1. Optimal on-line time-domain calibration of the
dual-recycled GEO 600
M. Hewitson for the GEO 600 team
Max-Planck-Institut für Gravitationsphysik
(Albert-Einstein-Institut)
Institut für Atom- und Molekülphysik
Introduction
GEO 600 uses dual recycling (power and signal recycling)
Gravitational wave signal appears in two output quadratures
One quadrature is used as control signal for Michelson
Transfer function from strain to main output signals is a function of frequency (see Figure 2); this includes:
Optical response
Michelson control servo response
Calibration to interpret sensitivity of GEO:
Provides one signal for analysis
Helpful during commissioning
Figure 2: Measured transfer functions from strain to two detector outputs.
Figure 1: A simplified schematic of the optical layout of GEO 600. The two main detector outputs, P(t) and Q(t), are shown.
Signal processing pipeline
On-line transfer function measurement
Injected calibration lines are used to induce known differential displacement (and hence, strain) – see Figure 3.
By observing the magnitude and phase of the calibration lines in the two detector outputs, we get measurements of the transfer functions at these spot frequencies
Measured once per second
Parameterised models (see Figure 4) of the transfer functions are fit to the measurements using an optimisation routine
Models are inverted and used to generate IIR filters
Detector outputs, P(t) and Q(t), are filtered to give strain outputs, hP(t) and hQ(t)
Figure 5: A summary of the signal processing tasks used to calibrate each of the detector outputs.
Calibration signal
P and Q
actuator
optical
Figure 3: Snap-shot amplitude spectral densities of the injected calibration lines and the two detector output signals.
Figure 6: A more detailed schematic of the signal processing pipeline used to calibrate each of the detector outputs to strain.
Figure 4: Parameterised model of the Michelson control servo including the two model optical transfer functions that give the two detector outputs, P(t) and Q(t).
Al density estimates
Optimal combination of the two calibrated strain outputs
Figure 7:
Left: Amplitude and cross-spectral density estimates of hP and hQ. Noise floor estimates are shown; these are used to estimate the various sigma terms in the maximum likelihood estimator of h.
Right: Combined responses of FIR filters made from maximum likelihood estimator of h.
sPP
sQQ
sPQ
Figure 8: Amplitude spectral density estimates of the two calibrated strain outputs of GEO and the combined strain signal, h(t). The induced strain from the injected calibration lines is shown for reference.
Both calibrated outputs of GEO contain the same strain signal but different noise
Form a maximum likelihood estimator for h(t) (Eq. 1)
Estimate sigma terms from variance of noise in the two calibrated data streams
Compute two sets of filters for hP(t) and hQ(t)
Equation 1: A maximum likelihood estimator for h(t)