Tara and I repositioned the QWP and PBS immediately preceding the periscope so that we could move the 64.4-mm ROC lens closer to the cavity. For space reasons, this lens is now forked directly to the table rather than mounted on a translation stage. I tried for a while to adjust this and the 38.6-mm ROC lens to improve the mode matching, but I can't seem to do much better than 80% visibility. We may have to adjust the 103-mm ROC lens directly after the PMC in order to go further.
In better news, we were able to couple some power into the fiber that runs into the ATF. The beam is picked off with the PBS immediately following the EAOM and then sent through two mode-matching lenses and a HWP before hitting the fiber. We're sending 10 mW in and currently getting 0.85 mW out. More work is needed to get the polarization correct and to improve the coupling efficiency. This setup will probably have to be redone at some point, since the current pickoff beam is downstream of the cavity EOM and therefore has sidebands on it. Also, we will have to redo the coupling if we touch the 103-mm ROC lens to improve the cavity mode matching.
I redid the mode matching for both refcav, the visibilities are up to ~ 93% and 95% for RCAV and ACAV.
I'll add the new layout for the current situation soon.
I installed the beat board back behind the cavities. I still have not finished aligning both beams to the 1811.
Both cavities are locked (not optimized yet). Since it has been awhile that both are locked, here is a picture.
Rcav is locked by Fast feedback only. I still have to check the polarity for PC feedback. I adjusted the phase between the LO and PD for RCAV loop to get a nice error signal. I noticed that there is an offset in the error signal, I will try to adjust the polarization of the beam in front of the EOM to see if I can reduce this offset from RFAM.
Since we are trying to optimize a layer structure for AlGaAs coatings. It is a good idea to summarize some notes about all the coatings details. Thanks Koji for the discussion about the coaitngs.
==some background about SiO2/Ta2O5 QWL with 1/2 wave cap coatings==
For quarter wave layer stack (QWL) SiO2/Ta2O5 coatings, SiO2 and Ta2O5 are the material with low (nl) and high refractive indices (nh), respectively. Due to the stronger structure of SiO2, we usually have a cap of SiO2 as a protective layer on top. This cap has thickness of 1/2 wave length. The reason is that the reflected beam from the interface between the cap and the next layer will be in phase with the first reflected beam at the air-coating surface, see the figure below (top).
If the SiO2 cap is 1/4 thick, the reflected beam from the interface between the cap and the next layer will destructively interfere, causing the reflectivity to go down (see the picture below, middle).
However, if the cap is Ta2O5 (nH) material, it can be QWL thickness, and the phase from every reflected beams still interferes constructively (picture below, bottom).
Note: As we can see, the incoming beam and the reflected beam are 180 degree out of phase. It means that the E field at the coatings surface will always be zero. This will prevent the burning on the surface of the coating. With this, the standing wave in the cavity will always have zero E field at the coating surface, see below picture.
This is not AR coat, since all the reflected beams interfere constructively. The reflected beams from AR coating will destructively interfere among each layer.
To sum up for the SiO2/Ta2O5 coatings:
For GaAs/Al0.92Ga0.08As (AlGaAs) coatings, the situation is a bit different from SiO2/Ta2O5. The cap has to be GaAs (nH) because Al0.92Ga0.08As will oxidize and change its material properties. Now that the cap will be nH, the thickness has to be 1/4 wavelength. The last layer next to the substrate has to be GaAs (nH) too (I think because of both the better reflectivity and the fabrication process).
There is an assumption about the layer structure used in the optimization code that the cap is nL(SiO2), 1/2 layer. The coatings layers are even number ( doublets of SiO2/Ta2O5). I'm making sure all the assumptions in the code are fixed. Here is a preliminary result.
above: Layer structure, the first layer (cap) is GaAs (nH). In the optimization, I keep the cap thickness to be 1/4, and vary the rest.
above: Noise budget of the optimized layer. TO noise is below BR noise from DC up to 1kHz.
The reflectivity of the coatings is -0.9997 + 0.0209i (reflection phase = 180 - 1.2 degree). I'm not sure if this is good enough, maybe better optimization can be done.
Note: My layer structure is really different from what rana did in T1200003. For my structure, the layers near the cap vary a lot before getting close to 0.25 when the layers are close to the substrate. The result from 1200003 is the opposite. The layers near the cap are about 0.25, and start to diverge when the layers are close to the substrate.
above: Optimized coatings result from T1200003. The optimization probably assume the cap of low index material, but the following layers evolution are opposite of what I got. That's why I'm not sure about my optimization.
I'll upload my codes soon so that people can check my optimization.
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
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:
This code calls on optAlGaAs.m when running fmincon.
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:
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
In preparation for getting the ISS up and running, Tara and I have been fooling around with the EOAM and associated half waveplates. Additionally, Tara inserted a quarter waveplate (mounted horizontally, for space reasons) after the EOAM in order to get linear amplitude modulation. The HWP before the EOAM is at 99 degrees and the QWP after the EOAM is at 51 degrees.
There's currently 8.0 mW going into the EOAM and 4.0 mW coming out after the EOAM + QWP + PBS. When 10 V dc is applied to the EOAM, the power drops to 3.7 mW. This gives a conversion factor of 3.0×10−5 W/V. The value expected from the manual is (π/2)(8 mW / 300 V) = 4×10−5 W/V, so we're not too far off.
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
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.
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==
Since the optimized layer structure is designed, I'm checking how the coatings properties change with error in layer thickness.
G.Cole said that they can control each layer thickness within 0.3%. So I tested the optimized coatings properties by adding some random number within +/- 0.5% on each layer thickness. The results are shown below for 10 000 test.
The error check does the following:
The figure below is an example of the varying layer thickness added by rand command. They are confined within 0.5%.
1) result from the error in thickness control
Above: histograms of the important values. top left, reflected phase. top right, ratio between PSD of Brownian noise and Thermo optic noise at 100 Hz. Bottom left, transmission. Bottom right, total coating thickness error.
comments: this test is chosen for 0.5% error which is almost a factor of 2 worse than what they claimed (0.3%), so the actual result should be better. I assumed 0.5% errof because of the irregular layer structure of the optimized coatings, there might be some more error in the manufacturing process.
2) result from different calculated Beta values:
Here I checked what happen if the beta calculation was wrong, and the error is still within 0.5% in each layer.
In Evans paper, the effect from "Thermo-refractive" comes from the phase changes of the wave travels in each layer. So it includes the effect from dn/dT and dz. The effective beta for each layer is given as
where alpha bar is
Where s denotes substrate, k denotes the material in each layer (high or low indices).
So my, calculation & optimization have been using these equations.
However, in the original GWINC code for TO calculation, the calculation [B8], alphabark( used in dTR) is not the same as A1, but rather.
alphaH * (1 + sigH) / (1 - sigH)
see getCoatLayerAGS.m. Line 16-17.
This is used in the calculation for beta effective in getCoatTOphase. Line73-74. Notice that for dTE, the alpha_bar_k is the same as used in Evans. (line 77).
the comment says "Yamamoto thermo-refractive correction". I emailed kazuhiro yamamoto, but never got a response back. So I keep using the same formula as in Evans because I don't see the reason why the expansion contribution should be different between TE and TR.
So this is the nb plot for TO noise from the optimized coating, if using yamamoto TR correction.
Above: nb from the optimized coatings, using Yamamoto TR correction. The cancellation becomes worse, but TO is still lower than other noise.
Finally, I repeat the same error analysis for random noise in the thickness (+/- 0.5%).
Most of the parameters behave similarly, except the cancellation (upper right plot). Now BR is only ~ x12 larger than TO noise because of the worse cancellation. Good news is, it still below Brownian noise, the cancellation still somehow works.
Here is a summary for how I verify the codes for TO calculation.
So far, we have been using a set of modified GWINC codes to calculate TO noise, but I have not mentioned how did I make sure that the codes were reliable. So I'll try to explain how I check the codes here.
==What do we compute?==
For the TO nosie calculation and the optimization, we are interested in:
==Beta calculation check==
For TR coefficient we can compare GWINC with an analytical result (see Gorodetsky,2008, and Evans 2008) (when # of layers ~ 50 or more), see psl:1181. I tried the solution with nH, 1/4 cap and nL, 1/4 and 1/2 cap. All results agree.
==Alpha calculation check==
There is no complication in this calculation. The effective alpha is just the sum of all layers. This calculation is quite straight forward.
This was done by reducing the coating layers to one or two layers and comparing with an analytical solution by hand. I checked this and the results agreed.
So I think the calculations for TO noise is valid, the noise estimated from the optimized coatings is done with some error check (previous entry). I think we should be ready to order.
I updated the optimization and error analysis. The error in optimized structure is comparable to that of a standard quarter wave length structure.
After a discussion with Rana, Garrett, and Matt, I fixed the thermo-optic calculation, and the error analysis done in PSL:PSL:1315. The modifications are
1) fix the TO calculation (Yamamoto TR correction): There is a modification for TR correction that is not in Evans etal 2008, paper. I contacted M. Evans to ask about the details of this correction which is done in GWINC.
2) Try another optimized coatings with the correct TO calculation: After the correction, I ran doAlGaAs.m code, cf PSL:1269 using fmincon function , to find another optimized structure. The result is shown below.
above) layer structure in optical thickness, the .fig and .mat file are attached below. Note .mat file contains 54 layers, you need to add 1/4 cap to the first entry to calculate the noise budget.
above) noise budget of the optimized coating.
3) Repeat the error analysis : This time I used the following assumptions (from G Cole)
Fig1: Above, percentage of error distribution between the two materials used in the calculation. nH(red) has 2 sigma = 1% and nL(blue) has 2sigma=1%.The same error distributions are used for both optimized layers and QWL layers in comparion, see fig2.
The section below is the algorithm used to distribute the error, this one makes the error between the two materials to be the same sign. The whole code can be found on svn.
mu1 = 0;
sigma1 = 0.5; % 2sigma is 1percent;
mu2 = 0;
sigma2 = 1;
run_num = 5e4; % how many test we want
errH = normrnd(mu1,sigma1,[run_num,1]); %errH in percent unit
errL = normrnd(mu2,sigma2,[run_num,1]); %errL in percent unit
errL = abs(errL).*sign(errH); %make sure that errH and errL have the same sign
dOpt = xout; % xout from doAlGaAs (optimized layer)
dOpt = [ 1/4 ; dOpt]; % got 54 layer no cap from doALGaAs, need to add the cap back
dOpt_e = zeros(length(dOpt),1);
for ii = 1:run_num;
dOpt_e(1:2:end)= dOpt(1:2:end)*(1+ errH(ii)/100 );
dOpt_e(2:2:end)= dOpt(2:2:end)*(1+ errL(ii)/100 );
This time I calculated the change in reflection phase (TOP left), the ratio between TO noise from the coatings with error and the coatings with no error(top right), transmission (bottom left), and ratio of BR noise ( bottom right). The result from the optimized coating(blue) is compared with the QWL coating (black).
Fig2: Error analysis, in 5e4 run. Blue: from optimized coatings Black:from 55 QWL coatings, from 5x10^4 runs.
Reflection phase: The reflection phase can be away up to ~6 degree. The power at the surface will be ~Finesse/pi * Power input * sin^2 (6degree) ~ 50mW. Seems high, but this is about a regular power used in the lab.
Ratio of PSD TO/TO_0 : At worse, it seems the PSD TO noise will be ~ a factor of 10 higher than the design. However, this will be still lower than BR noise. I plotted only the error distribution for optimized coatings because for QWL coatings, the ratio will be about the same, since TO is dominated by TE.
Transmission: Most of the results are within 197-200 ppm. The optimized coating has transmission ~ 197ppm. The QWL with 55 layers has transmission ~100ppm.
Ratio of BR: not much change here.
Coating optimization and error analysis are updated, see PSL:1320.
Optimized coatings structure.
Details for AlGaAs coatings order
Above, plot of ratio of power due to finite size mirror P(r) / P0, P(r) is the power of the beam at radius r from the center. G Cole said that the wafer can be made to 8mm diameter. diameter between 5-8 mm should be good for us.
I'm using Matt's code to do error analysis for AlGaAs coatings. This time I vary materials' parameters and compare the thermo optic noise, reflected phase and transmission. There is no alarming parameter that will be critical in TO optimization, but the values of refractive indices will change the transmission considerably.
Eric, Matt and I discussed about this to make sure that even with the errors in some parameters, the optimization will still work.
Parameters in calculation and one sigma estimated from Matt
% Coating stuff
betaL = 1.7924e-4 +/- 0.07e-4; %dn/dT
betaH = 3.66e-4 +/-0.07e-4 ;
CL = 1.6982e6 +/- 5% ; % Heat Capacity per volume
CH = 1.754445e6 +/- 5%;
kL = 69.8672 +/- 5% ; % Thermal Conductivity
kH = 55 +/- 5%;
alphaL = 5.2424e-6 +/- 5%; % Thermal expansion
alphaH = (5.73e-6 ) +/- 5%;
sigmaL = 0.32 +/- 10%; % Poisson Ratio
sigmaH = 0.32 +/- 10% ;
EL = 100e9 +/-20e9; % Young's modulus
EH = 100e9 +/-20e9;
nH = 3.51 +/-0.03 ; % Index of refraction
nL = 3.0 +/-0.03 ;
* Note: when I change nH and nL value, I keep the physical thickness of the layers constant. This is done under the assumption that the manufacturing process controls the physical thickness. The optical thickness in the calculation will be changed, as the actual dOpt = physical thickness * actual n / lambda. And averaged values of coatings will depend on physical thickness.
This is fixed in Line 120-180
== Effect on TO cancellation from each parameters==
First, I calculate the TO cancellation when one of the parameter changes. Some parameters, for examples, Poisson ratios, Young's moduli, are chosen to be the same for both AlAs and GaAs. In this test, I vary only one of them individually, to see which parameters are important. The numbers indicate the ratio between the PSD of TO noise with change in the parameter and the optimized TO noise . Not the standard deviation of the parameters.
Turns out that the change in Young's moduli and Poisson's ratios are quite important.
==Effect on TO cancellation, from all paramerters==
Then, I calculate the TO noise when all parameters vary in Gaussian distribution, similar to what I did before,all parameters are uncorrelated. The histograms from 1000 runs are shown below.
I'll try more run overnight. Each run takes about 1 second.
== combined effect from errors in layer thickness and material parameters==
Since the comparison does not need to calculate the thermal fluctuations and finite size correction all the time, I cut that calculation out and save some computation time. Now I compare errors from
Each simulation contains 5e4 runs. The Transmission varies a lot depending on the material parameters ( mostly refractive indices, see the cyan plot).
The cancellation seems still ok. Most of the time it will not be 10 times larger than the optimized one. Only the transmission that seems to be a problem, but this is highly depends on refractive indices. It's weird that the error makes the mean of the transmission smaller.
In our meeting, Eric mentioned that there might be some uncertainty in how the average coating properties are calculated.
To see how much it matters, I set the average properties to either that of the high-index (H) or low-index (L) material, and calculated the ratio of the new thermo-optic noise to the original calculation (TO'/TO) and the ratio of the new thermo-optic noise to the unchanged Brownian noise (TO'/Br) for Tara's optimized coating structure. The results are in the table below:
C = Heat Capacity/Volume, k = thermal conductivity, alpha/a = thermal expansion
alphaBar_c and alphaBar_k are more complicated, since they take into account the Poisson ratio and Young's modulus of the coating materials, and may be wildly different from the thermal expansion coefficient. alphaBar_c is an average of alphaBar_k values, and when I use "alphaBar_k = alphas", I'm indicating that alphaBar_k is an array, and I have replaced that array with an array of the corresponding thermal expansion coefficients. As we can see in the final four rows of the table, alphaBar_c has a much smaller affect if we use an alphaBar_k value with all its added moduli and ratios instead of just regular thermal expansion. alphaBar_k_TR is the array of values used in the "Yamamoto Correction" to calculate the appropriate alphaBar for the thremo-refractive noise.
This all indicates to me that while most of the averages won't have much effect on our cancellation, a mistake in the calculation of alphaBar_k will.
The difference between alphaB_k and alphaBar_k_TR (in the last two rows of the table) is also interesting. Kazuhiro Yamamoto tells us this equation is correct, and explains the correction here. It's apparently because there is no added strain in the substrate due to the change in the refractive index, while there is strain for the thermal expansion.
I don't understand these values for n.
How can nH be 3 or 11? Isn't just that nL is ~1.45 and nH is ~2 ? I would guess that the sigma for these is only ~1% of the mean values.
The numbers in the table are the ratio between the TO noise when the parameter is changed by 1sigma and the TO noise calculated form the nominal value.
About the Poisson's ratios, Matt asked me to check for the values between 0.024 to 0.32, and the TO cancellation becomes much worse. I looked up papers about AlGaAs' Poisson's ratios. Most of the literature report the value ~0.32. I think we don't have to worry about it that much.
Krieger etal 1995 Table2, and ref 16 17 thereof.
Wasilewski et al1997 page 6, also discuss about the calculation and the measurement of poisson value in GaAs and AlAs, the value is still in the range of 0.27-0.33, not 0.024. The value of 0.27 is already considered very low.
zhou and usher has a calculation for poisson's ratio of AlAs. they report ~0.32, see table 2. and there references.
So I don't think Poisson's ratios of the materials will be a problem for us, since the reported numbers agree quite well.
If that's true, then it means that a 1% deviation in the index of refraction of the low index material can by a 10x increase in the TO noise. Is this really true?
That surprises me too, but, that's what the calculation gives me. It is also strange that deviation in nH has smaller effect on to TO noise than nL does. I'm checking it. I ran the code one more time, and still got the same result.
Note: when I calculate the error in refractive indices, I assume that the physical thickness is constant = x * lambda/ n_0. Where x is the optical thicknesss. But if the the actual refractive index is not n_0, it means the optical length is not x, but x*n/n_0. I think this is a valid assumption, if they control the physical thickness during the manufacturing process.
update:Tue Sep 24 02:09:28 2013
The TO noise level does really change a lot when nL is nL + sigma (nL=3.0+ 0.03), dark green trace. Most of the change comes from TR noise level (not shown in the plot). TE noise remains about the same level. It might be worth a try to find another optimization that is less sensitive to the change in value of n. I'll spend sometime working on it.
I'm trying to find another optimization that is less sensitive to change in nH and nL. Here is a few thought and a few examples.
We have seen that uncertainties (withing +/- 1%)in nH and nL result in higher TO noise (up to 10 time as much) in the coating. So we are trying to see if there is another possible optimized structure that is less sensitive to the values of n. We estimate the value of nH to be 3.51 +/- 0.03, and nL to be 3.0 +/-0.03. (The numbers we have used so far are nH/nL = 3.51/3.0, while G.Cole etal use nH/nL = 3.48/2.977.
The algorithm is similar to what I did before[PSL]. But this time the cost function is taken from different values of refractive indices. The values of nH and nL used in this optimization are
The cost function is the sum of the TO noise level at 100Hz, Transmission, and reflected phase, calculated from 9 possible pairs of nH and nL values. The weight number from each parameters (which parameter is more important) are chosen to be 1, as a test run. I have not had time to try other values yet, but the prelim result seems to be ok.
[Details about the codes, attached codes]
Note about the calculation,
The calculation follows these facts:
==results from QWL (55layers) and 4 other optimized coatings.==
Each plot has three traces (blue, black, red) for different values of nH (3.48, 3.51, 3.54). nL is varied on x-axis from 2.97 to 3.03. The first result is from QWL coating, with 55 layers. This serves as a reference, to see how much each property changes with the uncertainty in nH and nL.
I tried to change the cost function in the optimization code and numbers of layer to see if better optimized structure can be done. The optimized structure (V3,4,5) seems to be less sensitive to the values of n, see below.
Above: from QWL coatings, 55 layers. nominal transmission = 100ppm. We can see that the transmission of QWL coatings is still quite sensitive to uncertainties in nH and nL.
Above: First optimization reported before, TO noise is larger by a factor of 10 in certain case, and transmission can be up to 500 ppm. This coating is very sensitive to the change in refractive indices.
Above: opt3, obtained from the code using the new cost function discussed above. 55 layers, nominal transmission = 150ppm. The TO noise is less dependent on nH and nL, but the transmission is still quite high.
Above: opt4, the weight parameter for transmission is changed to 3, 57 layers.
above opt5,the weight parameter for transmission is changed to 50, Lower/Upper thickness bound = 0.1/0.5 lambda, 59 layers
Above: Opt6, the weight parameter for transmission is changed to 500, Lower/Upper thickness bound = 0.1/1.2 lambda, 59 layers
From the results, optimized structure # 3,4,5 seem to be good candidates. So I ran another monte carlo error analysis on opt1 (as a reference), opt3, opt4, and opt5, assuming errors in both material properties and coating thickness. Each one has 5e4 runs. Surprisingly, the results from all designs are very similar (see the plot below). It is possible that, by making the coatings less sensitive to changes in nH/nL, it is more sensitive to other parameters (which I have to check like I did before). Or the properties are more dependent on coating thickness, not material parameters (this is not likely, see psl:1345). Or perhaps, there might be a mistake in the monte carlo run. I'll check this too.
I'll update the coating structure and forward it in google doc soon.
The new optimization is less sensitive to the values of refractive indices, but the overall error will not change much if other material parameters have the uncertainties as we estimate.
Summary: see update of error analysis in PSL:1356. The issues from the previous entry are cleared
1) show error analysis
I recalculated the coatings properties, with the values of nH and nL to be 3.48 and 2.977. Note about each optimization is included here. Transmission plots are added in google spread sheet. I'll finish the calculation for E field in each layer soon.
Note about each optimized coating version: different versions were obtained from different cost functions, and different number of layers.
Judging from TO noise level, Transmission and reflected phase, I think opt4 is the best choice for us. The structure consist of thick nH layers and thin nL layers. This is good for us in terms of thickness control.
Electric field in coating layer is calculated. This will be used in loss calculation in AlGaAs coatings.
1) average E field in layer is the transmitted E field in the layer.
I attached a short matlab file for a simulation of the combined field. Ein in each layer will be the transmitted beam through the layers. For a value of r close to 1, we get a standing wave. Try changing the value of r in test_refl.m to see the effect
2) Calculation for the transmitted field in each layer
I borrow the notation from Evns etal paper (rbar), the calculation code multidiel_rt.m is attached below. Note: the final transmission calculated in the code is the transmission from the coating to the substrate. To calculate the transmission to the air, multiply the last transmission by 2*n_sub/(n_sub + n_air) which is the transmission from sub to air. Since the thickness of the substrate is not known with the exact number, it will not be exact to the transmision calculated in GWINC or Matt A's code (which do not take the sub-air surface into account), but they will be close, because the reflected beam in the last interface will be small compare to those in the coatings.
The penetration of E field for QWL and different optimized coatings are shown here. The transmissions in the legend are calculated from MattA./GWINC and the values in the parenthesis are calculated from multidiel_rt.m which include the effect from the substrate-air surface. This makes the values in the parenthesis smaller (as more is reflected back and less is transmitted).
I checked the dependent of coatings properties with the uncertainty in x (amount of Al in Al_x Ga_(1-x) As). The effect is already within the uncertainties in materials parameters we did before and will not be a problem.
G. Cole told us about the variations in Al contents in the coatings. Right now the values are 92% +/- 0.6%.
(92.10, 91.43, 91.34, 91.57, 92.73, 92.67). Although the deviation is small, the Al content does not always hit 92%, but 92+/- sigma%. So I decided to check the effect of x on the optimization.
The materials properties that change with x are heat capacity, alpha, beta, heat conductivity and n. The values of those as functions of x can be found on ioffee except n. So I looked through a couple of sources ( rpi, sadao) to get n as a function of x, (Note: E0 and D0 are in eV, they have to be converted to Joules when you calculate chi and chi_so). GaAs (nH) has a well defined value ~ 3.48+-0.001, nL has a bit more uncertainty, but it is within the approximated standard deviation of 0.03 . The table below has numbers from the sources. For RPI, I use linear approximation to get nL for x = 0.92 @ 1064nm.
The dependent of n on x is about -0.578 *dx. The numbers from RPI and Sadao are about the same. This means that for the error of 0.6% in Al. nL can change by 0.578*0.006 = 0.0035. The number is almost a factor of ten smaller than the standard deviation of nL and nH I used in previous calculation (0.03). For examples,
This means that the uncertainty in nL/nH (+/- 0.03) we used are much larger than the effect coming from uncertainty in x. This is true for other parameters as well.
I revised the calculation for photo-thermal noise in AlGaAs coatings, the photo thermal noise should not be a limiting source.
photothermal noise arises from the fluctuation in the absorbed laser power (RIN + shot noise, mostly from RIN) on the mirror. The absorbed power heats up the coatings and the mirror. The expansion coefficient and refractive coefficients convert thermal change into phase change in the reflected beam which is the same effect as the change of the position of the mirror surface.
Farsi etal 2012, calculate the displacement noise from the effect. The methods are
When they solve the heat equation, the assume that all the heat is absorbed on the surface of the mirror. This assumption is ok for their case ( SiO2/Ta2O5) with Ta2O5 at the top surface, all QWL, as 74% of the power is absorbed in the first four layers (with the assumption that the absorbed power is proportional to the intensity of the beam, and all absorption in both materials are similar).
However, for AlGaAs coatings with (nH/nL) = (3.48/2.977) The E field goes in the coatings more that it does in SiO2/Ta2O5, see the previous entry. So we might want to look deeper in the calculation and make sure that photo thermal noise will not be a dominating noise source.
==calculation and a hand waving argument==
The plot below shows the intensity of the beam in AlGaAs Coatings, opt4, and the estimated intensity that decreases with exponential square A exp(-z^2/z0^2). X axis is plotted in nm (distance from surface into coatings). The thickness of opt4 is about 4500 nm. To simplify the problem, I use the exponential decay function as the heat source in the diff equation. But I have not been able to solve this differential equation yet. Finding particular solution is impossible. So I tried to solve it numerically with newton's method, see PSL:284. But the solution does not converge. I'm trying green function approach, but i'm still in the middle of it.
However, the coatings optimized for TO noise should still be working. Evans etal 2008 discuss about how the cancellation works because the thermal length is longer than the coating thickness. The calculation (TE and TR) treat that the temperature is coherent in all the coatings ( they also do the thick coatings correction where the heat is not all coherent, and the cancellation starts to fail at several kHz). So the clue here is that the cancellation works if the heat (temperature) in the coatings change coherently.
For photothermal calculation, if we follow the assumption that all heat is absorbed at the surface (as in Farsi etal), we get the result as shown in psl:1298, where most of the effect comes from substrate TE . In reality, where heat is absorbed inside the coatings as shown in the above plot, heat distribution in the coatings will be even more coherent, and the effect from TE and TR should be able to cancel each other better. Plus, higher thermal conductivity of AlGaAs will help distribute the heat through the coatings better.
This means that the whole coatings should see the temperature change more coherently, thus allowing the TO cancellation in the coatings to work. The assumption that heat is absorbed on the surface should put us on an upper limit of the photothermal noise.
This means that photothermal noise in the optimized coatings should be small and will not be a dominating source for the measurement.
I'm optimizing the setup, and clearing the table a little bit.
To do lists
above: old PBS, bad inter surface can be seen.
above: new PBS: all surfaces are clear
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.
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%.
We heard back from G. Cole about the thickness resolution in the AlGaAs coating manufacturing process will be around 0.5 A. So I'm checking how the noise budget will change by rounding up the physical thickness in opt V4 to the next 0.5A. The design will still work. The round up thickness is added in the google document (for opt v4 only).
The estimated growth rate of the crystal is 4.8A/s and shutter speed is assumed to have 0.1 sec time step. This means the smallest step of the thickness control is ~0.5A. So I round up the physical thickness to the next 0.5 A and calculate the coating properties.
1) Rounding up to the next 0.5 Angstrom. The truncating process acts like a random thickness variation in the optimized coatings with maximum error ~ 0.25 Angstrom. The averaged layer thickness is ~ 800 Angstrom.
2)Results when the layers physical thickness are round up to the closest 0.5 A. The noise budget does not change much.
The coatings properties still hold, even with random error in parameters, thickness.
Note: For the error calculation I did before I used 1 sigma to be 1% for AlGaAs, and 0.5% for GaAs. The thinnest layer is AlGaAs at 35 A, so its sigma is about 0.35 A. The average thickness is 90 Angstrom, so the average error is about 0.9 A. The estimated error in the calibration process is already larger than the error from the truncation(0.25A). That's why the error analysis results are still valid.
I"m packing the mirrors so that they are ready to be shipped to G. Cole. The mirrors are packed properly, see picasa.
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
Note about optimization:
To run the code:
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.
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.
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.
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.
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.
==measurement and data analysis==
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==
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==
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.
[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:
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.
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).
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:
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?
Do we need to improve this before moving on to the AlGaAs BRDF measurement?
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)
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)°.
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.
Yesterday we took a scatter measurement of AlGaAs mirror #143. Instead of one bright scattering center, we saw 3.
The procedure is identical to the procedure used for mirror #137, although the exposure settings and choice of angles are a bit different (see the attached plot). Also, we used 20 mW of incident power instead of 10 mW.
Total integrated scatter from 14° to 82° is 80(8) ppm.
Data, images, and plot-generating code are on the SVN at CTNlab/measurements/2014_07_28.
I used the setup to measure scattered loss from an REO mirror (mirror for iLIGO refcav, the one we measured coating thermal noise) and get 6 ppm. This number agrees quite well with the previous Finesse measurement.
Finesse measurement from REO mirrors = 9700 , see PSL:424 The absorption loss in each mirror is ~ 5 ppm ( from photo thermal measurement, see PSL:1375). The measured finesse infers that the roundtrip loss is ~ 24 ppm, see here. So each mirror has ~ 12 ppm loss. With ~ 5ppm absorption loss, we can expect ~ 6-7 ppm loss for scattered loss. So this measurement roughly says that our scattered light setup and calibration is ok.
Tara took a BRDF measurement yesterday of AlGaAs mirror #114.
In this measurement, the return beam is dumped using black anodized foil instead of a razor blade dump. This seems to make the peak at 20° disappear, and now we get a more or less monotonic falloff in scattered power.
TIS from 14° to 71° is 39(6) ppm.
Data and code are on the SVN at CTNlab/measurements/2014_07_30.
Tara also took a BRDF measurement of #114 after cleaning it.
After cleaning, TIS from 14° to 71° is 2.7(5) ppm. Much improved.
Data and code are on the SVN at CTNlab/measurements/2014_07_31.
Incident power: 20.0(1) mW
Exposure times used: 25 ms, 50 ms, 200 ms, 500 ms, 1000 ms
Transmitted power: 3.34(2) µW. This gives a transmission of 167(1) ppm for this mirror.
TIS from 16° to 73° is 18(1) ppm.
Data and code are on the SVN at CTNLab/measurements/2014_08_05.
Basically the same story with 132.