Compact Gravity Wave detector
Munawar Karim
Department of Physics
St. John Fisher College
Rochester, NY 14618
MICHELSON INTERFEROMETER
arms 10cm long• rigid mirrors
sampled interferometer• null detector configuration
• extension to 3-axis interferometer
• detector array
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h ≈ 10−23 / Hz
Detector design
• Each sample independent, unlike Herriot delay line or Fabry-Perot cavity
• Reflected beams recorded by photo-diode at end of each round-trip
• Ensures independence of each sample• Rigid mirrors facilitate vibration isolation• Vacuum environment 10-7 T • Dark fringe operation• Frequency response
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10−4 Hz to 104 Hz
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τ rt =2Lx
c+
12c
h11 ωt − k ⋅x( )0
Lx
∫ dx −12c
h11 ωt − k ⋅x( )Lx
0
∫ dx
Lx = L +ξ 1 ( proper length)
Excess time delay
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ε i ≡x i − L
L=
ξ i
L
∂ 2ε i
∂t 2+ ω0
2 +12
ω2h11 sinωt ⎛ ⎝ ⎜
⎞ ⎠ ⎟ε i = −
12
ω2h11 sinωt
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ξ1 −ξ 2
L≈ h11
ω2
ω02 −ω2
( )
ωω0
sinω0t − sinωt ⎛
⎝ ⎜
⎞
⎠ ⎟; ω0 ≠ ω
• Inhomogeneous Hill’s equation
Excess time delay
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Δτ i = hτ i 1+ω2
ω02 −ω2
( )
ωω0
sinωτ i − sinωτ i
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
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Δτ i =i=1
p
∑ pΔτ i
= h pτ i +ω2
ω02 −ω2
( )τ i
ωω0
sinω0τ i − sinωτ i
⎛
⎝ ⎜
⎞
⎠ ⎟
i=1
p
∑ ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
For p independent samples
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ω2
ω02 −ω2
( )τ i
ωω0
sinω0τ i − sinωτ i
⎛
⎝ ⎜
⎞
⎠ ⎟
i=1
p
∑
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→ω /ω0( )
2
1− ω /ω0( )2
( )dτ
0
0.01
∫ωω0
sinω0τ − sinωτ ⎛
⎝ ⎜
⎞
⎠ ⎟
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ω =2π 800;ω0 = 2π104 stiff mount
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ω /ω0( )2
1− ω /ω0( )2
( )dτ
0
0.01
∫ωω0
sinω0τ − sinωτ ⎛
⎝ ⎜
⎞
⎠ ⎟≈ −10−19
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ω /ω0( )2
1− ω /ω0( )2
( )dτ
0
0.01
∫ωω0
sinω0τ − sinωτ ⎛
⎝ ⎜
⎞
⎠ ⎟≈ −0.25
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ω =2π 800;ω0 = 2π soft mount
Shot noise uncertainty
• Time uncertainty due to shot noise for p round-trips
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σδτ =±hλ
Pin 4πc1τ i
1p
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pΔτ i = σ δτEquate
Shot-noise limited sensitivity
• Result valid for generalized mirror mount
• Depends on pulse duration only
• Independent of arm-length
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h ≥hλ
Pin 4πc1
pτ i
1pτ i
≥10−22
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