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  • Università degli studi di Napoli

    Federico II

    Facoltà di Scienze MM.FF.NN. Dipartimento di Scienze Fisiche

    Dottorato di Ricerca XIX Ciclo

    PhD Thesis in Fundamental and Applied Physics

    November 2006

    Adaptive Optics Techniques for

    Gravitational Wave Interferometers

    Coordinator:

    Prof. Gennaro Miele

    Advisors:

    Prof. Leopoldo Milano Author: Saverio Avino

    Dr. Enrico Calloni

  • To Isabel

    1

  • Contents

    Introduction 6

    1 Gravitational waves 9

    1.1 Einstein equations in the week-field approximation . . . . . . . . . . 9

    1.2 Propagation of gravitational waves . . . . . . . . . . . . . . . . . . . 11

    1.3 Sources of gravitational waves . . . . . . . . . . . . . . . . . . . . . . 13

    1.3.1 Gravitational waves generation . . . . . . . . . . . . . . . . . 13

    1.3.2 Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    1.3.3 Coalescent binaries . . . . . . . . . . . . . . . . . . . . . . . . 16

    1.3.4 Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    1.3.5 Stochastic background . . . . . . . . . . . . . . . . . . . . . . 17

    2 GW interferometric detection: the Virgo project 19

    2.1 Interferometric detection of gravitational waves . . . . . . . . . . . . 19

    2.2 Shot-noise and quantum limit . . . . . . . . . . . . . . . . . . . . . . 22

    2.3 Fabry-Perot cavities and recycling of light . . . . . . . . . . . . . . . 23

    2.4 Seismic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    2.5 Thermal noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    2.6 The VIRGO experiment . . . . . . . . . . . . . . . . . . . . . . . . . 30

    2

  • 2.6.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

    2.6.2 Optical layout . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    2.6.3 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    2.6.4 Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    2.6.5 The commissioning of the interferometer . . . . . . . . . . . . 37

    2.6.6 Status of Virgo . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    3 Adaptive Optics 42

    3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    3.2 Wavefront sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    3.3 Wavefront control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    3.4 Adaptive Optics applications . . . . . . . . . . . . . . . . . . . . . . . 49

    4 Adaptive optics system for the correction of aberrations of input

    laser beam in gravitational wave interferometers 52

    4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

    4.2 Effects of coupling between the interferometer asymmetries and the

    input beam jitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

    4.3 Implementation of an Adaptive Optics system in gravitational wave

    interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

    4.4 Adaptive Optics system based on interferometric wave-front sensing

    and membrane deformable mirror . . . . . . . . . . . . . . . . . . . . 61

    4.4.1 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . 61

    4.4.2 Wavefront sensing . . . . . . . . . . . . . . . . . . . . . . . . . 62

    4.4.3 Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    4.4.4 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . 72

    3

  • 4.4.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 73

    4.5 Implementation of a piezoelectric deformable mirror as actuator . . . 80

    4.5.1 Piezoelectric deformable mirror . . . . . . . . . . . . . . . . . 80

    4.5.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 81

    4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    5 Generation of non-Gaussian flat laser beams 85

    5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

    5.2 thermoelastic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

    5.3 Mexican-Hat cavities and generation of flat-beam . . . . . . . . . . . 86

    5.4 Generation of flat-beam by using deformable mirrors . . . . . . . . . 89

    5.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    6 Scanning wavefront sensor for the analysis of thermal deformation

    effects of the Virgo mirrors 97

    6.1 Thermal compensation problem . . . . . . . . . . . . . . . . . . . . . 97

    6.2 Wavefront signal extraction . . . . . . . . . . . . . . . . . . . . . . . 101

    6.2.1 Signals in DP and BS in the ideal case and introduction to

    approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 101

    6.2.2 Signals in DP and BS in the not ideal case of input mirrors

    aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

    6.3 Scanning wavefront sensor . . . . . . . . . . . . . . . . . . . . . . . . 110

    6.4 Characterization of the PZT tip-tilt mirror . . . . . . . . . . . . . . . 112

    6.4.1 Characterization of the PSD readout . . . . . . . . . . . . . . 113

    6.4.2 Characterization of the PZT mirror . . . . . . . . . . . . . . . 116

    6.4.3 Tests results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

    4

  • 6.5 Implementation in Virgo: first images acquisition . . . . . . . . . . . 123

    Conclusion 131

    Appendix 132

    A Calculation of the cij coefficients 133

    Bibliography 139

    Acknowledgments 145

    5

  • Introduction

    The present work has been carried out in the framework of Virgo experiment, a

    ground-based interferometric antenna for gravitational wave detection, located in

    Cascina (Pisa, Italy).

    The gravitational waves are perturbations of the space-time metric propagating at

    th speed of light, foreseen by the General Theory of Relativity. The existence of

    gravitational waves has been provided indirectly in 1992 by J.H. Taylor and R.A.

    Hulse. They observed an orbital period decay of the binary system PSR 1913+16,

    which was in agreement with the results predicted by gravitational wave emission.

    However, not yet direct detection of gravitational waves has been performed. In-

    deed, is not possible to produce detectable gravitational waves in laboratory, and

    also those produced by astronomical objects are very weak and difficult to detect.

    For this reason, very complex and high sensitivity ground-based detectors have been

    built, in order to obtain the firsts gravitational wave detections and to study and

    observe the universe through a new observational window, that is developing a Grav-

    itational Wave Astronomy.

    Different kinds of gravitational wave detector have been developed; among them the

    most important and diffused presently are the ground-based laser interferometric

    detectors. An interferometric detector consists of a long baseline Michelson inter-

    ferometer with a Fabry-Perot cavity implemented in each arm; the passage of a

    6

  • gravitational wave causes a differential variation of the length of the arms, which is

    detected at the output of the interferometer as variation of the interference pattern.

    To detect the very small length variations expected (of the order of 10−18m), it

    necessary to reduce all the noise sources that can mask the signal, as for example,

    the seismic noise, the thermal noise, the shot-noise.

    The various detectors presently in operation are near (or have already reached) the

    design curve of sensitivity. In this condition the interferometer, besides the isola-

    tion from the fundamental noises, must work in condition of extremely precision,

    especially in terms of alignments and wavefronts matching; for example, the angu-

    lar jitter of the Virgo input laser, cleaned by an input mode-cleaner at the level of

    10−11rad/ √

    Hz, imposes that the mirrors of the cavities must be aligned with an

    accuracy of the order of 3 × 10−9rad. In this thesis, it has been investigated the

    possibility to apply the Adaptive Optics techniques to improve the interferometer

    performances for these requirements. The first part of the work has been addressed

    to the problem of noise generated by the coupling of the laser input geometrical

    aberrations with the interferometer asymmetries. In particular, an adaptive optics

    system based on interferometric wavefront detection and deformable mirrors has

    been developed in order to correct the laser aberrations and to improve the beam

    matching to the interferometer. The system has shown performances in terms of

    efficiency, bandwidth and reintroduced noise which respect the specifics required for

    the application in the advanced Virgo interferometer.

    In the second part, the use of this system has been extended to generate, start-

    ing from standard Gaussian laser beams, the so called Flat-Beams, which are laser

    beams with particular shape proposed for future detectors in order to reduce the

    effects of thermoelastic noise. First flat-beams have

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