Abstract

This paper shows a novel method to precisely measure the laser power using an optomechanical system. By measuring a mirror displacement caused by the reflection of an amplitude modulated laser beam, the number of photons in the incident continuous-wave laser can be precisely measured. We have demonstrated this principle by means of a prototype experiment uses a suspended 25 mg mirror as an mechanical oscillator coupled with the radiation pressure and a Michelson interferometer as the displacement sensor. A measurement of the laser power with an uncertainty of less than one percent (1σ) is achievable.

© 2014 Optical Society of America

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  1. T. Maiman, R. Hoskins, I. D’Haenens, C. Asawa, V. Evtuhov, “Stimulated optical emission in fluorescent solids. II. spectroscopy and stimulated emission in ruby,” Phys. Rev. 123, 1151–1157 (1961).
    [CrossRef]
  2. T. Li, S. D. Sims, “A calorimeter for energy measurements of optical masers,” Appl. Opt. 1, 325 (1962).
    [CrossRef]
  3. E. D. West, K. L. Churney, “Theory of isoperibol calorimetry for laser power and energy measurements,” J. Appl. Phys. 41, 2705 (1970).
    [CrossRef]
  4. The web page of NIST, http://www.nist.gov/calibrations/lasers-optoelectronic-components.cfm
  5. D. J. Livigni, C. L. Cromer, T. R. Scott, B. Carol Johnson, Z. M. Zhang, “Thermal characterization of a cryogenic radiometer and comparison with a laser calorimeter,” Metrologia, 35, 819–827 (1998).
    [CrossRef]
  6. J. M. Houston, J. P. Rice, “NIST reference cryogenic radiometer designed for versatile performance,” Metrologia 43, S31–S35 (2006).
    [CrossRef]
  7. D. Deacon, L. Elias, J. Madey, G. Ramian, H. Schwettman, T. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
    [CrossRef]
  8. J. Blau, K. Cohn, W. B. Colson, C. Pogue, M. Stanton, A. Yilmaz, “Free electron lasers in 2011,” in Proceedings of FEL2011, (2011), pp. 274–278.
  9. M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
    [CrossRef]
  10. H. Kimble, Y. Levin, A. Matsko, K. Thorne, S. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 022002 (2001).
    [CrossRef]
  11. H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
    [CrossRef] [PubMed]
  12. R. Schnabel, N. Mavalvala, D. E. McClelland, P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nature Commun. 1, 121 (2010).
    [CrossRef]
  13. D. Clubley, G. Newton, K. Skeldon, J. Hough, “Calibration of the Glasgow 10 m prototype laser interferometric gravitational wave detector using photon pressure,” Phys. Lett. A 283, 85–88 (2001).
    [CrossRef]
  14. K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
    [CrossRef]
  15. K. Agatsuma, T. Mori, S. Ballmer, G. DeSalvo, S. Sakata, E. Nishida, S. Kawamura, “High accuracy measurement of the quantum efficiency using radiation pressure,” J. Phys. Conf. Ser. 363, 012002 (2012).
    [CrossRef]
  16. S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
    [CrossRef]
  17. Guide to the Expression of Uncertainty in Measurement (BIPM, IEC, IFCC, ISO, IUPAC, OIML, 1995).
  18. For example, the micro analytical balances (BM-20 A&D Inc.) has 0.0025 mg of the repeatability (standard deviation). http://www.aandd.jp/products/weighing/balance/analytical/bm.html
  19. S. Hild, Beyond the First Generation: Extending the Science Range of the Gravitational Wave Detector GEO 600 (Gottfried Wilhelm Leibniz Universitat Hannover, 2007), pp. 86–91.
  20. S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
    [CrossRef]

2012 (2)

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

K. Agatsuma, T. Mori, S. Ballmer, G. DeSalvo, S. Sakata, E. Nishida, S. Kawamura, “High accuracy measurement of the quantum efficiency using radiation pressure,” J. Phys. Conf. Ser. 363, 012002 (2012).
[CrossRef]

2010 (2)

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

R. Schnabel, N. Mavalvala, D. E. McClelland, P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nature Commun. 1, 121 (2010).
[CrossRef]

2008 (1)

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

2007 (1)

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

2006 (2)

K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
[CrossRef]

J. M. Houston, J. P. Rice, “NIST reference cryogenic radiometer designed for versatile performance,” Metrologia 43, S31–S35 (2006).
[CrossRef]

2001 (2)

D. Clubley, G. Newton, K. Skeldon, J. Hough, “Calibration of the Glasgow 10 m prototype laser interferometric gravitational wave detector using photon pressure,” Phys. Lett. A 283, 85–88 (2001).
[CrossRef]

H. Kimble, Y. Levin, A. Matsko, K. Thorne, S. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 022002 (2001).
[CrossRef]

1998 (1)

D. J. Livigni, C. L. Cromer, T. R. Scott, B. Carol Johnson, Z. M. Zhang, “Thermal characterization of a cryogenic radiometer and comparison with a laser calorimeter,” Metrologia, 35, 819–827 (1998).
[CrossRef]

1977 (1)

D. Deacon, L. Elias, J. Madey, G. Ramian, H. Schwettman, T. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

1970 (1)

E. D. West, K. L. Churney, “Theory of isoperibol calorimetry for laser power and energy measurements,” J. Appl. Phys. 41, 2705 (1970).
[CrossRef]

1962 (1)

1961 (1)

T. Maiman, R. Hoskins, I. D’Haenens, C. Asawa, V. Evtuhov, “Stimulated optical emission in fluorescent solids. II. spectroscopy and stimulated emission in ruby,” Phys. Rev. 123, 1151–1157 (1961).
[CrossRef]

Agatsuma, K.

K. Agatsuma, T. Mori, S. Ballmer, G. DeSalvo, S. Sakata, E. Nishida, S. Kawamura, “High accuracy measurement of the quantum efficiency using radiation pressure,” J. Phys. Conf. Ser. 363, 012002 (2012).
[CrossRef]

Asawa, C.

T. Maiman, R. Hoskins, I. D’Haenens, C. Asawa, V. Evtuhov, “Stimulated optical emission in fluorescent solids. II. spectroscopy and stimulated emission in ruby,” Phys. Rev. 123, 1151–1157 (1961).
[CrossRef]

Ballmer, S.

K. Agatsuma, T. Mori, S. Ballmer, G. DeSalvo, S. Sakata, E. Nishida, S. Kawamura, “High accuracy measurement of the quantum efficiency using radiation pressure,” J. Phys. Conf. Ser. 363, 012002 (2012).
[CrossRef]

Blau, J.

J. Blau, K. Cohn, W. B. Colson, C. Pogue, M. Stanton, A. Yilmaz, “Free electron lasers in 2011,” in Proceedings of FEL2011, (2011), pp. 274–278.

Brinkmann, M.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

Carol Johnson, B.

D. J. Livigni, C. L. Cromer, T. R. Scott, B. Carol Johnson, Z. M. Zhang, “Thermal characterization of a cryogenic radiometer and comparison with a laser calorimeter,” Metrologia, 35, 819–827 (1998).
[CrossRef]

Chelkowski, S.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

Churney, K. L.

E. D. West, K. L. Churney, “Theory of isoperibol calorimetry for laser power and energy measurements,” J. Appl. Phys. 41, 2705 (1970).
[CrossRef]

Clubley, D.

D. Clubley, G. Newton, K. Skeldon, J. Hough, “Calibration of the Glasgow 10 m prototype laser interferometric gravitational wave detector using photon pressure,” Phys. Lett. A 283, 85–88 (2001).
[CrossRef]

Cohn, K.

J. Blau, K. Cohn, W. B. Colson, C. Pogue, M. Stanton, A. Yilmaz, “Free electron lasers in 2011,” in Proceedings of FEL2011, (2011), pp. 274–278.

Colson, W. B.

J. Blau, K. Cohn, W. B. Colson, C. Pogue, M. Stanton, A. Yilmaz, “Free electron lasers in 2011,” in Proceedings of FEL2011, (2011), pp. 274–278.

Cromer, C. L.

D. J. Livigni, C. L. Cromer, T. R. Scott, B. Carol Johnson, Z. M. Zhang, “Thermal characterization of a cryogenic radiometer and comparison with a laser calorimeter,” Metrologia, 35, 819–827 (1998).
[CrossRef]

D’Haenens, I.

T. Maiman, R. Hoskins, I. D’Haenens, C. Asawa, V. Evtuhov, “Stimulated optical emission in fluorescent solids. II. spectroscopy and stimulated emission in ruby,” Phys. Rev. 123, 1151–1157 (1961).
[CrossRef]

Danzmann, K.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
[CrossRef]

Deacon, D.

D. Deacon, L. Elias, J. Madey, G. Ramian, H. Schwettman, T. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

DeSalvo, G.

K. Agatsuma, T. Mori, S. Ballmer, G. DeSalvo, S. Sakata, E. Nishida, S. Kawamura, “High accuracy measurement of the quantum efficiency using radiation pressure,” J. Phys. Conf. Ser. 363, 012002 (2012).
[CrossRef]

Elias, L.

D. Deacon, L. Elias, J. Madey, G. Ramian, H. Schwettman, T. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

Evtuhov, V.

T. Maiman, R. Hoskins, I. D’Haenens, C. Asawa, V. Evtuhov, “Stimulated optical emission in fluorescent solids. II. spectroscopy and stimulated emission in ruby,” Phys. Rev. 123, 1151–1157 (1961).
[CrossRef]

Franzen, A.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

Goßler, S.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

Grote, H.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
[CrossRef]

Hage, B.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

Hewitson, M.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
[CrossRef]

Hild, S.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
[CrossRef]

S. Hild, Beyond the First Generation: Extending the Science Range of the Gravitational Wave Detector GEO 600 (Gottfried Wilhelm Leibniz Universitat Hannover, 2007), pp. 86–91.

Hoskins, R.

T. Maiman, R. Hoskins, I. D’Haenens, C. Asawa, V. Evtuhov, “Stimulated optical emission in fluorescent solids. II. spectroscopy and stimulated emission in ruby,” Phys. Rev. 123, 1151–1157 (1961).
[CrossRef]

Hough, J.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

D. Clubley, G. Newton, K. Skeldon, J. Hough, “Calibration of the Glasgow 10 m prototype laser interferometric gravitational wave detector using photon pressure,” Phys. Lett. A 283, 85–88 (2001).
[CrossRef]

Houston, J. M.

J. M. Houston, J. P. Rice, “NIST reference cryogenic radiometer designed for versatile performance,” Metrologia 43, S31–S35 (2006).
[CrossRef]

Ishikawa, T.

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

Ishizaki, H.

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

Kato, M.

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

Kawamura, S.

K. Agatsuma, T. Mori, S. Ballmer, G. DeSalvo, S. Sakata, E. Nishida, S. Kawamura, “High accuracy measurement of the quantum efficiency using radiation pressure,” J. Phys. Conf. Ser. 363, 012002 (2012).
[CrossRef]

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

Kimble, H.

H. Kimble, Y. Levin, A. Matsko, K. Thorne, S. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 022002 (2001).
[CrossRef]

Kudo, T.

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

Kurosawa, T.

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

Lam, P. K.

R. Schnabel, N. Mavalvala, D. E. McClelland, P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nature Commun. 1, 121 (2010).
[CrossRef]

Lastzka, N.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

Levin, Y.

H. Kimble, Y. Levin, A. Matsko, K. Thorne, S. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 022002 (2001).
[CrossRef]

Li, T.

Livigni, D. J.

D. J. Livigni, C. L. Cromer, T. R. Scott, B. Carol Johnson, Z. M. Zhang, “Thermal characterization of a cryogenic radiometer and comparison with a laser calorimeter,” Metrologia, 35, 819–827 (1998).
[CrossRef]

Lück, H.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
[CrossRef]

Madey, J.

D. Deacon, L. Elias, J. Madey, G. Ramian, H. Schwettman, T. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

Maiman, T.

T. Maiman, R. Hoskins, I. D’Haenens, C. Asawa, V. Evtuhov, “Stimulated optical emission in fluorescent solids. II. spectroscopy and stimulated emission in ruby,” Phys. Rev. 123, 1151–1157 (1961).
[CrossRef]

Martin, I.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

Matsko, A.

H. Kimble, Y. Levin, A. Matsko, K. Thorne, S. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 022002 (2001).
[CrossRef]

Mavalvala, N.

R. Schnabel, N. Mavalvala, D. E. McClelland, P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nature Commun. 1, 121 (2010).
[CrossRef]

McClelland, D. E.

R. Schnabel, N. Mavalvala, D. E. McClelland, P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nature Commun. 1, 121 (2010).
[CrossRef]

Mehmet, M.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

Miyakawa, O.

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

Mori, T.

K. Agatsuma, T. Mori, S. Ballmer, G. DeSalvo, S. Sakata, E. Nishida, S. Kawamura, “High accuracy measurement of the quantum efficiency using radiation pressure,” J. Phys. Conf. Ser. 363, 012002 (2012).
[CrossRef]

Mossavi, K.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
[CrossRef]

Newton, G.

D. Clubley, G. Newton, K. Skeldon, J. Hough, “Calibration of the Glasgow 10 m prototype laser interferometric gravitational wave detector using photon pressure,” Phys. Lett. A 283, 85–88 (2001).
[CrossRef]

Nishida, E.

K. Agatsuma, T. Mori, S. Ballmer, G. DeSalvo, S. Sakata, E. Nishida, S. Kawamura, “High accuracy measurement of the quantum efficiency using radiation pressure,” J. Phys. Conf. Ser. 363, 012002 (2012).
[CrossRef]

Nishizawa, A.

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

Pogue, C.

J. Blau, K. Cohn, W. B. Colson, C. Pogue, M. Stanton, A. Yilmaz, “Free electron lasers in 2011,” in Proceedings of FEL2011, (2011), pp. 274–278.

Rainer, N.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

Ramian, G.

D. Deacon, L. Elias, J. Madey, G. Ramian, H. Schwettman, T. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

Reid, S.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

Rice, J. P.

J. M. Houston, J. P. Rice, “NIST reference cryogenic radiometer designed for versatile performance,” Metrologia 43, S31–S35 (2006).
[CrossRef]

Richter, M.

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

Saito, N.

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

Sakata, S.

K. Agatsuma, T. Mori, S. Ballmer, G. DeSalvo, S. Sakata, E. Nishida, S. Kawamura, “High accuracy measurement of the quantum efficiency using radiation pressure,” J. Phys. Conf. Ser. 363, 012002 (2012).
[CrossRef]

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

Schnabel, R.

R. Schnabel, N. Mavalvala, D. E. McClelland, P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nature Commun. 1, 121 (2010).
[CrossRef]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

Schwettman, H.

D. Deacon, L. Elias, J. Madey, G. Ramian, H. Schwettman, T. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

Scott, T. R.

D. J. Livigni, C. L. Cromer, T. R. Scott, B. Carol Johnson, Z. M. Zhang, “Thermal characterization of a cryogenic radiometer and comparison with a laser calorimeter,” Metrologia, 35, 819–827 (1998).
[CrossRef]

Seifert, F.

K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
[CrossRef]

Sims, S. D.

Skeldon, K.

D. Clubley, G. Newton, K. Skeldon, J. Hough, “Calibration of the Glasgow 10 m prototype laser interferometric gravitational wave detector using photon pressure,” Phys. Lett. A 283, 85–88 (2001).
[CrossRef]

Smith, J. R.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
[CrossRef]

Smith, T.

D. Deacon, L. Elias, J. Madey, G. Ramian, H. Schwettman, T. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

Sorokin, A. A.

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

Stanton, M.

J. Blau, K. Cohn, W. B. Colson, C. Pogue, M. Stanton, A. Yilmaz, “Free electron lasers in 2011,” in Proceedings of FEL2011, (2011), pp. 274–278.

Strain, K.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

Tanaka, T.

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

Thorne, K.

H. Kimble, Y. Levin, A. Matsko, K. Thorne, S. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 022002 (2001).
[CrossRef]

Tiedtke, K.

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

Tono, K.

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

Vahlbruch, H.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

Vyatchanin, S.

H. Kimble, Y. Levin, A. Matsko, K. Thorne, S. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 022002 (2001).
[CrossRef]

Weiland, U.

K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
[CrossRef]

Weinert, M.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

West, E. D.

E. D. West, K. L. Churney, “Theory of isoperibol calorimetry for laser power and energy measurements,” J. Appl. Phys. 41, 2705 (1970).
[CrossRef]

Willems, P.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

Willke, B.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
[CrossRef]

Winkler, W.

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

Yabashi, M.

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

Yilmaz, A.

J. Blau, K. Cohn, W. B. Colson, C. Pogue, M. Stanton, A. Yilmaz, “Free electron lasers in 2011,” in Proceedings of FEL2011, (2011), pp. 274–278.

Zhang, Z. M.

D. J. Livigni, C. L. Cromer, T. R. Scott, B. Carol Johnson, Z. M. Zhang, “Thermal characterization of a cryogenic radiometer and comparison with a laser calorimeter,” Metrologia, 35, 819–827 (1998).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. Kato, T. Tanaka, T. Kurosawa, N. Saito, M. Richter, A. A. Sorokin, K. Tiedtke, T. Kudo, K. Tono, M. Yabashi, T. Ishikawa, “Pulse energy measurement at the hard x-ray laser in Japan,” Appl. Phys. Lett. 101, 023503 (2012).
[CrossRef]

Classical Quant. Grav. (1)

S. Hild, M. Brinkmann, K. Danzmann, H. Grote, M. Hewitson, J. Hough, H. Lück, I. Martin, K. Mossavi, N. Rainer, S. Reid, J. R. Smith, K. Strain, M. Weinert, P. Willems, B. Willke, W. Winkler, “Photon-pressure-induced test mass deformation in gravitational-wave detectors,” Classical Quant. Grav. 24, 5681–5688 (2007).
[CrossRef]

J. Appl. Phys. (1)

E. D. West, K. L. Churney, “Theory of isoperibol calorimetry for laser power and energy measurements,” J. Appl. Phys. 41, 2705 (1970).
[CrossRef]

J. Phys. Conf. Ser. (1)

K. Agatsuma, T. Mori, S. Ballmer, G. DeSalvo, S. Sakata, E. Nishida, S. Kawamura, “High accuracy measurement of the quantum efficiency using radiation pressure,” J. Phys. Conf. Ser. 363, 012002 (2012).
[CrossRef]

Metrologia (2)

D. J. Livigni, C. L. Cromer, T. R. Scott, B. Carol Johnson, Z. M. Zhang, “Thermal characterization of a cryogenic radiometer and comparison with a laser calorimeter,” Metrologia, 35, 819–827 (1998).
[CrossRef]

J. M. Houston, J. P. Rice, “NIST reference cryogenic radiometer designed for versatile performance,” Metrologia 43, S31–S35 (2006).
[CrossRef]

Nature Commun. (1)

R. Schnabel, N. Mavalvala, D. E. McClelland, P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nature Commun. 1, 121 (2010).
[CrossRef]

Phys. Lett. A (2)

D. Clubley, G. Newton, K. Skeldon, J. Hough, “Calibration of the Glasgow 10 m prototype laser interferometric gravitational wave detector using photon pressure,” Phys. Lett. A 283, 85–88 (2001).
[CrossRef]

K. Mossavi, M. Hewitson, S. Hild, F. Seifert, U. Weiland, J. R. Smith, H. Lück, H. Grote, B. Willke, K. Danzmann, “A photon pressure calibrator for the GEO 600 gravitational wave detector,” Phys. Lett. A 353, 1–3 (2006).
[CrossRef]

Phys. Rev. (1)

T. Maiman, R. Hoskins, I. D’Haenens, C. Asawa, V. Evtuhov, “Stimulated optical emission in fluorescent solids. II. spectroscopy and stimulated emission in ruby,” Phys. Rev. 123, 1151–1157 (1961).
[CrossRef]

Phys. Rev. D (2)

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

H. Kimble, Y. Levin, A. Matsko, K. Thorne, S. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 022002 (2001).
[CrossRef]

Phys. Rev. Lett. (2)

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, R. Schnabel, “Observation of squeezed light with 10-dB quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

D. Deacon, L. Elias, J. Madey, G. Ramian, H. Schwettman, T. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977).
[CrossRef]

Other (5)

J. Blau, K. Cohn, W. B. Colson, C. Pogue, M. Stanton, A. Yilmaz, “Free electron lasers in 2011,” in Proceedings of FEL2011, (2011), pp. 274–278.

The web page of NIST, http://www.nist.gov/calibrations/lasers-optoelectronic-components.cfm

Guide to the Expression of Uncertainty in Measurement (BIPM, IEC, IFCC, ISO, IUPAC, OIML, 1995).

For example, the micro analytical balances (BM-20 A&D Inc.) has 0.0025 mg of the repeatability (standard deviation). http://www.aandd.jp/products/weighing/balance/analytical/bm.html

S. Hild, Beyond the First Generation: Extending the Science Range of the Gravitational Wave Detector GEO 600 (Gottfried Wilhelm Leibniz Universitat Hannover, 2007), pp. 86–91.

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Figures (11)

Fig. 1
Fig. 1

Sketch of the optomechanical system investigated in this work. By measuring the displacement of a suspended mirror the incident modulated laser power can be determined.

Fig. 2
Fig. 2

Block diagram for the MI control. The symbols S, A and HPD show each transfer function of the servo filter, actuator response and PD response, respectively.

Fig. 3
Fig. 3

Schematic view of the experimental setup. Acronym explanation: HWP, half wave plate; QWP, quarter wave plate; BS, beam splitter; PBS, polarized beam splitter; FI, faraday isolator; and AOM, acousto-optic modulator. L1 and L2 show length measurements for the incident angle. There are three photo detectors; Symmetric port, Asymmetric port and AOM port on an optical table. The solid red line indicates a beam trace for the MI and the dotted red line is that for an intensity modulation by AOM.

Fig. 4
Fig. 4

The tiny mirror suspension system. (a) photographs of the 25 mg mirror and the double pendulum. (b) model of the double pendulum.

Fig. 5
Fig. 5

The sensitivity of the MI and the displacement shaken by radiation pressure at 66, 72, 82 and 233 Hz. The orange dashed line shows f−2 response of the free mass from Eq. (2).

Fig. 6
Fig. 6

Deviation from the mid-fringe lock. (a) effect of k and rs. Here, p is an offset phase of an operation point, q an average phase of a residual error signal and rs the difference between an ideal linear response and an actual sinusoidal response; (b) required precision for the measurement of p from Eq. (11).

Fig. 7
Fig. 7

Feedback signal with modulated laser at 72 Hz. Left: covering a wide frequency. Right: zoomed-in around 70 Hz. The black arrow marks the peak at 72 Hz. The black line indicates the noise level without the laser modulation.

Fig. 8
Fig. 8

Analysis of the closed loop transfer function. (a) absolute value of the closed loop gain; (b) standard deviation of the closed loop gain. Black arrows indicate the measurement points.

Fig. 9
Fig. 9

Analysis of the actuator response. (a) comparison of A between two measurement methods (monolithic and broadband), which are absolute values and divided by each PD response. Four monochromatic measurements are superimposed as the red solid line. Black allows are the measurement points; (b) output signal from the MI without lock.

Fig. 10
Fig. 10

Effects from parameter deviations of the double pendulum on its mechanical response. (a) comparison between the free mass response and mechanical response with the default setting; (b–f) deviations from the default setting in case the possible differences exist in the parameters of the damping factor (Q) at the intermediate mass, the test mass, the intermediate mass, the length of the upper wire, and of the lower wire, respectively.

Fig. 11
Fig. 11

Illustration of the rotational effect. The red arrows show the displacement of the mirror.

Tables (1)

Tables Icon

Table 1 Uncertainty budget of the prototype experiment at 72 Hz.

Equations (18)

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

d X ˜ = 2 P m cm ω 2 .
d X ˜ = 2 P m α r cos ϕ cm ω 2 ( 1 + R c ) ,
d X ˜ = ( λ / 2 π V pp ) d V PD ,
P m = c λ 4 π V pp H m α r ( 1 + R c ) cos ϕ d V PD .
P m = c λ V f G CL T AH 4 π V pp H m α r ( 1 + R c ) cos ϕ .
σ Y = i = 1 s ( Y ξ i σ i ) 2 + 2 i = 1 s 1 j = i + 1 s Y ξ i Y ξ j σ i σ j r ( ξ i , ξ j ) ,
σ P P = ( σ d V d V PD ) 2 + ( σ V V pp ) 2 + ( σ H H m ) 2 + ( σ α α r ) 2 + ( σ ϕ tan ϕ ) 2 + ( σ R 1 + R c ) 2 .
V PD = 1 2 V pp sin [ 2 π ( x λ / 2 ) ] .
H PD = 2 π V pp λ k r s ,
( σ d V d V PD ) 2 = ( σ V f V f ) 2 + ( σ G C L G CL ) 2 + ( σ T T AH ) 2 + ( σ k k ) 2 + ( σ r s r s ) 2 .
( k σ k ) / k = cos ( p + E p ) / cos ( p ) 0.997 ( k + σ k ) / k = cos ( p E p ) / cos ( p ) 1.003 ,
r s = sin ( p + q ) sin ( p q ) 2 q cos ( p ) 0.997
m 1 x ¨ 1 = ( m 1 + m 2 ) g l 1 x 1 m 2 g l 2 ( x 1 x 2 ) Γ 1 x ˙ 1 ,
m 2 x ¨ 2 = m 2 g l 2 ( x 2 x 1 ) Γ 2 x ˙ 2 + F ,
A ( x ˜ 1 x ˜ 2 ) = ( 0 F ˜ ) , A 1 ( a m b m c m d m ) .
H m ( ω ) x ˜ 2 F ˜ = d m .
σ R = 2 [ d 0 ( r 2 / 4 + l 2 / 12 ) σ d ] 2 + [ d 0 2 ( r 2 / 4 + l 2 / 12 ) 2 r 2 σ r ] 2 + [ d 0 2 ( r 2 / 4 + l 2 / 12 ) 2 l 6 σ l ] 2 ,
[ σ R ( 1 + R c ) ] 2 1 ( 1 + R c ) 2 { [ d c ( r 2 / 4 + l 2 / 12 ) σ d c 3 ] 2 + [ d pp ( r 2 / 4 + l 2 / 12 ) σ d pp 3 ] 2 } ,

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