I measured the beat note between the "Alberto" NPRO laser and the PSL varying the PSL temperature and find the matching NPRO temperature that gave the beat.

I first switched off the FSS loop for the PSL, then I varied its temperature and switched on the loop back.

PSL temperature has been varied starting from 31.88 °C (its starting temperature) down to 23.88 by 1°C step, and then from 31.88 °C up to 36.92 °C, always with a 1°C step.

For each PSL temperature, the NPRO temperature was varied as well, in way to find the temperature to have a beat note between the two.

The trend of the NPRO laser temperature reminds the frequency change of the laser as a function of the crystal temperature continuous tuning.

I made measurements only for the first temperature of the NPRO laser which gave me the beat note. Tomorrow I'm going to find the beat note also for higher frequencies of the NPRO laser.

 The beat note between the PSL laser and the "Alberto" NPRO laser has been measured. In particular, for each PSL temperature, more than one Aux laser frequency has been found.

The second of the three curves seems to be more stable than the other two, even if a "step" trend can be found in all of them (maybe due to the frequency change of the NPRO laser as a function of the crystal temperature continuous tuning, as mentioned in the previous elog). This is the reason why the points are not perfectly aligned, and the errors on the fit parameters are so big.

​Summary:

Attachment #1 shows the RIN and phase noise requirements for the 40m BHD for measuring Ponderomotive squeezing.

Some details:

1. The interferometer topology is not systematically optimized - I just picked values which are likely close to what we will eventually choose. Namely, , , , , . Nevertheless, I think these requirements will not change by more than 30% for changes to the interferometer config.
2. The requirements are evaluated using the following criterion: assuming that the dominant noises are (i) coil driver at mid-frequencies and (ii) quantum noise at high frequencies, what do the RIN and phase noise on the LO have to be such that the equivalent displacement noise is a factor of 10 below? I opted for a safety factor of 10, this can be relaxed. 
3. An unknown is how much contrast defect light we will end up having due to the mismatch between arms. I assumed a few representative values.
4. The calculations were done analytically. This paper provides a good summary of the relations - although my RIN requirement is more stringent because of the safety factor of 10, and phase noise requirement is less stringent (despite the same safety factor) because we plan to read out at nearly the amplitude quadrature.
5. Since we are discussing the possibility of delivering the LO field using a fiber-coupled pickoff of the laser prior to RF sidebands being added, these requirements do not benefit from passive filtering from the cavity transfer functions. Consequently, the requirements are pretty challenging I think.

Conclusions:

1. The RIN requirement looks very challenging - we will need a shot noise limited ISS with 100 mW DC sensing light, and will likely have to relax the safety factor depending on how much contrast defect light we end up having. This actually sets some requirement on the amount of filtering we need from the OMC (next step).
2. The phase noise requirement also looks very challenging - I need to look up what is possible with the double-pass through fiber technique.

​Next steps:

1. Evaluate the pointing stability requirement on the LO field (IFO output is filtered by the OMC).
2. We still need to think of a control scheme for the LO phase - likely, I think we will need a suspended optic between the fiber collimator delivering the light to the BHD setup with some kind of length actuation capability. 
3. Numerical validation of this analytic study. I believe Finesse is still missing some capabilities that allow us to calculate these couplings, but I'll ask the experts to be sure.
4. Build up the requirements on the OMC cavity:
   • Backscatter requirement (related = OFI isolation requirement, relative length noise between SRM and OMC, OFI and SRM). Does the OFI also have to be suspended?
   • Filtering requirement
   • Pointing stability requirement
   • Length noise requirement 

Summary:

I did some more calculations based on our discussions at the meeting yesterday. Posting preliminary results here for comments.

Details:

Attachment #1 - Schematic illustration for the scattering scenarios. For all three scenarios, we would like for the scattered field to be lower than unsqueezed vacuum (safety factor to be debated).

Attachment #2 - Requirements on a fraction  of the counter-propagating resonant mode of the OMC scattering back into the antisymmetric port, as a function of RIN and phase noise on this field (y-axis) and amount of field (depends on the amount of contrast defect light which can become resonant in the counter propagating mode). I don't encode any frequency dependence here.

Attachment #3 - Requirements on the direct scatter from the arm cavity resonant field (assumed to dominate any contribution from the PRC) onto the OMC DCPDs, for some assumed phase noise (y-axis) and fraction of the field that makes it onto the OMC DCPDs. This is a pretty stringent requirement. But the probability is low (it is the product of three presumably small numbers, (i) probablity of the beam scattering out of the TEM00 mode, (ii) BRDF of the scattering surface, (iii) probability of scattering back towards the DCPDs), so maybe feasible? I didn't model any RIN on this field, which would be an additional noise term to contend with. The range of the y-axis was chosen because I think these are reasonable amplitudes for chamber wall  / other scattering surface motion at acoustic frequencies.

I drew out some idea of how we might use a single OMC to clean both paths of the BHD after mixing, without being susceptible to polarization-dependent effects within the OMC. Basically, can we send the two legs of the BHD into the OMC counterpropagating. I've attached a diagram.

I think one issue would be scattered light, since any backscatter directly couples into the counterpropagating mode, and thus directly to the PD. However, unless the polarization of the scattered light rotates it would not scatter back to the IFO. And, since the LO and signal mix before the OMC, this scattered light would not directly add phase noise.

Maybe more problematic would be that if the rejection at the PBS (or the polarization rotation) isn't perfect, light from the LO directly couples into the dark port. Can we get away with a Faraday isolator before the OMC? 

Diagram attached.

I think a Faraday rotator rotates the polarizations in a same way for both forward and backward beam, and it's not like in this figure.
And the transmission through multiple faradays will also be a big issue.

A question was raised as to how much passive filtering we benefit from if we pick off the local oscillator beam for BHD from the PRC. I did some simplified modeling of this. For the expected range of arm cavity round trip losses (20-50 ppm), I think that the 40m CARM pole will be between 75-85 Hz. The corresponding recycling gain will be 40-50, with the current PRM. I assumed 1000 ppm loss inside the PRC. The net result is that, assuming the single pole coupled cavity response, we will get ~8-9 dB of filtering at ~200 Hz of the intensity noise of the input laser field to the interferometer if we pick the LO beam off from the PRC (e.g. PR2 transmission), instead of picking it off before.

The next questions are: (i) can we do a sufficiently good job of achieving the required RIN stability on the LO field for BHD without relying on the passive filtering action of the PRC? and (ii) is the benefit of the PRC filtering ruined in the process of routing the LO field from wherever the pickoff happens to the BHD setup?

Summary:

We need to determine the geometry (= round-trip length and RoC of curved mirrors) of the OMC cavities for the 40m BHD experiment. Sticking to the aLIGO design of a 4 mirror bowite cavity with 2 flat mirrors and 2 curved mirrors, with a ~4deg angle of incidence, we need to modify the parameters for the 40m slightly on account of our different modulation frequencies. I've setup some infrastructure to do this analytically - even if we end up doing this with Finesse, it is useful to have an analytic calculation to validate against (also not sure if Finesse can calculate HOMs up to order 20 in a reasonable time, I've only seen maxtem 8). 

Attachment #1: Heatmap of the OMC transmission for the following fields:

• Carrier TEM00 is excluded, but HOMs up to m+n=20 included for both the horizontal and vertical modes of the cavity.
• f1 and f2 upper and lower sidebands, up to m+n=20 HOMs for both the horizontal and vertical modes of the cavity, including TEM00.
• Power law decay assumed for the HoM content incident on the OMC - this will need to be refined
• The white region is where the cavity isn't geometrically stable.
• Green dashed line indicates a possible operating point, white dashed line indicates the aLIGO OMC operating point. On the basis of this modeling, we would benefit from choosing a better operating point than the aLIGO OMC geometric parameters.

Algorithm:

1. Compute the round-trip Gouy phase, , for the cavity.
2. With the carrier TEM00 mode resonant, compute the round-trip propagation phase, , and the round-trip Gouy phase,  for the  mode of the field, with  specifying the offset from the carrier frequency (positive for the upper sideband, negative for the lower sideband). For the aLIGO cavity geometry, the 40m modulation sidebands acquire ~20% more propagation phase than the aLIGO modulation sidebands.
3. Compute the OMC transmission for this round-trip phase (propagation + Gouy).
4. Multiply the incident mode power (depending on the power law model assumed) by the cavity transmission.
5. Sum all the fields.

Next steps:

1. Refine the incident mode content (and power) assumption. Right now, I have not accounted for the fact that the f2 sideband is resonant inside the SRC while the f1 sideband is not. Can we somehow measure this for the 40m? I don't see an easy way as it's probably power dependent?
2. Make plots for the projection along the slices indicated by the dashed lines - which HOMs are close to resonating? Might give us some insight.
3. What is the requriement on transmitted power w.r.t. shot noise? i.e. the colorbar needs to be translated to dBVac.
4. If we were being really fancy, we could simultaneously also optimize for the cavity finesse and angle of incidence as well.
5. Question for Koji: how is the aLIGO OMC angle of incidence of ~4 degrees chosen? Presumably we want it to be as small as possible to minimize astigmatism, and also, we want the geometric layout on the OMC breadboard to be easy to work with, but was there a quantitative metric? Koji points out that the backscatter is also expected to get worse with smaller angles of incidence.

The code used for the ABCD matrix calcs have been uploaded to the BHD modeling GIT (but not the one for making this plot, yet, I need to clean it up a bit). Some design considerations have also been added to our laundry list on the 40m wiki.

4 deg is not an optimized number optimized for criteria, but to keep the cavity short width to 0.1m. But the justification of 4deg is found in Section 3 and 4 of T1000276 on Page 4.

Summary:

The requirement on the phase noise on the direct backscatter from the OMC back into the SRM is that it be less than  @ 100 Hz, for a safety factor (arbitrarily chosen) of 10 (= 20dB below unsqueezed vacuum). Assuming 5 optics between the OMC and SRM which contribute incoherently for a factor of sqrt(5), and assuming a total of 1 ppm of the LO power to be backscattered, we need the suspensions to be moving  @ 100 Hz. This seems possible to realize with single stage suspensions - I assume we get f^4 filtering from the pendulum at 100 Hz, and that there is an additional 80 dB attenuation (from the stack) of the assumed 1 micron/rtHz motion at 100 Hz, for an overall 160 dB attenutaiton, yielding 10^-14 m/rtHz at 100 Hz.

Details:

This is the same calculation as I had posted a couple of months ago (see elog that this is a reply to), except that Koji pointed out that the LO power is expected to dominate the (carrier) power incident on the OMC cavity(ies). So the more meaningful comparison to make is to have the x-axes of the plots denote the backscatter fraction,  rather than the LO power. One subtlety is that because the phase of the scattered field is random, the displacement-noise induced phase noise could show up in the amplitude quadrature. I think that in these quadrature field amplitude units, the RIN and phase noise are directly comparable but I might have missed a factor of 2*pi. But in the worst case, if all the phase noise shows up in the amplitude quadrature, we end up being only ~10dB below unsqueezed vacuum (for 1 ppm backscatter). 

For the requirement on the noise in the intensity quadrature - I think this is automatically satisfied because the RIN requirement on the incident LO field is in the mid 10^-9 1/rtHz regime.

Summary:

I did some more investigation of what the appropriate cavity geometry would be for the OMC. Unsurprisingly, depending on the incident mode content, the preferred operating point changes. So how do we choose what the "correct" model is? Is it accurate to model the output beam HOM content from NPROs (is this purely determined by the geometry of the lasing cavity?), which we can then propagate through the PMC, IMC, and CARM cavities? This modeling will be written up in the design document shortly.

*Colorbar label errata - instead of 1 W on BS, it should read 1 W on PRM. The heatmaps take a while to generate, so I'll fix that in a bit.

Update 230pm PDT: I realize there are some problems with these plots. The critically coupled f2 sideband getting transmitted through the T=10% SRM should have significantly more power than the transmission through a T=100ppm optic. For similar modulation depth (which we have), I think it is indeed true that there will be x1000 more f2 power than f1 power for both the IFO AS beam and the LO pickoff through the PRC. But if the LO is picked off elsewhere, we have to do the numbers again.

Details:

Attachment #1: Two candidate models. The first follows the power law assumption of G1201111, while in the second, I preserved the same scaling, but for the f1 sideband, I set the DC level by assuming a PRG of 45, modulation depth of 0.18, and 100 ppm pickoff from the PRC such that we get 50 mW of carrier light (to act as a local oscillator) for 10 W incident on the back of PRM. Is this a reasonable assumption?

Attachment #2: Heatmaps of the OMC transmission, assuming (i) 0 contrast defect light in the carrier TEM00 mode, (ii) PRG=45 and (iii) 1 W incident on the back of PRM. The color bar limits are preserved for both plots, so the "dark" areas of the plot, which indicate candidate operating points, are darker in the left-hand plot. Obviously, when there is more f1 power incident on the OMC, more of it is transmitted. But my point is that the "best operating point(s)" in both plots are different.

Why is this model refinement necessary? In the aLIGO OMC design, an assumption was made that the light level of the f1 sideband is 1/1000th that of the f2 sideband in the interferometer AS beam. This is justified as the RC lengths are chosen such that the f2 sideband is critically coupled to the AS port, but the f1 is not (it is not quite anti-resonant either). For the BHD application, this assumption is no longer true, as long as the LO beam is picked off after the RF sidebands are applied. There will be significant f1 content as well, and so the mode content of the f1 field is critical in determining the OMC filtering performance.

I've started a spreadsheet for the BHD optics specifications and populated it with my best initial guesses. There are a few open questions we still need to resolve, mostly related to mode-matching:

• PR2 replacement: What transmission do we need for a ~100 mW pickoff? Also, do we want to keep the current curvature of -700 m?
• LO mode-matching telescope: What are the curvatures of the two mirrors?
• Lenses: We have six of them in the current layout. What FLs do we need?

The spreadsheet is editable by anyone. If you can contribute any information, please do!

Last night Yehonathan and I located the two steel PMCs in the QIL, with help from Anchal. They are currently sitting on my desk in Bridge, inside a box that also contains optics and other OMC parts. I will bring them over to the 40m the next time I come.

To supplement the material presented during the BHD review, I've put together a projected noise budget for the 40m. Note these are the expected interferometer noises (originating in the IFO itself), not BHD readout noises. The key parameters for each case are listed in the figure title. Also attached is a tarball (attachment 4) containing the ipython notebook, data files, and rolled-back version of pygwinc which were used to generate these figures.

Attachment 1: Phase quadrature readout.

Attachment 2: Comparison to aLIGO design sensitivity (phase quadrature).

Attachment 3: Amplitude quadrature readout.

The quantum noise curves here are not correct. c.f. amplitude quadrature noise budget.

Updated noise budget curves, now computed using the latest version of pygwinc. This resolves the inconsistency between the gwinc quantum noise curves and Gautam's analytic calculations. As before, the key configuration parameters are listed in the figure titles.

Attachment 1: Phase quadrature

Attachment 2: Amplitude quadrature

Attachment 3: Comparison to aLIGO design (phase quadrature)

Revised noise estimates, correcting a couple of factor of 2 and factor of pi errors found in the coil driver noise calculation. Also resolves a strain vs. displacement units confusion using the new pygwinc. Gautam and I have checked these noises against the analytical predictions and believe they are now accurate. Attachments are again:

Attachment 1: Phase quadrature

Attachment 2: Amplitude quadrature

Attachment 3: Comparison to aLIGO design (phase quadrature)

I opened the packages send from Syracuse.

- The components are not vacuum clean. We need C&B.
- Some large parts are there, but many parts are missing to build complete SOSs.

- No OSEMs.
- Left and right panels for 6 towers
- 3 base blocks
- 1 suspension block
- 8 OSEM plates. (1 SOS needs 2 plates)

- The parts looks like old versions. The side panels needs insert pins to hold the OSEMs in place. We need to check what needs to be inserted there.

- An unrelated tower was also included.

Since there has been a proliferation of BHD Google docs recently, I've linked them all from the BHD wiki page. Let's continue adding any new docs to this central list.

I wanted to pass along a complication pointed out by K. Thorne re: our plan to use Gen1 (old) Dolphin IPC cards in the new real-time machines: c1bhd, c1sus2. The implication is that we may be forced to install a very old OS (e.g., Debian 8) for compatibility with the IPC card driver, which could lead to other complications like an incompatibility with the modern network interface.

I have a query out to Dolphin asking:

1. Have they done any testing of these old drivers on Linux kernel 4.x (e.g., Debian 9/10)?
2. Is there any way to buy modern IPC cards for the two new machines and interface them with our existing Gen1 network?

I'll add more info if I hear back from them.

Answers from Dolphin:

1. No, and kernel 4.x (modern Linux) definitely will not work with the Gen1 cards.
2. No, cards using different PCIe chipsets cannot be mixed.

Since upgrading every front end is out of the question, our only option is to install an old OS (Linux kernel < 3.x) on the two new machines. Based on Keith's advice, I think we should go with Debian 8. (Link to Keith's Debian 8 instructions.)

Summary

I've generated specifications for the new BHD optics. This includes the suspended relay mirrors as well as the breadboard optics (but not the OMCs).

To design the mode-matching telescopes, I updated the BHD mode-matching scripts to reflect Koji's draft layout (Dec. 2019) and used A La Mode to optimize ROCs and positions. Of the relay optics, only a few have an AOI small enough for curvature (astigmatism) and most of those do not have much room to move. This reduced the optimization considerably.

These ROCs should be viewed as a first approximation. Many of the distances I had to eyeball from Koji's drawings. I also used the Gaussian PRC/SRC modes from the current IFO, even though the recycling cavities will both slightly change. I set up a running list of items like these that we still need to resolve in the BHD README.

Optics Specifications

• 2" mirrors (E2000226-v1)
• 2" 50:50 beamsplitters (E2000227-v1)
• 2" lenses (E2000228-v1)
• PR2 replacement (E2000229-v1)

At a glance, all the specifications can be seen in the optics summary spreadsheet.

LO Telescope Design

The LO beam originates from the PR2 transmission (POP), near ITMX. It is relayed to the BHD beamsplitter (and mode-matched to the OMCs) via the following optical sequence:

• LM1 (ROC = +10 m, AOI ≈ 3°)
• LM2 (Flat, AOI ≈ 45°)
• MMT1 (Flat, AOI ≈ 5°)
• MMT2 (ROC = +3.5 m, AOI ≈ 5°)

The resulting beam profile is shown in Attachment 1.

AS Telescope Design

The AS beam is relayed from the SRM to the BHD beamsplitter (and mode-matched to the OMCs) via the following sequence:

• AS1 (ROC = +1.5 m, AOI ≈ 3°)
• AS2 (Flat, AOI ≈ 45°)
• Lens (FL = -125 mm)

A lens is used because there is not enough room on the BHD breadboard for a pair of (low-AOI) telescope mirrors, like there is in the LO path. The resulting beam profile is shown in Attachment 2.

[Jon, Hang]

After updating the 40 m finesse file to incorporate the new SRC length (and the removal of SR2), we find that the current SRM radius curvature is fine. Thus a replacement of SRM is NOT required. 

Basically, the new one-way SRC gouy phase is 11.1 deg according to Finesse, which is very close to the previous value of 10.8 deg. Thus the transmode spacing should be essentially the same. 

In the first attached plot is the mode content calculated with Finesse. Here we have first offset DARM by 1m deg and misaligned the SRM by 10 urad. From the top to bottom we show the amplitude of the carrier fields, f1, and f2 sidebands, respectively. The red vertical line is the nominal operating point (thanks Koji for pointing out that we do signal recycling instead of extraction now). No direct co-resonance for the low-order TEM modes. (Note that the HOMs appeared to also have peaks at \phi_srm = 0. This is just because the 00 mode is resonant and thus the seed for the HOMs is greater. )

We can also consider a clean case without mode interactions in the second plot. Indeed we don't see co-resonances of high order modes. 

Hang and I have reanalyzed the BHD telescope designs, with the goal of identifying sufficiently non-degenerate locations for ASC actuation. Given the limited room to reposition optics and the requirement to remain insensitive to small positioning errors, we conclude it is not possible put sufficient Gouy phase separation between the AS1/AS2 and LO1/LO2 locations. However, we can make the current layout work if we instead actuate AS1/AS4 and LO1/LO4. This would require actuating one optic on the breadboard for each relay path. If possible, we believe this offers the simplest solution (i.e., least modification to the current layout).

LO Telescope Design (Attachment 1)

Radius of curvatures:

• LO1: +10 m
• LO2: flat
• LO3: +15 m
• LO4: flat

AS Telescope Design (Attachment 2)

Radius of curvatures:

• AS1: +3 m
• AS2: flat
• AS3: -1 m
• AS4: flat

[Jon, Tega, Hang]

We proposed a few BHD mode-matching telescope designs and then preformed a few monte-carlo experiments to see how the imperfections would change the story. We assumed a 2 mm (1-sigma) error on the location of the components and 1% (1-sigma) fractional error on the RoC of the curved mirrors. The angle of incidence has not yet been taken into account (no astigmatism at the moment but will be included in the follow-up study.)

For the LO path things are mostly fine. We can use LO1 and LO2 as the actuators (Sec. 2.2 of the note), and when errors are taken into account more than 90% of times we can still achieve 98% mode matching. The gouy phase separation between LO1 and LO2 > 34 deg for 90% of the time, which corresponds to a condition number of the sensing matrix of ~ 3. 

The situation is more tricky for the AS path. While the telescopes are usually robust against 2 or 3 mm of positional error, the 1% RoC does affect the performance quite significantly. In the note we choose two best-performing ones but still only 50% of the time they can maintain a power-overlap of > 99%. In fact, the 1% RoC error assumed should be quite optimistic... Not sure if we could achieve this in reality. 

One potential way out is to ignore the MM for the first round of BHD. Here anyway we only need to test the ISC schemes. Then in the second round when we have the whole BHD board suspended, we can then use AS1 and the BHD board as the actuators. This might be able to make things more forgiving if we don't need to shrink the AS beam very fast so that it could be separated from AS4 in gouy phase.

It would be good to have a corner plot with all the distances/ RoCs. Also perhaps a Jacobian like done in this breathtaking and seminal work.

As Rana suggested, we present the scattering plot of the AS path mode matching for various variables. The plot is for the AS path, Plan 2 (whose params we summarize at the end of this entry).

In the corner plot, we color-coded each realization according to the mode matching. We use (purple, olive, grey) for (MM>0.99, 0.98<MM<=0.99, MM<=0.98), respectively. From the plot, we can see that it is most sensitive to the RoC of AS1. The plot also shows that we can compensate for some of the MM errors if we adjust the distance between AS1-AS3 (note that AS2 is a flat mirror). The telescope is quite robust to other errors.

The compensation requirement is further shown in the second plot. To correct for the 1% RoC error of AS1, we typically need to adjust AS1-AS3 distance by ~ 1 cm (if we want to go back to MM=1; the window for >0.99 MM spans also about 1 cm). This should be doable because the nominal distance between AS1-AS3 is 115 cm. 

The story for plan1 is similar and thus not shown here. 

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

AS path plan2 nominal params:

label     z (m)     type             parameters  
-----     -----     ----             ----------  
SRMAR          0    flat mirror      none:     
AS1       0.7192    curved mirror    ROC: 2.5000 
AS2       1.2597    flat mirror      none:     
AS3       1.8658    curved mirror    ROC: -0.5000
AS4       2.5822    curved mirror    ROC: 0.6000  
OMCBS1    3.3271    flat mirror      none:   

Please see the attached doc. 

I think the conclusion is that if the AS1 RoC error is not significantly more than 1%, then with some adjustment of the AS1-AS3 distance (~ 1 cm), we could find a solution that simultaneously makes the AS path mode-matching better than 99% for the t- and s-planes. 

The requirement of the LO path is less strict and the current plan using LO1-LO2 actuation should work. 

We computed the required actuation range for the telescope design in elog:15357. The result is summarized in the table below. Here we assume we misalign an IFO mirror by 1 urad, and then compute how many urad do we need to move the (AS1, AS4) or (LO1, LO2) mirrors to simultaneously correct for the two gouy phases. 

The most demanding ifo mirrors are the ETMs and the BS, for every 1 urad misalignment the telescope needs to move 10-15 urad to correct for that. However, it is unlikely for those mirrors to move more 100 nrad for a locked ifo with ASC engaged. Thus a few urad actuation should be sufficient. For the recycling mirrors, every 1 urad misalignment also requires ~ 1 urad actuation. 

As a result, if we could afford 10 urad actuation range for each telescope suspension, then the gouy phase separations we have should be fine. 

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

Edits:

We looked at the oplev spectra from gps 1274418500 for 512 sec. This should be a period when the ifo was locked in the PRFPMI state according to elog:15348. We just focused on the yaw data for now. Please see the attached plots. The solid traces are for the ASD, and the dotted ones are the cumulative rms. The total rms for each mirror is also shown in the legend. 

I am now confused... The ITMs looked somewhat reasonable in that at least the < 1 Hz motion was suppressed. The total rms is ~ 0.1 urad, which was what I would expect naively (~ x100 times worse than aLIGO). 

There seems to be no low-freq suppression on the ETMs though... Is there no arm ASC at the moment???

Summary

For the initial phase of BHD testing, we recently discussed whether the mode-matching telescopes could be built with 100% stock optics. This would allow the optical system to be assembled more quickly and cheaply at a stage when having ultra-low loss and scattering is less important. I've looked into this possibility and conclude that, yes, we do have a good stock optics option. It in fact achieves comprable performance to our optimized custom-curvature design [ELOG 15357]. I think it is certainly sufficient for the initial phase of BHD testing.

Vendor

It turns out our usual suppliers (e.g., CVI, Edmunds) do not have enough stock options to meet our requirements. This is for two reasons:

• For sufficient LO1-LO2 (AS1-AS4) Gouy phase separation, we require a very particular ROC range for LO1 (AS1) of 5-6 m (2-3 m).
• We also require a 2" diameter for the suspended optics, which is a larger size than most vendors stock for curved reflectors (for example, CVI has no stock 2" options).

However I found that Lambda Research Optics carries 1" and 2" super-polished mirror blanks in an impressive variety of stock curvatures. Even more, they're polished to comprable tolerances as I had specificied for the custom low-scatter optics [DCC E2000296]: irregularity < λ/10 PV, 10-5 scratch-dig, ROC tolerance ±0.5%. They can be coated in-house for 1064 nm to our specifications.

From modeling Lambda's stock curvature options, I find it still possible to achieve mode-matching of 99.9% for the AS beam and 98.6% for the LO beam, if the optics are allowed to move ±1" from their current positions. The sensitivity to the optic positions is slightly increased compared to the custom-curvature design (but by < 1.5x). I have not run the stock designs through Hang's full MC corner-plot analysis which also perturbs the ROCs [ELOG 15339]. However for the early BHD testing, the sensitivity is secondary to the goal of having a quick, cheap implementation.

Stock-Part Telescope Designs

The following tables show the best telescope designs using stock curvature options. It assumes the optics are free to move ±1" from their current positions. For comparison, the values from the custom-curvature design are also given in parentheses.

AS Path

The AS relay path is shown in Attachment 1:

• AS1-AS4 Gouy phase separation: 71°
• Mode-matching to OMC: 99.9%

LO Path

The LO relay path is shown in Attachment 2:

• LO1-LO2 Gouy phase separation: 67°
• Mode-matching to OMC: 98.6%

Ordering Information

I've created a new tab in the BHD procurement spreadsheet ("Stock MM Optics Option") listing the part numbers for the above telescope designs, as well as their fabrication tolerances. The total cost is $2.8k + the cost of the coatings (I'm awaiting a quote from Lambda for the coatings). The good news is that all the curved substrates will receive the same HR/AR coatings, so I believe they can all be done in a single coating run.

We consider the astigmatism effects of the stock options. The conclusions are:

1. For the AS path, the stock should work fine for the phase-one of BHD, if we could tolerate a few percent MM loss. The window for length adjustment to achieve >99% MM for both s and t is only 1 mm for 1% RoC error (compared to ~ 1 cm in the customized case). 

2. The LO path seemed tricky. As LO3 & LO4 are both significantly curved (RoC<=0.5 m), the non-zero angle of incidence makes the astigmatism quite sever. For the t-plane the nominal MM can be 0.98, yet for the s-plane, the nominal MM is only 0.72. We could move things around to achieve a MM ~ 0.85, which is probably fine for the phase-one implementation but not long term. 

Details:

Attachments 1-3 are for the AS path; 4-6 are for the LO path. 

1 & 4. Marginalized MM distribution for the AS/LO paths. Here we assumed 5 mm positional error and 1% fractional RoC error. Due to the astigmatism, the nominal s-plane MM is only 0.72 for the LO path. 

2 & 5. Scattering plots for the AS/LO paths. We color coded the points as the following: pink: MM>0.99; olive: 0.98<MM<=0.99; grey: MM<=0.98. For the AS path, MM is mostly sensitive to the AS1 RoC and can be adjusted by changing AS1-AS3 distance. For the LO path, the LO3 RoC and LO3-LO4 distance are most critical for the MM. 

3 & 6. Assuming +- 1% AS1 (LO3) fractional RoC error, how much can we compensate for it using AS1-AS3 (LO3-LO4) distance. For the AS path, there exists a ~ 1 mm window where the MM for s and t can simultaneously > 99%. For the LO path, the best we can do is to make s and t both ~ 85%. 

Can you describe the mode matching  in terms of the total MM? Is MM_total = sqrt(MM_vert * MM_horiz)?

MM_total = (MM_vert + MM_horiz) / 2. 

The large astigmatic MM loss in the LO case is mainly due to the strong LO4 curvature (R=0.15m) with a 10 deg AOI. I looked again at whether LO1 could be increased from R=5m to the next higher stock value of 7.5m, as this would allow weaker curvatures on LO3 and LO4. However, no, that is not possible---it reduces the LO1-LO2 Gouy phase separation to only 18 deg.

There is, however, a good stock-curvature option if we want to reconsider actuating LO4 instead of LO2 (attachment 1). It achieves 99.2% MM with the OMCs, allowing positions to vary +/-1" from the current design. The LO1-LO4 Gouy phase separation is 72 deg.

Alternatively, we could look at reducing the AOI on LO3 and LO4 (keeping LO1-LO2 actuation).

Hmm? T1300364 suggests MM_total = Sqrt(MM_Vert * MM_Horiz)

I don't think we ever discussed why the angular RMS of the ETMs is so much higher than the ITMs. Maybe that's a separate matter because, even assuming the worst case, the actuation range requirement is

(0.82 μrad RMS) x (15 μrad/μrad) x (10 safety factor) = 0.12 mrad

which is still only order 1% of the pitch/yaw pointing range of the Small Optic Suspensions, according to P1600178 (sec. IV. A). Can we check this requirement off the list?

Using the updated AOI's for the LO path: (4.8, 47.9, 2.9, 4.5) deg for (LO1, LO2, LO3, LO4), we obtain the following results. 

First two plots are scattering plots for the t and s planes, respectively. Note that here we have changed to 0.5% fractional RoC error and 3 mm positional error. We have also changed the meaning of the colors: pink:MM>0.98; olive 0.95<MM<=0.98, and grey MM<=0.95. It seems that both planes would benefit statistically if we make the LO3-LO4 distance longer by a few mm. 

We also consider how much we could compensate for the MM error in the last plot. We have a few mm window to make both planes better than 0.95. 

After further astigmatism/tolerance analysis [ELOG 15380, 15387] our conclusion is that the stock-optic telescope designs [ELOG 15379] are sufficient for the first round of BHD testing. However, for the final BHD hardware we should still plan to procure the custom-curvature optics [DCC E2000296]. The optimized custom-curvature designs are much more error-tolerant and have high probability of achieving < 2% mode-matching loss. The stock-curvature designs can only guarantee about 95% mode-matching.

Below are the final distances between optics in the relay paths. The base set of distances is taken from the 2020-05-21 layout. To minimize the changes required to the CAD model, I was able to achieve near-maximum mode-matching by moving only one optic in each relay path. In the AS path, AS3 moves inwards (towards the BHDBS) by 1.06 cm. In the LO path, LO4 moves backwards (away from the BHDBS) by 3.90 cm.

AS Path

LO Path

As suggested last week, Hang and I have reviewed the A+ BHD status (DRD, CDD, and reviewers' comments) and compiled a list of key unanswered questions which could be addressed through Finesse analysis.

In anticipation of others helping with this modeling effort, we've tried to break questions into self-contained projects and estimated their level of difficulty. As you'll see, they range from beginner to Finesse guru.

Summary:

We can probably learn something about the interferometer / top level BHD plan with an in-air BHD setup, even if the noise is bad. Here are some thoughts about how we would do it. 

LO delivery:

For this first attempt, we don't really care about the PRC filtering. So possible places to pick off an LO beam are:

In all cases, I think the easiest option to actually route whatever beam we choose into a fiber, and then bring it over to whatever cavity we choose to use for an OMC. I'm assuming whatever phase control technique we end up using can cancel the fiber phase noise at relevant frequencies.

LO phase control

• Stress the fiber? This will require us to purchase some custom hardware, and interface it to the CDS system.
• PZT mirror? We should have sufficient hardware available to drive a PI style PZT mirror.

There is a question about the range, but I think these are the only two realistic options we can implement on a reasonable time scale.

OMC:

Again, there are a few options. Here are some pros and cons that come to my mind.

If we can do a vent (we'd just need a single chamber open), I'd go for the option of getting the copper OMC out and using that. Attachment #1 shows the approximate sizes of the various components (OMMT, OMC cavity, DCPDs), while Attachment #2 shows a rough sketch of where things would go on the AP table, with the rectangles approximately to scale.

CDS:

I'd made a c1omc model sometime ago. Basically, I think we have sufficient ADC/DAC channels in the c1ioo machine for any of the options listed above - but using the copper OMC and associated peripherals would allow the easiest interfacing.

Criticisms/comments/thoughts please.

For the first pass, it's probably easiest to use the existing DCPD amplifier. Looking at the gain and noise performance in Attachment #1, seems totally fine, the electronics noise will not be limiting if we have ~10mW of LO power. I assumed a transimpedance resistor of 1 kohm, and all other numbers as on the schematic (though who knows if the schematic is accurate). The noise should be measured to confirm that the box is performing as expected...

I forgot about the pointing - probably we will need another actuator to control the pointing of the AS beam onto the DCPDs. I found a few old PI PZTs (model number is S-320, which is a retired part), one is labelled broken but the others don't indicate a-priori that they are broken. I'll post a more detailed hardware survey later.

You can activate all 3axis

Attachment #1 - The 80mW pickoff was getting clipped on a BNC cable, and not making it to the doubling oven. 😢 .

• Since the PSL doubled beam isn't used for locking these days, I just didn't notice.
• I blame the ringdown team, this crazy tee arrangement wasn't the case before.
• I fixed the situation by changing the cabling such that the beam clears the cables comfortably.

Attachment #2 - PSL green shutter removed. Alignment into the doubling oven is extremely tedious, and so I opted to preserve the capability of recovering the green beam by simply removing a single mirror.

Attachment #3 - The beam path for coupling the LO beam into a fiber.

• Primary goal was to have easy access to some steering mirrors so that I can optimize alignment into the fiber collimator.
• I opted to use the NW corner of the PSL table - that's where most of our existing fiber hardware is anyways, and there was sufficient space and easy access over there.
• 3 Y1 mirrors were installed, using the preferred Polaris mounts and 3/4" post + baseplate hardware. They were labelled Y1-1037-45P so that future workers need not be un-necessarily tortured. The third mirror is not visible in this photograph.
• Once the collimator arrives, I will mode match this beam into the fiber. Plan is to use the fiber originally used for the mode spectroscopy project. It needs to be moved to the NW corner of the PSL table, and the other end needs to be routed to the AP table (it was brought back to the PSL table to facilitate Anjali's fiber MZ experiment). 
• There is plenty of space in the beam path for mode-matching lens(es) and polarization control optics.

Attachment #4 shows the BHD photodiodes taken from QIL. 

• Unfortunately, we could not find the readout electronics. 
• In the worst case, we can just interface these PDs with the existing Satellite box (associated with the copper OMC).
• It might be that the OMC cavity can simply be placed on this breadboard, making the whole setup nice and portable.
• We may want to consider having an OFI between the OMC and the IFO AS beam at some point...

Attachment #1 shows the proposed wiring and CDS topology for the in air BHD setup. The PDF document has hyperlinks you can follow to the DCC entries. Main points:

1. I think we should run the realtime model on c1lsc. This will negate the need for any IPC between c1ioo and c1lsc machines.
2. I think we have most of the electronics we need already, though I am still in the process of testing the various boards, especially the HV ones.
3. We may choose to use the switchable whitening feature for the M2 ISS board
   • This would require some BIO channels
   • There are plenty spare in c1lsc, so it's not going to be a show stopper
   • This is why I've not explicitly included a whitening board for now...
4. The main job seems to be to make a whole bunch of custom cables. For the most part, I think we have the long (~20m) long D9 cables, so I propose just snipping off the connector at one end, and soldering on the appropriate connectors to the correct conductors.
5. For the homodyne phase control - the proposal is to use a PI PZT with 3 piezoelectric elements. We would drive the 3 elements with the same voltage, by shorting the conductors together (at least that's how I understood Koji's comment), so we'd only need a single DAC channel for this purpose.
   • Need to confirm that the parallel PZT capacitances (each element is ~300 nF so 3 in parallel would be ~900 nF) still allows sufficient actuation bandwidth.
   • If the relative actuation strength of the 3 elements needs to be individually tuned, we may have to use three DAC channels. The D980323 board will allow the driving of 3 independent channels. I have one of these boards in hand, but need to check if it works, and also implement the changes outlined here.
6. The alignment control has not yet been accounted for
   • We could consider using the in-vacuum PZTs, these were verified to be working ~2018.
   • If we use only 1 steering PZT mirror, we have sufficient free DAC channels available in c1lsc. But if we need both (to avoid clipping for example), then we need more DAC channels - we can either free up one DAFI channel, or install a DAC in the c1lsc expansion chassis. 
7. We may want to expand to have a second OMC at some point. In which case we'd need, at the very least
   • 1 more DAC card
   • A HV driver for the second OMC length (could use the Trek driver if we use D980323 for the homodyne phase control).

Please comment if I've overlooked something.

The LO pickoff has been coupled into a fiber with ~90% MM (8 mW / 9 mW input). While I wait for the DCPD electronics to be found in the Cryo lab, I want to monitor the stability of the pointing, polarization etc, so I'd like to clear some space on the AP table that was occupied for the mode spectroscopy project. If there are no objections before 2pm tomorrow July 21 2020, I will commence this work.

Attachment #1 shows the RIN of the local oscillator beam delivered to the AP table via fiber. I used a PDA520 to make this measurement, while the electronics for the DCPDs are pending. I don't really have an explanation for the difference between the locked IFO trace vs the not locked trace - we don't have an ISS running (but this first test suggests we should) and the beam is picked off before any cavities etc, so this is a reflection of the state of the FSS servo at the times of measurement?

Tried locking CARM using the hybrid REFL (for AO path) and POX 11 (for MCL path) scheme a bunch of times today, but I had no luck. When the CARM offset is zeroed, the PRMI lock is lost almost immediately. Maybe this is indicative of some excess noise in the POX data stream relative to the REFL signal? The one thing I haven't tried is to take the IFO all the way to the locked state, and then transition the MCL actuation from CM_SLOW to POX11_I.

An SR785 is sitting on the North side of the AP table in the walkway - I will clear it tomorrow.

I decided to use the old EY auxiliary optics table, which is now stored along the east arm about 10 m from the end, as a workspace for assembling the little PMCs. I wiped everything down with isopropanol for general cleanliness, removed the metal plate on the south edge of the table enclosure to allow access, covered the table with some clean Aluminium foil, and then moved the plastic box with PMC parts to the table - see Attachment #1. I haven't actually done any assembly just yet, waiting for more info (if available) on the procedure and implements available...

We explore bilinear SRCL to DARM noise coupling mechanisms, and show two cases that by doing BHD readout the noise performance can be improved. In the first case, the bilinear piece is due to residual DHARD motion (see also LHO:45823), and it matters mostly for the low-frequency (<100 Hz) part, and in the second piece the bilinear piece is due to residual SRCL fluctuation and it matters mostly for the a few x 100 Hz part. Details are below:

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General Model:

We can write the SRCL to DARM transfer function as (Evan Hall's thesis, eq. 2.29)

Z_s2d(f) = C_lf(f) * F^2 * x_D + C_hf(f) * F * dphi_S * x_D    ---- (1)

where

C_lf ~ 1/f^2 and C_hf ~ f are constants at each frequency unless there are major upgrades to the IFO,

F is the finesse of the arm cavity which depends on the alignment, spot position on the TMs, etc., 

dphi_S is the SRCL detuning (wrt the nominal 90 deg value), 

x_D is the DC DARM offset. 

The linear part of this can be removed with feedforward subtractions and it is the bilinear piece that matters, which reads

dZ_s2d = C_lf * <F>^2 * dx_D + C_hf * <F> * <dphi_S> * dx_D

             + 2C_lf * <F> * <x_D>  * dF + C_hf * <dphi_S> * <x_D> * dF

             + C_hf  * <F> * <x_D> * d(dphi_S).     ---- (2)

The first term in (2) is due to residual DARM motion dx_D. This term does not depends on the DC value of DARM offset <x_D> and thus does not depend on doing BHD or DC readout. On the other hand, the typical residual DARM motion is 1 fm << 1 pm of DARM offset. Since the current feedforward reduction factor is about 10 (see both Den Martynov's thesis and Evan Hall's thesis), clearly we are not limited by the residual DARM motion. 

The second term is due to the change in the arm finesse, which can be affected by, e.g., the alignment fluctuation (both increasing the loss due to scattering into 01/10 modes and affecting the spot positon and hence changing the losses), and is likely to be the reason why we see the effect being modulated by DHARD. 

The last term in (2) is due to the residual SRCL fluctuation and is important for the ~ a few x 100 Hz band.

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

DHARD effects. 

As argued above, the DHARD affects the SRCL -> DARM coupling as it changes the finesse in the arm cavity (through scattering into 01/10 modes; in finesse we cannot directly simulate the effects due to spot hitting a rougher location). 

Since in the second term of eq. (2) the LF part depends on the DARM DC offset <x_D>, this effect can be improved by going from DC readout to BHD. 

To simulate it in finesse, at a fixed DARM DC offset, we compute the SRCL->DARM transfer functions at different DHARD offsets, and then numerically compute the derivative \partial Z_s2d / \partial \theta_{DH}. Then multiplying this derivative with the rms value of DHARD fluctuation \theta_{DH} we then know the expected bilinear coupling piece. 

The result is shown in the first attached plot. Here we have assumed a flat SRCL noise of 5e-16 m/rtHz for simplicity (see PRD 93, 112004, 2016). We do not account for the loop effects which further reduces the high frequency components for now. The residual DHARD RMS is assumed to be 1 nrad. 

In the first plot, from top to bottom we show the SRCL noise projection at different DARM DC offsets of (0.1, 1, 10) pm. Since the DHARD alignment only affects the arm finesse starting at quadratic order, it thus matters what DC offset in DHARD we assume. In each pannel, the blue trace is for no DC offset in DHARD and the orange one for a 5 nrad DC offset. As a reference, the A+ sensitivity is shown in grey trace in each plot as a reference. 

We can see if there is a large DC offset in DHARD (a few nrad) and we still do DC readout with a few pm of DARM offset, then the bilinear piece of SRCL can still contaminate the sensitivity in the 10-100 Hz band (bottom panel; orange trace). On the other hand, if we do BHD, then the SRCL noise should be down by ~ x100  even compared to with the top panel. 

(A 5 nrad of DC offset in DHARD coupled with 1 nrad RMS would cause about 0.5% RIN in the arms. This is somewhat greater than the typically measured RIN which is more like <~ 0.2%. See the second plot). 

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

SRCL effect. 

Similarly we can consider the SRCL->DARM coupling due to residual SRCL rms. The approach is very similar to what we did above for DHARD. I.e., we compute Z_s2d at fixed DARM offset and for different SRCL offsets, then we numerically evaluate \partial Z_s2d / \partial dphi_S. A residual SRCL rms of 0.1 nm is then used to generate the projection shown in the third figure. 

Unlike the DHARD effect, the bilinear SRCL piece does not depend on the DC SRCL detuning (for the 50-500 Hz part). It does still depends on the DARM DC offset and therefore could be improved by BHD.

Since we do not include the LP of the SRCL loop in this plot, the HF noise at 1 kHz is artifical as it can be easily filtered out. However, the LP will not be very strong around 100-300 Hz for a SRCL UGF ~ 30 Hz, and thus doing BHD could still have some small improvements for this effect. 

3. I agree - this whitening will be handy to have for diagnostics.

4. I think in principle, we can ask a company to make the custom cables for us to save us some hand labor. Rich/Chub probably know the right companies to do small numbers of dirty cables.

5. Can't we just a single Noliac PZT in the same way that the OMC does? Or is the lead time too long?

6. Do we need active steering for this in-air test? I'm not even sure how we would get the alignment signal, so maybe that's a good reason to figure this out.

I've pushed an MCMC simulation to the A+ BHD repo (filename MCMC_TFs.ipynb). The idea is to show how random offsets around ideal IFO change the noise couplings of different DOFs to readout.

At each step of the simulation:

1. Random offsets for the different DOFs are generated from a normal distribution. The RMSs are taken from experimental data and some guesses and can be changed later. The laser frequency is tuned to match the CARM offset.

These are the current RMS detunings I use:

2. A transfer function is computed for the noisy DOFs.

3. Projected noise is calculated.

These are the noise level for the DOFs:

The attachments show the projected noise levels for the noisy DOFs. Each curve is a different instance of random offsets. The ideal case - "zero offsets" is also shown.

OMC Comm and OMC diff refer to the common and differential length change of the OMCs.