Abstract

Several km-scale gravitational-wave detectors have been constructed worldwide. These instruments combine a number of advanced technologies to push the limits of precision length measurement. The core devices are laser interferometers of a new kind; developed from the classical Michelson topology these interferometers integrate additional optical elements, which significantly change the properties of the optical system. Much of the design and analysis of these laser interferometers can be performed using well-known classical optical techniques; however, the complex optical layouts provide a new challenge. In this review, we give a textbook-style introduction to the optical science required for the understanding of modern gravitational wave detectors, as well as other high-precision laser interferometers. In addition, we provide a number of examples for a freely available interferometer simulation software and encourage the reader to use these examples to gain hands-on experience with the discussed optical methods.

Highlights

  • 1.1 The scope and style of the reviewThe historical development of laser interferometers for application as gravitationalwave detectors (Pitkin et al 2011) has involved the combination of relatively simple optical subsystems into more and more complex assemblies

  • The latter is usually true for the steady state approach: assuming that the interferometer is in a steady state, all solutions must be independent of time so that we can perform all computations at t = 0 without loss of generality

  • As these modulations usually have as their origin a change in optical path length, they are often phase modulations of the laser frequency, the radio frequency (RF) sidebands are utilised for optical readout purposes, while the signal sidebands carry the signal to be measured

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Summary

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Living Rev Relativ (2016) 19:3 layouts provide a new challenge. We give a textbook-style introduction to the optical science required for the understanding of modern gravitational wave detectors, as well as other high-precision laser interferometers. We provide a number of examples for a freely available interferometer simulation software and encourage the reader to use these examples to gain hands-on experience with the discussed optical methods. Keywords Gravitational waves · Gravitational-wave detectors · Laser interferometry · Optics · Simulations · Finesse

The scope and style of the review
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Plane-wave analysis
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Frequency domain analysis
Optical components: coupling of field amplitudes
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The two-mirror resonator
Coupling matrices
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Phase relation at a mirror or beam splitter
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Revised coupling matrices for space and mirrors
Length and tunings
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Light with multiple frequency components
Modulation of light fields
Phase modulation
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Frequency modulation
Amplitude modulation
Sidebands as phasors in a rotating frame
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Phase modulation through a moving mirror
Coupling matrices for beams with multiple frequency components
Modulation index
Mirror modulation
Optical readout
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Signal demodulation
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Optical beat
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Basic interferometers
The two-mirror cavity: a Fabry–Perot interferometer
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Michelson interferometer
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Michelson interferometer and the sideband picture
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Response of the Michelson interferometer to a gravitational waves signal
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Cavity power
Michelson power
Michelson gravitational wave response
Radiation pressure and quantum fluctuations of light
Quantum noise sidebands
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Vacuum noise and gravitational-wave detector readout schemes
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Quantum noise with non-linear optical effects or squeezed states
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Semi-classical Schottky shot-noise formula
Optical springs
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Optical spring
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Quantum-noise limited interferometer sensitivity
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Advancing the interferometer layout
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Michelson interferometers with power recycling
Michelson interferometers with arm cavities
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Sagnac interferometer
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Michelson interferometer with signal recycling
Interferometric length sensing and control
An overview of the control problem
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11 The typical light target displacement wavelength is noise spectral
Linear time-invariant control theory: introductory concepts
Digital signal processing for control
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Degrees of freedom and operating points
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Error signals and transfer functions
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Bode plots: traditional control theory for SISO loops
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Separating mixtures of the degrees of freedom: control matrices
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Modern control methods in gravitational-wave detectors
Fabry–Perot length sensing
8.10 The Pound–Drever–Hall length sensing scheme
8.11 Michelson length sensing
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8.12 Advanced LIGO: an example of a complex interferometer
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8.13 The Schnupp modulation scheme
8.14 Extending the Pound–Drever–Hall technique to more complicated optical systems
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8.17.1 Michelson modulation
8.17.2 Cavity power and slope
8.17.3 Michelson with Schnupp modulation
Beam shapes: beyond the plane wave approximation
A typical laser beam: the fundamental Gaussian mode
Describing beam distortions with higher-order modes
The paraxial approximation
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Transverse electromagnetic modes
Properties of Gaussian beams
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Astigmatic beams: the tangential and sagittal plane
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Higher-order Hermite–Gauss modes
The Gaussian beam parameter
Properties of higher-order Hermite–Gauss modes
9.10 Gouy phase
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9.11 Laguerre–Gauss modes
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9.13 ABCD matrices
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9.14 Computing a cavity eigenmode and stability
9.15 Round-trip Gouy phase and higher-order-mode separation
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9.16 Coupling of higher-order-modes
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9.17.2 Telescope and Gouy phase
9.17.3 LG33 mode
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10 Imperfect interferometers
10.1 Spatial modes in optical cavities
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10.2 Cavity alignment in the mode picture
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10.3 Mode mismatch
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10.4 Spatial defects
10.5 Operating cavities at high power
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10.6 The Michelson: differential imperfections
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10.7 Advanced LIGO: implications for design and commissioning
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10.8 Commissioning
10.9.1 Higher-order mode resonances
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10.9.2 Mode cleaner
10.9.4 Impact of thermal aberrations
11 Scattering into higher-order modes
11.1 Light scattering in interferometers
11.2 Mirror surface defects
11.3 Coupling between higher-order modes
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11.4 Simulation methods
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11.5 Mirror surface maps
11.6 Spectrum of surface distortions
11.7 Surface description with Zernike polynomials
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11.8 Mode coupling due to mirror surfaces defects
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11.9 Efficient coupling matrix computations with multiple distortions
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11.10 Clipping by finite apertures
11.11 Cavity modes of many shapes
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Findings
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