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Beam jitter coupling in advanced LIGO

Guido Mueller

Open Access Open Access

Abstract

Fluctuations in the position or propagation direction of the laser beam (beam jitter) is one of the most critical technical noise sources in an interferometric gravitational wave detector. These fluctuations couple to spurious misalignments of the mirrors forming the interferometer and potentially decrease the sensitivity. In this paper we calculate the transfer function of beam jitter into the gravitational wave channel for the Advanced LIGO detector and derive a first expression for the requirements on beam jitter for Advanced LIGO.

©2005 Optical Society of America

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Figures (12)
Fig. 1.
Fig. 1. The Advanced LIGO interferometer consists of two input test masses (ITM1,2) and two end test masses (ETM1,2) which form the two arm cavities. In addition a beam splitter (BS) is used to split the light. The power recycling (PR) mirror builds up the power in the interferometer and the signal recycling (SR) mirror builds up the signal.

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Fig. 2.
Fig. 2. Transfer function of a mode u 1 into a mode u 0 in reflection at a 4km long cavity with tilted input (ITM) and end (ETM) mirrors. The abscissa shows the audio frequency offset of the jitter sideband with respect to the carrier (left panel) and the 180MHz-sideband (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10-8rad.

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Fig. 3.
Fig. 3. Transfer function of a mode u 1 into a mode u 0 at the dark port of a cavity-enhanced MI with Schnupp asymmetry, symmetric arm cavities, and common and differential mirror tilts. The abscissa shows the audio frequency offset of the jitter sideband with respect to the carrier for tilted ITMs (left panel) and tilted ETMs (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10-8rad.

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Fig. 4.
Fig. 4. Transfer function of a mode u 1 into a mode u 0 at the dark port of a cavity-enhanced MI with Schnupp asymmetry, non-symmetric arm cavities, and common mirror tilts. The abscissa shows the audio frequency offset of the jitter sideband with respect to the carrier for tilted ITMs (left panel) and tilted ETMs (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10-8rad.

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Fig. 5.
Fig. 5. Transfer function of a mode u 1 into a mode u 0 at the dark port of a power-recycled, cavity-enhanced MI (LIGO-I configuration) with Schnupp asymmetry, non-symmetric arm cavities, and common and differential mirror tilts. The abscissa shows the audio frequency offset of the jitter sideband with respect to the carrier for tilted ITMs (left panel) and tilted ETMs (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10-8rad.

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Fig. 6.
Fig. 6. Transfer function of a mode u 1 into a mode u 0 at the dark port of a dual-recycled, cavity-enhanced MI (Advanced LIGO configuration) with Schnupp asymmetry, non-symmetric arm cavities, and common and differential mirror tilts. The abscissa shows the audio frequency offset of the jitter sideband with respect to the carrier for tilted ITMs (left panel) and tilted ETMs (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10-8rad.

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Fig. 7.
Fig. 7. Transfer function of a mode u 1 into a mode u 0 at the dark port of a dual-recycled, cavity-enhanced MI (Advanced LIGO configuration) with Schnupp asymmetry, non-symmetric arm cavities, nearly degenerated recycling cavities (z R = 189m) and common and differential mirror tilts. The abscissa shows the audio frequency offset of the jitter sideband with respect to the 180MHz sideband for tilted ITMs (left panel) and tilted ETMs (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10-8rad.

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Fig. 8.
Fig. 8. Transfer function of a mode u 1 into a mode u 0 at the dark port of a dualrecycled, cavity-enhanced MI (Advanced LIGO configuration) with Schnupp asymmetry, non-symmetric arm cavities, non-degenerated recycling cavities (z R = 10m) and common and differential mirror tilts. The abscissa shows the audio frequency offset of the jitter sideband with respect to the 180MHz sideband for tilted ITMs (left panel) and tilted ETMs (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10-8rad.

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Fig. 9.
Fig. 9. Transfer function of a mode u 1 into a mode u 0 at the dark port of a dual-recycled, cavity-enhanced MI (Advanced LIGO configuration) with Schnupp asymmetry, non-symmetric arm cavities, degenerated recycling cavities (z R = 189m) and tilted recycling mirrors. The abscissa shows the audio frequency offset of the jitter sideband with respect to the carrier (left panel) and the 180MHz sidebands (right panel) for tilted recycling mirrors. There is virtually no difference between a tilted PR mirror and a tilted SR mirror in the transfer functions in the right panel. Note that the transfer functions for the jitter sidebands around the RF sidebands would be about one order of magnitude smaller in non-degenerated recycling cavities. All tilt angles are Θ = 10-8rad.

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Fig. 10.
Fig. 10. The expected displacement sensitivity of the Advanced LIGO detector [11]. At low frequencies the detector will be limited by radiation pressure noise, one component of the unified quantum noise. In the medium frequency range internal thermal noise of the mirror substrates will limit our sensitivity. Finally, shot noise, the second component of the unified quantum noise, will limit the sensitivity at high frequencies. Contributions from technical noise sources like beam jitter should be one order of magnitude smaller than the contributions from these fundamental noise sources.

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Fig. 11.
Fig. 11. The amplitude of the signal sidebands at the expected Advanced LIGO sensitivity. The units of the sidebands are the natural units number of photons s .

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Fig. 12.
Fig. 12. The Advanced LIGO requirements for the relative amplitudes of the jitter sidebands a 10(±f) for tilted ITMs (left panel) and tilted Signal recycling mirror (right panel). The assumed tilt angles are 10-8rad.

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Tables (1)

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Table 1. Advanced LIGO parameters used in the calculation unless otherwise noted.

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Equations (31)

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x = x ̂ w ( z 0 ) ( 1 + i z 0 z R ) , α = α ̂ πw ( z 0 ) λ ,
z R = π w 0 2 λ
E in ( z 0 ) = E 0 exp ( i ω 0 t ) ( u ̂ 0 a 1 2 ( exp ( i Ω t ) + exp ( − i Ω t ) ) · u ̂ 1 )
M ̂ = ( 1 − 4 θ 2 − 2 iθ − 2 iθ 1 − 4 θ 2 )
with θ = πw ( z ) λ Θ .
E r = r M ̂ E in
M ̂ t = ( 1 − x 2 x − x 1 − x 2 )
x = D w n − 1 n Θ
P ̂ cav = r 1 r 2 M ̂ 1 L ̂ M ̂ 2 L ̂
L ̂ = ( exp ( i 2 πf L c ) 0 0 exp ( i 2 πf L c + i ϕ C ) )
E cav = it 1 E in · ∑ n = 0 ∞ P ̂ cav n
E cav = it 1 ( U ̂ − P ̂ cav ) − 1 E in
E ref = ( r 1 M ̂ 1 − 1 − t 1 2 r 2 ( U ̂ − P ̂ cav ) − 1 L ̂ M ̂ 2 L ̂ ) E in ≙ R ̂ C E in
f res − = 180 MHz − N c 2 L ≈ 12.1 kHz
E mi t = it bs r bs ( L ̂ 1 R ̂ c 1 L ̂ 1 + L ̂ 2 R ̂ c 2 L ̂ 2 ) E in ≙ T ̂ E in
L ̂ i = ( exp ( i 2 πf l i c ) 0 0 exp ( i 2 πf l i c + i ϕ C ) )
( r bs 2 L ̂ 1 R ̂ c 1 L ̂ 1 − t bs 2 L ̂ 2 R ̂ c 2 L ̂ 2 ) ≙ R ̂ b
E pr = it pr T L ̂ p ( U ̂ − P ̂ pr ) − 1 E in
P ̂ pr = r pr M ̂ pr L ̂ p R ̂ b L ̂ p
L ̂ p = ( exp ( i 2 πf l p c ) 0 0 exp ( i 2 πf l p c + i ϕ PR ) )
E mi sr = ( − t bs 2 L ̂ 1 R ̂ c 1 L ̂ 1 + r bs 2 L ̂ 2 R ̂ c 2 L ̂ 2 ) E in ≙ R ̂ d E in
E dp = − t sr L ̂ S N ̂ dp − 1 T ̂ t p E in
N ̂ dp = U ̂ − R ̂ d A ̂ S − T ̂ A ̂ p R ̂ b T ̂ − 1 + T ̂ A ̂ p ( R ̂ b T ̂ − 1 R ̂ d − T ̂ ) A ̂ S
A ̂ p ( s ) = L ̂ p ( s ) r p ( s ) r M ̂ p ( s ) r L ̂ p ( s )
E b = L ̂ p N ̂ bp − 1 [ ( U ̂ − T ̂ A ̂ S R ̂ d T ̂ − 1 ) R ̂ b + T ̂ ] L ̂ p it p E in
N ̂ bp = U ̂ − R ̂ b A ̂ p − T ̂ A ̂ S R ̂ d T ̂ − 1 + T ̂ A ̂ S ( R ̂ d T ̂ − 1 R ̂ b − T ̂ ) A ̂ p
a 00 sig ( f ) ≈ ( 80 Hz f ) 4 + 1
a 00 max ( f ) ≤ 1 20 ( 80 Hz f ) 4 + 1
a ˜ 10 max ( f ) = a 10 in a 00 in ≤ a 00 max ( f ) T 1 → 0 ( f ) hν P in maximum relative amplitude of the jitter sidebands
a ˜ 10 max ( f ) ≤ 7 · 10 − 10 Hz 1 + ( 230 Hz f ) 4 ( 10 − 8 rad ΔΘ ITM ) for tilted ITM mirrors
a ˜ 10 max ( f ) ≤ 6 · 10 − 9 Hz 1 + ( 230 Hz f ) 4 ( 10 − 8 rad ΔΘ SR ) for tilted SR mirror
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