Abstract

Laser ultrasonics is a powerful technique for contactless investigation of important material parameters such as Young’s modulus or thin layer thickness. However, the often employed Gaussian beams result in diverging sound fields of quickly decreasing intensity. Conventionally, changing the laser beam profile requires the slow movement or exchange of optical elements. We present a laser ultrasonics setup for the creation of arbitrary intensity distributions by holographic projection using a MEMS spatial light modulator. High-intensity ultrasound foci with a focus width of 1.6 mm are scanned axially in a sample into depths of up to 7.4 mm by projecting ring-shaped intensity distributions of varying diameter without any mechanical movements. This technique is promising for highly spatially resolved flaw detection or a fast scanning investigation of biological tissue.

© 2016 Optical Society of America

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Corrections

12 December 2016: A correction was made to Ref. 11.


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References

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  1. S. J. Davies, C. Edwards, G. S. Taylor, and S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D: Appl Phys 26, 329–348 (1993).
    [Crossref]
  2. D.A. Hutchins, “Ultrasonic Generation by Pulsed Lasers,” in Physical Acoustics, Warren P. Mason and R. N. Thurston, eds. (Academic, 1988).
    [Crossref]
  3. D. Lévesque, L. Dubourg, and A. Blouin, “Laser ultrasonics for defect detection and residual stress measurement of friction stir welds,” Nondestruct. Test. Eva. 26(3–4), 319–333 (2011).
    [Crossref]
  4. R. S. Lima, S. E. Kruger, G. Lamouche, and B. R. Marple, “Elastic modulus measurements via laser-ultrasonic and knoop indentation techniques in thermally sprayed coatings,” J. Therm. Spray Techn. 14(1), 52–60 (2004).
    [Crossref]
  5. P. Cielo, F. Nadeau, and M. Lamontagne, “Laser generation of convergent acoustic waves for materials inspection,” Ultrasonics 23, 55–62 (1985).
    [Crossref]
  6. A. D. W. McKie, J. W. Wagner, J. B. Spicer, and C. M. Penney, “Laser generation of narrow-band and directed ultrasound,” Ultrasonics 27, 323–330 (1989).
    [Crossref]
  7. Yves H. Berthelot and Jaeek Jarzynski, “Directional Laser Generation and Detection of Ultrasound with Arrays of Optical Fibers,” J. Nondestruct. Eval. 9(4), 271–277 (1990).
    [Crossref]
  8. K. C. Baldwin, T. P. Berndt, and M.J. Ehrlich, “Narrowband laser generation/air-coupled detection: ultrasonic system for on-line process control of composites,” Ultrasonics 37, 329–334 (1999).
    [Crossref] [PubMed]
  9. H. Radner, L. Büttner, and J. Czarske, “Interferometric velocity measurements through a fluctuating phase boundary using two Fresnel guide stars,” Opt. Lett. 40(16), 3766–3769 (2015).
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  15. P. W. T. Mash and T. D. Wilkinson, “Real-time hologram generation using iterative methods,” Proc. SPIE 6252, 625210 (2006).
  16. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
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  20. A. J. Cable, E. Buckley, P. Mash, N. A. Lawrence, T. D. Wilkinson, and W. A. Crossland, “Real-time Binary Hologram Generation for High-quality Video Projection Applications,” in SID Symposium Digest of Technical Papers, (2004), pp. 1431–1433.
    [Crossref]
  21. J. Weng, T. Shimobaba, N. Okada, H. Nakayama, M. Oikawa, N. Masuda, and T. Ito, “Generation of real-time large computer generated hologram using wavefront recording method,” Opt. Express 20(4), 4018–4023 (2012).
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  22. P. W. M. Tsang, A. S. M. Jiao, and T.-C. Poon, “Fast conversion of digital Fresnel hologram to phase-only hologram based on localized error diffusion and redistribution,” Opt. Express 22, 5060–5066 (2014).
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  23. O. K. Ersoy, Diffraction, Fourier Optics and Imaging (John Wiley and Sons, 2007).
    [Crossref]
  24. F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
    [Crossref]
  25. D. Wang, E. H. Zhou, J. Brake, H. Ruan, M. Jang, and C. Yang, “Focusing through dynamic tissue with millisecond digital optical phase conjugation,” Optica 2(8), 728–735 (2015).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2016 (1)

2015 (4)

2014 (3)

2013 (1)

2012 (2)

2011 (1)

D. Lévesque, L. Dubourg, and A. Blouin, “Laser ultrasonics for defect detection and residual stress measurement of friction stir welds,” Nondestruct. Test. Eva. 26(3–4), 319–333 (2011).
[Crossref]

2008 (1)

B. G. Henderson and J. D. Mansell, “Laser beam shaping with membrane deformable mirrors,” Proc. SPIE 7093, 709301 (2008).

2007 (1)

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[Crossref]

2006 (1)

P. W. T. Mash and T. D. Wilkinson, “Real-time hologram generation using iterative methods,” Proc. SPIE 6252, 625210 (2006).

2004 (1)

R. S. Lima, S. E. Kruger, G. Lamouche, and B. R. Marple, “Elastic modulus measurements via laser-ultrasonic and knoop indentation techniques in thermally sprayed coatings,” J. Therm. Spray Techn. 14(1), 52–60 (2004).
[Crossref]

1999 (1)

K. C. Baldwin, T. P. Berndt, and M.J. Ehrlich, “Narrowband laser generation/air-coupled detection: ultrasonic system for on-line process control of composites,” Ultrasonics 37, 329–334 (1999).
[Crossref] [PubMed]

1993 (1)

S. J. Davies, C. Edwards, G. S. Taylor, and S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D: Appl Phys 26, 329–348 (1993).
[Crossref]

1990 (1)

Yves H. Berthelot and Jaeek Jarzynski, “Directional Laser Generation and Detection of Ultrasound with Arrays of Optical Fibers,” J. Nondestruct. Eval. 9(4), 271–277 (1990).
[Crossref]

1989 (1)

A. D. W. McKie, J. W. Wagner, J. B. Spicer, and C. M. Penney, “Laser generation of narrow-band and directed ultrasound,” Ultrasonics 27, 323–330 (1989).
[Crossref]

1985 (1)

P. Cielo, F. Nadeau, and M. Lamontagne, “Laser generation of convergent acoustic waves for materials inspection,” Ultrasonics 23, 55–62 (1985).
[Crossref]

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Baldwin, K. C.

K. C. Baldwin, T. P. Berndt, and M.J. Ehrlich, “Narrowband laser generation/air-coupled detection: ultrasonic system for on-line process control of composites,” Ultrasonics 37, 329–334 (1999).
[Crossref] [PubMed]

Baumbach, T.

Bergmann, R. B.

M. Kalms, S. Hellmers, P. Huke, and R. B. Bergmann, “Beam shaping using liquid crystal-on-silicon spatial light modulators for laser ultrasound generation,” Opt. Eng. 53(4), 044110 (2014).
[Crossref]

Berndt, T. P.

K. C. Baldwin, T. P. Berndt, and M.J. Ehrlich, “Narrowband laser generation/air-coupled detection: ultrasonic system for on-line process control of composites,” Ultrasonics 37, 329–334 (1999).
[Crossref] [PubMed]

Berthelot, Yves H.

Yves H. Berthelot and Jaeek Jarzynski, “Directional Laser Generation and Detection of Ultrasound with Arrays of Optical Fibers,” J. Nondestruct. Eval. 9(4), 271–277 (1990).
[Crossref]

Blouin, A.

D. Lévesque, L. Dubourg, and A. Blouin, “Laser ultrasonics for defect detection and residual stress measurement of friction stir welds,” Nondestruct. Test. Eva. 26(3–4), 319–333 (2011).
[Crossref]

Brake, J.

Buckley, E.

A. J. Cable, E. Buckley, P. Mash, N. A. Lawrence, T. D. Wilkinson, and W. A. Crossland, “Real-time Binary Hologram Generation for High-quality Video Projection Applications,” in SID Symposium Digest of Technical Papers, (2004), pp. 1431–1433.
[Crossref]

Büttner, L.

Cable, A. J.

A. J. Cable, E. Buckley, P. Mash, N. A. Lawrence, T. D. Wilkinson, and W. A. Crossland, “Real-time Binary Hologram Generation for High-quality Video Projection Applications,” in SID Symposium Digest of Technical Papers, (2004), pp. 1431–1433.
[Crossref]

Chang, C.

Chen, J.

Cielo, P.

P. Cielo, F. Nadeau, and M. Lamontagne, “Laser generation of convergent acoustic waves for materials inspection,” Ultrasonics 23, 55–62 (1985).
[Crossref]

Crossland, W. A.

A. J. Cable, E. Buckley, P. Mash, N. A. Lawrence, T. D. Wilkinson, and W. A. Crossland, “Real-time Binary Hologram Generation for High-quality Video Projection Applications,” in SID Symposium Digest of Technical Papers, (2004), pp. 1431–1433.
[Crossref]

Czarske, J.

Czarske, J. W.

Davies, S. J.

S. J. Davies, C. Edwards, G. S. Taylor, and S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D: Appl Phys 26, 329–348 (1993).
[Crossref]

Drain, L. E

C. B Scruby and L. E Drain, Laser Ultrasonics: Techniques and Applications (CRC, 1990).

Dubourg, L.

D. Lévesque, L. Dubourg, and A. Blouin, “Laser ultrasonics for defect detection and residual stress measurement of friction stir welds,” Nondestruct. Test. Eva. 26(3–4), 319–333 (2011).
[Crossref]

Edwards, C.

S. J. Davies, C. Edwards, G. S. Taylor, and S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D: Appl Phys 26, 329–348 (1993).
[Crossref]

Ehrlich, M.J.

K. C. Baldwin, T. P. Berndt, and M.J. Ehrlich, “Narrowband laser generation/air-coupled detection: ultrasonic system for on-line process control of composites,” Ultrasonics 37, 329–334 (1999).
[Crossref] [PubMed]

Ersoy, O. K.

O. K. Ersoy, Diffraction, Fourier Optics and Imaging (John Wiley and Sons, 2007).
[Crossref]

Fienup, J.

Finkeldey, M.

Fischer, A.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gerhardt, N. C.

Haufe, D.

Hellmers, S.

M. Kalms, S. Hellmers, P. Huke, and R. B. Bergmann, “Beam shaping using liquid crystal-on-silicon spatial light modulators for laser ultrasound generation,” Opt. Eng. 53(4), 044110 (2014).
[Crossref]

Henderson, B. G.

B. G. Henderson and J. D. Mansell, “Laser beam shaping with membrane deformable mirrors,” Proc. SPIE 7093, 709301 (2008).

Hofmann, M. R.

Huke, P.

M. Kalms, S. Hellmers, P. Huke, and R. B. Bergmann, “Beam shaping using liquid crystal-on-silicon spatial light modulators for laser ultrasound generation,” Opt. Eng. 53(4), 044110 (2014).
[Crossref]

Hutchins, D.A.

D.A. Hutchins, “Ultrasonic Generation by Pulsed Lasers,” in Physical Acoustics, Warren P. Mason and R. N. Thurston, eds. (Academic, 1988).
[Crossref]

Hyland, D.

Ito, T.

Jang, M.

Jarzynski, Jaeek

Yves H. Berthelot and Jaeek Jarzynski, “Directional Laser Generation and Detection of Ultrasound with Arrays of Optical Fibers,” J. Nondestruct. Eval. 9(4), 271–277 (1990).
[Crossref]

Jiao, A. S. M.

Kalms, M.

M. Kalms, S. Hellmers, P. Huke, and R. B. Bergmann, “Beam shaping using liquid crystal-on-silicon spatial light modulators for laser ultrasound generation,” Opt. Eng. 53(4), 044110 (2014).
[Crossref]

Köhl, M.

Koukourakis, N.

Kruger, S. E.

R. S. Lima, S. E. Kruger, G. Lamouche, and B. R. Marple, “Elastic modulus measurements via laser-ultrasonic and knoop indentation techniques in thermally sprayed coatings,” J. Therm. Spray Techn. 14(1), 52–60 (2004).
[Crossref]

Lamontagne, M.

P. Cielo, F. Nadeau, and M. Lamontagne, “Laser generation of convergent acoustic waves for materials inspection,” Ultrasonics 23, 55–62 (1985).
[Crossref]

Lamouche, G.

R. S. Lima, S. E. Kruger, G. Lamouche, and B. R. Marple, “Elastic modulus measurements via laser-ultrasonic and knoop indentation techniques in thermally sprayed coatings,” J. Therm. Spray Techn. 14(1), 52–60 (2004).
[Crossref]

Lawrence, N. A.

A. J. Cable, E. Buckley, P. Mash, N. A. Lawrence, T. D. Wilkinson, and W. A. Crossland, “Real-time Binary Hologram Generation for High-quality Video Projection Applications,” in SID Symposium Digest of Technical Papers, (2004), pp. 1431–1433.
[Crossref]

Lei, W.

Leithold, C.

Lévesque, D.

D. Lévesque, L. Dubourg, and A. Blouin, “Laser ultrasonics for defect detection and residual stress measurement of friction stir welds,” Nondestruct. Test. Eva. 26(3–4), 319–333 (2011).
[Crossref]

Lima, R. S.

R. S. Lima, S. E. Kruger, G. Lamouche, and B. R. Marple, “Elastic modulus measurements via laser-ultrasonic and knoop indentation techniques in thermally sprayed coatings,” J. Therm. Spray Techn. 14(1), 52–60 (2004).
[Crossref]

Mansell, J. D.

B. G. Henderson and J. D. Mansell, “Laser beam shaping with membrane deformable mirrors,” Proc. SPIE 7093, 709301 (2008).

Marple, B. R.

R. S. Lima, S. E. Kruger, G. Lamouche, and B. R. Marple, “Elastic modulus measurements via laser-ultrasonic and knoop indentation techniques in thermally sprayed coatings,” J. Therm. Spray Techn. 14(1), 52–60 (2004).
[Crossref]

Mash, P.

A. J. Cable, E. Buckley, P. Mash, N. A. Lawrence, T. D. Wilkinson, and W. A. Crossland, “Real-time Binary Hologram Generation for High-quality Video Projection Applications,” in SID Symposium Digest of Technical Papers, (2004), pp. 1431–1433.
[Crossref]

Mash, P. W. T.

P. W. T. Mash and T. D. Wilkinson, “Real-time hologram generation using iterative methods,” Proc. SPIE 6252, 625210 (2006).

Masuda, N.

McKie, A. D. W.

A. D. W. McKie, J. W. Wagner, J. B. Spicer, and C. M. Penney, “Laser generation of narrow-band and directed ultrasound,” Ultrasonics 27, 323–330 (1989).
[Crossref]

Minkevich, A. A.

Nadeau, F.

P. Cielo, F. Nadeau, and M. Lamontagne, “Laser generation of convergent acoustic waves for materials inspection,” Ultrasonics 23, 55–62 (1985).
[Crossref]

Nakayama, H.

Oikawa, M.

Okada, N.

Osten, W.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[Crossref]

Palmer, S. B.

S. J. Davies, C. Edwards, G. S. Taylor, and S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D: Appl Phys 26, 329–348 (1993).
[Crossref]

Pedrini, G.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[Crossref]

Penney, C. M.

A. D. W. McKie, J. W. Wagner, J. B. Spicer, and C. M. Penney, “Laser generation of narrow-band and directed ultrasound,” Ultrasonics 27, 323–330 (1989).
[Crossref]

Poon, T.-C.

Radner, H.

Ruan, H.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Scruby, C. B

C. B Scruby and L. E Drain, Laser Ultrasonics: Techniques and Applications (CRC, 1990).

Shimobaba, T.

Spicer, J. B.

A. D. W. McKie, J. W. Wagner, J. B. Spicer, and C. M. Penney, “Laser generation of narrow-band and directed ultrasound,” Ultrasonics 27, 323–330 (1989).
[Crossref]

Stürmer, M.

Taylor, G. S.

S. J. Davies, C. Edwards, G. S. Taylor, and S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D: Appl Phys 26, 329–348 (1993).
[Crossref]

Trahan, R.

Tsang, P. W. M.

Vellekoop, Ivo M.

Wagner, J. W.

A. D. W. McKie, J. W. Wagner, J. B. Spicer, and C. M. Penney, “Laser generation of narrow-band and directed ultrasound,” Ultrasonics 27, 323–330 (1989).
[Crossref]

Wallrabe, U.

Wang, D.

Weng, J.

Wilkinson, T. D.

P. W. T. Mash and T. D. Wilkinson, “Real-time hologram generation using iterative methods,” Proc. SPIE 6252, 625210 (2006).

A. J. Cable, E. Buckley, P. Mash, N. A. Lawrence, T. D. Wilkinson, and W. A. Crossland, “Real-time Binary Hologram Generation for High-quality Video Projection Applications,” in SID Symposium Digest of Technical Papers, (2004), pp. 1431–1433.
[Crossref]

Xia, J.

Yang, C.

Yang, L.

Yang, Z.

Zhang, F.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[Crossref]

Zhou, E. H.

Appl. Opt. (3)

J. Nondestruct. Eval. (1)

Yves H. Berthelot and Jaeek Jarzynski, “Directional Laser Generation and Detection of Ultrasound with Arrays of Optical Fibers,” J. Nondestruct. Eval. 9(4), 271–277 (1990).
[Crossref]

J. Phys. D: Appl Phys (1)

S. J. Davies, C. Edwards, G. S. Taylor, and S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D: Appl Phys 26, 329–348 (1993).
[Crossref]

J. Therm. Spray Techn. (1)

R. S. Lima, S. E. Kruger, G. Lamouche, and B. R. Marple, “Elastic modulus measurements via laser-ultrasonic and knoop indentation techniques in thermally sprayed coatings,” J. Therm. Spray Techn. 14(1), 52–60 (2004).
[Crossref]

Nondestruct. Test. Eva. (1)

D. Lévesque, L. Dubourg, and A. Blouin, “Laser ultrasonics for defect detection and residual stress measurement of friction stir welds,” Nondestruct. Test. Eva. 26(3–4), 319–333 (2011).
[Crossref]

Opt. Eng. (1)

M. Kalms, S. Hellmers, P. Huke, and R. B. Bergmann, “Beam shaping using liquid crystal-on-silicon spatial light modulators for laser ultrasound generation,” Opt. Eng. 53(4), 044110 (2014).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Optica (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Rev. A (1)

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[Crossref]

Proc. SPIE (2)

B. G. Henderson and J. D. Mansell, “Laser beam shaping with membrane deformable mirrors,” Proc. SPIE 7093, 709301 (2008).

P. W. T. Mash and T. D. Wilkinson, “Real-time hologram generation using iterative methods,” Proc. SPIE 6252, 625210 (2006).

Ultrasonics (3)

K. C. Baldwin, T. P. Berndt, and M.J. Ehrlich, “Narrowband laser generation/air-coupled detection: ultrasonic system for on-line process control of composites,” Ultrasonics 37, 329–334 (1999).
[Crossref] [PubMed]

P. Cielo, F. Nadeau, and M. Lamontagne, “Laser generation of convergent acoustic waves for materials inspection,” Ultrasonics 23, 55–62 (1985).
[Crossref]

A. D. W. McKie, J. W. Wagner, J. B. Spicer, and C. M. Penney, “Laser generation of narrow-band and directed ultrasound,” Ultrasonics 27, 323–330 (1989).
[Crossref]

Other (4)

D.A. Hutchins, “Ultrasonic Generation by Pulsed Lasers,” in Physical Acoustics, Warren P. Mason and R. N. Thurston, eds. (Academic, 1988).
[Crossref]

A. J. Cable, E. Buckley, P. Mash, N. A. Lawrence, T. D. Wilkinson, and W. A. Crossland, “Real-time Binary Hologram Generation for High-quality Video Projection Applications,” in SID Symposium Digest of Technical Papers, (2004), pp. 1431–1433.
[Crossref]

C. B Scruby and L. E Drain, Laser Ultrasonics: Techniques and Applications (CRC, 1990).

O. K. Ersoy, Diffraction, Fourier Optics and Imaging (John Wiley and Sons, 2007).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 An exemplary set of sinusoidal phase patterns (top) and the corresponding camera pictures of the diffracted beam (bottom) used as constraints for phase retrieval.
Fig. 2
Fig. 2 Experimental setup. PBS: polarizing beam splitter, S: Aluminum sample, M: Mirror, LV: Laser vibrometer. The polarization Optics are used to reduce losses in the beam shaping part of the setup. The beam sampler is used to picture the beam onto the CCD camera to perform the phase retrieval and assess the result of the beam shaping. Note that sample and CCD are at the same distance from the modulator, this distance being twice the focal length of the hinged lens.
Fig. 3
Fig. 3 Concept of focusing ultrasound waves in solids. The main lobes of the shear waves created by two thermoelastic ultrasound point sources overlap at a depth depending on the distance between the point sources.
Fig. 4
Fig. 4 Far field input intensity distribution of the unmodulated laser beam captured with a CCD camera (left) and the phase distribution of the input laser beam at the modulator. (right)
Fig. 5
Fig. 5 Output of the beam shaping: The laser is shaped into rings with diameters of 1 mm (top left), 2 mm (top right), 3 mm (bottom left) and 4 mm (bottom right) and a width of 250 µm each. Laser ultrasonic excitation was investigated with rings of up to 10 mm in diameter.
Fig. 6
Fig. 6 Normalized maximum recorded surface displacement due to transverse ultrasound waves in dependency of the diameter of the ultrasound-exciting laser beam ring. A maximum surface displacement can be observed for a ring diameter of 4 mm, which corresponds to a propagation angle of 34° for the transverse waves.
Fig. 7
Fig. 7 Cross-section of the measured sound field at the back of the sample at the time of highest surface displacement. From the data a maximum focal width of 1.6 mm can be estimated.

Equations (4)

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Ψ ( x , y , z ) = [ Ψ 0 × ] ,
Ψ 0 ( x , y , z ) = A 0 ( x , y , z 0 ) × exp [ i ϕ 0 ( x , y , z ) ]
= m ( x , y , z 0 ) × exp [ i ϕ m ( x , y , z 0 ) ] .
N = λ f Δ x C × Δ x M

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