J. Kissel

Executive Summary: here are some cool time-frequency ways to look at how the sensing function evolves during thermalization. No new information on the cause, and no conclusion about what to do -- only hints at patterns and identification that more work and data is required to create the "even though the details are inconsistent from lock to lock, we could get away with doing this simply thing" solution that everyone wants.



Picking up from LHO:69478, where we last left off characterizing sensing function during thermalization -- we were left craving more from the bode plots of the sensing function as it evolved in time. We identified that, right after power up, we're actually *over* correcting for SRC detuning with the fixed SRCL offset that "re-tunes" the SRC after everything's thermalized. 

So the natural follow-up question -- given that we're adjusting the SQZ subsystem and PRCL loop gains during thermalization with the THERMALIZATION guardian node (see, e.g. LHO:69441 and LHO:68948) -- can we do something similar with the SRCL offset? 

In order to answer that question, we need to understand the time evolution of the thermalization better than heat-map 2D bode plots shown as the feature plots LHO:69478. Namely, one might expect that the evolution is (natural logarithm, base e) exponential, given the shape of the time evolution we typically see from the arm power. 

Here, I present a collection of plots of that same data that I'm calling "3D Bode Plots." Here, I repeat the info about each thermalization segment, and show the plot 3D bode collection associated with it:
Period  UTC Start             UTC Stop    GPS Start      GPS Stop     Segment             3D Bode Plot collection
        YYYY-MM-DD                                                    Duration (hours)
(A)     2023-05-09 01:15:00   04:30:00    1367630118     1367641818   3.25 (3h 15m)       sensingFunctionResidual_meas_over_model_wTDCFs_1367630178-1367641698.pdf
(B)     2023-05-10 08:41:00   10:12:00    1367743278     1367748738   1.51 (1h 30m)       sensingFunctionResidual_meas_over_model_wTDCFs_1367743338-1367748618.pdf

(C)     2023-05-10 11:19:00   14:00:00    1367762418     1367752758   2.68 (2h 40m)       sensingFunctionResidual_meas_over_model_wTDCFs_1367752818-1367762298.pdf


For each segment, I show 4 pages of the "normalized sensing function," i.e. ratio of 
    - the measured sensing function (from the 8.9 to 24 Hz pairs of calibration lines, continuously driven by the CAL_AWG_LINES guardian) to 
    - the modeled sensing function (which has *no* detuning, but is corrected for changes in optical gain and cavity pole from the 410.3 Hz calibration line informed TDCF):
in various different ways.
    :: Page 1: A replica of what you've seen as the feature plot in LHO:69478 -- a two-panel, 2D bode plot of the sensing function, where the "heat map" coloration indicates the 2 minute steps in time.

    :: Page 2: The first of the 3D bode plots, showing four panels. 
        In the left two panels, I show essentially the same bode plot as Page 1 -- the frequency axis "forward" and the transfer function magnitude / phase on the vertical axis -- but enhanced in a third, where the dimension "into the page" is time (in minutes). So, X axis is frequency (in Hz), Y axis is time, and Z axis is the sensing function transfer function ratio's magnitude (upper, in [ct/m / ct/m]) and phase (lower in [deg]). With this view, you can actually resolve the 2-minute time steps that we didn't see in the first page.
        In the right two panels, it's the exact same 3D Bode plot, just with the vertical, Z axis (transfer function magnitude and phase) zoomed in to the range that we typically care about for calibration, between 1.0 to 1.25 and -5 to 30 deg, i.e. from 0 to 25% systematic error in the sensing function. With that zoom, we can quantitatively read off the sensing function's systematic error *after* thermalization, and compare against thermalized data.

    :: Page 3: The second of the 3D bode plots, also showing the same four panels, but yawed by 90-ish degrees such that now the Y, time, axis is "forward." It's with this view that we can actually see the time evolution of the detuning clearly.

    :: Page 4: The third and final 3D bode plot, where the axes are rotated/showed with the same yaw as page 3, but with the time axis displayed as the natural logarithm of time (in "ln(min)"). with this view, one can eye-ball which regions of the time evolution is "exponential" and which are not. Just so you don't have to calculate it yourself, 
        ln(min)    min    approx hours
        2      7.5     ~1/8    
        3     20.1     ~1/3
        4     54.6     ~1 
        5    148.4     ~2.5
        



What I conclude from these collections of plots thus far :
    (1) The later parts of the data segments, between "20-25 minutes after achieving final input power" and "thermalized," where to-date we think it's "just" the substrate of the ITMs thermalizing, these data seem to suggest that the thermal process it isn't necessarily exponential, but it seems close. 

    (2) The earlier parts of data segments, between "just after achieving final power" and "20-25 minutes after achieving final input power" is confusing. My suspicion is that there's two things that are confused in these first 30 minutes (i.e. at around ):
        (i) The data is noisy / less coherent. The landing at NOMINAL_LOW_NOISE is not a soft landing. Shutters are closing, whitening is getting switched, laser noise suppression is getting adjusted, etc -- all within 30 seconds of the declaration of "nominal low noise," so the DARM sensitivity is full of transients and glitches (at least according to the 10 sec, 3x rolling average, ASD on the front wall). So would guess *at least* the first 2 minute data point is spoiled by this non-stationary noise.
        (ii) We know there are several different time-scales of thermalization in play -- point absorbers, front-surfaces of optics, and the last-and-longest the substrates of the test masses. All are thermal processes, each with their own exponential time-scales all intermixing. We've learned from O3 that most of the shorter time scale heating is in equilibrium by ~30 minutes in. So my guess is that we see the transition to a single "clean" "exponential" of the test mass substrate by 30 minutes. ("clean" in quotes because it's only one process, and "exponential" because of point (1) above that the evolution is not quite exponential.

    (3) I need to synchronize the time axes between the three data sets to be really quantitative about it -- other than it being a "pro" spring, where the magnitude has a sharp feature with high quality factor -- the evolution of "the spring" is not consistent during the early parts of the thermalization.



Things I really want next as a result of this study:
    (a) more, lower, frequency points. At the current level of detuning at the start of the thermalization, I suspect the 8.9 Hz data point is at the upper-frequency edge of the detuned optical spring frequency. However, given that there are other sensing function response features in play, I'd love to see more data below 8.9 Hz to really resolve "the whole spring" response. Especially because the phase seems to be doing an entirely different thing for each of the three data sets during the first 30 to 60 minutes. The standard sensing function sweeps takes data all the way down to 5.6 Hz, so that'd be a good place to start. There was no magic to these three frequency points, they were arbitrarily chosen, as documented in LHO:68947, because 
        (i) they were the four data points I could tune with a DTT template, and
        (ii) that was the limit of PCALY lines I could drive, at the amplitude chosen, and still have the other more critical PCALY lines going.
    (b) synchronizing the thermalization period to a point in time these three test cases were just the first couple of data sets that I had started this study. The length of segment was chosen, and start time of plotting was chosen only be what data was available. At this point, with how the data is displayed, and the few number of data points, it's still difficult to assess whether the thermalization behavior is consistent. This should be doable by finding the time, say, at which the arm cavity power surpassed a given power, or the moment the guardian declares that the PSL power has hit 76 W or something. Now need for new data here, just more sophisticated offline analysis.
    (c) more data sets. These should be in the can already, but the data is clearly disobeying the "three makes a pattern" rule.
    (d) *If* we decide to leave the IFO as is, and decide *against* "actively" tune the SRCL offset with the THERMALIZATION guardian, (again because running out of time before the run) then we must make a decision on how we want to incorporate this time-dependent, sensing function systematic error into the overall modeled response function systematic error. In O3, we "just" used GPR fitting because that's what infrastructure we had available to us at the time. It's a blunt weapon, and honestly, for the O3 IFO -- who's thermalization period lasted only 1 hour -- we figured "good enough." But now that
        (i) the systematic error as a result of this detuning is so large (at the 11.575 Hz data point, greater than 20% in magnitude and 15 degrees in phase for over an hour), and
        (ii) the detuning causes the "we don't have any detuning" model to be greater than 10% and 10 deg for more that 2.5 hours, and thus spanning several time periods of modeled, response function systematic error estimates used for data analysis
    I think we'll have to figure out more sophisticated logic for accounting for this error as a function of time than what we used in O3 ... which was "just" "look for the first hour of a NLN segment that had recently come from low input laser power, and apply *the* single, O3 model of that sensing function systematic error transfer function to the overall budget."




Stay tuned while I fulfill my own requests.