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Calibration of the Glasgow 10 m prototype laser interferometric gravitational wave detector using photon pressure
7 May 2001
Physics Letters A 283 (2001) 85–88 www.elsevier.nl/locate/pla
Calibration of the Glasgow 10 m prototype laser interferometric gravitational wave detector using photon pressure D.A. Clubley, G.P. Newton, K.D. Skeldon ∗ , J. Hough Received 23 February 2001; accepted 26 March 2001 Communicated by P.R. Holland
Abstract We show that the radiation pressure from an amplitude modulated, low power, Nd:YAG laser can be used to calibrate the displacement sensitivity of the Glasgow 10 m prototype gravitational wave detector. This demonstrates the possibility of radiation pressure being used as a standard method of calibrating long base line gravitational wave detectors. Further, this technique has the very important additional advantage that the test mass acted upon by the radiation pressure is not altered in any way, by, for example, the attachment of magnets, etc. The high Q-factor of the internal modes, required for good detector sensitivity, is therefore preserved. 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Long baseline laser interferometric gravitational wave detectors are currently being constructed at a number of locations worldwide [1–4]. The calibration of the displacement sensitivity of a laser interferometric gravitational wave detector is generally performed by moving one of the interferometer mirrors by means of an external force, thus introducing a differential displacement between the arms [5]. This is usually achieved by attaching small magnets to the rear of the mirror mass, positioned so that they interact with the magnetic field generate by a nearby set of solenoids. This technique has the disadvantage that the attached magnets can significantly lower the internal Q-factor of the internal modes of the mirror mass [6] and so impair the performance of the gravitational wave detector. Therefore a technique that allows the force to act directly on the mirror mass without altering its in-
trinsic properties has a very significant advantage. The radiation pressure produced by flashlight was shown to produce an observable mirror displacement with the prototype gravitational wave detector at the California Institute of Technology [7]. We will demonstrate that modulated radiation pressure allows the application of a force that produces an oscillatory displacement large enough for useful calibration to be achieved, without compromising the performance of the detector.
2. The experiment The radiation pressure was supplied by a low power, Nd:YAG nonplanar ring oscillator (NPRO) 1 manufactured by Lightwave Electronics. The operation of the prototype gravitational wave detector can be understood with reference to Fig. 1. The detector measures differential length changes between the two Fabry– Perot cavities labeled primary and secondary. A fre-
* Corresponding author.
E-mail address: [email protected] (K.D. Skeldon).
1 Lightwave Electronics model-126.
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 2 3 1 - 6
86
D.A. Clubley et al. / Physics Letters A 283 (2001) 85–88
Fig. 1. A schematic of the Glasgow prototype gravitational wave detector. Frequency stabilisation schemes are simplified for clarity.
quency stabilised laser is locked to an axial mode of the primary cavity such that the cavity length defines the laser frequency. Forces applied to mirror M2 allow the length of the secondary cavity to be locked to the laser frequency. Any change in the length of the primary cavity changes the laser frequency and this change can be measured by monitoring the feedback signal to M2, used to hold the secondary cavity resonant with the laser light. To apply the radiation pressure we direct light from the low power Lightwave NPRO on to the mirror labeled M1 in Fig. 1. The light is reflected at the rear of the highly reflecting coating applied to the front face of the mirror. The momentum change of the mirror is therefore twice that of the incident laser beam. The laser power is amplitude modulated at a particular frequency so that any displacement of the mirror mass M1 resulting from the radiation pressure, will be seen as a peak at this frequency in the spectral density of displacement sensitivity of the gravitational wave detector. Since the test mass is suspended as a
pendulum, as described in [5], the expected response to the modulated laser light is readily estimated. The displacement of the mass is obtained from x¨ +
F (t) γ x˙ + ω02 x = , m m
(1)
where γ is the damping constant, ω0 is the fundamental frequency of the pendulum, m is the mass of the test mass and F (t) is the time dependent force resulting from the modulated radiation pressure. F (t) is then given by F (t) =
2P0 1 + sin(ωm t) , c
(2)
where P0 is the unmodulated laser carrier power, c is the velocity of light and ωm is the angular modulation 2 γ ω /m, the frequency. For ωm ω0 and ωm m displacement x(t) has the form x(t) =
2P0 2P0 + sin(ωm t − φ), 2 2 mcω0 mcωm
(3)
D.A. Clubley et al. / Physics Letters A 283 (2001) 85–88
87
where φ is a phase shift of ∼π . The rms value of the time dependent component of the displacement is therefore 2P0 xrms = √ . 2 2mcωm
(4)
3. Measurement and results The displacement sensitivity of the interferometer is derived from the control signal applied to M2. This control signal, when measured at point X in Fig. 1, is a measure of the differential length change between the two Fabry–Perot cavities. The previously used methods of calibrating the detector are described in detail in [5] and involve two independent techniques. The first method is to move the mirror M1 of the primary cavity by means of a coil and magnet arrangement. Small magnets bonded to the rear of the mirror and coils mounted rigidly to ground allow a variable frequency force to be applied to the mirror. Since the laser frequency is locked to an axial mode of the primary cavity, the resulting length change due to the applied force can be determined from the change in the laser frequency. Over the range of frequencies of the applied force the dominant actuator controlling the laser frequency is an acousto-optic modulator (AOM). Knowledge of the transfer function of the AOM allows a measurement of the feedback signal to the AOM which in turn leads to the displacement of M1. The spectral density of the signal required to hold the secondary cavity resonant with the laser will exhibit a peak at the frequency at which the primary cavity length is modulated. The change in length of the secondary cavity is then calculated from L/L = f /f . The second calibration method involves knowledge of the transfer function of mirror M2 to signals applied to the control coils mounted on the adjacent reaction pendulum. The transfer function has been modeled [8] such that knowledge of the magnitude of motion resulting from the application of a static force to the mirror allows the feedback signal at any frequency to be converted to a displacement. These calibration techniques have been shown to agree to within 1 dB. To calibrate the radiation pressure induced displacement,the spectral density of the displacement of mirror M2 was first recorded while the calibration sig-
Fig. 2. The feedback signal to the secondary cavity end mirror with calibration peaks applied. Spectra are for application of peak via coil/magnet arrangement at 339 Hz (labeled A) and via radiation pressure at 342 Hz (labeled B). This demonstrates the possibility of applying calibration signals by the two mechanisms and comparing the magnitude of the resulting mirror displacement.
Fig. 3. Calibrated motion of mirror resulting from radiation pressure at different amplitude modulation frequencies. Also shown is the expected response.
nal at a known frequency was applied to M1. A second spectrum of the displacement sensitivity was then taken with the radiation pressure applied but without the calibration signal. A sample of the results is shown in Fig. 2. These spectra show the feedback signal at X with mirror M1 moved by the coil/magnet arrangement and by the radiation pressure scheme. The relative height of the two peaks allows the motion resulting from radiation pressure to be calculated. We performed the calibration described above at a number of different modulation frequencies. The resulting calibrated mirror motion is shown in Fig. 3 at each frequency. Also shown is the predicted response
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D.A. Clubley et al. / Physics Letters A 283 (2001) 85–88
based on Eq. (4). The radiation pressure induced displacement can be seen to follow the theoretical prediction within measurement errors, demonstrating its validity as a method of calibrating a laser interferometric gravitational wave detectors.
and also from PPARC and the University of Glasgow. K.D.S. was funded by a Glasgow Science Centre grant and a Royal Society of Edinburgh BP/RSE industrial fellowship.
References 4. Conclusions We have shown that the expected displacement resulting from our analysis of the modulated radiation pressure is in excellent agreement with the measured displacement. This shows this technique to be a very promising method for calibrating the displacement sensitivity of laser interferometric gravitational wave detectors. A further possibility arises if the laser power can be significantly increased. The increased power together with re-reflecting the light onto the test mass could allow the radiation pressure applied motion to be used as part of the control scheme to position the mirror mass. We should also point out that both the LIGO and GEO projects are both planning to use radiation pressure for interferometer calibration purposes.
Acknowledgements The authors would like to thank our colleagues in the Institute for Gravitational Research at Glasgow University and, in particular, K.A. Strain and C.I Torrie for useful discussions. We also acknowledge those at GEO 600 for their support and interest in this work. Financial support came from a SHEFC/JREI grant
[1] B. Willke, P. Aufmuth, R. Balasubramanian, O.S. Brozek, G. Cagnoli, M. Casey, D. Clubley, D. Churches, K. Danzmann, C. Fallnich, A. Freise, S. Goßler, A. Grado, H. Grote, J. Hough, M. Husman, K. Kawabe, H. Luck, P. McNamara, S. McIntosh, K. Mossavi, G.P. Newton, D. Palmer, M.A. Papa, M.V. Plissi, V. Quetschke, D.I. Robertson, N.A. Robertson, S. Rowan, A. Rudiger, B.S. Sathyaprakash, R. Schilling, B.F. Schutz, A. Sintes-Olives, K.D. Skeldon, P. Sneddon, K.A. Strain, I. Taylor, C.I. Torrie, A. Vecchio, H. Ward, H. Welling, W. Winkler, I. Zawischa, C. Zhao, The GEO 600 gravitational wave detector, in: Gravitational Wave Detection II, Universal Academic Press, Tokyo, 2000, pp. 25–34. [2] A. Lazzarini, The status of LIGO, in: Gravitational Wave Detection II, Universal Academic Press, Tokyo, 2000, pp. 1–13. [3] E. Majorana, VIRGO collaboration, Status of VIRGO, in: Gravitational Wave Detection II, Universal Academic Press, Tokyo, 2000, pp. 15–24. [4] M. Fujimoto, TAMA collaboration, An overview of the TAMA project, in: Gravitational Wave Detection II, Universal Academic Press, Tokyo, 2000, pp. 41–43. [5] K.D. Skeldon, D.A. Clubley, B.W. Barr, M.M. Casey, J. Hough, S.D. Killbourn, P.W. McNamara, G.P. Newton, M.V. Plissi, D.I. Robertson, N.A. Robertson, K.A. Strain, H. Ward, Phys. Lett. A 273 (2000) 277. [6] A.D. Gillespie, Ph.D. thesis, California Institute of Technology (1995). [7] S. Whitcomb, Private communication (2001). [8] Ph.D. thesis, University of Glasgow (1998).
Physics Letters A 283 (2001) 85–88 www.elsevier.nl/locate/pla
Calibration of the Glasgow 10 m prototype laser interferometric gravitational wave detector using photon pressure D.A. Clubley, G.P. Newton, K.D. Skeldon ∗ , J. Hough Received 23 February 2001; accepted 26 March 2001 Communicated by P.R. Holland
Abstract We show that the radiation pressure from an amplitude modulated, low power, Nd:YAG laser can be used to calibrate the displacement sensitivity of the Glasgow 10 m prototype gravitational wave detector. This demonstrates the possibility of radiation pressure being used as a standard method of calibrating long base line gravitational wave detectors. Further, this technique has the very important additional advantage that the test mass acted upon by the radiation pressure is not altered in any way, by, for example, the attachment of magnets, etc. The high Q-factor of the internal modes, required for good detector sensitivity, is therefore preserved. 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Long baseline laser interferometric gravitational wave detectors are currently being constructed at a number of locations worldwide [1–4]. The calibration of the displacement sensitivity of a laser interferometric gravitational wave detector is generally performed by moving one of the interferometer mirrors by means of an external force, thus introducing a differential displacement between the arms [5]. This is usually achieved by attaching small magnets to the rear of the mirror mass, positioned so that they interact with the magnetic field generate by a nearby set of solenoids. This technique has the disadvantage that the attached magnets can significantly lower the internal Q-factor of the internal modes of the mirror mass [6] and so impair the performance of the gravitational wave detector. Therefore a technique that allows the force to act directly on the mirror mass without altering its in-
trinsic properties has a very significant advantage. The radiation pressure produced by flashlight was shown to produce an observable mirror displacement with the prototype gravitational wave detector at the California Institute of Technology [7]. We will demonstrate that modulated radiation pressure allows the application of a force that produces an oscillatory displacement large enough for useful calibration to be achieved, without compromising the performance of the detector.
2. The experiment The radiation pressure was supplied by a low power, Nd:YAG nonplanar ring oscillator (NPRO) 1 manufactured by Lightwave Electronics. The operation of the prototype gravitational wave detector can be understood with reference to Fig. 1. The detector measures differential length changes between the two Fabry– Perot cavities labeled primary and secondary. A fre-
* Corresponding author.
E-mail address: [email protected] (K.D. Skeldon).
1 Lightwave Electronics model-126.
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 2 3 1 - 6
86
D.A. Clubley et al. / Physics Letters A 283 (2001) 85–88
Fig. 1. A schematic of the Glasgow prototype gravitational wave detector. Frequency stabilisation schemes are simplified for clarity.
quency stabilised laser is locked to an axial mode of the primary cavity such that the cavity length defines the laser frequency. Forces applied to mirror M2 allow the length of the secondary cavity to be locked to the laser frequency. Any change in the length of the primary cavity changes the laser frequency and this change can be measured by monitoring the feedback signal to M2, used to hold the secondary cavity resonant with the laser light. To apply the radiation pressure we direct light from the low power Lightwave NPRO on to the mirror labeled M1 in Fig. 1. The light is reflected at the rear of the highly reflecting coating applied to the front face of the mirror. The momentum change of the mirror is therefore twice that of the incident laser beam. The laser power is amplitude modulated at a particular frequency so that any displacement of the mirror mass M1 resulting from the radiation pressure, will be seen as a peak at this frequency in the spectral density of displacement sensitivity of the gravitational wave detector. Since the test mass is suspended as a
pendulum, as described in [5], the expected response to the modulated laser light is readily estimated. The displacement of the mass is obtained from x¨ +
F (t) γ x˙ + ω02 x = , m m
(1)
where γ is the damping constant, ω0 is the fundamental frequency of the pendulum, m is the mass of the test mass and F (t) is the time dependent force resulting from the modulated radiation pressure. F (t) is then given by F (t) =
2P0 1 + sin(ωm t) , c
(2)
where P0 is the unmodulated laser carrier power, c is the velocity of light and ωm is the angular modulation 2 γ ω /m, the frequency. For ωm ω0 and ωm m displacement x(t) has the form x(t) =
2P0 2P0 + sin(ωm t − φ), 2 2 mcω0 mcωm
(3)
D.A. Clubley et al. / Physics Letters A 283 (2001) 85–88
87
where φ is a phase shift of ∼π . The rms value of the time dependent component of the displacement is therefore 2P0 xrms = √ . 2 2mcωm
(4)
3. Measurement and results The displacement sensitivity of the interferometer is derived from the control signal applied to M2. This control signal, when measured at point X in Fig. 1, is a measure of the differential length change between the two Fabry–Perot cavities. The previously used methods of calibrating the detector are described in detail in [5] and involve two independent techniques. The first method is to move the mirror M1 of the primary cavity by means of a coil and magnet arrangement. Small magnets bonded to the rear of the mirror and coils mounted rigidly to ground allow a variable frequency force to be applied to the mirror. Since the laser frequency is locked to an axial mode of the primary cavity, the resulting length change due to the applied force can be determined from the change in the laser frequency. Over the range of frequencies of the applied force the dominant actuator controlling the laser frequency is an acousto-optic modulator (AOM). Knowledge of the transfer function of the AOM allows a measurement of the feedback signal to the AOM which in turn leads to the displacement of M1. The spectral density of the signal required to hold the secondary cavity resonant with the laser will exhibit a peak at the frequency at which the primary cavity length is modulated. The change in length of the secondary cavity is then calculated from L/L = f /f . The second calibration method involves knowledge of the transfer function of mirror M2 to signals applied to the control coils mounted on the adjacent reaction pendulum. The transfer function has been modeled [8] such that knowledge of the magnitude of motion resulting from the application of a static force to the mirror allows the feedback signal at any frequency to be converted to a displacement. These calibration techniques have been shown to agree to within 1 dB. To calibrate the radiation pressure induced displacement,the spectral density of the displacement of mirror M2 was first recorded while the calibration sig-
Fig. 2. The feedback signal to the secondary cavity end mirror with calibration peaks applied. Spectra are for application of peak via coil/magnet arrangement at 339 Hz (labeled A) and via radiation pressure at 342 Hz (labeled B). This demonstrates the possibility of applying calibration signals by the two mechanisms and comparing the magnitude of the resulting mirror displacement.
Fig. 3. Calibrated motion of mirror resulting from radiation pressure at different amplitude modulation frequencies. Also shown is the expected response.
nal at a known frequency was applied to M1. A second spectrum of the displacement sensitivity was then taken with the radiation pressure applied but without the calibration signal. A sample of the results is shown in Fig. 2. These spectra show the feedback signal at X with mirror M1 moved by the coil/magnet arrangement and by the radiation pressure scheme. The relative height of the two peaks allows the motion resulting from radiation pressure to be calculated. We performed the calibration described above at a number of different modulation frequencies. The resulting calibrated mirror motion is shown in Fig. 3 at each frequency. Also shown is the predicted response
88
D.A. Clubley et al. / Physics Letters A 283 (2001) 85–88
based on Eq. (4). The radiation pressure induced displacement can be seen to follow the theoretical prediction within measurement errors, demonstrating its validity as a method of calibrating a laser interferometric gravitational wave detectors.
and also from PPARC and the University of Glasgow. K.D.S. was funded by a Glasgow Science Centre grant and a Royal Society of Edinburgh BP/RSE industrial fellowship.
References 4. Conclusions We have shown that the expected displacement resulting from our analysis of the modulated radiation pressure is in excellent agreement with the measured displacement. This shows this technique to be a very promising method for calibrating the displacement sensitivity of laser interferometric gravitational wave detectors. A further possibility arises if the laser power can be significantly increased. The increased power together with re-reflecting the light onto the test mass could allow the radiation pressure applied motion to be used as part of the control scheme to position the mirror mass. We should also point out that both the LIGO and GEO projects are both planning to use radiation pressure for interferometer calibration purposes.
Acknowledgements The authors would like to thank our colleagues in the Institute for Gravitational Research at Glasgow University and, in particular, K.A. Strain and C.I Torrie for useful discussions. We also acknowledge those at GEO 600 for their support and interest in this work. Financial support came from a SHEFC/JREI grant
[1] B. Willke, P. Aufmuth, R. Balasubramanian, O.S. Brozek, G. Cagnoli, M. Casey, D. Clubley, D. Churches, K. Danzmann, C. Fallnich, A. Freise, S. Goßler, A. Grado, H. Grote, J. Hough, M. Husman, K. Kawabe, H. Luck, P. McNamara, S. McIntosh, K. Mossavi, G.P. Newton, D. Palmer, M.A. Papa, M.V. Plissi, V. Quetschke, D.I. Robertson, N.A. Robertson, S. Rowan, A. Rudiger, B.S. Sathyaprakash, R. Schilling, B.F. Schutz, A. Sintes-Olives, K.D. Skeldon, P. Sneddon, K.A. Strain, I. Taylor, C.I. Torrie, A. Vecchio, H. Ward, H. Welling, W. Winkler, I. Zawischa, C. Zhao, The GEO 600 gravitational wave detector, in: Gravitational Wave Detection II, Universal Academic Press, Tokyo, 2000, pp. 25–34. [2] A. Lazzarini, The status of LIGO, in: Gravitational Wave Detection II, Universal Academic Press, Tokyo, 2000, pp. 1–13. [3] E. Majorana, VIRGO collaboration, Status of VIRGO, in: Gravitational Wave Detection II, Universal Academic Press, Tokyo, 2000, pp. 15–24. [4] M. Fujimoto, TAMA collaboration, An overview of the TAMA project, in: Gravitational Wave Detection II, Universal Academic Press, Tokyo, 2000, pp. 41–43. [5] K.D. Skeldon, D.A. Clubley, B.W. Barr, M.M. Casey, J. Hough, S.D. Killbourn, P.W. McNamara, G.P. Newton, M.V. Plissi, D.I. Robertson, N.A. Robertson, K.A. Strain, H. Ward, Phys. Lett. A 273 (2000) 277. [6] A.D. Gillespie, Ph.D. thesis, California Institute of Technology (1995). [7] S. Whitcomb, Private communication (2001). [8] Ph.D. thesis, University of Glasgow (1998).