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

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81 sideband has been made to be exactly resonant in both the power and signal cavities, so its transmission function is likewise purely real. β S t = arctan � � ℑ {ts} ℜ {ts} (3.43) The subscript s will be used to indicate the transmission function corresponding to the SSB, similar to the way in which + and − indicate the transmission functions for the upper and lower PM RF sidebands. Optimization Astrophysical considerations have driven the choice of the optics as well as the carrier resonance in the interferometer, so methods of optimization must largely be restricted to things which affect the sidebands. Phases are the quantities which affect the transmission functions of the interferometer, so the “knobs” that can be turned are either modulation frequencies and/or path lengths. Attention in this thesis is restricted to the case in which the modulation frequencies remain fixed, once chosen, and the lower RF sideband remains resonant in both the power and signal cavities, as a function of detuning. This fixes the lengths of the power and signal cavity relative to the RF sidebands, which leaves the asymmetry as the only flexible length parameter. 3.3.1 Reflected and Pickoff Signals The simplified RSE interferometer of Figure 3.8 can be simplified even further to a single cavity, the power recycling cavity, when considering the fields actually incident on the various photodiodes. The front mirror is the power mirror, and the back mirror is the compound mirror formed by the signal cavity which has the Michelson mirror as its front mirror and the signal mirror as its back mirror. The fields then are the reflected, pickoff, and transmitted fields of a single cavity, with the appropriate substitutions. Sensitivity to a particular degree of freedom at a particular photodiode can then be formed from the simple relations found for a single cavity. For the reflected and pickoff signals, the derivatives with respect to the degrees of freedom take the
Laser �§�� §� rp ©§© � � ©§© ¢ £ ¡ 82 Compound signal cavity mirror rs(Φ+, Φ−, φ−, φs) PD1 PD2 ¦§¦ ¦§¦ ¨§¨ ¨§¨ ¨§¨ ¦§¦ + − φ+ PRM pp Figure 3.12: A single cavity representation of the RSE interferometer. The subscripts on the photodiode transmission functions rp, pp, and dp indicate application of a cavity reflectivity, pickoff, and transmitted fields of this power recycling cavity. The subscript s on the end mirror indicates the reflectivity of a cavity applied to the signal cavity. form of the following products. ∂rp ∂Φ+ ∂rp ∂φ+ ∂rp ∂φ− ∂rp ∂φs = ∂rp ∂rs = ∂rp ∂φ+ = ∂rp ∂rs ∂rs ∂rm ∂rs = ∂rp ∂rs ∂rs ∂φs ∂rm ∂rac ∂rm ∂rm ∂φ− ∂rac ∂Φ+ dp ¤ ¥ ¥ ¤ PD3 (3.44) Specifically, these are for the reflected signals at the photodiode characterized by the transmission function rp. The subscript p is to indicate that the transmission function is for the effective single power recycling cavity. Similarly, the subscript s refers to the signal cavity, formed by the Michelson mirror and the signal mirror. The subscript m indicates the reflectivity and transmissivity of the Michelson mirror. The pickoff signals are formed by replacing rp subscript with pp.
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Signal Extraction and Optical Desig
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Abstract iii The LIGO project is tw
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v Of course, one couldn’t last si
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vii 2.3.2 Optimized Broadband . . .
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ix 5.7.2 Narrowband RSE . . . . . .
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xi 3.11 Signal cavity spectral sche
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xiii D.7 Michelson compensation boa
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xv 4.2 Effective response of the mi
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2 into which a stone has been dropp
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4 but no smoke). These tricks have
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6 At the heart of passive technique
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8 The various terms are defined as
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Pendulum Thermal Noise 10 The test
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12 back-action force on the inciden
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14 Scatter and absorption of coatin
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16 Another feature of signal tuned
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18 A scheme for controlling the fiv
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Laser PD1 PD2 ¦¡¦ ¥¡¥ PRM +
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22 known as the detuning phase. The
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24 cavity induces a phase shift upo
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26 of the first cavity, then for a
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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
Page 59 and 60: 44 For high frequency response, bot
Page 61 and 62: 46 Although this is a tiny signal,
Page 63 and 64: Strain sensitivity (h(f)/rtHz) 10
Page 65 and 66: 50 sources the interferometer would
Page 67 and 68: Chapter 3 Signal Extraction 52 The
Page 69 and 70: photodiode is proportional to 54 |E
Page 71 and 72: 56 The demodulation phase β = 0 is
Page 73 and 74: 58 to the difference in the rates o
Page 75 and 76: 60 out of the amplitude of the audi
Page 77 and 78: 62 is non-zero. This is the basic f
Page 79 and 80: 64 further to generate the second o
Page 81 and 82: The source amplitude is set by the
Page 83 and 84: 68 of their sum and difference, or
Page 85 and 86: however, is given by 70 φ− = Ω
Page 87 and 88: 72 transmittance is then higher tha
Page 89 and 90: 74 resonance, for both upper and lo
Page 91 and 92: 76 due to the fluctuating length of
Page 93 and 94: 78 the hash marks corresponding to
Page 95: sensitive to differential signals,
Page 99 and 100: compound mirror. The derivatives of
Page 101 and 102: freedom. 86 N ′ = 1 ∂rc −irc
Page 103 and 104: 88 The φs degree of freedom is qui
Page 105 and 106: care must be applied to this questi
Page 107 and 108: 92 The magnitude likewise doesn’t
Page 109 and 110: 94 Φ+ Φ− φ+ φ− φs Refl 81
Page 111 and 112: 96 10 more than this. With a transm
Page 113 and 114: 98 factor to Φ+, which was ∼ 500
Page 115 and 116: 100 Degree of freedom Residual leng
Page 117 and 118: 3.5 Conclusions 102 Designing the R
Page 119 and 120: Chapter 4 104 The RSE Tabletop Prot
Page 121 and 122: 106 Two input test masses (ITM) and
Page 123 and 124: 108 a CVI W1-1064 window, AR coated
Page 125 and 126: 110 order diffracted beam. Although
Page 127 and 128: 112 Reflected 81 Reflected 54 Picko
Page 129 and 130: 114 combination of the PRM and SM1
Page 131 and 132: 116 from ETM2 and SM5. A box of ele
Page 133 and 134: 118 output roughly +18 dBm, while t
Page 135 and 136: 120 The reflected powers are used t
Page 137 and 138: 122 Waist size (mm) Waist position
Page 139 and 140: 124 characterized mirrors, alignmen
Page 141 and 142: 126 completely different mixer, and
Page 143 and 144: 128 on with minimal gain and using
Page 145 and 146: 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
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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
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Relative Magnitude Relative Phase (
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144 an offset can be estimated. The
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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
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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
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214 is replaced with a 1 µF capaci
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216 Figure D.2: PZT compensation sc
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Input Test In 1 kΩ 1 kΩ − + 1
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Figure D.10: RF frequency generatio
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Bibliography 226 [1] J. Weber. Dete
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228 [19] Yuk Tung Liu and Kip S. Th
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230 [38] T.T. Lyons, M.W. Regehr, a
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232 [60] J.E. Mason. Noise coupling
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