Abstract

Laser ultrasonics is a technique where lasers are employed to generate and detect ultrasound. A data collection method (full matrix capture) and a post processing imaging algorithm, the total focusing method, both developed for ultrasonic arrays, are modified and used in order to enhance the capabilities of laser ultrasonics for nondestructive testing by improving defect detectability and increasing spatial resolution. In this way, a laser induced ultrasonic phased array is synthesized. A model is developed and compared with experimental results from aluminum samples with side drilled holes and slots at depths of 5 – 20 mm from the surface.

© 2016 Optical Society of America

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References

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    [Crossref]
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  34. S. Raetz, T. Dehoux, and B. Audoin, “Effect of laser beam incidence angle on the thermoelastic generation in semi-transparent materials,” J. Acoust. Soc. Am. 130, 3691–3697 (2011).
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    [Crossref]
  46. P. Delaye, A. Blouin, D. Drolet, L. A. de Montmorillon, G. Roosen, and J. P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive inp:fe under an applied dc field,” J. Opt. Soc. Am. 14, 1723–1734 (1997).
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    [Crossref]
  49. A. I. Bowler, B. W. Drinkwater, and P. D. Wilcox, “An investigation into the feasibility of internal strain measurement in solids by correlation of ultrasonic images,” P. Roy. Soc. A-Math. Phy. 467, 2247–2270 (2011).
    [Crossref]

2013 (2)

C. Pei, K. Demachi, T. Fukuchi, K. Koyama, and M. Uesaka, “Cracks measurement using fiber-phased array laser ultrasound generation,” J. Appl. Phys. 113, 163101 (2013).
[Crossref]

C. Li, D. Pain, P. D. Wilcox, and B. W. Drinkwater, “Imaging composite material using ultrasonic arrays,” NDT&E Int. 53, 8–17 (2013).
[Crossref]

2012 (1)

2011 (2)

A. I. Bowler, B. W. Drinkwater, and P. D. Wilcox, “An investigation into the feasibility of internal strain measurement in solids by correlation of ultrasonic images,” P. Roy. Soc. A-Math. Phy. 467, 2247–2270 (2011).
[Crossref]

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

2009 (1)

Y.-J. Chen, “Relationship between ultrasonic characteristics and relative porosity in Al and Al-XSi alloys,” Mater. Trans. 50, 2308–2313 (2009).
[Crossref]

2008 (2)

P. D. W. C. Holmes and B. W. Drinkwater, “Advanced post-processing for scanned ultrasonic arrays: Application to defect detection and classification in non-destructive evaluation,” Ultrasonics 48, 636–642 (2008).
[Crossref] [PubMed]

J. Zhang, B. W. Drinkwater, and P. D. Wilcox, “Defect characterization using an ultrasonic array to measure the scattering coefïňĄcient matrix,” IEEE T. Ultrason. Ferr. 55, 2254–2265 (2008).
[Crossref]

2006 (2)

B. Mi and C. Ume, “Real-time weld penetration depth monitoring with laser ultrasonic sensing system,” J. Manuf. Sci. E. - T. ASME 128, 280–286 (2006).
[Crossref]

E. Glushkov, N. Glushkova, A. Ekhlakov, and E. Shapar, “An analytically based computer model for surface measurements in ultrasonic crack detection,” Wave Motion 43, 458–473 (2006).
[Crossref]

2005 (2)

P. D. W. C. Holmes and B. W. Drinkwater, “Post-processing of the full matrix of ultrasonic transmit–receive array data for non-destructive evaluation,” NDT&E Int. 38, 701–711 (2005)
[Crossref]

A. L. Lopez-Sanchez, H. J. Kim, L. W. Schmerr, and A. Sedov, “Measurement models and scattering models for predicting the ultrasonic pulse-echo response from side-drilled holes,” J. Nondestruct. Eval. 24, 83–96 (2005).
[Crossref]

2003 (1)

S. D. Sharples, M. Clark, and M. G. Somekh, “All-optical adaptive scanning acoustic microscope,” Ultrasonics 41, 295–299 (2003).
[Crossref] [PubMed]

2002 (2)

S. N. Hopko, I. C. Ume, and D. S. Erdahl, “Development of a flexible laser ultrasonic probe,” J. Manuf. Sci. E. - T. ASME 124, 351–357 (2002).
[Crossref]

D. Levesque, A. Blouin, C. Neron, and J.-P. Monchalin, “Performance of laser-ultrasonic F-SAFT imaging,” Ultrasonics 40, 1057–1063 (2002).
[Crossref] [PubMed]

2001 (1)

T. W. Murray and S. Krishnaswamy, “Multiplexed interferometer for ultrasonic imaging applications,” Opt. Eng. 40, 1321–1328 (2001).
[Crossref]

2000 (3)

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

M. Dubois, M. Militzer, A. Moreau, and J. F. Bussiere, “A new technique for the quantitative real-time monitoring of austenite grain growth in steel,” Scripta Mater. 42, 867–874 (2000).
[Crossref]

M. Clark, S. D. Sharples, and M. G. Somekh, “Diffractive acoustic elements for laser ultrasonics,” J. Acoust. Soc. Am. 107, 3179–3185 (2000).
[Crossref] [PubMed]

1999 (1)

S. C. Wooh and Y. Shi, “Optimum beam steering of linear phased arrays,” Wave Motion 29, 245–265 (1999).
[Crossref]

1998 (1)

1997 (1)

P. Delaye, A. Blouin, D. Drolet, L. A. de Montmorillon, G. Roosen, and J. P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive inp:fe under an applied dc field,” J. Opt. Soc. Am. 14, 1723–1734 (1997).
[Crossref]

1996 (1)

J. T. W. Murray, J. B. Deaton, and J. W. Wagner, “Experimental evaluation of enhanced generation of ultrasonic waves using an array of laser sources,” Ultrasonics 34, 69–77 (1996).
[Crossref]

1995 (2)

J. S. Steckenrider, T. W. Murray, J. B. Deaton, and J. W. Wagner, “Sensitivity enhancement in laser ultrasonics using a versatile laser array system,” J. Acoust. Soc. Am. 97, 273–279 (1995).
[Crossref]

M.-H. Noroy, D. Royer, and M. Fink, “Shear-wave focusing with a laser-ultrasound phased-array,” IEEE T. Ultrason. Ferr. 42, 981–988 (1995).
[Crossref]

1993 (2)

M.-H. Noroy, D. Royer, and M. Fink, “The laser-generated ultrasonic phased array: Analysis and experiments,” J. Acoust. Soc. Am. 94, 1934–1943 (1993).
[Crossref]

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)

K. L. Telschow and R. J. Conant, “Optical and thermal parameter effects on laser-generated ultrasound,” J. Acoust. Soc. Am. 88, 1494–1502 (1990).
[Crossref]

1989 (2)

J. Jarzynski and Y. H. Berthelot, “The use of optical fibers to enhance the laser generation of ultrasonic waves,” J. Acoust. Soc. Am. 8, 158–162 (1989).
[Crossref]

J. P. Monchalin and R. Heon, “Broadband optical detection of ultrasound by sideband stripping with a confocal Fabry-Pérot,” Appl. Phys. Lett. 55, 1612–1614 (1989).
[Crossref]

1988 (1)

A. J. A. Bruinsma and J. A. Vogel, “Ultrasonic noncontact inspection system with optical fiber,” Appl. Optics 27, 4690–4695 (1988).
[Crossref]

1987 (1)

J. Wagner and J. Spicer, “Theoretical noise-limited sensitivity of classical interferometry,” J. Opt. Soc. Am. 4, 1316–1326 (1987).
[Crossref]

1986 (2)

P. A. Doyle, “On optical waveform for laser-generated ultrasound,” J. Phys. D Appl. Phys. 19, 1613–1623 (1986).
[Crossref]

J. P. Monchalin, “Optical detection of ultrasound,” IEEE T. Ultrason. Ferr. 33, 485–499 (1986).
[Crossref]

1984 (1)

L. R. F. Rose, “Point-source representation for laser-generated ultrasound,” J. Acoust. Soc. Am. 75, 723–732 (1984).
[Crossref]

1981 (1)

A. M. Aindow, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Laser-generated ultrasonic pulses at free metal surfaces,” J. Acoust. Soc. Am. 69, 449–455 (1981).
[Crossref]

1968 (1)

R. E. Lee and R. M. White, “Excitation of surface elastic waves by transient surface heating,” Appl. Phys. Lett. 12, 12–14 (1968).
[Crossref]

1963 (1)

R. M. White, “Generation of elastic waves by transient surface heating,” J. Appl. Phys. 34, 3559–3567 (1963).
[Crossref]

Achenbach, J. D.

L. S. Wang, J. S. Steckenrider, and J. D. Achenbach, “A fiber-based laser ultrasonic system for remote inspection of limited access components,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 16 (Plenum, 1997), pp. 507–514.
[Crossref]

Aindow, A. M.

A. M. Aindow, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Laser-generated ultrasonic pulses at free metal surfaces,” J. Acoust. Soc. Am. 69, 449–455 (1981).
[Crossref]

Audoin, B.

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

Barina, C.

K. R. Yawn, M. A. Osterkamp, D. Kaiser, and C. Barina, “Improved laser ultrasonic systems for industry,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 1581 (AIP, 2014), pp. 397–404.

Bernstein, J. R.

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

Berthelot, Y. H.

J. Jarzynski and Y. H. Berthelot, “The use of optical fibers to enhance the laser generation of ultrasonic waves,” J. Acoust. Soc. Am. 8, 158–162 (1989).
[Crossref]

Blouin, A.

G. Rousseau and A. Blouin, “Hadamard multiplexing in laser ultrasonics,” Opt. Express 20, 25798–25816 (2012).
[Crossref] [PubMed]

D. Levesque, A. Blouin, C. Neron, and J.-P. Monchalin, “Performance of laser-ultrasonic F-SAFT imaging,” Ultrasonics 40, 1057–1063 (2002).
[Crossref] [PubMed]

A. Blouin, D. Levesque, C. Neron, D. Drolet, and J.-P. Monchalin, “Improved resolution and signal-to-noise ratio in laser-ultrasonics by saft processing,” Opt. Express 2, 531–539 (1998).
[Crossref] [PubMed]

P. Delaye, A. Blouin, D. Drolet, L. A. de Montmorillon, G. Roosen, and J. P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive inp:fe under an applied dc field,” J. Opt. Soc. Am. 14, 1723–1734 (1997).
[Crossref]

Bowler, A. I.

A. I. Bowler, B. W. Drinkwater, and P. D. Wilcox, “An investigation into the feasibility of internal strain measurement in solids by correlation of ultrasonic images,” P. Roy. Soc. A-Math. Phy. 467, 2247–2270 (2011).
[Crossref]

Bruinsma, A. J. A.

A. J. A. Bruinsma and J. A. Vogel, “Ultrasonic noncontact inspection system with optical fiber,” Appl. Optics 27, 4690–4695 (1988).
[Crossref]

Bussiere, J. F.

M. Dubois, M. Militzer, A. Moreau, and J. F. Bussiere, “A new technique for the quantitative real-time monitoring of austenite grain growth in steel,” Scripta Mater. 42, 867–874 (2000).
[Crossref]

Cawley, P.

J. Davies, F. Simonetti, M. Lowe, and P. Cawley, “Review of synthetically focused guided wave imaging techniques with application to defect sizing,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 25 (Plenum, 2006), pp. 142–149.
[Crossref]

Chen, Y.-J.

Y.-J. Chen, “Relationship between ultrasonic characteristics and relative porosity in Al and Al-XSi alloys,” Mater. Trans. 50, 2308–2313 (2009).
[Crossref]

Clark, M.

S. D. Sharples, M. Clark, and M. G. Somekh, “All-optical adaptive scanning acoustic microscope,” Ultrasonics 41, 295–299 (2003).
[Crossref] [PubMed]

M. Clark, S. D. Sharples, and M. G. Somekh, “Diffractive acoustic elements for laser ultrasonics,” J. Acoust. Soc. Am. 107, 3179–3185 (2000).
[Crossref] [PubMed]

T. Stratoudaki, M. Clark, and M. G. Somekh, “Cheap optical transducers (CHOTS) for generation and detection of longitudinal waves,” in 2012 IEEE International Ultrasonics Symposium (IEEE, 2012), pp. 961–964.

Conant, R. J.

K. L. Telschow and R. J. Conant, “Optical and thermal parameter effects on laser-generated ultrasound,” J. Acoust. Soc. Am. 88, 1494–1502 (1990).
[Crossref]

Davies, J.

J. Davies, F. Simonetti, M. Lowe, and P. Cawley, “Review of synthetically focused guided wave imaging techniques with application to defect sizing,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 25 (Plenum, 2006), pp. 142–149.
[Crossref]

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]

de Montmorillon, L. A.

P. Delaye, A. Blouin, D. Drolet, L. A. de Montmorillon, G. Roosen, and J. P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive inp:fe under an applied dc field,” J. Opt. Soc. Am. 14, 1723–1734 (1997).
[Crossref]

Deaton, J. B.

J. T. W. Murray, J. B. Deaton, and J. W. Wagner, “Experimental evaluation of enhanced generation of ultrasonic waves using an array of laser sources,” Ultrasonics 34, 69–77 (1996).
[Crossref]

J. S. Steckenrider, T. W. Murray, J. B. Deaton, and J. W. Wagner, “Sensitivity enhancement in laser ultrasonics using a versatile laser array system,” J. Acoust. Soc. Am. 97, 273–279 (1995).
[Crossref]

Dehoux, T.

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

Delaye, P.

P. Delaye, A. Blouin, D. Drolet, L. A. de Montmorillon, G. Roosen, and J. P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive inp:fe under an applied dc field,” J. Opt. Soc. Am. 14, 1723–1734 (1997).
[Crossref]

Demachi, K.

C. Pei, K. Demachi, T. Fukuchi, K. Koyama, and M. Uesaka, “Cracks measurement using fiber-phased array laser ultrasound generation,” J. Appl. Phys. 113, 163101 (2013).
[Crossref]

Dewhurst, R. J.

A. M. Aindow, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Laser-generated ultrasonic pulses at free metal surfaces,” J. Acoust. Soc. Am. 69, 449–455 (1981).
[Crossref]

Doyle, P. A.

P. A. Doyle, “On optical waveform for laser-generated ultrasound,” J. Phys. D Appl. Phys. 19, 1613–1623 (1986).
[Crossref]

Drain, L. E.

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

Drinkwater, B. W.

C. Li, D. Pain, P. D. Wilcox, and B. W. Drinkwater, “Imaging composite material using ultrasonic arrays,” NDT&E Int. 53, 8–17 (2013).
[Crossref]

A. I. Bowler, B. W. Drinkwater, and P. D. Wilcox, “An investigation into the feasibility of internal strain measurement in solids by correlation of ultrasonic images,” P. Roy. Soc. A-Math. Phy. 467, 2247–2270 (2011).
[Crossref]

J. Zhang, B. W. Drinkwater, and P. D. Wilcox, “Defect characterization using an ultrasonic array to measure the scattering coefïňĄcient matrix,” IEEE T. Ultrason. Ferr. 55, 2254–2265 (2008).
[Crossref]

P. D. W. C. Holmes and B. W. Drinkwater, “Advanced post-processing for scanned ultrasonic arrays: Application to defect detection and classification in non-destructive evaluation,” Ultrasonics 48, 636–642 (2008).
[Crossref] [PubMed]

P. D. W. C. Holmes and B. W. Drinkwater, “Post-processing of the full matrix of ultrasonic transmit–receive array data for non-destructive evaluation,” NDT&E Int. 38, 701–711 (2005)
[Crossref]

Drolet, D.

A. Blouin, D. Levesque, C. Neron, D. Drolet, and J.-P. Monchalin, “Improved resolution and signal-to-noise ratio in laser-ultrasonics by saft processing,” Opt. Express 2, 531–539 (1998).
[Crossref] [PubMed]

P. Delaye, A. Blouin, D. Drolet, L. A. de Montmorillon, G. Roosen, and J. P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive inp:fe under an applied dc field,” J. Opt. Soc. Am. 14, 1723–1734 (1997).
[Crossref]

Dubois, M.

M. Dubois, M. Militzer, A. Moreau, and J. F. Bussiere, “A new technique for the quantitative real-time monitoring of austenite grain growth in steel,” Scripta Mater. 42, 867–874 (2000).
[Crossref]

Dunning, G. J.

P. V. Mitchell, G. J. Dunning, S. W. McCahon, M. B. Klein, T. R. OâĂŹMeara, and D. M. Pepper, “Compensated high-bandwidth laser ultrasonic detector based on photo-induced emf in GaAs,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 15 (Plenum, 1996), pp. 2149–2155.
[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]

Ekhlakov, A.

E. Glushkov, N. Glushkova, A. Ekhlakov, and E. Shapar, “An analytically based computer model for surface measurements in ultrasonic crack detection,” Wave Motion 43, 458–473 (2006).
[Crossref]

Erdahl, D. S.

S. N. Hopko, I. C. Ume, and D. S. Erdahl, “Development of a flexible laser ultrasonic probe,” J. Manuf. Sci. E. - T. ASME 124, 351–357 (2002).
[Crossref]

Fink, M.

M.-H. Noroy, D. Royer, and M. Fink, “Shear-wave focusing with a laser-ultrasound phased-array,” IEEE T. Ultrason. Ferr. 42, 981–988 (1995).
[Crossref]

M.-H. Noroy, D. Royer, and M. Fink, “The laser-generated ultrasonic phased array: Analysis and experiments,” J. Acoust. Soc. Am. 94, 1934–1943 (1993).
[Crossref]

Fukuchi, T.

C. Pei, K. Demachi, T. Fukuchi, K. Koyama, and M. Uesaka, “Cracks measurement using fiber-phased array laser ultrasound generation,” J. Appl. Phys. 113, 163101 (2013).
[Crossref]

Glushkov, E.

E. Glushkov, N. Glushkova, A. Ekhlakov, and E. Shapar, “An analytically based computer model for surface measurements in ultrasonic crack detection,” Wave Motion 43, 458–473 (2006).
[Crossref]

Glushkova, N.

E. Glushkov, N. Glushkova, A. Ekhlakov, and E. Shapar, “An analytically based computer model for surface measurements in ultrasonic crack detection,” Wave Motion 43, 458–473 (2006).
[Crossref]

Heon, R.

J. P. Monchalin and R. Heon, “Broadband optical detection of ultrasound by sideband stripping with a confocal Fabry-Pérot,” Appl. Phys. Lett. 55, 1612–1614 (1989).
[Crossref]

Holmes, P. D. W. C.

P. D. W. C. Holmes and B. W. Drinkwater, “Advanced post-processing for scanned ultrasonic arrays: Application to defect detection and classification in non-destructive evaluation,” Ultrasonics 48, 636–642 (2008).
[Crossref] [PubMed]

P. D. W. C. Holmes and B. W. Drinkwater, “Post-processing of the full matrix of ultrasonic transmit–receive array data for non-destructive evaluation,” NDT&E Int. 38, 701–711 (2005)
[Crossref]

Hopko, S. N.

S. N. Hopko, I. C. Ume, and D. S. Erdahl, “Development of a flexible laser ultrasonic probe,” J. Manuf. Sci. E. - T. ASME 124, 351–357 (2002).
[Crossref]

Hutchins, D. A.

A. M. Aindow, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Laser-generated ultrasonic pulses at free metal surfaces,” J. Acoust. Soc. Am. 69, 449–455 (1981).
[Crossref]

Jarzynski, J.

J. Jarzynski and Y. H. Berthelot, “The use of optical fibers to enhance the laser generation of ultrasonic waves,” J. Acoust. Soc. Am. 8, 158–162 (1989).
[Crossref]

Kaiser, D.

K. R. Yawn, M. A. Osterkamp, D. Kaiser, and C. Barina, “Improved laser ultrasonic systems for industry,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 1581 (AIP, 2014), pp. 397–404.

Kim, H. J.

A. L. Lopez-Sanchez, H. J. Kim, L. W. Schmerr, and A. Sedov, “Measurement models and scattering models for predicting the ultrasonic pulse-echo response from side-drilled holes,” J. Nondestruct. Eval. 24, 83–96 (2005).
[Crossref]

Klein, M. B.

P. V. Mitchell, G. J. Dunning, S. W. McCahon, M. B. Klein, T. R. OâĂŹMeara, and D. M. Pepper, “Compensated high-bandwidth laser ultrasonic detector based on photo-induced emf in GaAs,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 15 (Plenum, 1996), pp. 2149–2155.
[Crossref]

Koyama, K.

C. Pei, K. Demachi, T. Fukuchi, K. Koyama, and M. Uesaka, “Cracks measurement using fiber-phased array laser ultrasound generation,” J. Appl. Phys. 113, 163101 (2013).
[Crossref]

Krishnaswamy, S.

T. W. Murray and S. Krishnaswamy, “Multiplexed interferometer for ultrasonic imaging applications,” Opt. Eng. 40, 1321–1328 (2001).
[Crossref]

Lee, R. E.

R. E. Lee and R. M. White, “Excitation of surface elastic waves by transient surface heating,” Appl. Phys. Lett. 12, 12–14 (1968).
[Crossref]

Levesque, D.

Li, C.

C. Li, D. Pain, P. D. Wilcox, and B. W. Drinkwater, “Imaging composite material using ultrasonic arrays,” NDT&E Int. 53, 8–17 (2013).
[Crossref]

Lopez-Sanchez, A. L.

A. L. Lopez-Sanchez, H. J. Kim, L. W. Schmerr, and A. Sedov, “Measurement models and scattering models for predicting the ultrasonic pulse-echo response from side-drilled holes,” J. Nondestruct. Eval. 24, 83–96 (2005).
[Crossref]

Lowe, M.

J. Davies, F. Simonetti, M. Lowe, and P. Cawley, “Review of synthetically focused guided wave imaging techniques with application to defect sizing,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 25 (Plenum, 2006), pp. 142–149.
[Crossref]

McCahon, S. W.

P. V. Mitchell, G. J. Dunning, S. W. McCahon, M. B. Klein, T. R. OâĂŹMeara, and D. M. Pepper, “Compensated high-bandwidth laser ultrasonic detector based on photo-induced emf in GaAs,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 15 (Plenum, 1996), pp. 2149–2155.
[Crossref]

Mi, B.

B. Mi and C. Ume, “Real-time weld penetration depth monitoring with laser ultrasonic sensing system,” J. Manuf. Sci. E. - T. ASME 128, 280–286 (2006).
[Crossref]

Militzer, M.

M. Dubois, M. Militzer, A. Moreau, and J. F. Bussiere, “A new technique for the quantitative real-time monitoring of austenite grain growth in steel,” Scripta Mater. 42, 867–874 (2000).
[Crossref]

Miller, G. F.

G. F. Miller and H. Pursey, “The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid,” in Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, vol. 223 (Royal Society, 1954), pp. 521–542.

Mitchell, P. V.

P. V. Mitchell, G. J. Dunning, S. W. McCahon, M. B. Klein, T. R. OâĂŹMeara, and D. M. Pepper, “Compensated high-bandwidth laser ultrasonic detector based on photo-induced emf in GaAs,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 15 (Plenum, 1996), pp. 2149–2155.
[Crossref]

Monchalin, J. P.

P. Delaye, A. Blouin, D. Drolet, L. A. de Montmorillon, G. Roosen, and J. P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive inp:fe under an applied dc field,” J. Opt. Soc. Am. 14, 1723–1734 (1997).
[Crossref]

J. P. Monchalin and R. Heon, “Broadband optical detection of ultrasound by sideband stripping with a confocal Fabry-Pérot,” Appl. Phys. Lett. 55, 1612–1614 (1989).
[Crossref]

J. P. Monchalin, “Optical detection of ultrasound,” IEEE T. Ultrason. Ferr. 33, 485–499 (1986).
[Crossref]

J. P. Monchalin, “Progress towards the application of laser-ultrasonics in industry,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 11 (Plenum, 1993), pp. 495–506.
[Crossref]

Monchalin, J.-P.

Moreau, A.

M. Dubois, M. Militzer, A. Moreau, and J. F. Bussiere, “A new technique for the quantitative real-time monitoring of austenite grain growth in steel,” Scripta Mater. 42, 867–874 (2000).
[Crossref]

Murray, J. T. W.

J. T. W. Murray, J. B. Deaton, and J. W. Wagner, “Experimental evaluation of enhanced generation of ultrasonic waves using an array of laser sources,” Ultrasonics 34, 69–77 (1996).
[Crossref]

Murray, T. W.

T. W. Murray and S. Krishnaswamy, “Multiplexed interferometer for ultrasonic imaging applications,” Opt. Eng. 40, 1321–1328 (2001).
[Crossref]

J. S. Steckenrider, T. W. Murray, J. B. Deaton, and J. W. Wagner, “Sensitivity enhancement in laser ultrasonics using a versatile laser array system,” J. Acoust. Soc. Am. 97, 273–279 (1995).
[Crossref]

Neron, C.

Noroy, M.-H.

M.-H. Noroy, D. Royer, and M. Fink, “Shear-wave focusing with a laser-ultrasound phased-array,” IEEE T. Ultrason. Ferr. 42, 981–988 (1995).
[Crossref]

M.-H. Noroy, D. Royer, and M. Fink, “The laser-generated ultrasonic phased array: Analysis and experiments,” J. Acoust. Soc. Am. 94, 1934–1943 (1993).
[Crossref]

OâAZMeara, T. R.

P. V. Mitchell, G. J. Dunning, S. W. McCahon, M. B. Klein, T. R. OâĂŹMeara, and D. M. Pepper, “Compensated high-bandwidth laser ultrasonic detector based on photo-induced emf in GaAs,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 15 (Plenum, 1996), pp. 2149–2155.
[Crossref]

Osterkamp, M. A.

K. R. Yawn, M. A. Osterkamp, D. Kaiser, and C. Barina, “Improved laser ultrasonic systems for industry,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 1581 (AIP, 2014), pp. 397–404.

Pain, D.

C. Li, D. Pain, P. D. Wilcox, and B. W. Drinkwater, “Imaging composite material using ultrasonic arrays,” NDT&E Int. 53, 8–17 (2013).
[Crossref]

Palik, E.

E. Palik, Handbook of Optical Constants of Solids (Academic, 1998).

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]

A. M. Aindow, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Laser-generated ultrasonic pulses at free metal surfaces,” J. Acoust. Soc. Am. 69, 449–455 (1981).
[Crossref]

Pei, C.

C. Pei, K. Demachi, T. Fukuchi, K. Koyama, and M. Uesaka, “Cracks measurement using fiber-phased array laser ultrasound generation,” J. Appl. Phys. 113, 163101 (2013).
[Crossref]

Pepper, D. M.

P. V. Mitchell, G. J. Dunning, S. W. McCahon, M. B. Klein, T. R. OâĂŹMeara, and D. M. Pepper, “Compensated high-bandwidth laser ultrasonic detector based on photo-induced emf in GaAs,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 15 (Plenum, 1996), pp. 2149–2155.
[Crossref]

Pursey, H.

G. F. Miller and H. Pursey, “The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid,” in Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, vol. 223 (Royal Society, 1954), pp. 521–542.

Raetz, S.

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

Roosen, G.

P. Delaye, A. Blouin, D. Drolet, L. A. de Montmorillon, G. Roosen, and J. P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive inp:fe under an applied dc field,” J. Opt. Soc. Am. 14, 1723–1734 (1997).
[Crossref]

Rose, L. R. F.

L. R. F. Rose, “Point-source representation for laser-generated ultrasound,” J. Acoust. Soc. Am. 75, 723–732 (1984).
[Crossref]

Rousseau, G.

Royer, D.

M.-H. Noroy, D. Royer, and M. Fink, “Shear-wave focusing with a laser-ultrasound phased-array,” IEEE T. Ultrason. Ferr. 42, 981–988 (1995).
[Crossref]

M.-H. Noroy, D. Royer, and M. Fink, “The laser-generated ultrasonic phased array: Analysis and experiments,” J. Acoust. Soc. Am. 94, 1934–1943 (1993).
[Crossref]

Schmerr, L. W.

A. L. Lopez-Sanchez, H. J. Kim, L. W. Schmerr, and A. Sedov, “Measurement models and scattering models for predicting the ultrasonic pulse-echo response from side-drilled holes,” J. Nondestruct. Eval. 24, 83–96 (2005).
[Crossref]

Scruby, C. B.

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

Sedov, A.

A. L. Lopez-Sanchez, H. J. Kim, L. W. Schmerr, and A. Sedov, “Measurement models and scattering models for predicting the ultrasonic pulse-echo response from side-drilled holes,” J. Nondestruct. Eval. 24, 83–96 (2005).
[Crossref]

Shapar, E.

E. Glushkov, N. Glushkova, A. Ekhlakov, and E. Shapar, “An analytically based computer model for surface measurements in ultrasonic crack detection,” Wave Motion 43, 458–473 (2006).
[Crossref]

Sharples, S. D.

S. D. Sharples, M. Clark, and M. G. Somekh, “All-optical adaptive scanning acoustic microscope,” Ultrasonics 41, 295–299 (2003).
[Crossref] [PubMed]

M. Clark, S. D. Sharples, and M. G. Somekh, “Diffractive acoustic elements for laser ultrasonics,” J. Acoust. Soc. Am. 107, 3179–3185 (2000).
[Crossref] [PubMed]

Shi, Y.

S. C. Wooh and Y. Shi, “Optimum beam steering of linear phased arrays,” Wave Motion 29, 245–265 (1999).
[Crossref]

Simonetti, F.

J. Davies, F. Simonetti, M. Lowe, and P. Cawley, “Review of synthetically focused guided wave imaging techniques with application to defect sizing,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 25 (Plenum, 2006), pp. 142–149.
[Crossref]

Somekh, M. G.

S. D. Sharples, M. Clark, and M. G. Somekh, “All-optical adaptive scanning acoustic microscope,” Ultrasonics 41, 295–299 (2003).
[Crossref] [PubMed]

M. Clark, S. D. Sharples, and M. G. Somekh, “Diffractive acoustic elements for laser ultrasonics,” J. Acoust. Soc. Am. 107, 3179–3185 (2000).
[Crossref] [PubMed]

T. Stratoudaki, M. Clark, and M. G. Somekh, “Cheap optical transducers (CHOTS) for generation and detection of longitudinal waves,” in 2012 IEEE International Ultrasonics Symposium (IEEE, 2012), pp. 961–964.

Spicer, J.

J. Wagner and J. Spicer, “Theoretical noise-limited sensitivity of classical interferometry,” J. Opt. Soc. Am. 4, 1316–1326 (1987).
[Crossref]

Spicer, J. B.

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

Steckenrider, J. S.

J. S. Steckenrider, T. W. Murray, J. B. Deaton, and J. W. Wagner, “Sensitivity enhancement in laser ultrasonics using a versatile laser array system,” J. Acoust. Soc. Am. 97, 273–279 (1995).
[Crossref]

L. S. Wang, J. S. Steckenrider, and J. D. Achenbach, “A fiber-based laser ultrasonic system for remote inspection of limited access components,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 16 (Plenum, 1997), pp. 507–514.
[Crossref]

Stratoudaki, T.

T. Stratoudaki, M. Clark, and M. G. Somekh, “Cheap optical transducers (CHOTS) for generation and detection of longitudinal waves,” in 2012 IEEE International Ultrasonics Symposium (IEEE, 2012), pp. 961–964.

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]

Telschow, K. L.

K. L. Telschow and R. J. Conant, “Optical and thermal parameter effects on laser-generated ultrasound,” J. Acoust. Soc. Am. 88, 1494–1502 (1990).
[Crossref]

Uesaka, M.

C. Pei, K. Demachi, T. Fukuchi, K. Koyama, and M. Uesaka, “Cracks measurement using fiber-phased array laser ultrasound generation,” J. Appl. Phys. 113, 163101 (2013).
[Crossref]

Ume, C.

B. Mi and C. Ume, “Real-time weld penetration depth monitoring with laser ultrasonic sensing system,” J. Manuf. Sci. E. - T. ASME 128, 280–286 (2006).
[Crossref]

Ume, I. C.

S. N. Hopko, I. C. Ume, and D. S. Erdahl, “Development of a flexible laser ultrasonic probe,” J. Manuf. Sci. E. - T. ASME 124, 351–357 (2002).
[Crossref]

Vogel, J. A.

A. J. A. Bruinsma and J. A. Vogel, “Ultrasonic noncontact inspection system with optical fiber,” Appl. Optics 27, 4690–4695 (1988).
[Crossref]

Wagner, J.

J. Wagner and J. Spicer, “Theoretical noise-limited sensitivity of classical interferometry,” J. Opt. Soc. Am. 4, 1316–1326 (1987).
[Crossref]

J. Wagner, “Breaking the sensitivity barrier: the challenge for laser ultrasonics,” in Proceedings of IEEE Ultrasonics Symposium, vol. 1 and 2 (IEEE, 1992), pp. 791–800.

Wagner, J. W.

J. T. W. Murray, J. B. Deaton, and J. W. Wagner, “Experimental evaluation of enhanced generation of ultrasonic waves using an array of laser sources,” Ultrasonics 34, 69–77 (1996).
[Crossref]

J. S. Steckenrider, T. W. Murray, J. B. Deaton, and J. W. Wagner, “Sensitivity enhancement in laser ultrasonics using a versatile laser array system,” J. Acoust. Soc. Am. 97, 273–279 (1995).
[Crossref]

Wang, L. S.

L. S. Wang, J. S. Steckenrider, and J. D. Achenbach, “A fiber-based laser ultrasonic system for remote inspection of limited access components,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 16 (Plenum, 1997), pp. 507–514.
[Crossref]

White, R. M.

R. E. Lee and R. M. White, “Excitation of surface elastic waves by transient surface heating,” Appl. Phys. Lett. 12, 12–14 (1968).
[Crossref]

R. M. White, “Generation of elastic waves by transient surface heating,” J. Appl. Phys. 34, 3559–3567 (1963).
[Crossref]

Wilcox, P. D.

C. Li, D. Pain, P. D. Wilcox, and B. W. Drinkwater, “Imaging composite material using ultrasonic arrays,” NDT&E Int. 53, 8–17 (2013).
[Crossref]

A. I. Bowler, B. W. Drinkwater, and P. D. Wilcox, “An investigation into the feasibility of internal strain measurement in solids by correlation of ultrasonic images,” P. Roy. Soc. A-Math. Phy. 467, 2247–2270 (2011).
[Crossref]

J. Zhang, B. W. Drinkwater, and P. D. Wilcox, “Defect characterization using an ultrasonic array to measure the scattering coefïňĄcient matrix,” IEEE T. Ultrason. Ferr. 55, 2254–2265 (2008).
[Crossref]

P. D. Wilcox, “Ultrasonic arrays in nde: Beyond the b-scan,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 1151 (AIP, 2013), pp. 33–50.

Wooh, S. C.

S. C. Wooh and Y. Shi, “Optimum beam steering of linear phased arrays,” Wave Motion 29, 245–265 (1999).
[Crossref]

Yawn, K. R.

K. R. Yawn, M. A. Osterkamp, D. Kaiser, and C. Barina, “Improved laser ultrasonic systems for industry,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 1581 (AIP, 2014), pp. 397–404.

Zhang, J.

J. Zhang, B. W. Drinkwater, and P. D. Wilcox, “Defect characterization using an ultrasonic array to measure the scattering coefïňĄcient matrix,” IEEE T. Ultrason. Ferr. 55, 2254–2265 (2008).
[Crossref]

Appl. Optics (1)

A. J. A. Bruinsma and J. A. Vogel, “Ultrasonic noncontact inspection system with optical fiber,” Appl. Optics 27, 4690–4695 (1988).
[Crossref]

Appl. Phys. Lett. (2)

R. E. Lee and R. M. White, “Excitation of surface elastic waves by transient surface heating,” Appl. Phys. Lett. 12, 12–14 (1968).
[Crossref]

J. P. Monchalin and R. Heon, “Broadband optical detection of ultrasound by sideband stripping with a confocal Fabry-Pérot,” Appl. Phys. Lett. 55, 1612–1614 (1989).
[Crossref]

IEEE T. Ultrason. Ferr. (3)

J. P. Monchalin, “Optical detection of ultrasound,” IEEE T. Ultrason. Ferr. 33, 485–499 (1986).
[Crossref]

M.-H. Noroy, D. Royer, and M. Fink, “Shear-wave focusing with a laser-ultrasound phased-array,” IEEE T. Ultrason. Ferr. 42, 981–988 (1995).
[Crossref]

J. Zhang, B. W. Drinkwater, and P. D. Wilcox, “Defect characterization using an ultrasonic array to measure the scattering coefïňĄcient matrix,” IEEE T. Ultrason. Ferr. 55, 2254–2265 (2008).
[Crossref]

J. Acoust. Soc. Am. (9)

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

M.-H. Noroy, D. Royer, and M. Fink, “The laser-generated ultrasonic phased array: Analysis and experiments,” J. Acoust. Soc. Am. 94, 1934–1943 (1993).
[Crossref]

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

J. S. Steckenrider, T. W. Murray, J. B. Deaton, and J. W. Wagner, “Sensitivity enhancement in laser ultrasonics using a versatile laser array system,” J. Acoust. Soc. Am. 97, 273–279 (1995).
[Crossref]

K. L. Telschow and R. J. Conant, “Optical and thermal parameter effects on laser-generated ultrasound,” J. Acoust. Soc. Am. 88, 1494–1502 (1990).
[Crossref]

L. R. F. Rose, “Point-source representation for laser-generated ultrasound,” J. Acoust. Soc. Am. 75, 723–732 (1984).
[Crossref]

M. Clark, S. D. Sharples, and M. G. Somekh, “Diffractive acoustic elements for laser ultrasonics,” J. Acoust. Soc. Am. 107, 3179–3185 (2000).
[Crossref] [PubMed]

J. Jarzynski and Y. H. Berthelot, “The use of optical fibers to enhance the laser generation of ultrasonic waves,” J. Acoust. Soc. Am. 8, 158–162 (1989).
[Crossref]

A. M. Aindow, R. J. Dewhurst, D. A. Hutchins, and S. B. Palmer, “Laser-generated ultrasonic pulses at free metal surfaces,” J. Acoust. Soc. Am. 69, 449–455 (1981).
[Crossref]

J. Appl. Phys. (2)

C. Pei, K. Demachi, T. Fukuchi, K. Koyama, and M. Uesaka, “Cracks measurement using fiber-phased array laser ultrasound generation,” J. Appl. Phys. 113, 163101 (2013).
[Crossref]

R. M. White, “Generation of elastic waves by transient surface heating,” J. Appl. Phys. 34, 3559–3567 (1963).
[Crossref]

J. Manuf. Sci. E. - T. ASME (2)

S. N. Hopko, I. C. Ume, and D. S. Erdahl, “Development of a flexible laser ultrasonic probe,” J. Manuf. Sci. E. - T. ASME 124, 351–357 (2002).
[Crossref]

B. Mi and C. Ume, “Real-time weld penetration depth monitoring with laser ultrasonic sensing system,” J. Manuf. Sci. E. - T. ASME 128, 280–286 (2006).
[Crossref]

J. Nondestruct. Eval. (1)

A. L. Lopez-Sanchez, H. J. Kim, L. W. Schmerr, and A. Sedov, “Measurement models and scattering models for predicting the ultrasonic pulse-echo response from side-drilled holes,” J. Nondestruct. Eval. 24, 83–96 (2005).
[Crossref]

J. Opt. Soc. Am. (2)

J. Wagner and J. Spicer, “Theoretical noise-limited sensitivity of classical interferometry,” J. Opt. Soc. Am. 4, 1316–1326 (1987).
[Crossref]

P. Delaye, A. Blouin, D. Drolet, L. A. de Montmorillon, G. Roosen, and J. P. Monchalin, “Detection of ultrasonic motion of a scattering surface by photorefractive inp:fe under an applied dc field,” J. Opt. Soc. Am. 14, 1723–1734 (1997).
[Crossref]

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

P. A. Doyle, “On optical waveform for laser-generated ultrasound,” J. Phys. D Appl. Phys. 19, 1613–1623 (1986).
[Crossref]

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]

Mater. Trans. (1)

Y.-J. Chen, “Relationship between ultrasonic characteristics and relative porosity in Al and Al-XSi alloys,” Mater. Trans. 50, 2308–2313 (2009).
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Supplementary Material (2)

NameDescription
» Visualization 1: MP4 (828 KB)      Effect of filter frequency (sample 1)
» Visualization 2: MP4 (534 KB)      Effect of filter frequency (sample 2)

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Figures (9)

Fig. 1
Fig. 1 (a) Directivity pattern of longitudinal (GL) and (b) shear waves (GT), in aluminum, in the thermoelastic regime. (c) Sensitivity of the detector to longitudinal (DL) and (d) shear waves (DT), in aluminum, as a function of wave angle.
Fig. 2
Fig. 2 Experimental setup, side view (a) and top view (b). The scan was parallel to the x-axis, on the xy-plane, and the defects were parallel to the y-axis. (c) Schematic diagram showing angles and path lengths. (d) The full matrix composed from all signals (sgd). The red, dotted line indicates the set of data used with the SAFT algorithm.
Fig. 3
Fig. 3 Schematic diagram of samples 1 (a) and sample 2 (b), side views. Sample 1 has through holes and sample 2 has through slots of orientations ranging from 0°–60°.
Fig. 4
Fig. 4 Images of sample 1: (a) TFM image using experimental data, (b) TFM image using the simulated response model, (c) the normalized TFM image of the experimental data, shown in (a), over the sensitivity, shown in (d), (d) the sensitivity image, (e) SAFT image using experimental data and (f) SAFT image using the simulated response model. The array aperture and the digital filter used in each case are marked on images (a), (b), (c) and (e), as well as the dynamic range used (dB scale).
Fig. 5
Fig. 5 B-scan of received signals at all detector positions, when generation is at the position of the first element, using the same data as in Fig. 4(a). The surface acoustic wave is marked as “R”. The dynamic range is in dB scale.
Fig. 6
Fig. 6 Images of sample 2: (a) TFM image using experimental data, (b) TFM image using the simulated response model, (c) the normalized TFM image of the experimental data, shown in (a), over the sensitivity, shown in (d), (d) the sensitivity image, (e) SAFT image using experimental data and (f) SAFT image using the simulated response model. The array aperture and the digital filter used in each case are marked on images (a), (b), (c) and (e), as well as the dynamic range used (dB scale).
Fig. 7
Fig. 7 Detail of the TFM images showing the 30° angle slot of sample 2 using digital filter of center frequency (a) 4 MHz, (b) 7 MHz, (c) 10 MHz and (d) 13 MHz. The dynamic range of all images is set at 20 dB to facilitate comparison.
Fig. 8
Fig. 8 The SNR of the TFM images, measured at reflectors no. 4 of sample 1 and sample 2, as a function of the digital filter central frequency. Visualization 1 (for sample 1) and Visualization 2 (for sample 2) present the results for all reflectors, as a movie composed of the TFM images using different digital filters.
Fig. 9
Fig. 9 TFM image of sample 1 using (a) a subset of the full matrix corresponding to array step of 310μm and 44 element array and (b) the full matrix corresponding to array step of 155μm and 89 element array. A digital filter of 5MHz was applied in post processing and the dynamic range is 20dB in both images.

Tables (1)

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Table 1 Details of side drilled holes and slots in test samples.

Equations (15)

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G L ( θ ) sin θ sin 2 θ ( κ 2 sin 2 θ ) 1 / 2 2 sin θ sin 2 θ ( κ 2 sin 2 θ ) 1 / 2 + ( κ 2 2 sin 2 θ ) 2
G T ( θ ) sin 2 θ cos 2 θ cos 2 2 θ + 2 sin θ sin 2 θ ( κ 2 sin 2 θ ) 1 / 2
D L ( θ ) = cos θ ( κ 2 2 sin 2 θ ) F 0 ( sin θ )
D T ( θ ) sin 2 θ ( κ 2 sin 2 θ 1 ) 1 / 2 F 0 ( κ sin θ )
F 0 ( ξ ) = ( 2 ξ 2 κ 2 ) 2 4 ξ 2 ( ξ 2 1 ) 1 / 2 ( ξ 2 κ 2 ) 1 / 2
H g d j α β ( ω ) = G α ( θ g j ) D β ( θ d j ) ( | d g j | | d dj | ) 1 / 2 exp [ i ω ( | d gj | c α + | d dj | c β ) ] A j α β ( θ g j , θ d j , ω )
H g d ( ω ) = h g d ( t ) exp ( i ω t ) d t
S g d ( ω ) = H g d ( ω t ) G ( ω )
s g d ( t ) = 0 S g d ( ω ) exp ( i ω t ) d ω
I ( r ) = | g = 1 n d = 1 n s g d ( t g d ( r ) ) |
t g d = d g ( r ) + d d ( r ) c T
h g d T T = G T ( θ g ( r ) ) D T ( θ d ( r ) ) [ d g ( r ) d d ( r ) ] 1 / 2 δ ( t d g ( r ) + d d ( r ) c T )
E ( r ) = | g = 1 n d = 1 n G T ( θ g ( r ) ) D T ( θ d ( r ) ) [ d g ( r ) d d ( r ) ] 1 / 2 | = | g = 1 n G T ( θ g ( r ) ) [ d g ( r ) ] 1 / 2 d = 1 n D T ( θ d ( r ) ) [ d d ( r ) ] 1 / 2 |
I d B = 20 log 10 I ( r ) I m a x
S N R δ P D B D

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