Abstract

The effect of asymmetry caused by oblique line-shaped laser ablation on the generation of ultrasonic waves in metal, especially the effect of transverse component of the ablation force source on the ultrasonic waves is analyzed. Due to the oblique force source, the displacements of shear wave increase obviously by the enhanced shear force, the energy concentration area of longitudinal wave deflects to the small range centered on the incident direction while that of shear wave is approximately perpendicular to incident direction. In addition, surface wave enhances in the direction of transverse power flow. Furthermore, some ultrasonic characteristics under vortex laser ablation condition are inferred.

© 2014 Optical Society of America

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  1. J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
    [CrossRef]
  2. W. Feng, D. X. Yang, Y. N. Guo, Y. Chang, “Finite element modeling of bulk ultrasonic waves generated by ring-shaped laser illumination in a diamond anvil cell,” Opt. Express 20(6), 6429–6438 (2012).
    [CrossRef] [PubMed]
  3. R. Coulette, E. Lafond, M.-H. Nadal, C. Gondard, F. Lepoutre, O. Petillon, “Laser-generated ultrasound applied to two-layered materials characterization: semi-analytical model and experimental validation,” Ultrasonics 36(1–5), 239–243 (1998).
    [CrossRef]
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  6. X. Zeng, X. L. Mao, R. Greif, R. E. Russo, “Experimental investigation of ablation efficiency and plasma expansion during femtosecond and nanosecond laser ablation of silicon,” Appl. Phys., A Mater. Sci. Process. 80(2), 237–241 (2005).
    [CrossRef]
  7. N. Zhang, X. Zhu, J. Yang, X. Wang, M. Wang, “Time-resolved shadowgraphs of material ejection in intense femtosecond laser ablation of aluminum,” Phys. Rev. Lett. 99(16), 167602 (2007).
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  10. R. R. Fang, D. M. Zhang, Z. H. Li, F. X. Yang, L. Li, X. Y. Tan, M. Sun, “Improved thermal model and its application in UV high-power pulsed laser ablation of metal target,” Solid State Commun. 145(11–12), 556–560 (2008).
    [CrossRef]
  11. T. W. Murray, J. W. Wagner, “Laser generation of acoustic waves in the ablative regime,” J. Appl. Phys. 85(4), 2031–2040 (1999).
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  12. B. Mi, I. C. Ume, “Parametric studies of laser generated ultrasonic signals in ablative regime: time and frequency domains,” J. Nondestruct. Eval. 21(1), 23–33 (2002).
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  13. S. J. Reese, Z. N. Utegulov, F. Farzbod, R. S. Schley, D. H. Hurley, “Examination of the epicentral waveform for laser ultrasound in the melting regime,” Ultrasonics 53(3), 799–802 (2013).
    [CrossRef] [PubMed]
  14. Z. H. Shen, B. Q. Xu, X. W. Ni, J. Lu, S. Y. Zhang, “Theoretical study on line source laser-induced surface acoustic waves in two-layer structure in ablative regime,” Opt. Laser Technol. 36(2), 139–143 (2004).
    [CrossRef]
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    [CrossRef]
  17. K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
    [CrossRef] [PubMed]
  18. C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–6216 (1980).
    [CrossRef]
  19. D. H. Hurley, “Laser-generated thermoelastic acoustic sources in anisotropic materials,” J. Acoust. Soc. Am. 115(5), 2054–2058 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
  21. S. J. Davies, C. Edwards, G. S. Taylor, S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D Appl. Phys. 26(3), 329–348 (1993).
    [CrossRef]
  22. S. Raetz, T. Dehoux, B. Audoin, “Effect of laser beam incidence angle on the thermoelastic generation in semi-transparent materials,” J. Acoust. Soc. Am. 130(6), 3691–3697 (2011).
    [CrossRef] [PubMed]
  23. C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
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  26. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [CrossRef] [PubMed]
  27. C. Hnatovsky, V. G. Shvedov, N. Shostka, A. V. Rode, W. Krolikowski, “Polarization-dependent ablation of silicon using tightly focused femtosecond laser vortex pulses,” Opt. Lett. 37(2), 226–228 (2012).
    [CrossRef] [PubMed]
  28. V. Caullet, N. Marsal, D. Wolfersberger, M. Sciamanna, “Vortex induced rotation dynamics of optical patterns,” Phys. Rev. Lett. 108(26), 263903 (2012).
    [CrossRef] [PubMed]
  29. J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, T. Omatsu, “Optical-vortex laser ablation,” Opt. Express 18(3), 2144–2151 (2010).
    [CrossRef] [PubMed]

2013 (1)

S. J. Reese, Z. N. Utegulov, F. Farzbod, R. S. Schley, D. H. Hurley, “Examination of the epicentral waveform for laser ultrasound in the melting regime,” Ultrasonics 53(3), 799–802 (2013).
[CrossRef] [PubMed]

2012 (4)

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[CrossRef] [PubMed]

W. Feng, D. X. Yang, Y. N. Guo, Y. Chang, “Finite element modeling of bulk ultrasonic waves generated by ring-shaped laser illumination in a diamond anvil cell,” Opt. Express 20(6), 6429–6438 (2012).
[CrossRef] [PubMed]

C. Hnatovsky, V. G. Shvedov, N. Shostka, A. V. Rode, W. Krolikowski, “Polarization-dependent ablation of silicon using tightly focused femtosecond laser vortex pulses,” Opt. Lett. 37(2), 226–228 (2012).
[CrossRef] [PubMed]

V. Caullet, N. Marsal, D. Wolfersberger, M. Sciamanna, “Vortex induced rotation dynamics of optical patterns,” Phys. Rev. Lett. 108(26), 263903 (2012).
[CrossRef] [PubMed]

2011 (1)

S. Raetz, T. Dehoux, B. Audoin, “Effect of laser beam incidence angle on the thermoelastic generation in semi-transparent materials,” J. Acoust. Soc. Am. 130(6), 3691–3697 (2011).
[CrossRef] [PubMed]

2010 (2)

J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, T. Omatsu, “Optical-vortex laser ablation,” Opt. Express 18(3), 2144–2151 (2010).
[CrossRef] [PubMed]

Y. Qin, J. J. Zhao, P. B. Zhang, B. Wen, “Two-dimensional numerical simulation of laser-ablation of aluminum material by nanosecond laser pulse,” Acta Phys. Sin. 59(10), 7120–7128 (2010) (in Chinese).

2008 (1)

R. R. Fang, D. M. Zhang, Z. H. Li, F. X. Yang, L. Li, X. Y. Tan, M. Sun, “Improved thermal model and its application in UV high-power pulsed laser ablation of metal target,” Solid State Commun. 145(11–12), 556–560 (2008).
[CrossRef]

2007 (2)

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[CrossRef]

N. Zhang, X. Zhu, J. Yang, X. Wang, M. Wang, “Time-resolved shadowgraphs of material ejection in intense femtosecond laser ablation of aluminum,” Phys. Rev. Lett. 99(16), 167602 (2007).
[CrossRef] [PubMed]

2006 (1)

I. Zinovik, A. Povitsky, “Dynamics of multiple plumes in laser ablation: modeling of the shielding effect,” J. Appl. Phys. 100(2), 024911 (2006).
[CrossRef]

2005 (1)

X. Zeng, X. L. Mao, R. Greif, R. E. Russo, “Experimental investigation of ablation efficiency and plasma expansion during femtosecond and nanosecond laser ablation of silicon,” Appl. Phys., A Mater. Sci. Process. 80(2), 237–241 (2005).
[CrossRef]

2004 (2)

Z. H. Shen, B. Q. Xu, X. W. Ni, J. Lu, S. Y. Zhang, “Theoretical study on line source laser-induced surface acoustic waves in two-layer structure in ablative regime,” Opt. Laser Technol. 36(2), 139–143 (2004).
[CrossRef]

D. H. Hurley, “Laser-generated thermoelastic acoustic sources in anisotropic materials,” J. Acoust. Soc. Am. 115(5), 2054–2058 (2004).
[CrossRef]

2002 (2)

B. Mi, I. C. Ume, “Parametric studies of laser generated ultrasonic signals in ablative regime: time and frequency domains,” J. Nondestruct. Eval. 21(1), 23–33 (2002).
[CrossRef]

R. J. Conant, K. L. Telschow, J. B. Walter, “Mathematical modeling of laser ablation in liquids with application to laser ultrasonics,” Ultrasonics 40(10), 1065–1077 (2002).
[CrossRef]

2000 (3)

M. C. Gower, “Industrial applications of laser micromachining,” Opt. Express 7(2), 56–67 (2000).
[CrossRef] [PubMed]

P. R. Willmott, J. R. Huber, “Pulsed laser vaporization and deposition,” Rev. Mod. Phys. 72(1), 315–328 (2000).
[CrossRef]

J. R. Bernstein, J. B. Spicer, “Line source representation for laser-generated ultrasound in aluminum,” J. Acoust. Soc. Am. 107(3), 1352–1357 (2000).
[CrossRef] [PubMed]

1999 (1)

T. W. Murray, J. W. Wagner, “Laser generation of acoustic waves in the ablative regime,” J. Appl. Phys. 85(4), 2031–2040 (1999).
[CrossRef]

1998 (2)

A. V. Bulgakov, N. M. Bulgakova, “Gas-dynamic effects of the interaction between a pulsed laser-ablation plume and the ambient gas: analogy with an underexpanded jet,” J. Phys. D Appl. Phys. 31(6), 693–703 (1998).
[CrossRef]

R. Coulette, E. Lafond, M.-H. Nadal, C. Gondard, F. Lepoutre, O. Petillon, “Laser-generated ultrasound applied to two-layered materials characterization: semi-analytical model and experimental validation,” Ultrasonics 36(1–5), 239–243 (1998).
[CrossRef]

1993 (1)

S. J. Davies, C. Edwards, G. S. Taylor, S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D Appl. Phys. 26(3), 329–348 (1993).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

1988 (1)

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

1980 (1)

C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–6216 (1980).
[CrossRef]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Anderson, G. K.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

Aoki, N.

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[CrossRef] [PubMed]

Audoin, B.

S. Raetz, T. Dehoux, B. Audoin, “Effect of laser beam incidence angle on the thermoelastic generation in semi-transparent materials,” J. Acoust. Soc. Am. 130(6), 3691–3697 (2011).
[CrossRef] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Bernstein, J. R.

J. R. Bernstein, J. B. Spicer, “Line source representation for laser-generated ultrasound in aluminum,” J. Acoust. Soc. Am. 107(3), 1352–1357 (2000).
[CrossRef] [PubMed]

Bulgakov, A. V.

A. V. Bulgakov, N. M. Bulgakova, “Gas-dynamic effects of the interaction between a pulsed laser-ablation plume and the ambient gas: analogy with an underexpanded jet,” J. Phys. D Appl. Phys. 31(6), 693–703 (1998).
[CrossRef]

Bulgakova, N. M.

A. V. Bulgakov, N. M. Bulgakova, “Gas-dynamic effects of the interaction between a pulsed laser-ablation plume and the ambient gas: analogy with an underexpanded jet,” J. Phys. D Appl. Phys. 31(6), 693–703 (1998).
[CrossRef]

Caullet, V.

V. Caullet, N. Marsal, D. Wolfersberger, M. Sciamanna, “Vortex induced rotation dynamics of optical patterns,” Phys. Rev. Lett. 108(26), 263903 (2012).
[CrossRef] [PubMed]

Chang, Y.

Chujo, K.

Conant, R. J.

R. J. Conant, K. L. Telschow, J. B. Walter, “Mathematical modeling of laser ablation in liquids with application to laser ultrasonics,” Ultrasonics 40(10), 1065–1077 (2002).
[CrossRef]

Corlis, X. F.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

Coulette, R.

R. Coulette, E. Lafond, M.-H. Nadal, C. Gondard, F. Lepoutre, O. Petillon, “Laser-generated ultrasound applied to two-layered materials characterization: semi-analytical model and experimental validation,” Ultrasonics 36(1–5), 239–243 (1998).
[CrossRef]

Davies, S. J.

S. J. Davies, C. Edwards, G. S. Taylor, S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D Appl. Phys. 26(3), 329–348 (1993).
[CrossRef]

Dehoux, T.

S. Raetz, T. Dehoux, B. Audoin, “Effect of laser beam incidence angle on the thermoelastic generation in semi-transparent materials,” J. Acoust. Soc. Am. 130(6), 3691–3697 (2011).
[CrossRef] [PubMed]

Dewhurst, R. J.

C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–6216 (1980).
[CrossRef]

Edwards, C.

S. J. Davies, C. Edwards, G. S. Taylor, S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D Appl. Phys. 26(3), 329–348 (1993).
[CrossRef]

Fang, R. R.

R. R. Fang, D. M. Zhang, Z. H. Li, F. X. Yang, L. Li, X. Y. Tan, M. Sun, “Improved thermal model and its application in UV high-power pulsed laser ablation of metal target,” Solid State Commun. 145(11–12), 556–560 (2008).
[CrossRef]

Farzbod, F.

S. J. Reese, Z. N. Utegulov, F. Farzbod, R. S. Schley, D. H. Hurley, “Examination of the epicentral waveform for laser ultrasound in the melting regime,” Ultrasonics 53(3), 799–802 (2013).
[CrossRef] [PubMed]

Feng, W.

Gondard, C.

R. Coulette, E. Lafond, M.-H. Nadal, C. Gondard, F. Lepoutre, O. Petillon, “Laser-generated ultrasound applied to two-layered materials characterization: semi-analytical model and experimental validation,” Ultrasonics 36(1–5), 239–243 (1998).
[CrossRef]

Gower, M. C.

Greif, R.

X. Zeng, X. L. Mao, R. Greif, R. E. Russo, “Experimental investigation of ablation efficiency and plasma expansion during femtosecond and nanosecond laser ablation of silicon,” Appl. Phys., A Mater. Sci. Process. 80(2), 237–241 (2005).
[CrossRef]

Guan, J. F.

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[CrossRef]

Guo, Y. N.

Hamazaki, J.

Harrison, R. F.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

Haynes, L. C.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

Hnatovsky, C.

Huber, J. R.

P. R. Willmott, J. R. Huber, “Pulsed laser vaporization and deposition,” Rev. Mod. Phys. 72(1), 315–328 (2000).
[CrossRef]

Hurley, D. H.

S. J. Reese, Z. N. Utegulov, F. Farzbod, R. S. Schley, D. H. Hurley, “Examination of the epicentral waveform for laser ultrasound in the melting regime,” Ultrasonics 53(3), 799–802 (2013).
[CrossRef] [PubMed]

D. H. Hurley, “Laser-generated thermoelastic acoustic sources in anisotropic materials,” J. Acoust. Soc. Am. 115(5), 2054–2058 (2004).
[CrossRef]

Hutchins, D. A.

C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–6216 (1980).
[CrossRef]

King, T. R.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

Kobayashi, Y.

Krolikowski, W.

Lafond, E.

R. Coulette, E. Lafond, M.-H. Nadal, C. Gondard, F. Lepoutre, O. Petillon, “Laser-generated ultrasound applied to two-layered materials characterization: semi-analytical model and experimental validation,” Ultrasonics 36(1–5), 239–243 (1998).
[CrossRef]

Lepoutre, F.

R. Coulette, E. Lafond, M.-H. Nadal, C. Gondard, F. Lepoutre, O. Petillon, “Laser-generated ultrasound applied to two-layered materials characterization: semi-analytical model and experimental validation,” Ultrasonics 36(1–5), 239–243 (1998).
[CrossRef]

Li, L.

R. R. Fang, D. M. Zhang, Z. H. Li, F. X. Yang, L. Li, X. Y. Tan, M. Sun, “Improved thermal model and its application in UV high-power pulsed laser ablation of metal target,” Solid State Commun. 145(11–12), 556–560 (2008).
[CrossRef]

Li, Z. H.

R. R. Fang, D. M. Zhang, Z. H. Li, F. X. Yang, L. Li, X. Y. Tan, M. Sun, “Improved thermal model and its application in UV high-power pulsed laser ablation of metal target,” Solid State Commun. 145(11–12), 556–560 (2008).
[CrossRef]

Lu, J.

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[CrossRef]

Z. H. Shen, B. Q. Xu, X. W. Ni, J. Lu, S. Y. Zhang, “Theoretical study on line source laser-induced surface acoustic waves in two-layer structure in ablative regime,” Opt. Laser Technol. 36(2), 139–143 (2004).
[CrossRef]

Mao, X. L.

X. Zeng, X. L. Mao, R. Greif, R. E. Russo, “Experimental investigation of ablation efficiency and plasma expansion during femtosecond and nanosecond laser ablation of silicon,” Appl. Phys., A Mater. Sci. Process. 80(2), 237–241 (2005).
[CrossRef]

Marsal, N.

V. Caullet, N. Marsal, D. Wolfersberger, M. Sciamanna, “Vortex induced rotation dynamics of optical patterns,” Phys. Rev. Lett. 108(26), 263903 (2012).
[CrossRef] [PubMed]

Mi, B.

B. Mi, I. C. Ume, “Parametric studies of laser generated ultrasonic signals in ablative regime: time and frequency domains,” J. Nondestruct. Eval. 21(1), 23–33 (2002).
[CrossRef]

Miyamoto, K.

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[CrossRef] [PubMed]

Morita, R.

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[CrossRef] [PubMed]

J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, T. Omatsu, “Optical-vortex laser ablation,” Opt. Express 18(3), 2144–2151 (2010).
[CrossRef] [PubMed]

Murray, T. W.

T. W. Murray, J. W. Wagner, “Laser generation of acoustic waves in the ablative regime,” J. Appl. Phys. 85(4), 2031–2040 (1999).
[CrossRef]

Nadal, M.-H.

R. Coulette, E. Lafond, M.-H. Nadal, C. Gondard, F. Lepoutre, O. Petillon, “Laser-generated ultrasound applied to two-layered materials characterization: semi-analytical model and experimental validation,” Ultrasonics 36(1–5), 239–243 (1998).
[CrossRef]

Ni, X. W.

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[CrossRef]

Z. H. Shen, B. Q. Xu, X. W. Ni, J. Lu, S. Y. Zhang, “Theoretical study on line source laser-induced surface acoustic waves in two-layer structure in ablative regime,” Opt. Laser Technol. 36(2), 139–143 (2004).
[CrossRef]

Omatsu, T.

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[CrossRef] [PubMed]

J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, T. Omatsu, “Optical-vortex laser ablation,” Opt. Express 18(3), 2144–2151 (2010).
[CrossRef] [PubMed]

Osborne, W. Z.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

Palmer, S. B.

S. J. Davies, C. Edwards, G. S. Taylor, S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D Appl. Phys. 26(3), 329–348 (1993).
[CrossRef]

C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–6216 (1980).
[CrossRef]

Petillon, O.

R. Coulette, E. Lafond, M.-H. Nadal, C. Gondard, F. Lepoutre, O. Petillon, “Laser-generated ultrasound applied to two-layered materials characterization: semi-analytical model and experimental validation,” Ultrasonics 36(1–5), 239–243 (1998).
[CrossRef]

Phipps, C. R.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

Povitsky, A.

I. Zinovik, A. Povitsky, “Dynamics of multiple plumes in laser ablation: modeling of the shielding effect,” J. Appl. Phys. 100(2), 024911 (2006).
[CrossRef]

Qin, Y.

Y. Qin, J. J. Zhao, P. B. Zhang, B. Wen, “Two-dimensional numerical simulation of laser-ablation of aluminum material by nanosecond laser pulse,” Acta Phys. Sin. 59(10), 7120–7128 (2010) (in Chinese).

Raetz, S.

S. Raetz, T. Dehoux, B. Audoin, “Effect of laser beam incidence angle on the thermoelastic generation in semi-transparent materials,” J. Acoust. Soc. Am. 130(6), 3691–3697 (2011).
[CrossRef] [PubMed]

Reese, S. J.

S. J. Reese, Z. N. Utegulov, F. Farzbod, R. S. Schley, D. H. Hurley, “Examination of the epicentral waveform for laser ultrasound in the melting regime,” Ultrasonics 53(3), 799–802 (2013).
[CrossRef] [PubMed]

Rode, A. V.

Russo, R. E.

X. Zeng, X. L. Mao, R. Greif, R. E. Russo, “Experimental investigation of ablation efficiency and plasma expansion during femtosecond and nanosecond laser ablation of silicon,” Appl. Phys., A Mater. Sci. Process. 80(2), 237–241 (2005).
[CrossRef]

Schley, R. S.

S. J. Reese, Z. N. Utegulov, F. Farzbod, R. S. Schley, D. H. Hurley, “Examination of the epicentral waveform for laser ultrasound in the melting regime,” Ultrasonics 53(3), 799–802 (2013).
[CrossRef] [PubMed]

Sciamanna, M.

V. Caullet, N. Marsal, D. Wolfersberger, M. Sciamanna, “Vortex induced rotation dynamics of optical patterns,” Phys. Rev. Lett. 108(26), 263903 (2012).
[CrossRef] [PubMed]

Scruby, C. B.

C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–6216 (1980).
[CrossRef]

Shen, Z. H.

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[CrossRef]

Z. H. Shen, B. Q. Xu, X. W. Ni, J. Lu, S. Y. Zhang, “Theoretical study on line source laser-induced surface acoustic waves in two-layer structure in ablative regime,” Opt. Laser Technol. 36(2), 139–143 (2004).
[CrossRef]

Shostka, N.

Shvedov, V. G.

Spicer, J. B.

J. R. Bernstein, J. B. Spicer, “Line source representation for laser-generated ultrasound in aluminum,” J. Acoust. Soc. Am. 107(3), 1352–1357 (2000).
[CrossRef] [PubMed]

Spicochi, K. C.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Steele, H. S.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

Sun, M.

R. R. Fang, D. M. Zhang, Z. H. Li, F. X. Yang, L. Li, X. Y. Tan, M. Sun, “Improved thermal model and its application in UV high-power pulsed laser ablation of metal target,” Solid State Commun. 145(11–12), 556–560 (2008).
[CrossRef]

Tan, X. Y.

R. R. Fang, D. M. Zhang, Z. H. Li, F. X. Yang, L. Li, X. Y. Tan, M. Sun, “Improved thermal model and its application in UV high-power pulsed laser ablation of metal target,” Solid State Commun. 145(11–12), 556–560 (2008).
[CrossRef]

Tanda, S.

Taylor, G. S.

S. J. Davies, C. Edwards, G. S. Taylor, S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D Appl. Phys. 26(3), 329–348 (1993).
[CrossRef]

Telschow, K. L.

R. J. Conant, K. L. Telschow, J. B. Walter, “Mathematical modeling of laser ablation in liquids with application to laser ultrasonics,” Ultrasonics 40(10), 1065–1077 (2002).
[CrossRef]

Toyoda, K.

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[CrossRef] [PubMed]

Turner, T. P.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

Ume, I. C.

B. Mi, I. C. Ume, “Parametric studies of laser generated ultrasonic signals in ablative regime: time and frequency domains,” J. Nondestruct. Eval. 21(1), 23–33 (2002).
[CrossRef]

Utegulov, Z. N.

S. J. Reese, Z. N. Utegulov, F. Farzbod, R. S. Schley, D. H. Hurley, “Examination of the epicentral waveform for laser ultrasound in the melting regime,” Ultrasonics 53(3), 799–802 (2013).
[CrossRef] [PubMed]

Wagner, J. W.

T. W. Murray, J. W. Wagner, “Laser generation of acoustic waves in the ablative regime,” J. Appl. Phys. 85(4), 2031–2040 (1999).
[CrossRef]

Walter, J. B.

R. J. Conant, K. L. Telschow, J. B. Walter, “Mathematical modeling of laser ablation in liquids with application to laser ultrasonics,” Ultrasonics 40(10), 1065–1077 (2002).
[CrossRef]

Wang, J. J.

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[CrossRef]

Wang, M.

N. Zhang, X. Zhu, J. Yang, X. Wang, M. Wang, “Time-resolved shadowgraphs of material ejection in intense femtosecond laser ablation of aluminum,” Phys. Rev. Lett. 99(16), 167602 (2007).
[CrossRef] [PubMed]

Wang, X.

N. Zhang, X. Zhu, J. Yang, X. Wang, M. Wang, “Time-resolved shadowgraphs of material ejection in intense femtosecond laser ablation of aluminum,” Phys. Rev. Lett. 99(16), 167602 (2007).
[CrossRef] [PubMed]

Wen, B.

Y. Qin, J. J. Zhao, P. B. Zhang, B. Wen, “Two-dimensional numerical simulation of laser-ablation of aluminum material by nanosecond laser pulse,” Acta Phys. Sin. 59(10), 7120–7128 (2010) (in Chinese).

Willmott, P. R.

P. R. Willmott, J. R. Huber, “Pulsed laser vaporization and deposition,” Rev. Mod. Phys. 72(1), 315–328 (2000).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Wolfersberger, D.

V. Caullet, N. Marsal, D. Wolfersberger, M. Sciamanna, “Vortex induced rotation dynamics of optical patterns,” Phys. Rev. Lett. 108(26), 263903 (2012).
[CrossRef] [PubMed]

Xu, B. Q.

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[CrossRef]

Z. H. Shen, B. Q. Xu, X. W. Ni, J. Lu, S. Y. Zhang, “Theoretical study on line source laser-induced surface acoustic waves in two-layer structure in ablative regime,” Opt. Laser Technol. 36(2), 139–143 (2004).
[CrossRef]

Yang, D. X.

Yang, F. X.

R. R. Fang, D. M. Zhang, Z. H. Li, F. X. Yang, L. Li, X. Y. Tan, M. Sun, “Improved thermal model and its application in UV high-power pulsed laser ablation of metal target,” Solid State Commun. 145(11–12), 556–560 (2008).
[CrossRef]

Yang, J.

N. Zhang, X. Zhu, J. Yang, X. Wang, M. Wang, “Time-resolved shadowgraphs of material ejection in intense femtosecond laser ablation of aluminum,” Phys. Rev. Lett. 99(16), 167602 (2007).
[CrossRef] [PubMed]

York, G. W.

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

Zeng, X.

X. Zeng, X. L. Mao, R. Greif, R. E. Russo, “Experimental investigation of ablation efficiency and plasma expansion during femtosecond and nanosecond laser ablation of silicon,” Appl. Phys., A Mater. Sci. Process. 80(2), 237–241 (2005).
[CrossRef]

Zhang, D. M.

R. R. Fang, D. M. Zhang, Z. H. Li, F. X. Yang, L. Li, X. Y. Tan, M. Sun, “Improved thermal model and its application in UV high-power pulsed laser ablation of metal target,” Solid State Commun. 145(11–12), 556–560 (2008).
[CrossRef]

Zhang, N.

N. Zhang, X. Zhu, J. Yang, X. Wang, M. Wang, “Time-resolved shadowgraphs of material ejection in intense femtosecond laser ablation of aluminum,” Phys. Rev. Lett. 99(16), 167602 (2007).
[CrossRef] [PubMed]

Zhang, P. B.

Y. Qin, J. J. Zhao, P. B. Zhang, B. Wen, “Two-dimensional numerical simulation of laser-ablation of aluminum material by nanosecond laser pulse,” Acta Phys. Sin. 59(10), 7120–7128 (2010) (in Chinese).

Zhang, S. Y.

Z. H. Shen, B. Q. Xu, X. W. Ni, J. Lu, S. Y. Zhang, “Theoretical study on line source laser-induced surface acoustic waves in two-layer structure in ablative regime,” Opt. Laser Technol. 36(2), 139–143 (2004).
[CrossRef]

Zhao, J. J.

Y. Qin, J. J. Zhao, P. B. Zhang, B. Wen, “Two-dimensional numerical simulation of laser-ablation of aluminum material by nanosecond laser pulse,” Acta Phys. Sin. 59(10), 7120–7128 (2010) (in Chinese).

Zhu, X.

N. Zhang, X. Zhu, J. Yang, X. Wang, M. Wang, “Time-resolved shadowgraphs of material ejection in intense femtosecond laser ablation of aluminum,” Phys. Rev. Lett. 99(16), 167602 (2007).
[CrossRef] [PubMed]

Zinovik, I.

I. Zinovik, A. Povitsky, “Dynamics of multiple plumes in laser ablation: modeling of the shielding effect,” J. Appl. Phys. 100(2), 024911 (2006).
[CrossRef]

Acta Phys. Sin. (1)

Y. Qin, J. J. Zhao, P. B. Zhang, B. Wen, “Two-dimensional numerical simulation of laser-ablation of aluminum material by nanosecond laser pulse,” Acta Phys. Sin. 59(10), 7120–7128 (2010) (in Chinese).

Appl. Phys., A Mater. Sci. Process. (1)

X. Zeng, X. L. Mao, R. Greif, R. E. Russo, “Experimental investigation of ablation efficiency and plasma expansion during femtosecond and nanosecond laser ablation of silicon,” Appl. Phys., A Mater. Sci. Process. 80(2), 237–241 (2005).
[CrossRef]

J. Acoust. Soc. Am. (3)

D. H. Hurley, “Laser-generated thermoelastic acoustic sources in anisotropic materials,” J. Acoust. Soc. Am. 115(5), 2054–2058 (2004).
[CrossRef]

J. R. Bernstein, J. B. Spicer, “Line source representation for laser-generated ultrasound in aluminum,” J. Acoust. Soc. Am. 107(3), 1352–1357 (2000).
[CrossRef] [PubMed]

S. Raetz, T. Dehoux, B. Audoin, “Effect of laser beam incidence angle on the thermoelastic generation in semi-transparent materials,” J. Acoust. Soc. Am. 130(6), 3691–3697 (2011).
[CrossRef] [PubMed]

J. Appl. Phys. (4)

C. R. Phipps, T. P. Turner, R. F. Harrison, G. W. York, W. Z. Osborne, G. K. Anderson, X. F. Corlis, L. C. Haynes, H. S. Steele, K. C. Spicochi, T. R. King, “Impulse coupling to targets in vacuum by KrF, HF, and CO2 single-pulse lasers,” J. Appl. Phys. 64(3), 1083–1096 (1988).
[CrossRef]

T. W. Murray, J. W. Wagner, “Laser generation of acoustic waves in the ablative regime,” J. Appl. Phys. 85(4), 2031–2040 (1999).
[CrossRef]

I. Zinovik, A. Povitsky, “Dynamics of multiple plumes in laser ablation: modeling of the shielding effect,” J. Appl. Phys. 100(2), 024911 (2006).
[CrossRef]

C. B. Scruby, R. J. Dewhurst, D. A. Hutchins, S. B. Palmer, “Quantitative studies of thermally generated elastic waves in laser-irradiated metals,” J. Appl. Phys. 51(12), 6210–6216 (1980).
[CrossRef]

J. Nondestruct. Eval. (1)

B. Mi, I. C. Ume, “Parametric studies of laser generated ultrasonic signals in ablative regime: time and frequency domains,” J. Nondestruct. Eval. 21(1), 23–33 (2002).
[CrossRef]

J. Phys. D Appl. Phys. (2)

A. V. Bulgakov, N. M. Bulgakova, “Gas-dynamic effects of the interaction between a pulsed laser-ablation plume and the ambient gas: analogy with an underexpanded jet,” J. Phys. D Appl. Phys. 31(6), 693–703 (1998).
[CrossRef]

S. J. Davies, C. Edwards, G. S. Taylor, S. B. Palmer, “Laser-generated ultrasound: its properties, mechanisms and multifarious applications,” J. Phys. D Appl. Phys. 26(3), 329–348 (1993).
[CrossRef]

Nano Lett. (1)

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Laser Technol. (2)

J. F. Guan, Z. H. Shen, X. W. Ni, J. Lu, J. J. Wang, B. Q. Xu, “Numerical simulation of the ultrasonic waves generated by ring-shaped laser illumination patterns,” Opt. Laser Technol. 39(6), 1281–1287 (2007).
[CrossRef]

Z. H. Shen, B. Q. Xu, X. W. Ni, J. Lu, S. Y. Zhang, “Theoretical study on line source laser-induced surface acoustic waves in two-layer structure in ablative regime,” Opt. Laser Technol. 36(2), 139–143 (2004).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

V. Caullet, N. Marsal, D. Wolfersberger, M. Sciamanna, “Vortex induced rotation dynamics of optical patterns,” Phys. Rev. Lett. 108(26), 263903 (2012).
[CrossRef] [PubMed]

N. Zhang, X. Zhu, J. Yang, X. Wang, M. Wang, “Time-resolved shadowgraphs of material ejection in intense femtosecond laser ablation of aluminum,” Phys. Rev. Lett. 99(16), 167602 (2007).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

P. R. Willmott, J. R. Huber, “Pulsed laser vaporization and deposition,” Rev. Mod. Phys. 72(1), 315–328 (2000).
[CrossRef]

Solid State Commun. (1)

R. R. Fang, D. M. Zhang, Z. H. Li, F. X. Yang, L. Li, X. Y. Tan, M. Sun, “Improved thermal model and its application in UV high-power pulsed laser ablation of metal target,” Solid State Commun. 145(11–12), 556–560 (2008).
[CrossRef]

Ultrasonics (3)

S. J. Reese, Z. N. Utegulov, F. Farzbod, R. S. Schley, D. H. Hurley, “Examination of the epicentral waveform for laser ultrasound in the melting regime,” Ultrasonics 53(3), 799–802 (2013).
[CrossRef] [PubMed]

R. Coulette, E. Lafond, M.-H. Nadal, C. Gondard, F. Lepoutre, O. Petillon, “Laser-generated ultrasound applied to two-layered materials characterization: semi-analytical model and experimental validation,” Ultrasonics 36(1–5), 239–243 (1998).
[CrossRef]

R. J. Conant, K. L. Telschow, J. B. Walter, “Mathematical modeling of laser ablation in liquids with application to laser ultrasonics,” Ultrasonics 40(10), 1065–1077 (2002).
[CrossRef]

Other (2)

R. J. Conant and S. E. Garwick, “Mathematical modeling of laser ablation in liquids with application to laser ultrasonics,” Review of Progress in Quantitative: Non-destructive Evaluation, D. O. Thompson, and D. E. Chimenti eds., vol. 16, Plenum, New York, 1997, pp. 491–498.

C. W. Sun, Effects of Laser Irradiation (National Defense Industry Press, Beijing, 2002), Chap. 3. (in Chinese)

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of system induced by line-shaped laser under different conditions: (a) normal incident condition; (b) oblique incident condition.

Fig. 2
Fig. 2

Temperature in the laser incident point in ablative regime.

Fig. 3
Fig. 3

The displacement of the bulk waves at epicenter.

Fig. 4
Fig. 4

Amplitudes of (a) longitudinal wave and (b) shear wave at different positions on the rear surface of specimen along the transverse direction.

Fig. 5
Fig. 5

The directivity patterns of bulk wave under different conditions (normalized): (a) the longitudinal wave induced by normal force source; (b) the shear wave induced by normal force source; (c) the longitudinal wave induced by oblique force source and (d) the shear wave induced by oblique force source.

Fig. 6
Fig. 6

Amplitudes of the surface wave on different points.

Fig. 7
Fig. 7

Ultrasonic displacement fields at 7ns under different conditions: (a) line-shaped laser normal incident condition; (b) line-shaped laser oblique incident.

Fig. 8
Fig. 8

The schematic diagram of (a) energy distribution and (b) the spiral phase.

Fig. 9
Fig. 9

The schematic diagram of shear wave field induced by vortex laser: (a) three-dimensional diagram; (b) two- dimensional diagram from the view of propagation direction.

Equations (8)

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

P= b 0 ( λ τ ) n I 0 1+n ,
C m =(P P 0 )/ I 0 ,
C m =5.56 ( I 0 λ t p ) 0.301 ,
δ= exp( (T T m ) 2 / (ΔT) 2 ) ΔT π ,
Δ C P = ΔH T .
ρ C P T( x,y,t ) t =k 2 T( x,y,t )+A I 0 βexp(βy)f( x )g( t )+L,
f( x )=exp( x 2 x 0 2 ), g( t )=exp[ ( t t 0 ) 2 τ 2 ],
sinθ= mλ ( 2πr ) 2 + (mλ) 2 ,

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