Abstract

We have designed, built and tested a high-performance phase camera, which can observe laser wavefronts in a large range of sideband frequencies. Our phase camera scans the laser beam over a pinhole diode and uses a heterodyne technique to independently assess the information in the upper and lower sidebands of up to five different modulation frequencies. Amplitude and phase images, consisting of 214 points each, are obtained every second for each of the 11 demodulated frequencies in parallel. The achieved sensitivity is about 4×103 rad (λ/1600 at λ = 1064 nm) at the center of the beam, corresponding to a wavefront deformation of 0.7 nm, and drops to about 3 nm over the beam size. This sensitivity is extremely useful for diagnostic purposes in gravitational wave detectors and fits the requirements for control loops in Advanced Virgo. We report on the design, realization and performance of our phase camera.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  10. A. M. Gretarsson, E. D’Ambrosio, V. Frolov, B. O’Reilly, and P. K. Fritschel, “Effects of mode degeneracy in the LIGO Livingston Observatory recycling cavity,” J. Opt. Soc. Am. B 24(11), 2821–2828 (2007).
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    [Crossref]
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  19. S. Bigotta, “phase camera 1 status and measurements,” Virgo internal working note (VIR-125A–09) (2009).
  20. A. Di Lieto, “phase camera,” Virgo internal working note (VIR-230A–09) (2009).
  21. B. Swinkels, “Phase Camera 1 signals,” Virgo internal working note (VIR-0300A–09) (2009).
  22. J. Marque, “Phase Camera Images & SSFS TF,” Virgo internal working note (VIR-0354A–09) (2009).
  23. R. Day, “Experience with the Virgo phase camera and ideas for AdV,” Virgo internal working note (VIR-0442A–09) (2009).
  24. R. Day, “Using the phase camera in advanced interferometers,” presented at Gravitational-wave Advanced Detector Workshop 2013 (VIR-0310A–13), Elba, Italy, 19-25 May 2013.
  25. R. Day, E. Genin, and A. Rocchi, “Phase Camera requirements from INJ and TCS Subsystems,” Virgo internal working note (VIR-0258A–12) (2012).
  26. R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).
  27. M. van Beuzekom, J. van den Brand, and D. S. Rabeling, “ADC and DAQ Constraints for Digital Demodulation for the Advanced Virgo Phase Camera Virgo Technical Documentation System,” Virgo internal working note (VIR-0304A–12) (2012).
  28. K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.
  29. K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
    [Crossref]
  30. L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).
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    [Crossref]
  32. International Bureau of Weights and Measures, “Guide to the Expression of Uncertainty in Measurement,” https://www.bipm.org/en/publications/guides/gum.html (1995).
  33. R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
    [Crossref]
  34. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing — 2nd ed.(Prentice Hall, 1998), pp. 204–205.

2017 (1)

LIGO Scientific Collaboration and Virgo Collaboration, “GW170814: A Three-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence,” Phys. Rev. Lett. 119, 141101 (2017).
[Crossref] [PubMed]

2016 (3)

LIGO Scientific Collaboration and Virgo Collaboration, “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref] [PubMed]

K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
[Crossref]

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

2015 (1)

The Virgo Collaboration, “Advanced Virgo: a second-generation interferometric gravitational wave detector,” Class. Quantum Grav. 32, 024001 (2015).
[Crossref]

2014 (1)

2012 (1)

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

2007 (2)

2006 (1)

R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

2004 (1)

1994 (2)

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Adhikari, R.

Adhikari, R. X.

Agatsuma, K.

K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
[Crossref]

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.

Arai, K.

Betzwieser, J.

J. Betzwieser, K. Kawabe, and L. Matone, “Study of the Ouput Mode Cleaner Prototype using the Phasecamera,” LIGO internal working note (LIGO-T040156–00-D) (2004).

J. Betzwieser, “Analysis of spatial mode sensitivity of a gravitational wave interferometer and a targeted search for gravitational radiation from the Crab pulsar,” PhD thesis, Massachusetts Institute of Technology (2007).

Bigotta, S.

S. Bigotta, “Development of a frequency resolving wavefront detector (Phase camera),” Virgo internal working note (VIR-0049A–10) (2006).

S. Bigotta, “phase camera 1 status and measurements,” Virgo internal working note (VIR-125A–09) (2009).

Brooks, A. F.

Buck, J. R.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing — 2nd ed.(Prentice Hall, 1998), pp. 204–205.

Coccia, E.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

Connelly, B.

D’Ambrosio, E.

Day, R.

R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).

R. Day, “Experience with the Virgo phase camera and ideas for AdV,” Virgo internal working note (VIR-0442A–09) (2009).

R. Day, “Using the phase camera in advanced interferometers,” presented at Gravitational-wave Advanced Detector Workshop 2013 (VIR-0310A–13), Elba, Italy, 19-25 May 2013.

R. Day, E. Genin, and A. Rocchi, “Phase Camera requirements from INJ and TCS Subsystems,” Virgo internal working note (VIR-0258A–12) (2012).

R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).

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 Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Fafone, V.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Fritschel, P. K.

Frolov, V.

Gebyehu, M.

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

Genin, E.

R. Day, E. Genin, and A. Rocchi, “Phase Camera requirements from INJ and TCS Subsystems,” Virgo internal working note (VIR-0258A–12) (2012).

Goda, K.

Gretarsson, A.

Gretarsson, A. M.

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Izumi, K.

Kawabe, K.

J. Betzwieser, K. Kawabe, and L. Matone, “Study of the Ouput Mode Cleaner Prototype using the Phasecamera,” LIGO internal working note (LIGO-T040156–00-D) (2004).

Kelly, T.-L.

Kokeyama, K.

Korth, W. Z.

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Lieto, A. Di

A. Di Lieto, “phase camera,” Virgo internal working note (VIR-230A–09) (2009).

Malvezzi, V.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

Marque, J.

J. Marque, “Phase Camera Images & SSFS TF,” Virgo internal working note (VIR-0354A–09) (2009).

Matone, L.

J. Betzwieser, K. Kawabe, and L. Matone, “Study of the Ouput Mode Cleaner Prototype using the Phasecamera,” LIGO internal working note (LIGO-T040156–00-D) (2004).

Mavalvala, N.

Meers, B. J.

Minenkov, Y.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

Morrison, E.

Munch, J.

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

O’Reilly, B.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing — 2nd ed.(Prentice Hall, 1998), pp. 204–205.

Ottaway, D.

Rabeling, D.

K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.

Rabeling, D. S.

M. van Beuzekom, J. van den Brand, and D. S. Rabeling, “ADC and DAQ Constraints for Digital Demodulation for the Advanced Virgo Phase Camera Virgo Technical Documentation System,” Virgo internal working note (VIR-0304A–12) (2012).

Robertson, D. l.

Rocchi, A.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

R. Day, E. Genin, and A. Rocchi, “Phase Camera requirements from INJ and TCS Subsystems,” Virgo internal working note (VIR-0258A–12) (2012).

Saleh, B.E.A.

B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics(John Wiley & Sons, 1991).
[Crossref]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing — 2nd ed.(Prentice Hall, 1998), pp. 204–205.

Sleeman, R.

R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

Smith-Lefebvre, N.

Sperandio, L.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

Swinkels, B.

B. Swinkels, “Phase Camera 1 signals,” Virgo internal working note (VIR-0300A–09) (2009).

Teich, M.C.

B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics(John Wiley & Sons, 1991).
[Crossref]

Trampert, J.

R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

van Beuzekom, M.

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
[Crossref]

M. van Beuzekom, J. van den Brand, and D. S. Rabeling, “ADC and DAQ Constraints for Digital Demodulation for the Advanced Virgo Phase Camera Virgo Technical Documentation System,” Virgo internal working note (VIR-0304A–12) (2012).

K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.

van den Brand, J.

K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
[Crossref]

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.

M. van Beuzekom, J. van den Brand, and D. S. Rabeling, “ADC and DAQ Constraints for Digital Demodulation for the Advanced Virgo Phase Camera Virgo Technical Documentation System,” Virgo internal working note (VIR-0304A–12) (2012).

van der Schaaf, L.

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
[Crossref]

van Wettum, A.

R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

Veitch, P. J.

Ward, H.

Wu, W.

W. Wu, “Instrumentation of the next generation gravitational wave detector: triple pendulum suspension and electro-optic modulator,” Ph.D. thesis, University of Florida (2007).

Appl. Opt. (2)

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

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Bull. Seismol. Soc. Am. (1)

R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

Class. Quantum Grav. (1)

The Virgo Collaboration, “Advanced Virgo: a second-generation interferometric gravitational wave detector,” Class. Quantum Grav. 32, 024001 (2015).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Phys.: Conf. Ser. (2)

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

Nucl. Instrum. Methods Phys. Res., Sect. A (1)

K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. Lett. (2)

LIGO Scientific Collaboration and Virgo Collaboration, “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref] [PubMed]

LIGO Scientific Collaboration and Virgo Collaboration, “GW170814: A Three-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence,” Phys. Rev. Lett. 119, 141101 (2017).
[Crossref] [PubMed]

Other (20)

W. Wu, “Instrumentation of the next generation gravitational wave detector: triple pendulum suspension and electro-optic modulator,” Ph.D. thesis, University of Florida (2007).

S. Bigotta, “Development of a frequency resolving wavefront detector (Phase camera),” Virgo internal working note (VIR-0049A–10) (2006).

S. Bigotta, “phase camera 1 status and measurements,” Virgo internal working note (VIR-125A–09) (2009).

A. Di Lieto, “phase camera,” Virgo internal working note (VIR-230A–09) (2009).

B. Swinkels, “Phase Camera 1 signals,” Virgo internal working note (VIR-0300A–09) (2009).

J. Marque, “Phase Camera Images & SSFS TF,” Virgo internal working note (VIR-0354A–09) (2009).

R. Day, “Experience with the Virgo phase camera and ideas for AdV,” Virgo internal working note (VIR-0442A–09) (2009).

R. Day, “Using the phase camera in advanced interferometers,” presented at Gravitational-wave Advanced Detector Workshop 2013 (VIR-0310A–13), Elba, Italy, 19-25 May 2013.

R. Day, E. Genin, and A. Rocchi, “Phase Camera requirements from INJ and TCS Subsystems,” Virgo internal working note (VIR-0258A–12) (2012).

R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).

M. van Beuzekom, J. van den Brand, and D. S. Rabeling, “ADC and DAQ Constraints for Digital Demodulation for the Advanced Virgo Phase Camera Virgo Technical Documentation System,” Virgo internal working note (VIR-0304A–12) (2012).

K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing — 2nd ed.(Prentice Hall, 1998), pp. 204–205.

B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics(John Wiley & Sons, 1991).
[Crossref]

International Bureau of Weights and Measures, “Guide to the Expression of Uncertainty in Measurement,” https://www.bipm.org/en/publications/guides/gum.html (1995).

J. Betzwieser, K. Kawabe, and L. Matone, “Study of the Ouput Mode Cleaner Prototype using the Phasecamera,” LIGO internal working note (LIGO-T040156–00-D) (2004).

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The Virgo Collaboration, “Advanced Virgo Technical Design Report,” Virgo internal working note (VIR-0128A–12) (2012).

R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).

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Figures (7)

Fig. 1
Fig. 1 Principle of the phase camera.
Fig. 2
Fig. 2 Schematic view of configuration. (a) One-beam scanning, (b) two-beam scanning, and (c) definition of angles.
Fig. 3
Fig. 3 Visibility reduction due to averaging the spatial fringe over the PD active area. In the case of Λ < dPD or even approaching Λ ≈ dPD, the maximum value of AC signal is reduced.
Fig. 4
Fig. 4 Setup of the prototype. ADC is the Analogue Digital Converter, and Hann* shows the Hann window function.
Fig. 5
Fig. 5 Frequency response of the piezo scanner (x direction). A value for the transfer function TFR is obtained from this plot. There are resonances of piezo elements above 2 kHz.
Fig. 6
Fig. 6 Amplitude and phase maps. The top row of plots shows amplitude maps and bottom row shows phase maps. From left to right the columns are USB (upper sideband), carrier (Heterodyne signal at 80 MHz), and LSB (lower sideband).
Fig. 7
Fig. 7 Residual phase of the relative measurement (absolute value). Before the subtraction process, DC offsets were removed and the calculation phases are unwrapped. Left: Sensitivity at the cross-section Y = 0. The residual phase (Δϕ = ϕcϕUSB) is converted to the displacement (zdisp) using |Δϕ| = kzdisp. The spikes in the measurement are reproduced by a noise correlation between I and Q signals. Right: Residual phase between LSB and USB (Δϕ = ϕLSBϕUSB).

Tables (2)

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Table 1 Pros and cons of the two scanning configurations.

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Table 2 Test of PD output.

Equations (22)

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E j ( x , y , z , t ) = U j e ^ j Ψ m n j ( x , y , z ) exp  ( i ω j t ) ,
Ψ m n j ( x , y , z ) = A m n j w j ( z ) H m ( 2 x w j ( z ) ) H n ( 2 y w j ( z ) ) × exp   [ x 2 + y 2 w j 2 ( z ) i k ( x 2 + y 2 ) 2 R j ( z ) i ( k z ζ m n j ( z ) ) ] .
ζ m n j ( z ) = ( m + n + 1 ) arctan  ( λ z π w 0 2 ) .
( O 1 O 2 ) = ( r BS t BS t BS r BS ) ( E c E h ) .
O 2 = t BS E c ( x , y , z , t ) + r BS E h ( x , y , z , t ) = U 0 exp   ( i ω c t ) { e ^ c Ψ m n c ( x , y , z ) + e ^ h α Ψ m n h ( x , y , z ) exp   ( i ω h t ) }
| O 2 | 2 = P tot e ^ c e ^ h { 1 + ν Re cos   ( ω h t ) + ν Im sin   ( ω h t ) } P tot e ^ c e ^ h + P PD _ AC ,
I p = P PD _ AC × G DAQ cos  ( ω h t ) low pass P tot G DAQ ν Re / 2
Q p = P PD _ AC × G DAQ sin  ( ω h t ) low pass P tot G DAQ ν Im / 2
A = I p 2 + Q p 2 = P tot G DAQ ν Re 2 + ν Im 2 / 2 = P tot G DAQ | ν | / 2 ,
ϕ = arctan  ( Q p / I p ) = arctan  ( ν Im / ν Re ) .
Ψ 00 j ( x , y , z ) = 2 π 1 w j exp   [ x 2 + y 2 w j 2 i k ( x 2 + y 2 ) 2 R j i ( k z ζ 00 j ) ] = A 00 j exp   [ i ( k ( x 2 + y 2 ) 2 R j k z + ζ 00 j ) ] .
ϕ 00 = arctan  ( Im [ Ψ 00 c Ψ 00 h * ] Re [ Ψ 00 c Ψ 00 h * ] ) = Δ R Δ G + Δ L ,
E test ( x , y , z , t ) = U t e ^ t exp  ( i ω c t ) { Ψ m n c ( x , y , z ) ( 1 s δ s 2 4 ) + s δ s 2 [ Ψ m n + s ( x , y , z ) exp   ( i ω s t ) Ψ m n s ( x , y , z ) exp   ( i ω s t ) ] } .
Λ = λ 2 s i n   θ .
θ < λ 4 d PD .
η = 1 ϵ | ϵ / 2 ϵ / 2 cos  x d x | .
η = 4 d PD | 0 d PD / 2 1 ϵ ϵ / 2 ϵ / 2 cos  x d x d y | , ϵ = 4 π Λ ( d P D 2 ) 2 y 2 .
d m 2 θ Smax T FR cos  θ in < L SP < d m 20 N pixel Δ θ s
V output = P PD _ AC R λ G TIA = P tot R λ G TIA { ν Re cos   ( ω h t ) + ν Im sin   ( ω h t ) } .
P tot = π ( d PD 2 ) 2 j 2 P j π w j 2 exp  [ 2 ( x 2 + y 2 ) w j 2 ]
ϕ pN = { 2 N total / I p 2 + Q p 2 = 2 N total / A p ( no correlation ) 2 N total ( I p Q p ) / A p 2 ( with correlation )
( N s ' N s ) /   [ ( δ s 0.21 ) ( w t 833 μ m ) 1 ( w r 1530 μ m ) 1 ( P t 4.6 mW ) ( P r 9.5 mW ) ] 1