Abstract

We present an optical technique to image the frequency-dependent complex mechanical response of a viscoelastic sample. Three-dimensional hyperspectral data, comprising two-dimensional B-mode images and a third dimension corresponding to vibration frequency, were acquired from samples undergoing external mechanical excitation in the audio-frequency range. We describe the optical coherence tomography (OCT) signal when vibration is applied to a sample and detail the processing and acquisition techniques used to extract the local complex mechanical response from three-dimensional data that, due to a wide range of vibration frequencies, possess a wide range of sample velocities. We demonstrate frequency-dependent contrast of the displacement amplitude and phase of a silicone phantom containing inclusions of higher stiffness. Measurements of an ex vivo tumor margin demonstrate distinct spectra between adipose and tumor regions, and images of displacement amplitude and phase demonstrated spatially-resolved contrast. Contrast was also observed in displacement amplitude and phase images of a rat muscle sample. These results represent the first demonstration of mechanical spectroscopy based on B-mode OCT imaging. Spectroscopic optical coherence elastography (S-OCE) provides a high-resolution imaging capability for the detection of tissue pathologies that are characterized by a frequency-dependent viscoelastic response.

© 2010 OSA

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  1. W. Drexler, and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications. (Springer, New York, 2009).
  2. J. M. Schmitt, “OCT elastography: imaging microscopic deformation and strain of tissue,” Opt. Express 3(6), 199–211 (1998).
    [CrossRef] [PubMed]
  3. A. S. Khalil, R. C. Chan, A. H. Chau, B. E. Bouma, and M. R. Mofrad, “Tissue elasticity estimation with optical coherence elastography: toward mechanical characterization of in vivo soft tissue,” Ann. Biomed. Eng. 33(11), 1631–1639 (2005).
    [CrossRef] [PubMed]
  4. G. van Soest, F. Mastik, N. de Jong, and A. F. W. van der Steen, “Robust intravascular optical coherence elastography by line correlations,” Phys. Med. Biol. 52(9), 2445–2458 (2007).
    [CrossRef] [PubMed]
  5. J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
    [CrossRef] [PubMed]
  6. H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
    [CrossRef] [PubMed]
  7. S. J. Kirkpatrick, R. K. Wang, and D. D. Duncan, “OCT-based elastography for large and small deformations,” Opt. Express 14(24), 11585–11597 (2006).
    [CrossRef] [PubMed]
  8. R. K. Wang, Z. H. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
    [CrossRef]
  9. R. K. Wang, S. J. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90(16), 164105 (2007).
    [CrossRef]
  10. J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
    [CrossRef] [PubMed]
  11. M. Fatemi, A. Manduca, and J. F. Greenleaf, “Imaging elastic properties of biological tissues by low- frequency harmonic vibration,” Proc. IEEE 91(10), 1503–1519 (2003).
    [CrossRef]
  12. M. Orescanin, K. S. Toohey, and M. F. Insana, “Material properties from acoustic radiation force step response,” J. Acoust. Soc. Am. 125(5), 2928–2936 (2009).
    [CrossRef] [PubMed]
  13. X. Liang, and S. A. Boppart, “Dynamic optical coherence elastography and applications,” in Asia Communications and Photonics Conference and Exhibition, Technical Digest (CD) (Optical Society of America, 2009), paper TuG2.
  14. X. Liang, A. L. Oldenburg, V. Crecea, E. J. Chaney, and S. A. Boppart, “Optical micro-scale mapping of dynamic biomechanical tissue properties,” Opt. Express 16(15), 11052–11065 (2008).
    [CrossRef] [PubMed]
  15. S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
    [CrossRef] [PubMed]
  16. B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
    [CrossRef] [PubMed]
  17. X. Liang, S. G. Adie, R. John, and S. A. Boppart, “Dynamic spectral-domain optical coherence elastography for tissue characterization,” Opt. Express 18(13), 14183–14190 (2010).
    [CrossRef] [PubMed]
  18. X. Liang, M. Orescanin, K. S. Toohey, M. F. Insana, and S. A. Boppart, “Acoustomotive optical coherence elastography for measuring material mechanical properties,” Opt. Lett. 34(19), 2894–2896 (2009).
    [CrossRef] [PubMed]
  19. X. Liang and S. A. Boppart, “Biomechanical properties of in vivo human skin from dynamic optical coherence elastography,” IEEE Trans. Biomed. Eng. 57(4), 953–959 (2010).
    [CrossRef]
  20. M. Fatemi and J. F. Greenleaf, “Ultrasound-stimulated vibro-acoustic spectrography,” Science 280(5360), 82–85 (1998).
    [CrossRef] [PubMed]
  21. M. Fatemi and J. F. Greenleaf, “Vibro-acoustography: an imaging modality based on ultrasound-stimulated acoustic emission,” Proc. Natl. Acad. Sci. U.S.A. 96(12), 6603–6608 (1999).
    [CrossRef] [PubMed]
  22. V. Crecea, A. L. Oldenburg, X. Liang, T. S. Ralston, and S. A. Boppart, “Magnetomotive nanoparticle transducers for optical rheology of viscoelastic materials,” Opt. Express 17(25), 23114–23122 (2009).
    [CrossRef]
  23. A. L. Oldenburg and S. A. Boppart, “Resonant acoustic spectroscopy of soft tissues using embedded magnetomotive nanotransducers and optical coherence tomography,” Phys. Med. Biol. 55(4), 1189–1201 (2010).
    [CrossRef] [PubMed]
  24. S. S. Rao, Mechanical Vibrations, (Addison-Wesley, Reading Massachusetts, 1986).
  25. B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35(13), 2192–2198 (1996).
    [CrossRef] [PubMed]
  26. C. J. Tay, C. Quan, and W. Chen, “Dynamic measurement by digital holographic interferometry based on complex phasor method,” Opt. Laser Technol. 41(2), 172–180 (2009).
    [CrossRef]
  27. C. Quan, C. J. Tay, and W. Chen, “Determination of displacement derivative in digital holographic interferometry,” Opt. Commun. 282(5), 809–815 (2009).
    [CrossRef]
  28. A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. 33(13), 1425–1427 (2008).
    [CrossRef] [PubMed]
  29. R. A. Leitgeb, and M. Wojtkowski, “Complex and coherence noise free Fourier domain optical coherence tomography,” in Optical Coherence Tomography: Technology and Applications, W. Drexler and J. G. Fujimoto, eds., (Springer, New York, 2008).
  30. B. W. Pogue and M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11(4), 041102 (2006).
    [CrossRef] [PubMed]
  31. P. J. Prendergast, “Finite element models in tissue mechanics and orthopaedic implant design,” Clin. Biomech. (Bristol, Avon) 12(6), 343–366 (1997).
    [CrossRef]

2010 (3)

X. Liang, S. G. Adie, R. John, and S. A. Boppart, “Dynamic spectral-domain optical coherence elastography for tissue characterization,” Opt. Express 18(13), 14183–14190 (2010).
[CrossRef] [PubMed]

X. Liang and S. A. Boppart, “Biomechanical properties of in vivo human skin from dynamic optical coherence elastography,” IEEE Trans. Biomed. Eng. 57(4), 953–959 (2010).
[CrossRef]

A. L. Oldenburg and S. A. Boppart, “Resonant acoustic spectroscopy of soft tissues using embedded magnetomotive nanotransducers and optical coherence tomography,” Phys. Med. Biol. 55(4), 1189–1201 (2010).
[CrossRef] [PubMed]

2009 (7)

V. Crecea, A. L. Oldenburg, X. Liang, T. S. Ralston, and S. A. Boppart, “Magnetomotive nanoparticle transducers for optical rheology of viscoelastic materials,” Opt. Express 17(25), 23114–23122 (2009).
[CrossRef]

C. J. Tay, C. Quan, and W. Chen, “Dynamic measurement by digital holographic interferometry based on complex phasor method,” Opt. Laser Technol. 41(2), 172–180 (2009).
[CrossRef]

C. Quan, C. J. Tay, and W. Chen, “Determination of displacement derivative in digital holographic interferometry,” Opt. Commun. 282(5), 809–815 (2009).
[CrossRef]

X. Liang, M. Orescanin, K. S. Toohey, M. F. Insana, and S. A. Boppart, “Acoustomotive optical coherence elastography for measuring material mechanical properties,” Opt. Lett. 34(19), 2894–2896 (2009).
[CrossRef] [PubMed]

M. Orescanin, K. S. Toohey, and M. F. Insana, “Material properties from acoustic radiation force step response,” J. Acoust. Soc. Am. 125(5), 2928–2936 (2009).
[CrossRef] [PubMed]

S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
[CrossRef] [PubMed]

B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
[CrossRef] [PubMed]

2008 (2)

X. Liang, A. L. Oldenburg, V. Crecea, E. J. Chaney, and S. A. Boppart, “Optical micro-scale mapping of dynamic biomechanical tissue properties,” Opt. Express 16(15), 11052–11065 (2008).
[CrossRef] [PubMed]

A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. 33(13), 1425–1427 (2008).
[CrossRef] [PubMed]

2007 (2)

G. van Soest, F. Mastik, N. de Jong, and A. F. W. van der Steen, “Robust intravascular optical coherence elastography by line correlations,” Phys. Med. Biol. 52(9), 2445–2458 (2007).
[CrossRef] [PubMed]

R. K. Wang, S. J. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90(16), 164105 (2007).
[CrossRef]

2006 (4)

H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
[CrossRef] [PubMed]

S. J. Kirkpatrick, R. K. Wang, and D. D. Duncan, “OCT-based elastography for large and small deformations,” Opt. Express 14(24), 11585–11597 (2006).
[CrossRef] [PubMed]

R. K. Wang, Z. H. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
[CrossRef]

B. W. Pogue and M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11(4), 041102 (2006).
[CrossRef] [PubMed]

2005 (1)

A. S. Khalil, R. C. Chan, A. H. Chau, B. E. Bouma, and M. R. Mofrad, “Tissue elasticity estimation with optical coherence elastography: toward mechanical characterization of in vivo soft tissue,” Ann. Biomed. Eng. 33(11), 1631–1639 (2005).
[CrossRef] [PubMed]

2004 (1)

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[CrossRef] [PubMed]

2003 (2)

J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
[CrossRef] [PubMed]

M. Fatemi, A. Manduca, and J. F. Greenleaf, “Imaging elastic properties of biological tissues by low- frequency harmonic vibration,” Proc. IEEE 91(10), 1503–1519 (2003).
[CrossRef]

1999 (1)

M. Fatemi and J. F. Greenleaf, “Vibro-acoustography: an imaging modality based on ultrasound-stimulated acoustic emission,” Proc. Natl. Acad. Sci. U.S.A. 96(12), 6603–6608 (1999).
[CrossRef] [PubMed]

1998 (2)

J. M. Schmitt, “OCT elastography: imaging microscopic deformation and strain of tissue,” Opt. Express 3(6), 199–211 (1998).
[CrossRef] [PubMed]

M. Fatemi and J. F. Greenleaf, “Ultrasound-stimulated vibro-acoustic spectrography,” Science 280(5360), 82–85 (1998).
[CrossRef] [PubMed]

1997 (1)

P. J. Prendergast, “Finite element models in tissue mechanics and orthopaedic implant design,” Clin. Biomech. (Bristol, Avon) 12(6), 343–366 (1997).
[CrossRef]

1996 (1)

B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35(13), 2192–2198 (1996).
[CrossRef] [PubMed]

Adie, S. G.

X. Liang, S. G. Adie, R. John, and S. A. Boppart, “Dynamic spectral-domain optical coherence elastography for tissue characterization,” Opt. Express 18(13), 14183–14190 (2010).
[CrossRef] [PubMed]

S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
[CrossRef] [PubMed]

Alexandrov, S. A.

S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
[CrossRef] [PubMed]

Armstrong, J. J.

S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
[CrossRef] [PubMed]

Boppart, S. A.

X. Liang and S. A. Boppart, “Biomechanical properties of in vivo human skin from dynamic optical coherence elastography,” IEEE Trans. Biomed. Eng. 57(4), 953–959 (2010).
[CrossRef]

X. Liang, S. G. Adie, R. John, and S. A. Boppart, “Dynamic spectral-domain optical coherence elastography for tissue characterization,” Opt. Express 18(13), 14183–14190 (2010).
[CrossRef] [PubMed]

A. L. Oldenburg and S. A. Boppart, “Resonant acoustic spectroscopy of soft tissues using embedded magnetomotive nanotransducers and optical coherence tomography,” Phys. Med. Biol. 55(4), 1189–1201 (2010).
[CrossRef] [PubMed]

X. Liang, M. Orescanin, K. S. Toohey, M. F. Insana, and S. A. Boppart, “Acoustomotive optical coherence elastography for measuring material mechanical properties,” Opt. Lett. 34(19), 2894–2896 (2009).
[CrossRef] [PubMed]

V. Crecea, A. L. Oldenburg, X. Liang, T. S. Ralston, and S. A. Boppart, “Magnetomotive nanoparticle transducers for optical rheology of viscoelastic materials,” Opt. Express 17(25), 23114–23122 (2009).
[CrossRef]

X. Liang, A. L. Oldenburg, V. Crecea, E. J. Chaney, and S. A. Boppart, “Optical micro-scale mapping of dynamic biomechanical tissue properties,” Opt. Express 16(15), 11052–11065 (2008).
[CrossRef] [PubMed]

H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
[CrossRef] [PubMed]

Bouma, B. E.

A. S. Khalil, R. C. Chan, A. H. Chau, B. E. Bouma, and M. R. Mofrad, “Tissue elasticity estimation with optical coherence elastography: toward mechanical characterization of in vivo soft tissue,” Ann. Biomed. Eng. 33(11), 1631–1639 (2005).
[CrossRef] [PubMed]

Brezinski, M. E.

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[CrossRef] [PubMed]

Chan, R. C.

A. S. Khalil, R. C. Chan, A. H. Chau, B. E. Bouma, and M. R. Mofrad, “Tissue elasticity estimation with optical coherence elastography: toward mechanical characterization of in vivo soft tissue,” Ann. Biomed. Eng. 33(11), 1631–1639 (2005).
[CrossRef] [PubMed]

Chaney, E. J.

X. Liang, A. L. Oldenburg, V. Crecea, E. J. Chaney, and S. A. Boppart, “Optical micro-scale mapping of dynamic biomechanical tissue properties,” Opt. Express 16(15), 11052–11065 (2008).
[CrossRef] [PubMed]

Chau, A. H.

A. S. Khalil, R. C. Chan, A. H. Chau, B. E. Bouma, and M. R. Mofrad, “Tissue elasticity estimation with optical coherence elastography: toward mechanical characterization of in vivo soft tissue,” Ann. Biomed. Eng. 33(11), 1631–1639 (2005).
[CrossRef] [PubMed]

Chen, W.

C. Quan, C. J. Tay, and W. Chen, “Determination of displacement derivative in digital holographic interferometry,” Opt. Commun. 282(5), 809–815 (2009).
[CrossRef]

C. J. Tay, C. Quan, and W. Chen, “Dynamic measurement by digital holographic interferometry based on complex phasor method,” Opt. Laser Technol. 41(2), 172–180 (2009).
[CrossRef]

Crecea, V.

V. Crecea, A. L. Oldenburg, X. Liang, T. S. Ralston, and S. A. Boppart, “Magnetomotive nanoparticle transducers for optical rheology of viscoelastic materials,” Opt. Express 17(25), 23114–23122 (2009).
[CrossRef]

X. Liang, A. L. Oldenburg, V. Crecea, E. J. Chaney, and S. A. Boppart, “Optical micro-scale mapping of dynamic biomechanical tissue properties,” Opt. Express 16(15), 11052–11065 (2008).
[CrossRef] [PubMed]

de Jong, N.

G. van Soest, F. Mastik, N. de Jong, and A. F. W. van der Steen, “Robust intravascular optical coherence elastography by line correlations,” Phys. Med. Biol. 52(9), 2445–2458 (2007).
[CrossRef] [PubMed]

Duncan, D. D.

S. J. Kirkpatrick, R. K. Wang, and D. D. Duncan, “OCT-based elastography for large and small deformations,” Opt. Express 14(24), 11585–11597 (2006).
[CrossRef] [PubMed]

Fatemi, M.

M. Fatemi, A. Manduca, and J. F. Greenleaf, “Imaging elastic properties of biological tissues by low- frequency harmonic vibration,” Proc. IEEE 91(10), 1503–1519 (2003).
[CrossRef]

J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
[CrossRef] [PubMed]

M. Fatemi and J. F. Greenleaf, “Vibro-acoustography: an imaging modality based on ultrasound-stimulated acoustic emission,” Proc. Natl. Acad. Sci. U.S.A. 96(12), 6603–6608 (1999).
[CrossRef] [PubMed]

M. Fatemi and J. F. Greenleaf, “Ultrasound-stimulated vibro-acoustic spectrography,” Science 280(5360), 82–85 (1998).
[CrossRef] [PubMed]

Fujimoto, J. G.

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[CrossRef] [PubMed]

Greenleaf, J. F.

M. Fatemi, A. Manduca, and J. F. Greenleaf, “Imaging elastic properties of biological tissues by low- frequency harmonic vibration,” Proc. IEEE 91(10), 1503–1519 (2003).
[CrossRef]

J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
[CrossRef] [PubMed]

M. Fatemi and J. F. Greenleaf, “Vibro-acoustography: an imaging modality based on ultrasound-stimulated acoustic emission,” Proc. Natl. Acad. Sci. U.S.A. 96(12), 6603–6608 (1999).
[CrossRef] [PubMed]

M. Fatemi and J. F. Greenleaf, “Ultrasound-stimulated vibro-acoustic spectrography,” Science 280(5360), 82–85 (1998).
[CrossRef] [PubMed]

Hillman, T. R.

B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
[CrossRef] [PubMed]

Hinds, M.

R. K. Wang, S. J. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90(16), 164105 (2007).
[CrossRef]

Insana, M.

J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
[CrossRef] [PubMed]

Insana, M. F.

X. Liang, M. Orescanin, K. S. Toohey, M. F. Insana, and S. A. Boppart, “Acoustomotive optical coherence elastography for measuring material mechanical properties,” Opt. Lett. 34(19), 2894–2896 (2009).
[CrossRef] [PubMed]

M. Orescanin, K. S. Toohey, and M. F. Insana, “Material properties from acoustic radiation force step response,” J. Acoust. Soc. Am. 125(5), 2928–2936 (2009).
[CrossRef] [PubMed]

John, R.

X. Liang, S. G. Adie, R. John, and S. A. Boppart, “Dynamic spectral-domain optical coherence elastography for tissue characterization,” Opt. Express 18(13), 14183–14190 (2010).
[CrossRef] [PubMed]

Kennedy, B. F.

B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
[CrossRef] [PubMed]

S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
[CrossRef] [PubMed]

Khalil, A. S.

A. S. Khalil, R. C. Chan, A. H. Chau, B. E. Bouma, and M. R. Mofrad, “Tissue elasticity estimation with optical coherence elastography: toward mechanical characterization of in vivo soft tissue,” Ann. Biomed. Eng. 33(11), 1631–1639 (2005).
[CrossRef] [PubMed]

Kirkpatrick, S. J.

R. K. Wang, S. J. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90(16), 164105 (2007).
[CrossRef]

S. J. Kirkpatrick, R. K. Wang, and D. D. Duncan, “OCT-based elastography for large and small deformations,” Opt. Express 14(24), 11585–11597 (2006).
[CrossRef] [PubMed]

R. K. Wang, Z. H. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
[CrossRef]

Ko, H. J.

H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
[CrossRef] [PubMed]

Kowalczyk, A.

A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. 33(13), 1425–1427 (2008).
[CrossRef] [PubMed]

Liang, X.

X. Liang and S. A. Boppart, “Biomechanical properties of in vivo human skin from dynamic optical coherence elastography,” IEEE Trans. Biomed. Eng. 57(4), 953–959 (2010).
[CrossRef]

X. Liang, S. G. Adie, R. John, and S. A. Boppart, “Dynamic spectral-domain optical coherence elastography for tissue characterization,” Opt. Express 18(13), 14183–14190 (2010).
[CrossRef] [PubMed]

X. Liang, M. Orescanin, K. S. Toohey, M. F. Insana, and S. A. Boppart, “Acoustomotive optical coherence elastography for measuring material mechanical properties,” Opt. Lett. 34(19), 2894–2896 (2009).
[CrossRef] [PubMed]

V. Crecea, A. L. Oldenburg, X. Liang, T. S. Ralston, and S. A. Boppart, “Magnetomotive nanoparticle transducers for optical rheology of viscoelastic materials,” Opt. Express 17(25), 23114–23122 (2009).
[CrossRef]

X. Liang, A. L. Oldenburg, V. Crecea, E. J. Chaney, and S. A. Boppart, “Optical micro-scale mapping of dynamic biomechanical tissue properties,” Opt. Express 16(15), 11052–11065 (2008).
[CrossRef] [PubMed]

Ma, Z. H.

R. K. Wang, Z. H. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
[CrossRef]

Manduca, A.

M. Fatemi, A. Manduca, and J. F. Greenleaf, “Imaging elastic properties of biological tissues by low- frequency harmonic vibration,” Proc. IEEE 91(10), 1503–1519 (2003).
[CrossRef]

Mastik, F.

G. van Soest, F. Mastik, N. de Jong, and A. F. W. van der Steen, “Robust intravascular optical coherence elastography by line correlations,” Phys. Med. Biol. 52(9), 2445–2458 (2007).
[CrossRef] [PubMed]

McLaughlin, R. A.

B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
[CrossRef] [PubMed]

Mofrad, M. R.

A. S. Khalil, R. C. Chan, A. H. Chau, B. E. Bouma, and M. R. Mofrad, “Tissue elasticity estimation with optical coherence elastography: toward mechanical characterization of in vivo soft tissue,” Ann. Biomed. Eng. 33(11), 1631–1639 (2005).
[CrossRef] [PubMed]

Oldenburg, A. L.

A. L. Oldenburg and S. A. Boppart, “Resonant acoustic spectroscopy of soft tissues using embedded magnetomotive nanotransducers and optical coherence tomography,” Phys. Med. Biol. 55(4), 1189–1201 (2010).
[CrossRef] [PubMed]

V. Crecea, A. L. Oldenburg, X. Liang, T. S. Ralston, and S. A. Boppart, “Magnetomotive nanoparticle transducers for optical rheology of viscoelastic materials,” Opt. Express 17(25), 23114–23122 (2009).
[CrossRef]

X. Liang, A. L. Oldenburg, V. Crecea, E. J. Chaney, and S. A. Boppart, “Optical micro-scale mapping of dynamic biomechanical tissue properties,” Opt. Express 16(15), 11052–11065 (2008).
[CrossRef] [PubMed]

Orescanin, M.

M. Orescanin, K. S. Toohey, and M. F. Insana, “Material properties from acoustic radiation force step response,” J. Acoust. Soc. Am. 125(5), 2928–2936 (2009).
[CrossRef] [PubMed]

X. Liang, M. Orescanin, K. S. Toohey, M. F. Insana, and S. A. Boppart, “Acoustomotive optical coherence elastography for measuring material mechanical properties,” Opt. Lett. 34(19), 2894–2896 (2009).
[CrossRef] [PubMed]

Patel, N. A.

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[CrossRef] [PubMed]

Patterson, M. S.

B. W. Pogue and M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11(4), 041102 (2006).
[CrossRef] [PubMed]

Pogue, B. W.

B. W. Pogue and M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11(4), 041102 (2006).
[CrossRef] [PubMed]

Prendergast, P. J.

P. J. Prendergast, “Finite element models in tissue mechanics and orthopaedic implant design,” Clin. Biomech. (Bristol, Avon) 12(6), 343–366 (1997).
[CrossRef]

Quan, C.

C. Quan, C. J. Tay, and W. Chen, “Determination of displacement derivative in digital holographic interferometry,” Opt. Commun. 282(5), 809–815 (2009).
[CrossRef]

C. J. Tay, C. Quan, and W. Chen, “Dynamic measurement by digital holographic interferometry based on complex phasor method,” Opt. Laser Technol. 41(2), 172–180 (2009).
[CrossRef]

Quirk, B. C.

B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
[CrossRef] [PubMed]

Ralston, T. S.

V. Crecea, A. L. Oldenburg, X. Liang, T. S. Ralston, and S. A. Boppart, “Magnetomotive nanoparticle transducers for optical rheology of viscoelastic materials,” Opt. Express 17(25), 23114–23122 (2009).
[CrossRef]

Rogowska, J.

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[CrossRef] [PubMed]

Sampson, D. D.

S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
[CrossRef] [PubMed]

B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
[CrossRef] [PubMed]

Schmitt, J. M.

J. M. Schmitt, “OCT elastography: imaging microscopic deformation and strain of tissue,” Opt. Express 3(6), 199–211 (1998).
[CrossRef] [PubMed]

Stack, R.

H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
[CrossRef] [PubMed]

Ströbel, B.

B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35(13), 2192–2198 (1996).
[CrossRef] [PubMed]

Szkulmowska, A.

A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. 33(13), 1425–1427 (2008).
[CrossRef] [PubMed]

Szkulmowski, M.

A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. 33(13), 1425–1427 (2008).
[CrossRef] [PubMed]

Tan, W.

H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
[CrossRef] [PubMed]

Tay, C. J.

C. Quan, C. J. Tay, and W. Chen, “Determination of displacement derivative in digital holographic interferometry,” Opt. Commun. 282(5), 809–815 (2009).
[CrossRef]

C. J. Tay, C. Quan, and W. Chen, “Dynamic measurement by digital holographic interferometry based on complex phasor method,” Opt. Laser Technol. 41(2), 172–180 (2009).
[CrossRef]

Toohey, K. S.

X. Liang, M. Orescanin, K. S. Toohey, M. F. Insana, and S. A. Boppart, “Acoustomotive optical coherence elastography for measuring material mechanical properties,” Opt. Lett. 34(19), 2894–2896 (2009).
[CrossRef] [PubMed]

M. Orescanin, K. S. Toohey, and M. F. Insana, “Material properties from acoustic radiation force step response,” J. Acoust. Soc. Am. 125(5), 2928–2936 (2009).
[CrossRef] [PubMed]

van der Steen, A. F. W.

G. van Soest, F. Mastik, N. de Jong, and A. F. W. van der Steen, “Robust intravascular optical coherence elastography by line correlations,” Phys. Med. Biol. 52(9), 2445–2458 (2007).
[CrossRef] [PubMed]

van Soest, G.

G. van Soest, F. Mastik, N. de Jong, and A. F. W. van der Steen, “Robust intravascular optical coherence elastography by line correlations,” Phys. Med. Biol. 52(9), 2445–2458 (2007).
[CrossRef] [PubMed]

Wang, R. K.

R. K. Wang, S. J. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90(16), 164105 (2007).
[CrossRef]

S. J. Kirkpatrick, R. K. Wang, and D. D. Duncan, “OCT-based elastography for large and small deformations,” Opt. Express 14(24), 11585–11597 (2006).
[CrossRef] [PubMed]

R. K. Wang, Z. H. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
[CrossRef]

Wojtkowski, M.

A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. 33(13), 1425–1427 (2008).
[CrossRef] [PubMed]

Ann. Biomed. Eng. (1)

A. S. Khalil, R. C. Chan, A. H. Chau, B. E. Bouma, and M. R. Mofrad, “Tissue elasticity estimation with optical coherence elastography: toward mechanical characterization of in vivo soft tissue,” Ann. Biomed. Eng. 33(11), 1631–1639 (2005).
[CrossRef] [PubMed]

Annu. Rev. Biomed. Eng. (1)

J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
[CrossRef] [PubMed]

Appl. Opt. (1)

B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35(13), 2192–2198 (1996).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

R. K. Wang, Z. H. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
[CrossRef]

R. K. Wang, S. J. Kirkpatrick, and M. Hinds, “Phase-sensitive optical coherence elastography for mapping tissue microstrains in real time,” Appl. Phys. Lett. 90(16), 164105 (2007).
[CrossRef]

Clin. Biomech. (Bristol, Avon) (1)

P. J. Prendergast, “Finite element models in tissue mechanics and orthopaedic implant design,” Clin. Biomech. (Bristol, Avon) 12(6), 343–366 (1997).
[CrossRef]

Heart (1)

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng. (1)

X. Liang and S. A. Boppart, “Biomechanical properties of in vivo human skin from dynamic optical coherence elastography,” IEEE Trans. Biomed. Eng. 57(4), 953–959 (2010).
[CrossRef]

J. Acoust. Soc. Am. (1)

M. Orescanin, K. S. Toohey, and M. F. Insana, “Material properties from acoustic radiation force step response,” J. Acoust. Soc. Am. 125(5), 2928–2936 (2009).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

B. W. Pogue and M. S. Patterson, “Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry,” J. Biomed. Opt. 11(4), 041102 (2006).
[CrossRef] [PubMed]

Opt. Commun. (1)

C. Quan, C. J. Tay, and W. Chen, “Determination of displacement derivative in digital holographic interferometry,” Opt. Commun. 282(5), 809–815 (2009).
[CrossRef]

Opt. Express (6)

S. J. Kirkpatrick, R. K. Wang, and D. D. Duncan, “OCT-based elastography for large and small deformations,” Opt. Express 14(24), 11585–11597 (2006).
[CrossRef] [PubMed]

V. Crecea, A. L. Oldenburg, X. Liang, T. S. Ralston, and S. A. Boppart, “Magnetomotive nanoparticle transducers for optical rheology of viscoelastic materials,” Opt. Express 17(25), 23114–23122 (2009).
[CrossRef]

X. Liang, A. L. Oldenburg, V. Crecea, E. J. Chaney, and S. A. Boppart, “Optical micro-scale mapping of dynamic biomechanical tissue properties,” Opt. Express 16(15), 11052–11065 (2008).
[CrossRef] [PubMed]

B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
[CrossRef] [PubMed]

X. Liang, S. G. Adie, R. John, and S. A. Boppart, “Dynamic spectral-domain optical coherence elastography for tissue characterization,” Opt. Express 18(13), 14183–14190 (2010).
[CrossRef] [PubMed]

J. M. Schmitt, “OCT elastography: imaging microscopic deformation and strain of tissue,” Opt. Express 3(6), 199–211 (1998).
[CrossRef] [PubMed]

Opt. Laser Technol. (1)

C. J. Tay, C. Quan, and W. Chen, “Dynamic measurement by digital holographic interferometry based on complex phasor method,” Opt. Laser Technol. 41(2), 172–180 (2009).
[CrossRef]

Opt. Lett. (2)

A. Szkulmowska, M. Szkulmowski, A. Kowalczyk, and M. Wojtkowski, “Phase-resolved Doppler optical coherence tomography--limitations and improvements,” Opt. Lett. 33(13), 1425–1427 (2008).
[CrossRef] [PubMed]

X. Liang, M. Orescanin, K. S. Toohey, M. F. Insana, and S. A. Boppart, “Acoustomotive optical coherence elastography for measuring material mechanical properties,” Opt. Lett. 34(19), 2894–2896 (2009).
[CrossRef] [PubMed]

Phys. Med. Biol. (3)

S. G. Adie, B. F. Kennedy, J. J. Armstrong, S. A. Alexandrov, and D. D. Sampson, “Audio frequency in vivo optical coherence elastography,” Phys. Med. Biol. 54(10), 3129–3139 (2009).
[CrossRef] [PubMed]

G. van Soest, F. Mastik, N. de Jong, and A. F. W. van der Steen, “Robust intravascular optical coherence elastography by line correlations,” Phys. Med. Biol. 52(9), 2445–2458 (2007).
[CrossRef] [PubMed]

A. L. Oldenburg and S. A. Boppart, “Resonant acoustic spectroscopy of soft tissues using embedded magnetomotive nanotransducers and optical coherence tomography,” Phys. Med. Biol. 55(4), 1189–1201 (2010).
[CrossRef] [PubMed]

Proc. IEEE (1)

M. Fatemi, A. Manduca, and J. F. Greenleaf, “Imaging elastic properties of biological tissues by low- frequency harmonic vibration,” Proc. IEEE 91(10), 1503–1519 (2003).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

M. Fatemi and J. F. Greenleaf, “Vibro-acoustography: an imaging modality based on ultrasound-stimulated acoustic emission,” Proc. Natl. Acad. Sci. U.S.A. 96(12), 6603–6608 (1999).
[CrossRef] [PubMed]

Science (1)

M. Fatemi and J. F. Greenleaf, “Ultrasound-stimulated vibro-acoustic spectrography,” Science 280(5360), 82–85 (1998).
[CrossRef] [PubMed]

Tissue Eng. (1)

H. J. Ko, W. Tan, R. Stack, and S. A. Boppart, “Optical coherence elastography of engineered and developing tissue,” Tissue Eng. 12(1), 63–73 (2006).
[CrossRef] [PubMed]

Other (4)

W. Drexler, and J. G. Fujimoto, Optical Coherence Tomography: Technology and Applications. (Springer, New York, 2009).

X. Liang, and S. A. Boppart, “Dynamic optical coherence elastography and applications,” in Asia Communications and Photonics Conference and Exhibition, Technical Digest (CD) (Optical Society of America, 2009), paper TuG2.

S. S. Rao, Mechanical Vibrations, (Addison-Wesley, Reading Massachusetts, 1986).

R. A. Leitgeb, and M. Wojtkowski, “Complex and coherence noise free Fourier domain optical coherence tomography,” in Optical Coherence Tomography: Technology and Applications, W. Drexler and J. G. Fujimoto, eds., (Springer, New York, 2008).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the OCE system.

Fig. 2
Fig. 2

Flowchart of the processing steps used to compute the spatially-localized complex mechanical response of the sample.

Fig. 3
Fig. 3

Frequency response spectra of the mechanical hardware calculated from M-mode chirp measurements. (a) Piezo actuator with coverslip affixed, and (b) Coupled mechanical response of PZT, mounting hardware and sample, for four single-layer silicone samples with elastic moduli of 36, 66, 101 and 141 kPa. Note: For clarity, the spectra for 101 kPa and 141 kPa are shown with 5 dB and 10 dB subtracted respectively.

Fig. 4
Fig. 4

Complex mechanical spectra of a silicone phantom containing an inclusion with closely matched optical scattering, but with a different elastic modulus of 195 kPa compared to the background matrix (66 kPa). (a) Displacement amplitude (top) and phase (bottom) of the background matrix, (b) OCT image (top) and displacement amplitude image corresponding to vibration frequency of 71 Hz (bottom), and (c) displacement amplitude (top) and phase (bottom) of the inclusion. Image dimensions are 750 μm (optical depth) by 2 mm (lateral). The solid lines in the displacement amplitude and phase plots represent a linear interpolation between the experimentally measured points.

Fig. 5
Fig. 5

Images of the complex mechanical response of a silicone phantom containing an inclusion at vibration frequencies of (a) 109 Hz, (b) 119 Hz, and (c) 125 Hz. Displacement amplitude images are shown in the top row, and phase images are shown in the bottom row. Image dimensions are 750 μm (optical depth) by 2 mm (lateral).

Fig. 6
Fig. 6

Complex mechanical spectra of a rat mammary tumor margin. (a) Adipose displacement and phase spectra averaged over the blue ROI box in the OCT image, (b) OCT image with nearby histology (adipose on the left and tumor on the right), and (c) tumor displacement and phase spectra averaged over the red ROI box in the OCT image. Image dimensions are 750 μm (optical depth) by 2 mm (lateral). The scale bar in the histology images is 200 μm. The solid lines in the displacement amplitude and phase plots represent a linear interpolation between the experimentally measured points.

Fig. 7
Fig. 7

Complex mechanical response images of rat mammary tumor adjacent to adipose tissue for vibration frequencies of (a) 83 Hz, (b) 313 Hz and (c) 385 Hz. Displacement amplitude images are shown in the top row, and phase images are shown in the bottom row. Image dimensions are 750 μm (optical depth) by 2 mm (lateral). Note that the maximum value of the displacement amplitude scale in (a) is ~0.12 μm, but is 0.20 μm in (b) and (c).

Fig. 8
Fig. 8

Complex mechanical response images for rat muscle tissue. (a) OCT image, and displacement amplitude (top) and phase images (bottom) at vibration frequencies of (b) 83 Hz and (c) 125 Hz. Image dimensions are 750 μm (optical depth) by 2 mm (lateral). Arrows in the displacement image at 83 Hz indicate regions with different OCE and OCT contrast.

Equations (10)

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

m     Δ z ( t ) + c Δ z ( t ) + k Δ z = F 0 e i Ω t ,
Δ z ( t ) = F 0 ( k m Ω 2 ) 2 + ( c Ω ) 2 e i ( Ω t Φ ) ,
Φ = tan 1 ( c Ω k m Ω 2 ) .
S ˜ ( x , z , Ω ) = S ˜ 0 ( x , z ) e i ϕ ( x , z , Ω ) ,
ϕ ( x , z , Ω ) = ( 4 π n / λ ¯ ) Re { | h ˜ ( x , z , Ω ) | e i Φ ( x , z , Ω ) Δ z Drive e i ( Ω x / v x + ϕ 0 ) } .
S ˜ ( x + d x , z ) S ˜ * ( x , z ) = e i [ ϕ ( x + d x , z , Ω ) ϕ ( x , z , Ω ) ] ,
Δ ϕ ( x , z , Ω ) = ϕ ( x + d x , z , Ω ) ϕ ( x , z , Ω ) .
d ( x , z , Ω ) = ( λ ¯ / 4 π n ) 0 x Δ ϕ ( x , z , Ω ) d x ,
d ˜ ( x , z , Ω ) = x Δ x / 2 x + Δ x / 2 d ( x , z , Ω ) e i ( Ω x / v x + ϕ 0 ) w ( x ) d x .
π < m     v z ( x , z , Ω ) f A-scan ( λ ¯ / 4 π n ) < π ,

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