P010010-00-R - LIGO - California Institute of Technology

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Contents vi Abstract iii Acknowledgements iv 1 Prologue 1 1.1 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Search for Gravitational Waves . . . . . . . . . . . . . . . . . . . 3 1.2.1 Resonant Bar Detectors . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Interferometer Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 Seismic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.2 Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.3 Optical Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.4 Interferometer Configuration . . . . . . . . . . . . . . . . . . . 12 1.4 The Goal of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Resonant Sideband Extraction 19 2.1 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.1 The Three Mirror Coupled Cavity . . . . . . . . . . . . . . . . 22 2.1.2 RSE Transfer Functions . . . . . . . . . . . . . . . . . . . . . 31 2.2 Practical Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1 Power Recycling . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.2 Transfer Function Limitations . . . . . . . . . . . . . . . . . . 39 2.2.3 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3 Detector Optimization for RSE . . . . . . . . . . . . . . . . . . . . . 44 2.3.1 Narrowband Response . . . . . . . . . . . . . . . . . . . . . . 45
vii 2.3.2 Optimized Broadband . . . . . . . . . . . . . . . . . . . . . . 48 3 Signal Extraction 52 3.1 Optically Heterodyned Signals . . . . . . . . . . . . . . . . . . . . . . 52 3.1.1 DC Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1.2 Frequency Dependent Signal . . . . . . . . . . . . . . . . . . . 58 3.1.3 Phasor Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.1.4 Twiddle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 The RSE Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2.1 Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.2 Detuned Interferometer . . . . . . . . . . . . . . . . . . . . . . 73 3.2.3 Additional Sidebands . . . . . . . . . . . . . . . . . . . . . . . 75 3.2.4 Proposed Signal Extraction Scheme . . . . . . . . . . . . . . . 76 3.3 Signal Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3.1 Reflected and Pickoff Signals . . . . . . . . . . . . . . . . . . . 81 3.3.2 Dark Port Signals . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.4 Matrix of Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.4.1 Optimized Broadband . . . . . . . . . . . . . . . . . . . . . . 93 3.4.2 Narrowband RSE . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4 The RSE Tabletop Prototype Experiment 104 4.1 Prototype Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.1.1 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.1.2 Lengths and Frequencies . . . . . . . . . . . . . . . . . . . . . 108 4.1.3 Electro-Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.1.4 Mirror Mounting . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.1.5 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2 Input Optics and the Mach-Zender . . . . . . . . . . . . . . . . . . . 118 4.3 Arm Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.3.1 Visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Page 1 and 2: Signal Extraction and Optical Desig
Page 3 and 4: Abstract iii The LIGO project is tw
Page 5: v Of course, one couldn’t last si
Page 9 and 10: ix 5.7.2 Narrowband RSE . . . . . .
Page 11 and 12: xi 3.11 Signal cavity spectral sche
Page 13 and 14: xiii D.7 Michelson compensation boa
Page 15 and 16: xv 4.2 Effective response of the mi
Page 17 and 18: 2 into which a stone has been dropp
Page 19 and 20: 4 but no smoke). These tricks have
Page 21 and 22: 6 At the heart of passive technique
Page 23 and 24: 8 The various terms are defined as
Page 25 and 26: Pendulum Thermal Noise 10 The test
Page 27 and 28: 12 back-action force on the inciden
Page 29 and 30: 14 Scatter and absorption of coatin
Page 31 and 32: 16 Another feature of signal tuned
Page 33 and 34: 18 A scheme for controlling the fiv
Page 35 and 36: Laser PD1 PD2 ¦¡¦ ¥¡¥ PRM +
Page 37 and 38: 22 known as the detuning phase. The
Page 39 and 40: 24 cavity induces a phase shift upo
Page 41 and 42: 26 of the first cavity, then for a
Page 43 and 44: Frequency (Hz) 5000 4000 3000 2000
Page 45 and 46: 0.8 0.6 0.4 0.2 0 −4 φ sec 1 6 5
Page 47 and 48: the mirror is 32 i2k(l+δl/2 cos(ω
Page 49 and 50: The shot noise current spectral den
Page 51 and 52: 36 depends on the choice of ITM, up
Page 53 and 54: 38 mirror is calculated based on th
Page 55 and 56: 40 This is not the transmissivity o
Page 57 and 58:
Cavity reflectivity 1 0.9 0.8 0.7 0
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44 For high frequency response, bot
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46 Although this is a tiny signal,
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Strain sensitivity (h(f)/rtHz) 10
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50 sources the interferometer would
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Chapter 3 Signal Extraction 52 The
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photodiode is proportional to 54 |E
Page 71 and 72:
56 The demodulation phase β = 0 is
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58 to the difference in the rates o
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60 out of the amplitude of the audi
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62 is non-zero. This is the basic f
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64 further to generate the second o
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The source amplitude is set by the
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68 of their sum and difference, or
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however, is given by 70 φ− = Ω
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72 transmittance is then higher tha
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74 resonance, for both upper and lo
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76 due to the fluctuating length of
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78 the hash marks corresponding to
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sensitive to differential signals,
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Laser �§�� §� rp
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compound mirror. The derivatives of
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freedom. 86 N ′ = 1 ∂rc −irc
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88 The φs degree of freedom is qui
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care must be applied to this questi
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92 The magnitude likewise doesn’t
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94 Φ+ Φ− φ+ φ− φs Refl 81
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96 10 more than this. With a transm
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98 factor to Φ+, which was ∼ 500
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100 Degree of freedom Residual leng
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3.5 Conclusions 102 Designing the R
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Chapter 4 104 The RSE Tabletop Prot
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106 Two input test masses (ITM) and
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108 a CVI W1-1064 window, AR coated
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110 order diffracted beam. Although
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112 Reflected 81 Reflected 54 Picko
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114 combination of the PRM and SM1
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116 from ETM2 and SM5. A box of ele
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118 output roughly +18 dBm, while t
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120 The reflected powers are used t
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122 Waist size (mm) Waist position
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124 characterized mirrors, alignmen
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126 completely different mixer, and
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128 on with minimal gain and using
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130 for the φ− servo, hand tunin
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132 matrix to imperfections in thes
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RF photodiodes L.O. BLP-5 5 MHz low
Page 151 and 152:
136 modeled matrix are the demodula
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138 the dual-recycled Michelson. Th
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140 4.5.3 Gravitational Wave Transf
Page 157 and 158:
Relative Magnitude Relative Phase (
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144 an offset can be estimated. The
Page 161 and 162:
146 Many aspects of the process of
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148 the detuned case is more compli
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150 different, and these terms do n
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152 where there (ideally) is no car
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154 equivalent to shot noise, isign
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156 The laser field can be expanded
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Φ+ = φ3 + φ4 Φ− = φ3 − φ4
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160 Eq. (5.15) will then be evaluat
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DC Carrier Term 162 It’s useful t
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164 of interest, then, the noise si
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166 transmissivity terms. Eq. (5.15
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168 match the detuning phase for th
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170 A second subscript, I or Q, wil
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each noise source. 5.7 Analysis 172
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174 the demodulation phase. This is
Page 191 and 192:
176 noise, the frequency noise spec
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Amplitude noise (arb. units) Freque
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180 ferential length RMS fluctuatio
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182 characterizes the fringe width
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A.1.1 Cavity Coupling 184 One of th
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186 A.1.2 Cavity Approximations The
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esonance. 188 rcanti−res. = rf +
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190 where the primed parameters ref
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Magnitude Phase (degrees) 10 0 10
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194 The magnitude of the transmitte
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196 Where As is 1 − Ls, or one mi
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198 The beamsplitter is typically v
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200 closed loop gain of the control
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202 Substitution and some algebra m
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204 Defining the phases for the RF
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Appendix C 206 Cross-Coupled 2 × 2
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208 the coupling due to the second
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to the disturbance at d1. 210 ⎛
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212 The measurement is also assumed
Page 229 and 230:
214 is replaced with a 1 µF capaci
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216 Figure D.2: PZT compensation sc
Page 233 and 234:
Page 235 and 236:
Page 237 and 238:
Input Test In 1 kΩ 1 kΩ − + 1
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Figure D.10: RF frequency generatio
Page 241 and 242:
Bibliography 226 [1] J. Weber. Dete
Page 243 and 244:
228 [19] Yuk Tung Liu and Kip S. Th
Page 245 and 246:
230 [38] T.T. Lyons, M.W. Regehr, a
Page 247:
232 [60] J.E. Mason. Noise coupling
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