Abstract

We analyze the electromagnetic fields in a Pound–Drever–Hall locked, marginally unstable, Fabry–Perot cavity as a function of small changes in the cavity length during resonance. More specifically, we compare the results of a detailed numerical model with the behavior of the recycling cavity of the Laser Interferometer Gravitational-wave Observatory (LIGO) detector located in Livingston, Louisiana. In the interferometer’s normal mode of operation, the recycling cavity is stabilized by inducing a thermal lens in the cavity mirrors with an external CO2 laser. During our study, this thermal compensation system was not operating, causing the cavity to be marginally optically unstable and cavity modes to become degenerate. In contrast to stable optical cavities, the modal content of the resonating beam in the uncompensated recycling cavity is significantly altered by very small cavity length changes. This modifies the error signals used to control the cavity length in such a way that the zero crossing point is no longer the point of maximum power in the cavity, nor is it the point where the input-beam mode in the cavity is maximized.

© 2007 Optical Society of America

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  1. B. Abbott, R. Abbott, and R. Adhikari, 'Detector description and performance for the first coincidence observations between LIGO and GEO,' Nucl. Instrum. Methods A517, 154-179 (2004), http://www.ligo.caltech.edu/docs/P/P030024-01/P030024-01.pdf.
    [CrossRef]
  2. A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
    [CrossRef] [PubMed]
  3. P. Fritschel, R. Bork, G. Gonzlez, N. Mavalvala, D. Ouimette, H. Rong, D. Sigg, and M. Zucker, 'Readout and control of a power-recycled interferometric gravitational-wave antenna,' Appl. Opt. 40, 4988-4998 (2001), http://resolver.caltech.edu/CaltechAUTHORS:FRIao01.
    [CrossRef]
  4. R. Weiss, 'Gravitational radiation,' Rev. Mod. Phys. 71S187-S196 (1999).
    [CrossRef]
  5. J. Hough, S. Rowan, and B. S. Sathyaprakash, 'The search for gravitational waves,' arXiv:gr-qc/0501007; http://arxiv.org/abs/gr-qc/0501007.
  6. R. Beusoleil, E. D'Ambrosio, B. Kells, J. Camp, E. Gustafson, and M. Fejer, 'Model of thermal wave-front distortion in interferometric gravitational-wave detectors. I. Thermal focusing,' J. Opt. Soc. Am. B 20, 1247-1268 (2003), http://arxiv.org/abs/gr-qc/0205124.
    [CrossRef]
  7. B. Kells and J. Camp, 'Absorption in the core optics and LIGO sensitivity,' LIGO Document Control Center, T970097-01 (1997), http://www.ligo.caltech.edu/docs/T/T970097-01.pdf.
  8. S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, 'Thermal compensation system description,' LIGO Document Control Center, T050064-00 (2005), http://www.ligo.caltech.edu/docs/T/T050064-00.pdf.
  9. P. Fritschel, N. Mavalvala, D. Shoemaker, D. Sigg, M. Zucker, and G. Gonzalez, 'Alignment of an interferometric gravitational wave detector,' Appl. Opt. 37, 6734-6747 (1998), http://www.ligo.caltech.edu/docs/P/P970017-A.pdf.
    [CrossRef]
  10. B. Bhawal, 'The effect of thermal lensing on wave-front sensor signals,' LIGO Document Control Center, T040066-00 (2004), http://www.ligo.caltech.edu/docs/T/T040066-00.pdf.
  11. E. D'Ambrosio and W. Kells, 'Carrier mode selective working point and side band imbalance in LIGO I,'Phys. Rev. D 73, 122002-26 (2006), http://authors.library.caltech.edu/3586/01/DAMprd06.pdf.
    [CrossRef]
  12. B. Bochner, 'Modelling the performance of interferometric gravitational-wave detectors with realistically imperfect optics,' Ph.D. thesis (Massachusetts Institute of Technology, 1998), http://www.ligo.caltech.edu/docs/P/P980004-00.pdf.
  13. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, 'Laser phase and frequency stabilization using an optical resonator,' Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
    [CrossRef]
  14. LIGO Livingston Obseratory, http://www.ligo-la.caltech.edu. The measurements described were made during March 2004.
  15. E. D'Ambrosio, 'Study of the different responsive behaviour of the sidebands in LIGO I,' Class. Quantum Grav. 21, S1113-S1120 (2004).
    [CrossRef]
  16. A. M. Gretarsson and V. Frolov, 'Excerpt from the LLO electronic detector logs,' LIGO Document Control Center, T070074-00 (2004), http://www.ligo.caltech.edu/docs/T/T070074-00.pdf.
  17. K. Goda, D. Ottaway, B. Connelly, R. Adhikari, N. Mavalvala, and A. Gretarsson, 'Frequency-resolving spatiotemporal wave-front sensor,' Opt. Lett. 29, 1452-1454 (2004), http://www.ligo.caltech.edu/docs/P/P030069-00.pdf.
    [CrossRef] [PubMed]
  18. B. Kells, 'Distorted PRM SB fields in Eikonal approximation,' LIGO Document Control Center, T070074-00(2004), http://www.ligo.caltech.edu/docs/T/T040195-02.pdf, and references to the LIGO Hanford interferometer logs therein.
  19. D. Ottaway, LIGO Project MIT, Massachusetts Institute of Technology, NW22-295, 185 Albany Street, Cambridge, Mass. 02139, USA. (personal communication, 2005).
  20. M. Regehr, 'Signal extraction and control for an interferometric gravitational wave detector,' Ph.d. thesis (California Institute of Technology, 1994), http://www.ligo.caltech.edu/docs/P/P940002-00.pdf.

2006 (1)

E. D'Ambrosio and W. Kells, 'Carrier mode selective working point and side band imbalance in LIGO I,'Phys. Rev. D 73, 122002-26 (2006), http://authors.library.caltech.edu/3586/01/DAMprd06.pdf.
[CrossRef]

2004 (3)

B. Abbott, R. Abbott, and R. Adhikari, 'Detector description and performance for the first coincidence observations between LIGO and GEO,' Nucl. Instrum. Methods A517, 154-179 (2004), http://www.ligo.caltech.edu/docs/P/P030024-01/P030024-01.pdf.
[CrossRef]

E. D'Ambrosio, 'Study of the different responsive behaviour of the sidebands in LIGO I,' Class. Quantum Grav. 21, S1113-S1120 (2004).
[CrossRef]

K. Goda, D. Ottaway, B. Connelly, R. Adhikari, N. Mavalvala, and A. Gretarsson, 'Frequency-resolving spatiotemporal wave-front sensor,' Opt. Lett. 29, 1452-1454 (2004), http://www.ligo.caltech.edu/docs/P/P030069-00.pdf.
[CrossRef] [PubMed]

2003 (1)

2001 (1)

1999 (1)

R. Weiss, 'Gravitational radiation,' Rev. Mod. Phys. 71S187-S196 (1999).
[CrossRef]

1998 (1)

1992 (1)

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, 'Laser phase and frequency stabilization using an optical resonator,' Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Abbott, B.

B. Abbott, R. Abbott, and R. Adhikari, 'Detector description and performance for the first coincidence observations between LIGO and GEO,' Nucl. Instrum. Methods A517, 154-179 (2004), http://www.ligo.caltech.edu/docs/P/P030024-01/P030024-01.pdf.
[CrossRef]

Abbott, R.

B. Abbott, R. Abbott, and R. Adhikari, 'Detector description and performance for the first coincidence observations between LIGO and GEO,' Nucl. Instrum. Methods A517, 154-179 (2004), http://www.ligo.caltech.edu/docs/P/P030024-01/P030024-01.pdf.
[CrossRef]

Abramovici, A.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Adhikari, R.

K. Goda, D. Ottaway, B. Connelly, R. Adhikari, N. Mavalvala, and A. Gretarsson, 'Frequency-resolving spatiotemporal wave-front sensor,' Opt. Lett. 29, 1452-1454 (2004), http://www.ligo.caltech.edu/docs/P/P030069-00.pdf.
[CrossRef] [PubMed]

B. Abbott, R. Abbott, and R. Adhikari, 'Detector description and performance for the first coincidence observations between LIGO and GEO,' Nucl. Instrum. Methods A517, 154-179 (2004), http://www.ligo.caltech.edu/docs/P/P030024-01/P030024-01.pdf.
[CrossRef]

Althouse, W. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Ballmer, S.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, 'Thermal compensation system description,' LIGO Document Control Center, T050064-00 (2005), http://www.ligo.caltech.edu/docs/T/T050064-00.pdf.

Beusoleil, R.

Bhawal, B.

B. Bhawal, 'The effect of thermal lensing on wave-front sensor signals,' LIGO Document Control Center, T040066-00 (2004), http://www.ligo.caltech.edu/docs/T/T040066-00.pdf.

Bochner, B.

B. Bochner, 'Modelling the performance of interferometric gravitational-wave detectors with realistically imperfect optics,' Ph.D. thesis (Massachusetts Institute of Technology, 1998), http://www.ligo.caltech.edu/docs/P/P980004-00.pdf.

Bork, R.

Camp, J.

Connelly, B.

D'Ambrosio, E.

E. D'Ambrosio and W. Kells, 'Carrier mode selective working point and side band imbalance in LIGO I,'Phys. Rev. D 73, 122002-26 (2006), http://authors.library.caltech.edu/3586/01/DAMprd06.pdf.
[CrossRef]

E. D'Ambrosio, 'Study of the different responsive behaviour of the sidebands in LIGO I,' Class. Quantum Grav. 21, S1113-S1120 (2004).
[CrossRef]

R. Beusoleil, E. D'Ambrosio, B. Kells, J. Camp, E. Gustafson, and M. Fejer, 'Model of thermal wave-front distortion in interferometric gravitational-wave detectors. I. Thermal focusing,' J. Opt. Soc. Am. B 20, 1247-1268 (2003), http://arxiv.org/abs/gr-qc/0205124.
[CrossRef]

Drever, R. W. P.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, 'Laser phase and frequency stabilization using an optical resonator,' Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Fejer, M.

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, 'Laser phase and frequency stabilization using an optical resonator,' Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Fritschel, P.

Frolov, V.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, 'Thermal compensation system description,' LIGO Document Control Center, T050064-00 (2005), http://www.ligo.caltech.edu/docs/T/T050064-00.pdf.

A. M. Gretarsson and V. Frolov, 'Excerpt from the LLO electronic detector logs,' LIGO Document Control Center, T070074-00 (2004), http://www.ligo.caltech.edu/docs/T/T070074-00.pdf.

Goda, K.

Gonzalez, G.

Gonzlez, G.

Gretarsson, A.

Gretarsson, A. M.

A. M. Gretarsson and V. Frolov, 'Excerpt from the LLO electronic detector logs,' LIGO Document Control Center, T070074-00 (2004), http://www.ligo.caltech.edu/docs/T/T070074-00.pdf.

Gursel, Y.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Gustafson, E.

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, 'Laser phase and frequency stabilization using an optical resonator,' Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, 'Laser phase and frequency stabilization using an optical resonator,' Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

J. Hough, S. Rowan, and B. S. Sathyaprakash, 'The search for gravitational waves,' arXiv:gr-qc/0501007; http://arxiv.org/abs/gr-qc/0501007.

Kawamura, S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Kells, B.

R. Beusoleil, E. D'Ambrosio, B. Kells, J. Camp, E. Gustafson, and M. Fejer, 'Model of thermal wave-front distortion in interferometric gravitational-wave detectors. I. Thermal focusing,' J. Opt. Soc. Am. B 20, 1247-1268 (2003), http://arxiv.org/abs/gr-qc/0205124.
[CrossRef]

B. Kells, 'Distorted PRM SB fields in Eikonal approximation,' LIGO Document Control Center, T070074-00(2004), http://www.ligo.caltech.edu/docs/T/T040195-02.pdf, and references to the LIGO Hanford interferometer logs therein.

B. Kells and J. Camp, 'Absorption in the core optics and LIGO sensitivity,' LIGO Document Control Center, T970097-01 (1997), http://www.ligo.caltech.edu/docs/T/T970097-01.pdf.

Kells, W.

E. D'Ambrosio and W. Kells, 'Carrier mode selective working point and side band imbalance in LIGO I,'Phys. Rev. D 73, 122002-26 (2006), http://authors.library.caltech.edu/3586/01/DAMprd06.pdf.
[CrossRef]

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, 'Thermal compensation system description,' LIGO Document Control Center, T050064-00 (2005), http://www.ligo.caltech.edu/docs/T/T050064-00.pdf.

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, 'Laser phase and frequency stabilization using an optical resonator,' Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Lawrence, R.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, 'Thermal compensation system description,' LIGO Document Control Center, T050064-00 (2005), http://www.ligo.caltech.edu/docs/T/T050064-00.pdf.

Mason, K.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, 'Thermal compensation system description,' LIGO Document Control Center, T050064-00 (2005), http://www.ligo.caltech.edu/docs/T/T050064-00.pdf.

Mavalvala, N.

Moreno, G.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, 'Thermal compensation system description,' LIGO Document Control Center, T050064-00 (2005), http://www.ligo.caltech.edu/docs/T/T050064-00.pdf.

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, 'Laser phase and frequency stabilization using an optical resonator,' Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Ottaway, D.

K. Goda, D. Ottaway, B. Connelly, R. Adhikari, N. Mavalvala, and A. Gretarsson, 'Frequency-resolving spatiotemporal wave-front sensor,' Opt. Lett. 29, 1452-1454 (2004), http://www.ligo.caltech.edu/docs/P/P030069-00.pdf.
[CrossRef] [PubMed]

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, 'Thermal compensation system description,' LIGO Document Control Center, T050064-00 (2005), http://www.ligo.caltech.edu/docs/T/T050064-00.pdf.

D. Ottaway, LIGO Project MIT, Massachusetts Institute of Technology, NW22-295, 185 Albany Street, Cambridge, Mass. 02139, USA. (personal communication, 2005).

Ouimette, D.

Raab, F. J.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Regehr, M.

M. Regehr, 'Signal extraction and control for an interferometric gravitational wave detector,' Ph.d. thesis (California Institute of Technology, 1994), http://www.ligo.caltech.edu/docs/P/P940002-00.pdf.

Rong, H.

Rowan, S.

J. Hough, S. Rowan, and B. S. Sathyaprakash, 'The search for gravitational waves,' arXiv:gr-qc/0501007; http://arxiv.org/abs/gr-qc/0501007.

Sathyaprakash, B. S.

J. Hough, S. Rowan, and B. S. Sathyaprakash, 'The search for gravitational waves,' arXiv:gr-qc/0501007; http://arxiv.org/abs/gr-qc/0501007.

Shoemaker, D.

P. Fritschel, N. Mavalvala, D. Shoemaker, D. Sigg, M. Zucker, and G. Gonzalez, 'Alignment of an interferometric gravitational wave detector,' Appl. Opt. 37, 6734-6747 (1998), http://www.ligo.caltech.edu/docs/P/P970017-A.pdf.
[CrossRef]

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Sievers, L.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Sigg, D.

Smith, M.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, 'Thermal compensation system description,' LIGO Document Control Center, T050064-00 (2005), http://www.ligo.caltech.edu/docs/T/T050064-00.pdf.

Spero, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Thorne, K. S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Vogt, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Vorvick, C.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, 'Thermal compensation system description,' LIGO Document Control Center, T050064-00 (2005), http://www.ligo.caltech.edu/docs/T/T050064-00.pdf.

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, 'Laser phase and frequency stabilization using an optical resonator,' Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Weiss, R.

R. Weiss, 'Gravitational radiation,' Rev. Mod. Phys. 71S187-S196 (1999).
[CrossRef]

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Whitcomb, S. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Willems, P.

S. Ballmer, V. Frolov, R. Lawrence, W. Kells, G. Moreno, K. Mason, D. Ottaway, M. Smith, C. Vorvick, P. Willems, and M. Zucker, 'Thermal compensation system description,' LIGO Document Control Center, T050064-00 (2005), http://www.ligo.caltech.edu/docs/T/T050064-00.pdf.

Zucker, M.

Zucker, M. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb and M. E. Zucker, 'LIGO: The Laser Interferometer Gravitational-wave Observatory,' Science 256, 325-333 (1992).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. B: Photophys. Laser Chem. (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, 'Laser phase and frequency stabilization using an optical resonator,' Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Arrangement of the core optics of the LIGO interferometers (not to scale). The recycling cavity is the cavity formed by the recycling mirror (RM), beamsplitter (BS), and the two input mirrors to the arm cavities (ITMx and ITMy). The recycling cavity length is the mean of the two optical paths between the RM and ITMx and ITMy. Note that since the interferometer is operated on a dark fringe, almost no carrier light escapes out the antisymmetric port of the beamsplitter to the photodiode at right.

Fig. 2
Fig. 2

The laser field Ψ LASER is phase modulated to produce Ψ IN consisting (approximately) of carrier light and upper and lower phase modulation sidebands, 24.5 MHz on each side of the carrier. The modulation index is Γ 0.34 . The reflected field Ψ REFL is the sum of three components: the sidebands (which are almost completely reflected back towards the laser since the cavity length makes them non-resonant), the prompt reflection of the carrier that is not interacting with the recycling cavity, and the leakage through the recycling mirror of the carrier field resonating inside the cavity. The cavity beam is split by the beam splitter into two optical paths of different lengths, represented in the diagram by L 1 and L 2 . The amplitude transmission and reflection coefficients are represented by t i and r i respectively. The position of the beam splitter is actively controlled, so that its antisymmetric port corresponds to the dark fringe of the carrier. The various instruments are schematically arranged next to the beams they interrogate. Thus, one CCD camera interrogates the beam at the reflected port (REFL). Another CCD camera interrogates the beam from the y-leg of the recycling cavity picked off via the wedged side of ITMy. (This port is known as POY.) The RF phase camera interrogates this same beam also. The DC photodiode interrogates the beam from the x-leg of the recycling cavity picked off via the wedged side of the beamsplitter. (This port is known as POB). Not shown is an RF photodiode registering the NSPOB signal (geometric mean of the upper and lower sideband powers) and located at the same port. Like the phase camera, this photodiode was only used during the full interferometer lock discussed in Subsection 4B.

Fig. 3
Fig. 3

Effect of changing the cavity length by several nanometers on the shape of the carrier resonating in the recycling cavity. (Arms are not resonant.) Comparison of actual image captures with simulation results. The center row shows false color photographs of the recycling cavity beam picked off at ITMy for different cavity length offsets from the nominal lock point. From left to right, offsets are: 8 nm , 4 nm , 0 nm , and + 4 nm . The colors represent intensity and correspond to the linear scale shown at right (arbitrary units). The top and bottom rows show the FFT model results for the same cavity length offsets. The top row shows the simulation results for the beam intensity. Approximately the same color scale is used for the simulated intensity results as for the false color photographs of the center row so that the images can be directly compared. The bottom row shows the cross-sectional intensity from the simulation plotted as a function of radius. The units of distance represented by the axes are left arbitrary because the actual physical dimensions of images rendered by the camera were not recorded. (In other words, a camera calibration was not available.) However, the relative sizes of the images in each individual row are accurate. A uniform zoom factor was applied to all the images in the top and bottom rows so that the size of the beam at 0 nm length offset (second from right) approximately matched the beam diameter in the corresponding image of the center row.

Fig. 4
Fig. 4

Power in the recycling cavity as a function of cavity length offset. The FFT model correctly predicts the length offset at which the cavity power is maximized. The circles show the total cavity power measured at ten different length offsets. The dark solid curve shows the FFT model prediction for the total cavity power. The light solid curve shows the FFT model prediction for the power in the mode of the input beam to the cavity. The scale on the right refers to the light dashed curve, representing the FFT model phase ϕ = arg ( Ψ LASER Ψ R E F L C R ) in the notation of Eq. (5). ϕ is zero at the locking point.

Fig. 5
Fig. 5

Effect of changing the cavity length by several nanometers on the shape of the sidebands resonating in the recycling cavity. The colors represent the geometric mean of the intensity of the upper and lower sidebands. Blue corresponds to the regions of greatest intensity with orange/red corresponding to the regions of lowest intensity. The second image from right represents the natural lock point (zero applied offset). Note that in these images, the center of the beam is in the upper half of the image. The asymmetric structure of the beam in some of the images, particularly the two at right, is due to pitch and yaw motion of the optics to which the instantaneous field distribution is very sensitive.

Fig. 6
Fig. 6

Sideband intensity as a function of uncalibrated length offset. The x axis shows the offset added to the error point in uncalibrated counts. The y axis is the value of the interferometer NSPOB signal, which measures the geometric mean of the upper and lower sideband powers. This graph contains data from two lock stretches distinguished by the two different shades.

Equations (7)

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Ψ I N = Ψ LASER exp [ i Γ cos ω t ] J 0 ( Γ ) Ψ LASER + i J 1 ( Γ ) Ψ LASER exp [ i ω t ] + i J 1 ( Γ ) Ψ LASER exp [ i ω t ] + Ψ I N C R + Ψ IN S B + exp [ i ω t ] + Ψ IN S B exp [ i ω t ] + ,
Ψ REFL S B Ψ IN S B ,
Ψ REFL S B + Ψ IN S B + ,
Ψ R E F L C R = D Ψ I N C R ,
P REFL = S Ψ REFL 2 d S = S { Ψ REFL C R 2 + Ψ REFL S B + 2 + Ψ REFL S B 2 + 2 R [ ( Ψ REFL C R Ψ REFL S B * + Ψ REFL S B + Ψ REFL C R * ) exp ( i ω t ) ] + 2 R [ Ψ REFL S B + Ψ REFL S B * exp ( 2 i ω t ) ] + } d S .
V I α S d S 0 T d t P REFL cos ( ω t ) T = α S R ( Ψ REFL C R Ψ REFL S B * + Ψ REFL S B + Ψ REFL C R * ) d S ,
V I R [ Ψ REFL C R Ψ REFL S B + Ψ REFL S B + Ψ REFL C R ] = 2 J 1 ( Γ ) I Ψ LASER Ψ R E F L C R ,

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