We're going to use this elog to store some of our lab work on blades that will be going on in the SUS lab.

As a first entry here is a useful document on the ALIGO blade design on the DCC:  LIGO T030107  by M.V.Plissi

 I've turned the main turbo back on and aligned the readout now.  The vacuum is already down to 1e-6torr.  It seems that the the offset pin on the bottom of the mass is causing the fiber to move a lot as the isolation mass rotates.  The movement does not appear much to the eye, but is taking up most of our readout range at the moment.

We can wait to see if this motion dies down.  If not we may be forced to replace this intermediate mass with one where the pin is in the center.

 This is some data measured in the ATF (by Zach) for 2 fixed mirrors on the bench, and gives some idea of what mechanical vibration noise you can expect.  You'll need to scale this for the number of mirrors in your own setup.  You can add this to your noise budget by including the following lines in your Matlab code:

load gyro_single-arm_noise_calibrated.txt;

f_mech_noise = gyro_single_arm_noise_calibrated(:,1); % frequency vector for plotting mechanical noise

SA_x_noise = gyro_single_arm_noise_calibrated(:,2); % gyro single-arm noise calibrated to meters

loglog plots please.  Use the matlab loglog function instead of plot.

 I have a laptop with a LabView license that you can use, that was bought for that setup.  There is also a National Instruments card that goes with it.

Measurement details:

• Connected New Focus 4004 broadband IR phase modulator to the output of the SN06 EOM driver.
• Splitted R output of AG4395A. One end went to the input of SN06 EOM driver.
• The other end goes to input R of Ag4395A.
• The RFMon of the driver is fed to input A of AG4395A.
• Used a tuning stick for the coilcraft inductor to tune the resonance peak to 36 MHz.
• Took measurements with the configuration files in this directory.

Data

• This is a repetition of SUS/1864.
• Used Data Ray Beam'R2-DD.
• Took 50 averages and recorded beam "diameters" at 10 different points after the laser head.
• Configuration file is BeamProfileConfiguration2um.ojf.
• Used fitBeamWidth function of ala mode to fit X and Y beams separately and then their geometric mean.
• We'll use the geometric mean as the seed profile for future calculations.

Data

• After installing a 400mm focal length plano-convex lens at 24" from the laser head at (20, 12), we found that higher-order modes are present in the beam.
• We installed an iris at 34" from laser head at (17, 5)
• Configuration file is BeamProfileConfiguration2um.ojf.
• Used fitBeamWidth function of ala mode to fit X and Y beams separately and then their geometric mean.
• We'll use the geometric mean as the seed profile for future calculations.
• Found a beam waist of 306 um at 58" from the laser head.
• Installing the EOM between 49" and 52" from the laser head where the beam waist is between 1 mm and 740 um.

Data

Goals and restrictions:

• Use the fewest lenses as possible.
• The beam widths in both onward and reflection direction should be such that there is a 5-inch space somewhere where we can put in the faraday isolator which has an aperture size of 3 mm and intensity limit of 500 W/cm².
• The lens should not be closer than 1.5 inches from each other or to the EOM mount or cavity edge.
• Choose a lens from a list of focal lengths available in west bridge labs.
• Find the best overlap with the target beam of 18 um at the cavity waist with the most sensitivity with respect to lens positions.

Analysis & Results

• CavityLens.m is run to try all possible lens combinations for 2-lens or 3-lens solutions using ../20201222_BeamProfileNeatEOM/SeedBeam.mat as the seed beam.
• Then save solutions with more than 70% overlap in CavityModeMatchingSolutions.mat.
• findBestSolutions.m increases the overlap threshold to 0.9, calculates reflected beam profile for the sideband reflection from the cavity (blue curves in the figures), and discards solution which does not have a 5-inch long area where we can place a faraday isolator with aperture of 3 mm.
• Black lines show the region where Thorlabs IO-3-1064-HP can be placed safely without clipping or exceeding the intensity limit with 1W power.
• All solutions in order of sensitivity are plotted here with details of lens choice and positions. In total 7 solutions were found which are stored in BestSolutions.mat.

Analysis & Data

Wed Jan 6 10:00:35 2021: This analysis was wrong. See SUS_Lab/1887.

Goals and restrictions:

• Use the fewest lenses as possible.
• The beam widths in both onward and reflection direction should be such that there is a 5-inch space somewhere where we can put in the faraday isolator which has an aperture size of 3 mm and intensity limit of 500 W/cm^2.
• The lens should not be closer than 1.5 inches from each other or to the EOM mount or cavity edge.
• Choose a lens from a list of focal lengths available in west bridge labs.
• Find the best overlap with the target beam of 18 um at the cavity waist with the most sensitivity with respect to lens positions.

Analysis & Results

• CavityLens.m is run to try all possible lens combinations for 2-lens or 3-lens solutions using …/20201222_BeamProfileNeatEOM/SeedBeam.mat as the seed beam.
• CavityLens4.m is run to try all possible lens combinations for 4-lens solutions using …/20201222_BeamProfileNeatEOM/SeedBeam.mat as the seed beam.
• Then save all possible solutions (in which placing a Faraday isolator is possible) with more than 90% overlap in AllPossibleSolutionsAbove90.mat.
• findBestSolutions.m increases the overlap threshold to 0.97 and plots the best solutions in order of positional sensitivity of the lens. These are stored in BestSolutions.mat.
• Black lines show the region where Thorlabs IO-3-1064-HP can be placed safely without clipping or exceeding the intensity limit with 1W power.

Analysis and Data

Wed Jan 6 10:00:35 2021: This analysis was wrong. See SUS_Lab/1887.

Errors in the previous analysis

• Previously, we wrongly assumed. that the reflected light from the cavity would be as if a reflection is happening from a flat mirror. It actually follows the same paths as incident light in the reflection path.
• Lens were restricted to non-overlapping regions but that meant that solutions where lens are close to each other can only happen near the boundaries of these regions. Removing this condition widens the search for a good solution.
• We collimated the beam to near 0.5 mm radius with a 229.1 mm focal length lens at 67" from laser head and put faraday isolator in front. So now the problem only remained to match the mode after this point to the cavity mode.

Goals and restrictions:

• Use the fewest lenses as possible after having used a fixed lens at 67" point before the faraday isolator.
• Choose a lens from a list of focal lengths available in west bridge labs.
• Find the best overlap with the target beam of 18 um at the cavity waist with the most sensitivity with respect to lens positions.
• The lens should not be closer than 1.5 inches from each other or to the EOM mount or cavity edge.
• The beam widths should not exceed 4mm in diameter anywhere to ensure small areas of lenses are used.

Analysis & Results

• CavityLens.m is run to try all possible lens combinations for 1-lens or 2-lens solutions using ../20201222_BeamProfileNeatEOM/SeedBeam.mat as the seed beam.
• CavityLens3.m and other versions of it are run to try all possible lens combinations for 3-lens solutions using ../20201222_BeamProfileNeatEOM/SeedBeam.mat as the seed beam.
• Then save all possible solutions with more than 90% overlap and where lenses are atleast 1.5" away from each other and the cavity in AllPossibleSolutionsAbove90.mat using findPossibleSolutions.m.
• findBestSolutions.m increases the overlap threshold to 0.995, allows maximum beam radius of 2mm anywhere and plots the best solutions in order of positional sensitivity of the lens. These are stored in BestSolutions.mat.

Analysis & Data

Error in previous calculations:

• We did not take into account the effect of cavity input mirror on the coupled light. It would act as a thick concave lens for the coupled light into the cavity.
• We did not take into account the divergence due to refraction at the crystal surface.

Goals and restrictions:

• Use the fewest lenses as possible after having used a fixed lens at 67" point before the faraday isolator.
• Choose a lens from a list of focal lengths available in west bridge labs.
• Find the best overlap with the target beam of 18 um at the cavity waist with the most sensitivity with respect to lens positions.
• The lens should not be closer than 1.5 inches from each other.
• The beam widths should not exceed 4mm in diameter anywhere to ensure small areas of lenses are used.
• Take into account the concave lens due to the input mirror.
• Take into account the refraction due to crystal surface.

Analysis & Results

• CavityLens.m is run to try all possible lens combinations for 1-lens or 2-lens solutions using ../20201222_BeamProfileNeatEOM/SeedBeam.mat as the seed beam.
• The cavity input mirror is modeled as two refracting surfaces separated by 6.5mm. The first surface is flat while the second has ROC of -25 mm.
• The crystal is modeled as two refracting flat surfaces separated by 20 mm.
• The target beam waist is kept at the center of the crystal with 35.578 um diameter.
• Then save all possible solutions with more than 90% overlap and where lenses are atleast 1.5" away from each other in AllPossibleSolutionsAbove90.mat using findPossibleSolutions.m.
• findBestSolutions.m increases the overlap threshold to 0.995, allows maximum beam radius of 2mm anywhere and plots the best solutions in order of positional sensitivity of the lens. These are stored in BestSolutions.mat.

Analysis & Data

Methods

• Use the actual measured beam profile in X and Y directions.
• Propagate them with the current position of lens.
• Assume the position of cavity mirror and crystal as given by the second solution in BestSolutions.mat in the Jan 8th analysis which is implemented currently.
• Calculate the overlap with target and position of waists in X and Y direction.

Conclusions

• Astigmatism should not be hurting us significantly.
• The mode matching in principle can be improved in the experiment.

Analysis

 This is an experiment I am working on to gain experience with some of the common techniques used here surrounding the detection of frequencies noise, the mechanics of setting up a michelson, and using lenses to adjust modes (beam size).

I began by creating a highly asymmetrical michelson on an optics table with arms of approximately 0.1m and 1.4m in length. After the laser, I used to mirrors to bring the laser down to a more useful height. This involved placing an iris both very close to the output of the second mirror, and very far from it. Adjusting the mirror orientations so that the beam passed through the center of the iris in both configurations yielded a beam parallel to the plane of the table.

I want to observe the interference pattern that will appear at the detector (which is placed at the output). This interference pattern will be able to give us a plot of V/√Hz as a function of frequency, but we are really looking for Hz/√Hz as a function of frequency (frequency noise). Luckily, we have just the tool to make this conversion. Eric has posted in "Calculating Frequency Noise" the constant of proportionality necessary to make the conversion. That being said, in order to measure the interference patterns, we must make the beams the "same size" when they hit the detector. Practically we will have to use lenses along the long arm of the michelson to manipulate the mode (or, as I understand it, change the beam size). On Monday, I'm going to learn how to use a program written in MATLAB called "a la mode" which will be able to do all sorts of cool calculations. By inputing the beam width at various points it will be able to extrapolate the position of the waist, and it will tell us where we need to place lenses in order to have the beam converge to a certain side at a certain point. Measuring these points accurately will be another important project. The laser beam, of course, is not just a defined cylinder with perfect boundaries - it is a gaussian distribution. We can put a razor blade on a micrometer platform and move it slowly through the beam, measuring the power output at each location. This will give us a distribution curve, and we can define a certain height on the curve to be the width.

Please let me know if I'm doing something wrong with ELOG since this is my first time posting!

-Ben

 Today I used a razor blade mounted on a moving platform to measure the width of my laser's beam at five points along the beam. The moving platform was controlled by a micrometer, so I could easily record the position of my razor blade. I mounted the detector such that the entire laser beam was hitting it, and then I placed the razor blade assembly between the laser and the detector such that by using the micrometer, the blade could be driven across the beam (causing part or all of it to not hit the detector). As I drove the razor farther and farther across the beam, the output voltage of the detector would decrease. I recorded an output voltage at each razor blade displacement coordinate.  I preformed this process with the razor positioned 0.75, 10.75, 22.75, 36.75, and 57.75 inches away from the output of the laser.

The immediate goal with this data is to find the characteristic beam width at each of the these 5 points, so that the "A La Mode" software for MATLAB can make a good fit for the shape of the beam. Finding the characteristic beam widths will involve using MATLAB to help me to work out the math - the beam is a Gaussian distribution, not a circle! I have made some progress with this, and will continue tomorrow.

 Today Eric and I worked to convert my raw data from yesterday into characteristic beam widths using A La Mode. The first step was to fit the equation a(erf(-(x-m)/2b)+1), (where "erf" is the Error Function) to the data points for each z-displacement. I used MATLAB's "Custom Equation" option in the curve fitting window five times in order to create a fits and find "b" values for each of the 5 z-displacements. I then used the fact that the characteristic width sigma = √2 * b. 

Next I plugged the sigma values into A La Mode, and after a lot of fussing around, we found out that the actual width is 2*sigma, so we had a scaling error. Finally, A La Mode fit a 2-dimensional beam profile to the points. The MATLAB output is below with the waist indicated.

With this done, I carefully measured the path length of the laser in both arms. By imagining the whole michelson as a strait path, I could plug in the position of the source and the target (the detector) into A La Mode, and I could also determine the z-displacement necessary in order to place a lens on the long arm. We then asked A La Mode to choose a position and focal length for the lens that would force the spot size of the beam from the long arm to match the spot size from the short arm at the detector. Unfortunately, A La Mode couldn't do it. We were not quite positive whether this reflects the physical impossibility of the problem, or just a computing problem, but we decided to turn to other means. 

We placed a lens that was in the ballpark of what should work 55cm from the end of the long arm (Eric calculated this), and indeed, the spot sizes were visually identical. Much to our frustration though, no fringing could be seen. The two red dots could be placed on top of each other without any fuzziness or black lines. Perhaps this was because the way I had created the michelson, the angle between the arms was nowhere near 45. We worked to "straighten out" the interferometer in order to remedy this, and I remeasured the distances. Eric also realized that the two output beams (from the long and short arms) were not parallel (vertically). Tomorrow I will work to set up another degree of freedom using another mirror so that this problem can be eliminated.

 Today I only had a little bit of time to work in the lab. With the current set up, A La Mode indicated that there was an offset between the beams from the short and long arms as can be seen in this graphic.

Using guess and check, I found that by decreasing the length of the short arm by 5 cm, this could be solved.

We realigned the michelson, and much to our dismay, no fringing appeared at the detector. To rectify this, we tried a few things. First, we replaced the round beam splitter with a cubical one. The round splitter had to be housed in a black mount which meant it was very susceptible to clipping. The cube did not have this problem. After changing this, we still could not see any fringing, but we noticed another problem. Horizontally, the two output beams were not parallel. I began trying to fix this today, and will continue tomorrow.

 Yesterday was very productive. I began by continuing to align the two output beams horizontally of the asymmetric Michelson. Unfortunately, it wasn't working - I was limited greatly in my range of motion by the fact that the long arm had such a narrow window in which it could both come back in perpendicularly, and still not be reflected straight back into the laser (which would be bad for the laser). Eric suggested that instead, I try to use the laser he had been using for his crackle experiment which travels through a long, fiber optic cable. This gave us two benefits. First, the fiber optic cable is supposed to act as a collimator, so our lens would be unnecessary. The spot size would automatically be the same everywhere along the beam, including at the detector. Also, the tiny radius of the fiber optic "groove" meant that there was basically no chance of shining the laser directly back into the tube and into the laser. This allowed me to align the beam out and back along the same path, which was less challenging than before. After some fiddling, it worked! We could see fringing effects clearly. (slightly hard to see in this photo)

After replacing some of the mirror mounts with more sturdy "Polaris" mounts, we were ready to take some data. Eric positioned the beam so that we were sitting right on a fringe (the PD output voltage would wander around with slight shakes of the table or loud noises), and he let the spectrum analyzer take its data. There was a lot of unexpected noise in the signal. We wondered if this was acoustic noise, so I covered the setup with a large blue plastic box (with a hole cut out for the long arm). Suddenly low frequency noise (less than 10Hz or so) dropped significantly, so we decided to use the box for our measurements. Eric took a noise spectrum both for the asymmetrical setup, and for a symmetrical setup with both arms just about 8.8 cm in length.

Here is an overhead view of the asymmetric setup. The source is at the bottom, and the two grey mounts are the end mirrors of the arms. The detector is on the right, and the other two mirrors are used to bring the beam to the correct vertical level. The small iris in the bottom right is irrelevant.

Here is the long arm of the Michelson which passes outside of the blue box in the asymmetric setup.

 Additionally, yesterday I learned how to plot the noise spectrum density using MATLAB's pwelch function. I also learned to make good-looking plots. Here is the PSD for 4 random signals that Eric came up with.

 I converted the raw Volts/rootHz data retrieved from the asymmetric Michelson experiment to frequency noise using the proportionality constant derived in Eric's June 19 Post. I then plotted frequency noise as a function of frequency on a log log plot. The noise for the asymmetric setup is shown in red (it is the noisiest). The other three data sets can be thought of as components of the total signal. The symmetric setup (in green) theoretically give us the noise for everything besides frequency noise from the asymmetry. The red line was found by taking the spectrum of the laser shining directly into the detector (the intensity noise component), and cyan curve is from when the laser was turned off.

 Minimizing the coherence of the intensity noise signal.

We need to use the difference between the the output voltages of the PDs at the symmetric and asymmetric michelson ports. This lets us eliminate intensity noise while not messing with the displacement data. The gains of these two signals (Vas & Vsy) may not be quite the same though, so a small level of coherence may still exist. I created a MATLAB script which takes in Vas and Vsy signals, and find the coefficient by which we should multiply one of the input voltages in order to minimize the coherence.

To do this, I first created a simple function (integralCoherence.m) to find the integral of the coherence for a given Coefficient and given Vas and Vsy inputs. The coherence is found using MATLAB's mscohere function.

Next, I created a function which uses Newton's root-finding method to detect which coefficient value would give integralCoherence the smallest value possible. In other words, this function (rootFinder.m) minimizes integralCoherence(Coefficient). To do this, I seed the function with a coefficient of 1, then ask MATLAB to find the first and second derivatives of integralCoherence at 1. The first divided by the second derivative is a quantity I call the "jump." The best coefficient must lie somewhere in the direction of the jump according to Newton's method. We add the jump to the original coefficient and start over. The process iterates until the jump size is less than 1e-6, which means that we are converging on a best value for the coefficient. That coefficient is returned.

Finally, I made a test script which loops over several possible offset values for the Vas signal (relative to the Vsy signal), and returns the best coefficient to minimize the hypothetical intensity noise in each case. I have included a plot of the percent error of each output coefficient with respect to the input value. Although we are seeing some trends in the error, this does not seem to be too much of an issue, as the relative deviation is generally well below 1e-3.

Tomorrow, I will create a function that takes real-life Vas and Vsy values as input and outputs the best coefficient.

 I have designed the following new shadow sensor mount, and the designs have been given to the machine shop - the parts should be here soon.

 We needed a new design because currently it is very difficult to adjust the position of the shadow sensors without messing with the position of the coil relative to the magnet. The whole apparatus is also currently too flimsy. The new assembly is going to be machined out of 6061 aluminum, and it will not have large thin surfaces, so it should be much less susceptible to vibrations. A simple thumb screw will drive the shadow sensors up and down relative to the razor blade.

The assembly is made of the 4 parts (shown in the first 4 attachments below). There is an assembly picture as well as an exploded view afterwards.

• Part 2 replaces the upper half of the current blade spring clamp. It is modified to be larger and to house a groove where the razor blade mounts. As usual, the magnet will be epoxied directly between the two holes.
• Part 1 has a hole pattern with which we will mount the coil (note that the coil is simply modeled as a large grey cylinder. This part also has a large track cut out in the middle which will house the mobile sensor mount piece (part 4). A through-hole on the top will allow for the thumb screw. The bread board with the electronics will sit on the front face of the part. The four front-facing holes are tapped. standoffs will separate the circuit board from the metal part.
• Part 4 is the sensor mount. The diodes sit on the inside of the cavity with their leads coming out of the small holes (two holes per side). The large threaded hole allows the thumb screw to drive the part up and down. Two set screws can be tightened down through the holes in part 1 onto part 4 so as to prevent further adjustment.
• Part 3 simply attaches the assembly to the large optical mounting post. It allows well over 180 degrees of freedom so that we can easily change the position of the mounting post relative to the optics when necessary.

Note that a spring on the thumb screw between parts 1 and 4 will maintain a force between those parts - this should make adjustment smoother. Also note that I will order washers to stack up on the thumb screw on top of part 1 to allow for an extended range.

I have made new drawings which better abide by engineering conventions. Here they are.

 Today I figured out the proper orientation for the SME2470 Emitter and the SMD2420 Detector which we will be using for the shadow sensor. A very simple diagram is shown in the first attachment (a photo from my notebook). My system is to always use a black wire for the ground side, which is in both cases the side with a flat (not half-moon-shaped) conductor. The emitter gets a white wire on the curved side, while the detector will get a red wire (so we can tell the two apart). Note that the wiring can be confusing because the Cathode and Anode side of the components are opposite for each, but the fact that they are placed in the reverse orientation un-does this.

The second attachments shows the emitter with two wires soldered onto it. This proved to be difficult. As noted in my notebook, the soldering iron can only be set to 500° F for 5 seconds before the part is ruined (according to its documentation), so making these connections strong took work. The final attachment shows the emitter mounted in the part I designed to hold it. (See my 7/16/13 ELOG).

 Today I finished wiring one emitter/receiver set (as shown in yesterday's post), and I mounted the two components in their location inside the shadow sensor mount. I coated the surface they sit on in nail polish in order to prevents shorts, and epoxied the diodes in place as shown below.

I tested the setup on a solderless breadboard. The wiring I used can be seen in the photo below, with one exception. In reality I used an oscilloscope to measure the output voltage across the 50 ohm resistor (which is sitting in the upper, right corner of the board). The scope read approximately 1.75 V above the dark level when nothing was between the two diodes.

Finally, I cut out a piece of bread board for the purpose of mounting that circuitry onto the face of the assembly. Here is a photo of the first completed assembly. The four wires that will be soldered in place along the top of the board already exist in the experiment. At time of installation, we will solder these wires in place. From left to right, they are Sensor +, Sensor -, +15V, Ground. I used the power supply and scope to check that the circuit still worked, and it did.

The thumb screw and spring system for adjusting the sensor height feels strong, and it will not move when you take your hand off of the screw. Tightening the set screws locks the height completely.

 Today I modified part 1 from my July 16 elog by couterboring the two rear coil mounting holes as shown in the photo below. This allows us to forgo the extra spacer, and it lets us use the same, short screws that we had already been using to attach the coil.

Next Eric removed one of the old shadow sensor apparatuses from the chamber, and I soldered the power and sensor cables in place (as described in my July 23 elog). Thus, the top half of the assembly was complete and ready to be installed. The following photo shows it mounted in place with the one remaining old mount in the background.

Another view (note that the screws are not yet tightened down):

Finally, I epoxied the magnet onto the new razor blade mount. It shouldn't take too much time to get this new assembly up and running tomorrow.

 Today we finished installing both shadow sensor mounts, and everything is wired up. The system successfully locked with the new hardware, although there is some tweaking that needs to be done to the transfer functions.  Adjusting the shadow sensor heights was a piece of cake compared with how it was before. Here is a photo of the new setup from outside of one of the windows.

Afterwards, I recorded the output voltage of each channel (A and B) at offsets from -15000 to 15000, and I used MATLAB to find a line of best fit. The plots with the two best fit lines are attachments 2 & 3. The slopes are printed on the plots.

Plot A: 

Plot B: 

Eric used the ratio of the slopes to compensate for the fact that a given input voltage into one coil will not necessarily lead to the same force on the on the blade spring as it will on the other. With the new proper offset ration (SlopeA/SlopeB = 1.9298), we can control the two blades as if they were identical.

 Over the past two days I have been busy simulating a signal like the one we might extract from our crackle experiment. This signal is made of two things: crackle and background noise. 

• the traditional spring constant with no crackle
• a force crackle term: this manifests as a random delta-k (I used a normal distribution) multiplied by the Force and a constant Alpha
• a jerk crackle term: this manifests as another random delta-k multiplied by the derivative of the force and a constant Beta

 This is a continuation of my post from yesterday. Today I was able to fit a polynomial to my Mean force Q signal with crackle. Here is the polynomial fit.

Visually, it is hard to tell the 2nd-order polynomial fit with the actual data points on this plot, but I wanted to make sure that there was no trend in the error, so I plotted the residual (polynomial - original points). As you can see below, there is no trend, and the residual is very small, so the 2nd order polynomial is a good approximation.

Satisfied with this, I used MATLAB's "root" function to find the intersection of the 2-sigma boundary line and the Man Qf line.  This is the Minimum value of alpha (so in a sense, the minimum amount of crackle noise) we need in order to be able to be 95% sure that we have seen crackle above the background noise. I then plotted this minimum alpha value as a function of the level of background noise.  The x-axis of this plot is in terms of a coefficient that I am using. The actual range of this plot goes from a power spectral density of  approximately 10^-15 to 10^-12 [sqrt(W/Hz)]. The circular points are the individual data points I found, and the line is a second order polynomial I fit to it.

In the future, I will figure out how to add in Jerk Crackle.

 Today I made several improvements to the code I had created yesterday. I improved my script for finding the lowest amount of crackle (smallest alpha value) at which we can see crackle as a function of background noise. Along with enhancements such as better spaced data points (using log space), and improved efficiency through removal of necessary random number generation, I also converted the x-axis to display power densities of the background noise instead of a meaningless coefficient. The graph is shown below. It has data taken at 50 different background noise power densities. There is also a handy, second order polynomial fit to it.

I also began creating a script to plot the Power Spectral density of the Noise Signal and the Crackle signal separately on the same axes. This should show that the background noise signal at a HIGHER power than the Crackle signal - even though we can detect that crackle. Unfortunately, as you can see in the plot below, something has gone wrong, and my Crackle Spectral Power Density is actually above that of the Background noise when the Background is set to 10^-15. I will continue to look into this to see what is wrong. Once that is complete I will add in a simple bandpass filter in order to simulate how we will only be taking crackle data from a small portion of the sensors' outputs (just like LIGO observatories).

Eric helped me to figure out why my crackle signal appeared to have more power than my background signal in the last plot from Friday's post. I was using 2 standard deviations from my force Q as the threshold above which we would be able to be 95% sure that we'd seen crackle, but in reality, we don't need that big of a threshold. Instead, I used 2 standard deviations from the time-averaged background noise signal (which converges quickly towards zero). This decreased the alpha value (amount of crackle) necessary for us to see crackle, and now we can clearly notice that this code allows us to detect crackle EVEN when the crackle's power density is less than that of the background noise! I also added a bandpass filter from 50 to 150Hz. The first plot below shows the minimum alpha values necessary to be 95% sure we've seen crackle as a function of background Power Spectral Density with this bandpass filter in place (note that the alpha values are lower than without the bandpass, as we would expect). The second plot shows the power as a function of frequency for the noise and crackle signals in this situation. Both of these have an integration time of 600 seconds.

Next, I checked that the computer was actually spitting out numbers that made sense by visually finding the minimum alpha value at which I could observe crackle in the signal, and comparing that to the results of the first graph above. I found that the observations made sense. Below is a plot of the signal (the sum of the background and crackle noises) as a function of time over a 20 second range. I chose to test at a background power spectral density of 1e-15.

At alpha = 1e-6 you can see that the peaks of the crackle signal have just become visible above the "fuzzy caterpillar" of background noise (there are 5 peaks). This agrees very well with the code. The computer claims we can be sure about seeing crackle at alpha = 0.7e-6 for this power.

Finally, I noticed that the integration time used to generate the power graph affected the crackle noise's apparent power. I decided to make a plot of the crackle power (mean of the blue line of the second plot on this page) as a function of integration time. Ideally, we will run this experiment for a very long time, but here I only took a few data points because longer integration times would cause my computer to lock up due to the huge number of calculations. As a result the plot doesn't really converge very well, but it give a good idea of the behavior.

With the recent realignment, it was time to re-calibrate the shadow sensor mounts. I wanted to find the conversion factor between the number of counts typed into the "offset" box and the displacement (caused by the coil/magnet actuator) for each blade. This required two steps. First I found the output voltage of the shadow sensors as a function of displacement. Then I found the output voltage of the photodetectors as a function of offset. When I had a conversion factor for each, I could multiply them to get my final conversion factor:

(displacement [microns] / volts)  *  (volts / counts)  = displacement / counts

I took each shadow sensor mount out of the pressure chamber, and measured its output voltage as I moved a razor blade between the diodes. I used a micrometer stage to precisely position the razor at specific displacements. I then fit the output voltages as a function of displacement to the error function using MATLAB's curve fitting app. Here are the parameters for each shadow sensor.

The function fit to is f(x) = a*(erf(-(x-m)/(2*b))+1)

Sensor A: 

Coefficients (with 95% confidence bounds):

a =      0.8901  (0.8782, 0.9019)

b =       172.4  (169, 175.9)

m =       325.7  (319.8, 331.6) 

Goodness of fit:    SSE: 0.000221        R-square: 0.9999        Adjusted R-square: 0.9999     RMSE: 0.004955

Sensor B:

Coefficients (with 95% confidence bounds):

a =      0.9585  (0.9256, 0.9914)

b =         151  (144.1, 157.9)

m =         260  (248.4, 271.6)

Goodness of fit:   SSE: 0.002777     R-square: 0.9991    Adjusted R-square: 0.999     RMSE: 0.01361

Then I plotted output voltage as a function of displacement, and fit a lien to the middle 50% of the voltage output range.  This gave me the following values. My plot with the original data points shown as circles, the fitted error function as a dashed curve, and the line as solid is shown below.

BladeA: 0.0027 ± 3.22e-5 V/micron

BladeB: 0.0034 ± 9.72e-5 V/micron

Next I set the offsets on a range from -15000 to 15000 for each blade, and recorded the output voltages. The relationship is a simple linear fit as you can see in the two plots below. The data points are shown as dots while the fit is a line. The conversion factors are:

BladeA: 1.12e-5 ± 1.55e-7 V/count

BladeB: 7.79e- 6 ± 1.85e-8 V/count

When multiplied, we get the following overall conversion factor:

BladeA: 4.0952e-3 ± 7.59e-5 microns/count

BladeB: 2.27408e-3 ± 6.57e-5 microns/count

 Eric has been able to take data with the pumped system. On Tuesday (6th) he collected 20 minutes of data, and yesterday (the 8th) he was able to take 2 hours of data with the drive on, and 1 hour with the drive off (the Michelson unlocked after an hour on the undriven run). I set out to look at the data. After doing the necessary conversions to turn the output voltages into a differential displacement, I took a bandpass filter of the 400-500Hz region of both signals. Then I observed the signals and noticed that the first 0.1 seconds or so were very strange and noisy while things stabilized, so I chopped off the first 400 samples. Next I plotted the driving force that had been used during the collection of the driven data, and I calculated (using a curve fit) the phase offset necessary to get that force in phase with the sine function. I then chopped off the appropriate number of samples from the data so that they matched up with the sine function themselves. Finally I was able to calculate the force Q and I values for both signals, and I plotted their cumulative sum (time average). The following plot shows the I and Q values for the driven and undriven cases. Note that the Undriven data ends early because we only had 1 hour of data as opposed to 2.  It is interesting how the driven Q seems to stay above the I and all the undriven plots. 

My computer was not up to the task of plotting I and Q (it would just freeze), so I instead plotted every 100th point. (To compensate, I multiplied the I and Q values by 100). The result is a graph that looks as it would with all the points, but please let me know if this is invalid for some reason I haven't thought of!

After this I tried to see what the effects of an unbalanced driving force on our ability to detect crackle. Unfortunately, a whole host of little coding setbacks kept me from coming to any big conclusions here, but I was able to observe the effect of imbalance on the minimum alpha value ("amount of crackle") at which we'd be able to observe the effect. I chose to look at a signal with a background noise of power spectral density 1e-12. Interestingly, the minimum value is much lower (easier to detect crackle) when there is no imbalance, but as you move away from no imbalance, the alpha value sharply increases before leveling off. At an imbalance on the order of 10^-8, crackle would be about 10 times harder to detect than if there were no imbalance.

I will continue later with a more thorough analysis of this.

 Today I refined my method of observing the driven and undriven I and Q values in order to find a faster way to find the final values after long integration times. I was able to reduce calculation times, and I found the follow I and Q values. The Driven data uses a 2 hour integration time, and the Undriven data uses a 1 hour integration time. These are from the August 8 data sets. As you might expect, the driven Q value does not have some obvious DC offset. After just 1 or 2 hours, it would be surprising to see crackle, I believe!

Note that I am only working with Force Crackle currently, and recall that we are defining I = sin(2*wd*t)*signal^2 and Q = cos(2*wd*t)*signal^2. wd is the driving frequency. In the presence of crackle, this demodulation scheme works because Q would have a DC offset relative to I.

----------

Final UN-Driven I Value: -3.1178e-30

Final UN-Driven Q Value: 4.5409e-31

Final Driven I Value: -5.5231e-31

Final Driven Q Value: 2.4821e-31

----------

Today I was able to plot I and Q for both the driven and undriven plots on one reliable plot. Unfortunately, what we saw is that a few gigantic outlier points (especially in the undriven signal) caused the I and Q traces to make wild jumps. As you can see in the first attachment, there there are weird steps in the green and black (undriven) traces.

Those can easily be matched up with the huge outliers that we see in the following plot: Each of the huge spikes in the green signal (for example, at 1756 seconds and 2964 seconds) corresponds to a step on the first plot!

It seems prudent to investigate what happens when we subtract out all the outliers. Basically, I am currently writing a code that picks out all the points with an absolute value greater than some multiple of the standard deviation, and proceeds with the data analysis in segments BETWEEN these points. This will give I and Q values for each segment. I will then be able to take a weighted average of those values and thus come to a better estimate of the final I and Q values.

An Optical Setup for Crackle Noise Detection

Due to the mechanical upconversion of the local deformations due to grain slippage or similar microscopic events. It has long been suspected that there are multiple instances of crackling noise in the LIGO Interferometers, and as such, a crackle experiment has been developed where the crackling noise in the maraging steel blades of a Michelson Interferometer will be measured. The steel blades present a straightforward mechanical system which can be driven and stressed easily. Since the crackling noise in LIGO is vertical noise and the coupling to the horizontal motion of the test masses is very small, the vertical noise in the blades are measured directly in the crackle experiment, to have maximum sensitivity to the crackling noise. A proposal for a crackle experiment with increased sensitivity has been proposed, and this project will be concerned with the construction and alignment of the optical setup for the upgraded crackle experiment. Following this, the response of the maraging steel blades will be characterized, and a control system will be developed and implemented to lock the Michelson Interferometer in order to keep it at resonance.

Report: LIGO-T1400511

Presentation: LIGO-G1400906

We measured the seismic noise again.

The result is the attached file.

Difference from yesterday (yesterday -> today):
    1. Input voltage (5V -> +-12V)
    2. Adjustment of the spring in the seismometer
    3. Consideration of preamp factor in the seismometer (1 -> 100)
The calibration factor of coil is still 2150 V/(m/s)

From the result, it looks like no change on the high frequency, but more noisy on the low frequency.

To damp the motion of the mass, I attached magnets on the plate below the mass, and set nonmagnetic metal around the magnets. (Attached file)
In the result Q goes down to ~20.
I had tried to control.
But I couldn't.
When I closed the loop, the error signal began to oscillate at 2-3 Hz. (Attached file)
It is about the resonance of the spring.
So I should check the transfer fumctiom of the loop.

[Seiji, Mingyuan, Dan]
Today, we measured the transfer function (TF) of the system with a shadow sensor (coil magnet actuator -> mass -> shadow sensor).
We also measured the TF with eddy current damping.
There are some peaks (@2.3Hz, 7.7Hz, 3.2Hz, 11Hz).
The peak at the 2.3Hz is the resonance of the vertical vibration.  (We could damp this peak.)
And the peak on the 3.2Hz may be the resonance of the twist vibration.
We are searching the sources of the other peaks.
 
We measured the noise spectrums of the mass with the shadow sensor.
(With damping and without damping, vertical component and horizontal component)
The result is consistent with the TF. (The noise at 2.3Hz lowers by the damping)

Yesterday, we knew the noise at 7Hz disturbed to control the Michelson.
So we searched the source.
We found the source was the yawing of the mass.
To suppress this motion we unclasp the mass and only put on the plate nipping a gum.
The noise at 7Hz vanished.
Although, a noise arose at 3.6Hz.
This noise originated from the violin mode of the plate.
We could damp this noise using eddy current damping. (fig and graph)
We measured the transfer function of this system. (Coil -> mass -> shadow sensor) (graph)
Next, we will control the Michelson.
 

We could not lock the Michelson by yesterday's configuration.
So we controlled the mass with shadow sensor to damp.
Then we tried to lock the Michelson in the state.
But we could not lock because of a noise around 40Hz.
We can see this noise on the seismic noise spectrum. (Graph)
So this may come from the motion of the LASER source.

Therefore, we isolate the LASER sources. (Fig)
The graph is the error signals from a shadow sensor. (Graph)
The noise around 40Hz vanished.
But the noise level went up totally.
This also disturbs to lock the Michelson.

Yesterday, there was a noise around 40Hz disturbing to lock the Michelson.
We put the lasers on the plate to isolate.
Then we found the noise went up totally.
Today we found the noise was because of the laser.
So we borrowed a laser from 40m.

We measured the seismic noise. (Graph: red line is on the table, green line is on the plate isolated from the table by gums)
From the result, we can suppress the 40Hz noise to use the isolated plate.

To synchronize the noise of two plats, we connect these plates with metal fittings. (Fig)

The signal from Michelson seems to be better in this configuration.
We will control the Michelson with damping by a shadow sensor.
 

Today, I used a current buffer to raise the efficiency of a coil actuator.
First, I soldered the circuit to put it into place.
Next, I tested the current buffer work or not.
The gain of this circuit is 10.
To apply 300mA current to a coil (10 ohm), we need apply 6Vpp to the coil.
This value should be the maximum voltage.
We will use this current buffer to control the Michelson.

Today, I adjusted the filter to control the Michelson.
After this operation, I could control the Michelson for several second. (Graph)
Following is the filter setup.
Shadow PD -> SR560(BP=10Hz, g=5) --------------------------> SR560(A-B, g=1) -> circuit(g=10) -> coil
Michelson PD ---> SR560(A-B, LP=300Hz, g=1) ------> SR560(A-B, g=2) ----^
                        |            ^----function_generator(0.07V)         ^
                        |-> SR560(BP=100Hz&300Hz, g=5, inv) --------|
I will write this setup legibly when the configration is decided.

I could lock the Michelson for several seconds.
When I try to lock the Michelson, it seems that a noise at 200Hz grows up and breaks the lock. (Oscilloscope signal)
I measured a noise spectrum over a short time when the Michelson was locked. (Graph)
There are peeks at 53Hz and 214Hz.

I had tried to lock the Michelson by tow masses.
I hung another mass in a same way. (Fig)
I adjusted the filter, but I could not control.
It seems that a noise at 200Hz disturbs the control.

I measured the transfer function of the current buffer. (Graph)
It shows this circuit has gain 10.

I could lock the Michelson interferometer which has one free end-mirror.
We applied the offset voltage to the point before LPF only up until yesterday.
(The control filter is built by a LPF and a BPF)
Today I applied the offset voltage to both of the points before LPF and BPF.
The figure I attach shows the Michelson signal (yellow) and feedback signal (purple) when Michelson is locked.

We measured the open loop transfer function of this control system. (Graph)
It has 30 degrees phase margin at unity gain frequency.
 

I measured the TF of@the control filter. (Graph)
I wrote a figure of this filter. (Fig)

The open loop TF measured in the last week had too high gain on the low frequency.
So it was like noise on the low frequency.
We need suppress seismic noises to decrease the gain.

We changed the fixed mirror on the end of the Michelson into a PZT mirror.
We tried to let the suspended mirror move on with the movement of this PZT.
But its range was not enough.
Next we need to lock the Michelson with two suspended mirrors.
That way, we will be able to keep the suspended mirror moving.

We measured the TF from the coil actuator to the displacement of the suspended mirror. (Graph)
 

Today, we could lock the Michelson with two suspended masses. (Graph: Yellow line=error signal, Purple line=feedback signal)
We used the same filter which I reported yesterday.
We applied feedback signals to one coil and invert feedback signals to the other coil.
We tried to use more simple filter circuit (Graph: filter0830.png) for the phase compensation.
But it could not lock the Michelson completely.
So we will use the filter which I reported yesterday.

I made a circuit to output the square of input signal. (The IC is AD734)
The output of this circuit is following.
   Vout = Vin * Vin / 10
I tested this circuit.
I attached a figure which shows the input signal (blue) and output signal (yellow).

We put the optics in a chamber. (Fig)
We stacked two plates to isolate the seismic noise in the chamber.
All optics is on the plate including laser source and PD.

We could lock the Michelson with two suspended masses. (We used two coils)
I will attach the open loop transfer function of this control. (Graph)

We measured the Michelson error signal. (Graph)
The green line in the graph meant that we closed all windows of the chamber.
The blue one is the noise from spectrum analyzer.
We could suppress the noise in the high frequency by a factor of 10.
This is considered the acoustic noise.
Now the limit at 200Hz is the noise from spectrum analyzer.
 

We found why we couldn't lock the Michelson yesterday.
The stack had been moving by little and little.
Therefore it touched chamber.
Today we reset the stack and adjusted the alignment.
Then we could lock again.

We could lock the Michelson with two blades moving.
We used a function generator to apply voltage to the coils and let the masses move as common mode.
We used the shadow sensor to measure the mass motion.
The mass shakes with amplitude 45um when we apply sin wave (300mVpp, 3Hz) to the coil.
We could lock the Michelson under this condition.
In order to let the two masses move just the same, we used a variable resistance to match the efficiency of actuators.