Abstract

The operating characteristics of a novel phase-shifting interferometer are presented. Interference arises by reflecting the light from a sample back into the cavity of a cw He–Ne laser. Changes in phase and fringe visibility are calculated from an overdetermined set of phase-shifted intensity measurements with the phase shifts being introduced with an electro-optic modulator. The interferometer is sensitive enough to measure displacements below 1 Hz with a rms error of approximately 1 nm from a sample that reflects only 3% of the 28 µW that is incident on its surface. The interferometer is applied to the determination of cantilever bending of a piezoelectric bimorph.

© 1999 Optical Society of America

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  1. D. Li, B. J. Schnapp, “Improved nm displacement detector for microscopic beads at frequencies below 10 Hz,” Rev. Sci. Instrum. 68, 2195–2199 (1997).
    [CrossRef]
  2. K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature (London) 365, 721–727 (1993).
    [CrossRef]
  3. M. H. Kiang, O. Solgaard, K. Y. Lau, R. S. Muller, “Electrostatic combdrive-actuated micromirrors for laser-beam scanning and positioning,” IEEE J. Microelectromech. Syst. 7, 27–37 (1998).
    [CrossRef]
  4. B. Chui, W. T. D. Stowe, Y. Sungtack, K. E. Goodson, T. W. Kenny, H. J. Mamin, B. D. Terris, R. P. Ried, D. Rugar, “Low-stiffness silicon cantilevers with integrated heaters and piezoresistive sensors for high density AFM thermomechanical data storage,” IEEE J. Microelectromech. Syst. 7, 69–78 (1998).
    [CrossRef]
  5. E. Calloni, L. D. Fiore, A. Grado, L. Milano, “An interferometric device to measure the mechanical transfer function of the VIRGO mirror suspensions,” Rev. Sci. Instrum. 69, 1882–1885 (1998).
    [CrossRef]
  6. Th. H. Peek, P. T. Bolwijn, C. Th. Alkemade, “Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967).
    [CrossRef]
  7. E. B. Hooper, G. Bekefi, “Laser interferometer for repetitively pulsed plasmas,” J. Appl. Phys. 37, 4083–4094 (1966), erratum, 38, 1998 (1967).
    [CrossRef]
  8. P. T. Bolwijn, “Single mode tuning dip in the modulated power output of gas lasers,” Phys. Lett. 19, 384–385 (1963).
    [CrossRef]
  9. R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
    [CrossRef]
  10. G. A. Acket, D. Lenstra, A. J. Den Boef, B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. QE-20, 1163–1169 (1984).
    [CrossRef]
  11. A. Bearden, M. P. O’Neill, L. C. Osborne, T. L. Wong, “Imaging and vibrational analysis with laser-feedback interferometry,” Opt. Lett. 18, 238–240 (1993).
    [CrossRef] [PubMed]
  12. R. Juskaitis, T. Wilson, N. P. Rea, “Compact confocal interference microscopy,” Opt. Commun. 109, 167–177 (1994).
    [CrossRef]
  13. D. Sarid, V. Weissenberger, D. A. Iams, J. T. Ingle, “Theory of the laser diode interaction in scanning force microscopy,” IEEE J. Quantum Electron. 25, 1968–1972 (1989).
    [CrossRef]
  14. D. Sarid, D. A. Iams, J. T. Ingle, V. Weissenberger, “Performance of a scanning force microscope using a laser diode,” J. Vac. Sci. Technol. A 8, 378–383 (1990).
    [CrossRef]
  15. B. Ovryn, J. H. Andrews, “Phase-shifted laser feedback interferometry,” Opt. Lett. 23, 1078–1080 (1998).
    [CrossRef]
  16. K. Creath, “Phase measurement interferometry techniques” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 349–393.
  17. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.
  18. A. Yariv, Quantum Electronics (Wiley, New York, 1989), p. 192ff.
  19. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
    [CrossRef] [PubMed]
  20. B. Ovryn, J. H. Andrews, S. J. Eppell, J. D. Khaydarov, “Phase shifted real time laser feedback interferometry,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takada, ed., Proc. SPIE2860, 263–274 (1996).
    [CrossRef]
  21. B. Ovryn, E. M. Haacke, “Temporal averaging of phase measurements in the presence of spurious phase drift: application to phase-stepped, real-time holographic interferometry,” Appl. Opt. 32, 147–154 (1993).
    [CrossRef] [PubMed]
  22. B. Ovryn, “Two camera phase measurements using phase stepped, real-time holographic interferometry,” in Holography, Interferometry and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 120–131 (1993).
    [CrossRef]

1998 (4)

M. H. Kiang, O. Solgaard, K. Y. Lau, R. S. Muller, “Electrostatic combdrive-actuated micromirrors for laser-beam scanning and positioning,” IEEE J. Microelectromech. Syst. 7, 27–37 (1998).
[CrossRef]

B. Chui, W. T. D. Stowe, Y. Sungtack, K. E. Goodson, T. W. Kenny, H. J. Mamin, B. D. Terris, R. P. Ried, D. Rugar, “Low-stiffness silicon cantilevers with integrated heaters and piezoresistive sensors for high density AFM thermomechanical data storage,” IEEE J. Microelectromech. Syst. 7, 69–78 (1998).
[CrossRef]

E. Calloni, L. D. Fiore, A. Grado, L. Milano, “An interferometric device to measure the mechanical transfer function of the VIRGO mirror suspensions,” Rev. Sci. Instrum. 69, 1882–1885 (1998).
[CrossRef]

B. Ovryn, J. H. Andrews, “Phase-shifted laser feedback interferometry,” Opt. Lett. 23, 1078–1080 (1998).
[CrossRef]

1997 (1)

D. Li, B. J. Schnapp, “Improved nm displacement detector for microscopic beads at frequencies below 10 Hz,” Rev. Sci. Instrum. 68, 2195–2199 (1997).
[CrossRef]

1994 (1)

R. Juskaitis, T. Wilson, N. P. Rea, “Compact confocal interference microscopy,” Opt. Commun. 109, 167–177 (1994).
[CrossRef]

1993 (3)

1990 (1)

D. Sarid, D. A. Iams, J. T. Ingle, V. Weissenberger, “Performance of a scanning force microscope using a laser diode,” J. Vac. Sci. Technol. A 8, 378–383 (1990).
[CrossRef]

1989 (1)

D. Sarid, V. Weissenberger, D. A. Iams, J. T. Ingle, “Theory of the laser diode interaction in scanning force microscopy,” IEEE J. Quantum Electron. 25, 1968–1972 (1989).
[CrossRef]

1987 (1)

1984 (1)

G. A. Acket, D. Lenstra, A. J. Den Boef, B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. QE-20, 1163–1169 (1984).
[CrossRef]

1980 (1)

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

1967 (1)

Th. H. Peek, P. T. Bolwijn, C. Th. Alkemade, “Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967).
[CrossRef]

1966 (1)

E. B. Hooper, G. Bekefi, “Laser interferometer for repetitively pulsed plasmas,” J. Appl. Phys. 37, 4083–4094 (1966), erratum, 38, 1998 (1967).
[CrossRef]

1963 (1)

P. T. Bolwijn, “Single mode tuning dip in the modulated power output of gas lasers,” Phys. Lett. 19, 384–385 (1963).
[CrossRef]

Acket, G. A.

G. A. Acket, D. Lenstra, A. J. Den Boef, B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. QE-20, 1163–1169 (1984).
[CrossRef]

Alkemade, C. Th.

Th. H. Peek, P. T. Bolwijn, C. Th. Alkemade, “Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967).
[CrossRef]

Andrews, J. H.

B. Ovryn, J. H. Andrews, “Phase-shifted laser feedback interferometry,” Opt. Lett. 23, 1078–1080 (1998).
[CrossRef]

B. Ovryn, J. H. Andrews, S. J. Eppell, J. D. Khaydarov, “Phase shifted real time laser feedback interferometry,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takada, ed., Proc. SPIE2860, 263–274 (1996).
[CrossRef]

Bearden, A.

Bekefi, G.

E. B. Hooper, G. Bekefi, “Laser interferometer for repetitively pulsed plasmas,” J. Appl. Phys. 37, 4083–4094 (1966), erratum, 38, 1998 (1967).
[CrossRef]

Block, S. M.

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature (London) 365, 721–727 (1993).
[CrossRef]

Bolwijn, P. T.

Th. H. Peek, P. T. Bolwijn, C. Th. Alkemade, “Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967).
[CrossRef]

P. T. Bolwijn, “Single mode tuning dip in the modulated power output of gas lasers,” Phys. Lett. 19, 384–385 (1963).
[CrossRef]

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

Calloni, E.

E. Calloni, L. D. Fiore, A. Grado, L. Milano, “An interferometric device to measure the mechanical transfer function of the VIRGO mirror suspensions,” Rev. Sci. Instrum. 69, 1882–1885 (1998).
[CrossRef]

Chui, B.

B. Chui, W. T. D. Stowe, Y. Sungtack, K. E. Goodson, T. W. Kenny, H. J. Mamin, B. D. Terris, R. P. Ried, D. Rugar, “Low-stiffness silicon cantilevers with integrated heaters and piezoresistive sensors for high density AFM thermomechanical data storage,” IEEE J. Microelectromech. Syst. 7, 69–78 (1998).
[CrossRef]

Creath, K.

K. Creath, “Phase measurement interferometry techniques” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 349–393.

Den Boef, A. J.

G. A. Acket, D. Lenstra, A. J. Den Boef, B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. QE-20, 1163–1169 (1984).
[CrossRef]

Eiju, T.

Eppell, S. J.

B. Ovryn, J. H. Andrews, S. J. Eppell, J. D. Khaydarov, “Phase shifted real time laser feedback interferometry,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takada, ed., Proc. SPIE2860, 263–274 (1996).
[CrossRef]

Fiore, L. D.

E. Calloni, L. D. Fiore, A. Grado, L. Milano, “An interferometric device to measure the mechanical transfer function of the VIRGO mirror suspensions,” Rev. Sci. Instrum. 69, 1882–1885 (1998).
[CrossRef]

Goodson, K. E.

B. Chui, W. T. D. Stowe, Y. Sungtack, K. E. Goodson, T. W. Kenny, H. J. Mamin, B. D. Terris, R. P. Ried, D. Rugar, “Low-stiffness silicon cantilevers with integrated heaters and piezoresistive sensors for high density AFM thermomechanical data storage,” IEEE J. Microelectromech. Syst. 7, 69–78 (1998).
[CrossRef]

Grado, A.

E. Calloni, L. D. Fiore, A. Grado, L. Milano, “An interferometric device to measure the mechanical transfer function of the VIRGO mirror suspensions,” Rev. Sci. Instrum. 69, 1882–1885 (1998).
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

Haacke, E. M.

Hariharan, P.

Hooper, E. B.

E. B. Hooper, G. Bekefi, “Laser interferometer for repetitively pulsed plasmas,” J. Appl. Phys. 37, 4083–4094 (1966), erratum, 38, 1998 (1967).
[CrossRef]

Iams, D. A.

D. Sarid, D. A. Iams, J. T. Ingle, V. Weissenberger, “Performance of a scanning force microscope using a laser diode,” J. Vac. Sci. Technol. A 8, 378–383 (1990).
[CrossRef]

D. Sarid, V. Weissenberger, D. A. Iams, J. T. Ingle, “Theory of the laser diode interaction in scanning force microscopy,” IEEE J. Quantum Electron. 25, 1968–1972 (1989).
[CrossRef]

Ingle, J. T.

D. Sarid, D. A. Iams, J. T. Ingle, V. Weissenberger, “Performance of a scanning force microscope using a laser diode,” J. Vac. Sci. Technol. A 8, 378–383 (1990).
[CrossRef]

D. Sarid, V. Weissenberger, D. A. Iams, J. T. Ingle, “Theory of the laser diode interaction in scanning force microscopy,” IEEE J. Quantum Electron. 25, 1968–1972 (1989).
[CrossRef]

Juskaitis, R.

R. Juskaitis, T. Wilson, N. P. Rea, “Compact confocal interference microscopy,” Opt. Commun. 109, 167–177 (1994).
[CrossRef]

Kenny, T. W.

B. Chui, W. T. D. Stowe, Y. Sungtack, K. E. Goodson, T. W. Kenny, H. J. Mamin, B. D. Terris, R. P. Ried, D. Rugar, “Low-stiffness silicon cantilevers with integrated heaters and piezoresistive sensors for high density AFM thermomechanical data storage,” IEEE J. Microelectromech. Syst. 7, 69–78 (1998).
[CrossRef]

Khaydarov, J. D.

B. Ovryn, J. H. Andrews, S. J. Eppell, J. D. Khaydarov, “Phase shifted real time laser feedback interferometry,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takada, ed., Proc. SPIE2860, 263–274 (1996).
[CrossRef]

Kiang, M. H.

M. H. Kiang, O. Solgaard, K. Y. Lau, R. S. Muller, “Electrostatic combdrive-actuated micromirrors for laser-beam scanning and positioning,” IEEE J. Microelectromech. Syst. 7, 27–37 (1998).
[CrossRef]

Kobayashi, K.

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

Lang, R.

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

Lau, K. Y.

M. H. Kiang, O. Solgaard, K. Y. Lau, R. S. Muller, “Electrostatic combdrive-actuated micromirrors for laser-beam scanning and positioning,” IEEE J. Microelectromech. Syst. 7, 27–37 (1998).
[CrossRef]

Lenstra, D.

G. A. Acket, D. Lenstra, A. J. Den Boef, B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. QE-20, 1163–1169 (1984).
[CrossRef]

Li, D.

D. Li, B. J. Schnapp, “Improved nm displacement detector for microscopic beads at frequencies below 10 Hz,” Rev. Sci. Instrum. 68, 2195–2199 (1997).
[CrossRef]

Mamin, H. J.

B. Chui, W. T. D. Stowe, Y. Sungtack, K. E. Goodson, T. W. Kenny, H. J. Mamin, B. D. Terris, R. P. Ried, D. Rugar, “Low-stiffness silicon cantilevers with integrated heaters and piezoresistive sensors for high density AFM thermomechanical data storage,” IEEE J. Microelectromech. Syst. 7, 69–78 (1998).
[CrossRef]

Milano, L.

E. Calloni, L. D. Fiore, A. Grado, L. Milano, “An interferometric device to measure the mechanical transfer function of the VIRGO mirror suspensions,” Rev. Sci. Instrum. 69, 1882–1885 (1998).
[CrossRef]

Muller, R. S.

M. H. Kiang, O. Solgaard, K. Y. Lau, R. S. Muller, “Electrostatic combdrive-actuated micromirrors for laser-beam scanning and positioning,” IEEE J. Microelectromech. Syst. 7, 27–37 (1998).
[CrossRef]

O’Neill, M. P.

Oreb, B. F.

Osborne, L. C.

Ovryn, B.

B. Ovryn, J. H. Andrews, “Phase-shifted laser feedback interferometry,” Opt. Lett. 23, 1078–1080 (1998).
[CrossRef]

B. Ovryn, E. M. Haacke, “Temporal averaging of phase measurements in the presence of spurious phase drift: application to phase-stepped, real-time holographic interferometry,” Appl. Opt. 32, 147–154 (1993).
[CrossRef] [PubMed]

B. Ovryn, “Two camera phase measurements using phase stepped, real-time holographic interferometry,” in Holography, Interferometry and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 120–131 (1993).
[CrossRef]

B. Ovryn, J. H. Andrews, S. J. Eppell, J. D. Khaydarov, “Phase shifted real time laser feedback interferometry,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takada, ed., Proc. SPIE2860, 263–274 (1996).
[CrossRef]

Peek, Th. H.

Th. H. Peek, P. T. Bolwijn, C. Th. Alkemade, “Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967).
[CrossRef]

Rea, N. P.

R. Juskaitis, T. Wilson, N. P. Rea, “Compact confocal interference microscopy,” Opt. Commun. 109, 167–177 (1994).
[CrossRef]

Ried, R. P.

B. Chui, W. T. D. Stowe, Y. Sungtack, K. E. Goodson, T. W. Kenny, H. J. Mamin, B. D. Terris, R. P. Ried, D. Rugar, “Low-stiffness silicon cantilevers with integrated heaters and piezoresistive sensors for high density AFM thermomechanical data storage,” IEEE J. Microelectromech. Syst. 7, 69–78 (1998).
[CrossRef]

Rugar, D.

B. Chui, W. T. D. Stowe, Y. Sungtack, K. E. Goodson, T. W. Kenny, H. J. Mamin, B. D. Terris, R. P. Ried, D. Rugar, “Low-stiffness silicon cantilevers with integrated heaters and piezoresistive sensors for high density AFM thermomechanical data storage,” IEEE J. Microelectromech. Syst. 7, 69–78 (1998).
[CrossRef]

Sarid, D.

D. Sarid, D. A. Iams, J. T. Ingle, V. Weissenberger, “Performance of a scanning force microscope using a laser diode,” J. Vac. Sci. Technol. A 8, 378–383 (1990).
[CrossRef]

D. Sarid, V. Weissenberger, D. A. Iams, J. T. Ingle, “Theory of the laser diode interaction in scanning force microscopy,” IEEE J. Quantum Electron. 25, 1968–1972 (1989).
[CrossRef]

Schmidt, C. F.

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature (London) 365, 721–727 (1993).
[CrossRef]

Schnapp, B. J.

D. Li, B. J. Schnapp, “Improved nm displacement detector for microscopic beads at frequencies below 10 Hz,” Rev. Sci. Instrum. 68, 2195–2199 (1997).
[CrossRef]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature (London) 365, 721–727 (1993).
[CrossRef]

Solgaard, O.

M. H. Kiang, O. Solgaard, K. Y. Lau, R. S. Muller, “Electrostatic combdrive-actuated micromirrors for laser-beam scanning and positioning,” IEEE J. Microelectromech. Syst. 7, 27–37 (1998).
[CrossRef]

Stowe, W. T. D.

B. Chui, W. T. D. Stowe, Y. Sungtack, K. E. Goodson, T. W. Kenny, H. J. Mamin, B. D. Terris, R. P. Ried, D. Rugar, “Low-stiffness silicon cantilevers with integrated heaters and piezoresistive sensors for high density AFM thermomechanical data storage,” IEEE J. Microelectromech. Syst. 7, 69–78 (1998).
[CrossRef]

Sungtack, Y.

B. Chui, W. T. D. Stowe, Y. Sungtack, K. E. Goodson, T. W. Kenny, H. J. Mamin, B. D. Terris, R. P. Ried, D. Rugar, “Low-stiffness silicon cantilevers with integrated heaters and piezoresistive sensors for high density AFM thermomechanical data storage,” IEEE J. Microelectromech. Syst. 7, 69–78 (1998).
[CrossRef]

Svoboda, K.

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature (London) 365, 721–727 (1993).
[CrossRef]

Terris, B. D.

B. Chui, W. T. D. Stowe, Y. Sungtack, K. E. Goodson, T. W. Kenny, H. J. Mamin, B. D. Terris, R. P. Ried, D. Rugar, “Low-stiffness silicon cantilevers with integrated heaters and piezoresistive sensors for high density AFM thermomechanical data storage,” IEEE J. Microelectromech. Syst. 7, 69–78 (1998).
[CrossRef]

Verbeek, B. H.

G. A. Acket, D. Lenstra, A. J. Den Boef, B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. QE-20, 1163–1169 (1984).
[CrossRef]

Weissenberger, V.

D. Sarid, D. A. Iams, J. T. Ingle, V. Weissenberger, “Performance of a scanning force microscope using a laser diode,” J. Vac. Sci. Technol. A 8, 378–383 (1990).
[CrossRef]

D. Sarid, V. Weissenberger, D. A. Iams, J. T. Ingle, “Theory of the laser diode interaction in scanning force microscopy,” IEEE J. Quantum Electron. 25, 1968–1972 (1989).
[CrossRef]

Wilson, T.

R. Juskaitis, T. Wilson, N. P. Rea, “Compact confocal interference microscopy,” Opt. Commun. 109, 167–177 (1994).
[CrossRef]

Wong, T. L.

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1989), p. 192ff.

Am. J. Phys. (1)

Th. H. Peek, P. T. Bolwijn, C. Th. Alkemade, “Axial mode number of gas lasers from moving-mirror experiments,” Am. J. Phys. 35, 820–831 (1967).
[CrossRef]

Appl. Opt. (2)

IEEE J. Microelectromech. Syst. (2)

M. H. Kiang, O. Solgaard, K. Y. Lau, R. S. Muller, “Electrostatic combdrive-actuated micromirrors for laser-beam scanning and positioning,” IEEE J. Microelectromech. Syst. 7, 27–37 (1998).
[CrossRef]

B. Chui, W. T. D. Stowe, Y. Sungtack, K. E. Goodson, T. W. Kenny, H. J. Mamin, B. D. Terris, R. P. Ried, D. Rugar, “Low-stiffness silicon cantilevers with integrated heaters and piezoresistive sensors for high density AFM thermomechanical data storage,” IEEE J. Microelectromech. Syst. 7, 69–78 (1998).
[CrossRef]

IEEE J. Quantum Electron. (3)

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

G. A. Acket, D. Lenstra, A. J. Den Boef, B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. QE-20, 1163–1169 (1984).
[CrossRef]

D. Sarid, V. Weissenberger, D. A. Iams, J. T. Ingle, “Theory of the laser diode interaction in scanning force microscopy,” IEEE J. Quantum Electron. 25, 1968–1972 (1989).
[CrossRef]

J. Appl. Phys. (1)

E. B. Hooper, G. Bekefi, “Laser interferometer for repetitively pulsed plasmas,” J. Appl. Phys. 37, 4083–4094 (1966), erratum, 38, 1998 (1967).
[CrossRef]

J. Vac. Sci. Technol. A (1)

D. Sarid, D. A. Iams, J. T. Ingle, V. Weissenberger, “Performance of a scanning force microscope using a laser diode,” J. Vac. Sci. Technol. A 8, 378–383 (1990).
[CrossRef]

Nature (London) (1)

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature (London) 365, 721–727 (1993).
[CrossRef]

Opt. Commun. (1)

R. Juskaitis, T. Wilson, N. P. Rea, “Compact confocal interference microscopy,” Opt. Commun. 109, 167–177 (1994).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. (1)

P. T. Bolwijn, “Single mode tuning dip in the modulated power output of gas lasers,” Phys. Lett. 19, 384–385 (1963).
[CrossRef]

Rev. Sci. Instrum. (2)

D. Li, B. J. Schnapp, “Improved nm displacement detector for microscopic beads at frequencies below 10 Hz,” Rev. Sci. Instrum. 68, 2195–2199 (1997).
[CrossRef]

E. Calloni, L. D. Fiore, A. Grado, L. Milano, “An interferometric device to measure the mechanical transfer function of the VIRGO mirror suspensions,” Rev. Sci. Instrum. 69, 1882–1885 (1998).
[CrossRef]

Other (5)

K. Creath, “Phase measurement interferometry techniques” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), pp. 349–393.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 501–598.

A. Yariv, Quantum Electronics (Wiley, New York, 1989), p. 192ff.

B. Ovryn, “Two camera phase measurements using phase stepped, real-time holographic interferometry,” in Holography, Interferometry and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 120–131 (1993).
[CrossRef]

B. Ovryn, J. H. Andrews, S. J. Eppell, J. D. Khaydarov, “Phase shifted real time laser feedback interferometry,” in Laser Interferometry VIII: Techniques and Analysis, M. Kujawinska, R. J. Pryputniewicz, M. Takada, ed., Proc. SPIE2860, 263–274 (1996).
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

(a) Error in OPL that is due to the extra reflections [Eq. (12)] calculated from the difference between the measured and the actual OPL. The visibilities shown in plots (b) and (c) represent the observed value [given by Eq. (13)] with an actual visibility equal to m. The ratio of the parameters m to b is 0.455 with m = 0.128 (dashed curve) or m = 0.064 (solid curve).

Fig. 2
Fig. 2

Step response of the interferometer (top trace) to two sets of five sequential voltage steps to the EOM (bottom trace). Each set of five phase shifts is required for a single phase measurement.

Fig. 3
Fig. 3

Laser intensity when a sinusoidal voltage of (a) 400 kHz, (b) 800 kHz, and (c) 1 MHz was applied to the EOM. Solid curve (time domain) represents the input voltage to the EOM.

Fig. 4
Fig. 4

(a) OPL and (b) visibility measurements obtained at moderately high visibility. Both the visibility and the OPL show systematic oscillations that depend on the value of the OPL. The solid curves in (b) and (c) represent fits using Eqs. (12) and (13) with m = 0.128 and b = 0.282.

Fig. 5
Fig. 5

Low-frequency systematic variation in the OPL (a) caused by environmental perturbations during a 50-s period; (b) the rms error in the OPL calculated from a linear fit to the last 30 s of data; and (c) visibility during the 30-s period.

Fig. 6
Fig. 6

Error in the measured OPL as a function of the visibility. The crosses represent the raw data, and the circles represent the data corrected for the effect of multiple reflections.

Fig. 7
Fig. 7

Visibility as a function of the transmission of the ND filter. At high visibility, the data were corrected for systematic errors that were due to multiple reflections. The solid curve is based on Eq. (7).

Fig. 8
Fig. 8

Measured OPL and visibility for a 5-nm ramp applied to the EOM. The rms OPL error, obtained from a linear fit to a single period, is 0.14 nm.

Fig. 9
Fig. 9

Linear change in OPL that is due to a 40-mV voltage ramp applied to the EOM. (a) The presence of a linear drift in the OPL during the 40 s is evident. (b) Variation in OPL and visibility during a 5-s period. (c) The rms error in OPL, obtained from a linear fit to the data, is 0.13 nm; a histogram indicates the error distribution.

Fig. 10
Fig. 10

Small sinusoidal change in the OPL that is due to a sinusoidal voltage applied to the EOM. The visibility is essentially flat, indicating the absence of systematic errors.

Fig. 11
Fig. 11

Sinusoidal change in the OPL that is due to a sinusoidal voltage applied to a piezoelectric translator that is moving parallel to the optical axis.

Fig. 12
Fig. 12

Reduction in error in OPL by averaging ten phase measurements; the error was reduced from approximately 7 to 2.2 nm.

Fig. 13
Fig. 13

Photodetector signal from the spectrum analyzer when the EOM was modulated at 2 kHz. The first harmonic was maximized (a) by changing the dc offset ϕdc, whereas there is residual first-harmonic response in (b) the nearly maximal second harmonic.

Fig. 14
Fig. 14

LFI response at the fundamental and second harmonic as a function of the amplitude of the ac oscillation of the EOM. The solid curves represent fits to the harmonics based on the first and second Bessel functions.

Fig. 15
Fig. 15

Bending curves for the bimorph when ±2.000 V were applied to the bimorph; there is no directional ambiguity in the displacement measurement. The multiple data points at each of the 250 µm increments are caused by superimposing the displacement measured in both directions of the scan, with the first scan beginning at the base of the bimorph (we corrected for the 2π ambiguity for the first data point acquired on the retrace).

Equations (19)

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Im, b, ϕ=Io1+m cosϕj=0-bj cosjϕ.
m=K1-R2R2Reff1/21-R1+1-R2=γReff1/2,
b=R2Reff1/2.
E1=a1 exp-iϕ1,
E2=a2 exp-iϕ2,
I=|E1+E2|2=Io1+m cosϕ,
m=2 a1a2a12+a22.
Iim, ϕ, ψ=Io1+m cosϕ+ψi.
tanϕ=2I2-I42I3-I1+I5,
m=32I2-I42+2I3-I5-I121/22I1+I2+2I3+I4+I5.
mj=1=-6-6+3bm+bm cos2ϕ m.
tanϕj=3,4=1-b2 cos2ϕ+b4 cos4ϕ1+b2 cos2ϕ+b4 cos4ϕtanϕ.
mj=3=-32 m2+2b2+b4+b22+b2cos4ϕ1/2-6+3bm+b+b3m cos2ϕ+3b3m cos4ϕ.
Idcm, ϕ=Io1+m cosϕdcJ0ϕac,
Iacm, ϕ, ω=Io2m cosϕdcn=1 J2nϕac×cos2nωt-2m sinϕdc×n=1 J2n-1ϕacsin2n-1ωt.
tanϕdc=-J2ϕacJ1ϕacIacωIac2ω.
M0=σ11h2b4,
σ11=d31YE3,
u3=-M02YI u12=-32d31ΔVh2 u12.

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