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ID Date Authorup Type Category Subject
  1109   Wed Mar 6 12:48:50 2013 taraDailyProgressVacuumthermal shield + peek supports are baked

I setup the small vacuum chamber to bake the shields and peek cavity supporting pieces. All pieces are baked and I'm assembling all the parts.

 photo(13).JPG

Details about how to use the pump and the chamber can be found in CTNwiki.

  1111   Wed Mar 6 21:56:12 2013 taraDailyProgressVacuumnote on feedthrough

Feedthrough channel (as seen from the connector outside of the chamber)

1-6: heater on cavity#98 : 85.4 Ohm

3-7: Temp sensor on cavity #98

 

4-8: heater on cavity#99: 156 Ohtm

5-9: Temp sensor on cavity#99

 

  1114   Tue Mar 12 18:20:21 2013 taraDailyProgressVacuumCavities inserted; chamber pumping down

After pumping the chamber down for two days, I disabled the turbo pump and turned on the ion pump. The initial current was 7mA. After a day now it is 0.38 mA. It was better before, see PSL842, (started the ionpump at 1mA, and operated at less than 0.1 uA). If this is true, this means the pressure will be about a factor of 0.38mA/0.1uA ~ 3800 higher than before. (the calculation is based on this equation, where the current is directly proportional to the pressure.

 

The flanges do not flatly touch each other. There's a tiny bit of something in the middle that causes a gap as shown below. This might be the reason why the ion pump current is high, or it might be some out-gasing problems of the in-vac materials. I think we can try to spray isopropanol around the flange to see if the current comes up, but I think it is a bad idea for the ion pump. I'll ask Steve or Koji for their opinions.

 IMG_0815.JPG

The pictures during the installation can be found onpicasapage

Quote:

 [Tara, Koji, Evan]

Tara and Koji spent the better part of yesterday afternoon inserting the new cavity assembly into the vacuum chamber. In the process of putting the window back on the chamber, the old copper gasket may or may not have hit the inner surface of the window, so Koji performed a drag wipe. Tara and Koji then inspected the window under a high-power bulb, and I think the consensus is that there's no visible damage.

The chamber is currently pumping down. Unfortunately, there appears to be a small speck of something trapped between the window and chamber flanges, so there's a small gap on one side of the joint. We'll see if we can achieve high vacuum.

 

  1117   Wed Mar 13 23:52:12 2013 taraDailyProgressVacuumCavities inserted; chamber pumping down

It's still improving, now the current is 0.3 mA (0.38mA yesterday). I'll wait until it stops improving and try to tighten the screws a bit. \

About the plastic pieces, they are peek. I think it is vacuum compatible, cf E960050-v11.

May be it is the temp sensor that I have to re-solder on the copper pieces. I did not bake them after soldering.

Quote:

The gap at the frange is OK as long as the gasket is evenly squeezed.

1. It the pressure still improving? => Your cavity is still out gassing.

2. Fasten the screws little more (not too much) for these two franges.
    => If there is any improvement there is some leakage.
    => If not, this is just a outgas from your cavity.

Do you think your black plastic pieces are vacuum compatible? Are they made of Delrin?

 

  1119   Thu Mar 14 18:20:04 2013 taraNotesopticfused silica pmc

Peter told me that  the fused silica pmc currently used in the lab is bonded by Vac-seal epoxy. So we don't need to polish any surfaces for optical contact.

IMG_0819.JPG

Traces of vac-seal can be seen between the mirror and the tip, the tip and the spacer bonded areas. Vac-Seal epoxy is chosen for its low out gasing, so that the mirrors won't be contaminated.

IMG_0820.JPG

  1121   Fri Mar 15 11:23:45 2013 taraNotesopticAlAs/GaAs

I'm thinking about the spec for AlAs/GaAs coatings. Here is the list of what I have:

  • coating on concave side of the mirror for 0.5m x6 (I'm not sure if they can do the transfer on 0.5m mirror now) for 1.0m x6 for flat mirror x3 -
  • for circularly polarized light, normal incidence
  • Transmission @1064 = 100ppm +/- 10ppm. 10% error is still within the acceptable value for 10ppm loss (T ~ 67-73%), see T1200057v11 -
  • Absorption + scatter loss < 10ppm, this is what Garrett told us. -
  • coatings diameter = 8mm (The number is from Garrett), the loss around the edge for our beam with diameter=364 um is less than 10^-10 ppm. -
  • Max scratch surface and point defects are not determined yet. I can look up the specs from our current SiO2/Ta2O5 mirror since they are ok for us. -
  • I think we are aiming for the thermo-optic optimized coatings. The layer structure can be found in T1200003-v1.

==Coating diamter for 0.5m ROC mirror==

About the coatings diameter, Garrett said it depends on the aperture size/ coating diameter. So I made a plot to estimate the loss due to the finite size coating vs Coating diameter for our spot radius of 182 um. The loss is simply calculated by the ratio of the power not falling on the coating = Ploss/Pin = (exp(-2*r0.^2./w0.^2))*1e6*26000/pi   

where r0 = coating radius, w0 = spot radius, a factor of 1e6 for showing the result in ppm, 26000/pi is the total loss due to the light bouncing in the cavity.

edgeloss.png

fig1: Loss vs coating diameter (in meter)

 

It seems we can go to 2mm coating diameter, and the loss is still much less than 1ppm (the expected loss from absorption and scatter is ~ 10ppm). However, we have to consider about how well they can center the film, how well we can assemble the cavity. So larger coating diameter is always better. If we assume that 1mm error is limiting us, coating diameter of 4-5 mm should be ok for us.

 ==for mirror with 1m ROC==

If the ROC is 1.0m, the coating diameter can be 8mm. For the cavity with 1.45" long, the spot radius on the mirror will be 215um (182um with 0.5m mirror). This changes the noise budget of the setup a little bit. The total noise level is lower by a factor of ~ 1.2. (see below figure) at 100 Hz.

noise_compare.png

fig2: Noise budget comparison between setup with 0.5 m and 1.0m RoC mirrors, plotted on top of each other. Noises that change with spotsize are coating brownian, substrate brownian, thermoelastic in substrate, and thermo-optic.

 

==What do we choose? 0.5m or 1.0m==

For both 0.5 and 1m, the cavity will be stable (see T1200057-v11, fig11). So either choice is fine

if we use 1.0 m,

  • we loss the signal level a bit,
  • but we are more certain that the coating will work. 
  • The procurement should be faster (as promised by Garrett)
  • have large area coating up to 8mm diamter
  • need to check if we can mode match or not (I'm positive that we can, but I'll check or let Evan check)

So at this point, I'm thinking about going with 1.0 m mirror.

 

 

  1124   Wed Mar 20 22:18:28 2013 taraNotesRefCavestimated beat frequency

At current temperature, the estimated beat frequency will be ~ 60-100MHz. This is not so bad, since we can use 1811 to measure the beat signal and use PLL to extract the beat noise.

 

We will need to use thermal expansion to tune the beat frequency. So, as a start, I try to figure out the beat frequency, and how much we have to heat up the cavity. The heaters on each cavity is off, only the heater around the vac chamber is on (but the servo is off). 

Right now we have one laser locked to one cavity, but the beam path to the first cavity has a beam splitter that we can borrow the beam and direct it to the 2nd cavity. I realigned the beam to have both beams into both cavities. By adjusting the temperature control on the NPRO (slow signal), I can bring the beam to resonant in each cavity.

1st cav is resonant @ (334/398) and (150/398). The numbers correspond to coarse and fine knobs of the slow feedback to the laser.

2nd cav is resonant @ (154/398). (I'll come up with a better name to call the cavities)

The FSR is 4.07 GHz (for 1.45" long cavity). This means 334-154 = 184 clicks on the coarse knob equals to 4.07 GHz, or 22MHz per coarse click. Both cavities resonant at ~3-5 clicks apart. So the beat frequency is ~ 60-100MHz. This is quite good, at least we are not close to half FSR apart. The power required to tune the cavity length should not be that high.

 

The next thing to do is try to see which cavity we need to heat up in order to bring both cavities resonant frequency closer together.

  1131   Wed Mar 27 01:52:34 2013 taraNotesPMCPMC Longitudinal Testing and Modal Analysis

I use COMSOL to find the first longitudinal mode of a stainless steel PMC,  it is about 16 kHz. I'll find an analytical solution and compare them to make sure that the FEA result gives us a reasonable answer or not.

The FEA result in psl:1088 does not show the right body mode of the PMC. The frequency of 440 Hz is from some weird mode as seen from the figure in the entry. Evan checked the body mode of a simplified steel PMC, and I also check independently. Our results agree quite well that the first longitudinal mode is at ~16kHz.

pmc_eigen.png

However, this does not answer what we measured in PSL:1097, where the longitudinal motion is around 300Hz. I checked the body frequency of the base blocked and it is even higher than the PMC body modes' frequencies (this should be expected since the base is even bulkier).

Note: I just learned from Zach that the PMC in GYRO setup does not have 3-point support. It just sits on the base block. But this has not given me any clues about the possible modes yet.

 I'm writing some background and requirement for the PMC[coming soon]

 

  1133   Sat Mar 30 01:42:16 2013 taraDailyProgressElectronics EquipmentFrequency readout noise of PDH loop

It is indeed really high. We expect something in the order of ten nV/rtHz level (like what we had before, see PSL:781.). We are investigating it.

One of the possible causes is the back reflected beam. When I reduced the power input from 6mW to 1mW (with common/fast gain = 999/720, boost on), the error noise was down to ~ 10nV/rtHz up to 5 kHz, with bumps and peaks around 50kHz up to 100kHz. Our mode matching this time is not very good.

 I replaced the PBS for PDH locking to a bigger one, since the beam spot at that position was quite large. This got rid off some high frequency peaks and bumps. I still have to align the beam to minimize back reflection.

 

Quote:

Is this really the input noise of the PDH servo (i.e., output noise / servo TF)? If so, it seems pretty high. With the right components, you should to be able to do ~100x better.

Quote:

Tara was able to tame the cavity servo loop. The third attachment is the result of several SR785 measurements of the error signal power spectrum. I converted this to a frequency noise power spectrum (fourth attachment) by extracting the voltage/frequency calibration factor from the error signal as follows.

 

 

  1134   Sun Mar 31 03:03:28 2013 taraNotesPMCPMC Longitudinal Testing and Modal Analysis

[see PSL:1135]

I compared results between COMSOL and analytical solution. The first longitudinal mode from both results are comparable.

Peter sent me a note from Dennis about PMC longitudinal mode calculation. Dennis mentioned about a book by Young&Roark (here), so I looked it up and see how to estimate body mode frequencies of a simple block/beam.  I tried a simple geometry, a 0.1x0.1x0.175 (m) block. According to the book, cf situation 7b, table16.1 page 771, the first longitudinal mode is

f1 = (1.57/2*pi) * sqrt ( E/ rho*L^2), ), rho is the mass density of the material (2202 kg/m^3, for SiO2), E is the Young's modulus (72 GPa), L is the length of the block ( I use L = 0.175/2 because 7b situation is a uniform bar vibrates along its longitudinal axis, with upper end fixed, lower end free. This is similar to a whole beam resonate freely on both end because its center will be fix. Thus, to use the formula for our case, we have to use half length of the beam).

The analytical solution and COMSOL give f1 ~ 16 kHz.

psl_log2.png

 It is very strange that, according to COMSOL simulation, when the cross sectional area of the block is changed to 0.01x0.01 m^2 instead of 0.1x0.1 m^2, the frequency of the longitudinal mode does not change that much (still close to 16kHz. However, from the analytical solution, the frequency should drop by a factor of 10 ( around 165 Hz).

I'm going to think about this a bit more, but at this point, I think my COMSOL model is not correct. Might be some kind of bdy conditions that I'm missing.

psl_log.png

 

 

 

  1136   Sun Mar 31 20:06:16 2013 taraNotesPMCPMC Longitudinal Testing and Modal Analysis

Quote:

I think the analytical formula in terms of rho is going to be (1.57/2*pi) * sqrt(E / rho * L^2), since the Roark formula is (1.57/2*pi) * sqrt(A* E * g / w * L^2) and the weight per unit length is w = m * g / L = rho * A * g. With your values for L, A, E, and rho, this gives f1 = 16 kHz. Since A does not appear in the analytical formula, this also explains why changing the area in the Comsol model doesn't change the frequency.

 

 

 good catch! Thanks. Then both analytical and FEA results are the same. So our COMSOL results for PMC should be valid, the first body for a stainless steel PMC, see psl:1131,at 16 kHz is reasonable.

  1137   Mon Apr 1 20:59:19 2013 taraNotesPMCPMC Longitudinal Testing and Modal Analysis

I calculated some requirement for the beam jitter at the output of the PMC. A rough estimate shows that we need the angular stability at the PMC about half nano radian so that the frequency noise of the beam locked to the refcav is less than 10-2 Hz/rtHz.

==Background==

PMC also reduces beam jitters from the laser, so that the beam alignment to the cavity is kept centered. Since the laser is locked to the reference cavity, any misalignment of the input beam will cause the beam to sense the change of the cavity length.

So vibration that shakes the PMC will change the alignment of the output beam. With stiff material, the seismic induced deformation of the PMC will be reduced.

==Calculation==

  • calculate the ray tracing matrices from the PMC to the cavity. I assume that only the angle of the output beam changes due to PMC sagging, because of a long distance from the PMC to the refcav, with several mirrors in between. This gives me the position and the angle of the beam going to the cavity.
  • find out what is the change of the cavity length (dL), when the input beam is translated by dx, with angle theta.
  • convert displacement noise to frequency nosie (dL -> df), as a rough estimate I choose the requirement for df to be less than 10-2 Hz/rtHz (about the level of the estimated coating noise). This step is not really necessary, but I feel that it is easier to compare the noise in Hz/rtHz unit rather than m/rtHz.
  • The required angular stability at the PMC is ~ 0.5 nano rad. This number seems to be too strict. I will double check it.

==next==

Eavn is working on COMSOL to find out the angular tilt of the output beam due to PMC sagging. Optimum support points will be determined to minimize beam jitter due to seismic.

  1139   Thu Apr 4 18:45:38 2013 taraNotesDAQTemperature control updated

[peter, tara]The temperature servo for the chamber is back on, the current setup is at 31.2 C.

There was a problem with C3:PSL-VAC_CHAMBERTEMP channel, and I could not run the script for temperature control of the chamber. Peter helped me figure out what happened. It turned out that one of the parenthesis in the database file (cavities.db) was missing due to an accidental delete, and the name of the channel was too long (it was working before, I don't know why).

Anyway, the channel was renamed to C3:PSL-VAC_CHMBRTMP, in 1)cavities.db, 2)rcav_PID_2012_06_15.pl, and 3) medm screen for controlling the servo. The temperature servo is working again.

Quote:

We switched the temperature readout channels used for temp feedback control to improve the signal. The new signal is significantly smoother.

      The signals from 4 thermostats around the vacuum chamber were acquired through 4 channels, C3:PSL-RCAV_SENSE(1-4). These channels were then connected to DAQ. This made the signal noisy because the resolution of analog to digital converter was low. In order to fix that we use an analog circuit to sum and average the signals from 4 sensors then amplify it before sending to DAQ,C3:PSL-RCAV_TEMP, then calibrated it to C3:PSL-VAC_CHAMBERTEMP by comparing RCAV_TEMP [V] to RCAV_TEMPAVG[C] which is calibrated to deg C already. 

      CHAMBERTEMP = (RCAV_TEMPx-0.495) + 34.957

     

     We corrected the perl script (in SUN machine) used for thermal feedback on the heater jacket. Now the script is named rcav_PID_2012_06_15.pl, see wiki. The servo is now back on.

 

  1142   Fri Apr 5 13:56:01 2013 taraNotesRefCavestimated beat frequency

Note:A test to check which cavity needs to be heated up.

1) when heaters (on the shields) are off, C3:PSL-VAC_CHMBRTMP =31.2

  • CAV1 is locked @ (slow out Coarse/fine) = 774/500
  • CAV2                 @ 769/500

2 when heater on CAV2(4V), C3:PSL-VAC_CHMBRTMP = 31.2

  • CAV1             @ 775/500
  • CAV2              @ 756/500

So to bring both cavity to be resonant at the same time, the heater on cav1 should be on.

I'm not sure if the thermometers on the shields are working or not, I'll check them.

Quote:

At current temperature, the estimated beat frequency will be ~ 60-100MHz. This is not so bad, since we can use 1811 to measure the beat signal and use PLL to extract the beat noise.

 

We will need to use thermal expansion to tune the beat frequency. So, as a start, I try to figure out the beat frequency, and how much we have to heat up the cavity. The heaters on each cavity is off, only the heater around the vac chamber is on (but the servo is off). 

Right now we have one laser locked to one cavity, but the beam path to the first cavity has a beam splitter that we can borrow the beam and direct it to the 2nd cavity. I realigned the beam to have both beams into both cavities. By adjusting the temperature control on the NPRO (slow signal), I can bring the beam to resonant in each cavity.

1st cav is resonant @ (334/398) and (150/398). The numbers correspond to coarse and fine knobs of the slow feedback to the laser.

2nd cav is resonant @ (154/398). (I'll come up with a better name to call the cavities)

The FSR is 4.07 GHz (for 1.45" long cavity). This means 334-154 = 184 clicks on the coarse knob equals to 4.07 GHz, or 22MHz per coarse click. Both cavities resonant at ~3-5 clicks apart. So the beat frequency is ~ 60-100MHz. This is quite good, at least we are not close to half FSR apart. The power required to tune the cavity length should not be that high.

 

The next thing to do is try to see which cavity we need to heat up in order to bring both cavities resonant frequency closer together.

 

  1146   Tue Apr 9 15:39:16 2013 taraNotesPMCAbout PMCs

Considerations for PMC design:

  1. Stiffness(Acoustic susceptibility) & heavy material: With heavier material, the pmc motion on the support becomes smaller.(RXA: please quantify with a formula)
  2. Filtering factor (Finesse/FSR/Cavity pole), g-factor: Filter out intensity noise around 10 MHz (RXA: please quantify with a formula)
  3. Design for thermal expansion cancellation between the spacer and the end cap: So that the PMC is less sensitive to ambient temperature
  4.  3 or 4 mirrors?  3 is polarization selective. For general lab use with power less than 1 W,  3 mirror design should be good. (RXA: I don't follow this logic at all)

RXA: In general, all of these considerations need some sort of quantitative detail. Make a DeBra Matrix so that we can evaluate. 

  1149   Mon Apr 15 10:59:35 2013 taraNotesPMCDebra matrix for PMC design

Quote:

Considerations for PMC design:

  1. Stiffness(Acoustic susceptibility) & heavy material: With heavier material, the pmc motion on the support becomes smaller.(RXA: please quantify with a formula)
  2. Filtering factor (Finesse/FSR/Cavity pole), g-factor: Filter out intensity noise around 10 MHz (RXA: please quantify with a formula)
  3. Design for thermal expansion cancellation between the spacer and the end cap: So that the PMC is less sensitive to ambient temperature
  4.  3 or 4 mirrors?  3 is polarization selective. For general lab use with power less than 1 W,  3 mirror design should be good. (RXA: I don't follow this logic at all)

RXA: In general, all of these considerations need some sort of quantitative detail. Make a DeBra Matrix so that we can evaluate. 

 Some requirements for the PMC:

==Cavity pole==

 For intensity filtering. The modulation frequencies for the refcavs is ~ 15-25 MHz, we want the intensity fluctuation at this frequency to be shot noise limited.  We have to determine what should be the frequency pole. Intensity noise around 1MHz - 30MHz will be ~ 1/f^2, see the paper by Harb etal, eq1 and fig9, get the paper from psl:1156. Under the assumption that RIN remains constant, at 20MHz the laser will already by shot noise limited (@ 1mW input).  laser intensity noise / shot noise ~ 0.16. (laser intensity noise here means intensity noise from spontaneous emission/ pump-source intensity noise/ dipole fluctuation noise/ noise from intra cavity losses, any thing except shot noise)

laser_rin.jpg

  Thie pole can change with the cavity length and Finesse, [ Finesse = FSR/(2*cavity Pole)] , so our choices for mirror reflectivity, cavity length will affect this number as well. So for a fixed set of mirrors (fixed finesse), longer perimeter means lower cavity pole, but the cavity will be more susceptible to acoustic coupling.

==First longitudinal body mode==

  It should be at high frequency ( for high UGF servo). The shorter the length, the higher the frequency. See PSL:1134.

== g-factor==

 For a stable cavity, g factor has to be between 0 and 1.  Another reason: We should choose g-factor such that HOMs do not coincide with other cavity axial modes (FSR apart). For a ring cavity with 2 curve mirror R1,and R2, g = (1- p/R1) x (1 - p/R2) where p is the round trip length. (For 3-mirror cavity, g = (1 - p/(R))^2 . See HOM calculation.

==Stiffness==

 we want a solid, bulk shape PMC, not thin long one. This will make the PMC less susceptible to acoustic noise.

==Higher order mode suppression== 

Other transverse modes will be suppressed by a factor of (1-r)^2 / (1 +r^2 -2rcos(2*pi* dfmn/ FSR)  where dfmn is the gouy phase shift of m+n mode, r =r1*r2*r3.. (reflectivity of each mirror in the cavity) see evan's note. Transverse modes of the output of the NPRO can be found by scanning the PMC and measure the transmitted beam. Other modes beside TEM00, will be reflected back from the refcav and incident on the RFPD. This will cause the mode mismatch and increase shot noise level. Usually, higher r (higher Finesse), will suppress more HOMs.

==Build up power==:

= Pin x Finesse/ pi. CVI mirrorsfor high damage threshold power have maximum power for cw around 10MW/cm2. So I use this number as an upper limit for the power threshold. Assuming the power input is ~ 30 mW, average spotsize is 350 um. This gives ~ 8W/cm2. So Finesse can be up to ~ 3e6.  (10 MW/cm2 > (Finesse/pi) x 8 W/cm2) .

 

Some assumptions:

  • Losses(scatter/absorption) on each mirror is ~ 100 ppm. It seems that a super polished mirrors in vacuum has ~ 10 ppm loss. This comes from a Finesse measurement of the previous 8" refcavs, see psl:1046. The calculation shows that loss in one cavity is 25 ppm (for 2 mirrors), and 160ppm for another cavity. Since the PMC mirrors will be in air, and probably not as good as refcav mirrors, dust in air might accumulate over time and causes extra loss on the mirrors, 100 ppm loss assumption might be ok for this calculation.
  • PZT range is about 15um @1000V, as shown in the catalog, see PSL 1052 for the details, (we can drive it with ~0-300 V, so ~ 4um displacement),see PSL:1052

 

Let's see some of the designs that are available. Then we can decide which one we should modify to suit our requirement.

  1. Design1 iLIGO PMC: Isosceles triangular PMC, fused silica, perimeter = 0.42m, flat-curve (1m ROC)-flat mirrors. Round Trip = 0.42m See T-080195,here (it says the pole is 7 MHz).
  2. Design2 (Dmass' PMC): stainless steel PMC, perimeter =0.4m , same mirrors as those of design1, so its finesse is the same.
  3. Design3, AdvLIGO PMC style (4 mirrors, bow-tie): stainless steel (see PSL:)
  Cavity pole /FSR/ Finesse g-factor Stiffness  1st Longitudinal body mode  Approximate dimension(height x width  x length)  Note
Design1  cav pole = 7MHz / FSR=714MHz / Finesse =50 0.34    14 kHz  2" x 2.4" x 7.1"  The values are for p-pol, waist radius = 370um.
Design2  cav pole =  9MHz  /FSR = 925MHz / Finesse = 50  0.46    16.6kHz [PSL:1134]  2 x 2.6 x 6  assuming similar mirrors from design 1, w0 = 353 um.
Design3            
             
             
             

 

  1151   Thu Apr 18 19:54:01 2013 taraNotesDAQEPICS channel for slow feedback to laser

I created a channel for feedback to slow DC to the laser head. The servo will be done digitally using a perl script similar to what we have for the vacuum can.

 

There are unused channels for temperature monitor, so I modify them for FAST MON instead.

In the database file "cavities.db" in the sun machine, I changed [C3:PSL-BOX_SENS1] to [C3:PSL-ACAV_FMON]. For input +/- 10 V.

The next thing is to create perl scripts for the servos. Then find an output channel for feeding back to the laser.

 I made an medm screen for controlling the slow feedback signals to both lasers.

C3PSL_LASER_SLOW.png

 Channels that will be created are:

input

  • C3:PSL-ACAV_FMON

soft channels

  • C3:PSL-ACAV_PID_KP
  • C3:PSL-ACAV_PID_KI
  • C3:PSL-ACAV_PID_KD

 output

  • C3:PSL-ACAV_SLOWOUT
  1155   Mon Apr 22 17:13:24 2013 taraNotesDAQslow feedback to laser via EPICS is on

The medm screen for the 1st laser is completed, the servo is on an stable. Refcav has been locked for a few hours as of now.

  • The output for slow feedback is on J9, slot 9/10 (VMIVME 4116 C2 S3). This is an unused channel previously assigned to PSL-ISS_ISET, I checked ISS.db file to look for the VMIVME address. For the slot number, I look up the channel name in D980535-C-C document.
  • I added low pass filters (~100mHz) to both input and outputs of EPICS.
  • All EPICS channels for slow feedback and perl scripts are in SLOW_LASER.db file.
  • The startup.cmd file is updated accordingly.
  • servo gain is optimized. ??? What does that mean??? How about some performance plots? (About the bode plot, I'm trying to get a transfer function of the NPRO slow input, with that I can estimate the bode plot of the loop. As of now I just adjust the PID gain so that the loop is stable)

KP.jpg

fig1: FAST feedback to the laser is shown in blue plot, vertical axis:1V/div, horizontal axis:  4 sec/ div. I adjusted proportional gain first, to get only a few overshoots with acceptable rise time.

KI.jpg

fig2: Then I adjusted integral gain to eliminate the offset, and Derivative gain to reduce overshoot. More about PID gain can be found here. Current Value KP = -0.0002, KI =-0.00015, KD = 0.

I set the output to be between -2 V to 9 V. Since we need to lock it to GYRO later, it has to be able to be tuned to match the gyro laser. Currently, Gyro laser is operated around 35 Deg C which is similar to 8V input to slow feedback.

 

I'm trying to draw a cartoon for DAQ wiring in CTN lab for future reference. This is what I have so far. I'll add it in WIKI page.

CTN_DAQ_wiring.png

  1157   Tue Apr 23 18:43:39 2013 taraDailyProgressRefCavHOM for new sideband frequencies

35.5 MHz and 38MHz sideband frequencies are chosen for  1.45 " refcav. These frequencies will be suitable for cavities formed by 0.5/0.5m RoC mirrors and 1.0/1.0m RoC mirrors.

 

HOM35.5MHz0.5m.png                           HOM38MHz0.5m.png

  a) For0.5m/0.5m RoC mirrors 1.45" cavity, f1 = 35.5MHz.           b) For0.5m/0.5m RoC mirrors 1.45" cavity, f2 = 38MHz

HOM35.5MHz1m.png                                HOM38MHz1m.png

 c) For 1m/1m RoC mirrors 1.45" cavity, f1 = 35.5MHz                d) For 1m/1m RoC mirrors 1.45" cavity, f2 = 38MHz

 

Since we will use a crystal oscillator to drive the EOMs, I have to check how much power we need for the sideband.

If the crystal oscillator can provide us with enough power, we can use the crystal to drive a broadband EOM directly. Otherwise we will need an EOM driver, or a resonant EOM.

 

==shot noise level vs mod index(Beta)==

shot_vs_beta.png

To see how much should the mod index be, I plot shot noise level vs Beta, with Power intpu = 1mW and 2 mW, and Finesse = 1e5 (for  T=300 ppm mirrors)and 2e5 (For AlAs/GaAs coatings), with mode match = 80%. It seems that for the lowest shot noise level, we need beta = 0.8.

For resonant EOM, mod depth = 0.2 rad/V, for BB EOM, mode depth = 15mrad/V , see psl:745.  These correspond to 4V (25dBm) and 53 V (47dBm) for the resonant and BB EOMs, respectively.

 

  1158   Tue Apr 23 22:28:53 2013 taraNotesTempCtrltemp sensor on heat shields

I checked both temp sensors on the heat shields. They are working. I can see the change in resistance when I the heater is on. It seems to be a wiring problem. I'm investigating it.

  1159   Thu Apr 25 10:57:52 2013 taraNotesPurchasesPomona BNC cable

I plan to order RG58 bnc cables from Pomona, here is a list of what I need

catalog

  1. Cables for Fast monitor (from TTFSS to electronic shelf) (~15ft , x2)
  2. For slow feedback (from the shelf to the laser controllers) (~20ft x2)
  3. For EOM temp control feedback (I'm not sure where the nim crate will be, this will be decided soon).
  1163   Tue Apr 30 01:15:26 2013 taraNotesNoiseBudgetAlAs/GaAs layer structure optimized for TO

I'm computing  coating Brownian and thermo optic noise (TO)  in AlGaAs coatings using GWINC code to compare it with the result reported by G Cole etal. Brownian noise from my result is similar to theirs, but TO noise is still not correct.  I'm working on it.

==Background==

We have talked about what kind optimizations should we go for AlGaAs coatings in order to minimize TO noise. There are two choices for us to consider

  1. using the layer structure as proposed in T1200003, or
  2.  adding a half wavelength cap on top of the quarter wave stack coatings as suggested by the authors.

Since the second option is more desirable in terms of manufacturing because of its simplicity, I decided to check if it really can  bring TO noise below coating Brownian noise. If it is true, we can use it for our mirrors.

 ==calculation==

  • I use GWINC code for TO noise and brownian noise calculations to verify the result if they are agree or not.
  • Materials parameters used in the calculation are taken from the paper. But most of the coatings material properties of an individual layer of AlxGa1-xAs are not provided. There are only the average values of thermal expansion, heat capacity, thermal conductivity, dn/dT.  There are refractive indices (nh/nl = 3.48/2.977) and layer structure (81 layers, starts with nh, ends with nh). So, as a start, the values for high index material and low index material are the same as the averaged values.

==result==

  • My Coating thermal noise level is 8.4*10^-35 m^2/Hz while their result is 9.8 *10^-35 m^2/Hz, @ 1Hz. This is not very bad, since there are some differences in the formulas between GWINC and their calcualtion.
  • However, my TO is off the roof, almost 2 orders of magnitude above their result. I'm checking if it is because of the code is wrong(typo in the parameters) or the fact that I used all the averaged values.

(I'll add more details about the calculations later)

  1165   Wed May 1 01:45:55 2013 taraNotesPMCDebra matrix for PMC design

Considerations for PMC design is corrected and updated

  1168   Thu May 2 03:03:48 2013 taraNotesNoiseBudgetAlAs/GaAs layer structure optimized for TO

[matt, tara] Got AlxGa1-xAs material parameters from Matt Abernathy. I plug the numbers (all in SI) in GWINC, but the result is still not quite similar to that in Cole etal paper.

 ioffe has materials parameters for TO noise calculation.

Specific heat: 0.33+0.12x J/gK
 Mass density rho = 5.3165-1.5875x g/cm^3
Thermal conductivity,kappa: 0.55-2.12x+2.48x^2 W/cmK  (There is also thermal diffusivity = kapp/(rho*specific heat) [m^2/s]. The results are the same)
Thermal Expansion: (5.73-0.53x)·10-6/K

dn/dT: 3.66-2.03x *10^-4/K
This is from a paper, "Thermal dependence of the refractive index of GaAs and AlAs measured using semiconductor multilayer optical cavities", by Talghader and Smith. Keep in mind that this paper has an important Erratum if you want use values from it.
Unfortunately, this paper measures dn/dT at a max wavelength of 1030nm, so it's not quite accurate, but probably good enough.

Note:

One of the variables in GWINC code is ThermalDiffusivity. But the numbers used in previous TO plot is thermal conductivity of materials. I'll check the TO calculation codes and see if it is just a naming error, or the calculation is actually wrong.

  1169   Thu May 2 23:40:46 2013 taraNotesNoiseBudgetAlAs/GaAs layer structure optimized for TO

I used GWINC code to calculate TO noise in AlGaAs coatings, with some modifications to the code I can get the result that is comparable to Cole etal's result. However, there seems to be some minor details that I have to check. The half wave cap solutions for TO cancellation is not verified by the current calculation yet.

 

 What I modified and checked in the code:

  • The variable called thermaldiffusivity in the code is actually treated as thermal conductivity in the calculation, so all calculations in the past are still correct.
  • The layer structure in the code was originally for Ta2O5/SiO2. The first layer started with SiO2 (low index material,nL), and ended with Ta2O5(high index material,nH) at the substrate surface. However, the AlGaAs coatings start with nH and ends with nH. I changed the calculation for effective thermal expansion accordingly. With the correct layer structure and materials parameters from Matt, the TO nosie is closer to JILA's result. However, the shape is still not the same, what reported in JILA is almost flat across 1-100 Hz. The calculated transmission from the layers is 1.8 ppm, but the paper says 4ppm. I'm looking into this.

RefCav_AlGaAs_TOnoise.png

cole.png

Above figures: top plot is the result from GWINC. Its title should be Al0.92Ga0.08As coatings, not SiO2/Ta2O5, bottom picture is taken from Cole, etal. TO noise crosses coating brownian noise around 3 Hz for both plots, however the slope is very different. NOTE: the y axes are in Hz^2 / Hz.

As a quick check for the proposed half wavelength cap solution to reduce TO noise, I modified the layer structure and computed TO noise. Since they did not mention what kind of material for the cap I tried:

  1. 81 layers, starts with nH, ends with nH, the first layer is 0.5 lambda thick. This is not working.RefCav_AlGaAs_TOnoise.png
  2. 82 layers, starts wit half wave nL, followed by the original 81 layers. This also does not work. Both cases have comparable TO noise, but transmissions are different. RefCav_AlGaAs_TOnoise.png

I'll check their formula and GWINC to see where the differences are.

 

Quote:

[matt, tara] Got AlxGa1-xAs material parameters from Matt Abernathy. I plug the numbers (all in SI) in GWINC, but the result is still not quite similar to that in Cole etal paper.

 ioffe has materials parameters for TO noise calculation.

Specific heat: 0.33+0.12x J/gK
 Mass density rho = 5.3165-1.5875x g/cm^3
Thermal conductivity,kappa: 0.55-2.12x+2.48x^2 W/cmK  (There is also thermal diffusivity = kapp/(rho*specific heat) [m^2/s]. The results are the same)
Thermal Expansion: (5.73-0.53x)·10-6/K

dn/dT: 3.66-2.03x *10^-4/K
This is from a paper, "Thermal dependence of the refractive index of GaAs and AlAs measured using semiconductor multilayer optical cavities", by Talghader and Smith. Keep in mind that this paper has an important Erratum if you want use values from it.
Unfortunately, this paper measures dn/dT at a max wavelength of 1030nm, so it's not quite accurate, but probably good enough.

Note:

One of the variables in GWINC code is ThermalDiffusivity. But the numbers used in previous TO plot is thermal conductivity of materials. I'll check the TO calculation codes and see if it is just a naming error, or the calculation is actually wrong.

 

  1170   Mon May 6 03:11:44 2013 taraNotesNoiseBudgetAlAs/GaAs layer structure optimized for TO

I checked the calculation for TO noise in Cole etal people and found a few problems that I didn't understand.

  • In the paper, they have two solutions for TO-noise, at low and high frequency. The solution for high frequency is similar to that in Evan etal paper, but I'm not sure where the solution for low frequency are from.  I don't see this kind of calculation in Evans etal paper.
  •  I repeated and plotted the TO noise calculation as used in Cole etal's paper. The TO noise plotted in their paper mostly came from the low frequency part.
  • Some parameters reported in the paper might not be accurate, for example their beam radius is 250 um. However, with their 35mm spacer, 1.0 m RoC mirrors, the spotradius on the mirror should be 212 um. I haven't checked how much their materials parameters and what I used in my codes differ.
  • For low frequency solution (solid blue line), with the materials parameter given in the paper, it is a factor of 1.5 higher than their result (I got 3e-3, they report ~2 e-3 around 1-10 Hz).
  • For high frequency solution (solid yellow line), with the materials parameters given in the paper, the result is about a factor of 10 higher than that from Gwinc code (dashed blue line). The formulas are the same, but I used different material parameters. The two lines should be close, but they are a factor of 10 apart, just because of the material parameters. We should really make sure that the numbers are correct. Before trying to do the optimization.

 

RefCav_AlGaAs_TOnoise.png

 

My GWINC code for TO calculation can be found here. (other modified functions are in /GwincDev/ ).The main code is plotTO_algaas.m. This code uses getCoatThermoOpticsAGS.m which calls out other other functions in /gwincdev/

  1. getcoatTOposAGS.m (calculated effective alpha and beta in coatings.) This function uses getcoatLayers.m to generate the layer structure. The original one started with nL, I modified it to start with nH, and end with nH.
  2. getcoatThickCorrAGS.m, which computes the correction factor (gamma TO).
  3. getcoatavgAGS.m, this code compute the average material parameters in coatings.
  4. in /coating/AlGaAs_Refcav, I created a database file for material parameters called algaasmodels.m.

 

  1172   Wed May 8 01:11:09 2013 taraNotesDAQslow feedback to laser via EPICS is on

Slow feedback for 2nd laser is ready.

EPICs channel:

  • C3:PSL-RCAV-FMON was created for fast mon to laser.
  • C3:PSL-RCAV_SLOWOUT was created for SLOW feedback. The channel was originally named C3:PSL-FSS_VCOMODLEVEL J9 input 11 and 12, VMIVME-4116, C2 S4.

The output of EPICS channels have capacitors installed in parallel for low pass filter.

 

 

  1173   Fri May 10 01:24:01 2013 taraNotesTempCtrltemp sensor on heat shields

AD590s on both thermal shields are not working. I was wrong when I checked them at the first time.

The temp sensors in the vacuum tank for monitoring temperature on heat shields are wired as shown in the picture. The resistor,R, is 30k ohms. According the the datasheet, the current from AD590 should be ~ 300uA, (30kx300uA = 9V). But what I read from the voltage across the readout R was 20V which was over the input range of EPICS (+/-10V). This happened on both of them. I compared the readout with a left over AD590, and got ~ 9.3 V readout which was expected at room temp.

At first I thought it might still be working linearly and useable if I just switched to lower R. However, with R=12 k, the readout voltage was 18V (I expected 20x(12/30) =8V). So certainly, this is not working.

I think the reasons they are broken is that they were overheated when I soldered them. I tried to be careful, but, apparently, that was not enough.

I'll check if there are spare AD590s in the lab or not, otherwise I'll order some more.

  1176   Tue May 14 02:06:15 2013 taraNotesNoiseBudgetAlAs/GaAs layer structure optimized for TO

[matt,tara]  We compared the TO result using GWINC, our results are similar (see PSL:1170). However, it still not agrees with result in Cole etal paper.

The result from GWINC and Cole etal's result are different in the following ways:

  • TO noise from GWINC is higher their result. This might be due to different values of the effective alpha, and effective beta in the calculation. We will check this next.
  • The calculated transmission for 81 layers is ~ 1.8 ppm, while they reported 10ppm. We are not sure what happen here.
  •  Half wave cap solution for TO noise cancellation is not shown in GWINC.
  • Thermal fluctuation as observed by a Guassian beam, SdTTO = const x kBT^2/ r0^2 sqrt(kappa x heat cap x 2*pi*f) depends on substrate parameters in GWINC, but their result use coatings' parameters. With coatings parameters, the thermal fluctuation will be lower, thus lower TO noise. It means that our TO result should be larger by an order of magnitude. However the results are about the same.   We think that the subsrate parameters should be used in the calculation, because thermal length in the coatings from dc up to 170kHz is smaller than the coatings thickness (~6 um).
  • The calculation in GWINC assumes adiabatic assumption. However, the assumption breaks down at 270 Hz for AlGaAs coatings, and 6 Hz in substrate. That explains why the TO noise in Cole paper is almost flat from DC to 100 Hz. Mike Martin's thesis explains the TO noise at all frequency, but I haven't yet quite understood all the equations.

 

  1177   Wed May 15 20:07:12 2013 taraNotesNoiseBudgetAlAs/GaAs layer structure optimized for TO

We checked the half wave cap solution for  minimizing TO noise. WIth a half wave cap of nl, the TO noise is smaller by ~ a factor of 2 in Hz^2/Hz unit.

Matt and I checked the  calculate the TO noise for a half wave cap solution. The noise goes down by a factor of 2. 

A few issues that we still have to investigate:

  • A thin layer of nh: we talked to Mike, he said that to prevent the oxidation that occurs on GaAs layer (nL), a thin layer of AlGaAs(nH) has to be applied on top. We are not sure how thick the layer will be, we should ask G Cole, so that we can estimated the effect before hand.
  • The TO noise with  half wave cap may already be lower than substrate thermoelastic (TE) noise. I'm checking the TE calculation and find out that the value for thermal expansion of fused silica is 3.9e-7 in Gwinc, but 5.5e-7 elsewhere (add sources). If it is really 5.5e-7, this will be higher than the current TO noise already. I'll look into it.
  • A factor of 2 : This comes from double sided PSD or either 2 mirrors. I'll change that to our standard here (1-sided PSD, with single mirror).
  •  The cancellation might change for different numbers of doublet. Since we plan to have ~ 100-200 ppm, the actual TO noise may be different than this calculation (2ppm). I try using 56 layers (1/2 lambda cap of nL included) which give us 100ppm, and TO noise is below coating brownian from DC to 200 Hz. This is a pretty good result which should be expected. Since we reduce the number of doublet, the effect from TE becomes smaller, (still larger than TR). Thus the different between the two (the total TO noise) is smaller.
  • Different cap thickness may bring down TO noise more than half wave cap does. I just try the cap with 0.1 wavelength of nL (for 40 doublet stack), and TO noise goes down by another factor of 2. This might apply for 56 stack as well. I'll check.

TO_compare_cap.png

  1178   Tue May 21 01:06:43 2013 taraNotesNoiseBudgetAlAs/GaAs layer structure optimized for TO

I checked all the discrepancies in the calculations between GWINC and that of Cole. The issues are almost cleared, only the value of effective beta, (dn/dT) that still remains.

  The PSD of TO noise in [m^2/Hz] is given by Sx(f)= ST (f) x (dTE + dTR).  See Evans etal Phys Rev D 78, 102003.  Where:

  • ST (f) = Temperature fluctuation as sensed by a Gaussian beam
  • dTE = dx/dT, or rate of change of mirror position with respect to temperature change due to thermoelastic mechanism.
  • dTR = rate of change of mirror position with respect to temp change due to thermo-refractive mechanism

ST(f) can be calculated analytically, see BGV, Phys Lett A 271 , (2000) 303-307 eq9, this also assumes adiabatic approximation. In Mike Martin's thesis, the temp fluctuation is generalized  to all frequency (by contour integral, I'll show the details later). The parameters for calculating ST(f) are taken from that of substrate (in GWINC), but Cole's paper and Mike's thesis use that of the coatings. That makes Cole's result about a factor of 7 higher than that from GWINC. Matt and I discussed this with Mike, he thought that the calculation should use the substrate's properties since the thermal length in the frequency of interest is much larger than the coating thickness.

The issue with which parameters should be used might be a less serious problem if (dTE + dTR) can partially cancel out making the whole TO noise much smaller. Basically dTE is ~ alpha* coatings thickness, where alpha is the thermal expansion coefficient of the coatings. dTR is ~ beta_eff * lambda.  The calculations for dTE from GWINC and Cole are about the same (1.1 x 10^-10) [m/K], where the effective beta are different by about and order of magnitude. Cole reports the value of beta effective to be -5.5 x10^-10 , meanwhile GWINC gives me 0.5x10^-10.

This means that the TE and TR,as calculated from GWINC are more comparable, and the TO result is reduced significantly. While the TO result from Cole is mostly TR. I calculated the TR following the 1/4 stack approximation in Evans paper and got the same result as in Cole. I'm checking what happen in GWINC code for TR calculation.

  1180   Wed May 22 00:04:48 2013 taraNotesNoiseBudgetAlAs/GaAs layer structure optimized for TO

I found out why the calculated values of the coatings' effective beta from GWINC and Cole etal paper are different. The order of Low/High refractive index material have something to do with the beta effective calculation.

Here are some facts about the coatings and calculation:

  • The AlGaAs coatings used in the paper have no 1/2 wave cap. The structure is consisted of only 1/4 wave layers. Start with nH on top, and end with nH at the substrate (SiO2).
  •  The PSD from thermo-refractive is SdT x Beta_eff x lambda. Where SdT is temperature fluctuation, Beta_effective is the overall dn/dT of the coatings,see the entry below for more details.
  • In GWINC, Beta_eff is calculated numerically, taking each layer and calculating the reflectivity, then sum all the effect together. The result for Beta_eff is different, if the first layer (the top one) is changed between nH or nL. ( 5e-5 and 5e-4, cf PSL1178).
  • The approximated Beta_eff, for 1/4 high reflective coatings, which is reported in BGV 2000, and Evans 2008 is given by B_eff ~ (nH^2 *BL + nL^2*BH) / (nH^2 - nL^2) (which was used in Cole's paper to calculated their TO noise). BGV gave a sketch of this calculation in their paper (which I have not yet thoroughly understood). One problem is that, the result for B_eff obtained from this formula is the same whether the coatings start with nH or nL. This should be wrong, since most of the TR effect comes from the very first layers. The order of nH/nL should matter.
  • Computed values of B_eff from Gwinc code and the simplified formula agree if both start with nL. This makes me think that there is some assumptions in the simplified B_eff formula that the first layer is nL (which is customary, in SiO2/Ta2O5 coatings ).

So, I believe that the calculation for TO noise I have right now is correct. And for 100 ppm transmission (56 layers) with 1/2 wave cap, the TO noise is significantly reduced (add plot).  We should be able to finalize what we want for the AlGaAs mirrors soon.

  1181   Fri May 24 04:04:58 2013 taraNotesNoiseBudgetAlAs/GaAs layer structure optimized for TO

Quote:

  • The approximated Beta_eff, for 1/4 high reflective coatings, which is reported in BGV 2000, and Evans 2008 is given by B_eff ~ (nH^2 *BL + nL^2*BH) / (nH^2 - nL^2) (which was used in Cole's paper to calculated their TO noise). BGV gave a sketch of this calculation in their paper (which I have not yet thoroughly understood). One problem is that, the result for B_eff obtained from this formula is the same whether the coatings start with nH or nL. This should be wrong, since most of the TR effect comes from the very first layers. The order of nH/nL should matter.

 Beff ~ (nH^2 *BL + nL^2*BH) / (nH^2 - nL^2) is valid only if the top layer is 1/4 layer of nL, [Gorodetsky, Phys Lett A 372 (2008)].  The complete calculation for general case is given in the reference. If the layer starts with nH, beta eff is = (BetaH + BetaL) / (4x(nH^2 - nL^2) ).  So, GWINC and analytical approximation agree, Yay! .

The effective beta reported in Cole's paper is 5e-4, but it should be ~ 5e-5 for coatings start with nH. The real thermo optic noise for their setup will be lower ( because TE is about the same level as TR). Their real TO noise should be a factor of 5.5 below the reported one (in Hz^2/Hz unit).

Note: There are still issues about the thermal fluctuation and the cut off frequency. These will greatly change the shape of the TO noise and the total noise level. I'm still investigating it.

The 1/2 wavelength cap with nL does reduce the TO noise. But we need to know exactly how thick the nH film on top will be, so the real TO effect can be estimated accurately.

 

 

 

 

  1183   Fri May 24 23:57:27 2013 taraNotesNoiseBudgetAlAs/GaAs layer structure optimized for TO

 

 The noise budgets below show noise from coating brownian, TO noise and TE in substrate. The three plots are from 52,54 and 56 Layer coatings.

 All the designs have 1/2 cap of nL, with nH ending on the substrate surface.  There are no significant differences in the noise level at low frequency, since TE noise in substrate starts to dominate. I used the substrate

parameters in thermal fluctuations, so the cut off frequency for TO calculation is low (~ 3 Hz instead of ~ 200 Hz). The design can go for 56 layers.

52lay.png

54Lay.png

56lay2.png

 

I'm thinking about another solution, where the top layer is nH, followed by 1/4 layers. If the first nH is 1/8 lambda thick, TO can be cancelled nicely (for 56Layer + nH cap). The transmission is 140 ppm , which is in the chosen range (100-200ppm). But I feel that the 1/8 cap is not good for a high reflectivity mirror, since the phase of the reflected light within  that layer is not really inphase or out of face with the light reflected at the air surface. I'll think about it more to see if it would be a good solution or not.

 

  1187   Fri May 31 13:21:37 2013 taraDailyProgressComputersTemporary south refcav autolocker

Quote:

 Special bonus settings: the common gain on the TTFSS is 404 clicks, the fast gain is 426 clicks, and the offset is 967 clicks. Tara pointed out that the frequency loop would catch lock easier if the gain settings were around these low values rather that what they were previously (~600).

 The offset should be ~ 500. I turned it back down.

  1190   Tue Jun 4 18:02:26 2013 taraDailyProgressRefCavbeam directed to 2nd refcav

The beam is sent to 2nd cavity (RCAV). The beam is mode match roughly, since there is no PMC, the exact beam size is hard to measure. The laser resonance when RCAV_SLOWOUT @ 0.2383V. There is enough transmitted beam for alignment the beat setup behind the cavity.

  1191   Wed Jun 5 22:25:28 2013 taraNotesNoiseBudgetTO calculation review

Since we have to review the calculation for Thermo-Optic noise (TO), I'll sketch an outline and some remarks here.

==TO noise overview==

To calculate TO noise, we have to calculate temperature fluctuations, then multiply by Thermoelastic (TE) and Thermorefractive (TR) coefficients to convert temperature fluctuation to displacement noise. Usually, in the frequency of interest, thermal length is much larger than coating thickness. Thermal fluctuations in coatings are uniform making the whole coatings expand/ contract uniformly. This assumption is important for cancellation between TE and TR. As TE effect comes from the whole coating thickness, while TR comes from only the first few layers (most of the power is reflected from these top layers). Modifying the first few layers can change TR effect significantly.

 ==Temperature fluctuations==

can be obtained from direct method (Levin 2008), by injecting heat with Gaussian beam profile. Example are done in Levin 2008, Evans etal 2008.

A few issues about these calculations:

  • heat flow in 1-D, under the assumption that temperature gradient is mostly in z direction coating thickness d << thermal length << beam radius. Where thermal length is ~ sqrt (  kappa/ (rho*C* 2pif) )  This is not true for AlGaAs coatings where kappa is ~ 60 W/mK which gives thermal length to be~ 2370 um  [sqrt (1Hz/f)], beam radius is ~ 200um.  Cerdonio 2001, and Mike Martin's thesis have the calculation in 3-D, however, heat diffusion in coatings is not taken into account.
  • Heat diffusion in coatings, is done in Fejer 2004, Somiya2009. (It is ignored in BGV1999/Liuthorne2000/cerdonio 2001)

At this point, I think Somiya paper is very good for us to look through. The calculation includes TE and TR. However, I don't quite get it yet. The calculation solve heat equation in 1-D, but has results for finite test mass. I need to spend more time on the paper.

 Heinert 2011, has calculation for TR in finite size substrate. I'm not sure how to connect the results to our setup yet. Plus, for our setup, the actual coatings will be ~ 8mm in diameter, with 1" diameter substrate, the boundary conditions will be non-trivial for us.

 ==TE +TR coefficients==

  [ coming soon]

  1192   Thu Jun 6 22:28:46 2013 taraNotesNoiseBudgetAlAs/GaAs layer structure optimized for TO

Here is an outline for TO calculation. I tried to summarize it and make it as simple to follow as possible.

  •  Use Levin's direct approach to calculate thermal fluctuations seen by the beam.
  • Apply power injection at the coating surface, with proper boundary condition, take coating into account. (Evans2008 see thick coating correction, Somiya2009)
  • To calculate the loss due to the dissipated heat, we need to solve heat equation. The loss associated with the injected heat is proportional to (gradient of temperature)2
  • The calculation for gradient of temperature has to be calculated in both longitudinal and transverse direction, as thermal length is comparable to the beam size [Cerdonio 2001]. Other papers usually approximate grad T = dT/dZ, which is 1-D treatment [Evans2008, Somiya2009]. The effect from Heat flow in transverse direction shows up at low frequency, where the noise level becomes lower.
  • When solve heat diffusion equation, apply boundary condition for finite size mirror (somiya2009).
  • Once we have thermal fluctuations, ST, we convert it to displacement noise with TE and TR coefficients. Sx = ST *(TE + TR)
  • TE and TR coefficients can be calculated from the layer structure. The cancellation will occur only at lower frequency where temp fluctuations in coatings are uniform. At higher frequency the effect from TE and TR will sum up in quadrature (if heat equation is solved in coatings), see thick coat correction section in Evans2008.

This means that for TO optimized coatings, we have to make sure that TE and TR coefficients are comparable for maximum cancellation. The calculation for TE and TR are quite well defined, [Fejer2004, Evans2008, Gorodetsky2008]. This part is independent from temperature fluctuation calculation outlined above. So we can choose the optimized design and then calculate the total TO noise level later. The proposed optimization can be found in psl:1183. (Here is the result for 1/8 cap of nH).

eighth.png

Note:

  1. Basically most of the calculations outlined above are done in Somiya2009, except transverse heat flow. If we consider transverse heat flow in coatings and substrate, the result will be valid at low frequency as well.
  2. The decision for G Cole etal to use substrate parameters in temperature fluctuations as suggested by Rana seems to be ok, since their calculation also include the thick coat correction (Evans2008), it means that temperature fluctuations in coatings are taken into account.  However, the cutoff frequency might be  off a bit, since the equation for transverse flow is only in substrate (BGV1999, cerdonio2001). I think the real cutoff frequency should be higher because kappa is larger in the coatings, and transverse heat flow becomes more significant at higher frequency. 
  3. Somiya paper also include Brownian noise in Coatings with finite size substrate/coatings (see fig2) which is not done in Harry etal 2002. Finite size effect increases the noise level by a lot, I think this might explain why the beat result we measured from 8" cavities is a bit higher than the estimated noise using the result from Harry etal. I'll check that later.
  4. I'm not quite sure about The TO calculation in Somiya. The injected heat from TO and TE are added independently, however, the result is similar to that of Evans (with half infinite limit). I'm checking it.
  1193   Tue Jun 11 00:45:48 2013 taraNotesNoiseBudgetnoisebudget for 8" SiO2/Ta2O5 cavity

Quote:

Note:

  1. Somiya paper also include Brownian noise in Coatings with finite size substrate/coatings (see fig2) which is not done in Harry etal 2002. Finite size effect increases the noise level by a lot, I think this might explain why the beat result we measured from 8" cavities is a bit higher than the estimated noise using the result from Harry etal. I'll check that later.

 Here I applied Somiya&Kazuhiro (SK)2009 coating brownian noise calculation to the previous 8" cavity setup. The estimated noise matches up with the measured result well.

 

sk2009nb.png

The result for coating Brownian noise presented in SK is for finite size mirror. They emphasize that the estimated noise diverges from Harry2002 result (half infinite mirror) in the case of a thin mirror (thickness is less than mirror radius) which is our case (radius = 0.5inch, thickness = 0.25 inch). 

I'll attached the calculation and explain some differences between the two calculations later. Here are some notes about the parameters:

  • Loss angle of the coatings used in the calculation is phi perpendicular (1.326 e-4), from GWINC, (phi parallel - 1.4e-4);
  • Young's modulus of the coatings is 93 GPa
  • Poisson's ratio  = 0.2
  • m = 45; (# of zeroes for besselj(1,x))
  • stepsize for radial integral = wspot/1000

 

  1194   Tue Jun 11 16:46:52 2013 taraNotesNoiseBudgetnoisebudget for 8" SiO2/Ta2O5 cavity

note about the calculation for coating Brownian noise in a finite size mirror .

==Coatings Parameters==

Young's modulus, Poisson's ratio, and loss angle are taken from the volume averaged value of the coatings (Yavg = d/ ( d1/Y1 + d2/Y2) , sigma avg = 1/2 (sigma1+sigma2 ). These are used for "perpendicular" direction in Harry2002 formula.

Loss angles

  • SiO2 loss angle  = 1e-4
  • Ta2O5 loss angle = 2.3e-4
  • coatings loss = 1.32e-4

Young's moduli

  • SiO2 Young's modulus = 72e9  Pa
  • Ta2O5                         =140e9  Pa
  • Coatings Young's modulus = 93e9 Pa

Coatings structure

  • 1/2 lambda cap of SiO2
  • 26 layers
  • 300 ppm transmission

 ==calculation codes==

  • I got the file for finding zeroes of the bessel function from Matlab exchange.
  • The code for calculating Br noise is attached below.
  • For the finite size bdy condition, the solutions include all the besselj function of all orders (m=1 to inf). I used m from 1 to 55 in the calculation since it converged quite fast after that.
  • For the integration to calculate all the elastic energy, I used Riemann sum, with stepsize of ~0.15 um. The result does not change much (less than 3%) if I go from 0.8*a to a where a is the radius of the mirror. This is important to note because our coatings do not cover the whole surface of the mirror. There is an annulus edge with ~3mm width for optical contact area. The result means that the elastic energy is still localized in the spot area.

 ==Implication to AdvLIGO coatings==

As noted in SK2009, the estimated values for half infinite and finite size analyses are about the same (~2.5% difference) (I have not verified this). Then, the result from GWINC using Harry2002 formula is still accurate.

==note/comments==

  • The calculation in SK2009 uses an overall loss angle of the coatings, while calculation in Harry2002 separates the elastic energy in two directions,parallel and perpendicular to the surface, and also loss angles in the associated directions.  I use the perpendicular average under the assumption that most energy/deformation occurs in that direction.
  • The result matches the measurement quite well. This reassures us that other noises introduced by the setup (i.e. noise in optical bonding/ noise from supporting structure/ thermoelastic/ brownian noise in the spacer) are not higher than coating thermal noise.

 

  1198   Fri Jun 14 02:36:22 2013 taraNotesNoiseBudgetnoisebudget for 8" SiO2/Ta2O5 cavity

I'm checking the result for the calculation.  I think it is too early to celebrate.

Nic suggested that I should use comsol to estimate the coating Brownian noise. There are a few problems:

  • For 3D model, I cannot mesh the geometry properly yet. The layer is too small for the rest of the mirror.
  • For 2D model, I cannot integrate the elastic energy from the coating layer. There is no choice to select the domain I want to integrate. I'll find out what happen.

I double checked the calculation code. I changed m to 65, and stepsize to w0/4000, the elastic energy (U) calculated is still ~ 1.5e-10 J. It did not change much from my last calculation. However, what I do not understand is the analytical result for half-infinite mirror (as given in SK2009 but different from GWINC), the number does not change that much.

I found a missing 2pi factor that causes the estimated noise to be higher and to match the measurement.  I'll check the calculation carefully again, but it might be that it's still not coating noise.

 

  1199   Tue Jun 18 11:53:17 2013 taraNotesNoiseBudgetnoisebudget for 8" SiO2/Ta2O5 cavity

After careful checks, the estimated result is still below the measurement. For our geometry, the result from SK2009 is similar to that from Harry2002.

 

 So here are the results

  • The calculation for coating Brownian noise from SK2009 (finite size mirror) is similar to Harry2002 result (half infinite mirror).
  • I double checked my code by changing my parameters to those used in SK2009 paper and got the same result. Apparently, their spotsize is very big compare to ours (1/3 of the mirror radius vs 1/40 of the mirror radius).
  • I revisited my comsol model again to check Brownian noise from substrate and spacer when the mirror is curve (more realistic model). The difference is small. (add fig).spacer_br_8_edge.png

The optical bonding area in this model is similar to the real cavity, compared to what is done before where the bonding area is everywhere on the mirror beside the bore hole. So it is quite certain that it is the noise from the substrate/spacer.

Since the measurement has 1/f slope, it is very likely to be Brownian thermal noise (Thermoelastic/ TO will have different slopes). It might be that the 1998 mirrors have high loss. We will see that with shorter cavity measurement.

  1204   Fri Jun 21 21:02:40 2013 taraNotesDrawingslayout for the new setup

The mode matching between the laser to PMC for the 2nd path was not very good before, so I fixed it.

 I used 3 lens to mode match before. The new one will use only 2, see figure. I move the laser by two inch. The new setup is shown in the figure.

With a more careful lens setup, I should be able to couple more light to the cavity.

=note=

I assume that the spotsize in the PMC is 370 um, similar to the current one. I'll revise the PMC drawing and have it made next week. The required optics are ready.

=next=

 I'm preparing the power supply for the second TTFSS. It requires +/- 180, +/- 24 and +/- 17V input. I can use high V power supply from the electronic rack for +/- 180V, and +/-24 V. I'll find a commercial power supply for +/-17V.

  1208   Mon Jun 24 21:27:02 2013 taraNotesRefCavsetup for 2nd refcav

I'm in the process of locking the 2nd cavity. The work is in progress. 

  •  The 2nd TTFSS is working fine. I tested it by using the 2nd TTFSS to lock the first refcav. The error signal was similar to what I got from the first TTFSS.
  •  Mode matching was revised (See Erica's entry).
  • Heatsink for the 2nd laser was ready, I added it on the laser.

To Do:

  • prepare for the 2nd EOM, I need to think about an oscillator driving and EOM. Since there is no resonant EOM, I'll use Rich's EOM driver on a BB EOM for sideband.
  1216   Fri Jun 28 14:52:11 2013 taraNotesECDLlaser noise requirement for locking to a refcav

I estimated some requirement for an ecdl such that it is possible to be locked to a high finesse refcav. For 1.45" cavity, finesse = 1e5, the frequency noise of the ecdl has to be less than 400Hz/rtHz (assuming flat noise from 1kHz to 1MHz).

 ==background==

 We are developing an ecdl, however, we have to check if it can be locked to a high finesse refcav. If so we can use an ecdl in CTN/ cryo style experiments where an ecdl is locked to a refcav. Cryo had some problems with locking a laser to their cavities because of the noise at high frequency, see CRYO elog)

==calculation==

 For a good error signal in PDH locking, the laser linewidth of the laser measured in 1ms - 1us should be smaller than the cavity width (2xcavity pole).

  • Linewidth of the cavity = FSR/Finesse, for the current cavity FSR = 4GHz, Finesse = 1e5-> cavity linewidth = 400kHz.
  • linewidth^2 = integrate frequency noise from 1kHz-1MHz ~ frequency noise PSD[Hz^2/Hz] x 1e6 [Hz] , so S has to be ~ 400 Hz/sqrtHz or lower. (watch out for the unit).

==comments==

The requirement of 400 Hz/rtHz or below seems to be do able, see Chloe's calculation. However, this number is from Finesse = 1e5, with 1.45" cavity length. If we use different cavity with different FInesse, the number will change as well.  The frequency noise requirement (assuming flat from 1kHz to 1MHz) is 400 x [FSR/4GHz] x [ 1e4/ Finesse] [Hz/sqrtHz]

  1228   Mon Jul 8 21:52:04 2013 taraDailyProgressopticmode matching to refcavs

 I redid the mode matching for both refcav, the visibilities are up to ~ 93% and 95% for RCAV and ACAV.

  • For RCAV (refcav with PMC), the visibility was ~ 80% before, now it is ~95%. (The numbers are measured from the reflected beam on the RFPD)
  • For ACAV (refcav without PMC), the visibility is now ~ 93%. This is pretty good, compared to ~ less than 85% from previous setup when we used an AOM.

I'll add the new layout for the current situation soon.

 

==Note==

  • We care about mode matching because we already saw that any light that was not coupled into the cavity was reflected back to the laser and caused extra noise.
  • By changing the lens, the beams for fiber optic (both for Gyro and Erica's experiment) have to be re calculated. I'm sorry about that .
  1231   Tue Jul 9 19:35:20 2013 taraDailyProgressopticbeat board is back

I installed the beat board back behind the cavities. I still have not finished aligning both beams to the 1811.

  • Note about ACAV ( this path has PMC on it): After new mode matching with more visibility (from 80% to 95%), I can increase more gain and the error noise is getting lower. However, there is a problem with the beam reflected from the window of the tank. It overlaps with the main beam and cannot be blocked. I think this is the reason why we cannot suppress the error noise down to what we had before. I still need to convert the error noise back to frequency noise to see if it is below the estimated coating noise or not. If not, we have to reopen the chamber and tilt the cavity a bit. Rcav does not have this problem, the back reflection is away from the main beam and can be dump properly.
  • Note about RCAV: Erica and I plan to finish the EOM driver test tomorrow. After that I'll use it to drive the broadband EOM for locking RCAV to the cavity. The plan is to use one marconi to drive two EOM at the same frequency (14.75 MHz). We use a 4-way splitter for 2 EOM and 2 demodulations. I don't know how using same frequency for EOM will turn out (cross talk problem), but I want to see the first beat measurement within this week.
  • Note about beat setup: Evan calculated the mode matcing for beat setup, but I had to modify it. The first lenses were moved out of the board and mounted between the vacuum tank and the board due to space limitation. This might add some extra resonant peaks in the beat setup due to the long posts for lenses. The spot diameter on the PD is about 130um, which should be fine because 1811's diameter is ~300 um.
  1233   Thu Jul 11 00:19:52 2013 taraDailyProgressopticboth cavities are locked

Both cavities are locked (not optimized yet). Since it has been awhile that both are locked, here is a picture.

photo_2013_07_10.JPG

 Rcav is locked by Fast feedback only. I still have to check the polarity for PC feedback.  I adjusted the phase between the LO and PD for RCAV loop to get a nice error signal. I noticed that there is an offset in the error signal, I will try to adjust the polarization of the beam in front of the EOM to see if I can reduce this offset from RFAM.

To do:

  • lock rcav with both fast and PC feedbacks
  • optimize the setup ( reducing RFAM, minimize back reflection)
  • setup the beat path (mode match + alignment)
  • setup the ISS path
  • check the beat frequency
  • re organizing the wiring on the table.
  • replace the current SMA cables with the semi-rigid ones, once all the equipments are in place.

 

 

 

  1239   Mon Jul 15 10:55:40 2013 taraNotesNoiseBudgetAlAs/GaAs layer structure optimized for TO

 After a discussion with Eric and Matt, here I'll summarize about thermo optic(TO) noise calculation plus some other important noise sources.

1) goal

     We aim to measure the limiting noise in AlGaAs coatings. If we order just 1/4 quarter wave stack, no optimization, the limiting noise source will be from TO noise due to high values of thermo elastic(TE) and thermo refractive(TR) coefficients of the materials. However, by optimizing the coatings structure to cancel TO noise we can:

  • Probe thermo-elastic (TE) noise in SiO2 substrate at low frequency and coating Brownian noise(BR) at higher frequency
  • Prove that TO cancellation can be done (according to Evans etal).

We can tell what kind of noise from the slope.  BR, TO noise or TE noise in substrate have different slopes at the interested band, see fig 1.

 

2) Is the calculation correct?

alGaAsnb.png

fig1: noise budget with some fundamental noise sources. The noise budget is for AlGaAs coatings on a mirror with ROC=1m. The cap is GaAs (high index material) with 1/8 lambda thickness. See explanation below for more details.

    The fundamental noise sources in our setup (1.45" cavity, 1m roc mirror, optimized AlGaAs coatings) will be:

==BR in coatings==:   

  • The calculation is taken from Harry2002, for half infinite mirror.
  • The result is compared with Somiya&Kazuhiro2009 for finite size mirror calculation (see solid blue line and dashed cyan line). The difference is small due to our small spotsize, so using either calculation is ok for us, but Harry's calculation is less time consuming.
  • The analytical result should be valid as it was verified by Numata and TNI measurements.

==BR in substrate==:

  • The calculation is taken from Levin1998, with finite size correction by Liu&Thorne(LT2001). 
  • The loss angle for bulk fused silica is frequency dependent ~ 10-11 x f0.8(Penn2006). This loss is much lower than conservative constant loss (10^-8) (number from DCC LIGO-T0900161) from dc upto 10kHz.
  • In this calculation, for constant loss of 10^-8, BR noise in substrate is still ~ a factor of 3 lower than BR in coatings.

==TE noise in substrate==:

  • BGV1999 gave a result for adiabatic limit (most of the heat flow is in 1-D heat diffusion length is much smaller than beamsize, sqrt(kappa/C * 2pi*f)<<r0 )for half infinite space mirror, Liu Thorne2001 verify the result. I used comsol to simulate the noise (with adiabatic assumption) and it agreed with the analytical solution.
  • However for our setup with a small spot size the assumption beaks down. Cerdonio2001, computed the noise that valid for low frequency and small beamsize which is a  case for our setup (cut off frquency ~ 10 Hz). All the factors and corrections are summarized in TNI2004 measurement and Nawrodt2012.  The calculation will be valid for our setup.

==TE and TR noise calculation:

  •   The temperature fluctuation sensed by the beam is taken from BGV1999 using Langevin approach, and Mike Martin Thesis (this takes care of the fluctuation at low frequency where adiabatic assumption breaks down. The calculation assume that coating thickness << thermal diffusion length. For AlGaAs, because of its high thermal conductivity, this assumption is still hold at the bandwidth of interest.
  • The thick coating calculation is given in Evans 2008.  It is important at high frequency and coatings with low thermal conductivity. This means that TE and TR effects won't be coherent in the coatings. This is not a problem for AlGaAs due to its high thermal conductivity.
  • TE and TR coefficients calculations are treated coherently in Evans2008.  The cancellation only depends on coating structure. With a cap of GaAs (nH) 1/8 lambda thickness, the cancellation is very good reducing the TO noise below other noise upto a few kHz.
  • The cap thickness has to be withing +/- 20Angstrom so that the TO is about a factor of2 below coating BR. G. Cole mentioned that each layer thickness varies about 0.3% or less which is about lambda/(4*n) * 0.3% = 2Angstrom. So the cancellation should be ok.

TR coefficients are calculated numerically (GWINC) and analytically (Gorodetsky2008). The results match up well (less than 1% difference), if all the parameters/ averaged values are from Evans.

In GWINC there is one correction noted as "Yamamoto thermo-refractive correction", this changes the Beta eff ~ 10% causing the cancellation to be not as good (still ok up to 1kHz). I emailed Kazuhiro Yamamoto asking him if he has anything to do with this. Otherwise all the calculations and optimization are in good shape.

  1263   Mon Jul 29 22:30:34 2013 taraNotesopticcoating optimization for AlGaAs

Since we are trying to optimize a layer structure for AlGaAs coatings. It is a good idea to summarize some notes about all the coatings details. Thanks Koji for the discussion about the coaitngs.

==some background about SiO2/Ta2O5 QWL with 1/2 wave cap coatings==

 For quarter wave layer stack (QWL) SiO2/Ta2O5 coatings, SiO2 and Ta2O5 are the material with low (nl) and high refractive indices (nh), respectively. Due to the stronger structure of SiO2, we usually have a cap of SiO2  as a protective layer on top. This cap has thickness of 1/2 wave length. The reason is that the reflected beam from the interface between the cap and the next layer will be in phase with the first reflected beam at the air-coating surface, see the figure below (top).

If the SiO2 cap is 1/4 thick, the reflected beam from the interface between the cap and the next layer will destructively interfere, causing the reflectivity to go down (see the picture below, middle). 

However, if the cap is Ta2O5 (nH) material, it can be QWL thickness, and the phase from every reflected beams still interferes constructively (picture below, bottom).

multilayer1.JPG

Note: As we can see, the incoming beam and the reflected beam are 180 degree out of phase. It means that the E field at the coatings surface will always be zero. This will prevent the burning on the surface of the coating. With this, the standing wave in the cavity will always have zero E field at the coating surface, see below picture.

This is not AR coat, since all the reflected beams interfere constructively. The reflected beams from AR coating will destructively interfere among each layer.

multilayer2.JPG

To sum up for the SiO2/Ta2O5 coatings:

  • SiO2 is stronger than Ta2O5, so we use it for the end cap.
  • Because SiO2 has lower n than that of Ta2O5, the cap thickness has to be 1/2 wave thick so that all the reflected beams interfere constructively.
  • We want the reflected phase to be 180 degree away from the incident beam so that the surface won't get burnt from the building up E field. (If the E field is non zero, it will be amplified by a factor of Finesse/pi).  My previous optimization for AlGaAs that used 1/8 cap was wrong because the reflection phase was not 180. This means that by adjusting the cap thickness to optimize the TO noise is not a good method, since the reflection phase is not close to 180 anymore. The optimization has to take the phase into account.

 

==AlGaAs coatings==

 For GaAs/Al0.92Ga0.08As (AlGaAs) coatings, the situation is a bit different from SiO2/Ta2O5. The cap has to be GaAs (nH) because Al0.92Ga0.08As will oxidize and change its material properties. Now that the cap will be nH, the thickness has to be 1/4 wavelength.  The last layer next to the substrate has to be GaAs (nH) too (I think because of both the better reflectivity and the fabrication process).

==optimization code==

 There is an assumption about the layer structure used in the optimization code that the cap is nL(SiO2), 1/2 layer. The coatings layers are even number ( doublets of SiO2/Ta2O5). I'm making sure all the assumptions in the code are fixed. Here is a preliminary result.

 

opt1_2013_07_29.png

above: Layer structure, the first layer (cap) is GaAs (nH). In the optimization, I keep the cap thickness to be 1/4, and vary the rest.

nb_opt1_2013_07_29.png

above: Noise budget of the optimized layer. TO noise is below BR noise from DC up to 1kHz.

The reflectivity of the coatings is -0.9997 + 0.0209i  (reflection phase = 180 - 1.2 degree). I'm not sure if this is good enough, maybe better optimization can be done.

Note: My layer structure is really different from what rana did in T1200003. For my structure, the layers near the cap vary a lot before getting close to 0.25 when the layers are close to the substrate. The result from 1200003 is the opposite. The layers near the cap are about 0.25, and start to diverge when the layers are close to the substrate.

T1200003_refcav.png

above:  Optimized coatings result from T1200003. The optimization probably assume the cap of low index material, but the following layers evolution are opposite of what I got. That's why I'm not sure about my optimization.

 

I'll upload my codes soon so that people can check my optimization.

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