Abstract

We present a discussion of the use of amplitude modulation techniques with regard to the length sensing and control of optical cavities for laser interferometric gravitational-wave detectors. Traditional radio-frequency amplitude modulation techniques automatically include phase modulation as a product of the modulation process, which can contaminate the signal after demodulation. In particular, with many length-sensing and control schemes the detected signals are demodulated in quadrature, which, in the case of a traditional amplitude modulation scheme, will result in offsets due to the additional phase modulation. We demonstrate this effect using a simple optical cavity configuration and show that minor adjustments to the modulator system can be used to compensate for the extra modulation components and provide additional flexibility.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  4. M. Ando and the TAMA collaboration, "Current status of TAMA," Class. Quantum Grav. 19, 1409-1419 (2002).
    [CrossRef]
  5. E. Gustafson, D. Shoemaker, K. Strain, and R. Weiss, "LSC white paper on detector research and development," LIGO Document T990080-00-D (1999).
  6. G. Müller, T. Delker, and D. H. Reitze, "Dual-recycled cavity-enhanced Michelson interferometer for gravitational-wave detection," Appl. Opt. 42, 1257-1268 (2003).
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  7. O. Miyakawa, R. Ward, R. Adhikari, M. Evans, B. Abbott, R. Bork, D. Busby, J. Heefner, A. Ivanov, M. Smith, R. Taylor, S. Vass, A. Weinstein, M. Varvella, S. Kawamura, F. Kawazoe, S. Sakata, and C. Mow-Lowry, "Measurement of optical response of a detuned resonant sideband extraction interferometer," Phys. Rev. D 74, 022001 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. J. Wilson and J. Hawkes, Optoelectronics: an Introduction, 3rd ed. (Prentice Hall, 1998), pp. 96-105.
  11. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, 1997), pp. 17-29.
  12. S. H. Huttner, B. W. Barr, M. V. Plissi, J. R. Taylor, B. Sorazu, and K. A. Strain, "Novel sensing and control schemes for a three mirror coupled cavity," Class. Quantum Grav. 24, 3825-3836 (2007).
    [CrossRef]
  13. B. W. Barr, O. Miyakawa, S. Kawamura, A. J. Weinstein, R. Ward, S. Vass, and K. A. Strain, "Control sideband generation for dual-recycled laser interferometric gravitational wave detectors," Class. Quantum Grav. 23, 5661-5666 (2006).
    [CrossRef]
  14. A. Freise, FINESSE--Frequency domain interferometer simulation software, http://www.rzg.mpg.de/~adf/.

2007

S. H. Huttner, B. W. Barr, M. V. Plissi, J. R. Taylor, B. Sorazu, and K. A. Strain, "Novel sensing and control schemes for a three mirror coupled cavity," Class. Quantum Grav. 24, 3825-3836 (2007).
[CrossRef]

2006

B. W. Barr, O. Miyakawa, S. Kawamura, A. J. Weinstein, R. Ward, S. Vass, and K. A. Strain, "Control sideband generation for dual-recycled laser interferometric gravitational wave detectors," Class. Quantum Grav. 23, 5661-5666 (2006).
[CrossRef]

O. Miyakawa, R. Ward, R. Adhikari, M. Evans, B. Abbott, R. Bork, D. Busby, J. Heefner, A. Ivanov, M. Smith, R. Taylor, S. Vass, A. Weinstein, M. Varvella, S. Kawamura, F. Kawazoe, S. Sakata, and C. Mow-Lowry, "Measurement of optical response of a detuned resonant sideband extraction interferometer," Phys. Rev. D 74, 022001 (2006).
[CrossRef]

2004

2003

2002

D. Sigg, "Commissioning of the LIGO detectors," Class. Quantum Grav. 19, 1429-1435 (2002).
[CrossRef]

B. Willke, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. W. Barr, S. Berukoff, S. Bose, G. Cagnoli, M. M. Casey, D. Churches, D. Clubley, C. N. Colacino, D. R. M. Crooks, C. Cutler, K. Danzmann, R. Davies, R. Dupuis, E. Elliffe, C. Fallnich, A. Freise, S. Goßler, A. Grant, H. Grote, G. Heinzel, A. Heptonstall, M. Heurs, M. Hewitson, J. Hough, O. Jennrich, K. Kawabe, K. Kötter, V. Leonhardt, H. Lück, M. Malec, P. W. McNamara, S. A. McIntosh, K. Mossavi, S. Mohanty, S. Mukherjee, S. Nagano, G. P. Newton, B. J. Owen, D. Palmer, M. A. Papa, M. V. Plissi, V. Quetschke, D. I. Robertson, N. A. Robertson, S. Rowan, A. Rüdiger, B. S. Sathyaprakash, R. Schilling, B. F. Schutz, R. Senior, A. M. Sintes, K. D. Skeldon, P. Sneddon, F. Stief, K. A. Strain, I. Taylor, C. I. Torrie, A. Vecchio, H. Ward, U. Weiland, H. Welling, P. Williams, W. Winkler, G. Woan, and I. Zawischa, "The GEO 600 gravitational wave detector," Class. Quantum Grav. 19, 1377-1387 (2002).
[CrossRef]

F. Acernese, P. Amico, N. Arnaud, C. Arnault, D. Babusci, G. Ballardin, F. Barone, M. Barsuglia, F. Bellachia, J. L. Beney, R. Bilhaut, M. A. Bizouard, C. Boccara, D. Boget, F. Bondu, C. Bourgoin, A. Bozzi, L. Bracci, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, D. Buskulic, J. Cachenaut, G. Calamai, E. Calloni, P. Canitrot, B. Caron, C. Casciano, C. Cattuto, F. Cavalier, S. Cavaliere, R. Cavalieri, R. Cecchi, G. Cella, R. Chiche, F. Chollet, F. Cleva, T. Cokelaer, S. Cortese, J. P. Coulon, E. Cuoco, S. Cuzon, V. Dattilo, P. Y. David, M. Davier, M. De Rosa, R. D. Rosa, M. Dehamme, L. D. Fiore, A. D. Virgilio, P. Dominici, D. Dufournaud, C. Eder, A. Eleuteri, D. Enard, A. Errico, G. Evangelista, L. Fabbroni, H. Fang, I. Ferrante, F. Fidecaro, R. Flaminio, J. D. Fournier, L. Fournier, S. Frasca, F. Frasconi, L. Gammaitoni, P. Ganau, F. Garufi, M. Gaspard, G. Gennaro, L. Giacobone, A. Giazotto, G. Giordano, C. Girard, G. Guidi, H. Heitmann, P. Hello, R. Hermel, P. Heusse, L. Holloway, M. Iannarelli, J. M. Innocent, E. Jules, P. L. Penna, J. C. Lacotte, B. Lagrange, M. Leliboux, B. Lieunard, O. Lodygenski, T. Lomtadze, V. Loriette, G. Losurdo, M. Loupias, J. M. Mackowski, E. Majorana, C. N. Man, B. Mansoux, F. Marchesoni, P. Marin, F. Marion, J. C. Marrucho, F. Martelli, A. Masserot, L. Massonnet, S. Mataguez, M. Mazzoni, M. Mencik, C. Michel, L. Milano, J. L. Montorio, N. Morgado, B. Mours, P. Mugnier, L. Nicolosi, J. Pacheco, C. Palomba, F. Paoletti, A. Paoli, A. Pasqualetti, R. Passaquieti, D. Passuello, M. Perciballi, L. Pinard, R. Poggiani, P. Popolizio, T. Pradier, M. Punturo, P. Puppo, K. Qipiani, J. Ramonet, P. Rapagnani, A. Reboux, T. Regimbau, V. Reita, A. Remillieux, F. Ricci, F. Richard, M. Ripepe, P. Rivoirard, J. P. Roger, J. P. Scheidecker, S. Solimeno, R. Sottile, R. Stanga, R. Taddei, M. Taurigna, J. M. Teuler, P. Tourrenc, H. Trinquet, E. Turri, M. Varvella, D. Verkindt, F. Vetrano, O. Veziant, A. Viceré, J. Y. Vinet, H. Vocca, M. Yvert, and Z. Zhang, "The present status of the VIRGO Central Interferometer," Class. Quantum Grav. 19, 1421-1428 (2002).
[CrossRef]

M. Ando and the TAMA collaboration, "Current status of TAMA," Class. Quantum Grav. 19, 1409-1419 (2002).
[CrossRef]

1999

E. Gustafson, D. Shoemaker, K. Strain, and R. Weiss, "LSC white paper on detector research and development," LIGO Document T990080-00-D (1999).

Appl. Opt.

Class. Quantum Grav.

D. Sigg, "Commissioning of the LIGO detectors," Class. Quantum Grav. 19, 1429-1435 (2002).
[CrossRef]

F. Acernese, P. Amico, N. Arnaud, C. Arnault, D. Babusci, G. Ballardin, F. Barone, M. Barsuglia, F. Bellachia, J. L. Beney, R. Bilhaut, M. A. Bizouard, C. Boccara, D. Boget, F. Bondu, C. Bourgoin, A. Bozzi, L. Bracci, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, D. Buskulic, J. Cachenaut, G. Calamai, E. Calloni, P. Canitrot, B. Caron, C. Casciano, C. Cattuto, F. Cavalier, S. Cavaliere, R. Cavalieri, R. Cecchi, G. Cella, R. Chiche, F. Chollet, F. Cleva, T. Cokelaer, S. Cortese, J. P. Coulon, E. Cuoco, S. Cuzon, V. Dattilo, P. Y. David, M. Davier, M. De Rosa, R. D. Rosa, M. Dehamme, L. D. Fiore, A. D. Virgilio, P. Dominici, D. Dufournaud, C. Eder, A. Eleuteri, D. Enard, A. Errico, G. Evangelista, L. Fabbroni, H. Fang, I. Ferrante, F. Fidecaro, R. Flaminio, J. D. Fournier, L. Fournier, S. Frasca, F. Frasconi, L. Gammaitoni, P. Ganau, F. Garufi, M. Gaspard, G. Gennaro, L. Giacobone, A. Giazotto, G. Giordano, C. Girard, G. Guidi, H. Heitmann, P. Hello, R. Hermel, P. Heusse, L. Holloway, M. Iannarelli, J. M. Innocent, E. Jules, P. L. Penna, J. C. Lacotte, B. Lagrange, M. Leliboux, B. Lieunard, O. Lodygenski, T. Lomtadze, V. Loriette, G. Losurdo, M. Loupias, J. M. Mackowski, E. Majorana, C. N. Man, B. Mansoux, F. Marchesoni, P. Marin, F. Marion, J. C. Marrucho, F. Martelli, A. Masserot, L. Massonnet, S. Mataguez, M. Mazzoni, M. Mencik, C. Michel, L. Milano, J. L. Montorio, N. Morgado, B. Mours, P. Mugnier, L. Nicolosi, J. Pacheco, C. Palomba, F. Paoletti, A. Paoli, A. Pasqualetti, R. Passaquieti, D. Passuello, M. Perciballi, L. Pinard, R. Poggiani, P. Popolizio, T. Pradier, M. Punturo, P. Puppo, K. Qipiani, J. Ramonet, P. Rapagnani, A. Reboux, T. Regimbau, V. Reita, A. Remillieux, F. Ricci, F. Richard, M. Ripepe, P. Rivoirard, J. P. Roger, J. P. Scheidecker, S. Solimeno, R. Sottile, R. Stanga, R. Taddei, M. Taurigna, J. M. Teuler, P. Tourrenc, H. Trinquet, E. Turri, M. Varvella, D. Verkindt, F. Vetrano, O. Veziant, A. Viceré, J. Y. Vinet, H. Vocca, M. Yvert, and Z. Zhang, "The present status of the VIRGO Central Interferometer," Class. Quantum Grav. 19, 1421-1428 (2002).
[CrossRef]

M. Ando and the TAMA collaboration, "Current status of TAMA," Class. Quantum Grav. 19, 1409-1419 (2002).
[CrossRef]

Class. Quantum Grav.

B. Willke, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. W. Barr, S. Berukoff, S. Bose, G. Cagnoli, M. M. Casey, D. Churches, D. Clubley, C. N. Colacino, D. R. M. Crooks, C. Cutler, K. Danzmann, R. Davies, R. Dupuis, E. Elliffe, C. Fallnich, A. Freise, S. Goßler, A. Grant, H. Grote, G. Heinzel, A. Heptonstall, M. Heurs, M. Hewitson, J. Hough, O. Jennrich, K. Kawabe, K. Kötter, V. Leonhardt, H. Lück, M. Malec, P. W. McNamara, S. A. McIntosh, K. Mossavi, S. Mohanty, S. Mukherjee, S. Nagano, G. P. Newton, B. J. Owen, D. Palmer, M. A. Papa, M. V. Plissi, V. Quetschke, D. I. Robertson, N. A. Robertson, S. Rowan, A. Rüdiger, B. S. Sathyaprakash, R. Schilling, B. F. Schutz, R. Senior, A. M. Sintes, K. D. Skeldon, P. Sneddon, F. Stief, K. A. Strain, I. Taylor, C. I. Torrie, A. Vecchio, H. Ward, U. Weiland, H. Welling, P. Williams, W. Winkler, G. Woan, and I. Zawischa, "The GEO 600 gravitational wave detector," Class. Quantum Grav. 19, 1377-1387 (2002).
[CrossRef]

S. H. Huttner, B. W. Barr, M. V. Plissi, J. R. Taylor, B. Sorazu, and K. A. Strain, "Novel sensing and control schemes for a three mirror coupled cavity," Class. Quantum Grav. 24, 3825-3836 (2007).
[CrossRef]

B. W. Barr, O. Miyakawa, S. Kawamura, A. J. Weinstein, R. Ward, S. Vass, and K. A. Strain, "Control sideband generation for dual-recycled laser interferometric gravitational wave detectors," Class. Quantum Grav. 23, 5661-5666 (2006).
[CrossRef]

LIGO Document T990080-00-D

E. Gustafson, D. Shoemaker, K. Strain, and R. Weiss, "LSC white paper on detector research and development," LIGO Document T990080-00-D (1999).

Phys. Rev. D

O. Miyakawa, R. Ward, R. Adhikari, M. Evans, B. Abbott, R. Bork, D. Busby, J. Heefner, A. Ivanov, M. Smith, R. Taylor, S. Vass, A. Weinstein, M. Varvella, S. Kawamura, F. Kawazoe, S. Sakata, and C. Mow-Lowry, "Measurement of optical response of a detuned resonant sideband extraction interferometer," Phys. Rev. D 74, 022001 (2006).
[CrossRef]

Other

A. Freise, FINESSE--Frequency domain interferometer simulation software, http://www.rzg.mpg.de/~adf/.

J. Wilson and J. Hawkes, Optoelectronics: an Introduction, 3rd ed. (Prentice Hall, 1998), pp. 96-105.

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, 1997), pp. 17-29.

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Figures (6)

Fig. 1
Fig. 1

AM∕PM∕SSB modulator configuration. This setup is comparable with a conventional AM modulation arrangement but with an additional half-wave plate (HWP) placed before the output polarizer. The polarizer axes are aligned along x and y while the crystal axes (along x and y ) are set at 45 deg to this reference, respectively. A, B, C, D, and E are included as points of reference for discussion. QWP, quarter-wave plate; PBS, polarizing beam splitter.

Fig. 2
Fig. 2

Evolution of the carrier and sideband frequency components through the modulator expressed as polarization. The points A, B, C, D, and E are in reference to the positions marked in Fig. 1. Note that this diagram indicates polarization as unit vectors only, while in practice the sideband vectors are actually much smaller in comparison with the carrier.

Fig. 3
Fig. 3

Optical test configuration for the modulator system. After first modulating using the new modulator configuration ( 14.525 MHz ), the beam is then passed through a phase modulator ( 10 MHz ) before interacting with the suspended test cavity. The light returning from the cavity is detected on a photodiode (tuned to 4.525 MHz ) and demodulated to give the cavity length sensing signals used to verify the modulator operation. An additional photodiode at the end of the cavity provides a measure of the power buildup in the cavity. NPRO, nonplanar ring oscillator.

Fig. 4
Fig. 4

Measured (left) and modeled (right) plots for a full sweep of the test cavity with the modulator set for AM sidebands. The main features of the trace from left to right are (a) carrier resonance, (b) upper 4.525 MHz resonance, (c) upper 10 MHz resonance, (d) both 14.525 MHz resonances, (e) lower 10 MHz resonance, (f) lower 4.525 MHz resonance, and finally (g) the next carrier resonance. FSR, free spectral range.

Fig. 5
Fig. 5

Measured (left) and modeled (right) plots for a full sweep of the test cavity with the modulator set for PM sidebands. The main features of the trace are in the same locations as for the AM case.

Fig. 6
Fig. 6

Measured (left) and modeled (right) plots for a full sweep of the test cavity with the modulator set for SSB modulation. Again, the main features are as for the AM setup but with the obvious reduction in sideband presence on one side of the trace. Additional peaks superimposed on the measured trace are due to higher-order mode resonances (these were minimized but not completely eliminated by careful alignment) and residual AM∕PM noncancellation.

Equations (13)

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

E A = [ E A x     E A y ] = E 0 exp ( i ω 0 t ) [ 1 0 ] ,
E g e n = E 0 exp ( i ω 0 t ) [ M C + i M exp ( i ω m t ) + i M exp ( i ω m t ) ] ,
V C = J 0 ( m ) ,
V ( α , θ 0 ) = J 1 ( m ) [ cos ( θ 0 ) sin ( θ 0 ) sin ( θ 0 ) cos ( θ 0 ) ] × [ exp ( i α ) 0 0 exp ( i α ) ] × [ cos ( θ 0 ) sin ( θ 0 ) sin ( θ 0 ) cos ( θ 0 ) ] ,
V ( α , θ 0 ) = J 1 ( m ) [ cos ( θ 0 ) sin ( θ 0 ) sin ( θ 0 ) cos ( θ 0 ) ] × [ exp ( i α ) 0 0 exp ( i α ) ] × [ cos ( θ 0 ) sin ( θ 0 ) sin ( θ 0 ) cos ( θ 0 ) ] ,
M C = exp ( i 3 π / 4 ) J 0 ( m ) 2 { [ 4 sin ( θ 1 ) cos ( θ 1 ) cos ( θ 2 ) 2 4 cos ( θ 1 ) 2 sin ( θ 2 ) cos ( θ 2 ) + 2 sin ( θ 2 ) cos ( θ 2 ) 2 sin ( θ 1 ) cos ( θ 1 ) ] 2 i sin ( θ 2 ) cos ( θ 2 ) } ,
M = | J 1 ( m ) 2 [ ( sin ( 2 θ 0 2 θ 2 ) sin ( α ) + sin ( 2 θ 1 2 θ 2 ) cos ( α ) ] [ sin ( 2 θ 0 2 θ 1 + 2 θ 2 ) sin ( α ) + sin ( 2 θ 2 ) cos ( α ) ] i } | ,
M = | J 1 ( m ) 2 { [ sin ( 2 θ 0 2 θ 2 ) sin ( α ) + sin ( 2 θ 1 2 θ 2 ) cos ( α ) ] + [ sin ( 2 θ 0 2 θ 1 + 2 θ 2 ) sin ( α ) sin ( 2 θ 2 ) cos ( α ) ] i } | .
θ 1 = 1 2 arctan [ sin ( 4 θ 0 ) ( cos ( 2 α 0 ) 1 ) cos ( 2 α 0 ) cos ( 4 θ 0 ) 1 cos ( 4 θ 0 ) cos ( 2 α 0 ) ] ,
θ 2 = 1 2 arctan [ tan ( α 0 ) sin ( 2 θ 0 2 θ 1 ) tan ( α 0 ) cos ( 2 θ 0 2 θ 1 ) + 1 ] .
θ 2 = 1 2 arctan [ tan ( α 0 ) sin ( 2 θ 0 2 θ 1 ) tan ( α 0 ) cos ( 2 θ 0 2 θ 1 ) 1 ] .
θ 2 = arctan [ sin ( θ 1 ) cos ( θ 1 ) + 1 ]
θ 2 = arctan [ sin ( θ 1 ) + 1 cos ( θ 1 ) ]     ( where π 2 < θ 1 < π 2 ) .

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