Fabry-Perot Arm Cavities Research Articles

Temperature Control for an Intra-Mirror Etalon in Interferometric Gravitational Wave Detector Fabry–Perot Cavities

The sensitivity of interferometric gravitational wave detectors is optimized, in part, by balanced finesse in the long Fabry–Perot arm cavities. The input test mass mirrors of Advanced Virgo feature parallel faces, which creates an etalon within the substrate, adding variability in the total mirror reflectivity, in order to correct imbalanced finesse due to manufacturing tolerances. Temperature variations in mirror substrate change the optical path length primarily through varying the index of refraction and are tuned to correct for a finesse imbalance of up to 2.8% by a full etalon fringe of 0.257 K. A negative feedback control system was designed to control the mirror temperature by using an electrical resistive heating belt actuator for a heat transfer process modeled as a two-pole plant. A zero controller filter was designed which achieves temperature control within 2.3% of the etalon fringe and recovers to within 10% of the working point within 32 hours after a step input of one etalon fringe. A preliminary unlock condition control designed to compensate when the interferometer unlocks shows that the control remains stable even after a drastic change in the plant due to the absence of the laser heating. Further improvements to the control must also consider the full heat transfer mechanisms by using modern control state space models.

Auxiliary lasers for Advanced Virgo Gravitational Wave detector using single pass Second Harmonic Generation in Periodically Poled Lithium Niobate crystal

The Advanced Virgo (AdV) detector is composed of different degrees of freedom (DOFs) i.e. Michelson interferometer, two Fabry-Perot arm cavities, signal recycling cavity, and power recycling cavity. These DOFs need to be locked to precise accuracy with robust, fast and reliable control systems. The control signals used to lock all the DOFs are mildly decoupled in frequency and the optical response of the DOFs are nonlinear, where the linear range of operation is just some percentage of the fringe for each DOF, thus posing difficulty in resonance lock at input laser wavelength for all the cavities. In particular, the control status of the arm cavities can alter the state of the detector’s operational configuration. Using Auxiliary Lasers to lock the arm cavities at different wavelength offers flexible and robust lock of the detector and more spatial margin on the control signals. Second harmonic generation offers the most direct way to have laser beam with different wavelength and phase locked to the AdV input laser beam. We generated upto 97 mW of SH beam in single configuration at 532 nm using fibered amplified laser source at 1064 nm in a 10 mm long Poled Lithium Niobate crystal.

Large mirror surface control by corrective coating

The Advanced Virgo gravitational wave detector aims at a sensitivity ten times better than the initial LIGO and Virgo detectors. This implies very stringent requirement on the optical losses in the interferometer arm cavities. In this paper we focus on the mirrors which form the interferometer arm cavities and that require a surface figure error to be well below one nanometre on a diameter of 150 mm. This ‘sub-nanometric flatness’ is not achievable by classical polishing on such a large diameter. Therefore we present the corrective coating technique which has been developed to reach this requirement. Its principle is to add a non-uniform thin film on top of the substrate in order to flatten its surface. In this paper we will introduce the Advanced Virgo requirements and present the basic principle of the corrective coating technique. Then we show the results obtained experimentally on an initial Virgo substrate. Finally we provide an evaluation of the round-trip losses in the Fabry–Perot arm cavities once the corrected surface is used.

All-reflective coupling of two optical cavities with 3-port diffraction gratings

The shot-noise limited sensitivity of Michelson-type laser interferometers with Fabry-Perot arm cavities can be increased by the so-called power-recycling technique. In such a scheme the power-recycling cavity is optically coupled with the interferometer's arm cavities. A problem arises because the central coupling mirror transmits a rather high laser power and may show thermal lensing, thermo-refractive noise and photo-thermo-refractive noise. Cryogenic cooling of this mirror is also challenging, and thus thermal noise becomes a general problem. Here, we theoretically investigate an all-reflective coupling scheme of two optical cavities based on a 3-port diffraction grating.We show that power-recycling of a high-finesse arm cavity is possible without transmitting any laser power through a substrate material. The power splitting ratio of the three output ports of the grating is, surprisingly, noncritical.

The Virgo automatic alignment system

The automatic alignment system of the Virgo interferometer differs substantially from those used in similar experiments, since it uses a variant of the Anderson technique. This implies a completely different control topology with respect to other detectors, and the main feature is a strong coupling of different degrees of freedom in the optical signals. It also provides two extra output ports in which differential wave-front sensors can be placed, namely the light transmitted by the Fabry–Perot arm cavities. We report on the first experimental demonstration of this technique on a large scale recycled interferometer, and on the present status of the automatic alignment system.

Sagnac interferometer as a speed-meter-type, quantum-nondemolition gravitational-wave detector

According to quantum measurement theory, "speed meters" -- devices that measure the momentum, or speed, of free test masses -- are immune to the standard quantum limit (SQL). It is shown that a Sagnac-interferometer gravitational-wave detector is a speed meter and therefore in principle it can beat the SQL by large amounts over a wide band of frequencies. It is shown, further, that, when one ignores optical losses, a signal-recycled Sagnac interferometer with Fabry-Perot arm cavities has precisely the same performance, for the same circulating light power, as the Michelson speed-meter interferometer recently invented and studied by P. Purdue and the author. The influence of optical losses is not studied, but it is plausible that they be fairly unimportant for the Sagnac, as for other speed meters. With squeezed vacuum (squeeze factor $e^{-2R} = 0.1$) injected into its dark port, the recycled Sagnac can beat the SQL by a factor $ \sqrt{10} \simeq 3$ over the frequency band $10 {\rm Hz} \alt f \alt 150 {\rm Hz}$ using the same circulating power $I_c\sim 820$ kW as is used by the (quantum limited) second-generation Advanced LIGO interferometers -- if other noise sources are made sufficiently small. It is concluded that the Sagnac optical configuration, with signal recycling and squeezed-vacuum injection, is an attractive candidate for third-generation interferometric gravitational-wave detectors (LIGO-III and EURO).

Model of thermal wave-front distortion in interferometric gravitational-wave detectors I Thermal focusing

We develop a steady-state analytical and numerical model of the optical response of power-recycled Fabry-Perot Michelson laser gravitational-wave detectors to thermal focusing in optical substrates. We assume that the thermal distortions are small enough that we can represent the unperturbed intracavity field anywhere in the detector as a linear combination of basis functions related to the eigenmodes of one of the Fabry-Perot arm cavities, and we take great care to preserve numerically the nearly ideal longitudinal phase resonance conditions that would otherwise be provided by an external servo-locking control system. We have included the effects of nonlinear thermal focusing due to power absorption in both the substrates and coatings of the mirrors and beamsplitter, the effects of a finite mismatch between the curvatures of the laser wavefront and the mirror surface, and the diffraction by the mirror aperture at each instance of reflection and transmission. We demonstrate a detailed numerical example of this model using the MATLAB program Melody for the initial LIGO detector in the Hermite-Gauss basis, and compare the resulting computations of intracavity fields in two special cases with those of a fast Fourier transform field propagation model. Additional systematic perturbations (e.g., mirror tilt, thermoelastic surface deformations, and other optical imperfections) can be included easily by incorporating the appropriate operators into the transfer matrices describing reflection and transmission for the mirrors and beamsplitter.

Analysis of light noise sources in a recycled Michelson interferometer with Fabry–Perot arms

We present a method by which the effect of laser field variations on the signal output of an interferometric gravitational wave detector is rigorously determined. Using the Laser Interferometer Gravitational Wave Observatory (LIGO) optical configuration of a power recycled Michelson interferometer with Fabry-Perot arm cavities as an example, we calculate the excess noise after the input filter cavity (mode cleaner) and the dependence of the detector strain sensitivity on laser frequency and amplitude noise, radio frequency oscillator noise, and scattered-light phase noise. We find that noise on the radio frequency sidebands generally limits the detector's sensitivity.

Spatiotemporal model of the LIGO interferometer

We develop a detailed spatiotemporal model in the adiabatic approximation of the optical response of the Laser Interferometer Gravitational-Wave Observatory (LIGO) optical system to environmental perturbations. We begin by deriving a first-order linear time-dependent evolution equation that models the electromagnetic field as a function of position within a Fabry–Perot interferometer. This model allows both the length of the resonator and the misalignment angles of the end mirrors to vary in time and describes both resonant and nonresonant phenomena. After defining a biorthogonality relation that must be satisfied by general unperturbed spatial eigenfunctions of the Fabry–Perot interferometer, we expand the intracavity field as a linear combination of these functions and convert the spatiotemporal evolution equation into a linear system of coupled time-dependent iteration and/or differential equations. We then calculate the adiabatic connection equations that link the two LIGO Fabry–Perot interferometers through the power recycling cavity, which comprises two mirrors (the power recycling input mirror and the beam splitter) that have the same mechanical degrees of freedom as those in the Fabry–Perot arm cavities. We develop a detailed instance of this model for the evolution of the intracavity field within a resonator with sufficiently small misalignment angles that a Hermite–Gauss basis set can be used. We develop a detailed general approach to signal demodulation for simulation of servo-control systems and describe its implementation in the Hermite–Gauss approximation for Cartesian split-plane detectors. Finally, we demonstrate the use of this small-angle model to simulate the effect of angular misalignment on the longitudinal response function.

Signal extraction in a power-recycled Michelson interferometer with Fabry–Perot arm cavities by use of a multiple-carrier frontal modulation scheme

We present a signal extraction scheme for longitudinal sensing and control of an interferometric gravitational-wave detector based on a multiple-frequency heterodyne detection technique. Gravitational-wave detectors use multiple-mirror resonant optical systems where resonance conditions must be satisfied for multiple degrees of freedom that are optically coupled. The multiple-carrier longitudinal-sensing technique provides sensitive signals for all interferometric lengths to be controlled and successfully decouples them. The feasibility of the technique is demonstrated on a tabletop-scale power-recycled Michelson interferometer with Fabry-Perot arm cavities, and the experimentally measured values of the length-sensing signals are in good agreement with theoretical calculations.