings structure that minmize TO noise. Some modifications include:

• (Line16-18 )Number of layer. For AlGaAs, the number of layer will be odd number (start with GaAs, end with GaAs), I fixed the layer structure to be odd number.
• (Line74) Cap. During the optimization, the first cap is kept constant. For a cap made with high refractive index material (nH), the layer thickness is 1/4 lambda, see previous entry.

This code calls on  optAlGaAs.m when running fmincon.

    2) optAlGaAs.m

        This file is the modification of optETM.m found in ../iscmodeling/coating/AlGaAs/optETM.m .It calculates the reflectivity and the TO coefficients from the given layer structure. The modifications are:

• (Line41-45) Layer structure, the cap start with nH. The material for substrate is SiO2 with nsub = 1.45.
• (Line60) Desired transmission, as a test, I chose 200 ppm.
• (Line88) Calculation for TO coefficients (StoZ), I switched from getCoatThermoOptics.m to getCoatThermoOPticsAGS.m. Codes with AGS suffix in /GwincDev folder are fixed for AlGaAs coatings structure. This code calls many functions in /GwincDev folder.

     2.1) multidiel1.m

       This code is used in optAlGaAs.m it calculates the reflectivity and impedance of the given coatinns structure. There is no modification to it. The code can be found in .../coating/coating_optimization_new/. 

    To run the codes

    check out .../iscmodeling/ folder from the svn. The optimization is in .../iscmodeling/coating/AlGaAs_TO_opt_CTN/ folder, but you need other functions in other folders.

    Once you run DoAlGaAs.m, the optimized layer will be in matlab workspace called xout. This is the layer structure withtout 1/4 cap. Check if there is a layer with thickness of 0.002 or not. I ran the code several times, sometime it shows up. Just rerun the code and get the layer that is around 0.1 or thicker. The 0.002 is just the lower bound used in fmincon search in doAlGaAs.m.

  Plotting noise budget

 The noise budget of the optimized layer can be plotted with /coating/AlGaAs_Refcav/nb_algaas.m . Currently, at line 38-39, the code will take xout  and create a layer structure with 1/4 cap on top of it. The reflectivity of the coatings is in rCoat workspace item after running the noise budget code. It should be close to -1 + 0i

I locked both cavities and trying to search for the beat signal, I have not succeeded yet.

I used lenses that could get the two transmitted beam to be close and small enough for the beat PD (new focus 1811) (we ordered  what we need but they are not here  yet).

I locked ACAV at a fixed SLOW DC level (1.207 V), and varied RCAV's SLOW DC level from 1.199V, 0.33V, -0.554V, -1.477V (1FSR ~ 4GHz is about 1 V). The slider for RCAV slow is set to +/- 2V so I have not tried other values yet. It can be changed to -2V to 9 V, but I have to restart the crate which will disturb the temperature servo, so I'll try to adjust RCAV slow value using a voltage calibrator instead.

I talked to Evan about the beat measurement in GYRO lab, the SLOW DC for both lasers can be different up to 6 V (for ~100MHz beat). see gyro1832

I varied RCAV's SLOW DC first because this path does not have a PMC, so I don't have to worry about locking the PMC.

From PSl:1124 ,the beat frequency should be ~60-100 MHz, without the heater on any cavity.  I'll try the same method to check the beat frequency between the two cavities one more time. If it is still ~ 100 MHz, I'll increase the range of SLOWDC, and see if the beat will show up of not.  The setpoint was not changed that much (31.2 to 31.25), So I expect the beat frequency should still be close.

If the beat still not show up, I'll try to realign the beam.

Current setup

Vac chamber Setpoint = 31.25

Vheat for RCAV =  0

Vheat for ACAV = 0

Found the beat @ 116 MHz. RCAV SLOW =5.762V, ACAV SLOW = 1.209 V.

beat 1kHz input range, calibration  = 718 Hz/V

above, beat signal with 1kHz input range on Marconi.

Plenty of things that I need to optimize and add:

input optics (ACAV/RCAV):

• beam alignment
• optimizing quarter wave plates in front of the cavities.
• block all the reflected beams properly
• fixing the back reflection from vac window for ACAV.
• measure error point noise from both servos and compare them with beat
• optimizing TTFSS servo gain

Beat setup:

•  mode matching lens
• power on beat PD
• optimizing PLL servo
• implementing ISS

Seismic isolation

• new table legs ( I have not ordered the new set yet). The current set is broken
•  

I turned off the hepa fans over the table over the night. I came back this morning and the temperature (measured on the vacuum tank) was very stable(within 2mK) over 2 hrs.

above:BLUE Temperature measured on the can, the Y scale is in degree C. The temperature variation is within 2mK over 150 mins.

So I looked at the PD for Erica's fringe measurement, the fringe wrapping was slow, so with better temperature insulation, we should be able to hold the fringe for at least a minute.

above: The fringe signal from PD, the cursors show the max/min signal from the fringe. The signal drifts from min to max over ~ 60 seconds compared to ~10seconds as before.

So the drift we saw before was very likely to be from the temperature drift (1mK per second for 20second fringe wrap). More thermal insulation on the optic should reduce the temperature drift.

I measured the slope of the error signal for ACAV path to be 200 kHz/V. This will be used for calibration the error point noise to frequency noise.

 See some details about the error signal's slope and calibration in psl:562.

THe setup for ACAV path is

• Input power to the cavity 1 mW.
• RF power on marconi for the driver = 13dB, to 4-way splitter then 14.75 MHz resonant EOM.
• The error point noise was measured at COMMON channel out1 on RCAV TTFSS.

Next: Measure the slope at RCAV path, measure error noise from both loops, compare to beat signal.

Plan for opening the chamber:

I'm certain that the beam reflected from the window that overlaps with the reflected beam from the cavity going to the RFPD causes a lot of noise. This should show up in the error noise. So to avoid the reflection from the window, I have open the chamber to turn the cavity axis a bit. I need to:

• calculate how much the cavity has to be turned if we will dump the beam at the lens for the RFPD.
• see if the beam path is still ok for the rotated cavities.
• Replace the cavity mount wall. The current one is too short due to the mistake in the design. I needed to use  nuts to raise the height, see pic.  Without the gap on the side, temperature control between the two cavities will be better due to smaller coupling. The walls will be ready on Monday, I might need a day or two to clean and bake them before the installation.
• use screws to hold the cavities down firmly, instead of resting on four point supports.
• I don't plan to replace the AD590s on the thermal shield. This will take too much work to remove the feed through, fix the cable. Otherwise, we can just slide the stack half way out of the chamber for replacing the wall and rotating the cavities. Plus, we can use beat signal as an error signal for Temp control.

Better TO optimized coatings calculation is done. Now the Transmission, phase reflection, and TO noise are optimized.

From previous elog, these are explanation about the optimization codes.

 So optAlGaAs.m calculates a parameter y which is the cost function that is minimized in fmincon in doAlGaAs.m code.  Originally the cost function y includes the difference between the expected transmission and the transmission from the given layer, and the level of TO noise which are:

y = [(T - <T>) / <T>]^2   + sTO (f0).   The goal is to minimize y.   Where

• T = transmission of the mirror with the optimized layers
• <T> is the required Transmission
• sTO(f0) is TO noise at f0
• Each effect is weighted differently

This cost function does not care about the total phase of the reflected beam. T is the absolute value of the transmission, so the information about the phase is removed, and the optmized coatings calculated from this cost function won't have phase close to 180 degree. The previous result showed 180-1.2 degree.

So I added the phase of the reflection in the cost function, with appropriate weight, and ran the optimization.

==Phase calculation==

rCoat is the reflectivity of the coatings, by using atan(imag(rCoat)/real(rCoat)), we obtain the phase of the reflectivity. I tried to you atan2(y,x) to get the phase of 180, but it does not work well with the optimization. I'm not sure why. So I use atan function, and check the value of rCoat after the optimization to make sure that rCoat is close to -1 + 0i. The result is shown below.

above: the layer structure, optimized for 200ppm, y axis is in unit of lambda in the layer. The first layer is the 1/4 wave cap, the last layer is the layer just before the substrate.

above: noise budget for the optmized structure, the reflection phase is 180- 1e-6 degree.

 The layer structure is attached below in .mat format. Note: the structure does not include 1/4 cap on top.

== summary of the modifications of optAlGaAs.m==

• (line 90 - 95) add calculation of the phase of the reflectivity
• line 97 the cost function includes phase of the reflectivity that is close to 180 degree (r is close to -1 + 0i). The weigh functions  from TO noise/transmission/phase are chosen so that each factor are about the same, and the result looks reasonable ( coating thickness ~0.1 - 0.3 lamda, correct reflectivity, correct transmission).

I wrote a code to calculate thermo-refractive noise in a finite-sized cylindrical substrate as given in Heinert etal 2011. The noise is very small ~10^-7 [^Hz/_rtHz] compared to other noise in the cavity ( no surprise here). The code can be used to estimate the TR noise in fiber optic. The calculation should be correct as I double checked with the calculation by Koji and Deep.

I followed the calculation for TR noise in cylindrical substrate [Heinert etal 2011] for our setup (1" diameter , 0.25" thick, fused silica). The result is in [m/rtHz].

To convert it to frequency noise of the laser:

1. Convert the displacement noise to phase noise in the beam first,  S_phi = S_x * 2*pi*n/lambda_0  (n is index of refraction).
2. S_f = S_phi * f   (f is fourier frequency), multiply by 4 to get the contribution from 4 mirrors.

above: TR noise in substrate. It just so small compared to other noise sources in the noise budget(~ 10^-3 - 10^-1 Hz/rtHz level that I don't see the need to add it in the complete noise budget.

Since we will use the same substrate, the noise level will be the same for short and long cavities. The different in beamsize will vary the noise level a bit.

Note: this calculation is for a Gaussian beam profile in a cylindrical substrate, to use this calculation for  fiber optic TR noise, some assumption about the mode of the beam is required.

I rechecked the TF between power fluctuation and frequency noise in beat measurement that I did last year. The estimated result agrees more with the measured result. This can be used to estimate the requirement for ISS for SiO2/Ta2O5 and AlGaAs coatings.

The calculation is taken from Farsi etal 2012 (J. Appl. Phys. 111, 043101), and compared with the measurement from 8" cavities, SiO2/Ta2O5 QWL with SiO2 1/2 wave cap. The code I wrote before has several mistakes, so I fixed them.

Mistakes in the original code:

1.  Beta effective was for 1/4 cap of nL: I changed it to the right one (1/2 cap of nL). This can be done by GWINC or an analytical result.
2.  Cut off frequency w_s, w_c in the paper, I divided by a factor of 2*pi make them in Hz.
3. Missing a factor of imaginary in thermoelastic in coatings calculation.
4. r0 in the paper is where the power is dropped by 1/e, so r0 = w0/sqrt(2) where w0 is the radius of the beam when the power is dropped by 1/e^2.

Above: Measurement(purple) from SiO2/Ta2O5 coatings and analytical result (cyan) in comparison. Finesse = 7500 (old ACAV), absorbtion = 5ppm.  The slope at high frequency seems to be real TO noise. Notice that phases from TE and TR have different sign and cancel one another.

==for TO optimized AlGaAs coatings==

Above: Calculation for RIN induced thermo noise for optimized AlGaAs coatings in Hz/Watt unit. The calculation is for 200 ppm transmission,-> Finesse ~14 000. 1.45" cavity. The cancellation in coatings will reduce the noise. The estimated effect is plot against the measurement from 8" cavity, T=300ppm, SiO2,Ta2O5 cavity.

We might have to make sure that RIN is small enough, since this time we will have no common mode rejection like what we had with just a single laser. I'll add the estimated requirement later.

I estimated the requirement for laser RIN for AlGaAs coatings. The result is a factor of 5 more stringent from what we need for SiO2/Ta2O5 cavity.

See some calculation about RIN requirement PSL:1270.

I estimated the RIN induced TO noise in AlGaAs cavities. Due to the TO optimization, the effect will be small and we will see only the effect from the substrate, see RIN induced noise estimate.

 This will be quite serious, if we do not have a good ISS, since we will not have common mode rejection like what we had with the single laser setup anymore. I'll look up what was the RIN performance we had before.

 Noise hunting is in progress, I checked the error noise from ACAV and RCAV loops and compared them to the beat. The beat is about an order of magnitude higher than the sum of error noise.

 NOte: slope of error signal RCAV = 1.57 MHz/V (13 dBm from Marconi, throug 4-way splitter, to BB EOM, 1mW input power).

ABOVE: beat signal in comparison with noise at error points from ACAV and RCAV loops. The beat signal is about an order of magnitude higher than the error noise.

I'm working on optimization and noise characterization of the setup. Before measuring the beat I have to make sure that:

• The beams to the cavities are aligned
• The power input is 1mW for both cavities
• I aligned the polarization of the beams into EOM for side band ( minimizing RFAM)
• The gains for TTFSS are adjusted and recorded
• Beams in the beat setup are aligned, and dumped properly.
• The PD is not saturated.
• PLL is setup properly.

I think the gain in the TTFSS is the problem. For ACAV, the scattered light from the window interferes with the main beam and causes the loop to oscillate when the gain is up. For RCAV, the EOM is a broadband one and does not have enough gain. The bump in the frquency lower than 100Hz is probably the contribution from scattered light. I have not properly dumped all beams yet.

Also I noticed that the beat signal has weird sidebands at +/- 100kHz, 200kHz, and 300kHz, see the figure below. I'm not sure why, I have not seen it before. I might saturate the PD making it distorted from a perfect sine wave. I'll investigate this.

Noise hunting is in progress. Today I identified that scattered light from the window is one of the problem.

I spent sometime making sure that all the beams in the input optic and the beat areas were dumped properly. I also tightened all the screws on the optics and the mounts on the table.

I mentioned in the previous entry that for RCAV, the reflected beams from the cavity and the vacuum window overlapped a little bit. The window beam was much smaller and actually closer to the edge of the main beam, so I used an iris to remove the outer path, and let only the beam in the center area go through to the RFPD. With that I could increase the gain in RCAV loop to Common/Fast = 624/750, where they used to be ~ 600/600 before. The iris might introduce some extra scattered lights, since it clips a part of the beam.

The scattered noise around DC to 100 Hz is reduced a bit, see the below figure. However, not much improvement in the flat region (100Hz and above). Plus, some mechanical peaks around 1kHz appear with higher level than before.

I expected the scattered noise will be even lower if the cavities are tilted a bit to avoid the beams overlapping. At higher frequency, it might be the gain limit from RCAV loop where the modulation depth is very small.

Next thing to do is to increase more power in the modulation depth for RCAV.

==Note==

I found out that the sidebands in the beat signal mentioned in the previous entry changed with the gain of the TTFSS (both ACAV and RCAV). With higher gain, the sidebands are suppressed more. It might have to do with the PZT resonant of the NPRO. 

I installed an electro-optic amplitude modulator (EOAM) in RCAV path. Better optimization will be needed to reduce extra noise.

above, the setup for ISS actuator, the first 1/2 wave plate rotates p-beam to s-beam, EOAM, 1/4 Wave plate that tuned so that the output beam is 45 degree so the power transmitted through the final PBS is reduced in half.

After the EOAM was added, I checked the beat noise and saw a bump at ~ 2 kHz, see the figure below(blue plot). This was from the EOAM even though there was no input drive. It disappeared after I changed the EOAM position by rotating it a bit( yellow plot). I have not finished with optimizing it yet. I'm thinking about what kind of mechanism that causes the noise here.

 I went through all the code with Evan and found another mistake. This time the code should be correct, and the result is close to what we measured a year ago.

 The calculation in PSL:1014 is wrong. There should be no square root for the absorption power (Finesse/pi * absorption).  With that correction, and an assumption of absorption of 18ppm in the mirrors (9ppm on each) with Finesse of 7000, see PSL:425. The result matches with the calculation quite well.

The validity of this result depends on the absorption factor and cavity finesse. The finesse was measured, but the absorption measurement has never been done. So it might be good to think about how to measure that.

We did the same measurement with the current ACAV 1.45" cavity. Evan will post the result later.

The bump at 2kHz in the beat signal that I saw before was also from RFAM. By adjusting the 1/2 waveplate in front of the sideband EOM, the bump disappears. I still don't understand why adjusting the EOAM can reduce the bump from RFAM.

 As I planned to add the eom driver to the BB EOM for sideband in RCAV path, I wanted to see the improvement without worrying about the EOAM optimzation. So I removed the EOAM, but I still saw the bump I observed before. This time it came from the RFAM. By adjusting the wave plate to match the polarization of the input beam to the EOM axis, the bump is gone.

above: From right to left, 1) laser for RCAV, 2)&3) 1/2 and 1/4 wave plates, 4) lense, 5) Faraday isolator, 6) 1/2 wave plate, 7)BB EOM for frequency locking, 8) BB EOM for side band, the EOM driver is attached to the side (in aluminum foil wrapped box). RFAM is minimized by adjusting (6) 1/2 wave plate.

I added the EOM driver, however it was not yet modified for 14.75 MHz, so the amplification is small, see PSL:1234 . After adjusting the phase of the demodulated sigmal, the error signal slope is increased by a factor of 2. Then I remeasured the beat signal, and the beat was better by ~ a factor of 2 at high frequency. So I think now the signal is gain limited (in RCAV loop) at high frequency. This makes me confused why the error noise from RCAV loop does not match the beat signal in PSL:1307. I have to re check my work.

The next few things to do are:

• minimize RFAM (by temp control on both EOMs )
• re-install EOAM in RCAV path, think about alignment
• now scattered light at low frequency might come from seismic noise as well. I'll order the new floating table legs soon.
• check other noise limit to make sure that it will not be dominating (shot noise, electronic noise)
• modify the EOM driver, so that we have more gain in RCAV path.

I closed the chamber. The turbo pump is on and pumping down.

 I realigned the beams so the visibilities for both cavities were 80% or more. This made sure that the beams' path would be close to the optimized path.

Now, the window reflection won't overlap with the cavity reflection, and can be dumped properly.

Note about a few things to do:

• The beam holes on the foam might have to be fixed, the beams slightly clip at the openings. I have to check if the beams are clipped at the periscope or not.
• modification of the seismic stack as suggested by Koji. The teflon pieces at the bottom plate are not screwed down to the stack making it hard to put the stack in the chamber. I think this should be fixed after the SiO2/Ta2O5 measurement is done and we have to reopen/ installed AlGaAs cavities.
• There is some strayed beam from the PBS in RCAV path. This is from left over beam in S-light that reflected off, and bounced back at the PBS surface before going to the PD. This might have to be fixed too.

FYI for torque wrench setting for CTN cavity:

 The CTN cavity is 10" OD,  the Torque required is 190 InchPound.

Since the optimized layer structure is designed, I'm checking how the coatings properties change with error in layer thickness.

G.Cole said that they can control each layer thickness within 0.3%. So I tested the optimized coatings properties by adding some random number within +/- 0.5% on each layer thickness. The results are shown below for 10 000 test.

The error check does the following:

• start from the optimized coating structure reported in PSL:1291.
• add random thickness to each layer, within 0.5% of each layer
• calculate the values of interest, then histogram them.

The figure below is an example of the varying layer thickness added by rand command. They are confined within 0.5%.

 1) result from the error in thickness control

Above: histograms of the important values. top left, reflected phase. top right, ratio between PSD of Brownian noise and Thermo optic noise at 100 Hz. Bottom left, transmission. Bottom right, total coating thickness error.

 comments: this test is chosen for 0.5% error which is almost a factor of 2 worse than what they claimed (0.3%), so the actual result should be better. I assumed 0.5% errof because of the irregular layer structure of the optimized coatings, there might be some more error in the manufacturing process.

• Reflected phase: we want the reflected phase to be close to 180, so that the E-field at the coating surface is close to 0. more than 50% of the results are within 179.5degree, this means that the power build up will be ~ Finesse/pi * Power input * sin^2 (0.5degree)  ~ less than 0.4 mW, so there should be no problem about burning at the surface.
• ratio between PSD of Brownian/Thermo optic noise. This plot imply how well the cancellation works. Since Brownian noise will almost not change (both materials have the same loss, total thickness varies less than 1%), the ratio of Br/TO noise (at 100Hz) tells how much TO cancellation is. From the histogram we are quite sure that cancellation will work most of the time.
• Transmission is good around 200+/- 10ppm this is ok with the requirement.
• total physical error is ~5nm while the coatings thickness is ~ 4um. so the total error is <0.1% Brownian noise calculation will not change much.

2) result from different calculated Beta values:

Here I checked what happen if the beta calculation was wrong, and the error is still within 0.5% in each layer.

In Evans paper, the effect from "Thermo-refractive" comes from the phase changes of the wave travels in each layer. So it includes the effect from dn/dT and dz. The effective beta for each layer is given as

[evan B8],

where alpha bar is

[evans A1]

Where s denotes substrate, k denotes the material in each layer (high or low indices).

So my, calculation & optimization have been using these equations.

However, in the original GWINC code for TO calculation, the calculation [B8], alphabark( used in dTR) is not the same as A1, but rather.

alphaH * (1 + sigH) / (1 - sigH)

see getCoatLayerAGS.m.  Line 16-17.

This is used in the calculation for beta effective in getCoatTOphase. Line73-74. Notice that for dTE, the alpha_bar_k is the same as used in Evans. (line 77).

the comment says "Yamamoto thermo-refractive correction". I emailed kazuhiro yamamoto, but never got a response back. So I keep using the same formula as in Evans because I don't see the reason why the expansion contribution should be different between TE and TR.

So this is the nb plot for TO noise from the optimized coating, if using yamamoto TR correction.

Above: nb from the optimized coatings, using Yamamoto TR correction. The cancellation becomes worse, but TO is still lower than other noise.

Finally, I repeat the same error analysis for random noise in the thickness (+/- 0.5%).

Most of the parameters behave similarly, except the cancellation (upper right plot). Now BR is only ~ x12 larger than TO noise because of the worse cancellation. Good news is, it still below Brownian noise, the cancellation still somehow works.

=summary=

• From the optimized coating structure (T=200ppm), thickness control within 0.5% in each layer will make the coating work as expected.
• The yamamoto TR correction is still an unresolved issue, but the optimized coating will still work.
• we should be ready to order soon.

We modified the EOM driver, so that the resonant frequency is now~ 14.75MHz. The full test will be done later.

As mentioned in PSL:1311, the resonant frequency on the EOM driver was not at 14.75MHz. Evan and I discussed about how to modify it and decided tof change L4 from 1.4uH to 3 uH, see the schematic here.

above, the driver after the inductor was replaced. The new one has a shield to reduce any magnetic field leakage. The legs are not fit with the footprint on the PCB, so I had to solder it to another wire to reach the footprint.

above: the TF of the driver measured between the drive and the mon output. Red trace shows the TF before the modification. Yellow  trace shows the TF after the modification, notice the peak is at 14.75MHz, the Q is about the same.

Here is a summary for how I verify the codes for TO calculation.

So far, we have been using a set of modified GWINC codes to calculate TO noise, but I have not mentioned how did I make sure that the codes were reliable. So I'll try to explain how I check the codes here.

==What do we compute?==

  For the TO nosie calculation and the optimization, we are interested in:

• effective dn/dT (TR coefficient) of the coatings
• effective alpha (TE coefficient) of the coatings
• total reflectivity of the coatings (including the phase), and transmissivity

==Beta calculation check==

For TR coefficient we can compare GWINC with an analytical result (see Gorodetsky,2008, and Evans 2008) (when # of layers ~ 50 or more), see psl:1181. I tried the solution with nH, 1/4 cap and nL, 1/4 and 1/2 cap. All results agree.

==Alpha calculation check==

 There is no complication in this calculation. The effective alpha is just the sum of all layers. This calculation is quite straight forward.

==reflectivity check==

   This was done by reducing the coating layers to one or two layers and comparing with an analytical solution by hand. I checked this and the results agreed.

So I think the calculations for TO noise is valid, the noise estimated from the optimized coatings is done with some error check (previous entry). I think we should be ready to order.

The turbo pump is removed, and the ion pump is on. The initial value is ~7mA.

I removed the turbo pump and turn on the ion pump, see the procedure on wiki page. The initial value on the ion pump is ~ 7mA, similar to the last time we opened the chamber although this time I left the turbo pump on 4 days instead of 2 days. So I think this is the limit of this turbo pump.

I updated the optimization and error analysis. The error in optimized structure is comparable to that of a standard quarter wave length structure.

      After a discussion with Rana, Garrett, and Matt, I fixed the thermo-optic calculation, and the error analysis done in PSL:PSL:1315.  The modifications are

       1)  fix the TO calculation (Yamamoto TR correction): There is a modification for TR correction that is not in Evans etal 2008, paper. I contacted M. Evans to ask about the details of this correction which is done in GWINC.  

       2)  Try another optimized coatings with the correct TO calculation:  After the correction, I ran doAlGaAs.m code, cf PSL:1269  using fmincon function , to find another optimized structure. The result is shown below.

above) layer structure in optical thickness, the .fig and .mat file are attached below. Note .mat file contains 54 layers, you need to add 1/4 cap to the first entry to calculate the noise budget.

above) noise budget of the optimized coating.

       3)  Repeat the error analysis : This time I used the following assumptions (from G Cole)

• the error is not random among each layer
• the error is constant in each layer type, ie all the layers from the same material (nH or nL) have the same percentage of error,
• error from nH and nL have the same sign. If one is thicker, another one is thicker, but the magnitude are uncorrelated.
• nH (GaAs) has better thickness control with 2sigma = 1percent, while nL(AlGaAs), has 2sigma = 2 percent.

Fig1: Above, percentage of error distribution between the two materials used in the calculation. nH(red) has 2 sigma = 1% and nL(blue) has 2sigma=1%.The same error distributions are used for both optimized layers and QWL layers in comparion, see fig2.

The section below is the algorithm used to distribute the error, this one makes the error between the two materials to be the same sign. The whole code can be found on svn.

===Result==

This time I calculated the change in reflection phase (TOP left), the ratio between TO noise from the coatings with error and the coatings with no error(top right), transmission (bottom left), and ratio of BR noise ( bottom right). The result from the optimized coating(blue) is compared with the QWL coating (black).

Fig2: Error analysis, in 5e4 run. Blue: from optimized coatings Black:from 55 QWL coatings, from 5x10^4 runs.

Reflection phase: The reflection phase can be away up to ~6 degree. The power at the surface will be ~Finesse/pi * Power input * sin^2 (6degree) ~ 50mW. Seems high, but this is about a regular power used in the lab.

Ratio of PSD TO/TO_0 : At worse, it seems the PSD TO noise will be ~ a factor of 10 higher than the design. However, this will be still lower than BR noise. I plotted only the error distribution for optimized coatings because for QWL coatings, the ratio will be about the same, since TO is dominated by TE.

Transmission: Most of the results are within 197-200 ppm. The optimized coating has transmission ~ 197ppm. The QWL with 55 layers has transmission ~100ppm.

Ratio of BR: not much change here.

It's a quiet night, so I went down the lab to measure the beat signal. We are getting close. I think I have to review my noise budget calculation and estimate the error in the measurement carefully.

So after a few things Evan and I did a few days ago:

1. rotate the stack to get rid off the reflected beam from the window
2. fix the insulation so the beam is not clipped on the opening.
3. add more modulation depth to RCAV path with the EOM driver (tuned to 14.75MHz)
4. Minimize some RFAM, by rotating the half wave plate in front of the sideband EOM

Then I measured the beat signal.

We reduce some noise from scattered light at frequency below 100 Hz, we are limited by some white noise at high frequency ~ above 1 kHz.

fig1: measurement vs noise budget

fig2: zoom in. The slope of the measured signal agrees well with the slope of thermal noise.

ToDo

• Estimate/measure shot noise PD noise and electronic noise in the setup. See if they match up with the measurement.
• Review the noise budget calculation. Looking at the slope of the signal around 1kHz, I think the calculated brownian noise is lower than what it should be.
• noise hunting, seems like scattered light at frequency below 100Hz. There are many mechanical peaks, and harmonic lines at higher frequency.
• add the contribution from RIN induced TO noise in the nb.

Coating optimization and error analysis are updated, see PSL:1320.

Short note from tonight measurement:

1) scattered bump from dc to 100Hz is mostly from seismic. It is worse during the day. It gets smaller at around 3-4 am. Unless we have a better seismic isolation, we might not be able to see anything below 100Hz.

2) RIN shape from RCAV changes, reasons still unknown. (DC level 0.7 V)

3) I might see the effect from RIN induced TO noise at frequency ~ 1-3 kHz. (compare RIN and beat).

I'll get into details tomorrow.

The measured RIN is measured and converted to frequency noise via photo thermal effect then compared to beat. The effect seems to be significant now since we lost the common mode rejection.

I measured RIN after ACAV (there is only one PD behind ACAV right now. we will add another one for RCAV soon). The magnitude is comparable from what we measured before but the peaks seem to change, see PSLPSL:1326, :PSL:1308,  (8"cavity) PSL:742 .

The peaks around kHz  are more clear. I'm not sure where they are from, but I think it is associated with vibration on mirror mounts that causes beam jitter. Because the peaks look like mechanical peaks, and this time the cavities are shorter, the beamsize is smaller from 8" cavities, the same beam misalignment will cause the power coupled into the cavities to change more compared to that of 8" cavity. We can check that by mis-aligning the input beam a bit, and see if RIN becomes larger or not.

The coupling from RIN to frequency noise is discussed in PSL:1328

I applied that to the measurement and here is the result. Note, only the effect from one cavity (ACAV) is taken into account.

The peaks seems to match up, especially around 20-30Hz and around 1kHz, see the zoomed in picture below. This makes me think that we might be limited by RIN noise now.

To Do next:

• Install ISS system on RCAV (PD behind the cavity / EOAM)
• Re-measure the coupling from RIN to frequency noise of the cavity
• Measure RIN and apply it to the noise budget.
• Find out what causes RIN to change

I'm trying to understand the measured RIN in the setup. The evidence suggests that the measured RIN in 100Hz- 6kHz, is real intensity noise and not associated with alignment + jitter.

==Problem==

As seen in PSL:1329 that we might be limited by RIN at high frequency, I tried to figure out what cause the shape of the RIN around kHz to be mechanical -like peaks. So the problem can be minimized, and does not have to rely on ISS that much.

==Assumption==

My assumption was that they were from mirror mounts, because

1. the frequencies were close to mechanical resonances of the mounts, see PSL:818, PSL:824 for examples. The power coupled into the cavity would reduce, and thus causing the peaks in the RIN measured behind the cavity. And
2. the shape changes during the time of measurement. During the day, the shape is like a big bumb, while during the night, around 2-3am, the level is smaller and the individual peaks shows up instead (may be because of the lower seismic)

==Measurement==

So to test this, I measured RIN before and after ACAV (NOTE:ACAV path has PMC in it),  when

1. The beam was well aligned to the cavity (DC from REFL PD =94mV, total level ~ 1.6V)
2. The beam was misaligned a bit (DC from REFL PD = 175mV)

above, beam path in front of ACAV, before the beam enters ACAV. The PD for RIN measurement is circled in blue.

above, beam path behind ACAV.

If the measured RIN was from the jitter, RIN after the cavity should change with the alignment, and RIN before the cavity should not change much. I made sure that the spotsize on both PDs are significantly smaller than the PD to make sure that any jitter in front of the cavity should not change the power level that much.

==comments about the result==

• The result agrees with the assumption at low frequency (DC - 30Hz),
  
• However, from 100Hz and above, the measured RIN from four cases are very similar.
• The measured coherence between the two PDs are similar in both cases (aligned and misaligned), I plotted the one from the misaligned beam. It shows that at low frequency, RIN behind and after the cavity are not caused by the same mechanism, the one behind PD might suffer more from jitter. However, at 100Hz and above, anything observed before the cavity is seen behind the cavity as well. This rules out the assumption that the alignment change due to the motions(resonant peaks) from optics.
  
• As a comparison, I plotted RIN measured around 3am in brown trace, see PSL:1329, the level is smaller than those measured in the evening. It still makes me think that it is related to seismic but not alignment. I have to think about what other seismic driven mechanism might cause this intensity noise.
• The level of RIN seems to change as well, I looked at Evan's measurement, earlier today[/PSL:1330]. The bump around 1kHz is ~ 2e-6 1/rtHz, while for my measurement, it is close to 1e-5.  I'll try to investigate more to find out what change the level of RIN.

==To do next==

• Pick up the beam some where before PMC to check the RIN level and see where the peaks occur
• Install PD behind RCAV, see RIN from RCAV
• update noise budget: add RIN induced noise from ACAV and RCAV.

The cause of the peaks around 1kHz in RIN is solved, PMC is the reason. After damping it, the peaks disappears.

Short notes from tonight measurement:

• RIN in ACAV is better after PMC is damped, no peaks around 1kHz anymore.
• The peaks in the beat measurement also disappear, so we really see photo thermal noise.
• PD for RIN measurement behind RCAV is added, RIN is measured.
• I Will add the effect from RIN and compare with beat measurement soon.

need to buy:

•    new PBS for PDH locking. the one for RCAV is not good because there is an unwanted reflected beam going to the RFPD.
•  other optics for EOAM for RCAV

 I checked the calculation. I think most of the discrepancies are from the thick coating correction calculation (from Evans etal paper). The error is frequency dependent, and the calculations that involve frequency dependence are temperature fluctuation and thick coating correction. The temperature fluctuations are the same from our results. So it is most likely the thick coating correction. I checked and the corrections did differ at high frequency.

 I need to take a closer look to tell exactly where the errors are. Since the error is small and only at high frequency (around the shot noise limit, 10kHz),  I don't think it will be a problem for us.

Optimized coatings structure.

Tara noticed an accidental re-definition in my old code. I fixed it, and updated the svn. This fixes most of the discrepancies, but shifts the difference in thermo-optic to the low-frequency region.

Attachment 1 is the comparison from case 3 between mine and Tara's calculations of his optimized coating structure.

Attachment 2 is the comparison from case 2 between mine and Tara's calculations of a 55-layer 1/4-wavelength stack.

 I discussed the calculation with Matt. The error in TO noise is large because it is a fraction of something small. Mostly it comes from TE part. The error in TO noise appears large (10%-20%) because the TO level is small.  Otherwise, the rests are in good agreement, and I think we should be able to order soon. 

 Below, summary of the calculation, dTE is alpha_effective * coating thickness, dTO is beta effective * lambda. 0.2% difference in dTE and 0% difference in dTR can cause error upto 40% in dTO when dTE and dTR cancel each other really well. But this will be insignificant, since the final TO levels are still in the same magnitude.

The summary of the TO cancellation is in wiki page AlGaAs

Details for AlGaAs coatings order

• Coating structure can be found in http://nodus.ligo.caltech.edu:8080/PSL_Lab/1340, 55 layers, T = 197ppm.
• Coatings for 4 mirrors plane/concave, 1” diameter, 1/4” thick, with radius of curvature = 1.0m.
• AlGaAs coatings will be applied on the concave side of the mirror.
• Flat side is already AR coated
• absorption loss 6-10ppm / scattered loss 3-4ppm
• Spot radius (1/e^2 power) will be 215 um.
• The mirrors have an annulus on the rim for optical contact with thickness ~ 3mm. This area should be kept clean.
• The coating wafer should be inside the mirror sagitta to make sure that it will not obstruct the optical bond area. By calculation, the wafer with 8mm diameter, 4.5um thick should be ok. The maximum diameter that makes the coating to be above the sagitta is about 16mm, for 4 um thickness.

• Required coating diameter = 5-8mm, Power loss due to clipping is less than 0.1 ppm, see below figure.

Above, plot of ratio of power due to finite size mirror P(r) / P0,  P(r) is the power of the beam at radius r from the center. G Cole said that the wafer can be made to 8mm diameter. diameter between 5-8 mm should be good for us.

I'm using Matt's code to do error analysis for AlGaAs coatings. This time I vary materials' parameters and compare the thermo optic noise, reflected phase and transmission. There is no alarming parameter that will be critical in TO optimization, but the values of refractive indices will change the transmission considerably.

Eric, Matt and I discussed about this to make sure that even with the errors in some parameters, the optimization will still work.

Parameters in calculation and one sigma estimated from Matt

% Coating stuff
betaL = 1.7924e-4 +/- 0.07e-4; %dn/dT
betaH = 3.66e-4  +/-0.07e-4 ;
CL = 1.6982e6   +/- 5%  ; % Heat Capacity per volume
CH = 1.754445e6   +/- 5%;
kL = 69.8672   +/- 5%   ; % Thermal Conductivity
kH = 55           +/- 5%;
alphaL = 5.2424e-6 +/- 5%; % Thermal expansion
alphaH = (5.73e-6 ) +/- 5%;
sigmaL = 0.32      +/- 10%; % Poisson Ratio
sigmaH = 0.32     +/- 10% ;
EL = 100e9    +/-20e9; % Young's modulus
EH = 100e9    +/-20e9;
nH = 3.51  +/-0.03   ; % Index of refraction
nL = 3.0     +/-0.03 ;

* Note: when I change nH and nL value, I keep the physical thickness of the layers constant. This is done under the assumption that the manufacturing process controls the physical thickness. The optical thickness in the calculation will be changed, as the actual dOpt = physical thickness * actual n / lambda.  And averaged values of coatings will depend on physical thickness.

 This is fixed in Line 120-180

== Effect on TO cancellation from each parameters==

 First, I calculate the TO cancellation when one of the parameter changes. Some parameters, for examples, Poisson ratios, Young's moduli, are chosen to be the same for both AlAs and GaAs. In this test, I vary only one of them individually, to see which parameters are important. The numbers indicate the ratio between the PSD of TO noise with change in the parameter and the optimized TO noise . Not the standard deviation of the parameters.

• A) + value for Young modulus is 142 Gpa, and - value is 83 Gpa, the value in the section below is 100 +/- 20 GPa
• B) Young's moduli and Poisson's ratios for the two materials are the same value in the calculation, Young HL row calculate the TO noise when both materials have the same value of Young's modulus, while YoungH and Young L row calculate the TO noise under the assumption that only nH material or nL material has different Young's mod.
• C) + value for Poisson is the nominal value, and - value is 0.024  the value in the section below is 0.32 +/- 10%

 Turns out that the change in Young's moduli and Poisson's ratios are quite important.

==Effect on TO cancellation, from all paramerters==

 Then, I calculate the TO noise when all parameters vary in Gaussian distribution, similar to what I did before,all parameters are uncorrelated. The histograms from 1000 runs are shown below.

1. Top, ratio of PSD of TO noise at 100Hz. The cancellation should still work well.
2. Bottom left, reflected phase. It is still close to 180 degree.
3. Bottom fight, transmission. The design is 200ppm, the result spread out in a big range from  10-500ppm.

I'll try more run overnight. Each run takes about 1 second.

== combined effect from errors in layer thickness and material parameters==

Since the comparison does not need to calculate the thermal fluctuations and finite size correction all the time, I cut that calculation out and save some computation time.  Now I compare errors from

1. Error in both layer thickness and materials parameters (red)
2. Error in layer thickness only (green)
3. Error in materials parameters only (blue)
4.  Error in refractive indices only (cyan)

Each simulation contains 5e4 runs.   The Transmission varies a lot depending on the material parameters ( mostly refractive indices,  see the cyan plot).

The cancellation seems still ok. Most of the time it will not be 10 times larger than the optimized one. Only the transmission that seems to be a problem, but this is highly depends on refractive indices. It's weird that the error makes the mean of the transmission smaller.

 That surprises me too, but, that's what the calculation gives me. It is also strange that deviation in nH has smaller effect on to TO noise than nL does. I'm checking it. I ran the code one more time, and still got the same result.

Note: when I calculate the error in refractive indices, I assume that the physical thickness is constant = x * lambda/ n_0. Where x is the optical thicknesss. But if the the actual refractive index is not n_0, it means the optical length is not x, but x*n/n_0. I think this is a valid assumption, if they control the physical thickness during the manufacturing process.

update:Tue Sep 24 02:09:28 2013

The TO noise level does really change a lot when nL is nL + sigma (nL=3.0+ 0.03), dark green trace. Most of the change comes from TR noise level (not shown in the plot). TE noise remains about the same level.  It might be worth a try to find another optimization that is less sensitive to the change in value of n. I'll spend sometime working on it.

I'm trying to find another optimization that is less sensitive to change in nH and nL. Here is a few thought and a few examples.

 ==problem==

We have seen that uncertainties (withing +/- 1%)in nH and nL result in higher TO noise (up to 10 time as much) in the coating. So we are trying to see if there is another possible optimized structure that is less sensitive to the values of n. We estimate the value of nH to be 3.51 +/- 0.03, and nL to be 3.0 +/-0.03. (The numbers we have used so far are nH/nL = 3.51/3.0,  while G.Cole etal use nH/nL = 3.48/2.977.

==Optimization method==

The algorithm is similar to what I did before[PSL]. But this time the cost function is taken from different values of refractive indices. The values of nH and nL used in this optimization are

• nH = 3.48, 3.51, 3.54
• nL = 2.97, 3.00, 3.03.

The cost function is the sum of the TO noise level at 100Hz, Transmission, and reflected phase, calculated from 9 possible pairs of nH and nL values. The weight number from each parameters (which parameter is more important) are chosen to be 1, as a test run. I have not had time to try other values yet, but the prelim result seems to be ok.

[Details about the codes, attached codes]

Note about the calculation,

The calculation follows these facts:

• The nominal values of nH/nL are 3.51/3.00
• The optical thickness is designed based on the above nH and nL
• The optimized design is reported in optical thickness which is converted to physical thickness with the nominal values of nH/nL
• The procurement of coatings control the physical thickness (with error in thickness discussed before PSL:)
• If the values of nH/nL changes from the nominal values, this will affect in the coatings properties because of the change in optical thickness.

 ==results from  QWL (55layers) and 4 other optimized coatings.==

1. Left plot shows  TO noise at 100Hz in m^2/Hz unit,
2. Middle plot:Transmission [ppm]
3. Right plot: reflection phase away from 180 degree.

Each plot has three traces (blue, black, red) for different values of nH (3.48, 3.51, 3.54). nL is varied on x-axis from 2.97 to 3.03. The first result is from QWL coating, with 55 layers. This serves as a reference, to see how much each property changes with the uncertainty in nH and nL.

   I tried to change the cost function in the optimization code and numbers of layer to see if better optimized structure can be done. The optimized structure (V3,4,5) seems to be less sensitive to the values of n, see below.

Above: from QWL coatings, 55 layers. nominal transmission = 100ppm.  We can see that the transmission of QWL coatings is still quite sensitive to uncertainties in nH and nL.

Above: First optimization reported before, TO noise is larger by a factor of 10 in certain case, and transmission can be up to 500 ppm. This coating is very sensitive to the change in refractive indices.

Above: opt3, obtained from the code using the new cost function discussed above.  55 layers, nominal transmission = 150ppm. The TO noise is less dependent on nH and nL, but the transmission is still quite high.

Above: opt4, the weight parameter for transmission is changed to 3, 57 layers.

above opt5,the weight parameter for transmission is changed to 50, Lower/Upper thickness bound = 0.1/0.5 lambda, 59 layers

 Above: Opt6, the weight parameter for transmission is changed to 500, Lower/Upper thickness bound = 0.1/1.2 lambda, 59 layers

From the results, optimized structure # 3,4,5 seem to be good candidates. So I ran another monte carlo error analysis on opt1 (as a reference), opt3, opt4, and opt5, assuming errors in both material properties and coating thickness. Each one has 5e4 runs. Surprisingly, the results from all designs are very similar (see the plot below). It is possible that, by making the coatings less sensitive to changes in nH/nL, it is more sensitive to other parameters (which I have to check like I did before). Or the properties are more dependent on coating thickness, not material parameters (this is not likely, see psl:1345). Or perhaps, there might be a mistake in the monte carlo run. I'll check this too.

I'll update the coating structure and forward it in google doc soon.

The new optimization is less sensitive to the values of refractive indices, but the overall error will not change much if other material parameters have the uncertainties as we estimate.

Summary: see update of error analysis in PSL:1356. The issues from the previous entry are cleared

• I made sure that the monte carlo tests were correct
• The new optimization (called opt4, and opt5) will make the TO noise level/Transmission less sensitive to nH and nL values. But with the current estimate of uncertainties in other parameters, the performance will be about the same to that of the original optimization (called opt1).

1) show error analysis

New legs were installed.  The table is floated. The cables for signals/ power supplies will be reconnected later

 Evan and I replaced the old legs. I made sure that the leak was not in the connections and the tube. After the legs replacement, the air pump can reach 25 psi within ~25 minutes and the table can be floated.

The regulating valves are adjusted and the table is leveled.

I recalculated the coatings properties, with the values of nH and nL to be 3.48 and 2.977. Note about each optimization is included here. Transmission plots are added in google spread sheet. I'll finish the calculation for E field in each layer soon.

Note about each optimized coating version: different versions were obtained from different cost functions, and different number of layers.

opt1

• 55 Layers
• T = 210 ppm
• TO noise and transmission is too sensitive to the change in nH and nL
• 1/4 cap of nH. I did not fix the cap thickness for other coatings. Since there is no reason to keep the thickness of the cap constant.
• TO noise and transmission of this one changes a lot with uncertainty in nH/nL

opt3

• 57 Laayers
• T = 150 ppm
• Transmission is still too sensitive to the change in nH and nL
• TO noise/ transmission is less susceptible to change in nH/nL.
• First layer is 0.1 lambda thick (~285 nm) I'm not sure if this will be a problem for a cap or not.

opt4

• 57 Layers
• T = 150 ppm
• TO noise and Transmission are less sensitive to nH and nL
• less amount of nL material, should be less sensitive to error in thickness control

opt5

• 59Layers
• T= 144 ppm
• TO noise and Transmission are less sensitive to nH and nL
• reflected phase is more sensitive compared to opt4
• use less nL material
• 0.1 lambda thick

Judging from TO noise level, Transmission and reflected phase, I think opt4 is the best choice for us. The structure consist of thick nH layers and thin nL layers. This is good for us in terms of thickness control.

Electric field in coating layer is calculated. This will be used in loss calculation in AlGaAs coatings.

• In each coating layer, there are two E waves, transmitted and reflected  waves. The two interfere and become an effective field.
• The averaged electric field will depend only on the transmitted beam inside each layer, see the calculation.
• The effective transmissivity can be calculated, for coatings with N layers between air and substrate, there will be an N+1 vector representing the effective transmission, called tbar in the code. This tbar(n) is the transmissivity in the nth layer, similar to rbar in Evans etal calculation.
• The ratio of E field/ E input in nth layer will be tbar(1)*tbar(2)*...tbar(n)
•  |E field/ E input |^2 of the final transmitted beam is the transmission of the coatings.  The numbers from this calculation agrees to the calculation from before.

==supplementary information==

1) average E field in layer is the transmitted E field in the layer.

 I attached a short matlab file for a simulation of the combined field. Ein in each layer will be the transmitted beam through the layers. For a value of r close to 1, we get a standing wave. Try changing the value of r in test_refl.m to see the effect

2) Calculation for the transmitted field in each layer

I borrow the notation from Evns etal paper (rbar), the calculation code multidiel_rt.m is attached below. Note: the final transmission calculated in the code is the transmission from the coating to the substrate. To calculate the transmission to the air, multiply the last transmission by 2*n_sub/(n_sub + n_air) which is the transmission from sub to air. Since the thickness of the substrate is not known with the exact number, it will not be exact to the transmision calculated in GWINC or Matt A's code (which do not take the sub-air surface into account), but they will be close, because the reflected beam in the last interface will be small compare to those in the coatings.

==result==

The penetration of E field for QWL and different optimized coatings are shown here. The transmissions in the legend are calculated from MattA./GWINC and the values in the parenthesis are calculated from multidiel_rt.m which include the effect from the substrate-air surface. This makes the values in the parenthesis smaller (as more is reflected back and less is transmitted).

I checked the dependent of coatings properties with the uncertainty in x (amount of Al in Al_x Ga_(1-x) As). The effect is already within the uncertainties in materials parameters we did before and will not be a problem.

G. Cole told us about the variations in Al contents in the coatings. Right now the values are 92% +/- 0.6%. 

(92.10, 91.43, 91.34, 91.57, 92.73, 92.67).  Although the deviation is small, the Al content does not always hit 92%, but 92+/- sigma%. So I decided to check the effect of x on the optimization.

The materials properties that change with x are heat capacity, alpha, beta, heat conductivity and n. The values of those as functions of x can be found on ioffee  except n. So I looked through a couple of sources ( rpi, sadao)  to get n as a function of x, (Note: E0 and D0 are in eV, they have to be converted to Joules when you calculate chi and chi_so).  GaAs (nH) has a well defined value ~ 3.48+-0.001, nL has a bit more uncertainty, but it is within the approximated standard deviation of 0.03 . The table below has numbers from the sources. For RPI, I use linear approximation to get nL for x = 0.92 @ 1064nm.

The dependent of n on x is about -0.578 *dx. The numbers from RPI and Sadao are about the same. This means that for the error of 0.6% in Al. nL can change by 0.578*0.006 = 0.0035. The number is almost a factor of ten smaller than the standard deviation of nL and nH I used in previous calculation (0.03). For examples,

• x = 0.914, nL = 2.993,
• x=0.92,     nL = 2.989
• x=0.926    nL = 2.986  (From Sadao's fit)

This means that the uncertainty in nL/nH (+/- 0.03) we used are much larger than the effect coming from uncertainty in x. This is true for other parameters as well.

After installing the table legs, I have been trying to measure the beat. However, there is an unknown scattered light noise up to 400 Hz. I'm still trying to fix that.

  Here are some bullets about what happened, I'll add the details later.

• Extra noise that looks like scattered light goes up to 400Hz ( was around 100 Hz before, not from floating the table)
• One of the air spring supporting the vac tank has a leak. But it is unlikely to be the source of the extra noise mentioned above.
• The finesse of the cavities may be less than the designed value (10 000) because of the not so clean isoprop I used on the mirrors.

Note: check if the beams in the tank is blocked by wires or not.

I revised the calculation for photo-thermal noise in AlGaAs coatings, the photo thermal noise should not be a limiting source.

==review==

photothermal noise arises from the fluctuation in the absorbed laser power (RIN + shot noise, mostly from RIN) on the mirror. The absorbed power heats up the coatings and the mirror. The expansion coefficient and refractive coefficients  convert thermal change into phase change in the reflected beam which is the same effect as the change of the position of the mirror surface.

Farsi etal 2012, calculate the displacement noise from the effect. The methods are

• Solving heat equation to get temperature profile in the mirror.
• Use elastic equation to calculate the displacement noise due to the temperature change (thermoelastic)
• For TR, the effect is estimated from effective beta (from QWL stack) and the temperature at the surface ,as most of the TR effect comes from only the first few layers

When they solve the heat equation, the assume that all the heat is absorbed on the surface of the mirror. This assumption is ok for their case ( SiO2/Ta2O5) with Ta2O5 at the top surface, all QWL, as 74% of the power is absorbed in the first four layers (with the assumption that the absorbed power is proportional to the intensity of the beam, and all absorption in both materials are similar).

However, for AlGaAs coatings with (nH/nL) = (3.48/2.977) The E field goes in the coatings more that it does in SiO2/Ta2O5, see the previous entry. So we might want to look deeper in the calculation and make sure that photo thermal noise will not be a dominating noise source.

==calculation and a hand waving argument==

 The plot below shows the intensity of the beam in AlGaAs Coatings, opt4, and the estimated intensity that decreases with exponential square A exp(-z^2/z0^2). X axis is plotted in nm (distance from surface into coatings). The thickness of opt4 is about 4500 nm. To simplify the problem, I use the exponential decay function as the heat source in the diff equation. But I have not been able to solve this differential equation yet. Finding particular solution is impossible.  So I tried to solve it numerically with newton's method, see PSL:284. But the solution does not converge. I'm trying green function approach, but i'm still in the middle of it.

However, the coatings optimized for TO noise should still be working. Evans etal 2008 discuss about how the cancellation works because the thermal length is longer than the coating thickness. The calculation (TE and TR)  treat that the temperature is coherent in all the coatings ( they also do the thick coatings correction where the heat is not all coherent, and the cancellation starts to fail at several kHz). So the clue here is that the cancellation works if the heat (temperature) in the coatings change coherently.

For photothermal calculation, if we follow the assumption that all heat is absorbed at the surface (as in Farsi etal), we get the result as shown in psl:1298, where most of the effect comes from substrate TE . In reality, where heat is absorbed inside the coatings as shown in the above plot, heat distribution in the coatings will be even more coherent, and the effect from TE and TR should be able to cancel each other better. Plus, higher thermal conductivity of AlGaAs will help distribute the heat through the coatings better.

This means that  the whole coatings should see the temperature change more coherently, thus allowing the TO cancellation in the coatings to work. The assumption that heat is absorbed on the surface should put us on an upper limit of the photothermal noise.

This means that photothermal noise in the optimized coatings should be small and will not be a dominating source for the measurement.

I'm optimizing the setup, and clearing the table a little bit.

• Self homodyne setup in ACAV path is removed. This is from Erica's setup and it is not used. The input part is left, since I might use it for fiber distribution system
• optics on RCAV path, all polarization are optimized. This includes, the input and output polarization for EOAM, and quarter wave plate before the periscope. The input polarization for sideband EOM is left intact after the last adjustment, and it should be good. With+/- 4V input, I can change the power by +/-10%, (1.0 +-0.1 mW is the current setup). For Evan: Do not touch anything before discussing with me!!!
• I replaced a new PBS for PDH locking in RCAV path. The old one is bad. The surface between the prisms is milky, see the pictures below for comparison. There is also beams from multiple reflection within the cube. The new one is much better. There is no ghost beam anymore.
• I blocked all the scattered light I could find in RCAV path with Irises and beam dumps. For ACAV, I just blocked the scattered lights from the laser to the PMC. I will finish the whole setup later.
• I rechecked the height of the beam through EOMs/EOAMs. Since it is a little tricky to center the beam through the openings. The EOMs in RCAV path are all checked. For ACAV, only those between the laser and the PMC are checked(BB for phase locking and 21.5 for PMC sideband). The 14.75Mhz sideband and EOAM will be done later. The EOAM and wave plates are removed temporarily.
• I modified the TTFSS for RCAV to have a gain reduction switch to help locking the laser. I tried to lock RCAV, but I cannot turn up the gain. I'm not sure what I did wrong but this has to be investigated.

To do lists

• put optics back in ACAV path and optimize them (alignment + polarization).
• fix RCAV TTFSS . Check by measuring the TF of the modified stage/ scanning laser + checking error signal

above: old PBS, bad inter surface can be seen.

above: new PBS: all surfaces are clear

Evan found that when common gain is changed, DC offset also changes as well. I'm still looking into the problem.

a part of schematic, the driving signal was sent in through test port (the switch was flipped from off to test), so the signal came through PD line in this page.

 We still cannot lock RCAV with TTFSS, so I'm checking the box 2009007 (#7).

• The modifications I did two days ago were 1) adding a push switch for gain reduction and 2) replacing one resistor. The rests were changed before we got the TTFSS.
• I checked the TF between TP1 and TP5, it works as it should be (20log( 390/100) ~ 12 dB). So this stage does not have any problem. I checked both TF and time domain signals. So the modifications are ok.
•  Note, when I measured the TF of TP1/TP4 or TP5/TP4, the signals oscillated and became very noisy. I don't understand why, but this problem disappeared when I used TP4 and out2. Both boxes (#5,#7) have this same problem.

Common Gain - DC offset problem

• When Common gain is increased (CG signal to U2A chip), there is an offset observed in TP4 which is after the variable gain stage. Both boxes behave similarly. <- This surprises me, as we haven't seen this (or haven't noticed this) before.
• I checked if the offset varied with the input drive or not. I changed the input from 20mV to 40mV, with constant gain = 1000(25dB). The behavior is nonlinear (see the plot below). I checked this only on box#5.

DC offset vs input drive. DC offset is calculated from (Vmax + Vmin) /2  from a sinusoidal signal input. The signal was taken from TP4. The behavior is very non linear and it is impossible to make a table for an appropriate offset level vs common gain setting.

 What to do next?

• This seems to be an important clue about why the loop behaves badly when common gain is increased. From today test, both boxes behave the same, so I think it might be the chips' problem.
• Fortunately, we can use the offset adjustment(OS)
• to cancel the offset introduced by the common gain, but we might need to add a port some where (we might be able use fast mon channels during laser scan).  So that when we increase the gain, we can adjust the offset accordingly.
• Box#7 that I modified should be working. I don't know why I could not lock the laser before, more checking has to be done.

I'm putting EOAM back on ACAV path. The setup is roughly optimized.

(14.75 MHz) EOM , EOAM, quarter waveplate and PBS in ACAV path are put back together. I used a half waveplate in front of the EOM to adjust the beam to S- polarization. Right now all the polarizations optimization (to all EOMs, both ACAV/RCAV path) are adjusted to S-polarization with respect to the table. We may have to fine tune it later to match the E field in the EOMs.  The EOAM setup is optimized. With +/-4 V, the output power can be adjusted to 1mW +/- 0.09 mW (+/- 9%). The performance is comparable to RCAV EOAM. (10%) . I have not add another half waveplate before the EOAM yet. We can add it back later if we need to adjust the input polariztion to the EOAM.

I checked scattered light in the area between PMC and ACAV.  There is a reflection from EOAM back to EOM, but I cannot really block it with an iris. It probably bounces of the case of the EOM or going back to the crystal. Anyway I'll block the beam around this path later.

I have not aligned the beam to the cavity yet, since the temperature was changing because I removed the insulation  caps to patch them with black out material.

 I put black out material (R @1064 ~0.4-0.6%)on the vac tank insulation caps to minimize any possible scattered light source inside the tank that might leak out.  It also keep the surface cleaner from all the foam dust.

 Not yet, we will add this later. but we measured the noise at error point before and it is well below the estimated coating noise.

Plan for this week

Mon: (See Evan's entry for more detail)

• minimize RFAM,
• bring back the beat signal, optimized gain setup. A lot of improvement around 10 - 50 Hz.
• measured error noise from each PDH loop, with slope of the error signal to calibrate it to frequency noise
• measure RIN

Tue:

• measure photothermal coupling from RCAV (may remeasure ACAV)
• measure cavity pole, to check the cavities' finesses
• check calibration on PLL

Wed

• replace the broken air spring with a new one.
• fix harmonic lines in the beat
• update the calculation for the noise budget ( electronic noise, etc)
• Turn on ISS

Thur

• beat measurement

We made a mistake by choosing the input power to the cavities to be 0.25 mW, so today I turned them back to 1mW and measure the beat.

Setup:

• input power: 1mW
• TTFSS gain (C/F): RCAV (760/980), ACAV (630/650)

Note about the measurement:

• The noise at error point from ACAV is pretty high (~up to 100 nV/rtHz around 6 kHz). Better characterization will be done later to see if the suppression is enough or not. I made sure that this measurement is good up to ~2kHz, (this was done by changing the gain level a bit and beat level did not change).
• Table was floated, but the air springs was not activated. I hope we will get better signal around 20-100 Hz once the air spring is re-installed.
•  Intensity noise->photothermal around a few hundred mHz, cause the PLL to drift away from the input range. This effect becomes worse when the input power is increased.
• RIN from ACAV is about a factor of 10 higher than that of RCAV.
• I measured the beat with ISS on/off on ACAV, nothing were significantly different. So maybe it is not a problem for now.

To do next:

• I'll compare this measurement with the one from 8" cavity to see if the results agree or not. They are different mirrors, but from the same coating run. I suspect that the loss in these mirrors are higher than what we estimate (Ta2O5=2.5e-4, SiO2 1e-4).
• Think about the error in the measurement (calibration, spotsize etc).

I add the photo thermal noise effect in the noise budget. With ISS, photothermal noise should be sufficiently small.

What I did

• Measure beat
• Measure RIN after ACAV and RCAV
• Measure TF between TRANSPD and beat, compare the result with Farsi's calculation to determine the absorption (8ppm, with Finesse = 1e4) [add more details]
• Apply the measured RIN to Farsi calculation to get the conversion from RIN to frequency noise ( I did not use the measured TF because I have not measured the whole range yet, and the calculation matches the measurement quite well).

Comment about the beat

• At DC -30 Hz, the noise seems to be a combination of photothermal noise, and seismic induced scattered light. Air spring might not help as much as I thought.
• Above 2kHz, it's not clear if it is gain limited on ACAV loop or not, but this is likely. We can check by measure the PSD of the error signal and convert it to frequency noise.
• Frequency stabilization of ACAV is significantly inferior than that of RCAV. I don't know if it is the result from PMC or not. More investigation is needed.

Note about RIN measurement

• RIN (measured behind the cavities) depends considerably on the TTFSS gain, luckily, at optimum gain level, RIN is pushed down enough.
• RIN from ACAV is almost a factor of 10 worse than that of RCAV @ the optimum gain setting
• There might be coupling from BB EOM to RIN (due to the mismatches E field between the EOM and the beam). This may explains why RIN is getting worse if common gain is increased a bit before the loop oscillate. Will check that.

Note about loss angles: For  SiO2 and Ta2O5 loss angles = 1e-4 and 7.5e-4 (a factor of 3 above the regular number), the noise budget matches the measurement well. I'll see if it is the same for the data from 8" cavities or not.

I'm re-arranging the optics in PMC path a bit. The work is in progress, so ACAV path is still down.

I'm investigating why ACAV TTFSS performance is worse than that of RCAV. One thing is that ACAV has the PMC. This area has not been optimized for awhile, so I'm checking everything.

PMC path is back, I aligned the polarization of the input beam to the BB EOM for TTFSS. The visibility of PMC is now ~ 80%.

I created an svn folder for my thesis on CTN measurement.

It can be found here