α, so thatCdarmat a given timetis replaced byα(t)Cdarmin equation (7.9). Indeed, for calibration of initial LIGO Cdarm is considered to be basically constant and is not dynamically corrected; it is measured perhaps several times over the course of a run, in contrast to α(t) which is measured constantly and tracked in order to later reconstruct the calibration. A second calibration line can be applied as a consistency check and to get a better measurement of changes in the open loop gain Gdarm.
7.1.4.2 Comment
In this discussion, we have glossed over actually understanding the behavior of the actuators and sensors, in favor of simply measuring them. Part of the work of the LIGO scientific collaboration detector calibration subgroup is to also understand in detail all the smaller pieces of the model (i.e., the ADCs considered individually, the coil drivers, etc.), and make sure the subparts fit together properly. This is significantly more work, but gives added confidence to the calibration.
Filter (F)
Sensor (S)
Plant (P)
Actuator (A)
err ctrl
+ +
h
Fields (E) Mirror/mass
(M) Optical Spring
(K) Digital
Figure 7.2: The DARM feedback loop model for a detuned RSE interferomter. The plant represents the opto-mechanical response, described in equation (3.83). A gravitational wave signal h is modeled as an injection into the loop at the optical field E.
7.2.2 DARM Calibration
Referring to figure7.2, we then measure the open loop gainG(we will now omit the subscripts when referring to DARM; alsoAM will refer to theAetmMetm, measured previously using the method in section7.1).
G=AP SF, (7.10)
whereP is now the whole plant (we can no longer separate E andM, since they are both inside a feedback loop due to the radiation pressure). We can also measure
H= AP S
1 +G. (7.11)
Now, with some algebra, assumptions, and recalling that we measured the productsAetmMetmand EcavScav in section7.1, we can isolate the response of the plantP,
P = H
AetmScava, (7.12)
where we have introduced an arbitrary scaling factora, the value of which we also just measured.
The assumptions for this last step were thatMetmis a pure (damped) pendulum response, thatEcav
is a simple pole (cf. section 3.6.2), and that S can be written as aScav. These are all reasonable assumptions, that could also be verified (although they have not all been verified at the 40 m). The same photodetector is used to measure S and Scav, so barring huge non-linearities in the PD or electronics the response should only differ by the optical power on the diode and the optical phase gain, both of which are included in the factora. Ecav is the cavity response; this is simple enough that we can be confident in its behavior; it has also recently been measured. AssumingMetmreally behaves like a pendulum at the frequencies where radiation pressure matters (below 100 Hz) should not be controversial; it is the behavior of the actuators (i.e., the OSEMs) that are really in question, and that is mainly above a kHz.
The sensing function forhis then
C= P Scav
Metm
a. (7.13)
We can now calibrateDARMerr into meters:
h=DARMerr
1 +G
C . (7.14)
7.2.3 Modeling
As discussed in section7.2, the plantPand consequently the sensing functionCcan vary significantly with interferometer operation, with changes that cannot be described as simply an overall scaling.
We can compensate for this by measuring the open loop gain G and using that measurement to determineC. Of course, for tracking purposes we cannot measureGat a large number of frequencies, since those frequencies would then have to be excluded from any analysis. We can track the behavior ofP by measuringGat just enough frequencies to be able to fit the isolatedP (cf. equation (7.12)) to a model (equation (3.83)).
We measure G at several frequencies (indexed by i), and then isolate the plant Pi at those frequencies:
Pi= Gi
AetmSF. (7.15)
SinceF is a digital filter, we know it precisely and do not have to measure it. The measurements of Aetm andS have been covered above.
We then minimize the function,
"
X
i
| Pi
Metm −RSEresp(fi, Ibs, φ, ζ, g)|2
#1/2
, (7.16)
where RSEresp is simply equation (3.83) with hardcoded values for things that do not vary (like
rs) and with an overall scale g included. Metm is the theoretical pendulum response. The output from the minimization routine is then a set of estimates for the circulating powerIbs, detune phase φ, detection quadratureζ, and overall gain g.
A model of the open loop gainGis then built using the output parameters from the fit and the theoretical opto-mechanical response (equation (3.83)), and the measured actuation/sensing product AetmS. An example model, and the measured points, can be seen in Figure7.3.
Gmod=RSEresp(f, Ibs, φ, ζ, g)AetmMetmSF. (7.17)
Since the both the Pi and RSEresp are complex valued, this routine fits both magnitude and phase, and so a model of the overall delay must be included when removing the electronics (cf.
equation7.15) or the routine will get lost trying to match an impossible phase profile. The delay is digital, and so should be constant (it really is not, but it flips between several discrete values–this will be fixed in a future code upgrade). The minimization routine used can also easily get trapped in local minima, and so it must primed with good guesses. One particular trap is that a πphase shift in the detection quadratureζ does not yield a simple sign change, as might be expected with a simpler system.
7.2.4 Tracking
Doing occasional detailed measurements of the open loop gain such as in figure7.3is good to verify the validity of the modeling. The regular calibration, however, cannot be done with such detailed measurements of the open loop gain, as such measurements take many minutes to complete. For Advanced LIGO, as in initial LIGO, the calibration will be continuously tracked; the number of calibration lines must be kept as small as possible to avoid spoiling the data.
At the 40 m, we do not apply calibration lines continuously; instead, we just measure the open loop gain whenever a calibrated measurement needs to be made (such as a noise spectrum). We use 7 frequencies to determine the response. This number was determined by running the routine with fewer and fewer measured frequencies until the fit to the response was poor, and then adding back three points. The actual frequencies used can be seen in figure7.5; these particular frequencies were selected to be at points where errors due to parameter misestimation should be large (cf.
figure7.4) while avoiding known mechanical resonances and frequencies with particularly poor noise performance.
In principle, these seven measurements are enough. In practice, at the 40 m, we add an additional measurement to get an overall scaling factor. This is because there are several length sensing and control parameters (gain settings) between the measurements for the DARM loop gainG and the
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40m DARM open loop gain
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Model Data
Figure 7.3: An example DARM open loop transfer such as is used for verification of modeling tools at the 40 m prototype. The squares represent measured data points, and the line is a model with four free parameters determined by a fit by eye. The large discrepancies at low frequencies are due to poor measurement coherence resulting from the large loop gain, and the isolated scattered points between 100 Hz and 1 kHz are due to unmodeled mechanical resonances (these points also have poor measurement coherence).
optic drive signal. To avoid tracking these, we use a separate excitation point whose calibration should not change. After building the open loop gain model, a calibrated excitation is performed by driving a ETM at a single frequency (fmeas), and measuring the response inDARMerr. The response to this known excitation at one frequency can then be used to appropriately scale the sensing functionC, by ensuring thatC(fmeas) is equal to the measured response.
A set of MATLAB functions has been written which automatically carries out the procedure described in this section by retrieving a set of data and taking the amplitude spectral density, measuring the open loop gain, applying a calibration line, and fitting a model to the measured open loop gain data. The model is then used to calibrate the amplitude spectral density. The calibration line and the open loop gain measurement excitations are not visible in the spectra because the data is not taken during these excitations, but rather approximately one minute before. Such a spectrum is in figure7.6. The current version of this automated calibration routine actually applies an excitation to the ITM, simply leaving out the step of bootstrapping the ETM calibration. This leads to an error (due to the ITM also being part of the signal recycling cavities) of less than 0.2%, which is certainly negligible compared to other statistical and measurement errors.
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Power 2%
Detune 1%
Quadrature 5%
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Figure 7.4: Ratio of the opto-mechanical response to nominal with given parameter variation, for a 40 m like RSE configuration. The red solid curves show the result of a +/- 2% power variation, the blue dash-dot that of +/- 1% detune phase variation, and the green dashed curves that of a +/- 5% variation.