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

Read more

Summary

Page 2 of 217

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
Page 6 of 217
Page 8 of 217
Plane-wave analysis
Page 10 of 217
Frequency domain analysis
Optical components: coupling of field amplitudes
Page 12 of 217
The two-mirror resonator
Coupling matrices
Page 14 of 217
Phase relation at a mirror or beam splitter
Page 16 of 217
Page 18 of 217
Page 20 of 217
Page 22 of 217
Revised coupling matrices for space and mirrors
Length and tunings
Page 24 of 217
Light with multiple frequency components
Modulation of light fields
Phase modulation
Page 28 of 217
Frequency modulation
Amplitude modulation
Sidebands as phasors in a rotating frame
Page 30 of 217
Phase modulation through a moving mirror
Coupling matrices for beams with multiple frequency components
Modulation index
Mirror modulation
Optical readout
Page 36 of 217
Signal demodulation
Page 38 of 217
Optical beat
Page 40 of 217
Basic interferometers
The two-mirror cavity: a Fabry–Perot interferometer
Page 42 of 217
Michelson interferometer
Page 46 of 217
Michelson interferometer and the sideband picture
Page 48 of 217
Page 50 of 217
Response of the Michelson interferometer to a gravitational waves signal
Page 52 of 217
Page 54 of 217
Cavity power
Michelson power
Michelson gravitational wave response
Radiation pressure and quantum fluctuations of light
Quantum noise sidebands
Page 58 of 217
Page 60 of 217
Vacuum noise and gravitational-wave detector readout schemes
Page 62 of 217
Page 64 of 217
Page 66 of 217
Quantum noise with non-linear optical effects or squeezed states
Page 68 of 217
Page 70 of 217
Semi-classical Schottky shot-noise formula
Optical springs
Page 72 of 217
Page 74 of 217
Optical spring
Page 76 of 217
Quantum-noise limited interferometer sensitivity
Page 78 of 217
Advancing the interferometer layout
Page 80 of 217
Michelson interferometers with power recycling
Michelson interferometers with arm cavities
Page 82 of 217
Page 84 of 217
Sagnac interferometer
Page 86 of 217
Michelson interferometer with signal recycling
Interferometric length sensing and control
An overview of the control problem
Page 90 of 217
11 The typical light target displacement wavelength is noise spectral
Linear time-invariant control theory: introductory concepts
Digital signal processing for control
Page 94 of 217
Degrees of freedom and operating points
Page 96 of 217
Page 98 of 217
Error signals and transfer functions
Page 100 of 217
Bode plots: traditional control theory for SISO loops
Page 102 of 217
Separating mixtures of the degrees of freedom: control matrices
Page 104 of 217
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
Page 108 of 217
8.12 Advanced LIGO: an example of a complex interferometer
Page 110 of 217
Page 112 of 217
8.13 The Schnupp modulation scheme
8.14 Extending the Pound–Drever–Hall technique to more complicated optical systems
Page 114 of 217
Page 116 of 217
Page 118 of 217
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
Page 126 of 217
Transverse electromagnetic modes
Properties of Gaussian beams
Page 128 of 217
Astigmatic beams: the tangential and sagittal plane
Page 130 of 217
Higher-order Hermite–Gauss modes
The Gaussian beam parameter
Properties of higher-order Hermite–Gauss modes
9.10 Gouy phase
Page 134 of 217
9.11 Laguerre–Gauss modes
Page 136 of 217
Page 138 of 217
9.13 ABCD matrices
Page 140 of 217
9.14 Computing a cavity eigenmode and stability
9.15 Round-trip Gouy phase and higher-order-mode separation
Page 144 of 217
9.16 Coupling of higher-order-modes
Page 146 of 217
Page 148 of 217
9.17.2 Telescope and Gouy phase
9.17.3 LG33 mode
Page 150 of 217
10 Imperfect interferometers
10.1 Spatial modes in optical cavities
Page 152 of 217
10.2 Cavity alignment in the mode picture
Page 154 of 217
10.3 Mode mismatch
Page 156 of 217
10.4 Spatial defects
10.5 Operating cavities at high power
Page 158 of 217
Page 160 of 217
Page 162 of 217
10.6 The Michelson: differential imperfections
Page 164 of 217
10.7 Advanced LIGO: implications for design and commissioning
Page 168 of 217
Page 170 of 217
Page 172 of 217
Page 174 of 217
10.8 Commissioning
10.9.1 Higher-order mode resonances
Page 176 of 217
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
Page 182 of 217
11.4 Simulation methods
Page 184 of 217
11.5 Mirror surface maps
11.6 Spectrum of surface distortions
11.7 Surface description with Zernike polynomials
Page 188 of 217
11.8 Mode coupling due to mirror surfaces defects
Page 192 of 217
Page 194 of 217
Page 196 of 217
Page 198 of 217
11.9 Efficient coupling matrix computations with multiple distortions
Page 200 of 217
11.10 Clipping by finite apertures
11.11 Cavity modes of many shapes
Page 204 of 217
Page 206 of 217
Page 208 of 217
Page 212 of 217
Page 214 of 217
Findings
Page 216 of 217
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call