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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  1. B. Abbott et al., ”Detector description and performance for the first coincidence observations between LIGO and GEO,” Nucl. Instr. and Meth. in Phys. Res. A 517 (2004) 154–179. See also: www.ligo.caltech.edu
    [CrossRef]
  2. Advanced LIGO is the first major upgrade of the current LIGO detectors. Informations about the planned upgrade are available at: http://www.ligo.caltech.edu/advLIGO/
  3. Nergis Mavalvala, David Shoemaker, and Daniel Sigg, ”Experimental Test of an Alignment-Sensing Scheme for a Gravitational-Wave Interferometer,” Applied Optics,  37 (1998) 7743
    [CrossRef]
  4. Peter Fritschel, Nergis Mavalvala, David Shoemaker, Daniels Sigg, Michael Zucker, and Gabriela Gonzales, ”Align-ment of an interferometric gravitational wave detector,” Appl. Opt.,  37 (1998) 6734
    [CrossRef]
  5. Kenneth Strain et al., ”Sensing and control in dual-recycled laser interferometer gravitational-wave detectors,” Appl. Opt.,  42 (2003) 1244
    [CrossRef] [PubMed]
  6. Guido Mueller et al., ”Dual-recycled cavity-enhanced Michelson interferometer for gravitational-wave detection,” Appl. Opt.,  42 (2003) 1257
    [CrossRef]
  7. A. Siegman, Lasers (University Science, Sausalito, Calif.1986)
  8. Y. Hefetz, N. Mavalvala, and D. Sigg, ”Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B14 (1997) 1597
  9. D. Sigg and N. Mavalvala, ”Principles of calculating the dynamical response of misaligned complex resonant optical interferometers,” J. Opt. Soc. Am. A17 (2000) 1642
    [CrossRef]
  10. N.A. Robertson et al., ”Seismic isolation and suspension systems for Advanced LIGO,” in Gravitational Wave and Particle Astrophysics Detectors, Proc. of SPIE, Vol.  5500, ed. James Hough and Gary Sanders (2004) 81–91
    [CrossRef]
  11. This sensitivity curve was calculated using Bench. Informations about Bench is available at: http://cosmos.nirvana.phys.psu.edu/˜lsf/Benchmarks/main.html. Details of the sensitivity curve depend on the fine-tuning of several parameters like tuning of the signal recycling mirror position, reflectivity of the mirrors, and the internal damping coefficients of the eigenmodes of the substrates and suspension systems. However, the sensitivity will not change by more than a factor of two at any signal frequency. The requirements for technical noise sources like beam jitter will not change significantly.
  12. The transfer function of the signal amplitude with respect to the displacement was calculated with Finesse. Finesse is an optical modeling program written by Andreas Freise. Finesse is available at http://www.rzg.mpg.de/˜adf/.
  13. S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

2004 (2)

B. Abbott et al., ”Detector description and performance for the first coincidence observations between LIGO and GEO,” Nucl. Instr. and Meth. in Phys. Res. A 517 (2004) 154–179. See also: www.ligo.caltech.edu
[CrossRef]

N.A. Robertson et al., ”Seismic isolation and suspension systems for Advanced LIGO,” in Gravitational Wave and Particle Astrophysics Detectors, Proc. of SPIE, Vol.  5500, ed. James Hough and Gary Sanders (2004) 81–91
[CrossRef]

2003 (2)

2000 (1)

D. Sigg and N. Mavalvala, ”Principles of calculating the dynamical response of misaligned complex resonant optical interferometers,” J. Opt. Soc. Am. A17 (2000) 1642
[CrossRef]

1998 (2)

Peter Fritschel, Nergis Mavalvala, David Shoemaker, Daniels Sigg, Michael Zucker, and Gabriela Gonzales, ”Align-ment of an interferometric gravitational wave detector,” Appl. Opt.,  37 (1998) 6734
[CrossRef]

Nergis Mavalvala, David Shoemaker, and Daniel Sigg, ”Experimental Test of an Alignment-Sensing Scheme for a Gravitational-Wave Interferometer,” Applied Optics,  37 (1998) 7743
[CrossRef]

1997 (1)

Y. Hefetz, N. Mavalvala, and D. Sigg, ”Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B14 (1997) 1597

Abbott, B.

B. Abbott et al., ”Detector description and performance for the first coincidence observations between LIGO and GEO,” Nucl. Instr. and Meth. in Phys. Res. A 517 (2004) 154–179. See also: www.ligo.caltech.edu
[CrossRef]

Adhikari, R.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Camp, J.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Delker, T.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Fritschel, P.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Fritschel, Peter

Gonzales, Gabriela

Heefner, J.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Hefetz, Y.

Y. Hefetz, N. Mavalvala, and D. Sigg, ”Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B14 (1997) 1597

Kells, B.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Mavalvala, N.

D. Sigg and N. Mavalvala, ”Principles of calculating the dynamical response of misaligned complex resonant optical interferometers,” J. Opt. Soc. Am. A17 (2000) 1642
[CrossRef]

Y. Hefetz, N. Mavalvala, and D. Sigg, ”Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B14 (1997) 1597

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Mavalvala, Nergis

Peter Fritschel, Nergis Mavalvala, David Shoemaker, Daniels Sigg, Michael Zucker, and Gabriela Gonzales, ”Align-ment of an interferometric gravitational wave detector,” Appl. Opt.,  37 (1998) 6734
[CrossRef]

Nergis Mavalvala, David Shoemaker, and Daniel Sigg, ”Experimental Test of an Alignment-Sensing Scheme for a Gravitational-Wave Interferometer,” Applied Optics,  37 (1998) 7743
[CrossRef]

Mueller, G.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Mueller, Guido

Ouimette, D.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Reitze, D.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Robertson, N.A.

N.A. Robertson et al., ”Seismic isolation and suspension systems for Advanced LIGO,” in Gravitational Wave and Particle Astrophysics Detectors, Proc. of SPIE, Vol.  5500, ed. James Hough and Gary Sanders (2004) 81–91
[CrossRef]

Rong, H.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Shoemaker, David

Nergis Mavalvala, David Shoemaker, and Daniel Sigg, ”Experimental Test of an Alignment-Sensing Scheme for a Gravitational-Wave Interferometer,” Applied Optics,  37 (1998) 7743
[CrossRef]

Peter Fritschel, Nergis Mavalvala, David Shoemaker, Daniels Sigg, Michael Zucker, and Gabriela Gonzales, ”Align-ment of an interferometric gravitational wave detector,” Appl. Opt.,  37 (1998) 6734
[CrossRef]

Shu, Q.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Siegman, A.

A. Siegman, Lasers (University Science, Sausalito, Calif.1986)

Sigg, D.

D. Sigg and N. Mavalvala, ”Principles of calculating the dynamical response of misaligned complex resonant optical interferometers,” J. Opt. Soc. Am. A17 (2000) 1642
[CrossRef]

Y. Hefetz, N. Mavalvala, and D. Sigg, ”Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B14 (1997) 1597

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Sigg, Daniel

Nergis Mavalvala, David Shoemaker, and Daniel Sigg, ”Experimental Test of an Alignment-Sensing Scheme for a Gravitational-Wave Interferometer,” Applied Optics,  37 (1998) 7743
[CrossRef]

Sigg, Daniels

Strain, Kenneth

Tanner, D.B.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Yoshida, S.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Zucker, M.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Zucker, Michael

Appl. Opt. (3)

Applied Optics (1)

Nergis Mavalvala, David Shoemaker, and Daniel Sigg, ”Experimental Test of an Alignment-Sensing Scheme for a Gravitational-Wave Interferometer,” Applied Optics,  37 (1998) 7743
[CrossRef]

J. Opt. Soc. Am. (2)

Y. Hefetz, N. Mavalvala, and D. Sigg, ”Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B14 (1997) 1597

D. Sigg and N. Mavalvala, ”Principles of calculating the dynamical response of misaligned complex resonant optical interferometers,” J. Opt. Soc. Am. A17 (2000) 1642
[CrossRef]

Nucl. Instr. and Meth. in Phys. Res. A (1)

B. Abbott et al., ”Detector description and performance for the first coincidence observations between LIGO and GEO,” Nucl. Instr. and Meth. in Phys. Res. A 517 (2004) 154–179. See also: www.ligo.caltech.edu
[CrossRef]

Proc. of SPIE (1)

N.A. Robertson et al., ”Seismic isolation and suspension systems for Advanced LIGO,” in Gravitational Wave and Particle Astrophysics Detectors, Proc. of SPIE, Vol.  5500, ed. James Hough and Gary Sanders (2004) 81–91
[CrossRef]

Other (5)

This sensitivity curve was calculated using Bench. Informations about Bench is available at: http://cosmos.nirvana.phys.psu.edu/˜lsf/Benchmarks/main.html. Details of the sensitivity curve depend on the fine-tuning of several parameters like tuning of the signal recycling mirror position, reflectivity of the mirrors, and the internal damping coefficients of the eigenmodes of the substrates and suspension systems. However, the sensitivity will not change by more than a factor of two at any signal frequency. The requirements for technical noise sources like beam jitter will not change significantly.

The transfer function of the signal amplitude with respect to the displacement was calculated with Finesse. Finesse is an optical modeling program written by Andreas Freise. Finesse is available at http://www.rzg.mpg.de/˜adf/.

S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

Advanced LIGO is the first major upgrade of the current LIGO detectors. Informations about the planned upgrade are available at: http://www.ligo.caltech.edu/advLIGO/

A. Siegman, Lasers (University Science, Sausalito, Calif.1986)

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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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 .

Fig. 12.
Fig. 12.

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

Tables (1)

Tables Icon

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

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 2 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 ) 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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