Abstract

We report on amplitude and phase imaging of out-of-plane sinusoidal surface vibration at nanometer scales with a heterodyne holographic interferometer. The originality of the proposed method is to make use of a multiplexed local oscillator to address several optical sidebands into the temporal bandwidth of a sensor array. This process is called coherent frequency-division multiplexing. It enables simultaneous recording and pixel-to-pixel division of sideband holograms, which permits quantitative wide-field mapping of optical phase-modulation depths. Additionally, a linear frequency chirp ensures the retrieval of the local mechanical phase shift of the vibration with respect to the excitation signal. The proposed approach is validated by quantitative motion characterization of the lamellophone of a musical box, behaving as a group of harmonic oscillators, under weak sinusoidal excitation. Images of the vibration amplitude versus excitation frequency show the resonance of the nanometric flexural response of one individual cantilever, at which a phase hop is measured.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Bosseboeuf and S. Petitgrand, “Characterization of the static and dynamic behaviour of M(O)EMS by optical techniques: status and trends,” J. Micromech. Microeng. 13, 1–16 (2003).
    [CrossRef]
  2. C. Rembe and A. Drabenstedt, “Laser-scanning confocal vibrometer microscope: theory and experiments,” Rev. Sci. Instrum. 77, 083702 (2006).
    [CrossRef]
  3. K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett. 92, 063502 (2008).
    [CrossRef]
  4. S. Bramhavar, B. Pouet, and T. W. Murray, “Superheterodyne detection of laser generated acoustic waves,” Appl. Phys. Lett. 94, 114102 (2009).
    [CrossRef]
  5. R. L. Whitman and A. Korpel, “Probing of acoustic surface perturbations by coherent light,” Appl. Opt. 8, 1567–1576 (1969).
    [CrossRef]
  6. R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic surface wave amplitude and phase measurements using laser probes,” Proc. IEEE 119, 117–126 (1972).
  7. G. Stegeman, “Optical probing of surface waves and surface wave devices,” IEEE Trans. Sonics Ultrason. 23, 33–63 (1976).
    [CrossRef]
  8. J.-P. Monchalin, “Heterodyne interferometric laser probe to measure continuous ultrasonic displacements,” Rev. Sci. Instrum. 56, 543–546 (1985).
    [CrossRef]
  9. J. W. Wagner and J. B. Spicer, “Theoretical noise-limited sensitivity of classical interferometry,” J. Opt. Soc. Am. B 4, 1316–1326 (1987).
    [CrossRef]
  10. D. Royer and E. Dieulesaint, “Mesures optiques de déplacements d’amplitude 10^-4 à 10^ 2 Angström: application aux ondes élastiques,” Rev. Phys. Appl. 24, 833–846 (1989).
    [CrossRef]
  11. D. Royer and E. Dieulesaint, “Optical probing of the mechanical impulse response of a transducer,” Appl. Phys. Lett. 49, 1056–1058 (1986).
    [CrossRef]
  12. X. Jia, A. Boumiz, and G. Quentin, “Laser interferometric detection of ultrasonic waves propagating inside a transparent solid,” Appl. Phys. Lett. 63, 2192–2194 (1993).
    [CrossRef]
  13. D. Royer and V. Kmetik, “Measurement of piezoelectric constants using an optical heterodyne interferometer,” Electron. Lett. 28, 1828–1830 (1992).
    [CrossRef]
  14. A. Kimachi, “Real-time heterodyne speckle pattern interferometry using the correlation image sensor,” Appl. Opt. 49, 6808–6815 (2010).
    [CrossRef]
  15. R. Patel, S. Achamfuo-Yeboah, R. Light, and M. Clark, “Widefield heterodyne interferometry using a custom CMOS modulated light camera,” Opt. Express 19, 24546–24556 (2011).
    [CrossRef]
  16. R. L. Powell and K. A. Stetson, “Interferometric vibration analysis by wavefront reconstruction,” J. Opt. Soc. Am. 55, 1593–1597 (1965).
    [CrossRef]
  17. C. C. Aleksoff, “Time average holography extended,” Appl. Phys. Lett. 14, 23–27 (1969).
    [CrossRef]
  18. K. A. Stetson, “Effects of beam modulation on fringe loci and localization in time-average hologram interferometry,” J. Opt. Soc. Am. 60, 1378–1388 (1970).
    [CrossRef]
  19. J. A. Levitt and K. A. Stetson, “Mechanical vibrations: mapping their phase with hologram interferometry,” Appl. Opt. 15, 195–199 (1976).
    [CrossRef]
  20. M. Ueda, S. Miida, and T. Sato, “Signal-to-noise ratio and smallest detectable vibration amplitude in frequency-translated holography: an analysis,” Appl. Opt. 15, 2690–2694 (1976).
    [CrossRef]
  21. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time-averaged digital holography,” Opt. Lett. 28, 1900–1902 (2003).
    [CrossRef]
  22. P. Psota, V. Ledl, R. Dolecek, J. Erhart, and V. Kopecky, “Measurement of piezoelectric transformer vibrations by digital holography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59, 1962–1968 (2012).
    [CrossRef]
  23. N. Verrier and M. Atlan, “Absolute measurement of small-amplitude vibrations by time-averaged heterodyne holography with a dual local oscillator,” Opt. Lett. 38, 739–741 (2013).
    [CrossRef]
  24. G. Pedrini, W. Osten, and M. E. Gusev, “High-speed digital holographic interferometry for vibration measurement,” Appl. Opt. 45, 3456–3462 (2006).
    [CrossRef]
  25. C. Perez-Lopez, M. H. De la Torre-Ibarra, and F. M. Santoyo, “Very high speed cw digital holographic interferometry,” Opt. Express 14, 9709–9715 (2006).
    [CrossRef]
  26. O. J. Løkberg and K. Høgmoen, “Vibration phase mapping using electronic speckle pattern interferometry,” Appl. Opt. 15, 2701–2704 (1976).
    [CrossRef]
  27. S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, and J. P. Gilles, “3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope,” Opt. Lasers Eng. 36, 77–101 (2001).
    [CrossRef]
  28. J. Leval, P. Picart, J. P. Boileau, and J. C. Pascal, “Full-field vibrometry with digital Fresnel holography,” Appl. Opt. 44, 5763–5772 (2005).
    [CrossRef]
  29. N. Verrier, M. Gross, and M. Atlan, “Phase-resolved heterodyne holographic vibrometry with a strobe local oscillator,” Opt. Lett. 38, 377–379 (2013).
    [CrossRef]
  30. M. Atlan and M. Gross, “Spatiotemporal heterodyne detection,” J. Opt. Soc. Am. A 24, 2701–2709 (2007).
    [CrossRef]
  31. M. Paturzo, P. Memmolo, A. Tulino, A. Finizio, and P. Ferraro, “Investigation of angular multiplexing and de-multiplexing of digital holograms recorded in microscope configuration,” Opt. Express 17, 8709–8718 (2009).
    [CrossRef]
  32. T. Kiire, D. Barada, J. I. Sugisaka, Y. Hayasaki, and T. Yatagai, “Color digital holography using a single monochromatic imaging sensor,” Opt. Lett. 37, 3153–3155 (2012).
    [CrossRef]
  33. T. Tahara, A. Maeda, Y. Awatsuji, T. Kakue, P. Xia, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Single-shot dual-illumination phase unwrapping using a single wavelength,” Opt. Lett. 37, 4002–4004 (2012).
    [CrossRef]
  34. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
    [CrossRef]

2013 (2)

2012 (3)

2011 (1)

2010 (1)

2009 (2)

2008 (1)

K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett. 92, 063502 (2008).
[CrossRef]

2007 (1)

2006 (3)

2005 (1)

2003 (2)

P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time-averaged digital holography,” Opt. Lett. 28, 1900–1902 (2003).
[CrossRef]

A. Bosseboeuf and S. Petitgrand, “Characterization of the static and dynamic behaviour of M(O)EMS by optical techniques: status and trends,” J. Micromech. Microeng. 13, 1–16 (2003).
[CrossRef]

2001 (1)

S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, and J. P. Gilles, “3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope,” Opt. Lasers Eng. 36, 77–101 (2001).
[CrossRef]

2000 (1)

1993 (1)

X. Jia, A. Boumiz, and G. Quentin, “Laser interferometric detection of ultrasonic waves propagating inside a transparent solid,” Appl. Phys. Lett. 63, 2192–2194 (1993).
[CrossRef]

1992 (1)

D. Royer and V. Kmetik, “Measurement of piezoelectric constants using an optical heterodyne interferometer,” Electron. Lett. 28, 1828–1830 (1992).
[CrossRef]

1989 (1)

D. Royer and E. Dieulesaint, “Mesures optiques de déplacements d’amplitude 10^-4 à 10^ 2 Angström: application aux ondes élastiques,” Rev. Phys. Appl. 24, 833–846 (1989).
[CrossRef]

1987 (1)

1986 (1)

D. Royer and E. Dieulesaint, “Optical probing of the mechanical impulse response of a transducer,” Appl. Phys. Lett. 49, 1056–1058 (1986).
[CrossRef]

1985 (1)

J.-P. Monchalin, “Heterodyne interferometric laser probe to measure continuous ultrasonic displacements,” Rev. Sci. Instrum. 56, 543–546 (1985).
[CrossRef]

1976 (4)

1972 (1)

R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic surface wave amplitude and phase measurements using laser probes,” Proc. IEEE 119, 117–126 (1972).

1970 (1)

1969 (2)

1965 (1)

Achamfuo-Yeboah, S.

Aleksoff, C. C.

C. C. Aleksoff, “Time average holography extended,” Appl. Phys. Lett. 14, 23–27 (1969).
[CrossRef]

Ash, E. A.

R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic surface wave amplitude and phase measurements using laser probes,” Proc. IEEE 119, 117–126 (1972).

Atlan, M.

Awatsuji, Y.

Barada, D.

Boileau, J. P.

Bosseboeuf, A.

A. Bosseboeuf and S. Petitgrand, “Characterization of the static and dynamic behaviour of M(O)EMS by optical techniques: status and trends,” J. Micromech. Microeng. 13, 1–16 (2003).
[CrossRef]

S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, and J. P. Gilles, “3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope,” Opt. Lasers Eng. 36, 77–101 (2001).
[CrossRef]

Boumiz, A.

X. Jia, A. Boumiz, and G. Quentin, “Laser interferometric detection of ultrasonic waves propagating inside a transparent solid,” Appl. Phys. Lett. 63, 2192–2194 (1993).
[CrossRef]

Bramhavar, S.

S. Bramhavar, B. Pouet, and T. W. Murray, “Superheterodyne detection of laser generated acoustic waves,” Appl. Phys. Lett. 94, 114102 (2009).
[CrossRef]

Clark, M.

Cuche, E.

Danaie, K.

S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, and J. P. Gilles, “3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope,” Opt. Lasers Eng. 36, 77–101 (2001).
[CrossRef]

De La Rue, R. M.

R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic surface wave amplitude and phase measurements using laser probes,” Proc. IEEE 119, 117–126 (1972).

De la Torre-Ibarra, M. H.

Depeursinge, C.

Dieulesaint, E.

D. Royer and E. Dieulesaint, “Mesures optiques de déplacements d’amplitude 10^-4 à 10^ 2 Angström: application aux ondes élastiques,” Rev. Phys. Appl. 24, 833–846 (1989).
[CrossRef]

D. Royer and E. Dieulesaint, “Optical probing of the mechanical impulse response of a transducer,” Appl. Phys. Lett. 49, 1056–1058 (1986).
[CrossRef]

Dolecek, R.

P. Psota, V. Ledl, R. Dolecek, J. Erhart, and V. Kopecky, “Measurement of piezoelectric transformer vibrations by digital holography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59, 1962–1968 (2012).
[CrossRef]

Drabenstedt, A.

C. Rembe and A. Drabenstedt, “Laser-scanning confocal vibrometer microscope: theory and experiments,” Rev. Sci. Instrum. 77, 083702 (2006).
[CrossRef]

Erhart, J.

P. Psota, V. Ledl, R. Dolecek, J. Erhart, and V. Kopecky, “Measurement of piezoelectric transformer vibrations by digital holography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59, 1962–1968 (2012).
[CrossRef]

Ferraro, P.

Finizio, A.

Gilles, J. P.

S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, and J. P. Gilles, “3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope,” Opt. Lasers Eng. 36, 77–101 (2001).
[CrossRef]

Gougeon, S.

Gross, M.

Gusev, M. E.

Hayasaki, Y.

Høgmoen, K.

Humphryes, R. F.

R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic surface wave amplitude and phase measurements using laser probes,” Proc. IEEE 119, 117–126 (1972).

Jia, X.

X. Jia, A. Boumiz, and G. Quentin, “Laser interferometric detection of ultrasonic waves propagating inside a transparent solid,” Appl. Phys. Lett. 63, 2192–2194 (1993).
[CrossRef]

Kaivola, M.

K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett. 92, 063502 (2008).
[CrossRef]

Kakue, T.

Kiire, T.

Kimachi, A.

Kmetik, V.

D. Royer and V. Kmetik, “Measurement of piezoelectric constants using an optical heterodyne interferometer,” Electron. Lett. 28, 1828–1830 (1992).
[CrossRef]

Kokkonen, K.

K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett. 92, 063502 (2008).
[CrossRef]

Kopecky, V.

P. Psota, V. Ledl, R. Dolecek, J. Erhart, and V. Kopecky, “Measurement of piezoelectric transformer vibrations by digital holography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59, 1962–1968 (2012).
[CrossRef]

Korpel, A.

Kubota, T.

Ledl, V.

P. Psota, V. Ledl, R. Dolecek, J. Erhart, and V. Kopecky, “Measurement of piezoelectric transformer vibrations by digital holography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59, 1962–1968 (2012).
[CrossRef]

Leval, J.

Levitt, J. A.

Light, R.

Løkberg, O. J.

Maeda, A.

Marquet, P.

Mason, I. M.

R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic surface wave amplitude and phase measurements using laser probes,” Proc. IEEE 119, 117–126 (1972).

Matoba, O.

Memmolo, P.

Miida, S.

Monchalin, J.-P.

J.-P. Monchalin, “Heterodyne interferometric laser probe to measure continuous ultrasonic displacements,” Rev. Sci. Instrum. 56, 543–546 (1985).
[CrossRef]

Mounier, D.

Murray, T. W.

S. Bramhavar, B. Pouet, and T. W. Murray, “Superheterodyne detection of laser generated acoustic waves,” Appl. Phys. Lett. 94, 114102 (2009).
[CrossRef]

Nishio, K.

Osten, W.

Pascal, J. C.

Patel, R.

Paturzo, M.

Pedrini, G.

Perez-Lopez, C.

Petitgrand, S.

A. Bosseboeuf and S. Petitgrand, “Characterization of the static and dynamic behaviour of M(O)EMS by optical techniques: status and trends,” J. Micromech. Microeng. 13, 1–16 (2003).
[CrossRef]

S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, and J. P. Gilles, “3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope,” Opt. Lasers Eng. 36, 77–101 (2001).
[CrossRef]

Picart, P.

Pouet, B.

S. Bramhavar, B. Pouet, and T. W. Murray, “Superheterodyne detection of laser generated acoustic waves,” Appl. Phys. Lett. 94, 114102 (2009).
[CrossRef]

Powell, R. L.

Psota, P.

P. Psota, V. Ledl, R. Dolecek, J. Erhart, and V. Kopecky, “Measurement of piezoelectric transformer vibrations by digital holography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59, 1962–1968 (2012).
[CrossRef]

Quentin, G.

X. Jia, A. Boumiz, and G. Quentin, “Laser interferometric detection of ultrasonic waves propagating inside a transparent solid,” Appl. Phys. Lett. 63, 2192–2194 (1993).
[CrossRef]

Rembe, C.

C. Rembe and A. Drabenstedt, “Laser-scanning confocal vibrometer microscope: theory and experiments,” Rev. Sci. Instrum. 77, 083702 (2006).
[CrossRef]

Royer, D.

D. Royer and V. Kmetik, “Measurement of piezoelectric constants using an optical heterodyne interferometer,” Electron. Lett. 28, 1828–1830 (1992).
[CrossRef]

D. Royer and E. Dieulesaint, “Mesures optiques de déplacements d’amplitude 10^-4 à 10^ 2 Angström: application aux ondes élastiques,” Rev. Phys. Appl. 24, 833–846 (1989).
[CrossRef]

D. Royer and E. Dieulesaint, “Optical probing of the mechanical impulse response of a transducer,” Appl. Phys. Lett. 49, 1056–1058 (1986).
[CrossRef]

Santoyo, F. M.

Sato, T.

Spicer, J. B.

Stegeman, G.

G. Stegeman, “Optical probing of surface waves and surface wave devices,” IEEE Trans. Sonics Ultrason. 23, 33–63 (1976).
[CrossRef]

Stetson, K. A.

Sugisaka, J. I.

Tahara, T.

Tulino, A.

Ueda, M.

Ura, S.

Verrier, N.

Wagner, J. W.

Whitman, R. L.

Xia, P.

Yahiaoui, R.

S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, and J. P. Gilles, “3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope,” Opt. Lasers Eng. 36, 77–101 (2001).
[CrossRef]

Yatagai, T.

Appl. Opt. (8)

Appl. Phys. Lett. (5)

C. C. Aleksoff, “Time average holography extended,” Appl. Phys. Lett. 14, 23–27 (1969).
[CrossRef]

K. Kokkonen and M. Kaivola, “Scanning heterodyne laser interferometer for phase-sensitive absolute-amplitude measurements of surface vibrations,” Appl. Phys. Lett. 92, 063502 (2008).
[CrossRef]

S. Bramhavar, B. Pouet, and T. W. Murray, “Superheterodyne detection of laser generated acoustic waves,” Appl. Phys. Lett. 94, 114102 (2009).
[CrossRef]

D. Royer and E. Dieulesaint, “Optical probing of the mechanical impulse response of a transducer,” Appl. Phys. Lett. 49, 1056–1058 (1986).
[CrossRef]

X. Jia, A. Boumiz, and G. Quentin, “Laser interferometric detection of ultrasonic waves propagating inside a transparent solid,” Appl. Phys. Lett. 63, 2192–2194 (1993).
[CrossRef]

Electron. Lett. (1)

D. Royer and V. Kmetik, “Measurement of piezoelectric constants using an optical heterodyne interferometer,” Electron. Lett. 28, 1828–1830 (1992).
[CrossRef]

IEEE Trans. Sonics Ultrason. (1)

G. Stegeman, “Optical probing of surface waves and surface wave devices,” IEEE Trans. Sonics Ultrason. 23, 33–63 (1976).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

P. Psota, V. Ledl, R. Dolecek, J. Erhart, and V. Kopecky, “Measurement of piezoelectric transformer vibrations by digital holography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59, 1962–1968 (2012).
[CrossRef]

J. Micromech. Microeng. (1)

A. Bosseboeuf and S. Petitgrand, “Characterization of the static and dynamic behaviour of M(O)EMS by optical techniques: status and trends,” J. Micromech. Microeng. 13, 1–16 (2003).
[CrossRef]

J. Opt. Soc. Am. (2)

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

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

Opt. Express (3)

Opt. Lasers Eng. (1)

S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, and J. P. Gilles, “3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope,” Opt. Lasers Eng. 36, 77–101 (2001).
[CrossRef]

Opt. Lett. (5)

Proc. IEEE (1)

R. M. De La Rue, R. F. Humphryes, I. M. Mason, and E. A. Ash, “Acoustic surface wave amplitude and phase measurements using laser probes,” Proc. IEEE 119, 117–126 (1972).

Rev. Phys. Appl. (1)

D. Royer and E. Dieulesaint, “Mesures optiques de déplacements d’amplitude 10^-4 à 10^ 2 Angström: application aux ondes élastiques,” Rev. Phys. Appl. 24, 833–846 (1989).
[CrossRef]

Rev. Sci. Instrum. (2)

C. Rembe and A. Drabenstedt, “Laser-scanning confocal vibrometer microscope: theory and experiments,” Rev. Sci. Instrum. 77, 083702 (2006).
[CrossRef]

J.-P. Monchalin, “Heterodyne interferometric laser probe to measure continuous ultrasonic displacements,” Rev. Sci. Instrum. 56, 543–546 (1985).
[CrossRef]

Supplementary Material (2)

» Media 1: AVI (5387 KB)     
» Media 2: AVI (2533 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

Experimental setup.

Fig. 2.
Fig. 2.

(a) Magnitude of a statically scattered light hologram |H˜0|. Magnitude of sideband holograms (b) |H˜1| and (c) |H˜1|. (d) Amplitude map of the flexural response, z0, of the first cantilever for the first resonance at ω/(2π)=541Hz. Phase images, ψ, calculated from 128 raw interferograms around (e) 540.9 Hz and (f) 541.8 Hz, in the neighborhood of the first resonance. Movies of the amplitude and phase maps of the out-of-plane vibration are reported in Media 1 and Media 2.

Fig. 3.
Fig. 3.

Magnitude of the FFT-spectrum [Eqs. (12) and (13)], where three modulation sidebands are addressed.

Fig. 4.
Fig. 4.

Vibration amplitude z0 versus excitation frequency ω/(2π), averaged over the (a) 1st, (b) 5th, and 17th cantilevers. Insets: vibration amplitude maps at (a) 541 Hz, (b) 1006 Hz, and (c) 2211 Hz.

Fig. 5.
Fig. 5.

Vibration amplitude z0 (a) and phase ψ (b) averaged over the first cantilever, arrowed in Fig. 2(b), versus excitation frequency ω/(2π). Insets: retrieved vibration amplitude and phase maps in the neighborhood of the resonance, reported in Figs. 2(b)2(f). The points were obtained from a sequential measurement, the lines were obtained with the linear chirp [Eq. (16)]. The theoretical resonance lines in gray were determined from Eqs. (3) and (4).

Tables (1)

Tables Icon

Table 1. Frequency Shifts Used in Experiments

Equations (16)

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

m2zt2+czt+kz=F(t).
z(t)=z0sin(ωt+ψ).
z0=F0mω[(2ω0ζ)2+1ω2(ω02ω2)2]1/2.
ψ=arctan(2ωω0ζω2ω02).
E=Eexp[iωLt+iϕ(t)].
E=Eexp(iωLt)nJn(ϕ0)exp[in(ωt+ψ)],
En=EJn(ϕ0)exp(inψ),
E1E02πλz0exp(iψ).
ELO=ELOexp(iωLt)n=13αnexp(iΔωnt),
H(t)=E(t)ELO*(t),
H(t)=n=13En2ELO*αnexp(iδωnt),
H˜(ωn)=p=1NH(2πp/ωS)exp(2inpπ/N),
H˜m=H˜(δωm+2)=AEm,
z0λ2π|H˜1/H˜0|.
ψψ0=Arg(H˜1/H˜0),
ω(t)=ωI+(ωFωI)t/T.

Metrics