This optical coupling represents a problem, since noise in these auxiliary degrees of freedom can pollute the DARM signal. The control system, of course is supposed to suppressdisplacement noise for these auxiliary DOFs, but it will not suppress sensing noise: it will instead impose the sensing noise on the auxiliary degree of freedom, and this will pollute the DARM signal via the optical coupling (M).
The solution, when the auxiliary degrees of freedom are sensing noise limited (usually the most important sensing noise in this case is shot noise, but we will see in chapter 9 that light source noise can also be a problem), is to simultaneously send the feedback signal to DARM, appropriately filtered and scaled. This is called feedforward loop correction, and is done in LIGO for MICH and PRCL. It has also been done at the 40 m for MICH in a power-recycled FPMI state, where it is especially simple, as the coupling factor is a constantπ/2F.
Figure6.4shows that in a detuned interferometer, the feedforward paths will need some filtering in order to be effective, especially for the SRCL and PRCL loops. These were never implemented because these length couplings never limited the noise of the interferometer, as other noise sources were dominant (specifically, digital to analog conversion noise for the test mass actuators).
Although sophisticated frequency-dependent feedforward loop corrections have not been tested for a detuned interferometer at the 40 m, they will prove crucial in Advanced LIGO, not just to reduce the effects shot of noise in the auxiliary loops, but as the measurements in chapter 9 will show, also for laser and oscillator noise.
The 40 m digital control system is currently being significantly upgraded (cf. section 5.10); the implementation of feedforward techniques should be a high priority. One technique that will be tested is the use of Wiener analysis (see [85] for a description) to determine the optimal filters for these correction paths.
a broadband RSE interferometer, and will thus not be plagued by the sideband imbalances.
Multiple modes of operation. Although the baseline for initial operation is broadband RSE, the LSC scheme for Advanced LIGO will actually allow multiple modes, including detuned operation. This has been accomplished by designing the resonance profile such that the error signal for the SRCL has a broad linear range; an electronic offset can be added to change the signal cavity tuning. It is even conceivable that this detuning change could happen during a single lock, allowing the interferometer to track a coalescing binary.
Lowering the arm cavity finesse from 1250 to 450. Although this will increase the thermal load on the substrates in the recycling cavity, this will lessen the effect of anomalous losses in the optics, and simplify lock acquisition. At the 40 m, the interferometer as a whole is undercoupled because the high finesse arm cavities greatly increase the impact of optical losses in the arm cavities.
No Mach Zehnder. The LSC scheme has been designed to minimize the effects of sidebands on sidebands, and will thus work without the need for a Mach Zehnder (cf. section5.6.1.1), which adds complexity to interferometer operation.
Lower modulation frequencies. This was not discussed in detail, but the experience with this scheme has shown that RF modulation frequencies above ∼100 MHz present significant tech- nical challenges in electronics and photodetector design, particularly for the detection of sideband-sideband beats, which are at even higher frequencies than the modulation frequencies.
Fast photodetectors must have a small active area; working with such small (1 mm diameter) photodetectors in a high-power, suspended interferometer proved challenging—several were de- stroyed during lock loss events. In addition, double demodulation places particular demands on photodetector frequency response; the previous Advanced LIGO design would have required a photodetector with resonances at 171 MHz and 189 MHz, and a notch at 180 MHz. Such a requirement is not practical. Furthermore, position sensitive, segmented photodetectors which operate well at such high frequencies were also believed to be very difficult to implement. The Advanced LIGO design was thus modified to limit the highest frequencies detected to less than∼100 MHz.
Lock acquisition. The Advanced LIGO LSC scheme has been designed with lock acquisition in mind; a significant advantage over the scheme described here. In addition, there are plans for significant additional optical hardware whose primary purpose is to ease lock acquisition (more in chapter 8).
Chapter 7
Calibration of a Detuned RSE Interferometer
Calibration of the interferometer output requires a thorough understanding, through a combina- tion of modeling and measurement, of the transduction process by which displacement (or strain) is converted to volts (and ultimately digital counts, here denoted by [cnt]). For Initial and En- hanced LIGO, this process is already fairly complicated, given the complexity of interferometric gravitational wave detectors. This situation will only be exacerbated by the use of a detuned optical configuration, with its more richly featured opto-mechanical response. Here we concern ourselves only with calibration of the DARM error signal in the frequency domain.
7.1 Calibration in Initial LIGO
One method used for calibrating the Initial LIGO detectors [86] involves a multistep process, based on the Schnupp sensing technique employed for the Michelson length DOF sensing. After demodulation and digital acquisition, the Schnupp technique yields an error signal for the Michelson length
M ICHerr =A0sin(4πl−
λ ), (7.1)
where the amplitude A0 depends on many factors and so the easiest (and most accurate method) is to just measure it, using the laser wavelengthλ as a reference. This procedure can be summed up as: ‘calibrate the actuator using a simpler system than the full interferometer, then excite the calibrated actuator and measure the response of the full system.’ This section outlines the calibration procedure.
Filter (F)
Sensor (S)
Plant (P)
Actuator (A)
err ctrl
+ +
h
Fields (E) Mirror/mass
(M) Digital
Figure 7.1: The MICH/XARM/YARM/DARM feedback loop model for non-signal-recycled interferometer.
The transfer function E for these simple subsystems is just the arm cavity pole (or unity for the MICH). A gravitational wave signal h is modeled as an injection into the loop at the optical field E.
7.1.1 Free Swinging Michelson
With the interferometer optics aligned to create a simple Michelson, excite the optics in length and measure the peak to peak amplitude of the signal as the Michelson swings through complete fringes. The maximum amplitude isA0. At the dark fringe, the Michelson error signal gain is then A04π
λ[cntm]; this is the calibration for the field+sensor transfer functionEmichS in figure7.1, at DC.
We assume S does not vary with frequency. Emich is also constant with frequency. Thus, for a MichelsonEmichS can be used to write the error signalM ICHerr in meters at all frequencies.
7.1.2 ITM Calibration
Lock the Michelson to a dark fringe, and excite the Michelson length by driving an input test mass (ITM) with a sine wave. Measure the swept sine transfer function fromM ICHctrltoM ICHerr. De- modulate the response in the (now calibrated) Michelson length sensing output. This step measures the transfer functionHmich/itm,
Hmich/itm= M ICHerr
M ICHctrl
= AitmMitmEmichSmich
1 +Gmich
, (7.2)
where we have used the open loop gainG=F AM ES. Since we already knowEmichS, after simply measuring the open loop gainG, we now also know AitmMitm.
AitmMitm = Hmich/itm
EmichSmich
(1 +Gmich). (7.3)
7.1.3 ETM Calibration
Lock a single arm cavity (say XARM), and measure the open loop gain Gcav and the transfer functions,
Hcav/itm=AitmMitmEcavScav
1 +Gcav
, (7.4)
by exciting the ITM and
Hcav/etm= AetmMetmEcavScav
1 +Gcav
, (7.5)
by exciting the ETM. SinceEcavScav is common to both transfer functions, and we already know AitmMitm, we can now easily determineAetmMetm:
AetmMetm= Hcav/etm Hcav/1tm
AitmMitm. (7.6)
7.1.4 DARM calibration
Lock the full interferometer, and measure the open loop gainGdarm and Hdarm= AetmMetmEdarmSdarm
1 +Gdarm
. (7.7)
Then the sensing function is
Cdarm=EdarmSdarm=Hdarm(1 +Gdarm) AetmMetm
, (7.8)
in units of [cntm].
WithCdarm, we can determine the magnitude of any signalhin meters by applying the calibration to the DARM error signal:
h=DARMerr
1 +Gdarm
Cdarm
. (7.9)
The strain calibration is then just dividing by the arm length.
7.1.4.1 Tracking
As the state of the interferometer changes (e.g., due to alignment fluctuations), Cdarm andGdarm
will in general only be affected in overall magnitude; the actual shape of their frequency responses will not change. It is thus sufficient to apply one calibration line which monitors this overall scaling
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.