ID 
Date 
Author 
Type 
Category 
Subject 
55

Fri Jul 12 00:09:17 2013 
Emory Brown  Optics  General  Analysis of substrate Brownian noise in a silicon test mass with changing ratioR value 
I performed the same analysis we previously did on a test mass made of silicon (using the builtin silicon within COMSOL) and obtained the attached plots. The plots show more of a difference between the two solutions optimal points than for fused silica, but they are still quite close. All data is included in the attached tarball. The values for ratioR=1 or a cylindrical test mass place the frequency of the lowest real eigenmode at 8491 Hz and the Umax value of 6.5484*10^11 J. 
Attachment 1: SiliconEigenfrequencyPlot.png


Attachment 2: UmaxPlotSilicon.png


Attachment 3: Silicon.tar.gz

56

Fri Jul 12 00:31:11 2013 
Emory Brown  Optics  General  Reversed face mirror design 
We had noticed that when we ran an eigenfrequency analysis, there was often a node (see attached screenshot) on the face opposite the applied force. This indicated that the noise might be lowered by reversing the face the force was applied to, but it also seems like this would be equivalent to increasing the ratioR value and making the primary face smaller while retaining the mass of the test mass. It seems worthwhile to see how the Umax values compare for a beam incident on either face of a test mass in the optimal region with Umax=0.71. For the initial configuration, Umax=1.47228*10^10 J and changing the side of the mirror the force is applied to we get Umax=1.47148*10^10 J, which is about the same value. 
Attachment 1: node.png


59

Mon Jul 15 13:14:30 2013 
Emory Brown  Optics  General  Umax curves for varying height test masses 
I ran sweeps over the ratioR range from 0.5 to 0.9 for test masses with modified height, and thus mass, which were otherwise equivalent to our previous test mass. The attached plot was generated and indicates that the optimal ratioR value for the test mass was shifted significantly, for the half mass test mass into lower frequency regions, and for the 10x mass test mass into higher frequency ranges. I took the data collected from the half mass run and fit a polynomial of degree 2 to it in Matlab using the command p=polyfit(x,y,2) and obtained p=1.089*10^10*x^2  1.024*10^10*x +1.774*10^10 which has a minimum value for ratioR=0.47. This fits with the plot generated as it appears to be near its minumum value, but I should run the simulation more around this value to see if it fits the expected behaviour. I also did this fitting for the 10x mass curve, but based on the greater variance in the curve over this region of ratioR values I don't expect the predicted minimum to be as accurate, so if we want to find it we will have to search that region of ratioR values. For the 10x mass test mass, p=4.459*10^10*x^2  8.859*10^10*x + 6.289*10^10 which has a minimum of about 1.
I will run simulations in these regions overnight, and edit this post with the plots/results. 
Attachment 1: MassComparison.png


61

Tue Jul 23 18:26:04 2013 
Deep Chatterjee  Optics  General  Comparison between Liu and Thorne Results and COMSOL results for TE noise 
In this post I report of the results of TE noise simulated by COMSOL for the TE noise of Infinite test masses.
The aim was to follow the procedure by Liu and Thorne in their analytic calculations so that the same model could be used for the other
geometries.
The simulation is done in a different way than the TR simulations. It was observed that the output given by COMSOL by the use of commands
like mphinterp() or taking an export resulted in certain discrepancies between the results computed in COMSOL and that read by MATLAB.
Thus, the volume integration of the temperature gradient is performed in COMSOL itself and the results of the integration for each time
are sent to files. Matlab read these values and time averages them to get the result as in the paper (Sec. 2 of Liu and Thorne).
The errors expected are
> Fourier analysis is not done at all. This would have involved exporting data which, as mentioned before is giving errors
> The numerical errors by COMSOL are therefore not filtered off.
> The plot differs from the analytic solution for larger frequencies over 3000 Hz.
> It is to be noted from the paper by Liu and Thorne that the TE noise for the finite and infinite case are not very different. In
fact the correction factor goes as O(1). Thus, differences between finite and inifinte cases are unlikely to be prominent
in the log scale plots
The codes are put as a zip file. Corrections made to the codes will be uploaded as a reply.

Attachment 2: TE_calc_Jul23.zip

Attachment 3: Jul_23.eps


67

Tue Jul 23 20:53:37 2013 
Deep Chatterjee  Optics  General  Comparison between Liu and Thorne Results and COMSOL results for TE noise 
Quote: 
In this post I report of the results of TE noise simulated by COMSOL for the TE noise of Infinite test masses.
The aim was to follow the procedure by Liu and Thorne in their analytic calculations so that the same model could be used for the other
geometries.
The simulation is done in a different way than the TR simulations. It was observed that the output given by COMSOL by the use of commands
like mphinterp() or taking an export resulted in certain discrepancies between the results computed in COMSOL and that read by MATLAB.
Thus, the volume integration of the temperature gradient is performed in COMSOL itself and the results of the integration for each time
are sent to files. Matlab read these values and time averages them to get the result as in the paper (Sec. 2 of Liu and Thorne).
The errors expected are
> Fourier analysis is not done at all. This would have involved exporting data which, as mentioned before is giving errors
> The numerical errors by COMSOL are therefore not filtered off.
> The plot differs from the analytic solution for larger frequencies over 3000 Hz.
> It is to be noted from the paper by Liu and Thorne that the TE noise for the finite and infinite case are not very different. In
fact the correction factor goes as O(1). Thus, differences between finite and inifinte cases are unlikely to be prominent
in the log scale plots
The codes are put as a zip file. Corrections made to the codes will be uploaded as a reply.

Here is another plot with a mesh size slightly finer than the default Extra fine mesh in COMSOL.
One may notice that the value for the final frequency i.e. 10000Hz is different from the previous plot.
It maybe that the error for the higher frequencies is a result of the FEA. However, it may also be that
the appropriate boundary conditions required for an infinite model break down at high frequencies. 
Attachment 1: Jul_23_bettermesh2.eps


66

Tue Jul 23 20:53:37 2013 
Deep Chatterjee  Optics  General  Comparison between Liu and Thorne Results and COMSOL results for TE noise 
Quote: 
In this post I report of the results of TE noise simulated by COMSOL for the TE noise of Infinite test masses.
The aim was to follow the procedure by Liu and Thorne in their analytic calculations so that the same model could be used for the other
geometries.
The simulation is done in a different way than the TR simulations. It was observed that the output given by COMSOL by the use of commands
like mphinterp() or taking an export resulted in certain discrepancies between the results computed in COMSOL and that read by MATLAB.
Thus, the volume integration of the temperature gradient is performed in COMSOL itself and the results of the integration for each time
are sent to files. Matlab read these values and time averages them to get the result as in the paper (Sec. 2 of Liu and Thorne).
The errors expected are
> Fourier analysis is not done at all. This would have involved exporting data which, as mentioned before is giving errors
> The numerical errors by COMSOL are therefore not filtered off.
> The plot differs from the analytic solution for larger frequencies over 3000 Hz.
> It is to be noted from the paper by Liu and Thorne that the TE noise for the finite and infinite case are not very different. In
fact the correction factor goes as O(1). Thus, differences between finite and inifinte cases are unlikely to be prominent
in the log scale plots
The codes are put as a zip file. Corrections made to the codes will be uploaded as a reply.

Here is another plot with a mesh size slightly finer than the default Extra fine mesh in COMSOL.
One may notice that the value for the final frequency i.e. 10000Hz is different from the previous plot.
It maybe that the error for the higher frequencies is a result of the FEA. However, it may also be that
the appropriate boundary conditions required for an infinite model break down at high frequencies. 
Attachment 1: Jul_23_bettermesh2.eps


65

Tue Jul 23 20:53:37 2013 
Deep Chatterjee  Optics  General  Comparison between Liu and Thorne Results and COMSOL results for TE noise 
Quote: 
In this post I report of the results of TE noise simulated by COMSOL for the TE noise of Infinite test masses.
The aim was to follow the procedure by Liu and Thorne in their analytic calculations so that the same model could be used for the other
geometries.
The simulation is done in a different way than the TR simulations. It was observed that the output given by COMSOL by the use of commands
like mphinterp() or taking an export resulted in certain discrepancies between the results computed in COMSOL and that read by MATLAB.
Thus, the volume integration of the temperature gradient is performed in COMSOL itself and the results of the integration for each time
are sent to files. Matlab read these values and time averages them to get the result as in the paper (Sec. 2 of Liu and Thorne).
The errors expected are
> Fourier analysis is not done at all. This would have involved exporting data which, as mentioned before is giving errors
> The numerical errors by COMSOL are therefore not filtered off.
> The plot differs from the analytic solution for larger frequencies over 3000 Hz.
> It is to be noted from the paper by Liu and Thorne that the TE noise for the finite and infinite case are not very different. In
fact the correction factor goes as O(1). Thus, differences between finite and inifinte cases are unlikely to be prominent
in the log scale plots
The codes are put as a zip file. Corrections made to the codes will be uploaded as a reply.

Here is another plot with a mesh size slightly finer than the default Extra fine mesh in COMSOL.
One may notice that the value for the final frequency i.e. 10000Hz is different from the previous plot.
It maybe that the error for the higher frequencies is a result of the FEA. However, it may also be that
the appropriate boundary conditions required for an infinite model break down at high frequencies. 
Attachment 1: Jul_23_bettermesh2.eps


64

Tue Jul 23 20:53:37 2013 
Deep Chatterjee  Optics  General  Comparison between Liu and Thorne Results and COMSOL results for TE noise 
Quote: 
In this post I report of the results of TE noise simulated by COMSOL for the TE noise of Infinite test masses.
The aim was to follow the procedure by Liu and Thorne in their analytic calculations so that the same model could be used for the other
geometries.
The simulation is done in a different way than the TR simulations. It was observed that the output given by COMSOL by the use of commands
like mphinterp() or taking an export resulted in certain discrepancies between the results computed in COMSOL and that read by MATLAB.
Thus, the volume integration of the temperature gradient is performed in COMSOL itself and the results of the integration for each time
are sent to files. Matlab read these values and time averages them to get the result as in the paper (Sec. 2 of Liu and Thorne).
The errors expected are
> Fourier analysis is not done at all. This would have involved exporting data which, as mentioned before is giving errors
> The numerical errors by COMSOL are therefore not filtered off.
> The plot differs from the analytic solution for larger frequencies over 3000 Hz.
> It is to be noted from the paper by Liu and Thorne that the TE noise for the finite and infinite case are not very different. In
fact the correction factor goes as O(1). Thus, differences between finite and inifinte cases are unlikely to be prominent
in the log scale plots
The codes are put as a zip file. Corrections made to the codes will be uploaded as a reply.

Here is another plot with a mesh size slightly finer than the default Extra fine mesh in COMSOL.
One may notice that the value for the final frequency i.e. 10000Hz is different from the previous plot.
It maybe that the error for the higher frequencies is a result of the FEA. However, it may also be that
the appropriate boundary conditions required for an infinite model break down at high frequencies. 
Attachment 1: Jul_23_bettermesh2.eps


63

Tue Jul 23 20:53:37 2013 
Deep Chatterjee  Optics  General  Comparison between Liu and Thorne Results and COMSOL results for TE noise 
Quote: 
In this post I report of the results of TE noise simulated by COMSOL for the TE noise of Infinite test masses.
The aim was to follow the procedure by Liu and Thorne in their analytic calculations so that the same model could be used for the other
geometries.
The simulation is done in a different way than the TR simulations. It was observed that the output given by COMSOL by the use of commands
like mphinterp() or taking an export resulted in certain discrepancies between the results computed in COMSOL and that read by MATLAB.
Thus, the volume integration of the temperature gradient is performed in COMSOL itself and the results of the integration for each time
are sent to files. Matlab read these values and time averages them to get the result as in the paper (Sec. 2 of Liu and Thorne).
The errors expected are
> Fourier analysis is not done at all. This would have involved exporting data which, as mentioned before is giving errors
> The numerical errors by COMSOL are therefore not filtered off.
> The plot differs from the analytic solution for larger frequencies over 3000 Hz.
> It is to be noted from the paper by Liu and Thorne that the TE noise for the finite and infinite case are not very different. In
fact the correction factor goes as O(1). Thus, differences between finite and inifinte cases are unlikely to be prominent
in the log scale plots
The codes are put as a zip file. Corrections made to the codes will be uploaded as a reply.

Here is another plot with a mesh size slightly finer than the default Extra fine mesh in COMSOL.
One may notice that the value for the final frequency i.e. 10000Hz is different from the previous plot.
It maybe that the error for the higher frequencies is a result of the FEA. However, it may also be that
the appropriate boundary conditions required for an infinite model break down at high frequencies. 
Attachment 1: Jul_23_bettermesh2.eps


62

Tue Jul 23 20:53:37 2013 
Deep Chatterjee  Optics  General  Comparison between Liu and Thorne Results and COMSOL results for TE noise 
Quote: 
In this post I report of the results of TE noise simulated by COMSOL for the TE noise of Infinite test masses.
The aim was to follow the procedure by Liu and Thorne in their analytic calculations so that the same model could be used for the other
geometries.
The simulation is done in a different way than the TR simulations. It was observed that the output given by COMSOL by the use of commands
like mphinterp() or taking an export resulted in certain discrepancies between the results computed in COMSOL and that read by MATLAB.
Thus, the volume integration of the temperature gradient is performed in COMSOL itself and the results of the integration for each time
are sent to files. Matlab read these values and time averages them to get the result as in the paper (Sec. 2 of Liu and Thorne).
The errors expected are
> Fourier analysis is not done at all. This would have involved exporting data which, as mentioned before is giving errors
> The numerical errors by COMSOL are therefore not filtered off.
> The plot differs from the analytic solution for larger frequencies over 3000 Hz.
> It is to be noted from the paper by Liu and Thorne that the TE noise for the finite and infinite case are not very different. In
fact the correction factor goes as O(1). Thus, differences between finite and inifinte cases are unlikely to be prominent
in the log scale plots
The codes are put as a zip file. Corrections made to the codes will be uploaded as a reply.

Here is another plot with a mesh size slightly finer than the default Extra fine mesh in COMSOL.
One may notice that the value for the final frequency i.e. 10000Hz is different from the previous plot.
It maybe that the error for the higher frequencies is a result of the FEA. However, it may also be that
the appropriate boundary conditions required for an infinite model break down at high frequencies. 
Attachment 1: Jul_23_bettermesh2.eps


68

Tue Jul 23 20:53:45 2013 
Deep Chatterjee  Optics  General  Comparison between Liu and Thorne Results and COMSOL results for TE noise 
Quote: 
In this post I report of the results of TE noise simulated by COMSOL for the TE noise of Infinite test masses.
The aim was to follow the procedure by Liu and Thorne in their analytic calculations so that the same model could be used for the other
geometries.
The simulation is done in a different way than the TR simulations. It was observed that the output given by COMSOL by the use of commands
like mphinterp() or taking an export resulted in certain discrepancies between the results computed in COMSOL and that read by MATLAB.
Thus, the volume integration of the temperature gradient is performed in COMSOL itself and the results of the integration for each time
are sent to files. Matlab read these values and time averages them to get the result as in the paper (Sec. 2 of Liu and Thorne).
The errors expected are
> Fourier analysis is not done at all. This would have involved exporting data which, as mentioned before is giving errors
> The numerical errors by COMSOL are therefore not filtered off.
> The plot differs from the analytic solution for larger frequencies over 3000 Hz.
> It is to be noted from the paper by Liu and Thorne that the TE noise for the finite and infinite case are not very different. In
fact the correction factor goes as O(1). Thus, differences between finite and inifinte cases are unlikely to be prominent
in the log scale plots
The codes are put as a zip file. Corrections made to the codes will be uploaded as a reply.

Here is another plot with a mesh size slightly finer than the default Extra fine mesh in COMSOL.
One may notice that the value for the final frequency i.e. 10000Hz is different from the previous plot.
It maybe that the error for the higher frequencies is a result of the FEA. However, it may also be that
the appropriate boundary conditions required for an infinite model break down at high frequencies. 
Attachment 1: Jul_23_bettermesh2.eps


69

Wed Jul 24 21:08:24 2013 
Deep Chatterjee  Optics  General  TR results for different dimensions 
In this post I simulate the procedure of calculating the TR noise for finite cavities as proposed by Heinert and check for a
match.
The technique of performing all necessary calculations in COMSOL and exporting the results was applied to the TR codes.
It was seen that the codes gives similar output as the technique of extraction of Fourier coefficients in place of time averaging
as has been done in the codes of Koji Arai. One can see the output as the present code runs to be similar to the previous ones
found in the SVN.
However, the results in the present case were off by a constant factor close to 100. This maybe due to some 'm'  'cm' or similar difference between
analytic calculations and COMSOL values of parameters. Although, it has not been found yet, the correction is hopeful to be
found soon.
The codes give results similar to the analytic result for other values of the mirror radius and beam radii (apart from the constant
factor I have mentioned above). One may have a look at the trend of the graphs between analytic and simulated values in the plots
attached. These plots are for the case when the mirror radius = 25m while the beam radius = 9 cm i.e. the original radii were 0.25m
and 9cm respectively i.e. the ratio has been changed by a order of 2.
As mentioned before the reason for the constant factor difference will be looked into. 
Attachment 1: Jul24.pdf


Attachment 2: Jul24_acalc.pdf


70

Thu Jul 25 11:36:17 2013 
rana  Optics  General  TR results for different dimensions 
PDF please instead of EPS or BMP or JFIF or TARGA or GIF or ascii art. 
71

Thu Jul 25 13:21:46 2013 
Deep Chatterjee  Optics  General  TR results for different dimensions 
Quote: 
In this post I simulate the procedure of calculating the TR noise for finite cavities as proposed by Heinert and check for a
match.
The technique of performing all necessary calculations in COMSOL and exporting the results was applied to the TR codes.
It was seen that the codes gives similar output as the technique of extraction of Fourier coefficients in place of time averaging
as has been done in the codes of Koji Arai. One can see the output as the present code runs to be similar to the previous ones
found in the SVN.
However, the results in the present case were off by a constant factor close to 100. This maybe due to some 'm'  'cm' or similar difference between
analytic calculations and COMSOL values of parameters. Although, it has not been found yet, the correction is hopeful to be
found soon.
The codes give results similar to the analytic result for other values of the mirror radius and beam radii (apart from the constant
factor I have mentioned above). One may have a look at the trend of the graphs between analytic and simulated values in the plots
attached. These plots are for the case when the mirror radius = 25m while the beam radius = 9 cm i.e. the original radii were 0.25m
and 9cm respectively i.e. the ratio has been changed by a order of 2.
As mentioned before the reason for the constant factor difference will be looked into.

The discrepancy related to the difference between the analytic and COMSOL results has been partially addressed. Attached is another
plot showing the comparison. The ratio this time between the COMSOL results and the analytic results is between 0.7  0.8. This difference
will be looked into. It is, however, observed that the difference is not a constant factor  it has to do with the model file. 
Attachment 1: Jul_25.pdf


72

Thu Jul 25 15:54:58 2013 
Deep Chatterjee  Optics  General  TR results for different dimensions 
Quote: 
Quote: 
In this post I simulate the procedure of calculating the TR noise for finite cavities as proposed by Heinert and check for a
match.
The technique of performing all necessary calculations in COMSOL and exporting the results was applied to the TR codes.
It was seen that the codes gives similar output as the technique of extraction of Fourier coefficients in place of time averaging
as has been done in the codes of Koji Arai. One can see the output as the present code runs to be similar to the previous ones
found in the SVN.
However, the results in the present case were off by a constant factor close to 100. This maybe due to some 'm'  'cm' or similar difference between
analytic calculations and COMSOL values of parameters. Although, it has not been found yet, the correction is hopeful to be
found soon.
The codes give results similar to the analytic result for other values of the mirror radius and beam radii (apart from the constant
factor I have mentioned above). One may have a look at the trend of the graphs between analytic and simulated values in the plots
attached. These plots are for the case when the mirror radius = 25m while the beam radius = 9 cm i.e. the original radii were 0.25m
and 9cm respectively i.e. the ratio has been changed by a order of 2.
As mentioned before the reason for the constant factor difference will be looked into.

The discrepancy related to the difference between the analytic and COMSOL results has been partially addressed. Attached is another
plot showing the comparison. The ratio this time between the COMSOL results and the analytic results is between 0.7  0.8. This difference
will be looked into. It is, however, observed that the difference is not a constant factor  it has to do with the model file.

The issue related to the difference between the analytic and simulated values has been resolved. The codes seems to give reasonable match
between the analytic and simulated case. There is, however, a difference between the formulas being used from the previous cases. Note that
the 1/2 in front of Eq.(15) of Heinert is a because the time average has already been considered. However, in the present codes, the volume
integral of grad_T is evaluated in COMSOL and exported as a function of time. It is then averaged in MATLAB. Thus the factor of 1/2 is to be
omitted in this case(see Liu and Thorne Eq.(5). The presence of this extra factor of 1/2 was giving error in the last upoaded plots. From the
relative difference plot, one can see the maximum difference between COMSOL and analytic results go upto 7% but for most of the graph
it is close to 1% which is a fair result. 
Attachment 1: Jul_25_1.pdf


Attachment 2: Jul_25_2.pdf


Attachment 3: relative_plot.pdf


73

Mon Jul 29 22:42:57 2013 
Deep Chatterjee  Optics  General  Avoiding transient solutions in the Computation of TE/TR noise 
An error was being encountered in the computation of the TR noise lately. It was observed that while running the simulations in the case of the materials which have a lower value of the thermal diffusivity (silicon / sapphire at room temperature), the simulated result were slightly off from the analytic result. On the other hand, if the simulation was run with a material of higher diffusivity(same materials at lower temperature), the match would be better. The reason being the transient solution not dying off significantly during the period of the simulation. Since a time average was being taken of the quantity integral{grad_T ^2}, the transient contributed to the integral. To get the correct value, the fourier coefficient of the time signal of integral{grad_T ^2}, was extracted at twice the frequency of the pressure oscillation. The reason being that the signal was squared. Extracting the response at this frequency after the integration is logical since the integration is over space while the response we extract is over time.
The same procedure was also applied to the TE noise calculation. However, this time we obtained similar result as the case where this procedure was not applied but a simple time averaging was performed. The tail of the plot, at high frequency, is still seen to deviate from the analytic result of Liu and Thorne as was the case previously. A plot is attached showing the spectrum for Fused silica at 290K. The conclusion being that the transients do not affect the TE noise calculation  the plot stayed the same even after filtering them out. This is probably because unlike the TR case which has a heat source present along the cylinder axis, the TE noise calculation involves applying pressure only on the face of the cylinder, and the transient do not contribute much to the volume integral.

Attachment 1: Jul29_2.pdf


74

Wed Jul 31 15:39:11 2013 
Deep Chatterjee  Optics  General  First try with paramter optimization for TE and TR noise profiles 
After the simulations have been found to match to a fair extent with the analytic results by Heinert and Liu and Thorne, the attempt is check out the parameters for which the TE and TR noise are
close to each other. This was done with the analytic results. The frequency range 10  1000 Hz was looked into. In between this range, the quantity that was minimized was the absolite value
of the logarithm of the ratio between the TR and TE noise. The fminsearch function was used to minimize the mentioned quantity. The parameters that were changed were  conductivity, thermo
optic coefficient and coefficient of linear expansion. The reason for choosing these three were 
> TR noise is independent of coefficient of linear expansion
> TE noise is independent of thermo optic coefficient
> The power law dependence on conductivity is different for the TE and TR cases as can be seen from the analytic expressions
once the code returned the optimized parameters, these values were plugged in and the results were plotted.
**Note that the minimization was done for frequencies between 10 to 1000 Hz 
Attachment 1: Jul31_param_opt.pdf


75

Thu Aug 8 17:17:19 2013 
Deep Chatterjee  Optics  General  Something like cancellation 
For the material parameters of Sapphire at 300K, the TE and TR Noise profiles, though not very close, lie close to within an order of magnitude. Sapphire has a positive coefficient of linear expansion. We just inverted the sign of this quantity
and the ran the codes that puts heat and pressure simultaneously to the test mass. The total noise looks to be lower than the TR noise which is greater.
Mirror radius 0.25[m]
Mirror height 0.46[m]
Beam Radius 0.09[m]
If we have physical parameters which make the two Noise sources come closer to each other and then flip the sign of alpha, we may be able to see some noise reduction to a greater extent. 
Attachment 1: suspected_cance_AUg8l.pdf


77

Wed Nov 13 23:55:44 2013 
Chris Couste  Optics  Analysis  Optical Mount Vibrational Analysis 
Project: Vibrational Analysis of Optical Mounts
Goal: Use COMSOL to run finite element analysis on simplified models of different types of optical mounts available to us, in order to find which shape/style/material reacts the least to external sound pollution. Once the few best candidates have been identified, develop test to experimentally determine optimal mount configuration and material.
Part 1: FEM
Process: Simultaneously build models in SolidWorks and design analysis in COMSOL
Solidworks: base/shaft models based on measurements of actual optical mounts, optic holder models matching mass and moment of inertia from downloaded models from their manufacturers. ~Progress: base/shaft done, optic holders almost done.
COMSOL: Design analysis that first does a general eigenfrequency analysis to find general vicinity of modes, then does a full modal frequency analysis. ~Progress: finished, ready for models to be imported. 
78

Mon Nov 18 15:56:59 2013 
Chris Couste  Optics  Analysis  Representative Models 
The simpler models of the optical mounts are finished, they will be run through the comsol analysis software soon.
see pictures below:

Attachment 1: Custom_Closed.JPG


Attachment 2: custom_open.JPG


Attachment 3: Stock_Closed.JPG


Attachment 4: Stock_Open.JPG


79

Fri Nov 22 21:04:53 2013 
Chris Couste  Optics  Analysis  analysis 
The analysis is making nice eigenmode and stress mode models, but the displacement experiment needs work. Should be fixed by monday. 
80

Fri Jan 24 17:26:38 2014 
Chris Cousté  Optics  Analysis  Mount Analysis Functional! 
The comsol eigenmode analysis is complete, and the only thing left to do on this part of the project is to run the analysis on a range of different configurations of optical mounts as well as a range of materials. This compilation will be posted on this elog in the next few weeks, due to the fairly long runtime of the analysis software. 
81

Sun Feb 16 21:10:56 2014 
Chris Cousté  Optics  Analysis  Optical Mount data compilation 1: Aluminum 
The time is finally here! this is the highest displacement of each mount in its lowest few eigenfrequencies, using 6061 Aluminum as a material. pictures will be added in a future log, because I'm going to make them into one file. Other materials will also be tested to see if there is variance in these findings, but only relevant data will be posted.
tl;dr findings: thickopen is the best combination because it has less displacement in eigenmodes and its eigenmodes are all very high/spread apart in frequency.
long:
(key) Eigenfrequency (Hz), highest displacement (mm), direction
Thin base, open mount:
315, 8.07, leaning back (laser will angle up)
316, 7.94, leaning to the side (laser won't change direction if the optic is flat, i.e. point of contact moves along surface of optic)
924, 10.50, twist to the side (laser will angle to the side)
1949, 8.89, twist about optic, slight lean forward (point of contact will move slightly, laser may angle down)
2144, 10.47, another leaning forward mode
thin base, open mount:
322, 13.29, leaning forward and translating to the side
328, 13.36, same as above
994, 17.66, twisting to the side (laser will angle to the side)
1943, 13.86, intense leaning forward
2100, 14.27, twist about optic
thick base, closed mount:
1329, 13.23, leaning back
1347, 13.32, leaning back and to the side
3577, 15.46, twisting to the side
thick base, open mount:
1326, 9.69, leaning to the side
1335, 9.77, leaning back
3504, 13.88, twisting to the side

All of these frequencies were checked for all four models, and the max displacements in those cases ranged from 100 picometers to 10 femtometers, so it's pretty reasonable to base decisions off of the displacements at the eigenmodes.
Based on this, it seems that for both thick and thin, open is the best, considering it displaces significantly less than closed (which is reasonable because there is less mass at the top to be thrown around). The choice between thick and thin depends on what frequencies we believe the mounts will be subjected to. If we are probably not going to have much sound at frequencies above 1000Hz, then thick is the better option (which is nice because it is the stock stalk instead of the custom made one). However, if there will be plenty of high noise, the thick is still probably the better option because it has fewer eigenmodes in the same range of frequencies.
here's a sample of two pictures of the relevant eigenmodes of the winner.

Attachment 1: aluminum2.png


Attachment 2: aluminum3.png


83

Sun Jun 22 23:44:56 2014 
Sam Moore  Optics   Going through Heinert's 'TR noise of cylindrical test masses' paper 
At this point, my are goals are to 1) convert the timedependent heat equation into stationary, complex form, 2) use the Levin approach to calculate the TR noise given this stationary PDE, and 3) verify the results in COMSOL
I have looked at the Heinert paper and converted the timedependent partial differential Heat equation to a stationary, complex one. I did this by assuming a temperature distribution of the form T(\vec{r},t) = T_0 ( \vec{r} ) e^{i \omega t }, where \omega is the frequency of oscillation of the input heat field: q( \vec{r} , t) = q( \vec{r} ) e^{i \omega t}. Hence, I am assuming the steadystate solution of the temperature field. From this ansatz, the PDE becomes
C_p T(\vec{r}) + \frac{ i \kappa}{ \omega } \nabla^2 T ( \vec{r} ) = q ( \vec{ r} ).
This complex conversion already seems to have been done in the paper, since the stationary solution of the form
T(r, z) = \sum_{n = 0}^{\infty} \sum_{m = 0}^{ \infty} T_{n m} J_0 (k_n r) \cos ( \ell_m z) is assumed.
k_n and \ell_m are obtained from the adiabatic boundary conditions. Moreover, if I plug this form for T(r, z) into the stationary PDE, I get Heinert's eq. 12. ( although I'm not quite sure how to take the second derivative of a zerothorder bessel function).
I've started to try out the cylindrical geometry in COMSOL, using the structural mechanics and heat transfer modules. At this point, I'm not quite sure how to specify the PDE that I'm interested in, nor am I sure about how to specify the boundary conditions.

84

Mon Jun 23 11:40:17 2014 
Sam Moore  Optics   Going through Heinert's 'TR noise of cylindrical test masses' paper 
Quote: 
At this point, my are goals are to 1) convert the timedependent heat equation into stationary, complex form, 2) use the Levin approach to calculate the TR noise given this stationary PDE, and 3) verify the results in COMSOL
I have looked at the Heinert paper and converted the timedependent partial differential Heat equation to a stationary, complex one. I did this by assuming a temperature distribution of the form T(\vec{r},t) = T_0 ( \vec{r} ) e^{i \omega t }, where \omega is the frequency of oscillation of the input heat field: q( \vec{r} , t) = q( \vec{r} ) e^{i \omega t}. Hence, I am assuming the steadystate solution of the temperature field. From this ansatz, the PDE becomes
C_p T(\vec{r}) + \frac{ i \kappa}{ \omega } \nabla^2 T ( \vec{r} ) = q ( \vec{ r} ).
This complex conversion already seems to have been done in the paper, since the stationary solution of the form
T(r, z) = \sum_{n = 0}^{\infty} \sum_{m = 0}^{ \infty} T_{n m} J_0 (k_n r) \cos ( \ell_m z) is assumed.
k_n and \ell_m are obtained from the adiabatic boundary conditions. Moreover, if I plug this form for T(r, z) into the stationary PDE, I get Heinert's eq. 12. ( although I'm not quite sure how to take the second derivative of a zerothorder bessel function).
I've started to try out the cylindrical geometry in COMSOL, using the structural mechanics and heat transfer modules. At this point, I'm not quite sure how to specify the PDE that I'm interested in, nor am I sure about how to specify the boundary conditions.

I carried out the calculations for Heinert's simple model in greater detail and was able to obtain equation 12 (with the exception of a plus sign instead of a minus sign in the denominator; I guess the Fourier transform method gives a minus sign. In the end, that doesn't matter, since the modulus square yields the same result in the denominator.) I obtained the same spectral density S_z (\omega), with the exception of a missing factor of R^2. I don't know where this R^2 went, or why it is that the spectral density is associated with the extrinsic variable z (instead of r, say). It's supposed to be a "homogeneous [noise] readout along the z direction." 
85

Mon Jun 23 16:10:17 2014 
Matt A.  Optics   Going through Heinert's 'TR noise of cylindrical test masses' paper 
Quote: 
At this point, my are goals are to 1) convert the timedependent heat equation into stationary, complex form, 2) use the Levin approach to calculate the TR noise given this stationary PDE, and 3) verify the results in COMSOL
I have looked at the Heinert paper and converted the timedependent partial differential Heat equation to a stationary, complex one. I did this by assuming a temperature distribution of the form T(\vec{r},t) = T_0 ( \vec{r} ) e^{i \omega t }, where \omega is the frequency of oscillation of the input heat field: q( \vec{r} , t) = q( \vec{r} ) e^{i \omega t}. Hence, I am assuming the steadystate solution of the temperature field. From this ansatz, the PDE becomes
C_p T(\vec{r}) + \frac{ i \kappa}{ \omega } \nabla^2 T ( \vec{r} ) = q ( \vec{ r} ).
This complex conversion already seems to have been done in the paper, since the stationary solution of the form
T(r, z) = \sum_{n = 0}^{\infty} \sum_{m = 0}^{ \infty} T_{n m} J_0 (k_n r) \cos ( \ell_m z) is assumed.
k_n and \ell_m are obtained from the adiabatic boundary conditions. Moreover, if I plug this form for T(r, z) into the stationary PDE, I get Heinert's eq. 12. ( although I'm not quite sure how to take the second derivative of a zerothorder bessel function).
I've started to try out the cylindrical geometry in COMSOL, using the structural mechanics and heat transfer modules. At this point, I'm not quite sure how to specify the PDE that I'm interested in, nor am I sure about how to specify the boundary conditions.

Hi Sam,
I know it seems a bit distracting, but it's sometimes nice to write up your elog entries in a more readable form. I've reedited your comments to make them more easily read.
I used http://www.sciweavers.org/freeonlinelatexequationeditor to convert your latex to png files for easy reading.
At this point, my are goals are to:
1) convert the timedependent heat equation into stationary, complex form,
2) use the Levin approach to calculate the TR noise given this stationary PDE, and
3) verify the results in COMSOL
I have looked at the Heinert paper and converted the timedependent partial differential Heat equation to a stationary, complex one.
I did this by assuming a temperature distribution of the form:
where ω is the frequency of oscillation of the input heat field:
Hence, I am assuming the steadystate solution of the temperature field.
From this ansatz, the PDE becomes:
This complex conversion already seems to have been done in the paper, since the stationary solution of the form:
is assumed. k_{n} and ℓ_{m} are obtained from the adiabatic boundary conditions.
Moreover, if I plug this form for T(r, z) into the stationary PDE, I get Heinert's eq. 12.
( although I'm not quite sure how to take the second derivative of a zerothorder bessel function).
I've started to try out the cylindrical geometry in COMSOL, using the structural mechanics and heat transfer modules.
At this point, I'm not quite sure how to specify the PDE that I'm interested in, nor am I sure about how to specify the boundary conditions.

86

Tue Jun 24 14:35:42 2014 
Sam Moore  Optics  General  Trying to Verify the Heinert Model 

Attachment 1: 06_23_14.pdf


87

Tue Jun 24 17:05:24 2014 
Sam Moore  Optics  General  Trying to Verify the Heinert Model 
It does appear that the simplified model is only relevant for the simulations. To quote Heinert: "An efficient computation is only possible for the simple model, as the advanced model would require an element of size more than 10^{6 }." I have run Koji's code that replicates Heinert's figure 3. I have attached the resulting temperature distribution and noise amplitude curve. In the noise amplitude curve, the red line is the analytical result, while the dots are from COMSOL.
The next step is to convert this code to an efficient complex timeindependent solution. As stated before, my main concern here is whether COMSOL actually solves the right equation in the stationary case. 
Attachment 1: temperature.png


Attachment 2: noiseAmplitude_agreement.png


88

Sat Jun 28 21:59:11 2014 
Sam Moore  Optics  General  Difficulty with the COMSOL stationary module; Test Cases 
Here, I describe some test cases to see if COMSOL's solutions are agreeing with some simple analytical solutions. Right now, I have two plots showing COMSOL's solution and my analytical solution on separate plots. I will be plotting there difference to see if they really match up.

Attachment 1: 6_27_14.pdf


89

Sun Jun 29 15:37:18 2014 
Sam Moore  Optics  General  Difficulty with the COMSOL stationary module; Test Cases 
Quote: 
Here, I describe some test cases to see if COMSOL's solutions are agreeing with some simple analytical solutions. Right now, I have two plots showing COMSOL's solution and my analytical solution on separate plots. I will be plotting there difference to see if they really match up.

The following document shows the relative difference between these two plots. 
Attachment 1: 6_29_14.pdf


90

Sun Jun 29 20:25:44 2014 
Koji  Optics  General  Difficulty with the COMSOL stationary module; Test Cases 
What about this example? The result is easier to understand intuitively.
Consider a bar with the length of L.
Let's say there is no body heat applied, but the temperature of the bar at x=L is kept at T=0
and at x=0 is kept at T=T0 Exp[I w t].
The equation for the bar is
...(1)
Consider the solution with the form of T(x, t) = T(x) T0 Exp(I w t), where T(x) is the position dependent transfer function.
T(x) is a complex function.
Eq.1 is modified with T(x) as
With the boundary condition of
This can be analytically solved in the following form
where alpha is defined by
So kappa/Cp is the characteristic (angular) frequency of the system.
Here is the example plot for L=1 and alpha = 1 (red), 10 (yellow green), 100 (turquoise), 1000 (blue)
If the oscillation is slow enough, the temperature decay length is longer than the bar length and thus the temperature is linear to the position.
If the oscillation is fast, the decay length is significantly shorter than the bar length and the temperature dependence on the position is exponential.
Now what we need is to solve this in COMSOL 
91

Sat Jul 5 13:04:32 2014 
Sam Moore  Optics   Heinert Model TR Noise Verification 
Agreement with Heinert's paper for cylindrical TR noise has now been achieved. Using the stationary state assumption to calculate the temperature profile, the computation time was reduced compared to the previous timedependent approach. Here are the plots showing the agreement. I have shown the plots for a 1D axisymmetric model, in addition to a full 3D model in COMSOL. Both give the same result.
What went wrong? In the 1D axisymmetric case, it turns out that COMSOL has the incorrect cylindrical coordinate Laplacian for the coefficient form PDE interface. I corrected for this by expanding the Laplacian with the product rule, giving an extra kappa/r ''convection coefficient'' in the PDE interface. Meanwhile, the 3D case worked from the beginning, since its Laplacian was in cartesian coordinates.
The next goal is to use a stationary state method to calculate TE noise for a test case.

Attachment 1: threeD_cylinderTRnoise.png


Attachment 2: oneDcylinderTRnoise_copy.png


92

Mon Jul 7 19:47:00 2014 
Koji  Optics   Heinert Model TR Noise Verification 
How close are these FEA calculations with the analytical values?
Can you plot residual too? (Put analytical values, 1D, abs(1D  analytical), 3D, and abs(3D  analytical) all together.) 
93

Thu Jul 10 16:51:14 2014 
Sam Moore  Optics  General  Duan and Heinert Comparison 
(See Plots in attached document)
My plan has been to replicate Duan's numerical thermoconductive (TE + TR) phase noise plot presented in his paper (section V). I am trying to match Duan's analytical expression with Heinert's analytical expression. This requires some rescaling of Heinert's TR displacement noise. (I also needed to divide Heinert's expression by 4 pi to match the Fourier Transform convention. ) Duan's analytical expression for the phase noise is obtained by evaluating the triple integral given in equation 13 of the Duan paper "General Treatment of Thermal Noise in Optical Fibers".
It turns out that an additional factor of 2 multiplies the phase noise because Duan's Fourier Transform only takes into account positive frequencies; there are also negative frequencies that occur in equal amplitude.
This integral was evaluated in Mathematica due to numerical noise in MATLAB's calculation. The calculation in Mathematica was very slow, so the upper limits on the integral were truncated. The following plots in the attached document show the resulting noise profile agreements for two different upper limits.
If the residual for the highest upper limit is considered acceptable for a match between the two plots, then I will use Heinert's plot as a reference when using the COMSOL steadystate method for Duan's numerical case (Heinert's plot runs much faster). 
Attachment 1: 7_10_14.pdf


94

Fri Jul 11 10:58:18 2014 
not Koji  Optics   Heinert Model TR Noise Verification 
Quote: 
How close are these FEA calculations with the analytical values?
Can you plot residual too? (Put analytical values, 1D, abs(1D  analytical), 3D, and abs(3D  analytical) all together.)

Here are the plots with their fractional residuals: abs(S_COMSOL  S_analytical)/S_analytical 
Attachment 1: heinert_analyticalTest_residual_threeD.eps


Attachment 2: heinert_analyticTest_residual_oneD.eps


95

Mon Jul 14 19:09:14 2014 
Sam Moore  Optics  General  Using Heinert's Solution for Duan's Parameters 
I have plotted Heinert's analytical solution for TR noise using Duan's parameters. Since TO and TE noise can be found by simply rescaling TR noise, these have been included in the plot as well. The solid curve represents the analytical solution, while the tick marks represent COMSOL's solution. I have used COMSOL for both a 1D axisymmetric and a 3D model. Since Duan's cylinder has a radius of 125 microns, but a length of 1 m, the meshing was difficult for the 3D model. I ended up shortening the length of the cylinder, converting to the actual length when finally calculating the thermal noise. 
Attachment 1: oneD_duanParams_residualepsconvertedto.pdf


Attachment 2: threeD_duanParams_residualepsconvertedto.pdf


96

Mon Jul 14 19:14:31 2014 
Sam Moore  Optics  General  Duan and Heinert Comparison 
Quote: 
(See Plots in attached document)
My plan has been to replicate Duan's numerical thermoconductive (TE + TR) phase noise plot presented in his paper (section V). I am trying to match Duan's analytical expression with Heinert's analytical expression. This requires some rescaling of Heinert's TR displacement noise. (I also needed to divide Heinert's expression by 4 pi to match the Fourier Transform convention. ) Duan's analytical expression for the phase noise is obtained by evaluating the triple integral given in equation 13 of the Duan paper "General Treatment of Thermal Noise in Optical Fibers".
It turns out that an additional factor of 2 multiplies the phase noise because Duan's Fourier Transform only takes into account positive frequencies; there are also negative frequencies that occur in equal amplitude.
This integral was evaluated in Mathematica due to numerical noise in MATLAB's calculation. The calculation in Mathematica was very slow, so the upper limits on the integral were truncated. The following plots in the attached document show the resulting noise profile agreements for two different upper limits.
If the residual for the highest upper limit is considered acceptable for a match between the two plots, then I will use Heinert's plot as a reference when using the COMSOL steadystate method for Duan's numerical case (Heinert's plot runs much faster).

I have now plotted the DuanHeinert comparison for the case of an infinite upper bound. It turns out that the curves differ by a maximum of 10 percent for low frequencies. Such a discrepancy has been attributed to lack of experimental investigation into this regime (according to Duan). For our purposes, such a discrepancy is acceptable. We will therefore use the Heinert curve for subsequent calculations due to its faster computation time.

Attachment 1: duan_heinert_comparisonInfiniteepsconvertedto.pdf


97

Thu Jul 31 20:55:38 2014 
Sam Moore  Optics  General  Finding the Right Meshing for the TIR cavity 
In this document, I try to identify I good mesh by comparing the numerical solution from that mesh with my analytical model. Since there are problems with carrying out the analytical calculation, it is still not entirely clear which mesh should be used.

Attachment 1: 7_30_14.pdf


98

Sat Aug 2 00:22:34 2014 
Sam Moore  Optics  General  Finding the Right Meshing for the TIR cavity 
Quote: 
In this document, I try to identify I good mesh by comparing the numerical solution from that mesh with my analytical model. Since there are problems with carrying out the analytical calculation, it is still not entirely clear which mesh should be used.

I have refined the analytical calculation procedure, as outlined in this new document. The procedure indicates that the discrepancy between the analytical and numerical solutions are more likely attributed to meshing inaccuracies. 
Attachment 1: 8_1_14.pdf


124

Thu Jan 18 21:13:59 2018 
aaron  Optics  PonderSqueeze  modifications to Gautam's 40m finesse model 
I made a copy of Gautam's 40m model to add the unstable filter cavity for the ponderomotive squeezing project. I wanted to make a more explanatory record of the changes I've made because I think some of them might be necessary for other scripts using gautam's original model, but I have not implemented them in that file (also just for my own paper trail).
Changes:
 swapped the role of nXBS and nYBS; before I think it was sending the reflected beam at the main BS to the X arm, and the transmitted beam to the Y arm
 I kept nPOY, but I am a bit confused by itthis is the beam returning from the xarm that is transmitted in to the BS, but it isn't followed out of the BS; rather it seems this pick off beam is detected inside the BS substrate, which is odd. Anyway we aren't using it for now.
 Changed some labels to distinguish between the beamsplitter already in the IFO (IBS) and the BS used in the quantum phase compensator (QBS)
 Added a quantum phase compensator cavity, consisting of QP1 and QP2, as well as a mirror (QP0) to direct light transmitted from the QPC through QBS back to QBS

143

Fri Jul 21 16:42:57 2023 
Rajashik  Optics  Analysis  
Initially, ideal conditions were considered, without any deformations or thermal heating effects in the mirrors. This offered a baseline model from which subsequent tests could draw comparisons.
Following this, more complex scenarios were introduced into the model: mirrorshape defects caused by suspension and thermal lensing due to heat absorption. These heat absorption rates were set at 0.5 ppm in the ETM coating and 0.3 ppm in the ITM coating.
The lensing calculations for these scenarios were performed using the power observed in the nondeformed mirror case. This lensing needs to be made dynamic in future codes to get the modeling correct. Nonetheless, introducing these conditions resulted in the emergence of higherorder modes (HOMs). Unfortunately, there seems to be a loss of power in the cavity that I haven't been able to resolve yet. Notably, this power loss persisted across the range of mirror position scans showing that this is not due to just a change in resonance length.
Also, the Hello Vinet Function in Finesse 3 seems to yield a higher degree of lensing than typically reported in the literature. I need to check with someone about what assumption I am taking incorrectly here. It's plausible that the heightened lensing effect is contributing to the observed power loss in the cavity, possibly due to high curvatures.
Looking forward, my focus will be on determining the source of this lensing discrepancy first. This analysis will involve a more detailed examination of the variables contributing to thermal lensing and potential causes for the higherthanexpected lensing observed in the simulation. Ultimately, the aim is to create a dynamic deformation model that can more accurately represent these phenomena within the aLIGO setup before introducing ring heaters and controls. 
Attachment 1: Cavity_Scan_Before_Adding_Defects.png


Attachment 2: Transmitted_Power_Before_Adding_Defects.png


Attachment 3: Cavity_Scan_After_Adding_Defects.png


Attachment 4: Transmitted_Power_After_Adding_Defects.png


144

Sat Jul 29 00:19:02 2023 
Rajashik  Optics  General  
I have coded a rudimentary triangular model using parameters provided by Aaron. In this model, I have kept a simple PDH signal which is the REFL power demodulated with the EO frequency. The PDH signal is then fed to the laser to modulate the frequency.
There are some points of confusion I have in this model such as the high loop gain required to achieve locking. I think this may be due to the fact that the reflective power in SI units is incredibly small while the frequency shift is considerably larger in SI units.
Then I took this model through basic operations such as determining the optimal phase angle for the PDH signal and assessing if lock can be achieved. They went as required for a sanity check.
Then I introduced a mismatch for light incident on the first mirror. This resulted in the creation of higherorder modes in the cavity as expected.
I am still analyzing the effect of this mismatch for the length control offsets. Also, I am not sure why we need a sensing matrix in this situation, because we're currently dealing with a single signal source and one degree of freedom. Or maybe a more complex readout and control scheme may be more helpful. This remains to be seen,
Nonetheless, I've attached plots that illustrate the cavity modes that have emerged due to the mismatch. 
Attachment 1: cavitymodesmin.png


Attachment 2: transmin.png


Attachment 3: download_(1)min.png

