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ID Date Author Type Category Subject
12130   Tue May 24 22:49:02 2016 gautamUpdateCOCFinesse modelling - mode overlap scans

Summary:

Having played around with a toy finesse model, I went about setting up a model in which the RC folding mirrors are not flipped. I then repeated the low-level tests detailed in the earlier elog, after which I ran a few spatial mode overlap analyses, the results of which are presented here. It remains to do a stability analysis.

Overview of model parameters (more details to follow):

• PRC length = 6.7727m (chosen using $\dpi{80} l_{PRC} = (N+\frac{1}{2})\frac{c}{2f_1}$, N=0 - I adjusted the position of the PRM to realize this length in the model, while leaving all the other vertex optics in the same positions as in elog 9590
• SRC length = 5.4182 (chosen using $\dpi{80} l_{SRC} = M\frac{c}{2f_2}$ but not $\dpi{80} l_{SRC} = N\frac{c}{2f_1}$, M and N being integers, for M=2 - as above, I adjusted the position of the SRM to realize this in the model, while leaving all other vertex optics in the same positions as in elog 9590. It remains to be verified if it is physically possible to realize these dimensions in vacuum without any beam clipping etc but I think it should be possible seeing as the PRM and SRM had to be moved by less than 2cm from their current positions..
• For the losses, I used the most recent numbers we have where applicable, and put in generic 25ppm loss for all the folding mirrors/BS/AR surfaces of arm cavity mirrors/PRM/SRM. Arm round trip loss was equally distributed between ITMs and ETMs
• Arm lengths used: L_X = 37.79m, L_Y = 37.81m
• To set the "tunings" of the various mirrors, I played around with a few configurations to see where the various fields resonated - it turns out that for PRM, ITMX, ITMY, ETMX and ETMY, the "phase" in the .kat file can be set as 0. while that for the SRM can be set as 90. In the full L1/H1 interferometer .kat files, these are tuned even further to the (tenth?!) decimal place, but I think these values suffice for out purposes.

Results (general note: positive RoC in these plots mean a concave surface as seen by the beam):

• Attachments #1, #2 and #3 reproduce the low-level tests performed earlier for this updated model - i.e. I look at the arm transmission with no PRM/SRM, circulating PRC power with no ETMs, and circulating SRC power with no ETMs. Everything looks consistent here... In Attachment #2, there is no legend, but the (almost overlapping) red and green lines are meant to denote the +f1 and +f2 sidebands.
• Attachments #4 and #5 are a summary of the mode-overlap scans for the PRC and SRC. What I did was to vary the radius of curvature of the RC mirrors (finesse only allows you to vary Rcx and Rcy, so I varied both simultaneously) and calculate the mode overlap between the appropriate pairs of cavities (e.g. PRX and XARM) in the tangential and saggital planes. The take-away here is that there is ~5% mode-mismatch going from an RoC of 1000m to 300m. I've also indicated the sag corresponding to a given RoC - these are pretty tiny, I wonder if it is possible to realize a sag of 1um? I suppose it is given that I've regularly seen specs of surface roughness of lambda/10?
• Attachment #6 shows the PRC gain (calculated as T_PRC * (transmitted arm power with PRM / transmitted arm power without PRM) as a function of the RoC of PR2 and PR3. As a sanity check, I repeated this calculation with lossless HR surfaces (but with nominal 25ppm losses for AR surfaces of ITMs, and BS etc), shown in Attachment #7. I think these make sense too...
• Attachment #8 - in order to investigate possible mode mismatch between the arm modes due to different radii of curvature of the ETMs, I kept the ETMY RoC fixed at 57.6m and varied the ETMY RoC between 50m and 70m (here, I've plotted the mode matching efficiency as a function of the RoC of the ETM in the X and Y directions separately - the mode overlap is computed as $\dpi{80} \frac{1}{\sqrt{2}}(x^2 + y^2)$ where x and y denote the overlap in the tangential and saggital planes respectively. It would seem that we only lose at most a couple of percent even if the RoCs are mismatched by up to 10m...
• Attachment #9 - .kat file and the various pykat scripts used to generate these plots...

Next step is to carry out a stability analysis...

Attachment 1: armTransmission.pdf
Attachment 2: prcFSR.pdf
Attachment 3: srcTransmission.pdf
Attachment 4: modeMatchPRX.pdf
Attachment 5: modeMatchSRX.pdf
Attachment 6: PRCgainScan.pdf
Attachment 7: PRCgainLossless.pdf
Attachment 8: armModeMatchScan.pdf
Attachment 9: Finesse_files.zip
12131   Tue May 24 23:17:37 2016 ericqUpdateCOCFinesse modelling - mode overlap scans

I think you should use the current actual PRC & SRC cavity lengths as measured, as it would be simplest to simply replace the folding mirror optics without changing the macroscopic lengths / optic positions. (EDIT: Gautam rightly points out that we have to move things around regardless, since our current lengths include propagation through the folding mirror subtrates)

Moreover, the recycling cavity lengths you posted are not the right "ideal" lengths to use, as they do not account for the complex reflectivities of the sidebands off of the arm cavities (I have made this mistake myself). See this 40m wiki page for details.

In short, given our current modulation frequency, the ideal lengths to use would be:

• Ideal arm length of 37.795 m
• Ideal PRC length of 6.753 m
• Ideal SRC length of 5.399 m

These are the lengths that the recycling cavity optics were positioned for (though we did not achieve them perfectly). If you do a finer PRC/SRC length scan around the DRFPMI resonance of your model, you would presumably see some undesired sideband splitting.

12190   Thu Jun 16 15:57:46 2016 gautamUpdateCOCContrast as a function of RoC of ETMX

Summary

In a previous elog, I demonstrated that the RoC mismatch between ETMX and ETMY does not result in appreciable degradation in the mode overlap of the two arm modes. Koji suggested also checking the effect on the contrast defect. I'm attaching the results of this investigation (I've plotted the contrast, $C = \frac{P\mathrm{_{max}}-P\mathrm{_{min}}}{P\mathrm{_{max}}+P\mathrm{_{min}}}$  rather than the contrast defect 1-C).

Details and methodology

• I used the same .kat file that I had made for the current configuration of the 40m, except that I set the reflectivities of the PRM and the SRM to 0.
• Then, I traced the Y arm cavity mode back to the node at which the laser sits in my .kat file to determine what beam coming out of the laser would be 100% matched to the Y arm (code used to do this attached)
• I then set the beam coming out of the laser for the subsequent simulations to the value thus determined using the gauss command in finesse.
• I then varied the RoC of ETMX (I varied the sagittal and tangential RoCs simultaneously) between 50m and 70m. As per the wiki page, the spare ETMs have an RoC between 54 and 55m, while the current ETMs have an RoC of 60.26m and 59.48m for the Y and X arms respectively (I quote the values in the "ATF" column). Simultaneously, at each value of the RoC of ETMX, I swept the microscopic position of the ETMX HR surface through 2pi radians (-180 degrees to 180 degrees) using the phi functionalilty of finesse, while monitoring the power at the AS port of this configuration using a pd in finesse. This guarantees that I sweep through all the resonances. I then calculate the contrast using the above formula. I divided the parameter space into a grid of 50 points for the RoC of ETMX and 1000 points for the microscopic position of ETMX.
• I fixed the RoC of ETMY as 57.6m in the simulations... Also, the maxtem option in the .kat file is set to 4 (i.e. higher order modes with indices m+n<=4 are accounted for...)

Result:

Attachment #1 shows the result of this scan (as mentioned earlier, I plot the contrast C and not the contrast defect 1-C, sorry for the wrong plot title but it takes ~30mins to run the simulation which is why I didn't want to do it agian). If the RoC of the spare ETMs is about 54m, the loss in contrast is about 0.5%. This is in good agreement with this technical note by Koji - it tells us to expect a contrast defect in the region of 0.5%-1% (depending on what parameter you use as the RoC of ETMY).

Conclusion:

It doesn't seem that switching out the current ETM with one of the spare ETMs will result in dramatic degradation of the contrast defect...

Misc notes:

1. Regarding the phase command in Finesse - EricQ pointed out that the default value of this is 3, which as per the manual could give unphysical results sometimes. The flags "0" or "2" are guaranteed to yield physical results always according to the manual, so it is best to set this flag appropriately for all future Finesse simulaitons.
2. I quickly poked around inside the cabinet near the EX table labelled "clean optics" to see if I could locate the spare ETMs. In my (non-exhaustive) search, I could not find it in any of the boxes labelled "2010 upgrade" or something to that effect. I did however find empty boxes for ETMU05 and ETMU07 which are the ETMs currently in the IFO... Does anyone know if I should look elsewhere for these?
EDIT 17Jun2016: I have located ETMU06 and ETMU08, they are indeed in the cabinet at the X end...
3. I'm attaching a zip file with all the code used to do this simulation. The phase flag has been appropriately set in the (only) .kat file. setLaserQparam.py was used to determine what beam parameter to assign to be perfectly matched to the Y arm. modeMatchCheck_ETM.py was used to generate the contrast as a function of the RoC of ETMX.
4. With regards to the remaining checks to be done - I will post results of my investigations into the HOM scans as a function of the RoC of the ETMs and also the folding mirrors shortly...
Attachment 1: contrastDefect.pdf
Attachment 2: finesseCode.zip
12193   Thu Jun 16 18:42:12 2016 ranaUpdateCOCContrast as a function of RoC of ETMX

That sounds weird. If the ETMY RoC is 60 m, why would you use 57.6 m in the simulation? According to the phase map web page, it really is 60.2 m.

12194   Thu Jun 16 23:02:57 2016 gautamUpdateCOCContrast as a function of RoC of ETMX
 Quote: That sounds weird. If the ETMY RoC is 60 m, why would you use 57.6 m in the simulation? According to the phase map web page, it really is 60.2 m.

This was an oversight on my part. I've updated the .kat file to have all the optics have the RoC as per the phase map page. I then re-did the tracing of the Y arm cavity mode to determine the appropriate beam parameters at the laser in the simulation, and repeated the sweep of RoC of ETMX while holding RoC of ETMY fixed at 60.2m. The revised contrast defect plot is attached (this time it is the contrast defect, and not the contrast, but since I was running the simulation again I thought I may as well change the plot).

As per this plot, if the ETMX RoC is ~54.8m (the closer of the two spares to 60.2m), the contrast defect is 0.9%, again in good agreement with what the note linked in the previous elog tells us to expect...

Attachment 1: contrastDefect.pdf
12197   Mon Jun 20 01:38:04 2016 ranaUpdateCOCContrast as a function of RoC of ETMX

So, it seems that changing the ETMX for one of the spares will change the contrast defect from ~0.1% to 0.9%. True? Seems like that might be a big deal.

12204   Mon Jun 20 18:07:15 2016 gautamUpdateCOCContrast as a function of RoC of ETMX
 Quote: So, it seems that changing the ETMX for one of the spares will change the contrast defect from ~0.1% to 0.9%. True? Seems like that might be a big deal.

That is what the simulation suggests... I repeated the simulation for a PRFPMI configuration (i.e. no SRM, everything else  as per the most up to date 40m numbers), and the conclusion is roughly the same - the contrast defect degrades from ~0.1% to ~1.4%... So I would say this is significant. I also attempted to see what the contribution of the asymmetry in loss in the arms is, by running over the simulation with the current loss numbers of 230ppm for Yarm and 484ppm for the X arm, split equally between the ITMs and ETMs for both cases, and then again with lossless arms - see attachment #1. While this is a factor, this plot seems to suggest that the RoC mismatch effect dominates the contrast defect...

Attachment 1: contrastDefectComparison.pdf
12219   Tue Jun 28 16:06:09 2016 gautamUpdateCOCRC folding mirrors - further checks

Having investigated the mode-overlap as a function of RoC of the PRC and SRC folding mirrors, I've now been looking into possible stability issues, with the help of some code that EricQ wrote some time back for a similar investigation, but using Finesse to calculate the round trip Gouy phase and other relevant parameters for our current IFO configuration.

To do so, I've been using:

1. Most up to date arm length measurements of 37.81m for the Y arm and 37.79m for the X arm
2. RoCs of all the mirrors from the phase map summary page
3. Loss numbers from our November investigations

As a first check, I used flat folding mirrors to see what the HOM coupling structure into the IFO is like (the idea being then to track the positions of HOM resonances in terms of CARM offset as I sweep the RoC of the folding mirror).

However, just working with the flat folding mirror configuration suggests that there are order 2 22MHz and order 4 44MHz HOM resonances that are really close to the carrier resonance (see attached plots). This seems to be originating from the fact that the Y-arm length is 37.81m (while the "ideal" length is 37.795m), and also the fact that the ETM RoCs are ~3m larger than the design specification of 57m. Interestingly, this problem isn't completely mitigated if we use the ideal arm lengths, although the order 2 resonances do move further away from the carrier resonance, but are still around a CARM offset of +/- 2nm. If we use the design RoC for the ETMs of 57m, then the HOM resonances move completely off the scale of these plots...

Attachment 1: C1_HOMcurves_Y.pdf
Attachment 2: C1_HOMcurves_DR.pdf
12223   Tue Jun 28 20:43:23 2016 KojiSummaryCOCFirst Contact cleaning practice

Made a dry run of the in-situ cleaning for a 3inch optic.

Attachment 1: The Al dummy mass is clamped in the suspension cage.
Attachment 2: The front surface was painted. The nominal brush with the FC bottle was used.
Attachment 3: Zoom in of the front surface.
Attachment 4: The back surface was painted.
Attachment 5: The back surface was peeled.
Attachment 6: The front surface was peeled too.
Attachment 7: The peeled layers.

Findings:

1. To paint a thick layer (particlarly on the rim) is the key to peel it nicely.

2. It was helpful for easier peeling to have mutiple peek tabs. Two tabs were sufficient for ~1" circle.

3. The nominal brush with the bottle was OK although one has to apply the liquid many times to cover such a large area. A larger brush may cause dripping.

4. The nominal brush was sufficiently long once the OSEMs are removed. In any case it is better to remove the OSEMs.

Attachment 1: IMG_20160628_170335196.jpg
Attachment 2: IMG_20160628_171547769.jpg
Attachment 3: IMG_20160628_171607802.jpg
Attachment 4: IMG_20160628_172328190.jpg
Attachment 5: IMG_20160628_174541960.jpg
Attachment 6: IMG_20160628_174556004.jpg
Attachment 7: IMG_20160628_174617198.jpg
12234   Thu Jun 30 16:21:32 2016 gautamUpdateCOCSideband HOMs resonating in arms

[EricQ, gautam]

Last night, we set about trying to see if we could measure and verify the predictions of the simulations, and if there are indeed HOM sidebands co-resonating with the carrier. Koji pointed out that if we clip the transmitted beam from the arm incident on a PD, then the power of the higher order HG modes no longer integrate to 0 (i.e. the orthogonality is broken), and so if there are indeed some co-resonating modes, we should be able to see the beat between them on a spectrum analyzer. The procedure we followed was:

1. Choose a suitable PD to measure the beat. We chose to use the Thorlabs PDA10CF because it has ~150MHz bandwidth, and also the responsivity is reasonable at 1064nm.
2. We started our measurements at the Y-end. There was a sufficiently fast lens in the beam path between the transmon QPD and the high gain PD at the Y end, so we went ahead and simply switched out the high gain thorlabs PDA520 for the PDA10CF. To power the PDA10CF, we borrowed the power cable from the green REFL PD temporarily.
3. We maximized the DC power of the photodiode signal using an oscilloscope. Then to introduce the above-mentioned clipping and orthogonality-breaking, we misaligned the beam on the PD until the DC power was ~2/3 the maximum value.
4. We then hooked up the PD output to the Agilent network analizyer (with a DC block).
5. We measured the spectrum of the PD signal around 11.066MHz (with 100kHz span) and higher harmonics up to 55MHz and used a narrow bandwidth (100Hz) and long integration time (64 averages) to see if we could find any peaks. More details in the results section.
6. Having satisfied ourselves with the Y-end measurements, we
• restored the power cable to the green beat PD
• re-installed the thorlabs PDA520
• verified that both IR and green could be locked to the arm

We then repeated the above steps at the X-end (but here, an additional lens had to be installed to focus the IR beam onto the PDA10CF - there was, however, sufficient space on the table so we didn't need to remove the PDA520 for this measurement).

Results:

Y-end: DC power on the photodiode at optimal alignment ~ 200mV => spectra taken by deliberately misaligning the beam incident on the PD till the DC power was ~120mV (see remarks about these values).

RF sideband (Y-arm) Peak height (uV) Beat power (nW) RF sideband (X-arm) Peak height (uV) Beat Power (nW)
11 1.55 0.52 11 1.2 0.4
22 10.6 3.53 22 none seen N.A.
33 none seen N.A. 33 none seen N.A.
44 22.0 7.33 44 7 2.33
55 8.6 2.97 55 5 1.67

I converted the peak heights seen on the spectrum analyzer in volts to power by dividing by transimpedance (=5*10^3 V/A into a 50ohm load) * responsivity at 1064nm (~0.6A/W for PDA10CF).

Remarks:

1. This effect flagged by the simulations seems to be real. Unfortunately I can't get a more quantitative picture because we can't quantify the mode-overlap between the carrier 00 mode and any higher order mode on the beat PD (as we know nothing about the profile of these modes), but the simulations did suggets that the 2nd order 22MHz and 4th order 44MHz HOMs are the ones closest to the carrier 00 resonance (see Attachments #2 and #3), which is kind of borne out by these results.
2. I disbelieve the conversions into power that I have done above, but have just put them in for now, because a DC power of 200mW at the Y-end suggests that there is >160uW of light transmitted from the arm, which is at least twice what we expect from a simple FP cavity calculation with the best-known parameters. If I've missed out something obvious in doing this conversion, please let me know!
3. For the Y-arm, the region around 55MHz had a peak (presumably from the sideband HOM beating with the carrier) but also a bunch of other weird sub-structures. I'm attaching a photo of the analyzer screen. Not sure what to make of this...
Attachment 1: image.jpeg
Attachment 2: C1_HOMcurves_Y.pdf
Attachment 3: C1_HOMcurves_X.pdf
12325   Fri Jul 22 03:02:37 2016 KojiUpdateCOCFC painting

[Koji Gautam]

We have worked on the FC painting on ITMX and ITMY. We also replaced the OSEM fixing screws with the ones with a hex knob.
This was done except for the SD OSEM as the new screw was not long enough. We left an allen-key version of the screw for the SD OSEM.

All the full-resolution photos can be found on g-photo.

ITMY

Attachment1: The barrel was pretty dusty. Some dusts were observed on the HR face but it was not so terrible. The barrel and the HR face were blown with the ionized N2 and then wiped with IPA. The face wiping was done n a similar way as the drag wiping.

Attachment2: FC was applied to the HR surface.

Attachment3: The AR surface was also painted with FC. The brush touched the coil holder.

Attachment4: The brush touched the coil holder. Another PEEK tab was applied to remove this FC stain on the metal holder.

Attachment5: This is the result of successful removal of the FC stain.

ITMX

Attachment6: The OSEM arrangement before removal. We confirmed that the OSEM arrangement was as described on Wiki.

Attachment7/8: The ITMX was obviously a lot dirtier than ITMY. The barrel accumulated dusts.

Attachment9: This is the HR face picture with large dusts on it.

Attachment10: The HR surface was painted with FC.

Attachment11: This is the AR surface with FC painted.

Attachment 1: ITMY_barrel_dust.jpg
Attachment 2: ITMY_HR_FC.jpg
Attachment 3: ITMY_AR_FC.jpg
Attachment 4: ITMY_drip_removal.jpg
Attachment 5: ITMY_drip_removed.jpg
Attachment 6: ITMX_OSEMS.jpg
Attachment 7: ITMX_barrel_dust1.jpg
Attachment 8: ITMX_barrel_dust2.jpg
Attachment 9: ITMX_HR_dusty.jpg
Attachment 10: ITMX_HR_FC.jpg
Attachment 11: ITMX_AR_FC.jpg
12407   Sat Aug 13 18:25:22 2016 gautamUpdateCOCRC folding mirrors - Numerical review

This elog is meant to summarize my numerical simulations for looking into the effects of curvature on the RC mirrors. I've tried to go through my reasoning (which may or may not be correct) and once this gets a bit more refined, I will put all of this into a technical note.

Motivation:

• Both the G&H (PR2, SR2) and Laseroptik (PR3 SR3) are convex on the HR side with RoCs of approximately -600m and -700m (though as stated in the linked elog, I'm not actually sure if there are measurements of this number) EDIT AUG15: There are measurements for the Laseroptik mirrors here
GV April 8 2017: This elog by Jenne suggests that the installed PR2 has an RoC of approximately -700m. Koji has uploaded the phase map data for the RC TT mirrors to
/users/public_html/40m_phasemap/40m_TT and
/users/public_html/40m_phasemap/40m_TT2. The G&H mirror data seems to be in the former folder, and it looks like there are two mirrors, one with RoC of ~ -700m and the other with RoC of ~ -500m. Does this mean PR2 has RoC -700m and SR2 has RoC -500m?
• As a result, both the PRC and SRC were close to instability
• By flipping the folding mirrors, the instability has been mitigated, but at the expense of the non-ideal situation where the AR coated side and the substrate are now inside the recycling cavity
• We would like to order some new folding mirrors. In order to avoid receiving convex mirrors from the vendor, we want to specify a concave curvature for the HR side
• The aim of this investigation is to look at how concave we should make these mirrors, because although the cavity stability improves with concavity of the HR side, possible disadvantages of having too convex mirrors are:
• ​Mode-mismatch between the recycling cavities and the arms
• Astigmatism

The study:

• I've built a Finesse model for the 40m, which has been used for all the numerical studies quoted here
• In constructing this Finesse model, I've used the following sources to specify various paramaters:
• ​RoCs, R, T and physical dimensions of 4 test-masses, PRM, SRM and BS: Core optics wiki page
• Losses - arm losses from Yutaro's measurements in elog11857 and elog11818 (distributed equally between ITM and ETM). For other optics, a generic value of 25ppm was used
• "Ideal" lengths for our current modulation frequency were used for the various cavities (37.795m for the arms, 6.753m for PRC, 5.399 for SRC)
• The folding mirrors (PR2, PR3, SR2, SR3) are initialized as flat in the model
• I performed some low-level checks (e.g. arm linewidth, PRC FSR etc) to check that the model was sensible
• I then proceeded to investigate the effects of curvature on the folding mirrors. Specifically, I investigated the following:
• What is the mode mismatch between the recycling cavity mode and the arm as a function of the RoC of the folding mirror?
• What is the effect of the RoC of the folding mirrors on the round-trip gouy phase accumulated (and hence the transverse mode spacing) in the recycling cavities?
• For now, the parameter space explored is from 300m concave to 1000m concave. An RoC of 1km for a 2" optic corresponds to a sag of ~0.3 microns. I will explore the 1km-10km concave space and update the results shortly

Results:

• Attachments #1 and #2 show the mode mismatch between the recycling cavity and the arm for various curvatures. The colorbars have been normalized to span the same range in all the plots
• For both the PRC and the SRC, if we have folding mirrors with an RoC of 1000m concave, we will have a mode mismatch of 2-3%. The number gets worse the more convex the mirror
• Attachments #3 and #4 show the one-way accumulated Gouy phase. Here, I have varied the curvature of the folding mirrors along a specific axis at a time (i.e. I've assumed that the folding mirrors are identical). I've also added the transverse mode spacing as a second y-axis. I have yet to check how these numbers compare with the linewidth of the 00-mode for the various fields, but for 1km concave folding mirrors, the TMS is in the region of 2MHz

To do:

• I will extend the range of RoCs explored to 10km concave and post results - but I will have to check with EricG to make sure that it is feasible for us to specify curvatures in this range
• I was trying to use the RT gouy phase as calcluated by my Finesse simulations to plug into some analytical expressions to try and generate plots like this for various RoCs of the folding mirrors, but if the TMS calculations suffice, I will abandon these efforts
• What are the other specifications we need to worry about before placing an order? Some thoughts from Rana's earlier elog:
• The coatings need to be dichroic to allow extraction of the green beam (but only PR3/SR3 is currently dichroic?)
• Wedge angle on the AR side?
• Are there any other obvious sanity checks I should carry out?

Attachment 1: PRX_consolidated.pdf
Attachment 2: SRX_consolidated.pdf
Attachment 3: Gouy_PRC.pdf
Attachment 4: Gouy_SRC.pdf
12413   Tue Aug 16 11:51:43 2016 gautamUpdateCOCRC folding mirrors - Numerical review

Summary of roundtable meeting yesterday between EricG, EricQ, Koji and Gautam:

We identified two possible courses of action.

1. Flip the G&H mirror (PR2/SR2) back such that the (convex) HR face is the right way round. We want to investigate what are the requirements on a new PR3/SR3 optic that will guarantee cavity stability and also give good mode matching.
2. Order two new sets of mirrors (i.e. replace all 4 folding mirrors). In this case, we want to spec a flat (how flat is reasonable to specify? EricG will update us) PR3/SR3, and design a PR2/SR2 with some concavity that will guarantee cavity stability in the event PR3/SR3 deviates from flatness (but still within what we spec). The choice to make PR3 as close to flat as possible is because the angle of incidence in our arrangement means that any curvature on PR3 dominates astigmatism.

I have done some calculations to evaluate the first alternative.

• Based on yesterday's preliminary discussion, we felt it is not reasonable to spec mirrors with RoC > 4km (sag of ~80nm). So I restrict my analyses to the range 300m-4km
• Koji has a measurement of the phase maps for the G&H mirrors. The measured curvature is ~-500m. In my simulations, I've tried to allow for error in this measurement, so I look at the range -450m to -700m for the G&H mirror.
• The Gouy phase analysis suggests we should look for an RoC of +500m (concave) for the new PR3/SR3 to have a TMS of ~1.5 MHz. Anything flatter (but still concave) means the TMS gets smaller.
• The mode-matching in this region also looks pretty good, between 98% and 99%
• I will post results of the analysis for the second alternative here for comparison

Something else that came up in yesterdays meeting was if we should go in for 1" optics rather than 2", seeing as the beam spot is only ~3mm on these. It is not clear what (if any) advantages this will offer us (indeed, for the same RoC, the sag is smaller for a 1" optic than a 2").

Attachments:

Attachment #1: Mode-matching maps between PRX and Xarm cavities, PRY and Yarm cavities with some contours overlaid.

Attachment #2: Mode-matching maps between SRX and Xarm cavities, SRY and Yarm cavities with some contours overlaid.

Attachment #3: Gouy phase calculations for the PRC

Attachment #3: Gouy phase calculations for the SRC

Attachment 1: PRC_consolidated.pdf
Attachment 2: SRC_consolidated.pdf
Attachment 3: GouyPRC.pdf
Attachment 4: GouySRC.pdf
12414   Tue Aug 16 16:38:00 2016 gautamUpdateCOCRC folding mirrors - Numerical review

Here are the results for case 2: (flat PR3/SR3, for purpose of simulation, I've used a concave mirror with RoC in the range 5-15km, and concave PR2/SR2 - I've looked at the RoC range 300m-4km).

• This is where we order two new sets of mirrors, one for use as PR2/SR2, and the other for use as PR3/SR3.
• RoC of flat PR3/SR3 in simulation explored in the range 5km-15km (concave)
• RoC of concave PR2/SR2 in simulation explored in the range 300m-4km (concave)

Attachment #1: Mode matching between PRC cavities and arm cavities with some contour plots

Attachment #2: Mode matching between SRC cavities and arm cavities with some contour plots

Attachment #3: Gouy phase and TMS for the PRC. I've plotted two sets of curves, one for a PR3 with RoC 5km, and the other for a PR3 with RoC 15km

Attachment #4: Gouy phase and TMS for the SRC. Two sets of curves plotted, as above.

Hopefully EricG will have some information with regards to what is practical to spec at tomorrow's meeting.

EDIT: Added 9pm, 16 Aug 2016

A useful number to have is the designed one-way Gouy phase and TMS for the various cavities. To calculate these, I assume flat folding mirrors, and that the PRM has an RoC of 115.5m, SRM has an RoC of 148m (numbers taken from the wiki). The results may be summarized as:

Cavity One-way Gouy phase [rad]           TMS [MHz]
PRX 0.244 1.730
PRY 0.243 1.716
SRX 0.197 1.743
SRY 0.194 1.717

So, there are regions in parameter space for both options (i.e. keep current G&H mirrors, or order two new sets of folding mirrors) that get us close to the design numbers...

Attachment 1: PRC_consolidated.pdf
Attachment 2: SRC_consolidated.pdf
Attachment 3: GouyPRC.pdf
Attachment 4: GouySRC.pdf
12417   Wed Aug 17 14:37:36 2016 gautamUpdateCOCRC folding mirrors - Numerical review
Quote:

Cavity One-way Gouy phase [rad]           TMS [MHz]
PRX 0.244 1.730
PRY 0.243 1.716
SRX 0.197 1.743
SRY 0.194 1.717

So, there are regions in parameter space for both options (i.e. keep current G&H mirrors, or order two new sets of folding mirrors) that get us close to the design numbers...

Keeping these design numbers in mind, here are a few possible scenarios. The "designed" TMS numbers from my previous elog are above for quick reference.

Case 1: Keep existing G&H mirror, flip it back the right way, and order new PR3/SR3.

• Spec PR3 to be concave with RoC 600 +/- 50m
• This means the TMS in the PRC is in the range 1.4 MHz - 1.6 MHz [see this plot]
• The mode matching efficiency for the PRC is > 98.5% [see this plot]
• The TMS in the SRC is in the range 1.6 MHz - 1.8 MHz [see this plot]
• Mode matching efficiency for SRC is > 98.5% [see this plot]
• PRG between 34-38, depending on uncertainty in measurement of RoC of existing G&H mirror [see Attachment #1, added Nov 11 2016]

Case 2: Order two new sets of folding mirrors

• Spec PR3/SR3 to be flat - for purposes of simulation, let's make it concave with RoC 10 +/- 5 km
• Spec PR2/SR2 to be concave with RoC 1500 +/- 500m
• The TMS in the PRC is between 1.7 MHz and 1.85 MHz [see this plot]
• Mode matching efficiency is >98.5% in the PRC [see this plot]
• TMS in the SRC is between 1.7 MHz and 2 MHz
• Mode matching efficiency >99.0% in the SRC

At first glance, it looks like the tolerances are much larger for Case 2, but we also have to keep in mind that for such large RoCs in the km range, it may be impractical to specify as tight tolerances as in the 100s of metres range. So these are a set of numbers to keep in mind, that we can re-iterate once we hear back from vendors as to what they can do.

For consolidation purposes, here are the aLIGO requirements for the coatings on the RC folding mirrors: PR2, PR3, SR2, SR3

Attachment 1: PRG.pdf
12418   Wed Aug 17 16:28:46 2016 KojiUpdateCOCRC folding mirrors - Numerical review

For the given range of the PR3/SR3 RoCs for both cases, all the resulting numbers such as TMSs/mode matching ratios look reasonable to me.

12631   Mon Nov 21 15:34:24 2016 gautamUpdateCOCRC folding mirrors - updated specs

Following up on the discussion from last week's Wednesday meeting, two points were raised:

1. How do we decide what number we want for the coating on the AR side for 532nm?
2. Do we want to adjust T@1064nm on the HR side to extract a stronger POP beam?

With regards to the coating on the AR side, I've put in R<300ppm@1064nm and R<1000ppm@532nm on the AR side. On the HR side, we have T>97% @ 532nm (copied from the current PR3/SR3 spec), and T<50ppm @1064nm. What are the ghost beams we need to be worried about?

• Scattered light the AR side interfering with the main transmitted green beam possibly making our beat measurement noisier
• With the above numbers, accounting for the fact that we ask for a 2 degree wedge on PR3, the first ghost beam from reflection on the AR side will have an angular separation from the main beam of ~7.6 degrees. So over the ~4m the green beam travels before reaching the PSL table, I think there is sufficient angular separation for us to catch this ghost and dump it.
• Moreover, the power in this first ghost beam will be ~30ppm relative to the main green beam. If we can get R<100ppm @532nm on the AR side, the number becomes 3ppm
• Prompt reflection from the HR surface of PR3 scattering green light back into the arm cavity mode
• The current spec has T>97% @532nm. So 3% is promptly reflected at the HR side of PR3
• I'm not sure how much of a problem this really will be - I couldn't find the reflectivities of PR2 and PRM @532nm (were these ever measured?)
• In any case, if we can have T<50ppm @1064nm and R>99.9% @532nm, that would be better

So in conclusion, with the specs as they are now, I don't think the ALS noise performance is adversely affected. I have updated the spec to have the following numbers now.

HR side: T < 50ppm @1064nm, T>99.9% @532nm

AR side: R < 100ppm @1064nm and @532nm

As for the POP question, if we want to extract a stronger POP beam, we will have to relax the requirement on the transmission @1064nm on the HR side. But recall that the approach we are now considering is to replace only PR3, and flip PR2 back the right way around. Currently, POP is extracted at PR2, so if we want to stick with the idea of getting a new PR3 and extracting a stronger POP beam, there needs to be a major optical layout reshuffle in the BS/PRM chamber. Koji suggested that in the interest of keeping things moving along, we don't worry about POP for the time being...

Alternatively, if it turns out that the vendor can meet the specs for our second requirement (which requires 1.5% of lambda @632nm measurement precision to meet the 10+/-5km RoC tolerance on PR3), then we can ast for T<1000ppm @1064nm for the HR coating on PR2, and keep the coating specs on PR3 as above.

Attached is a pdf with the specs updated to reflect all the above considerations...

Attachment 1: Recycling_Mirrors_Specs_Nov2016.pdf
12847   Thu Feb 23 10:59:53 2017 gautamUpdateCOCRC folding mirrors - coating optimization

I've now made a DCC page for the mirror specifications, all revisions should be reflected there.

Over the last couple of days, I've been playing around with Rana's coating optimization code to come up with a coating design that will work for us. The basic idea is a to use MATLAB's particle swarm constrained optimization tool to minimize an error function that is a composite of four penalties:

1. Thermal noise - we use the proxy function from E0900068-v3 to do this
2. Deviation from target T @1064nm, p-pol
3. Deviation from target T @532nm, p and s-pol
4. HR Surface field

On the AR side, I only considered 2 and 3. The weighting of these four components were set somewhat arbitrarily, but I seem to be able to get reasonable results so I am going with this for now.

From my first pass at it, the numbers I've been able to get, for 19 layer pairs, are (along with some plots):

HR Side:

• T = 50ppm, 1064nm p-pol
• T = 99%, 532nm s and p-pol

(in this picture, the substrate is to the right of layer 38)

AR Side:

• R ~50ppm for 532nm, s and p-pol

(substrate to the right of layer 38)

These numbers are already matching the specs we have on the DCC page currently. I am not sure how much better we can get the specs on the HR side keeping with 19 layer pairs...

All of this data, plus the code used to generate them, is on the gitlab coatings page...

Attachment 1: PR3_R_170222_2006.pdf
Attachment 2: PR3_123_TOnoise_170222_2203.pdf
Attachment 3: PR3_123_Layers_170222_2203.pdf
Attachment 4: PR3AR_R_170222_2258.pdf
Attachment 5: PR3AR_123_Layers_170222_2258.pdf
12887   Tue Mar 14 10:56:33 2017 gautamUpdateCOCRC folding mirrors - coating optimization

Rana suggested including some additional terms to the cost function to penalize high sensitivity to deviations in the layer thickness (L). So the list of terms contributing to the cost function now reads:

1. Thermal noise - we use the proxy function from E0900068-v3 to do this
2. Deviation from target T @1064nm, p-pol
3. Deviation from target T @532nm, p and s-pol
4. HR Surface field
5. The ratio $\frac{d\mathcal{T}/\mathcal{T}}{dL/L}$ with dL/L = 1%, evaluated at 1064nm p-pol and 532nm p and s-pol (only the latter two for the AR side)

I did not include other sensitivity terms, like sensitivity to the refractive index values for the low and high index materials (which are just taken from GWINC).

There is still some arbitrariness in how I chose to weight the relative contributions to the cost function, but after some playing around, I think I have a solution that I think will work. Here are the spectral reflectivity and layer thickness plots for the HR and AR sides respectively.

HR side: for a 1% increase in the thickness of all layers, the transmission changes by 5% @ 1064nm p-pol and 0.5% @ 532nm s and p-pol

AR sidefor a 1% change in the thickness of all layers, the transmission changes by <0.5% @ 532nm s and p-pol

(substrate to the right of layer 38)

I've also checked that we need 19 layer pairs to meet the spec requirements, running the code with fewer layer pairs leads to (in particular) large deviations from the target value of 50ppm @ 1064nm p-pol.

Do these look reasonable?

Attachment 1: PR3_R_170313_1701.pdf
Attachment 2: PR3AR_123_Layers_170313_1701.pdf
Attachment 3: PR3AR_R_170313_1752.pdf
Attachment 4: PR3AR_123_Layers_170313_1752.pdf
12936   Mon Apr 10 15:37:11 2017 gautamUpdateCOCRC folding mirrors - v3 of specs uploaded

Koji and I have been going over these calculations again before we send a list of revised requirements to Ramin. I've uploaded v3 of the specs to the DCC page. Here is a summary of important changes.

1. Change in RoC specification - I condensed the mode-matching information previously in 8 plots into the following 2 plots. Between tangential and saggital planes, the harmonic mean was taken. Between X and Y cavities, the arithmetic mean was taken. Considering the information in the following plots, we decided to change the spec RoC from 600 +/- 50m to 1000 +/- 150m. The required sensitivity in sag measurement is similar to the previous case, so I think this should be feasible.

Why this change? From the phase map information at  /users/public_html/40m_phasemap/40m_TTI gather that we have 2 G&H mirrors, one with curvature ~ -700m and the other with curvature ~ -500m. An elog search suggests that the installed PR2 has RoC ~ -700m, so this choice of RoC for PR3 should give us the best chance of achieving optimal modematching between the RCs and arms as per the plots below.

2. Cavity stability checks - these plots confirm that the cavity remains stable for this choice of RoC on PR3...

3. Coating design - I've been playing around with the code and my understanding of the situation is as follows. to really hit low AR of 10s of ppms, we need many dielectric layer pairs. But by adding more pairs, we essentially become more susceptible to errors in layer thickness etc, so that even though the code may tell us we can achieve R_AR(532nm) < 50ppm, the minima is pretty sharp so even small perturbations can lead to much higher R of the order of a few percent. On the HR side, we need a large number of layer pairs to achieve T_HR(1064nm)=50ppm. Anyways, the MC studies suggest that for the HR coating design, with 19 layer pairs, we can be fairly certain of T_HR(1064nm)<100ppm and R_HR(532nm)>97% for both polarizations, which seems reasonable. In order to make the R_HR(532nm) less susceptible to errors, we need to reduce the number of layer pairs, but then it becomes difficult to achieve the 50ppm T_HR(1064nm) requirement. Now, I tried using very few layer pairs on the AR side - the best result seems to be with 3 layer pairs, for which we get R_AR(532nm)<1% and T_AR(1064nm)>95%, both numbers seem reasonable to me. In the spectral reflectivity, we also see that the minima are much broader than with large number of layer pairs.

First row below is for the HR side, second row is for the AR side. For the MC studies, I perturbed the layer thicknesses and refractive indices by 1%, and the angle of incidence by 5%.

If there are no objections, I would like to send this version of the specs to Ramin and get his feedback. Specifically, I have assumed values for the refractive indices of SiOand Ta2O5 from google, Garilynn tells me that we should get these values from Ramin. Then we can run the code again if necessary, but these MC studies already suggest this coating design is robust to small changes in assumed values of the parameters...

Attachment 1: PRC_modematch.pdf
Attachment 2: SRC_modematch.pdf
Attachment 3: TMS_PRC.pdf
Attachment 4: TMS_SRC.pdf
Attachment 5: PR3_HR_spectralRefl.pdf
Attachment 6: PR3_HR_MC_CDF_revised.pdf
Attachment 7: PR3_AR_spectralRefl_new.pdf
Attachment 8: PR3_AR_MC_CDF_new.pdf
13155   Mon Jul 31 23:39:02 2017 ranaUpdateCOCCavity Scan Simulation Code

Hiro Yamamoto has updated SIS (Static Interferometer Simulation) to allow us to do the MCMC based inference of the 40m arm cavity mirror maps.

In the examples directory I have put 3 files:

1. mcmcCavityScans.m - runs many cavity scans using parfor and saves the data
2. plotCavityScans.m - loads the .mat file with the data and plots it
3. plotCavityScans.py - python file which also loads & plots, but nicer since python has a transparency option for the traces.

Attached is the plots and the data. The first attached plot is a low resolution one: 200 scans of 100 frequency points each. Second plot is 200 scans of 300 points each.

The run was done assuming perfect LIGO arm params with a random set of Zernike perturbations for each run. The amplitude of each Zernike was chosen from a Normal distribution with a standard deviation of 10 nm.

We need to come up with a better guess for the initial distribution from which to sample, and also to use the more smart sampling that one does using the MCMC Hammer.

Attachment 1: manyCavityScans-SIS.pdf
Attachment 2: manyCavityScans-SIS.pdf
Attachment 3: MonteCarlo_CavityScans.mat
14148   Thu Aug 9 02:12:13 2018 gautamUpdateCOCSouth East or West?

Summary:

For operating the SRC in the "Signal-Recycled" tuning, the SRC macroscopic length needs to be ~4.04m (compared to the current value of ~5.399m), assuming we don't do anything fancy like change the modulation frequencies and not transmit through the IMC. We're putting together a notebook with all the calculations, but today I was thinking about what the signal extraction path should be, specifically which chamber the SRM should be in. Just noting down the thoughts I had here while they're fresh in my head, all this has to be fleshed out, maybe I'm making this out to be more of a problem than it actually is.

Details:

• For the current modulation frequencies, if we want the reosnance conditions such that the f2 sideband is resonant in the SRC (but not f1, i.e. small Schnupp asymmetry regime) while the carrier is resonant in the arms (required for good sensing of the SRC length), the macroscopic length of the SRC needs to be changed to ~4.04m.
• Practically, this means that the folded SRC would only have one folding mirror (SR2).
• There is a shorter SRC length of ~1.something metres which would work, but that would involve changing the relative position between ITMs and BS (currently ~2.3m) so I reject that option for now.
• So the SR2 would be roughly where it is right now, ~20cm from the BS.
• The question then becomes, where do we direct the reflection from the SR2? We need an optical path length of ~1.5m from SR2. So options are
• ITMY table (East)
• ITMX table (South)
• IMC table (West)
• Moreover, after the SRM, we have to accommodate:
• Some kind of pickoff for in-air PDs.
• OFI.
• OMC MMT.
• OMC.
• Some kind of CBA (as of now I think going to the ITMY table is the best option):
ITMY
• Easy to direct beam from BS/PRM chamber to the ITMY table (i.e. we don't have to worry too much about avoiding other optics in the path etc).
• ITMY table probably has the most room to work out an OFI + OMC MMT + OMC solution.
• AS beam extraction to air will be more complicated, possibly have to do it on ITMY optical table.
• Not sure if the ITMY table can accommodate all of the output optics subsystems I listed above.
• Routing the LO beam to this table would be tricky I guess.
ITMX
• Routing the LO beam for homodyne detection is probably easiest in this chamber.
• Allows for small AoI on folding mirror, reducing the impact of astigmatism.
• Pain to work in this chamber because of IMC tube.
• Steering beam from SR2 to ITMX table means threading the needle between PRM and PR3 possibly.
IMC
• Probably allows the use of (almost) the entire existing OMC chamber for the output optics (OFI, OMC MMT, OMC).
• IMC table is crowded (2 SOS towers, several steering optics for the input beam, input faraday).
• Not sure what is the performance of the seismic isolation stacks on these tables vs the larger optical tables.
• Painful to work in these smaller chambers.
14164   Wed Aug 15 12:15:24 2018 gautamUpdateCOCMacroscopic SRC length for SR tuning

Summary:

It looks like we can have a stable SRC of length 4.044 m without getting any new mirrors, so this is an option to consider in the short-term.

Details:

• The detailed calculations are in the git repo
• The optical configuration is:
• A single folding mirror approximately at the current SR3 location.
• An SRM that is ~1.5m away from the above folding mirror. Which table the SRM goes on is still an open question, per the previous elog in this thread.
• The SRC length is chosen to be 4.044 m, which is what the modeling tells us we need for operating in the SR tuning instead of RSE.
• Using this macroscopic length, I found that we could use a single folding mirror in the SRC, and that the existing (convex) G&H folding mirrors, which have a curvature of -700m, happily combine with our existing SRM (concave with a curvature of 142m) to give reasonable TMS and mode-matching to the arm cavity.
• The existing SRM transmission of 10% may not be optimal but Kevin's calculations say we should still be able to see some squeezing (~0.8 dB) with this SRM.
• Attachment #1 - corner plot of the distribution of TMS for the vertical and horizontal modes, as well as the mode-matching (averaged between the two modes) between the SRC and arm cavity.
• Attachment #2 - histograms of the distributions of RoCs and lengths used to generate Attachment #1. The distributions were drawn from i.i.d Gaussian pdfs.

gautam 245pm: Koji pointed out that the G&H mirrors are coated for normal incidence, but looking at the measurement, it looks like the optic has T~75ppm at 45 degree incidence, which is maybe still okay. Alternatively, we could use the -600m SR3 as the single folding mirror in the SRC, at the expense of slightly reduced mode-matching between the arm cavity and SRC.

Attachment 1: SRC_MCMC_shortTerm.pdf
Attachment 2: SRC_dists_shortTerm.pdf
14314   Wed Nov 21 16:48:11 2018 gautamUpdateCOCEY mini cleanroom setup

With Chub's help, I've setup a mini cleanroom at EY - Attachment #1. The HEPA unit is running on high now. All surfaces were wiped with isopropanol, we can wipe everything down again on Monday and replace the foil.

Attachment 1: IMG_7174.JPG
15374   Thu Jun 4 00:21:28 2020 KojiSummaryCOCITM spares and New PR3 mirrors transported to Downs for phasemap measurement

GariLynn worked on the measurement of E1800089 mirrros.

The result of the data analysis, as well as the data and the codes, have been summarized here:
https://nodus.ligo.caltech.edu:30889/40m_phasemap/#E1800089

15401   Tue Jun 16 13:05:36 2020 KojiUpdateCOCITM spares and New PR3 mirrors transported to Downs for phasemap measurement

ITMU01 / ITMU02 as well as the five E1800089 mirrors came back to the 40m. Instead, the two ETM spares (ETMU06 / ETMU08) were delivered to GariLynn.
Jordan worked on transportation.

Note that the E1800089 mirrors are together with the ITM container in the precious optics cabinet.

Attachment 1: 40m_Optics.jpg
15625   Wed Oct 14 13:28:04 2020 KojiUpdateCOCITM/ETM spares in Downs

The two ITM spares and two ETM spares are together stored in the optic storage (B110) at Downs. c/o Liyuan and GariLynn

Attachment 1: IMG_3073.jpeg
3193   Mon Jul 12 11:20:56 2010 Gopal HowToCOMSOL TipsIntrusions (Negative Extrusions)

For the sake of future users, I have decided to periodically add tips and tricks in using COMSOL that I have figured out, most probably after hours of circuitous efforts. They will always be listed under the new COMSOL Tips category.

Today's topic: Intrusions

COMSOL has a very user-friendly interface for taking objects from 2D to 3D using the "extrusion" feature. But suppose one wants to design an object which contains screw holes or some other indentation. I've found that creating "punctures" in COMSOL is either impossible or very complicated.

Instead, COMSOL encourages users to always "add" to the object. In other words, one must form the lowest level first, then build layers sequentially on top using new work plane and boolean difference operators. This will probably be a bit clearer with an example:

1) First, create the planar projection in a work plane:

2) Extrude the first layer only in the regular fashion:

3) Add a new work plane which is offset in the z-direction to the deepest point of the intrusion.

4) Now, create the shape of the intrusion in this new work plane.

5) Use the Boolean "Difference" to let COMSOL know that, on this plane, the object has a hole.

6) Extrude once more from the second work plane to complete the intrusion.

3194   Mon Jul 12 12:16:50 2010 DmassHowToCOMSOL TipsIntrusions (Negative Extrusions)

An entry on the 40m wiki page might serve you better, and be easier to sift through once all is said and done

3291   Mon Jul 26 11:15:23 2010 GopalHowToCOMSOL TipsPictures from Transfer Function Tutorial on the Wiki

The attached pictures give a brief overview of my transfer function measurement procedure in COMSOL. For more details, please see the Wiki.

3322   Thu Jul 29 17:11:16 2010 GopalUpdateCOMSOL TipsIncluding Gravity in COMSOL

[Gopal, Jan]

For the past couple of days, Jan and I have been discussing a major issue in COMSOL involving modeling both oscillatory and non-oscillatory forces simultaneously while using FDA. It turns out that he and I had run into the same problem at different times and with different projects. After discussing with an expert, Jan had decided in the past that this simple task was impossible via direct means.

The issue could still be resolved if there was a way for us to work on the Weak Form of the differential equations describing the system:

• Usually, one must define weight as a body load in the negative-z direction. However, this problematically instantiates a new force in COMSOL, which is automatically driven over the range of frequencies during FDA.
• Instead, we could define gravity as an anti-restoring force, since we assume that the base of the stack is fixed.
• In other words, Fg = (ρ*g/L)*x + (ρ*g/L)*y for a point mass which is constrained on the bottom (for small angles).
• Working in Weak Form then, we'd never have to define an explicit gravity load-- this could just be an extra couple of terms in the differential equation which are related entirely to the x- and y-vectors (well-defined for each mesh point). This would fool COMSOL into never tacking on the oscillatory term during FDA.

According to current documentation however, Weak Form analysis is not yet possible in COMSOL 4.0. Jan suggested moving my work over to ANSYS or waiting for the 4.0 upgrade, but there's probably not enough time left in my SURF for either of these options. I suggested attempting a backwards-compatibility test to COMSOL 3.5; Jan and I will be exploring this option some time next week.

3536   Tue Sep 7 20:44:54 2010 YoichiHowToCOMSOL TipsCOMSOL example for calculating mechanical transfer functions

I added COMSOL example files to the 40m svn to demonstrate how to make transfer function measurements in COMSOL.

https://nodus.ligo.caltech.edu:30889/svn/trunk/comsol/MechanicalTF/

The directory also contains an (incomplete) explanation of the method in a PDF file.

8190   Wed Feb 27 19:27:29 2013 AnnalisaHowToCOMSOL TipsMirror support Eigenfrequency

I studied the eigenfrequencies of a mirror support using COMSOL.

Attachment 1: IronSupport.png
Attachment 2: IronSupportEigenfreq.png
8226   Mon Mar 4 20:03:42 2013 AnnalisaHowToCOMSOL TipsStudy of mirror mount eigenfrequencies

I studied the eigenfrequencies of a mirror mount designed with COMSOL.

I imposed fixed constraints for the base screws and for the screw connecting the base with the pedestal. Note that the central screw is connected to the base only for a small thickness, and the pedestal touches the base only with a thin annulus. This is in way to make a better model of the actual stress.

Shown in fig. 2 is the lowest eigenfrequency of the mount.

I' going to change the base and study the way the eigenfrequency vary, in way to find the configuration which minimizes the lowest eigenfrequency.

Attachment 1: MirrorSupport1.png
Attachment 2: MirrorSupportEig1.png
Attachment 3: pedestal.png
Attachment 4: Base2.png
8437   Wed Apr 10 15:49:22 2013 AnnalisaConfigurationCOMSOL TipsYend table eigenfrequency simulation with COMSOL

I made a Simulation with COMSOL for the Yend table. Mainly, I tried to see how the lower eigenmode changes with the number and the size of the posts inside.

The lateral frame is just sitting on the table, it is fixed by its weight. I also put a couple of screws to fix it better, but the resulting eigenfrequency didn't change so much (less than 1 Hz).

In Fig. 1 I didn't put any post. Of course, the lowest eigenfrequency is very low (around 80 Hz).

Then I added 2 posts, one per side (Fig. 2 and Fig. 3), with different diameter.

In some cases posts don't have a base, but they are fixed to the table only by a screw. It is just a condition to keep them fixed to the table

Eventually I put 4 posts, 2 per side.

The lowest eigenfrequency is always increasing.

At the end I also put a simulation for 4 1.6 inch diameter posts without base, and the eigenfrequency is slightly higher. I want to check it again, because I would expect that the configuration shown in Fig.5a could be more stable.

P.S.: All the post are stainless steel.

Attachment 1: Pics_end_table.pdf
15650   Thu Oct 29 09:50:12 2020 anchalSummaryCalibrationPreliminary calibration measurement taken

I went to 40m yesterday at around 2:30 pm and Koji showed me how to acquire lock in different arms and for different lasers. Finally, we took a preliminary measurement of shaking the ETMX at some discrete frequencies and looking at the beatnote frequency spectrum of X-end laser's fiber-coupled IR and Main laser's IR pick-off.

### Basic controls and measurement 101 at 40m

• I learned a few things from Koji about how to align the cavity mirrors for green laser or IR laser.
• I learned how to use ASS and how to align the green end laser to the cavity. I also found out about the window at ETMX chamber where we can directly see the cavity mode, cool stuff.
• Koji also showed me around on how to use diaggui and awggui for taking measurements with any of the channels.

### Preliminary measurement for calibration scheme

We verified that we can send discrete frequency excitation signals to ETMX actuators directly and see a corresponding peak in the spectrum of beatnote frequency between fiber-coupled X-end IR laser and main laser IR pickoff.

• I sent excitation signal at 200 Hz, 250 Hz and 270 Hz at C1:SUS-ETMX_LSC_EXC channel using awggui with an amplitude of 100 cts and gain of 2.
• I measured corresponding peaks in the beatnote spectrum using diaggui.
• Page 1 shows the ASD data for the 4 measurements taken with Hanning window and averaging of 10.
• Page 2 shows close up Spectrum data for the 4 measurements taken with flattop window and averaging of 10.
• I converted this frequency signal into displacement by using conversion factor $\nu_{FSR}/\frac{\lambda}{2}$ or $\frac{L \lambda}{c}$.

If full interferometer had been locked, we could have used the DARM error signal output to calibrate it against this measurement.

Data

Attachment 1: PreliminaryCalibrationData.pdf
16128   Mon May 10 10:57:54 2021 Anchal, PacoSummaryCalibrationUsing ALS beatnote for calibration, test

### Test details:

• We locked both arms and opened the shutter for Yend green laser.
• After toggling the shutter on.off, we got a TEM00 mode of green laser locked to YARM.
• We then cleared the phase Y history by clicking "CLEAR PHASE Y HISTROY" on C1LSC_ALS.adl (opened from sitemap > ALS > ALS).
• We sent excitation signal at ITMY_LSC_EXC using awggui at 43Hz, 77Hz and 57Hz.
• We measured the power spectrum and coherence of C1:ALS-BEATY_FINE_PHASE_OUT_HZ_DQ and C1:SUS-ITMY_LSC_OUT_DQ.
• The BEATY_FINE_PHASE_OUT_HZ is already calibrated in Hz. This we assume is done by multip[lying the VCO slope in Hz/cts to the error signal of the digital PLL loop that tracks the phase of beatnote.
• We calibrated C1:SUS-ITMY_LSC_OUT_DQ by multiplying with
$\dpi{150} \large 3 \times \frac{2.44 \, nm/cts}{f^2} \times \frac{c}{1064\,nm \times 37.79\, m} = \frac{54.77}{f^2} kHz/cts$ where f is in Hz.
The 2.44/f2 nm/cts is taken from 13984.
• We added the calibration as Poles/zeros option in diaggui using gain=54.577e3 and poles as "0, 0".
• We found that ITMY_LSC_OUT_DQ calibration matches well at 57Hz but overshoots (80 vs 40) at 43 Hz and undershoots (50 vs 80) at 77Hz.

### Conclusions:

• If we had DRFPMI locked, we could have used the beatnote spectrum as independent measurement of arm lengths to calibrate the interferometer output.
• We can also use the beatnote to confirm or correct the ITM actuator calibrations. Maybe shape is not exactly 1/f2 unless we did something wrong here or the PLL bandwidth is too short.
Attachment 1: BeatY_ITMY_CalibrationAt57Hz.pdf
Attachment 2: BeatY_ITMY_CalibrationAt43Hz.pdf
Attachment 3: BeatY_ITMY_CalibrationAt77Hz.pdf
16315   Tue Sep 7 18:00:54 2021 TegaSummaryCalibrationSystem Identification via line injection

[paco]

This morning, I spent some time restoring the jupyter notebook server running in allegra. This server was first set up by Anchal to be able to use the latest nds python API tools which is handy for the calibration stuff. The process to restore the environment was to run "source ~/bashrc.d/*" to restore some of the aliases, variables, paths, etc... that made the nds server work. I then ran ssh -N -f -L localhost:8888:localhost:8888 controls@allegra from pianosa and carry on with the experiment.

[paco, hang, tega]

We started a notebook under /users/paco/20210906_XARM_Cal/XARM_Cal.ipynb on which the first part was doing the following;

• Set up list of excitations for C1:LSC-XARM_EXC (for example three sine waveforms) using awg.py
• Make sure the arm is locked
• Read a reference time trace of the C1:LSC-XARM_IN2 channel for some duration
• Start excitations (one by one at the moment, ramptime ~ 3 seconds, same duration as above)
• Get data for C1:LSC-XARM_IN2 for an equal duration (raw data in Attachment #1)
• Generate the excitation sine and cosine waveforms using numpy and demodulate the raw timeseries using a 4th order lowpass filter with fc ~ 10 Hz
• Estimate the correct demod phase by computing arctan(Q / I) and rerunning the demodulation to dump the information into the I quadrature (Attachment #2).
• Plot the estimated ASD of all the quadratures (Attachment #3)

[paco, hang, tega]

Estimation of open loop gain:

• Grab data from the C1:LSC-XARM_IN1 and C1:LSC-XARM_IN2 test points
• Infer excitation from their differnce, i.e. C1:LSC-XARM_EXC = C1:LSC-XARM_IN2 - C1:LSC-XARM_IN1
• Compute the open loop gain as follows : G(f) = csd(EXC,IN1)/csd(EXC,IN2), where csd computes the cross spectra density of the input arguments
• For the uncertainty in G, dG, we repeat steps (1) to (3) with & without signal injection in the C1:LSC-XARM_EXC channel. In the absence of signal injection, the signal in C1:LSC-XARM_IN2 is of the form: Y_ref = Noise/(1-G), whereas with nonzero signal injection, the signal in C1:LSC-XARM_IN2 has the form: Y_cal = EXC/(1-G) + Noise/(1-G), so their ratio, Y_cal/Y_ref = EXC/Noise, gives the SNR, which we can then invert to give the uncertainty in our estimation of G, i.e dG = Y_ref/Y_cal.
• For the excitation at 53 Hz, our measurtement for the open loop gain comes out to about 5 dB whiich is consistent with previous measurement.
• We seem to have an SNR in excess of 100 at measurement time of 35 seconds and 1 count of amplitude which gives a relative uncertainty of G of 0.1%
• The analysis details are ongoing. Feedback is welcome.
Attachment 1: raw_timeseries.pdf
Attachment 2: demod_signals.pdf
Attachment 3: cal_noise_asd.pdf
16352   Tue Sep 21 11:13:01 2021 PacoSummaryCalibrationXARM calibration noise

Here are some plots from analyzing the C1:LSC-XARM calibration. The experiment is done with the XARM (POX) locked, a single line is injected at C1:LSC-XARM_EXC at f0 with some amplitude determined empirically using diaggui and awggui tools. For the analysis detailed in this post, f0 = 19 Hz, amp = 1 count, and gain = 300 (anything larger in amplitude would break the lock, and anything lower in frequency would not show up because of loop supression). Clearly, from Attachment #3 below, the calibration line can be detected with SNR > 1.

We read the test point right after the excitation C1:LSC-XARM_IN2 which, in a simplified loop will carry the excitation suppressed by 1 - OLTF, the open loop transfer function. The line is on for 5 minutes, and then we read for another 5 minutes but with the excitation off to have a reference. Both the calibration and reference signal time series are shown in Attachment #1 (decimated by 8). The corresponding ASDs are shown in Attachment #2. Then, we demodulate at 19 Hz and a 30 Hz, 4th-order butterworth LPF, and get an I and Q timeseries (shown in Attachment #3). Even though they look similar, the Q is centered about 0.2 counts, while the I is centered about 0.0. From this time series, we can of course show the noise ASDs in Attachment #3.

The ASD uncertainty bands in the last plot are statistical estimates and depend on the number of segments used in estimating the PSD. A thing to note is that the noise features surrounding the signal ASD around f0 are translated into the ASD in the demodulated signals, but now around dc. I guess from Attachment #3 there is no difference in the noise spectra around the calibration line with and without the excitation. This is what I would have expected from a linear system. If there was a systematic contribution, I would expect it to show at very low frequencies.

Attachment 1: XARM_signal_asd.pdf
Attachment 2: XARM_demod_timeseries.pdf
Attachment 3: XARM_demod_asds.pdf
Attachment 4: XARM_cal_0921_timeseries.pdf
16353   Wed Sep 22 11:43:04 2021 ranaSummaryCalibrationXARM calibration noise

I would expect to see some lower frequency effects. i.e. we should look at the timeseries of the demod with the excitation on and off.

I would guess tat the exc on should show us the variations in the optical gain below 3 Hz, whereas the exc off would not show it.

Maybe you should do some low pass filtering on the time series you have to see the ~DC effects? Also, reconsider your AA filter design: how do you quantitatively choose the cutoff frequency and stopband depth?

16363   Tue Sep 28 16:31:52 2021 PacoSummaryCalibrationXARM OLTF (calibration) at 55.511 Hz

[anchal, paco]

Here is a demonstration of the methods leading to the single (X)arm calibration with its budget uncertainty. The steps towards this measurement are the following:

1. We put a single line excitation through the C1:SUS-ETMX_LSC_EXC at 55.511 Hz, amp = 1 counts, gain = 300 (ramptime=10 s).
2. With the arm locked, we grab a long timeseries of the C1:LSC-XARM_IN1_DQ (error point) and C1:SUS-ETMX_LSC_OUT_DQ (control point) channels.
3. We assume the single arm loop to have the four blocks shown in Attachment #1, A (actuator + sus), plant (mainly the cavity pole), D (detection + electronics), and K (digital control).
1. At this point, Anchal made a model of the single arm loop including the appropriate filter coefficients and other parameters. See Attachments #2-3 for the split and total model TFs.
2. Our line would actually probe a TF from point b (error point) to point d (control point). We multiplied our measurement with open loop TF from b to d from model to get complete OLTF.
3. Our initial estimate from documents and elog made overall loop shape correct but it was off by an overall gain factor. This could be due to wrong assumption on RFPD transimpedance or analog gains of AA or whitening filters. We have corrected for this factor in the RFPD transimpedance, but this needs to be checked (if we really care).
4. We demodulate decimated timeseries (final sampling rate ~ 2.048 kHz) and I & Q for both the b and d signals. From this and our model for K, we estimate the OLTF. Attachment #4 shows timeseries for magnitude and phase.
5. Finally, we compute the ASD for the OLTF magnitude. We plot it in Attachment #5 together with the ASD of the XARM transmission (C1:LSC-TRX_OUT_DQ) times the OLTF to estimate the optical gain noise ASD (this last step was a quick attempt at budgeting the calibration noise).
1. For each ASD we used N = 24 averages, from which we estimate rms (statistical) uncertainties which are depicted by error bands ($\pm \sigma$) around the lines.

** Note: We ran the same procedure using dtt (diaggui) to validate our estimates at every point, as well as check our SNR in b and d before taking the ~3.5 hours of data.

Attachment 1: OLTF_Calibration_Scheme.jpg
Attachment 2: XARM_POX_Lock_Model_TF.pdf
Attachment 3: XARM_OLTF_Total_Model.pdf
Attachment 4: XARM_OLTF_55p511_Hz_timeseries.pdf
Attachment 5: Gmag_55p511_Hz_ASD.pdf
16369   Thu Sep 30 18:04:31 2021 PacoSummaryCalibrationXARM OLTF (calibration) with three lines

[anchal, paco]

We repeated the same procedure as before, but with 3 different lines at 55.511, 154.11, and 1071.11 Hz. We overlay the OLTF magnitudes and phases with our latest model (which we have updated with Koji's help) and include the rms uncertainties as errorbars in Attachment #1.

We also plot the noise ASDs of calibrated OLTF magnitudes at the line frequencies in Attachment #2. These curves are created by calculating power spectral density of timeseries of OLTF values at the line frequencies generated by demodulated XARM_IN and ETMX_LSC_OUT signals. We have overlayed the TRX noise spectrum here as an attempt to see if we can budget the noise measured in values of G to the fluctuation in optical gain due to changing power in the arms. We multiplied the the transmission ASD with the value of OLTF at those frequencies as the transfger function from normalized optical gain to the total transfer function value.

It is weird that the fluctuations in transmission power at 1 mHz always crosses the total noise in the OLTF value in all calibration lines. This could be an artificat of our data analysis though.

Even if the contribution of the fluctuating power is correct, there is remaining excess noise in the OLTF to be budgeted.

Attachment 1: XARM_OLTF_Model_and_Meas.pdf
Attachment 2: Gmag_ASD_nb_withTRX.pdf
16373   Mon Oct 4 15:50:31 2021 HangUpdateCalibrationFisher matrix estimation on XARM parameters

[Anchal, Hang]

What: Anchal and I measured the XARM OLTF last Thursday.

Goal: 1. measure the 2 zeros and 2 poles in the analog whitening filter, and potentially constrain the cavity pole and an overall gain.

2. Compare the parameter distribution obtained from measurements and that estimated analytically from the Fisher matrix calculation.

3. Obtain the optimized excitation spectrum for future measurements.

How: we inject at C1:SUS-ETMX_LSC_EXC so that each digital count should be directly proportional to the force applied to the suspension. We read out the signal at C1:SUS-ETMX_LSC_OUT_DQ. We use an approximately white excitation in the 50-300 Hz band, and intentionally choose the coherence to be only slightly above 0.9 so that we can get some statistical error to be compared with the Fisher matrix's prediction. For each measurement, we use a bandwidth of 0.25 Hz and 10 averages (no overlapping between adjacent segments).

The 2 zeros and 2 poles in the analog whitening filter and an overall gain are treated as free parameters to be fitted, while the rest are taken from the model by Anchal and Paco (elog:16363). The optical response of the arm cavity seems missing in that model, and thus we additionally include a real pole (for the cavity pole) in the model we fit. Thus in total, our model has 6 free parameters, 2 zeros, 3 poles, and 1 overall gain.

The analysis codes are pushed to the 40m/sysID repo.

===========================================================

Results:

Fig. 1 shows one measurement. The gray trace is the data and the olive one is the maximum likelihood estimation. The uncertainty for each frequency bin is shown in the shaded region. Note that the SNR is related to the coherence as

SNR^2 = [coherence / (1-coherence)] * (# of average),

and for a complex TF written as G = A * exp[1j*Phi], one can show the uncertainty is given by

\Delta A / A = 1/SNR,  \Delta \Phi = 1/SNR [rad].

Fig. 2. The gray contours show the 1- and 2-sigma levels of the model parameters using the Fisher matrix calculation. We repeated the measurement shown in Fig. 1 three times, and the best-fit parameters for each measurement are indicated in the red-crosses. Although we only did a small number of experiments, the amount of scattering is consistent with the Fisher matrix's prediction, giving us some confidence in our analytical calculation.

One thing to note though is that in order to fit the measured data, we would need an additional pole at around 1,500 Hz. This seems a bit low for the cavity pole frequency. For aLIGO w/ 4km arms, the single-arm pole is about 40-50 Hz. The arm is 100 times shorter here and I would naively expect the cavity pole to be at 3k-4k Hz if the test masses are similar.

Fig. 3. We then follow the algorithm outlined in Pintelon & Schoukens, sec. 5.4.2.2, to calculate how we should change the excitation spectrum. Note that here we are fixing the rms of the force applied to the suspension constant.

Fig. 4 then shows how the expected error changes as we optimize the excitation. It seems in this case a white-ish excitation is already decent (as the TF itself is quite flat in the range of interest), and we only get some mild improvement as we iterate the excitation spectra (note we use the color gray, olive, and purple for the results after the 0th, 1st, and 2nd iteration; same color-coding as in Fig. 3).

Attachment 1: tf_meas.pdf
Attachment 2: fisher_est_vs_data.pdf
Attachment 3: Pxx_evol.pdf
Attachment 4: fisher_evol.pdf
16399   Wed Oct 13 15:36:38 2021 HangUpdateCalibrationXARM OLTF

We did a few quick XARM oltf measurements. We excited C1:LSC-ETMX_EXC with a broadband white noise upto 4 kHz. The timestamps for the measurements are: 1318199043 (start) - 1318199427 (end).

We will process the measurement to compute the cavity pole and analog filter poles & zeros later.

Attachment 1: Screenshot_2021-10-13_15-32-16.png
16957   Tue Jun 28 17:07:47 2022 AnchalUpdateCalibrationAdded Beatnote channels in demodulation of c1cal

I added today demodulation of C1:LSC-BEATX/Y_FINE_I/Q in the c1cal demodulation where different degrees of freedom can be dithered. For McCal (formerly soCal), we'll dither the arm cavity for which we can use any of the DOFs (like DARM) to send the dither to ETMX/ETMY. Then with green laser locked as well, we'll get the calibration signal from the beatnotes in the demodulaed channels. We can also read right after the mixing in c1cal model and try differnt poles for integration .

I've also added medm screens in the sensing matrix part of LSC screen. These let you see demodulation of beatnote frequency signals.

17010   Mon Jul 18 04:42:54 2022 AnchalUpdateCalibrationError propagation to astrophysical parameters from detector calibration uncertainty

We can calculate how much detector calibration uncertainty affects the estimation of astrophysical parameters using the following method:

Let $\overrightarrow{\Theta}$ be set of astrophysical parameters (like component masses, distance etc), $\overrightarrow{\Lambda}$be set of detector parameters (like detector pole, gain or simply transfer function vaue for each frequency bin). If true GW waveform is given by $h(f; \overrightarrow{\Theta})$, and the detector transfer function is given by $\mathcal{R}(f; \overrightarrow{\Lambda})$, then the detected gravitational waveform becomes:
$g(f; \Theta, \Lambda) = \frac{\mathcal{R}(f; \overrightarrow{\Lambda_t})}{\mathcal{R}(f; \overrightarrow{\Lambda})} h(f; \overrightarrow{\Theta})$

One can calculate a derivative of waveform with respect to the different parameters and calculate Fisher matrix as (see correction in 40m/17017):

$\Gamma_{ij} = \left( \frac{\partial g}{\partial \mu_i} | \frac{\partial g}{\partial \mu_j}\right )$

where the bracket denotes iner product defined as:

$\left( k_1 | k_2 \right) = 4 Re \left( \int df \frac{k_1(f)^* k_2(f))}{S_{det}(f)}\right)$

where $S_{det}(f)$ is strain noise PSD of the detector.

With the gamma matrix in hand, the error propagation from detector parameter fractional errors $\frac{\Delta \Lambda_j}{\Lambda_j}$to astrophysical paramter fractional errors $\frac{\Delta \Theta_i}{\Theta_i}$is given by (eq 26 in Evan et al 2019 Class. Quantum Grav. 36 205006):

$\frac{\Delta \Theta_j}{\Theta_j} = - \mathbf{H}^{-1} \mathbf{M} \frac{\Delta \Lambda_j}{\Lambda_j}$

where $\mathbf{H}_{ij} = \left( \frac{\partial g}{\partial \Theta_i} | \frac{\partial g}{\partial \Theta_j}\right )$ and $\mathbf{M}_{ij} = \left( \frac{\partial g}{\partial \Lambda_i} | \frac{\partial g}{\partial \Theta_j}\right )$.

Using the above mentioned formalism, I looked into two ways of calculating error propagation from detector calibration error to astrophysical paramter estimations:

## Using detector response function model:

If we assume detector response function as a simple DC gain (4.2 W/nm) and one pole (500 Hz) transfer function, we can plot conversion of pole frequency error into astrophysical parameter errors. I took two cases:

• Binary Neutron Star merger with star masses of 1.3 and 1.35 solar masses at 100 Mpc distance with a $\tilde{\Lambda}$ of 500. (Attachment 1)
• Binary black hole merger with black masses of 35 and 30 at 400 MPc distance with spin along z direction of 0.5 and 0.8. (I do not fully understand the meaning of these spin components but a pycbc waveform generation model still lets me calculate the effect of detector errors) (Attachment 2)

The plots are plotted in both loglog and linear plots to show the order of magnitude effect and how the error propsagation slope is different for different parameters. 'm still not sure which way is the best to convey the information. The way to read this plot is for a given error say 4% in pole frequency determination, what is the expected error in component masses, merger distance etc. I

Note that the overall gain of detector response is not sensitive to astrophysical error estimation.

## Using detector transfer function as frequency bin wise multi-parameter function

Alternatively, we can choose to not fit any model to the detector transfer function and simply use the errors in magnitude and phase at each frequency point as an independent parameter in the above formalism. This then lets us see what is the error propagation slope for each frequency point. The hope is to identify which parts of the calibration function are more important to calibrate with low uncertainty to have the least effect on astrophysical parameter estimation. Attachment 3 and 4 show these plots for BNS and BBH cases mentioned above. The top panel is the error propagation slope at each frequency due to error in magnitude of the detector transfer function at that frequency and the bottom panel is the error propagation slope at each frequency due to error in phase of the detector transfer function.

The calibration error in magnitude and phase as a function of frequency would be multiplied by the curves and summed together, to get total uncertainty in each parameter estimation.

This is my first attempt at this problem, so I expect to have made some mistakes. Please let me know if you can point out any. Like, do the order of magnitude and shape of error propagation makes sense? Also, comments/suggestions on the inference of these plots would be helpful.

Finally, I haven't yet tried seeing how these curves change for different true values of the merger event parameters. I'm not yet sure what is the best way to extract some general information for a variety of merger parameters.

Future goals are to utilize this information in informing system identification method i.e. multicolor calibration scheme parameters like calibration line frequencies and strength.

Code location

Attachment 1: BNSparamsErrorwrtfdError-merged.pdf
Attachment 2: BBHparamsErrorwrtfdError-merged.pdf
Attachment 3: BNSparamsEPSwrtCalError.pdf
Attachment 4: BBHparamsEPSwrtCalError.pdf
17011   Mon Jul 18 15:17:51 2022 HangUpdateCalibrationError propagation to astrophysical parameters from detector calibration uncertainty

1. In the error propogation equation, it should be \Delta \Theta = -H^{-1} M \Delta \Lambda, instead of the fractional error.

2. For the astro parameters, in general you would need t_c for the time of coalescence and \phi_c for the phase. See, e.g., https://ui.adsabs.harvard.edu/abs/1994PhRvD..49.2658C/abstract.

3. Fig. 1 looks very nice to me, yet I don't understand Fig. 3... Why would phase or amplitude uncertainties at 30 Hz affect the tidal deformability? The tide should be visible only > 500 Hz.

4. For BBH, we don't measure individual spin well but only their mass-weighted sum, \chi_eff = (m_1*a_1 + m_2*a_2)/(m_1 + m_2). If you treat S1z and S2z as free parameters, your matrix is likely degenerate. Might want to double-check. Also, for a BBH, you don't need to extend the signal much higher than \omega ~ 0.4/M_tot ~ 10^4 Hz * (Ms/M_tot). So if the total mass is ~ 100 Ms, then the highest frequency should be ~ 100 Hz. Above this number there is no signal.

17017   Tue Jul 19 07:34:46 2022 AnchalUpdateCalibrationError propagation to astrophysical parameters from detector calibration uncertainty

1. Yeah, that's correct, that equation normally $\Delta \Theta = -\mathbf{H}^{-1} \mathbf{M} \Delta \Lambda$ but it is different if I define $\Gamma$ bit differently that I did in the code, correct my definition of $\Gamma$ to :
$\Gamma_{ij} = \mu_i \mu_j \left( \frac{\partial g}{\partial \mu_i} | \frac{\partial g}{\partial \mu_j} \right )$
then the relation between fractional errors of detector parameter and astrophysical parameters is:
$\frac{\Delta \Theta}{\Theta} = - \mathbf{H}^{-1} \mathbf{M} \frac{\Delta \Lambda}{\Lambda}$
I prefer this as the relation between fractional errors is a dimensionless way to see it.
2. Thanks for pointing this out. I didn't see these parameters used anywhere in the examples (in fact there is no t_c in documentation even though it works). Using these did not affect the shape of error propagation slope function vs frequency but reduced the slope for chirped Mass $M_c$ by a couple of order of magnitudes.
1. I used the get_t_merger(f_gw, M1, M2) function from Hang's work to calculate t_c by assuming $f_{gw}$ must be the lowest frequency that comes within the detection band during inspiral. This function is:
$t_c = \frac{5}{256 \pi^{8/3}} \left(\frac{c^3}{G M_c}\right)^{5/3} f_{gw}^{-8/3}$
For my calculations, I've taken $f_{gw}$ as 20 Hz.
2. I used the get_f_gw_2(f_gw_1, M1, M2, t) function from Hang's work to calculate the evolution of the frequency of the IMR defined as:
$f_{gw}(t) = \left( f_{gw0}^{-8/3} - \frac{768}{15} \pi^{8/3} \left(\frac{G M_c}{c^3}\right)^{5/3} t \right)^{-3/8}$
where $f_{gw0}$ is the frequency at t=0. I integrated this frequency evolution for t_c time to get the coalescence phase phi_c as:
$\phi_c = \int^{t_c}_0 2 \pi f_{gw}(t) dt$
3. In Fig 1, which representation makes more sense, loglog of linear axis plot? Regarding the affect of uncertainties on Tidal amplitude below 500 Hz, I agree that I was also expecting more contribution from higher frequencies. I did find one bug in my code that I corrected but it did not affect this point. Maybe the SNR of chosen BNS parameters (which is ~28) is too low for tidal information to come reliably anyways and the curve is just an inverse of the strain noise PSD, that is all the information is dumped below statistical noise. Maybe someone else can also take a look at get_fisher2() function that I wrote to do this calculation.
4. Now, I have made BBH parameters such that the spin of the two black holes would be assumed the same along z. You were right, the gamma matrix was degenerate before. To your second point, I think the curve also shows that above ~200 Hz, there is not much contribution to the uncertainty of any parameter, and it rolls-off very steeply. I've reduced the yspan of the plot to see the details of the curve in the relevant region.
 Quote: 1. In the error propogation equation, it should be \Delta \Theta = -H^{-1} M \Delta \Lambda, instead of the fractional error.  2. For the astro parameters, in general you would need t_c for the time of coalescence and \phi_c for the phase. See, e.g., https://ui.adsabs.harvard.edu/abs/1994PhRvD..49.2658C/abstract. 3. Fig. 1 looks very nice to me, yet I don't understand Fig. 3... Why would phase or amplitude uncertainties at 30 Hz affect the tidal deformability? The tide should be visible only > 500 Hz. 4. For BBH, we don't measure individual spin well but only their mass-weighted sum, \chi_eff = (m_1*a_1 + m_2*a_2)/(m_1 + m_2). If you treat S1z and S2z as free parameters, your matrix is likely degenerate. Might want to double-check. Also, for a BBH, you don't need to extend the signal much higher than \omega ~ 0.4/M_tot ~ 10^4 Hz * (Ms/M_tot). So if the total mass is ~ 100 Ms, then the highest frequency should be ~ 100 Hz. Above this number there is no signal.

Attachment 1: BNSparamsErrorwrtfdError.pdf
Attachment 2: BBHparamsErrorwrtfdError.pdf
Attachment 3: BNSparamsEPSwrtCalError.pdf
Attachment 4: BBHparamsEPSwrtCalError.pdf
17029   Sun Jul 24 08:56:01 2022 HangUpdateCalibrationError propagation to astrophysical parameters from detector calibration uncertainty

Sorry I forgot to put tc & phic in the example.

I modified astroFisherLib.py to include these parameters. Please note that their meaning is that we don't know when the signal happens and at which phase it merges.

It does not mean the time & phase from a reference frequency to the merger. This part is not free to vary because it is fixed by the intrinsic parameters.

It might be good to have a quick scan through the Cutler & Flanagan 94 paper to better understand their physical meanings.

10436   Thu Aug 28 11:02:53 2014 SteveUpdateCalibration-RepairSR785 repair

SN 46,795 of 2003 is back.

Attachment 1: 08281401.PDF
ELOG V3.1.3-