Abstract

We observe power coupling from the fundamental mode to frequency-degenerate higher-order spatial modes in optical resonators illuminated with a 30W laser. Thermally-induced modal frequency degeneracy facilitates power transfer from the fundamental mode to higher-order modes, reduces power coupling into the cavity, and triggers power fluctuations. Modeling thermoelastic deformation of a mirror’s surface shows predicted modal frequency degeneracy to be in reasonable agreement with experimental observations. Predictions for the Laser Interferometer Gravitational-wave Observatory (LIGO) show that the circulating fundamental-mode power necessary for gravitational-wave detection is compromised at coating absorptions of 3.8 and 0.44ppm for Enhanced and Advanced LIGO Fabry–Pérot cavities, respectively.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Kimura and K. Otsuka, “Thermal effects of a continuously pumped Nd3+:YAG laser,” IEEE J. Quantum Electron. 7, 403-407 (1971).
    [CrossRef]
  2. T. Kimura, K. Otsuka, and M. Saruwatari, “Spatial hole-burning effects in a Nd3+:YAG laser,” IEEE J. Quantum Electron. 7, 225-230 (1971).
    [CrossRef]
  3. T. Klaassen, J. de Jong, M. van Exter, and J. P. Woerdman, “Transverse mode coupling in an optical resonator,” Opt. Lett. 30, 1959-1961 (2005).
    [CrossRef] [PubMed]
  4. P. Fritschel, “The second generation LIGO interferometers,” in Proceedings of Astrophysical Sources for Ground-Based Gravitational Wave Detectors, J. M. Centrella, ed. (American Institute of Physics, 2001), pp. 15-23.
  5. R. Adhikari, P. Fritschel, and S. Waldman, “Enhanced LIGO,” http://www.ligo.caltech.edu/docs/T/T060156-01.pdf.
  6. E. D'Ambrosio, LIGO Laboratory, California Institute of Technology, MS 18-34, Pasadena, CA 91125, USA, and A. M. Gretarsson, V. Frolov, B. O'Reilly, and P. K. Fritschel, are preparing a manuscript to be called “Effects of mode degeneracy in the LIGO Livingston Observatory recycling cavity.”
  7. V. Loriette and C. Boccara, “Absorption of low-loss optical materials measured at 1064 nm by a position-modulated collinear photothermal detection technique,” Appl. Opt. 42, 649-656(2003).
    [CrossRef] [PubMed]
  8. C. Janke, “Thermal loading of optical components in interferometric systems,” presented at the LIGO Scientific Collaboration Conference, Baton Rouge, Louisiana, (March 2001).
  9. A. E. Siegman, Lasers (University Science, 1986). Errata URL: http://www.stanford.edu/siegman/lasers_book_errata.pdf.
  10. 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, Laser Opt. 31, 97-105 (1983).
    [CrossRef]
  11. W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022-7036 (1991).
    [CrossRef] [PubMed]
  12. N. Uehara and K. Ueda, “Accurate measurement of the radius of curvature of a concave mirror and the power dependence in a high-finesse Fabry-Pérot interferometer,” Appl. Opt. 34, 5611-5619 (1995).
    [CrossRef] [PubMed]
  13. R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions,” Opt. Express 14, 6069-6074(2006).
    [CrossRef] [PubMed]
  14. P. T. Beyersdorf, S. Zappe, M. M. Fejer, and M. Burkhardt, “Cavity with a deformable mirror for tailoring the shape of the eigenmode,” Appl. Opt. 45, 6723-6728 (2006).
    [CrossRef] [PubMed]
  15. H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Proceedings of the Symposium on Quasi-Optics, (Polytechnic Press, 1964), pp. 333-347.
  16. S. Saraf, R. L. Byer, and P. J. King, “High-extinction-ratio resonant cavity polarizer for quantum-optics measurements,” Appl. Opt. 46, 3850-3855 (2007).
    [CrossRef] [PubMed]
  17. N. Uehara, E. K. Gustafson, M. M. Fejer, and R. L. Byer, “Modeling of efficient mode matching and thermal-lensing effect on a laser-beam coupling into a mode-cleaner cavity,” Proc. SPIE 2989, 57-68 (1997).
    [CrossRef]
  18. K. An, B. A. Sones, C. Fang-Yen, R. R. Dasari, and M. S. Feld, “Optical bistability induced by mirror absorption: measurement of absorption coefficients at the sub-ppm level,” Opt. Lett. 22, 1433-1435 (1997).
    [CrossRef]
  19. D. Tanner, Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida, 32611 (personal communication, 2007).
  20. R. Adhikari, Department of Physics, California Institute of Technology, Physics Department 103-33, Pasadena, California, 91125 (personal communication, 2007).
  21. P. Fritschel, “Advanced LIGO interferometer parameters,” http://emvogil3.mit.edu/~pf/advligo/SYS/ALparameters.htm.
  22. K. X. Sun and R. L. Byer, “All-reflective Michelson, Sagnac, and Fabry-Pérot interferometers based on grating beam splitters,” Opt. Lett. 23, 567-569 (1998).
    [CrossRef]
  23. P. T. Beyersdorf, R. L. Byer, and M. M. Fejer, “Results from the Stanford 10 m Sagnac interferometer,” Class. Quantum Grav. 19, 1585-1589 (2002).
    [CrossRef]
  24. S. Rowan, J. Hough, and D. R. M. Crooks, “Thermal noise and material issues for gravitational wave detectors,” Phys. Lett. A 347, 25-32 (2005).
    [CrossRef]

2007 (1)

2006 (2)

2005 (2)

T. Klaassen, J. de Jong, M. van Exter, and J. P. Woerdman, “Transverse mode coupling in an optical resonator,” Opt. Lett. 30, 1959-1961 (2005).
[CrossRef] [PubMed]

S. Rowan, J. Hough, and D. R. M. Crooks, “Thermal noise and material issues for gravitational wave detectors,” Phys. Lett. A 347, 25-32 (2005).
[CrossRef]

2003 (1)

2002 (1)

P. T. Beyersdorf, R. L. Byer, and M. M. Fejer, “Results from the Stanford 10 m Sagnac interferometer,” Class. Quantum Grav. 19, 1585-1589 (2002).
[CrossRef]

1998 (1)

1997 (2)

N. Uehara, E. K. Gustafson, M. M. Fejer, and R. L. Byer, “Modeling of efficient mode matching and thermal-lensing effect on a laser-beam coupling into a mode-cleaner cavity,” Proc. SPIE 2989, 57-68 (1997).
[CrossRef]

K. An, B. A. Sones, C. Fang-Yen, R. R. Dasari, and M. S. Feld, “Optical bistability induced by mirror absorption: measurement of absorption coefficients at the sub-ppm level,” Opt. Lett. 22, 1433-1435 (1997).
[CrossRef]

1995 (1)

1991 (1)

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022-7036 (1991).
[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, Laser Opt. 31, 97-105 (1983).
[CrossRef]

1971 (2)

T. Kimura and K. Otsuka, “Thermal effects of a continuously pumped Nd3+:YAG laser,” IEEE J. Quantum Electron. 7, 403-407 (1971).
[CrossRef]

T. Kimura, K. Otsuka, and M. Saruwatari, “Spatial hole-burning effects in a Nd3+:YAG laser,” IEEE J. Quantum Electron. 7, 225-230 (1971).
[CrossRef]

Adhikari, R.

R. Adhikari, P. Fritschel, and S. Waldman, “Enhanced LIGO,” http://www.ligo.caltech.edu/docs/T/T060156-01.pdf.

R. Adhikari, Department of Physics, California Institute of Technology, Physics Department 103-33, Pasadena, California, 91125 (personal communication, 2007).

An, K.

Beyersdorf, P. T.

P. T. Beyersdorf, S. Zappe, M. M. Fejer, and M. Burkhardt, “Cavity with a deformable mirror for tailoring the shape of the eigenmode,” Appl. Opt. 45, 6723-6728 (2006).
[CrossRef] [PubMed]

P. T. Beyersdorf, R. L. Byer, and M. M. Fejer, “Results from the Stanford 10 m Sagnac interferometer,” Class. Quantum Grav. 19, 1585-1589 (2002).
[CrossRef]

Boccara, C.

Burkhardt, M.

Byer, R. L.

S. Saraf, R. L. Byer, and P. J. King, “High-extinction-ratio resonant cavity polarizer for quantum-optics measurements,” Appl. Opt. 46, 3850-3855 (2007).
[CrossRef] [PubMed]

P. T. Beyersdorf, R. L. Byer, and M. M. Fejer, “Results from the Stanford 10 m Sagnac interferometer,” Class. Quantum Grav. 19, 1585-1589 (2002).
[CrossRef]

K. X. Sun and R. L. Byer, “All-reflective Michelson, Sagnac, and Fabry-Pérot interferometers based on grating beam splitters,” Opt. Lett. 23, 567-569 (1998).
[CrossRef]

N. Uehara, E. K. Gustafson, M. M. Fejer, and R. L. Byer, “Modeling of efficient mode matching and thermal-lensing effect on a laser-beam coupling into a mode-cleaner cavity,” Proc. SPIE 2989, 57-68 (1997).
[CrossRef]

Crooks, D. R. M.

S. Rowan, J. Hough, and D. R. M. Crooks, “Thermal noise and material issues for gravitational wave detectors,” Phys. Lett. A 347, 25-32 (2005).
[CrossRef]

D'Ambrosio, E.

E. D'Ambrosio, LIGO Laboratory, California Institute of Technology, MS 18-34, Pasadena, CA 91125, USA, and A. M. Gretarsson, V. Frolov, B. O'Reilly, and P. K. Fritschel, are preparing a manuscript to be called “Effects of mode degeneracy in the LIGO Livingston Observatory recycling cavity.”

Danzmann, K.

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022-7036 (1991).
[CrossRef] [PubMed]

Dasari, R. R.

de Jong, J.

Drever, R. W. P.

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, Laser Opt. 31, 97-105 (1983).
[CrossRef]

Fang-Yen, C.

Fejer, M. M.

P. T. Beyersdorf, S. Zappe, M. M. Fejer, and M. Burkhardt, “Cavity with a deformable mirror for tailoring the shape of the eigenmode,” Appl. Opt. 45, 6723-6728 (2006).
[CrossRef] [PubMed]

P. T. Beyersdorf, R. L. Byer, and M. M. Fejer, “Results from the Stanford 10 m Sagnac interferometer,” Class. Quantum Grav. 19, 1585-1589 (2002).
[CrossRef]

N. Uehara, E. K. Gustafson, M. M. Fejer, and R. L. Byer, “Modeling of efficient mode matching and thermal-lensing effect on a laser-beam coupling into a mode-cleaner cavity,” Proc. SPIE 2989, 57-68 (1997).
[CrossRef]

Feld, M. S.

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, Laser Opt. 31, 97-105 (1983).
[CrossRef]

Fritschel, P.

P. Fritschel, “The second generation LIGO interferometers,” in Proceedings of Astrophysical Sources for Ground-Based Gravitational Wave Detectors, J. M. Centrella, ed. (American Institute of Physics, 2001), pp. 15-23.

R. Adhikari, P. Fritschel, and S. Waldman, “Enhanced LIGO,” http://www.ligo.caltech.edu/docs/T/T060156-01.pdf.

P. Fritschel, “Advanced LIGO interferometer parameters,” http://emvogil3.mit.edu/~pf/advligo/SYS/ALparameters.htm.

Fritschel, P. K.

E. D'Ambrosio, LIGO Laboratory, California Institute of Technology, MS 18-34, Pasadena, CA 91125, USA, and A. M. Gretarsson, V. Frolov, B. O'Reilly, and P. K. Fritschel, are preparing a manuscript to be called “Effects of mode degeneracy in the LIGO Livingston Observatory recycling cavity.”

Frolov, V.

E. D'Ambrosio, LIGO Laboratory, California Institute of Technology, MS 18-34, Pasadena, CA 91125, USA, and A. M. Gretarsson, V. Frolov, B. O'Reilly, and P. K. Fritschel, are preparing a manuscript to be called “Effects of mode degeneracy in the LIGO Livingston Observatory recycling cavity.”

Gretarsson, A. M.

E. D'Ambrosio, LIGO Laboratory, California Institute of Technology, MS 18-34, Pasadena, CA 91125, USA, and A. M. Gretarsson, V. Frolov, B. O'Reilly, and P. K. Fritschel, are preparing a manuscript to be called “Effects of mode degeneracy in the LIGO Livingston Observatory recycling cavity.”

Gustafson, E. K.

N. Uehara, E. K. Gustafson, M. M. Fejer, and R. L. Byer, “Modeling of efficient mode matching and thermal-lensing effect on a laser-beam coupling into a mode-cleaner cavity,” Proc. SPIE 2989, 57-68 (1997).
[CrossRef]

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, Laser Opt. 31, 97-105 (1983).
[CrossRef]

Hough, J.

S. Rowan, J. Hough, and D. R. M. Crooks, “Thermal noise and material issues for gravitational wave detectors,” Phys. Lett. A 347, 25-32 (2005).
[CrossRef]

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, Laser Opt. 31, 97-105 (1983).
[CrossRef]

Janke, C.

C. Janke, “Thermal loading of optical components in interferometric systems,” presented at the LIGO Scientific Collaboration Conference, Baton Rouge, Louisiana, (March 2001).

Kimura, T.

T. Kimura and K. Otsuka, “Thermal effects of a continuously pumped Nd3+:YAG laser,” IEEE J. Quantum Electron. 7, 403-407 (1971).
[CrossRef]

T. Kimura, K. Otsuka, and M. Saruwatari, “Spatial hole-burning effects in a Nd3+:YAG laser,” IEEE J. Quantum Electron. 7, 225-230 (1971).
[CrossRef]

King, P. J.

Klaassen, T.

Kogelnik, H.

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Proceedings of the Symposium on Quasi-Optics, (Polytechnic Press, 1964), pp. 333-347.

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, Laser Opt. 31, 97-105 (1983).
[CrossRef]

Loriette, V.

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, Laser Opt. 31, 97-105 (1983).
[CrossRef]

O'Reilly, B.

E. D'Ambrosio, LIGO Laboratory, California Institute of Technology, MS 18-34, Pasadena, CA 91125, USA, and A. M. Gretarsson, V. Frolov, B. O'Reilly, and P. K. Fritschel, are preparing a manuscript to be called “Effects of mode degeneracy in the LIGO Livingston Observatory recycling cavity.”

Otsuka, K.

T. Kimura, K. Otsuka, and M. Saruwatari, “Spatial hole-burning effects in a Nd3+:YAG laser,” IEEE J. Quantum Electron. 7, 225-230 (1971).
[CrossRef]

T. Kimura and K. Otsuka, “Thermal effects of a continuously pumped Nd3+:YAG laser,” IEEE J. Quantum Electron. 7, 403-407 (1971).
[CrossRef]

Paschotta, R.

Rowan, S.

S. Rowan, J. Hough, and D. R. M. Crooks, “Thermal noise and material issues for gravitational wave detectors,” Phys. Lett. A 347, 25-32 (2005).
[CrossRef]

Rudiger, A.

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022-7036 (1991).
[CrossRef] [PubMed]

Saraf, S.

Saruwatari, M.

T. Kimura, K. Otsuka, and M. Saruwatari, “Spatial hole-burning effects in a Nd3+:YAG laser,” IEEE J. Quantum Electron. 7, 225-230 (1971).
[CrossRef]

Schilling, R.

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022-7036 (1991).
[CrossRef] [PubMed]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986). Errata URL: http://www.stanford.edu/siegman/lasers_book_errata.pdf.

Sones, B. A.

Sun, K. X.

Tanner, D.

D. Tanner, Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida, 32611 (personal communication, 2007).

Ueda, K.

Uehara, N.

N. Uehara, E. K. Gustafson, M. M. Fejer, and R. L. Byer, “Modeling of efficient mode matching and thermal-lensing effect on a laser-beam coupling into a mode-cleaner cavity,” Proc. SPIE 2989, 57-68 (1997).
[CrossRef]

N. Uehara and K. Ueda, “Accurate measurement of the radius of curvature of a concave mirror and the power dependence in a high-finesse Fabry-Pérot interferometer,” Appl. Opt. 34, 5611-5619 (1995).
[CrossRef] [PubMed]

van Exter, M.

Waldman, S.

R. Adhikari, P. Fritschel, and S. Waldman, “Enhanced LIGO,” http://www.ligo.caltech.edu/docs/T/T060156-01.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, Laser Opt. 31, 97-105 (1983).
[CrossRef]

Winkler, W.

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022-7036 (1991).
[CrossRef] [PubMed]

Woerdman, J. P.

Zappe, S.

Appl. Opt. (4)

Appl. Phys. B, Laser Opt. (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, Laser Opt. 31, 97-105 (1983).
[CrossRef]

Class. Quantum Grav. (1)

P. T. Beyersdorf, R. L. Byer, and M. M. Fejer, “Results from the Stanford 10 m Sagnac interferometer,” Class. Quantum Grav. 19, 1585-1589 (2002).
[CrossRef]

IEEE J. Quantum Electron. (2)

T. Kimura and K. Otsuka, “Thermal effects of a continuously pumped Nd3+:YAG laser,” IEEE J. Quantum Electron. 7, 403-407 (1971).
[CrossRef]

T. Kimura, K. Otsuka, and M. Saruwatari, “Spatial hole-burning effects in a Nd3+:YAG laser,” IEEE J. Quantum Electron. 7, 225-230 (1971).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Lett. A (1)

S. Rowan, J. Hough, and D. R. M. Crooks, “Thermal noise and material issues for gravitational wave detectors,” Phys. Lett. A 347, 25-32 (2005).
[CrossRef]

Phys. Rev. A (1)

W. Winkler, K. Danzmann, A. Rudiger, and R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022-7036 (1991).
[CrossRef] [PubMed]

Proc. SPIE (1)

N. Uehara, E. K. Gustafson, M. M. Fejer, and R. L. Byer, “Modeling of efficient mode matching and thermal-lensing effect on a laser-beam coupling into a mode-cleaner cavity,” Proc. SPIE 2989, 57-68 (1997).
[CrossRef]

Other (9)

C. Janke, “Thermal loading of optical components in interferometric systems,” presented at the LIGO Scientific Collaboration Conference, Baton Rouge, Louisiana, (March 2001).

A. E. Siegman, Lasers (University Science, 1986). Errata URL: http://www.stanford.edu/siegman/lasers_book_errata.pdf.

P. Fritschel, “The second generation LIGO interferometers,” in Proceedings of Astrophysical Sources for Ground-Based Gravitational Wave Detectors, J. M. Centrella, ed. (American Institute of Physics, 2001), pp. 15-23.

R. Adhikari, P. Fritschel, and S. Waldman, “Enhanced LIGO,” http://www.ligo.caltech.edu/docs/T/T060156-01.pdf.

E. D'Ambrosio, LIGO Laboratory, California Institute of Technology, MS 18-34, Pasadena, CA 91125, USA, and A. M. Gretarsson, V. Frolov, B. O'Reilly, and P. K. Fritschel, are preparing a manuscript to be called “Effects of mode degeneracy in the LIGO Livingston Observatory recycling cavity.”

D. Tanner, Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida, 32611 (personal communication, 2007).

R. Adhikari, Department of Physics, California Institute of Technology, Physics Department 103-33, Pasadena, California, 91125 (personal communication, 2007).

P. Fritschel, “Advanced LIGO interferometer parameters,” http://emvogil3.mit.edu/~pf/advligo/SYS/ALparameters.htm.

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Proceedings of the Symposium on Quasi-Optics, (Polytechnic Press, 1964), pp. 333-347.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Schematic of a ring resonator (mode cleaner). M 1 and M 2 are 1 in . in diameter, and M 3 , attached to a piezo-electric transducer (PZT), is 0.5 in. in diameter with a 1 m radius of curvature. Round-trip perimeter, p m , is 42 cm .

Fig. 2
Fig. 2

Normalized modal resonance frequency, ω ¯ σ q , versus g 1 g 2 for the qth axial TEM 11 , 0 and ( q th + 2 ) TEM 00 modes of a modecleaner. A vertical line at g = 0.79 indicates a modecleaner’s undistorted (cold) g 1 g 2 . The resonance frequency of TEM 11 , 0 changes significantly compared to TEM 00 , overlapping in resonance with the fundamental mode for g 1 g 2 = 0.8274 . Since σ is odd for TEM 11 , 0 , where σ = m + n for a TEM m n mode, this mode’s resonance frequency is shifted by half the cavity’s axial mode spacing.

Fig. 3
Fig. 3

Experimental layout for testing a ring resonator (modecleaner) under thermal load. The filter modecleaner with a finesse of 50 filters the incident 30 W Nd:YAG master oscillator power amplifier (MOPA) laser beam to provide a spatially filtered, fundamental-mode input beam to illuminate the modecleaner under test. Output from the modecleaner under test is incident on the mode analyzer and data acquisition system for modal analysis and transmission monitoring. All cavities are locked to resonance using the Pound–Drever–Hall (PDH) technique.

Fig. 4
Fig. 4

Fraction of transmitted fundamental-mode power relative to total transmitted power of the absorbing modecleaner locked to resonance in p polarization versus absorbed power. Power enhancement in p polarization is 105 with a maximum input power of 6 W . Beyond 40 mW of absorbed power, the fraction of fundamental-mode power in the absorbing modeclean er’s transmitted beam rolls off sharply from coupling to the frequency-degenerate the L G 12 mode. Fundamental-mode power coupling to the frequency-degenerate L G 23 and TEM 11 , 0 modes occurs at 9 mW and 35 mW of absorbed power, respectively.

Fig. 5
Fig. 5

(a) The CCD image shows that the transmitted beam of the absorbing modecleaner contains both TEM 00 and TEM 11 , 0 modes at 35 mW of absorbed power. The CCD camera is allowed to be slightly saturated to fully resolve the higher-order mode. (b) Image shows the TEM 11 , 0 mode when the beam from image (a) is filtered with the mode analyzer. (c) and (d) Images show the absorbing modecleaner’s transmitted beam and resultant filtering via the mode analyzer at 47 mW of absorbed power, respectively, clearly showing the higher-order mode is L G 12 .

Fig. 6
Fig. 6

Transmitted power versus input power for the absorbing modecleaner. Beyond 40 mW of absorbed power ( 5 W of input power), strong coupling from the fundamental mode to the frequency-degenerate L G 12 mode causes the transmitted power to no longer increase linearly with input power.

Fig. 7
Fig. 7

Periodic fluctuation in the transmitted and the reflected power from the absorbing modecleaner at 6 W of incident power. For 47 mW of absorbed power, the transmission of the absorbing modecleaner fluctuates at a frequency of 29 Hz with a fluctuation depth of 41%.

Fig. 8
Fig. 8

Absorbing modecleaner’s g-factor product versus absorbed power. Predicted mode overlaps are highlighted along the curve with the appropriate mode-index sums [defined as m + n for TEM m n modes and 2 p + l for L G p l modes satisfying Eqs. (21, 22)]. This plot applies for both p and s polarization, where the intracavity power enhancement is 105 and 1200, respectively. The range of each overlap is estimated from the change in the g-factor product needed for a higher-order mode to overlap within the modecleaner’s undistorted full width at half-maximum (FWHM) linewidth.

Fig. 9
Fig. 9

A close-up of an oscilloscope trace of the transmitted power spectrum from the mode analyzer when scanned with the absorbing modecleaner’s transmitted beam at 10 mW of absorbed power ( 200 mW input power) in s polarization. The triangle wave shows the PZT drive signal for the mode analyzer. Coupling to multiple higher-order modes is visible with strong coupling to the L G 15 mode, which is closely spaced to the fundamental mode.

Fig. 10
Fig. 10

Enhanced LIGO arm-cavity g-factor product versus single-optic coating absorbed power for 100 kW of circulating power. Predicted mode-index sums giving higher-order modal frequency degeneracy are highlighted along the curve similar to that shown in Fig. 8. Vertical lines mark the maximum absorbed power at 100 kW for the specified coating absorption. The greater the absorption, the larger the number of possible frequency-degenerate higher-order modes.

Fig. 11
Fig. 11

Advanced LIGO arm-cavity g-factor product versus single-optic coating absorbed power for 800 kW of circulating power. Predicted degenerate mode-index sums are indicated with dots on the graph and vertical lines mark the maximum absorbed power at 800 kW for the specified coating absorption. Advanced LIGO arm cavities become more susceptible to modal frequency degeneracy at low coating absorption because of high circulating power.

Tables (4)

Tables Icon

Table 1 Properties of Modecleaners for Thermal Loading Experiment: Polarization Dependence, Absorption and Power Enhancement

Tables Icon

Table 2 Calculated and Measured Absorbed Power in the Absorbing Modecleaner for Three Observed Higher-Order Modes and Calculated Hot Radius of Curvature (ROC)

Tables Icon

Table 3 Enhanced and Advanced LIGO Suspended Modecleaner Properties and Frequency-Degenerate Higher-Order Modes for 1 ppm Coating Absorption Loss

Tables Icon

Table 4 Enhanced and Advanced LIGO Fabry–Pérot Arm Cavity Properties

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

E circ E in = j T 1 1 R 1 R 2 R 3 exp ( l R T / 2 j ω p m / c ) ,
E refl E in = R 1 R 2 R 3 exp ( l R T / 2 - j ω p m / c ) 1 - R 1 R 2 R 3 exp ( l R T / 2 - j ω p m / c ) ,
ω p m / c = 2 π .
ω ¯ σ q = q + σ + 1 π cos 1 [ ( g 1 g 2 ) 1 / 2 ] ,
σ H G = m + n ,
σ L G = 2 p + l ,
ω ¯ σ * q = q + σ * + 1 π cos 1 [ ( g 1 g 2 ) 1 / 2 ] + 1 2 ,
1 R 1 R 2 R 3 exp ( l R T ) 1 1 l R T l R T / 2
P circ P in 4 T 1 l R T 2 .
P circ P in 2 T 1 π l R T F .
2 T 1 = l R T .
l R T = T 1 + T 2 + a c + l c ,
a c T 3 ,
P a b s = a c P circ .
P a b s = T 3 T 2 P trans .
δ s = α 4 π κ P abs ,
R cold = w cold 2 2 s cold ,
δ w 1 , M 3 w 1 = π 2 λ δ s 3 [ g 1 g 2 ( 1 g 1 g 2 ) ] 1 / 2 ,
w hot = w cold δ w ,
s hot = s cold δ s .
g 1 g 2 , hot = 1 L m R hot , M 3 .
ω ¯ σ q ω ¯ 0 q = σ π cos 1 [ ( g 1 g 2 , hot ) 1 / 2 ] + ( q q ) = k
ω ¯ σ * q ω ¯ 0 q = σ * π cos 1 [ ( g 1 g 2 , hot ) 1 / 2 ] + 1 2 + ( q q ) = j ,
c 00 m n = u 00 * ( x , y ) d ( x , y ) u m n ( x , y ) d A ,
κ 00 m n = | c 00 m n | 2 .
l R T = T 1 + T 2 + a c + l c + a eff ,
a eff = Σ i = 1 k κ 00 m n , i
T > T 3 + l c + a eff ,

Metrics