Abstract

Acoustic vibrations in tissue are often difficult to image, requiring high-speed scanning, high sensitivity and nanometer-scale axial resolution. Here we use spectrally encoded interferometry to measure the vibration pattern of two-dimensional surfaces, including the skin of a volunteer, at nanometric resolution, without the need for rapid lateral scanning and with no prior knowledge of the driving acoustic waveform. Our results demonstrate the feasibility of this technique for measuring tissue biomechanics using simple and compact imaging probes.

© 2013 OSA

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  1. A. M. Huber, C. Schwab, T. Linder, S. J. Stoeckli, M. Ferrazzini, N. Dillier, and U. Fisch, “Evaluation of Eardrum Laser Doppler Interferometry as a Diagnostic Tool,” Laryngoscope 111(3), 501–507 (2001).
    [Crossref] [PubMed]
  2. J. J. Rosowski, H. H. Nakajima, and S. N. Merchant, “Clinical utility of laser-Doppler vibrometer measurements in live normal and pathologic human ears,” Ear Hear. 29(1), 3–19 (2008).
    [PubMed]
  3. K. R. Whittemore, S. N. Merchant, B. B. Poon, and J. J. Rosowski, “A normative study of tympanic membrane motion in humans using a laser Doppler vibrometer (LDV),” Hear. Res. 187(1-2), 85–104 (2004).
    [Crossref] [PubMed]
  4. E. W. Chang, J. B. Kobler, and S. H. Yun, “Subnanometer optical coherence tomographic vibrography,” Opt. Lett. 37(17), 3678–3680 (2012).
    [Crossref] [PubMed]
  5. G. Liu, M. Rubinstein, A. Saidi, W. Qi, A. Foulad, B. Wong, and Z. Chen, “Imaging vibrating vocal folds with a high speed 1050 nm swept source OCT and ODT,” Opt. Express 19(12), 11880–11889 (2011).
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  6. A. Burkhardt, L. Kirsten, M. Bornitz, T. Zahnert, and E. Koch, “Investigation of the human tympanic membrane oscillation ex vivo by Doppler optical coherence tomography,” J. Biophotonics (2012).
    [Crossref] [PubMed]
  7. E. W. Chang, J. B. Kobler, and S. H. Yun, “Triggered optical coherence tomography for capturing rapid periodic motion,” Sci. Rep. 1, 48 (2011).
    [Crossref] [PubMed]
  8. J. T. Cheng, A. A. Aarnisalo, E. Harrington, M. S. Hernandez-Montes, C. Furlong, S. N. Merchant, and J. J. Rosowski, “Motion of the surface of the human tympanic membrane measured with stroboscopic holography,” Hear. Res. 263(1-2), 66–77 (2010).
    [Crossref] [PubMed]
  9. J. J. Rosowski, I. Dobrev, M. Khaleghi, W. Lu, J. T. Cheng, E. Harrington, and C. Furlong, “Measurements of three-dimensional shape and sound-induced motion of the chinchilla tympanic membrane,” Hear. Res. 301, 44–52 (2013).
    [Crossref] [PubMed]
  10. G. J. Tearney, R. H. Webb, and B. E. Bouma, “Spectrally encoded confocal microscopy,” Opt. Lett. 23(15), 1152–1154 (1998).
    [Crossref] [PubMed]
  11. D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
    [Crossref] [PubMed]
  12. D. Yelin, B. E. Bouma, J. J. Rosowsky, and G. J. Tearney, “Doppler imaging using spectrally-encoded endoscopy,” Opt. Express 16(19), 14836–14844 (2008).
    [Crossref] [PubMed]
  13. K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
    [Crossref] [PubMed]
  14. M. Merman, A. Abramov, and D. Yelin, “Theoretical analysis of spectrally encoded endoscopy,” Opt. Express 17(26), 24045–24059 (2009).
    [Crossref] [PubMed]
  15. R. Onodera, H. Watanabe, and Y. Ishii, “Interferometric Phase-Measurement Using a One-Dimensional Discrete Hilbert Transform,” Opt. Rev. 12(1), 29–36 (2005).
    [Crossref]
  16. Y. Zhao, Z. Chen, C. Saxer, S. Xiang, J. F. de Boer, and J. S. Nelson, “Phase-resolved optical coherence tomography and optical Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. 25(2), 114–116 (2000).
    [Crossref] [PubMed]
  17. R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson, R. I. Mills, and T. D. Rossing, “Effects of air loading on timpani membrane vibrations,” J. Acoust. Soc. Am. 76(5), 1336–1345 (1984).
    [Crossref]
  18. J. Bishop, G. Poole, M. Leitch, and D. B. Plewes, “Magnetic resonance imaging of shear wave propagation in excised tissue,” J. Magn. Reson. Imaging 8(6), 1257–1265 (1998).
    [Crossref] [PubMed]
  19. A. P. Sarvazyan, O. V. Rudenko, S. D. Swanson, J. B. Fowlkes, and S. Y. Emelianov, “Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics,” Ultrasound Med. Biol. 24(9), 1419–1435 (1998).
    [Crossref] [PubMed]
  20. S. Chen, M. Fatemi, and J. F. Greenleaf, “Quantifying elasticity and viscosity from measurement of shear wave speed dispersion,” J. Acoust. Soc. Am. 115(6), 2781–2785 (2004).
    [Crossref] [PubMed]
  21. A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
    [Crossref] [PubMed]
  22. P. G. Stelmachowicz, K. A. Beauchaine, A. Kalberer, and W. Jesteadt, “Normative thresholds in the 8- to 20-kHz range as a function of age,” J. Acoust. Soc. Am. 86(4), 1384–1391 (1989).
    [Crossref] [PubMed]
  23. C. Boudoux, S. Yun, W. Oh, W. White, N. Iftimia, M. Shishkov, B. Bouma, and G. Tearney, “Rapid wavelength-swept spectrally encoded confocal microscopy,” Opt. Express 13(20), 8214–8221 (2005).
    [Crossref] [PubMed]

2013 (1)

J. J. Rosowski, I. Dobrev, M. Khaleghi, W. Lu, J. T. Cheng, E. Harrington, and C. Furlong, “Measurements of three-dimensional shape and sound-induced motion of the chinchilla tympanic membrane,” Hear. Res. 301, 44–52 (2013).
[Crossref] [PubMed]

2012 (3)

A. Burkhardt, L. Kirsten, M. Bornitz, T. Zahnert, and E. Koch, “Investigation of the human tympanic membrane oscillation ex vivo by Doppler optical coherence tomography,” J. Biophotonics (2012).
[Crossref] [PubMed]

E. W. Chang, J. B. Kobler, and S. H. Yun, “Subnanometer optical coherence tomographic vibrography,” Opt. Lett. 37(17), 3678–3680 (2012).
[Crossref] [PubMed]

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (1)

J. T. Cheng, A. A. Aarnisalo, E. Harrington, M. S. Hernandez-Montes, C. Furlong, S. N. Merchant, and J. J. Rosowski, “Motion of the surface of the human tympanic membrane measured with stroboscopic holography,” Hear. Res. 263(1-2), 66–77 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (2)

J. J. Rosowski, H. H. Nakajima, and S. N. Merchant, “Clinical utility of laser-Doppler vibrometer measurements in live normal and pathologic human ears,” Ear Hear. 29(1), 3–19 (2008).
[PubMed]

D. Yelin, B. E. Bouma, J. J. Rosowsky, and G. J. Tearney, “Doppler imaging using spectrally-encoded endoscopy,” Opt. Express 16(19), 14836–14844 (2008).
[Crossref] [PubMed]

2006 (1)

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

2005 (2)

R. Onodera, H. Watanabe, and Y. Ishii, “Interferometric Phase-Measurement Using a One-Dimensional Discrete Hilbert Transform,” Opt. Rev. 12(1), 29–36 (2005).
[Crossref]

C. Boudoux, S. Yun, W. Oh, W. White, N. Iftimia, M. Shishkov, B. Bouma, and G. Tearney, “Rapid wavelength-swept spectrally encoded confocal microscopy,” Opt. Express 13(20), 8214–8221 (2005).
[Crossref] [PubMed]

2004 (2)

S. Chen, M. Fatemi, and J. F. Greenleaf, “Quantifying elasticity and viscosity from measurement of shear wave speed dispersion,” J. Acoust. Soc. Am. 115(6), 2781–2785 (2004).
[Crossref] [PubMed]

K. R. Whittemore, S. N. Merchant, B. B. Poon, and J. J. Rosowski, “A normative study of tympanic membrane motion in humans using a laser Doppler vibrometer (LDV),” Hear. Res. 187(1-2), 85–104 (2004).
[Crossref] [PubMed]

2001 (2)

A. M. Huber, C. Schwab, T. Linder, S. J. Stoeckli, M. Ferrazzini, N. Dillier, and U. Fisch, “Evaluation of Eardrum Laser Doppler Interferometry as a Diagnostic Tool,” Laryngoscope 111(3), 501–507 (2001).
[Crossref] [PubMed]

A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
[Crossref] [PubMed]

2000 (1)

1998 (3)

J. Bishop, G. Poole, M. Leitch, and D. B. Plewes, “Magnetic resonance imaging of shear wave propagation in excised tissue,” J. Magn. Reson. Imaging 8(6), 1257–1265 (1998).
[Crossref] [PubMed]

A. P. Sarvazyan, O. V. Rudenko, S. D. Swanson, J. B. Fowlkes, and S. Y. Emelianov, “Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics,” Ultrasound Med. Biol. 24(9), 1419–1435 (1998).
[Crossref] [PubMed]

G. J. Tearney, R. H. Webb, and B. E. Bouma, “Spectrally encoded confocal microscopy,” Opt. Lett. 23(15), 1152–1154 (1998).
[Crossref] [PubMed]

1989 (1)

P. G. Stelmachowicz, K. A. Beauchaine, A. Kalberer, and W. Jesteadt, “Normative thresholds in the 8- to 20-kHz range as a function of age,” J. Acoust. Soc. Am. 86(4), 1384–1391 (1989).
[Crossref] [PubMed]

1984 (1)

R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson, R. I. Mills, and T. D. Rossing, “Effects of air loading on timpani membrane vibrations,” J. Acoust. Soc. Am. 76(5), 1336–1345 (1984).
[Crossref]

Aarnisalo, A. A.

J. T. Cheng, A. A. Aarnisalo, E. Harrington, M. S. Hernandez-Montes, C. Furlong, S. N. Merchant, and J. J. Rosowski, “Motion of the surface of the human tympanic membrane measured with stroboscopic holography,” Hear. Res. 263(1-2), 66–77 (2010).
[Crossref] [PubMed]

Abramov, A.

Adam, J.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Amromin, E.

A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
[Crossref] [PubMed]

Anderson, C. A.

R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson, R. I. Mills, and T. D. Rossing, “Effects of air loading on timpani membrane vibrations,” J. Acoust. Soc. Am. 76(5), 1336–1345 (1984).
[Crossref]

Ayazi, A.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Beauchaine, K. A.

P. G. Stelmachowicz, K. A. Beauchaine, A. Kalberer, and W. Jesteadt, “Normative thresholds in the 8- to 20-kHz range as a function of age,” J. Acoust. Soc. Am. 86(4), 1384–1391 (1989).
[Crossref] [PubMed]

Bishop, J.

J. Bishop, G. Poole, M. Leitch, and D. B. Plewes, “Magnetic resonance imaging of shear wave propagation in excised tissue,” J. Magn. Reson. Imaging 8(6), 1257–1265 (1998).
[Crossref] [PubMed]

Bornitz, M.

A. Burkhardt, L. Kirsten, M. Bornitz, T. Zahnert, and E. Koch, “Investigation of the human tympanic membrane oscillation ex vivo by Doppler optical coherence tomography,” J. Biophotonics (2012).
[Crossref] [PubMed]

Boudoux, C.

Bouma, B.

Bouma, B. E.

Brown, R.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Burkhardt, A.

A. Burkhardt, L. Kirsten, M. Bornitz, T. Zahnert, and E. Koch, “Investigation of the human tympanic membrane oscillation ex vivo by Doppler optical coherence tomography,” J. Biophotonics (2012).
[Crossref] [PubMed]

Chang, E. W.

E. W. Chang, J. B. Kobler, and S. H. Yun, “Subnanometer optical coherence tomographic vibrography,” Opt. Lett. 37(17), 3678–3680 (2012).
[Crossref] [PubMed]

E. W. Chang, J. B. Kobler, and S. H. Yun, “Triggered optical coherence tomography for capturing rapid periodic motion,” Sci. Rep. 1, 48 (2011).
[Crossref] [PubMed]

Chen, E.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Chen, S.

S. Chen, M. Fatemi, and J. F. Greenleaf, “Quantifying elasticity and viscosity from measurement of shear wave speed dispersion,” J. Acoust. Soc. Am. 115(6), 2781–2785 (2004).
[Crossref] [PubMed]

Chen, Z.

Cheng, J. T.

J. J. Rosowski, I. Dobrev, M. Khaleghi, W. Lu, J. T. Cheng, E. Harrington, and C. Furlong, “Measurements of three-dimensional shape and sound-induced motion of the chinchilla tympanic membrane,” Hear. Res. 301, 44–52 (2013).
[Crossref] [PubMed]

J. T. Cheng, A. A. Aarnisalo, E. Harrington, M. S. Hernandez-Montes, C. Furlong, S. N. Merchant, and J. J. Rosowski, “Motion of the surface of the human tympanic membrane measured with stroboscopic holography,” Hear. Res. 263(1-2), 66–77 (2010).
[Crossref] [PubMed]

Christian, R. S.

R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson, R. I. Mills, and T. D. Rossing, “Effects of air loading on timpani membrane vibrations,” J. Acoust. Soc. Am. 76(5), 1336–1345 (1984).
[Crossref]

Davis, R. E.

R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson, R. I. Mills, and T. D. Rossing, “Effects of air loading on timpani membrane vibrations,” J. Acoust. Soc. Am. 76(5), 1336–1345 (1984).
[Crossref]

de Boer, J. F.

Di Carlo, D.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Dillier, N.

A. M. Huber, C. Schwab, T. Linder, S. J. Stoeckli, M. Ferrazzini, N. Dillier, and U. Fisch, “Evaluation of Eardrum Laser Doppler Interferometry as a Diagnostic Tool,” Laryngoscope 111(3), 501–507 (2001).
[Crossref] [PubMed]

Dobrev, I.

J. J. Rosowski, I. Dobrev, M. Khaleghi, W. Lu, J. T. Cheng, E. Harrington, and C. Furlong, “Measurements of three-dimensional shape and sound-induced motion of the chinchilla tympanic membrane,” Hear. Res. 301, 44–52 (2013).
[Crossref] [PubMed]

Dresner, M. A.

A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
[Crossref] [PubMed]

Ehman, R. L.

A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
[Crossref] [PubMed]

Emelianov, S. Y.

A. P. Sarvazyan, O. V. Rudenko, S. D. Swanson, J. B. Fowlkes, and S. Y. Emelianov, “Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics,” Ultrasound Med. Biol. 24(9), 1419–1435 (1998).
[Crossref] [PubMed]

Fard, A.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Fatemi, M.

S. Chen, M. Fatemi, and J. F. Greenleaf, “Quantifying elasticity and viscosity from measurement of shear wave speed dispersion,” J. Acoust. Soc. Am. 115(6), 2781–2785 (2004).
[Crossref] [PubMed]

Felmlee, J. P.

A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
[Crossref] [PubMed]

Ferrazzini, M.

A. M. Huber, C. Schwab, T. Linder, S. J. Stoeckli, M. Ferrazzini, N. Dillier, and U. Fisch, “Evaluation of Eardrum Laser Doppler Interferometry as a Diagnostic Tool,” Laryngoscope 111(3), 501–507 (2001).
[Crossref] [PubMed]

Fisch, U.

A. M. Huber, C. Schwab, T. Linder, S. J. Stoeckli, M. Ferrazzini, N. Dillier, and U. Fisch, “Evaluation of Eardrum Laser Doppler Interferometry as a Diagnostic Tool,” Laryngoscope 111(3), 501–507 (2001).
[Crossref] [PubMed]

Foulad, A.

Fowlkes, J. B.

A. P. Sarvazyan, O. V. Rudenko, S. D. Swanson, J. B. Fowlkes, and S. Y. Emelianov, “Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics,” Ultrasound Med. Biol. 24(9), 1419–1435 (1998).
[Crossref] [PubMed]

Furlong, C.

J. J. Rosowski, I. Dobrev, M. Khaleghi, W. Lu, J. T. Cheng, E. Harrington, and C. Furlong, “Measurements of three-dimensional shape and sound-induced motion of the chinchilla tympanic membrane,” Hear. Res. 301, 44–52 (2013).
[Crossref] [PubMed]

J. T. Cheng, A. A. Aarnisalo, E. Harrington, M. S. Hernandez-Montes, C. Furlong, S. N. Merchant, and J. J. Rosowski, “Motion of the surface of the human tympanic membrane measured with stroboscopic holography,” Hear. Res. 263(1-2), 66–77 (2010).
[Crossref] [PubMed]

Goda, K.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Gossett, D. R.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Greenleaf, J. F.

S. Chen, M. Fatemi, and J. F. Greenleaf, “Quantifying elasticity and viscosity from measurement of shear wave speed dispersion,” J. Acoust. Soc. Am. 115(6), 2781–2785 (2004).
[Crossref] [PubMed]

A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
[Crossref] [PubMed]

Harrington, E.

J. J. Rosowski, I. Dobrev, M. Khaleghi, W. Lu, J. T. Cheng, E. Harrington, and C. Furlong, “Measurements of three-dimensional shape and sound-induced motion of the chinchilla tympanic membrane,” Hear. Res. 301, 44–52 (2013).
[Crossref] [PubMed]

J. T. Cheng, A. A. Aarnisalo, E. Harrington, M. S. Hernandez-Montes, C. Furlong, S. N. Merchant, and J. J. Rosowski, “Motion of the surface of the human tympanic membrane measured with stroboscopic holography,” Hear. Res. 263(1-2), 66–77 (2010).
[Crossref] [PubMed]

Hasan, T.

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

Hernandez-Montes, M. S.

J. T. Cheng, A. A. Aarnisalo, E. Harrington, M. S. Hernandez-Montes, C. Furlong, S. N. Merchant, and J. J. Rosowski, “Motion of the surface of the human tympanic membrane measured with stroboscopic holography,” Hear. Res. 263(1-2), 66–77 (2010).
[Crossref] [PubMed]

Huber, A. M.

A. M. Huber, C. Schwab, T. Linder, S. J. Stoeckli, M. Ferrazzini, N. Dillier, and U. Fisch, “Evaluation of Eardrum Laser Doppler Interferometry as a Diagnostic Tool,” Laryngoscope 111(3), 501–507 (2001).
[Crossref] [PubMed]

Iftimia, N.

Ishii, Y.

R. Onodera, H. Watanabe, and Y. Ishii, “Interferometric Phase-Measurement Using a One-Dimensional Discrete Hilbert Transform,” Opt. Rev. 12(1), 29–36 (2005).
[Crossref]

Jalali, B.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Jesteadt, W.

P. G. Stelmachowicz, K. A. Beauchaine, A. Kalberer, and W. Jesteadt, “Normative thresholds in the 8- to 20-kHz range as a function of age,” J. Acoust. Soc. Am. 86(4), 1384–1391 (1989).
[Crossref] [PubMed]

Kalberer, A.

P. G. Stelmachowicz, K. A. Beauchaine, A. Kalberer, and W. Jesteadt, “Normative thresholds in the 8- to 20-kHz range as a function of age,” J. Acoust. Soc. Am. 86(4), 1384–1391 (1989).
[Crossref] [PubMed]

Khaleghi, M.

J. J. Rosowski, I. Dobrev, M. Khaleghi, W. Lu, J. T. Cheng, E. Harrington, and C. Furlong, “Measurements of three-dimensional shape and sound-induced motion of the chinchilla tympanic membrane,” Hear. Res. 301, 44–52 (2013).
[Crossref] [PubMed]

Kirsten, L.

A. Burkhardt, L. Kirsten, M. Bornitz, T. Zahnert, and E. Koch, “Investigation of the human tympanic membrane oscillation ex vivo by Doppler optical coherence tomography,” J. Biophotonics (2012).
[Crossref] [PubMed]

Kobler, J. B.

E. W. Chang, J. B. Kobler, and S. H. Yun, “Subnanometer optical coherence tomographic vibrography,” Opt. Lett. 37(17), 3678–3680 (2012).
[Crossref] [PubMed]

E. W. Chang, J. B. Kobler, and S. H. Yun, “Triggered optical coherence tomography for capturing rapid periodic motion,” Sci. Rep. 1, 48 (2011).
[Crossref] [PubMed]

Koch, E.

A. Burkhardt, L. Kirsten, M. Bornitz, T. Zahnert, and E. Koch, “Investigation of the human tympanic membrane oscillation ex vivo by Doppler optical coherence tomography,” J. Biophotonics (2012).
[Crossref] [PubMed]

Kruse, S. A.

A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
[Crossref] [PubMed]

Leitch, M.

J. Bishop, G. Poole, M. Leitch, and D. B. Plewes, “Magnetic resonance imaging of shear wave propagation in excised tissue,” J. Magn. Reson. Imaging 8(6), 1257–1265 (1998).
[Crossref] [PubMed]

Linder, T.

A. M. Huber, C. Schwab, T. Linder, S. J. Stoeckli, M. Ferrazzini, N. Dillier, and U. Fisch, “Evaluation of Eardrum Laser Doppler Interferometry as a Diagnostic Tool,” Laryngoscope 111(3), 501–507 (2001).
[Crossref] [PubMed]

Liu, G.

Liu, Y.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Lu, W.

J. J. Rosowski, I. Dobrev, M. Khaleghi, W. Lu, J. T. Cheng, E. Harrington, and C. Furlong, “Measurements of three-dimensional shape and sound-induced motion of the chinchilla tympanic membrane,” Hear. Res. 301, 44–52 (2013).
[Crossref] [PubMed]

Mahjoubfar, A.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Mahowald, J. L.

A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
[Crossref] [PubMed]

Malik, O.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Manduca, A.

A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
[Crossref] [PubMed]

Merchant, S. N.

J. T. Cheng, A. A. Aarnisalo, E. Harrington, M. S. Hernandez-Montes, C. Furlong, S. N. Merchant, and J. J. Rosowski, “Motion of the surface of the human tympanic membrane measured with stroboscopic holography,” Hear. Res. 263(1-2), 66–77 (2010).
[Crossref] [PubMed]

J. J. Rosowski, H. H. Nakajima, and S. N. Merchant, “Clinical utility of laser-Doppler vibrometer measurements in live normal and pathologic human ears,” Ear Hear. 29(1), 3–19 (2008).
[PubMed]

K. R. Whittemore, S. N. Merchant, B. B. Poon, and J. J. Rosowski, “A normative study of tympanic membrane motion in humans using a laser Doppler vibrometer (LDV),” Hear. Res. 187(1-2), 85–104 (2004).
[Crossref] [PubMed]

Merman, M.

Mills, R. I.

R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson, R. I. Mills, and T. D. Rossing, “Effects of air loading on timpani membrane vibrations,” J. Acoust. Soc. Am. 76(5), 1336–1345 (1984).
[Crossref]

Motz, J. T.

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

Nakajima, H. H.

J. J. Rosowski, H. H. Nakajima, and S. N. Merchant, “Clinical utility of laser-Doppler vibrometer measurements in live normal and pathologic human ears,” Ear Hear. 29(1), 3–19 (2008).
[PubMed]

Nelson, J. S.

Oh, W.

Oliphant, T. E.

A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
[Crossref] [PubMed]

Onodera, R.

R. Onodera, H. Watanabe, and Y. Ishii, “Interferometric Phase-Measurement Using a One-Dimensional Discrete Hilbert Transform,” Opt. Rev. 12(1), 29–36 (2005).
[Crossref]

Plewes, D. B.

J. Bishop, G. Poole, M. Leitch, and D. B. Plewes, “Magnetic resonance imaging of shear wave propagation in excised tissue,” J. Magn. Reson. Imaging 8(6), 1257–1265 (1998).
[Crossref] [PubMed]

Poole, G.

J. Bishop, G. Poole, M. Leitch, and D. B. Plewes, “Magnetic resonance imaging of shear wave propagation in excised tissue,” J. Magn. Reson. Imaging 8(6), 1257–1265 (1998).
[Crossref] [PubMed]

Poon, B. B.

K. R. Whittemore, S. N. Merchant, B. B. Poon, and J. J. Rosowski, “A normative study of tympanic membrane motion in humans using a laser Doppler vibrometer (LDV),” Hear. Res. 187(1-2), 85–104 (2004).
[Crossref] [PubMed]

Qi, W.

Rizvi, I.

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

Rosowski, J. J.

J. J. Rosowski, I. Dobrev, M. Khaleghi, W. Lu, J. T. Cheng, E. Harrington, and C. Furlong, “Measurements of three-dimensional shape and sound-induced motion of the chinchilla tympanic membrane,” Hear. Res. 301, 44–52 (2013).
[Crossref] [PubMed]

J. T. Cheng, A. A. Aarnisalo, E. Harrington, M. S. Hernandez-Montes, C. Furlong, S. N. Merchant, and J. J. Rosowski, “Motion of the surface of the human tympanic membrane measured with stroboscopic holography,” Hear. Res. 263(1-2), 66–77 (2010).
[Crossref] [PubMed]

J. J. Rosowski, H. H. Nakajima, and S. N. Merchant, “Clinical utility of laser-Doppler vibrometer measurements in live normal and pathologic human ears,” Ear Hear. 29(1), 3–19 (2008).
[PubMed]

K. R. Whittemore, S. N. Merchant, B. B. Poon, and J. J. Rosowski, “A normative study of tympanic membrane motion in humans using a laser Doppler vibrometer (LDV),” Hear. Res. 187(1-2), 85–104 (2004).
[Crossref] [PubMed]

Rosowsky, J. J.

Rossing, T. D.

R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson, R. I. Mills, and T. D. Rossing, “Effects of air loading on timpani membrane vibrations,” J. Acoust. Soc. Am. 76(5), 1336–1345 (1984).
[Crossref]

Rubinstein, M.

Rudenko, O. V.

A. P. Sarvazyan, O. V. Rudenko, S. D. Swanson, J. B. Fowlkes, and S. Y. Emelianov, “Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics,” Ultrasound Med. Biol. 24(9), 1419–1435 (1998).
[Crossref] [PubMed]

Saidi, A.

Sarkhosh, N.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Sarvazyan, A. P.

A. P. Sarvazyan, O. V. Rudenko, S. D. Swanson, J. B. Fowlkes, and S. Y. Emelianov, “Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics,” Ultrasound Med. Biol. 24(9), 1419–1435 (1998).
[Crossref] [PubMed]

Saxer, C.

Schwab, C.

A. M. Huber, C. Schwab, T. Linder, S. J. Stoeckli, M. Ferrazzini, N. Dillier, and U. Fisch, “Evaluation of Eardrum Laser Doppler Interferometry as a Diagnostic Tool,” Laryngoscope 111(3), 501–507 (2001).
[Crossref] [PubMed]

Shishkov, M.

Sollier, E.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Stelmachowicz, P. G.

P. G. Stelmachowicz, K. A. Beauchaine, A. Kalberer, and W. Jesteadt, “Normative thresholds in the 8- to 20-kHz range as a function of age,” J. Acoust. Soc. Am. 86(4), 1384–1391 (1989).
[Crossref] [PubMed]

Stoeckli, S. J.

A. M. Huber, C. Schwab, T. Linder, S. J. Stoeckli, M. Ferrazzini, N. Dillier, and U. Fisch, “Evaluation of Eardrum Laser Doppler Interferometry as a Diagnostic Tool,” Laryngoscope 111(3), 501–507 (2001).
[Crossref] [PubMed]

Swanson, S. D.

A. P. Sarvazyan, O. V. Rudenko, S. D. Swanson, J. B. Fowlkes, and S. Y. Emelianov, “Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics,” Ultrasound Med. Biol. 24(9), 1419–1435 (1998).
[Crossref] [PubMed]

Tearney, G.

Tearney, G. J.

Tubis, A.

R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson, R. I. Mills, and T. D. Rossing, “Effects of air loading on timpani membrane vibrations,” J. Acoust. Soc. Am. 76(5), 1336–1345 (1984).
[Crossref]

Wang, C.

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

Watanabe, H.

R. Onodera, H. Watanabe, and Y. Ishii, “Interferometric Phase-Measurement Using a One-Dimensional Discrete Hilbert Transform,” Opt. Rev. 12(1), 29–36 (2005).
[Crossref]

Webb, R. H.

White, W.

White, W. M.

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

Whittemore, K. R.

K. R. Whittemore, S. N. Merchant, B. B. Poon, and J. J. Rosowski, “A normative study of tympanic membrane motion in humans using a laser Doppler vibrometer (LDV),” Hear. Res. 187(1-2), 85–104 (2004).
[Crossref] [PubMed]

Wong, B.

Xiang, S.

Yelin, D.

Yun, S.

Yun, S. H.

E. W. Chang, J. B. Kobler, and S. H. Yun, “Subnanometer optical coherence tomographic vibrography,” Opt. Lett. 37(17), 3678–3680 (2012).
[Crossref] [PubMed]

E. W. Chang, J. B. Kobler, and S. H. Yun, “Triggered optical coherence tomography for capturing rapid periodic motion,” Sci. Rep. 1, 48 (2011).
[Crossref] [PubMed]

Zahnert, T.

A. Burkhardt, L. Kirsten, M. Bornitz, T. Zahnert, and E. Koch, “Investigation of the human tympanic membrane oscillation ex vivo by Doppler optical coherence tomography,” J. Biophotonics (2012).
[Crossref] [PubMed]

Zhao, Y.

Ear Hear. (1)

J. J. Rosowski, H. H. Nakajima, and S. N. Merchant, “Clinical utility of laser-Doppler vibrometer measurements in live normal and pathologic human ears,” Ear Hear. 29(1), 3–19 (2008).
[PubMed]

Hear. Res. (3)

K. R. Whittemore, S. N. Merchant, B. B. Poon, and J. J. Rosowski, “A normative study of tympanic membrane motion in humans using a laser Doppler vibrometer (LDV),” Hear. Res. 187(1-2), 85–104 (2004).
[Crossref] [PubMed]

J. T. Cheng, A. A. Aarnisalo, E. Harrington, M. S. Hernandez-Montes, C. Furlong, S. N. Merchant, and J. J. Rosowski, “Motion of the surface of the human tympanic membrane measured with stroboscopic holography,” Hear. Res. 263(1-2), 66–77 (2010).
[Crossref] [PubMed]

J. J. Rosowski, I. Dobrev, M. Khaleghi, W. Lu, J. T. Cheng, E. Harrington, and C. Furlong, “Measurements of three-dimensional shape and sound-induced motion of the chinchilla tympanic membrane,” Hear. Res. 301, 44–52 (2013).
[Crossref] [PubMed]

J. Acoust. Soc. Am. (3)

R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson, R. I. Mills, and T. D. Rossing, “Effects of air loading on timpani membrane vibrations,” J. Acoust. Soc. Am. 76(5), 1336–1345 (1984).
[Crossref]

S. Chen, M. Fatemi, and J. F. Greenleaf, “Quantifying elasticity and viscosity from measurement of shear wave speed dispersion,” J. Acoust. Soc. Am. 115(6), 2781–2785 (2004).
[Crossref] [PubMed]

P. G. Stelmachowicz, K. A. Beauchaine, A. Kalberer, and W. Jesteadt, “Normative thresholds in the 8- to 20-kHz range as a function of age,” J. Acoust. Soc. Am. 86(4), 1384–1391 (1989).
[Crossref] [PubMed]

J. Biophotonics (1)

A. Burkhardt, L. Kirsten, M. Bornitz, T. Zahnert, and E. Koch, “Investigation of the human tympanic membrane oscillation ex vivo by Doppler optical coherence tomography,” J. Biophotonics (2012).
[Crossref] [PubMed]

J. Magn. Reson. Imaging (1)

J. Bishop, G. Poole, M. Leitch, and D. B. Plewes, “Magnetic resonance imaging of shear wave propagation in excised tissue,” J. Magn. Reson. Imaging 8(6), 1257–1265 (1998).
[Crossref] [PubMed]

Laryngoscope (1)

A. M. Huber, C. Schwab, T. Linder, S. J. Stoeckli, M. Ferrazzini, N. Dillier, and U. Fisch, “Evaluation of Eardrum Laser Doppler Interferometry as a Diagnostic Tool,” Laryngoscope 111(3), 501–507 (2001).
[Crossref] [PubMed]

Med. Image Anal. (1)

A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
[Crossref] [PubMed]

Nature (1)

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (3)

Opt. Rev. (1)

R. Onodera, H. Watanabe, and Y. Ishii, “Interferometric Phase-Measurement Using a One-Dimensional Discrete Hilbert Transform,” Opt. Rev. 12(1), 29–36 (2005).
[Crossref]

Sci. Rep. (2)

K. Goda, A. Mahjoubfar, C. Wang, A. Fard, J. Adam, D. R. Gossett, A. Ayazi, E. Sollier, O. Malik, E. Chen, Y. Liu, R. Brown, N. Sarkhosh, D. Di Carlo, and B. Jalali, “Hybrid dispersion laser scanner,” Sci. Rep. 2, 445 (2012).
[Crossref] [PubMed]

E. W. Chang, J. B. Kobler, and S. H. Yun, “Triggered optical coherence tomography for capturing rapid periodic motion,” Sci. Rep. 1, 48 (2011).
[Crossref] [PubMed]

Ultrasound Med. Biol. (1)

A. P. Sarvazyan, O. V. Rudenko, S. D. Swanson, J. B. Fowlkes, and S. Y. Emelianov, “Shear wave elasticity imaging: a new ultrasonic technology of medical diagnostics,” Ultrasound Med. Biol. 24(9), 1419–1435 (1998).
[Crossref] [PubMed]

Supplementary Material (5)

» Media 1: MOV (548 KB)     
» Media 2: MOV (512 KB)     
» Media 3: MOV (2544 KB)     
» Media 4: MOV (1597 KB)     
» Media 5: MOV (879 KB)     

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Figures (5)

Fig. 1
Fig. 1

Schematic of the spectrally encoded imaging system. PC – polarization controller, G – diffraction grating, ND – neutral density filter, L1, L2, L3 - achromatic lenses.

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Fig. 2
Fig. 2

(a) Calculating axial sample displacement from the raw interferometric data. The grey rectangle corresponds to the fractional bandwidth reflected from a single resolvable point on the sample. (b) Axial displacement of a single lateral line on the surface of a paper card attached to a piezoelectric actuator. Red and blue curves denote positions of extremum sample displacements. Solid (dashed) curves correspond to maximum (minimum) axial card displacements. See the full sample motion during a single oscillation period (3.33 ms total duration) in movies Media 1 and Media 2.

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Fig. 3
Fig. 3

(a) Schematic of the experimental arrangement for imaging vibrations of a circular rubber membrane. (b) Single frames, captured from Media 3, of different vibration modes of the membrane that was excited by single harmonic waveforms of different frequencies. Numbers at the bottom-right of each panel denote the dominant vibration modes. Scale bar represents 2 mm. Total movie duration is 1.67 ms.

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Fig. 4
Fig. 4

(a) A photograph of the imaged palm and the distal optics of the sample arm. The experimental configuration was similar to that illustrated in Fig. 1(b). Selected frames from Media 4 of the vibrating tissue at constant phase intervals. Scale bar represents 2 mm. Total movie duration is 2 ms.

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Fig. 5
Fig. 5

(a) A photograph showing the imaged nail plate and nail fold behind the vibrating aluminum ring. (b) Selected frames from Media 5, showing wave propagation across the tissue surface. Scale bars represents 2 mm. Total movie duration is 0.67 ms.

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Equations (3)

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

I k ( k∈[ k i − δk 2 , k i + δk 2 ] )= | E r | 2 + | E i | 2 +2 E r E i cos( 2k[ z r −z( k i ;t ) ] ),
φ i = tan −1 [ H{ I cos ( k,t ) } I cos ( k,t ) ],
Δ z i = φ i ( t+dt )− φ i ( t ) 2 k i ,

Metrics