Abstract

We report on the analysis and prototype characterization of a dual-electrode electro-optic modulator that can generate both amplitude and phase modulations with a selectable relative phase, termed a quadrature variable modulator (QVM). All modulation states can be reached by tuning only the electrical inputs, facilitating real-time tuning, and the device has shown good suppression and stability properties. A mathematical analysis is presented, including the development of a geometric-phase representation for modulation. The experimental characterization of the device shows that relative suppressions of 38, 39, and 30 dB for phase, single sideband, and carrier-suppressed modulations, respectively, can be obtained as well as that the device is well behaved when scanning continuously through the parameter space of modulations. The QVM is compared with existing optical configurations that can produce amplitude and phase-modulation combinations in the context of applications such as the tuning of lock points in optical-locking schemes, single-sideband applications, modulation fast-switching applications, and applications requiring combined modulations.

© 2004 Optical Society of America

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  1. B. Davies, J. Conradi, “Hybrid modulator structures for subcarrier and harmonic subcarrier optical single sideband,” IEEE Photonics Technol. Lett. 10, 600–602 (1998).
    [CrossRef]
  2. See, for example, JDS Uniphase, www.jdsu.com .
  3. G. Smith, D. Novak, Z. Ahmed, “Technique for optical SSB generation to overcome dispersion penalties in fiber-radio systems,” Electron. Lett. 33, 74–75 (1997).
    [CrossRef]
  4. G. G. Stokes, Mathematical and Physical Papers, 5 vols. (Cambridge U. Press, Cambridge, UK, 1880–1985).
  5. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  6. L. Schnupp, conference presentation at the European Collaboration Meeting on Interferometric Detection of Gravitational Waves, Sorrento, Italy, 1988.
  7. LIGO Scientific Collaboration, “Advanced LIGO systems design,” LIGO Tech. Note T010075, P. Fritschel, ed. (LIGO Laboratory, Pasadena, Calif., 2001), http://antares.ligo.caltech.edu/dcc/default.htf .
  8. J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors,” Phys. Lett. A 175, 273–276 (1993).
    [CrossRef]
  9. K. A. Strain, J. Hough, “Experimental demonstration of the use of a Fabry-Perot cavity as a mirror of variable reflectivity,” Rev. Sci. Instrum. 64, 799–802 (1994).
    [CrossRef]
  10. G. de Vine, D. A. Shaddock, D. E. McClelland, “Experimental demonstration of variable reflectivity signal recycling for interferometric gravitational wave detectors,” Opt. Lett. 27, 1507–1509 (2002).
    [CrossRef]
  11. A. Loayssa, C. Lim, A. Nirmalathas, D. Benito, “Simple optical single-sideband modulator for fibre-radio applications,” Electron. Lett. 39, 97–99 (2003).
    [CrossRef]
  12. K. Yonenaga, N. Takachio, “A fiber chromatic dispersion compensation technique with an optical SSB transmission in optical homodyne detection systems,” IEEE Photonics Technol. Lett. 5, 949–951 (1993).
    [CrossRef]
  13. A. Loayssa, D. Benito, M. J. Garde, “Single-sideband suppressed-carrier modulation using a single-electrode electro-optic modulator,” IEEE Photonics Technol. Lett. 13, 869–971 (2001).
    [CrossRef]
  14. S. Shimotsu, S. Oikawa, T. Saitou, N. Mitsugi, K. Kubodera, T. Kawanishi, M. Izutsu, “Single sideband modulation performance of a LiNbO3 integrated modulator consisting of four-phase modulator waveguides,” IEEE Photonics Technol. Lett. 13, 364–366 (2001).
    [CrossRef]
  15. F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, Ph. Grangier, “Quantum key distribution using Gaussian-modulated coherent states,” Nature (London) 421, 238–241 (2003).
    [CrossRef]
  16. W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 32302-1-4 (2003).
    [CrossRef]
  17. A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
    [CrossRef] [PubMed]

2003 (3)

A. Loayssa, C. Lim, A. Nirmalathas, D. Benito, “Simple optical single-sideband modulator for fibre-radio applications,” Electron. Lett. 39, 97–99 (2003).
[CrossRef]

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, Ph. Grangier, “Quantum key distribution using Gaussian-modulated coherent states,” Nature (London) 421, 238–241 (2003).
[CrossRef]

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 32302-1-4 (2003).
[CrossRef]

2002 (1)

2001 (2)

A. Loayssa, D. Benito, M. J. Garde, “Single-sideband suppressed-carrier modulation using a single-electrode electro-optic modulator,” IEEE Photonics Technol. Lett. 13, 869–971 (2001).
[CrossRef]

S. Shimotsu, S. Oikawa, T. Saitou, N. Mitsugi, K. Kubodera, T. Kawanishi, M. Izutsu, “Single sideband modulation performance of a LiNbO3 integrated modulator consisting of four-phase modulator waveguides,” IEEE Photonics Technol. Lett. 13, 364–366 (2001).
[CrossRef]

1998 (2)

B. Davies, J. Conradi, “Hybrid modulator structures for subcarrier and harmonic subcarrier optical single sideband,” IEEE Photonics Technol. Lett. 10, 600–602 (1998).
[CrossRef]

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

1997 (1)

G. Smith, D. Novak, Z. Ahmed, “Technique for optical SSB generation to overcome dispersion penalties in fiber-radio systems,” Electron. Lett. 33, 74–75 (1997).
[CrossRef]

1994 (1)

K. A. Strain, J. Hough, “Experimental demonstration of the use of a Fabry-Perot cavity as a mirror of variable reflectivity,” Rev. Sci. Instrum. 64, 799–802 (1994).
[CrossRef]

1993 (2)

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors,” Phys. Lett. A 175, 273–276 (1993).
[CrossRef]

K. Yonenaga, N. Takachio, “A fiber chromatic dispersion compensation technique with an optical SSB transmission in optical homodyne detection systems,” IEEE Photonics Technol. Lett. 5, 949–951 (1993).
[CrossRef]

1983 (1)

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

Ahmed, Z.

G. Smith, D. Novak, Z. Ahmed, “Technique for optical SSB generation to overcome dispersion penalties in fiber-radio systems,” Electron. Lett. 33, 74–75 (1997).
[CrossRef]

Assche, G. V.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, Ph. Grangier, “Quantum key distribution using Gaussian-modulated coherent states,” Nature (London) 421, 238–241 (2003).
[CrossRef]

Bachor, H.-A.

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 32302-1-4 (2003).
[CrossRef]

Benito, D.

A. Loayssa, C. Lim, A. Nirmalathas, D. Benito, “Simple optical single-sideband modulator for fibre-radio applications,” Electron. Lett. 39, 97–99 (2003).
[CrossRef]

A. Loayssa, D. Benito, M. J. Garde, “Single-sideband suppressed-carrier modulation using a single-electrode electro-optic modulator,” IEEE Photonics Technol. Lett. 13, 869–971 (2001).
[CrossRef]

Bowen, W. P.

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 32302-1-4 (2003).
[CrossRef]

Braunstein, S. L.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Brouri, R.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, Ph. Grangier, “Quantum key distribution using Gaussian-modulated coherent states,” Nature (London) 421, 238–241 (2003).
[CrossRef]

Buchler, B. C.

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 32302-1-4 (2003).
[CrossRef]

Cerf, N. J.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, Ph. Grangier, “Quantum key distribution using Gaussian-modulated coherent states,” Nature (London) 421, 238–241 (2003).
[CrossRef]

Chen, J. M.

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors,” Phys. Lett. A 175, 273–276 (1993).
[CrossRef]

Conradi, J.

B. Davies, J. Conradi, “Hybrid modulator structures for subcarrier and harmonic subcarrier optical single sideband,” IEEE Photonics Technol. Lett. 10, 600–602 (1998).
[CrossRef]

Danzmann, K.

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors,” Phys. Lett. A 175, 273–276 (1993).
[CrossRef]

Davies, B.

B. Davies, J. Conradi, “Hybrid modulator structures for subcarrier and harmonic subcarrier optical single sideband,” IEEE Photonics Technol. Lett. 10, 600–602 (1998).
[CrossRef]

de Vine, G.

Drever, R. W. P.

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

Ford, G. M.

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

Fuchs, C. A.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Furusawa, A.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Garde, M. J.

A. Loayssa, D. Benito, M. J. Garde, “Single-sideband suppressed-carrier modulation using a single-electrode electro-optic modulator,” IEEE Photonics Technol. Lett. 13, 869–971 (2001).
[CrossRef]

Grangier, Ph.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, Ph. Grangier, “Quantum key distribution using Gaussian-modulated coherent states,” Nature (London) 421, 238–241 (2003).
[CrossRef]

Grosshans, F.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, Ph. Grangier, “Quantum key distribution using Gaussian-modulated coherent states,” Nature (London) 421, 238–241 (2003).
[CrossRef]

Hall, J. L.

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

Hough, J.

K. A. Strain, J. Hough, “Experimental demonstration of the use of a Fabry-Perot cavity as a mirror of variable reflectivity,” Rev. Sci. Instrum. 64, 799–802 (1994).
[CrossRef]

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

Izutsu, M.

S. Shimotsu, S. Oikawa, T. Saitou, N. Mitsugi, K. Kubodera, T. Kawanishi, M. Izutsu, “Single sideband modulation performance of a LiNbO3 integrated modulator consisting of four-phase modulator waveguides,” IEEE Photonics Technol. Lett. 13, 364–366 (2001).
[CrossRef]

Kawanishi, T.

S. Shimotsu, S. Oikawa, T. Saitou, N. Mitsugi, K. Kubodera, T. Kawanishi, M. Izutsu, “Single sideband modulation performance of a LiNbO3 integrated modulator consisting of four-phase modulator waveguides,” IEEE Photonics Technol. Lett. 13, 364–366 (2001).
[CrossRef]

Kimble, H. J.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Kowalski, F. V.

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

Kubodera, K.

S. Shimotsu, S. Oikawa, T. Saitou, N. Mitsugi, K. Kubodera, T. Kawanishi, M. Izutsu, “Single sideband modulation performance of a LiNbO3 integrated modulator consisting of four-phase modulator waveguides,” IEEE Photonics Technol. Lett. 13, 364–366 (2001).
[CrossRef]

Lam, P. K.

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 32302-1-4 (2003).
[CrossRef]

Lim, C.

A. Loayssa, C. Lim, A. Nirmalathas, D. Benito, “Simple optical single-sideband modulator for fibre-radio applications,” Electron. Lett. 39, 97–99 (2003).
[CrossRef]

Loayssa, A.

A. Loayssa, C. Lim, A. Nirmalathas, D. Benito, “Simple optical single-sideband modulator for fibre-radio applications,” Electron. Lett. 39, 97–99 (2003).
[CrossRef]

A. Loayssa, D. Benito, M. J. Garde, “Single-sideband suppressed-carrier modulation using a single-electrode electro-optic modulator,” IEEE Photonics Technol. Lett. 13, 869–971 (2001).
[CrossRef]

McClelland, D. E.

Mitsugi, N.

S. Shimotsu, S. Oikawa, T. Saitou, N. Mitsugi, K. Kubodera, T. Kawanishi, M. Izutsu, “Single sideband modulation performance of a LiNbO3 integrated modulator consisting of four-phase modulator waveguides,” IEEE Photonics Technol. Lett. 13, 364–366 (2001).
[CrossRef]

Mizuno, J.

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors,” Phys. Lett. A 175, 273–276 (1993).
[CrossRef]

Munley, A. J.

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

Nelson, P. G.

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors,” Phys. Lett. A 175, 273–276 (1993).
[CrossRef]

Nirmalathas, A.

A. Loayssa, C. Lim, A. Nirmalathas, D. Benito, “Simple optical single-sideband modulator for fibre-radio applications,” Electron. Lett. 39, 97–99 (2003).
[CrossRef]

Novak, D.

G. Smith, D. Novak, Z. Ahmed, “Technique for optical SSB generation to overcome dispersion penalties in fiber-radio systems,” Electron. Lett. 33, 74–75 (1997).
[CrossRef]

Oikawa, S.

S. Shimotsu, S. Oikawa, T. Saitou, N. Mitsugi, K. Kubodera, T. Kawanishi, M. Izutsu, “Single sideband modulation performance of a LiNbO3 integrated modulator consisting of four-phase modulator waveguides,” IEEE Photonics Technol. Lett. 13, 364–366 (2001).
[CrossRef]

Polzik, E. S.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Ralph, T. C.

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 32302-1-4 (2003).
[CrossRef]

Rudiger, A.

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors,” Phys. Lett. A 175, 273–276 (1993).
[CrossRef]

Saitou, T.

S. Shimotsu, S. Oikawa, T. Saitou, N. Mitsugi, K. Kubodera, T. Kawanishi, M. Izutsu, “Single sideband modulation performance of a LiNbO3 integrated modulator consisting of four-phase modulator waveguides,” IEEE Photonics Technol. Lett. 13, 364–366 (2001).
[CrossRef]

Schilling, R.

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors,” Phys. Lett. A 175, 273–276 (1993).
[CrossRef]

Schnabel, R.

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 32302-1-4 (2003).
[CrossRef]

Schnupp, L.

L. Schnupp, conference presentation at the European Collaboration Meeting on Interferometric Detection of Gravitational Waves, Sorrento, Italy, 1988.

Shaddock, D. A.

Shimotsu, S.

S. Shimotsu, S. Oikawa, T. Saitou, N. Mitsugi, K. Kubodera, T. Kawanishi, M. Izutsu, “Single sideband modulation performance of a LiNbO3 integrated modulator consisting of four-phase modulator waveguides,” IEEE Photonics Technol. Lett. 13, 364–366 (2001).
[CrossRef]

Smith, G.

G. Smith, D. Novak, Z. Ahmed, “Technique for optical SSB generation to overcome dispersion penalties in fiber-radio systems,” Electron. Lett. 33, 74–75 (1997).
[CrossRef]

Sorensen, J. L.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Stokes, G. G.

G. G. Stokes, Mathematical and Physical Papers, 5 vols. (Cambridge U. Press, Cambridge, UK, 1880–1985).

Strain, K. A.

K. A. Strain, J. Hough, “Experimental demonstration of the use of a Fabry-Perot cavity as a mirror of variable reflectivity,” Rev. Sci. Instrum. 64, 799–802 (1994).
[CrossRef]

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors,” Phys. Lett. A 175, 273–276 (1993).
[CrossRef]

Symul, T.

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 32302-1-4 (2003).
[CrossRef]

Takachio, N.

K. Yonenaga, N. Takachio, “A fiber chromatic dispersion compensation technique with an optical SSB transmission in optical homodyne detection systems,” IEEE Photonics Technol. Lett. 5, 949–951 (1993).
[CrossRef]

Treps, N.

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 32302-1-4 (2003).
[CrossRef]

Ward, H.

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

Wenger, J.

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, Ph. Grangier, “Quantum key distribution using Gaussian-modulated coherent states,” Nature (London) 421, 238–241 (2003).
[CrossRef]

Winkler, W.

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors,” Phys. Lett. A 175, 273–276 (1993).
[CrossRef]

Yonenaga, K.

K. Yonenaga, N. Takachio, “A fiber chromatic dispersion compensation technique with an optical SSB transmission in optical homodyne detection systems,” IEEE Photonics Technol. Lett. 5, 949–951 (1993).
[CrossRef]

Appl. Phys. B (1)

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

Electron. Lett. (2)

G. Smith, D. Novak, Z. Ahmed, “Technique for optical SSB generation to overcome dispersion penalties in fiber-radio systems,” Electron. Lett. 33, 74–75 (1997).
[CrossRef]

A. Loayssa, C. Lim, A. Nirmalathas, D. Benito, “Simple optical single-sideband modulator for fibre-radio applications,” Electron. Lett. 39, 97–99 (2003).
[CrossRef]

IEEE Photonics Technol. Lett. (4)

K. Yonenaga, N. Takachio, “A fiber chromatic dispersion compensation technique with an optical SSB transmission in optical homodyne detection systems,” IEEE Photonics Technol. Lett. 5, 949–951 (1993).
[CrossRef]

A. Loayssa, D. Benito, M. J. Garde, “Single-sideband suppressed-carrier modulation using a single-electrode electro-optic modulator,” IEEE Photonics Technol. Lett. 13, 869–971 (2001).
[CrossRef]

S. Shimotsu, S. Oikawa, T. Saitou, N. Mitsugi, K. Kubodera, T. Kawanishi, M. Izutsu, “Single sideband modulation performance of a LiNbO3 integrated modulator consisting of four-phase modulator waveguides,” IEEE Photonics Technol. Lett. 13, 364–366 (2001).
[CrossRef]

B. Davies, J. Conradi, “Hybrid modulator structures for subcarrier and harmonic subcarrier optical single sideband,” IEEE Photonics Technol. Lett. 10, 600–602 (1998).
[CrossRef]

Nature (London) (1)

F. Grosshans, G. V. Assche, J. Wenger, R. Brouri, N. J. Cerf, Ph. Grangier, “Quantum key distribution using Gaussian-modulated coherent states,” Nature (London) 421, 238–241 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors,” Phys. Lett. A 175, 273–276 (1993).
[CrossRef]

Phys. Rev. A (1)

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 32302-1-4 (2003).
[CrossRef]

Rev. Sci. Instrum. (1)

K. A. Strain, J. Hough, “Experimental demonstration of the use of a Fabry-Perot cavity as a mirror of variable reflectivity,” Rev. Sci. Instrum. 64, 799–802 (1994).
[CrossRef]

Science (1)

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Other (4)

L. Schnupp, conference presentation at the European Collaboration Meeting on Interferometric Detection of Gravitational Waves, Sorrento, Italy, 1988.

LIGO Scientific Collaboration, “Advanced LIGO systems design,” LIGO Tech. Note T010075, P. Fritschel, ed. (LIGO Laboratory, Pasadena, Calif., 2001), http://antares.ligo.caltech.edu/dcc/default.htf .

See, for example, JDS Uniphase, www.jdsu.com .

G. G. Stokes, Mathematical and Physical Papers, 5 vols. (Cambridge U. Press, Cambridge, UK, 1880–1985).

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic of the QVM. Two modulating crystals are positioned in series with the modulating axes at 90°. A laser beam of elliptical polarization passes through each crystal and then through a vertically polarizing element. (b) The subset of input beam polarization states considered here is all states where there is an equal optical-field amplitude in both the left- and the right-diagonal components. The σ term equals the phase between these components: σ = 0 (vertical), π/4, π/2 (circular), 3π/4, and π (horizontal). (c) Phasor diagram representing the transfer function given by Eq. (3). If the input electrical signals are represented as rotating phasors (δ̃1 and δ̃2), the corresponding phasors that represent the output phase and amplitude modulations ( and Ã) are proportional to (δ̃1 + δ̃2) and (δ̃1 - δ̃2), respectively.

Fig. 2
Fig. 2

(a) Q-space diagram. Axis labels are visual reminders of relative amplitudes and phases of input electrical signals (try to visualize the display of a cathode-ray oscilloscope run in the XY mode with δ1 and δ2 as the X and Y inputs). (b) M-space diagram. Axis labels are optical amplitude phasors whose end points oscillate at the modulation frequency. For σ = π/2 there is a graphic transfer correspondence between (a) and (b) according to Eqs. (9). (c) In general the transfer function of the QVM, Eqs. (8), takes a sphere of constant input power in Q space and returns an ellipse in M space. A cross section is shown for the selection of values for σ. SSB, single sideband.

Fig. 3
Fig. 3

Heterodyne characterization experiment layout of the QVM. A spectrum analyzer provides frequency line data (of the direct AM beat and heterodyned copies of sidebands) for establishing the purity of the QVM, while two parallel double-demodulation circuits provide the complete set of modulation measurables for testing the variability of the QVM: BS, dielectric beam splitter; λ/2, half-wave plate; λ/4, quarter-wave plate; PBS, polarizing beam splitter; PZT, piezoelectric actuator; AOM, acousto-optic modulator; FB, feedback servo and HV amp; 90°, electronic phase shifter; DC/HF, bias-T; SA, spectrum analyzer. The 5-MHz signal generators are electronically phase-locked.

Fig. 4
Fig. 4

Spectrum analyzer traces of heterodyne measurements of four pure operating points. Frequencies of interest are 5 MHz (direct AM beat), 80 MHz (heterodyne-carrier beat), and 75 and 85 MHz (heterodyne-sideband beats): solid curve, data averaged over a few seconds; dashed curve, max-hold data acquired over approximately 1 h; arrows, curves that have been suppressed. (a) Upper single sideband (lower sideband suppressed by 34.9 dB), (b) lower single sideband (upper sideband suppressed by 39.5 dB), (c) pure PM (AM beat suppressed by around 38 dB), (d) carrier suppression (heterodyne beat suppressed by around 30 dB).

Fig. 5
Fig. 5

More detailed spectrum analyzer trace of sideband suppression with a 35.2-dB difference between sidebands. Frequency lines at 72.1, 86.1, and 87.7 MHz are from radio interference with the electronic equipment.

Fig. 6
Fig. 6

(a) Modulation ellipse showing the trajectory of the measurement sweep through parameter space where the phase difference between the electrical signals, ϕ = ϕ1 - ϕ2, is varied through 360°. [The ellipse has an (exaggerated) σ of ≈53°.] (b) Double-demodulation measurements of PM (squares and diamonds, in-phase and quadrature components, respectively) and AM (plusses and crosses, in-phase and quadrature components, respectively) with corresponding theoretical predictions (solid curves). The majority of modulation present is either PM or quadrature AM as predicted. This plot corresponds to σ ≈ 75°, a value derived by measuring, from corresponding spectrum-analyzer data, the phase ϕ at which the single-sideband operating points occur. When we use a result from Table 1, phase ϕ between the pure AM operating point and either of the single-sideband operating points is precisely equal to σ.

Fig. 7
Fig. 7

(a) Modulation ellipse showing the trajectory of the measurement sweep through parameter space, where the strength of one electrical signal δ2 is varied from +5.1 to -5.1 V. The sweep involves varying the overall input electrical power, so that the modulation trajectory does not stay on the ellipse surface and instead traces out a parabolic curve. [The ellipse has an (exaggerated) σ of ≈53°.] (b) Double-demodulation measurements of PM (squares and diamonds, in-phase and quadrature components, respectively) and AM (plusses and crosses, in-phase and quadrature components, respectively) with corresponding theoretical predictions (solid lines). There is diverging agreement in the PM that may be caused by imperfectly matched electrical impedances for the two crystals. As in Fig. 6 the location of the correlated PM and AM point is shifted toward the pure AM point by ∼0.7 V out of 5.1 V, and, referring to Table 1, this corresponds to a value of σ ≈ 75°.

Fig. 8
Fig. 8

(a) Modulation ellipse showing short-range trajectories designed to map the local region near the PM operating point. The electrical parameters, ϕ = ϕ1 - ϕ2 and δ2, were varied to achieve these results with units worth 5° and 0.44 V, respectively. (b), (c), (d), (e) Double-demodulation measurements of PM (squares, diamonds, in-phase and quadrature components, respectively) and AM (plusses and crosses, in-phase and quadrature components, respectively) with corresponding theoretical predictions (solid lines). Parameters varied were (b) ϕ, (c) δ2, (d) ϕ and δ2 together with the same polarity, and (e) ϕ and δ2 together with opposite polarities. Note: the in-phase PM data points (squares) have been scaled down by a factor of 10 to fit in the diagram. Also the systematic error in the quadrature PM data is probably due to an electronic phase drift between signal generators, causing a small amount of in-phase PM data (which is an order of magnitude stronger) to couple across.

Tables (1)

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Table 1 Details of Significant Operating Points

Equations (33)

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tan2σ2=PhPv
Eout=Eincosσ/2+iP˜ expiωmt+àexpiωmt=Eincosσ/2+iP cosωmt+ϕP+A cosωmt+ϕA,
P˜=12cosσ2δ˜1+δ˜2,Ã=12sinσ2δ˜1-δ˜2.
|Ã|2|P˜|2=PhPv.
Eout=Eincosσ/2+12Ã+iP˜expiωmt+12Ã-iP˜* exp-iωmt,
M1=P2-A2,M2=2PA cosϕP-ϕA=2P˜Ã*,M3=2PA sinϕP-ϕA=2P˜Ã*.
Q1=δ12-δ22,Q2=2δ1δ2 cosϕ1-ϕ2=2δ˜1δ˜2*,Q3=2δ1δ2 sinϕ1-ϕ2=2δ˜1δ˜2*,
M0=14Q0+Q2 cosσ,M1=14Q0 cosσ+Q2,M2=14sinσQ1,M3=-14sinσQ3.
M0=14 Q0, M1=14 Q2,M2=14 Q1, M3=-14 Q3.
Pdet=Ein2Pdc+Pωm+Pωh+Pωh-ωm+Pωh+ωm,Pdc=cos2σ2,Pωm=2 cosσ2àexpiωmt,Pωh=2 cosσ2γ expiωht,Pωh-ωm=2γÃ+iP˜* expiωh-ωmt,Pωh+ωm=2γÃ-iP˜expiωh+ωmt,
Pωh-ωm+Pωh+ωm=2 cosσ2×P˜ expiωmtγ expiωht+àexpiωmtγ expiωht.
EexitingQVM=L˜ expiδ1 expiωmtLˆ+R˜ expiδ˜2 expiωmtRˆexpiωt.
Eout=12L˜1+iδ˜1 expiωmt+R˜1+iδ˜2 expiωmt.
Eout=L˜+R˜21+iL2+L˜R˜*|L˜+R˜|2 δ˜1 expiωmt+iR˜L˜*+R2|L˜+R˜|2 δ˜2 expiωmt.
Eout=L˜+R˜21+R˜L˜*δ˜1-δ˜2|L˜+R˜|2expiωmt+iR˜L˜*δ˜1+δ˜2+L2δ˜1+R2δ˜2|L˜+R˜|2×expiωmt.
Eout=|L˜+R˜|2+àexpiωmt+iP˜ expiωmt
P˜= R˜L˜*δ˜1+δ˜2+L2δ˜1+R2δ˜22|L˜+R˜|,Ã= R˜L˜*δ˜1-δ˜22|L˜+R˜|,
Eout=Ein cosσ2+àexpiωmt+iP˜ expiωmt,
P˜=Ein2cosσ2δ˜1+δ˜2,Ã=Ein2sinσ2δ˜1-δ˜2.
P2=P˜*P˜=R˜L˜*2Q0+Q2+2L2R2Q22|L˜+R˜|2+12L4+R4Q0+L4-R4Q12|L˜+R˜|2+R˜L˜*L2+R2Q0+L2-R2Q1+L2+R2Q22|L˜+R˜|2,
A2=Ã*Ã=R˜L˜*2Q0-Q22|L˜+R˜|2,
P˜Ã*=R˜L˜*R˜L˜*Q12|L˜+R˜|2+12 R˜L˜*L2-R2Q0+L2+R2Q1-L2-R2Q22|L˜+R˜|2,
P˜Ã*=-R˜L˜*R˜L˜*+12L2+R2Q32|L˜+R˜|2.
S0=R2+L2,S1=R2-L2,S2=2RL cosσ=2R˜L˜*,S3=2RL sinσ=2R˜L˜*,
P2=14S0Q0-S1Q1+S2Q2-S322S0+S2×Q0-Q2,A2=18S32S0+S2Q0-Q2,P˜Ã*=18S3Q1-S3S1S0+S2Q0-Q2,P˜Ã*=-18 S3Q3.
M0M1M2M3=14S0-S1S20S2+S12S0+S2-S1S0-S12S0+S20-S1S3S0+S2S3S1S3S0+S20000-S3×Q0Q1Q2Q3.
M0M1M2M3=Ein2410cosσ0cosσ0100sinσ00000-sinσ×Q0Q1Q2Q3.
0, 4M0cosσ+1, 0
0, 4M0cosσ-1, 0
4M0sinσ,-4M0 cosσsin2σ, 0
-4M0sinσ,-4M0 cosσsin2σ, 0
0,-4M0 cosσsin2σ,-4M0sinσ
0,-4M0 cosσsin2σ, 4M0sinσ

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