Schematic diagram of a Michelson interferometer. There are two input beams, I D and I L , and two output ones, O D and O L . and represent the transfer functions of each arm, which are the result of the propagation inside the dashed boxes, while R BS , T BS are the reflectivity and the transmissivity of the beam splitter. and depend on the specific optical setup of the arm, in the Michelson case they are just free propagations. Whatever they were we get the relations shown between the input and the output beams. Note that if both input fields are completely reflected back.
A simplified optical scheme of the Virgo interferometer. The input laser is modulated by an EOM and locked at a RFC, then filtered by a mode cleaner cavity (IMC). It is then injected in the main detector through a power recycling mirror (PR), separated in two parts by a BS. Each split beam resonates inside a Fabry-Perot cavity (WI-WE and NI-NE). The beam is recombined on the beam splitter and is then filtered by an OMC cavity. Small secondary beams are obtained in several points, for example, at the WE and NE mirrors which are not completely reflective and analyzed with photo-diodes (labeled with the letter B) and quadrant photo-diodes (labeled with the letter Q).
The simplified optical scheme of the GEO600 interferometer.
The effect of a gravitational wave propagating along the z axis on a circular ring of free test masses with radius L. (Left) The displacement induced by the polarization +. (Right) The displacement induced by the polarization ×. The two polarizations differ for a rotation around the z axis by the angle π/4.
The Virgo design noise budget. The sensitivity is limited by seismic noise below 2 Hz, by the thermal noise of the suspension between 2 Hz and 50 Hz, by the thermal noise of the mirror between 50 Hz and 100 Hz and by the shot noise above 100 Hz.
The design sensitivities for LIGO 4 km, LIGO 2 km,Virgo, GEO600, and TAMA300. For GEO600 the sensitivity can be tuned by changing the parameters of signal recycling (see Subsection ??? and Ref. 50). The plotted curve correspond to a detuning of the recycling cavity of 550 Hz.
The relation between cavity length and output phase for a resonant cavity. The relation between φ = 2ωℓδL/c and ϕ (see Eq. (29)) is represented in the center plot, for values of the input mirror's reflectivity which correspond to a cavity finesse of (r = 0.5), (r = 0.9), and (r = 0.99). (Left) A qualitative representation of a typical random motion of the cavity, which we suppose is dominated by the pendular mode of the attenuation chain (arbitrary horizontal units, vertical units of λℓ/4π). (Right) The resulting phase shift for the chosen values of R (arbitrary horizontal units, vertical units of radians).
The amplitude (in arbitrary units) of the field in the plane transverse to the beam axis for Hermite-Gauss modes with m = 0, 1, 2 (rows) and n = 0, 1, 2 (columns). We set w 0 = 1 and z = 0.
Scheme of the high power laser used in Virgo, based on injection locking techniques. The master laser's beam (continuous orange) of frequency f ℓ is phase modulated by the electro-optical modulator EOM, which introduces two sidebands f ℓ ± f m (dashed green). The interaction between the master's beam and the mirrors inside the cavity introduce a further modulation (dashed-dotted, pink), which is the noise to be reduced. A fraction of the light produced is transmitted through the reflective mirror M s , and photodetected. The feedback signal is obtained using a standard Pound-Drever-Hall technique, and used to drive the piezoelectric actuator PZ.
The schematization of a resonant cavity with moving mirrors, with the naming conventions used in the text.
The optical strain equivalent noise spectral amplitude as a function of the frequency, for selected values of the laser power at the beam splitter. The mirror mass is m = 20 kg and the cavity finesse , the values chosen in Virgo. Taking into account the power recycling the laser power at the beam splitter is I 0 = 103 W.
The evolution of a probability distribution for the relative distance between two masses in the phase space during a repeated measurement of position (a.1-a.2) and velocity (b.1-b.2). In the upper part, there is a large indetermination δP and a small one in δL. In the lower part the situation is reversed.
The homodyne detection technique. The continuous line corresponds to the carrier field, the dashed one to the modulation sidebands induced by the differential motion of the cavities. In this case the interferometer is maintained in the “dark fringe” condition, so the carrier entering the light port is completely reflected back. In the output field the carrier frequency f ℓ and the sidebands corresponding to a gravitational signal f ℓ ± f gw are present, and the photo-diode detects the beat (black arrows) between carrier and sidebands.
The heterodyne detection technique. The continuous lines corresponds to the carrier field, the dashed ones to the modulation sidebands induced by the differential motion of the cavities, the dotted-dashed ones to the radio frequency sidebands generated by the EOM. The radio frequency sidebands must be transmitted by the interferometer, so a small asymmetry is introduced between the two arms. The photo-diode senses the beat between the signal sidebands and the modulation sidebands (black arrows). The resulting radio frequency signal is further demodulated in the mixer.
The dc detection technique. The continuous line represents the carrier field, the dashed one to the modulation sidebands induced by the differential motion of the cavities. In the output the sidebands at the frequencies f ℓ ± f gw are present, together with some carrier field at a frequency f ℓ. In order to allow the transmission of the carrier a small offset from the “dark fringe” working point is introduced.
Polar plot of the quadrature error σ(θ) defined in Eq. (100) (continuous line) compared with the corresponding error ellipse (dashed line). The plots correspond to (from left to right).
Signal-to-noise ratio in units of , for different values of the parameter (0, 1/2, 1, 2, 10) as a function of the measured quadrature angle.
(Left) A simplified version of the speed meter proposed in Ref. 98. The input laser light passes through the power recycling mirror PRM, and enters into a standard Michelson interferometer built with the beams splitter BS and the N and the E mirrors. The dark port of this interferometer is coupled with an additional “sloshing” cavity. See the text for explanations. (Right) The schematic of a Sagnac interferometer. The input beam incoming from the left is split in two by the semitransparent mirror BS and recombined on it after a round trip.
The ratio between the strain equivalent noise amplitude and its standard quantum limit for a traditional interferometer. Here λ = 2π × 123.2 Hz, ɛ = 2π × 13.8 Hz, and ξ = 0. The different plots correspond to ı c = i 1 = (2π × 120 Hz)3, ı c = i 2 = (2π × 80Hz)3, ı c = i 3 = (2π × 40 Hz)3, and ı c = i 4 = (2π × 20 Hz)3.
A possible scheme for the subtraction of radiation pressure noise in a cavity. Cavity A of unperturbed length L A is the main one, while cavity B of unperturbed length L B is used to monitor the position of the central mirror. The g i are gains for the actuators driven by the error signal and the detuned cavity is used to measure the appropriate quadrature of the output beam. The variation of the length of the two cavities used in the text are given by Δ21 = ΔX 2 − ΔX 1 and by Δ32 = ΔX 3 − ΔX 2, where X i are the coordinates of the mirrors.
The simple model of photo-diode with quantum efficiency η discussed in Subsection ???. Only a fraction η of the incoming energy is converted in photoelectrons, and this is equivalent to an amplitude loss in the input beam. The beam incoming on the detector gets an additional contribution from fluctuations.
The modulus of the transfer function of Eq. (133) for γ = 1 (dotted line), γ = 10 (dashed line), and γ = 30 (continuous line). The frequency on the horizontal axis is expressed in unit of .
The modulus of the transfer function for a chain of N physical pendula for N = 1 (dotted line), N = 2 (dashed line), N = 3 (dotted-dashed line), and N = 4 (continuous line) for γ = 60. The frequency on the horizontal axis is expressed in unit of , where L is the total length of the chain. All the pendula have the same mass and length. The functions are plotted (thin lines) for reference.
The scheme of a Virgo superattenuator (see Ref. 138).
The distribution of elastic energy inside an infinite half space, when a pressure is applied on its boundary surface, in the static limit. The shape of the pressure distribution is a Gaussian with spot size r 0 depicted on the left. It is evident that the energy is localized in a region of typical linear size r 0. It follows that the ratio between the total energy stored in the coating and in the bulk region is given approximately by E C /E B ∼ d/r 0. In each region the dissipated energy will be proportional to the stored one multiplied by the appropriate loss angle. For a real mirror there will be corrections when the mirror's dimension becomes comparable with r 0.
Mechanical loss angle for a mirror coating at different temperatures (see Refs. 180 and 181). The data corresponds to the mirror “sample 1” in Ref. 181. The loss angle is measured at the frequencies specified, which correspond to the first and third internal modes of the mirror. The usual reduction of loss angle at low temperature occurring in many materials unfortunately is absent here.
Plots of the dependence of the coupling between a gravitational wave and an interferometer with an arm length L = 4 km. The functions plotted are (top) and (bottom) for f = 0 (left) and f = 16 kHz (right). The angles θ (vertical axis) and ϕ (horizontal axis), both given in radians, are spherical coordinates parameterizing the direction of the gravitational wave. The polarization tensors are defined as in Ref. 2. The arms of the detector are in the x and y direction.
The main parameters of the LIGO, Virgo, GEO600, and TAMA300 detectors. The light source in each case is a Nd:YAG with .
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