Abstract

The detection characteristics for photoacoustic imaging of microcracks in silicon wafers were theoretically and quantitatively investigated using a numerical simulation. The simulation is based on a one-dimensional multilayered thermal diffusion model coupled with the thermal-wave impedance of each layer, the layer structures of which are constructed along the wafer surface and are variable according to the scanning position of the point heat source. As the modulation frequency was reduced, the spatial resolution of the temperature amplitude profile at the cracks decreased, showing good agreement with the experimentally obtained photoacoustic amplitude images. At a modulation frequency of 200 kHz, for cracks with narrow air gaps of up to 20nm, which is much smaller than both the beam spot size of 1.5μm and the thermal diffusion length of 12μm, the temperature amplitude is twice that of regions without cracks, and the temperature contrast increased with an increase in the modulation frequency. These calculation results suggest the effectiveness of using a high modulation frequency, making it possible to detect microcracks of the order of 10 nm.

© 2007 Optical Society of America

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  1. A. Rosencwaig, Photoacoustics and Photoacoustic Spectroscopy (Wiley Interscience, 1980), pp. 170-173, 295-296.
  2. Y. H. Wong, R. L. Thomas, and J. J. Pouch, "Subsurface structures of solids by scanning photoacoustic microscopy," Appl. Phys. Lett. 35, 368-369 (1979).
    [CrossRef]
  3. T. Nakata, Y. Kembo, T. Kitamori, and T. Sawada, "Detection and imaging of subsurface microcracks in silicon wafers using photoacoustic microscope," Jpn. J. Appl. Phys. Suppl. 31-1, 146-148 (1992).
  4. B. Witowski, W. L. Smith, and D. L. Willenborg, "Nondestructive technique for the detection of dislocations and stacking faults on silicon wafers," Appl. Phys. Lett. 52, 640-642 (1988).
    [CrossRef]
  5. W. L. Smith, A. Rosencwaig, and D. L. Willenborg, "Ion implant monitoring with thermal wave technology," Appl. Phys. Lett. 47, 584-586 (1985).
    [CrossRef]
  6. H. I. Ringermacher and C. A. Kittredge, "Laser-in/laser-out photoacoustics using Doppler heterodyne interferometry," in Proceedings of the 1986 Ultrasonic Symposium (IEEE, 1986), pp. 407-410.
  7. K. R. Grice, L. J. Inglehart, L. D. Favro, P. K. Kuo, and R. L. Thomas, "Thermal wave imaging of closed cracks in opaque solids," J. Appl. Phys. 54, 6245-6255 (1983).
    [CrossRef]
  8. I. Kaufman, P. T. Chang, H. S. Hsu, W. Y. Huang, and D. Y. Shyong, "Photothermal radiometric detection and imaging of surface cracks," J. Nondestruct. Eval. 6, 87-100 (1987).
    [CrossRef]
  9. J. Hartikainen, "Fast photothermal measurement system for inspection of weak adhesion defects," Appl. Phys. Lett. 55, 1188-1190 (1989).
    [CrossRef]
  10. H. Endoh, N. Ohtaki, and T. Hoshimiya, "Nondestructive detection of tilted surface defect with wedge shape by photoacoustic microscopy," Jpn. J. Appl. Phys. 45, 4609-4611 (2006).
    [CrossRef]
  11. A. Minamide, S. Shida, and Y. Tokunaga, "Improved method for three-dimensional imaging in photoacoustic microscope," Jpn. J. Appl. Phys. 38, 3189-3190 (1999).
    [CrossRef]
  12. A. Rosencwaig and A. Gersho, "Theory of the photoacoustic effect with solids," J. Appl. Phys. 47, 64-69 (1976).
    [CrossRef]
  13. F. A. McDonald and G. C. Wetsel, Jr., "Generalized theory of the photoacoustic effect," J. Appl. Phys. 49, 2313-2322 (1978).
    [CrossRef]
  14. W. B. Jackson and N. M. Amer, "Piezoelectric photoacoustic detection: theory and experiment," J. Appl. Phys. 51, 3343-3353 (1980).
    [CrossRef]
  15. J. Opsal and A. Rosencwaig, "Thermal-wave depth profiling: theory," J. Appl. Phys. 53, 4240-4246 (1982).
    [CrossRef]
  16. H. Nabeta, K. Yamanaka, Y. Nagata, T. Koda, Y. Kanemitsu, and Y. Masumoto, "Characterization of multilayered structures by piezoelectric photoacoustic imaging," Jpn. J. Appl. Phys. Suppl. 30-1, 289-291 (1991).
  17. M. Hangyo, S. Nakashima, S. Sugimoto, T. Yamaguchi, and A. Mitsuishi, "Study of edge effect in photoacoustic microscopy," Jpn. J. Appl. Phys. 25, 376-379 (1986).
    [CrossRef]

2006 (1)

H. Endoh, N. Ohtaki, and T. Hoshimiya, "Nondestructive detection of tilted surface defect with wedge shape by photoacoustic microscopy," Jpn. J. Appl. Phys. 45, 4609-4611 (2006).
[CrossRef]

1999 (1)

A. Minamide, S. Shida, and Y. Tokunaga, "Improved method for three-dimensional imaging in photoacoustic microscope," Jpn. J. Appl. Phys. 38, 3189-3190 (1999).
[CrossRef]

1992 (1)

T. Nakata, Y. Kembo, T. Kitamori, and T. Sawada, "Detection and imaging of subsurface microcracks in silicon wafers using photoacoustic microscope," Jpn. J. Appl. Phys. Suppl. 31-1, 146-148 (1992).

1991 (1)

H. Nabeta, K. Yamanaka, Y. Nagata, T. Koda, Y. Kanemitsu, and Y. Masumoto, "Characterization of multilayered structures by piezoelectric photoacoustic imaging," Jpn. J. Appl. Phys. Suppl. 30-1, 289-291 (1991).

1989 (1)

J. Hartikainen, "Fast photothermal measurement system for inspection of weak adhesion defects," Appl. Phys. Lett. 55, 1188-1190 (1989).
[CrossRef]

1988 (1)

B. Witowski, W. L. Smith, and D. L. Willenborg, "Nondestructive technique for the detection of dislocations and stacking faults on silicon wafers," Appl. Phys. Lett. 52, 640-642 (1988).
[CrossRef]

1987 (1)

I. Kaufman, P. T. Chang, H. S. Hsu, W. Y. Huang, and D. Y. Shyong, "Photothermal radiometric detection and imaging of surface cracks," J. Nondestruct. Eval. 6, 87-100 (1987).
[CrossRef]

1986 (1)

M. Hangyo, S. Nakashima, S. Sugimoto, T. Yamaguchi, and A. Mitsuishi, "Study of edge effect in photoacoustic microscopy," Jpn. J. Appl. Phys. 25, 376-379 (1986).
[CrossRef]

1985 (1)

W. L. Smith, A. Rosencwaig, and D. L. Willenborg, "Ion implant monitoring with thermal wave technology," Appl. Phys. Lett. 47, 584-586 (1985).
[CrossRef]

1983 (1)

K. R. Grice, L. J. Inglehart, L. D. Favro, P. K. Kuo, and R. L. Thomas, "Thermal wave imaging of closed cracks in opaque solids," J. Appl. Phys. 54, 6245-6255 (1983).
[CrossRef]

1982 (1)

J. Opsal and A. Rosencwaig, "Thermal-wave depth profiling: theory," J. Appl. Phys. 53, 4240-4246 (1982).
[CrossRef]

1980 (1)

W. B. Jackson and N. M. Amer, "Piezoelectric photoacoustic detection: theory and experiment," J. Appl. Phys. 51, 3343-3353 (1980).
[CrossRef]

1979 (1)

Y. H. Wong, R. L. Thomas, and J. J. Pouch, "Subsurface structures of solids by scanning photoacoustic microscopy," Appl. Phys. Lett. 35, 368-369 (1979).
[CrossRef]

1978 (1)

F. A. McDonald and G. C. Wetsel, Jr., "Generalized theory of the photoacoustic effect," J. Appl. Phys. 49, 2313-2322 (1978).
[CrossRef]

1976 (1)

A. Rosencwaig and A. Gersho, "Theory of the photoacoustic effect with solids," J. Appl. Phys. 47, 64-69 (1976).
[CrossRef]

Amer, N. M.

W. B. Jackson and N. M. Amer, "Piezoelectric photoacoustic detection: theory and experiment," J. Appl. Phys. 51, 3343-3353 (1980).
[CrossRef]

Chang, P. T.

I. Kaufman, P. T. Chang, H. S. Hsu, W. Y. Huang, and D. Y. Shyong, "Photothermal radiometric detection and imaging of surface cracks," J. Nondestruct. Eval. 6, 87-100 (1987).
[CrossRef]

Endoh, H.

H. Endoh, N. Ohtaki, and T. Hoshimiya, "Nondestructive detection of tilted surface defect with wedge shape by photoacoustic microscopy," Jpn. J. Appl. Phys. 45, 4609-4611 (2006).
[CrossRef]

Favro, L. D.

K. R. Grice, L. J. Inglehart, L. D. Favro, P. K. Kuo, and R. L. Thomas, "Thermal wave imaging of closed cracks in opaque solids," J. Appl. Phys. 54, 6245-6255 (1983).
[CrossRef]

Gersho, A.

A. Rosencwaig and A. Gersho, "Theory of the photoacoustic effect with solids," J. Appl. Phys. 47, 64-69 (1976).
[CrossRef]

Grice, K. R.

K. R. Grice, L. J. Inglehart, L. D. Favro, P. K. Kuo, and R. L. Thomas, "Thermal wave imaging of closed cracks in opaque solids," J. Appl. Phys. 54, 6245-6255 (1983).
[CrossRef]

Hangyo, M.

M. Hangyo, S. Nakashima, S. Sugimoto, T. Yamaguchi, and A. Mitsuishi, "Study of edge effect in photoacoustic microscopy," Jpn. J. Appl. Phys. 25, 376-379 (1986).
[CrossRef]

Hartikainen, J.

J. Hartikainen, "Fast photothermal measurement system for inspection of weak adhesion defects," Appl. Phys. Lett. 55, 1188-1190 (1989).
[CrossRef]

Hoshimiya, T.

H. Endoh, N. Ohtaki, and T. Hoshimiya, "Nondestructive detection of tilted surface defect with wedge shape by photoacoustic microscopy," Jpn. J. Appl. Phys. 45, 4609-4611 (2006).
[CrossRef]

Hsu, H. S.

I. Kaufman, P. T. Chang, H. S. Hsu, W. Y. Huang, and D. Y. Shyong, "Photothermal radiometric detection and imaging of surface cracks," J. Nondestruct. Eval. 6, 87-100 (1987).
[CrossRef]

Huang, W. Y.

I. Kaufman, P. T. Chang, H. S. Hsu, W. Y. Huang, and D. Y. Shyong, "Photothermal radiometric detection and imaging of surface cracks," J. Nondestruct. Eval. 6, 87-100 (1987).
[CrossRef]

Inglehart, L. J.

K. R. Grice, L. J. Inglehart, L. D. Favro, P. K. Kuo, and R. L. Thomas, "Thermal wave imaging of closed cracks in opaque solids," J. Appl. Phys. 54, 6245-6255 (1983).
[CrossRef]

Jackson, W. B.

W. B. Jackson and N. M. Amer, "Piezoelectric photoacoustic detection: theory and experiment," J. Appl. Phys. 51, 3343-3353 (1980).
[CrossRef]

Kanemitsu, Y.

H. Nabeta, K. Yamanaka, Y. Nagata, T. Koda, Y. Kanemitsu, and Y. Masumoto, "Characterization of multilayered structures by piezoelectric photoacoustic imaging," Jpn. J. Appl. Phys. Suppl. 30-1, 289-291 (1991).

Kaufman, I.

I. Kaufman, P. T. Chang, H. S. Hsu, W. Y. Huang, and D. Y. Shyong, "Photothermal radiometric detection and imaging of surface cracks," J. Nondestruct. Eval. 6, 87-100 (1987).
[CrossRef]

Kembo, Y.

T. Nakata, Y. Kembo, T. Kitamori, and T. Sawada, "Detection and imaging of subsurface microcracks in silicon wafers using photoacoustic microscope," Jpn. J. Appl. Phys. Suppl. 31-1, 146-148 (1992).

Kitamori, T.

T. Nakata, Y. Kembo, T. Kitamori, and T. Sawada, "Detection and imaging of subsurface microcracks in silicon wafers using photoacoustic microscope," Jpn. J. Appl. Phys. Suppl. 31-1, 146-148 (1992).

Kittredge, C. A.

H. I. Ringermacher and C. A. Kittredge, "Laser-in/laser-out photoacoustics using Doppler heterodyne interferometry," in Proceedings of the 1986 Ultrasonic Symposium (IEEE, 1986), pp. 407-410.

Koda, T.

H. Nabeta, K. Yamanaka, Y. Nagata, T. Koda, Y. Kanemitsu, and Y. Masumoto, "Characterization of multilayered structures by piezoelectric photoacoustic imaging," Jpn. J. Appl. Phys. Suppl. 30-1, 289-291 (1991).

Kuo, P. K.

K. R. Grice, L. J. Inglehart, L. D. Favro, P. K. Kuo, and R. L. Thomas, "Thermal wave imaging of closed cracks in opaque solids," J. Appl. Phys. 54, 6245-6255 (1983).
[CrossRef]

Masumoto, Y.

H. Nabeta, K. Yamanaka, Y. Nagata, T. Koda, Y. Kanemitsu, and Y. Masumoto, "Characterization of multilayered structures by piezoelectric photoacoustic imaging," Jpn. J. Appl. Phys. Suppl. 30-1, 289-291 (1991).

McDonald, F. A.

F. A. McDonald and G. C. Wetsel, Jr., "Generalized theory of the photoacoustic effect," J. Appl. Phys. 49, 2313-2322 (1978).
[CrossRef]

Minamide, A.

A. Minamide, S. Shida, and Y. Tokunaga, "Improved method for three-dimensional imaging in photoacoustic microscope," Jpn. J. Appl. Phys. 38, 3189-3190 (1999).
[CrossRef]

Mitsuishi, A.

M. Hangyo, S. Nakashima, S. Sugimoto, T. Yamaguchi, and A. Mitsuishi, "Study of edge effect in photoacoustic microscopy," Jpn. J. Appl. Phys. 25, 376-379 (1986).
[CrossRef]

Nabeta, H.

H. Nabeta, K. Yamanaka, Y. Nagata, T. Koda, Y. Kanemitsu, and Y. Masumoto, "Characterization of multilayered structures by piezoelectric photoacoustic imaging," Jpn. J. Appl. Phys. Suppl. 30-1, 289-291 (1991).

Nagata, Y.

H. Nabeta, K. Yamanaka, Y. Nagata, T. Koda, Y. Kanemitsu, and Y. Masumoto, "Characterization of multilayered structures by piezoelectric photoacoustic imaging," Jpn. J. Appl. Phys. Suppl. 30-1, 289-291 (1991).

Nakashima, S.

M. Hangyo, S. Nakashima, S. Sugimoto, T. Yamaguchi, and A. Mitsuishi, "Study of edge effect in photoacoustic microscopy," Jpn. J. Appl. Phys. 25, 376-379 (1986).
[CrossRef]

Nakata, T.

T. Nakata, Y. Kembo, T. Kitamori, and T. Sawada, "Detection and imaging of subsurface microcracks in silicon wafers using photoacoustic microscope," Jpn. J. Appl. Phys. Suppl. 31-1, 146-148 (1992).

Ohtaki, N.

H. Endoh, N. Ohtaki, and T. Hoshimiya, "Nondestructive detection of tilted surface defect with wedge shape by photoacoustic microscopy," Jpn. J. Appl. Phys. 45, 4609-4611 (2006).
[CrossRef]

Opsal, J.

J. Opsal and A. Rosencwaig, "Thermal-wave depth profiling: theory," J. Appl. Phys. 53, 4240-4246 (1982).
[CrossRef]

Pouch, J. J.

Y. H. Wong, R. L. Thomas, and J. J. Pouch, "Subsurface structures of solids by scanning photoacoustic microscopy," Appl. Phys. Lett. 35, 368-369 (1979).
[CrossRef]

Ringermacher, H. I.

H. I. Ringermacher and C. A. Kittredge, "Laser-in/laser-out photoacoustics using Doppler heterodyne interferometry," in Proceedings of the 1986 Ultrasonic Symposium (IEEE, 1986), pp. 407-410.

Rosencwaig, A.

W. L. Smith, A. Rosencwaig, and D. L. Willenborg, "Ion implant monitoring with thermal wave technology," Appl. Phys. Lett. 47, 584-586 (1985).
[CrossRef]

J. Opsal and A. Rosencwaig, "Thermal-wave depth profiling: theory," J. Appl. Phys. 53, 4240-4246 (1982).
[CrossRef]

A. Rosencwaig and A. Gersho, "Theory of the photoacoustic effect with solids," J. Appl. Phys. 47, 64-69 (1976).
[CrossRef]

A. Rosencwaig, Photoacoustics and Photoacoustic Spectroscopy (Wiley Interscience, 1980), pp. 170-173, 295-296.

Sawada, T.

T. Nakata, Y. Kembo, T. Kitamori, and T. Sawada, "Detection and imaging of subsurface microcracks in silicon wafers using photoacoustic microscope," Jpn. J. Appl. Phys. Suppl. 31-1, 146-148 (1992).

Shida, S.

A. Minamide, S. Shida, and Y. Tokunaga, "Improved method for three-dimensional imaging in photoacoustic microscope," Jpn. J. Appl. Phys. 38, 3189-3190 (1999).
[CrossRef]

Shyong, D. Y.

I. Kaufman, P. T. Chang, H. S. Hsu, W. Y. Huang, and D. Y. Shyong, "Photothermal radiometric detection and imaging of surface cracks," J. Nondestruct. Eval. 6, 87-100 (1987).
[CrossRef]

Smith, W. L.

B. Witowski, W. L. Smith, and D. L. Willenborg, "Nondestructive technique for the detection of dislocations and stacking faults on silicon wafers," Appl. Phys. Lett. 52, 640-642 (1988).
[CrossRef]

W. L. Smith, A. Rosencwaig, and D. L. Willenborg, "Ion implant monitoring with thermal wave technology," Appl. Phys. Lett. 47, 584-586 (1985).
[CrossRef]

Sugimoto, S.

M. Hangyo, S. Nakashima, S. Sugimoto, T. Yamaguchi, and A. Mitsuishi, "Study of edge effect in photoacoustic microscopy," Jpn. J. Appl. Phys. 25, 376-379 (1986).
[CrossRef]

Thomas, R. L.

K. R. Grice, L. J. Inglehart, L. D. Favro, P. K. Kuo, and R. L. Thomas, "Thermal wave imaging of closed cracks in opaque solids," J. Appl. Phys. 54, 6245-6255 (1983).
[CrossRef]

Y. H. Wong, R. L. Thomas, and J. J. Pouch, "Subsurface structures of solids by scanning photoacoustic microscopy," Appl. Phys. Lett. 35, 368-369 (1979).
[CrossRef]

Tokunaga, Y.

A. Minamide, S. Shida, and Y. Tokunaga, "Improved method for three-dimensional imaging in photoacoustic microscope," Jpn. J. Appl. Phys. 38, 3189-3190 (1999).
[CrossRef]

Wetsel, G. C.

F. A. McDonald and G. C. Wetsel, Jr., "Generalized theory of the photoacoustic effect," J. Appl. Phys. 49, 2313-2322 (1978).
[CrossRef]

Willenborg, D. L.

B. Witowski, W. L. Smith, and D. L. Willenborg, "Nondestructive technique for the detection of dislocations and stacking faults on silicon wafers," Appl. Phys. Lett. 52, 640-642 (1988).
[CrossRef]

W. L. Smith, A. Rosencwaig, and D. L. Willenborg, "Ion implant monitoring with thermal wave technology," Appl. Phys. Lett. 47, 584-586 (1985).
[CrossRef]

Witowski, B.

B. Witowski, W. L. Smith, and D. L. Willenborg, "Nondestructive technique for the detection of dislocations and stacking faults on silicon wafers," Appl. Phys. Lett. 52, 640-642 (1988).
[CrossRef]

Wong, Y. H.

Y. H. Wong, R. L. Thomas, and J. J. Pouch, "Subsurface structures of solids by scanning photoacoustic microscopy," Appl. Phys. Lett. 35, 368-369 (1979).
[CrossRef]

Yamaguchi, T.

M. Hangyo, S. Nakashima, S. Sugimoto, T. Yamaguchi, and A. Mitsuishi, "Study of edge effect in photoacoustic microscopy," Jpn. J. Appl. Phys. 25, 376-379 (1986).
[CrossRef]

Yamanaka, K.

H. Nabeta, K. Yamanaka, Y. Nagata, T. Koda, Y. Kanemitsu, and Y. Masumoto, "Characterization of multilayered structures by piezoelectric photoacoustic imaging," Jpn. J. Appl. Phys. Suppl. 30-1, 289-291 (1991).

Appl. Phys. Lett. (4)

Y. H. Wong, R. L. Thomas, and J. J. Pouch, "Subsurface structures of solids by scanning photoacoustic microscopy," Appl. Phys. Lett. 35, 368-369 (1979).
[CrossRef]

B. Witowski, W. L. Smith, and D. L. Willenborg, "Nondestructive technique for the detection of dislocations and stacking faults on silicon wafers," Appl. Phys. Lett. 52, 640-642 (1988).
[CrossRef]

W. L. Smith, A. Rosencwaig, and D. L. Willenborg, "Ion implant monitoring with thermal wave technology," Appl. Phys. Lett. 47, 584-586 (1985).
[CrossRef]

J. Hartikainen, "Fast photothermal measurement system for inspection of weak adhesion defects," Appl. Phys. Lett. 55, 1188-1190 (1989).
[CrossRef]

J. Appl. Phys. (5)

K. R. Grice, L. J. Inglehart, L. D. Favro, P. K. Kuo, and R. L. Thomas, "Thermal wave imaging of closed cracks in opaque solids," J. Appl. Phys. 54, 6245-6255 (1983).
[CrossRef]

A. Rosencwaig and A. Gersho, "Theory of the photoacoustic effect with solids," J. Appl. Phys. 47, 64-69 (1976).
[CrossRef]

F. A. McDonald and G. C. Wetsel, Jr., "Generalized theory of the photoacoustic effect," J. Appl. Phys. 49, 2313-2322 (1978).
[CrossRef]

W. B. Jackson and N. M. Amer, "Piezoelectric photoacoustic detection: theory and experiment," J. Appl. Phys. 51, 3343-3353 (1980).
[CrossRef]

J. Opsal and A. Rosencwaig, "Thermal-wave depth profiling: theory," J. Appl. Phys. 53, 4240-4246 (1982).
[CrossRef]

J. Nondestruct. Eval. (1)

I. Kaufman, P. T. Chang, H. S. Hsu, W. Y. Huang, and D. Y. Shyong, "Photothermal radiometric detection and imaging of surface cracks," J. Nondestruct. Eval. 6, 87-100 (1987).
[CrossRef]

Jpn. J. Appl. Phys. (3)

H. Endoh, N. Ohtaki, and T. Hoshimiya, "Nondestructive detection of tilted surface defect with wedge shape by photoacoustic microscopy," Jpn. J. Appl. Phys. 45, 4609-4611 (2006).
[CrossRef]

A. Minamide, S. Shida, and Y. Tokunaga, "Improved method for three-dimensional imaging in photoacoustic microscope," Jpn. J. Appl. Phys. 38, 3189-3190 (1999).
[CrossRef]

M. Hangyo, S. Nakashima, S. Sugimoto, T. Yamaguchi, and A. Mitsuishi, "Study of edge effect in photoacoustic microscopy," Jpn. J. Appl. Phys. 25, 376-379 (1986).
[CrossRef]

Jpn. J. Appl. Phys. Suppl. (2)

H. Nabeta, K. Yamanaka, Y. Nagata, T. Koda, Y. Kanemitsu, and Y. Masumoto, "Characterization of multilayered structures by piezoelectric photoacoustic imaging," Jpn. J. Appl. Phys. Suppl. 30-1, 289-291 (1991).

T. Nakata, Y. Kembo, T. Kitamori, and T. Sawada, "Detection and imaging of subsurface microcracks in silicon wafers using photoacoustic microscope," Jpn. J. Appl. Phys. Suppl. 31-1, 146-148 (1992).

Other (2)

H. I. Ringermacher and C. A. Kittredge, "Laser-in/laser-out photoacoustics using Doppler heterodyne interferometry," in Proceedings of the 1986 Ultrasonic Symposium (IEEE, 1986), pp. 407-410.

A. Rosencwaig, Photoacoustics and Photoacoustic Spectroscopy (Wiley Interscience, 1980), pp. 170-173, 295-296.

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Figures (5)

Fig. 1
Fig. 1

Thermal propagation and isolationprocesses at the microcrack.

Fig. 2
Fig. 2

One-dimensional four-layered model for thermal propagation and isolationprocesses at the microcrack.

Fig. 3
Fig. 3

One-dimensional multilayered thermal diffusion model including a plane heat source.

Fig. 4
Fig. 4

Dependence of temperature profile on crack width and modulation frequency: (a) calculation results of temperature profile obtained with a laser spot size of 1.5 μm , for modulation frequencies of 200, 40, and 10 kHz with thermal diffusion lengths of 12, 27, and 55 μm , respectively; (b) PA amplitude images obtained by PAM system using the same frequencies and laser spot sizes as those used in the calculations.

Fig. 5
Fig. 5

Dependence of temperature contrast on both crack width and modulation frequency.

Tables (1)

Tables Icon

Table 1 Physical Constants for Silicon and Air Used in Calculations

Equations (30)

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

d 2 T n ( x ) d x 2 q n 2 T n ( x ) = 0 ,
q n = ( 1 + j ) ( π f E ρ n c n κ n ) 1 / 2 ,
= 1 + j μ n ,
μ n = ( κ n π f E ρ n c n ) 1 / 2 .
T n ( x ) = A n exp ( q n x ) + B n exp ( q n x ) ,
T n | x = x n = T n + 1 | x = x n ,
κ n | d T n d x | x = x n = κ n + 1 | d T n + 1 d x | x = x n ,
T i | x = x i = T i + 1 | x = x i ,
Q 0 = κ i | d T i d x | x = x i κ i + 1 | d T i + 1 d x | x = x i .
Z n = 1 κ n q n ,
Z n in = | T n κ n d T n d x | x = x n 1 .
T i ( x i ) = Q 0 1 Z i in + 1 Z i + 1 in .
B n A n = ( Z n in Z n Z n in + Z n ) exp ( 2 q n x n 1 ) = ( Z n + 1 in Z n Z n + 1 in + Z n ) exp ( 2 q n x n ) .
Z n in = Z n [ Z n + 1 in + Z n tanh ( q n d n ) Z n + Z n + 1 in tanh ( q n d n ) ] ,
T 0 ( x 0 ) = Q 0 1 Z 0 + 1 Z 1 in ,
Z 1 in = Z 1 { Z 2 in + Z 1 tanh [ q 1 ( x 1 x 0 ) ] Z 1 + Z 2 in tanh [ q 1 ( x 1 x 0 ) ] } ,
Z 2 in = Z 2 { Z 3 + Z 2 tanh ( q 2 d 2 ) Z 2 + Z 3 tanh ( q 2 d 2 ) } .
T 3 ( x 3 ) = Q 0 1 Z 3 in + 1 Z 4 ,
Z 3 in = Z 3 { Z 4 + Z 3 tanh [ q 3 ( x 3 x 2 ) ] Z 3 + Z 4 tanh [ q 3 ( x 3 x 2 ) ] } .
T = x i p s / 2 x i + p s / 2 | T i ( x i ) | d x p s .
C T = T A T B T B × 100 .
T B = lim d 0 d 1 T
= lim d 0 d 1 x 0 p s / 2 x 0 + p s / 2 | T 0 ( x 0 ) | d x p s
= x 0 p s / 2 x 0 + p s / 2 | Q 0 1 / Z 0 + 1 / Z 1 | d x p s
= x 0 p s / 2 x 0 + p s / 2 | Z 0 Q 0 2 | d x p s ,
T A = lim d 0 d 1 0 T
= lim d 0 d 1 0 x 0 p s / 2 x 0 + p s / 2 | T 0 ( x 0 ) | d x p s
= x 0 p s / 2 x 0 + p s / 2 | Q 0 1 / Z 0 + 1 / Z 2 in | d x p s .
C T = x 0 p s / 2 x 0 + p s / 2 | ( Z 2 2 Z 1 2 ) tanh ( q 2 d 2 ) 2 Z 1 Z 2 + ( Z 2 2 + Z 1 2 ) tanh ( q 2 d 2 ) | d x p s × 100 .
C T = x 0 p s / 2 x 0 + p s / 2 | tanh ( q 2 d 2 ) 2 Z 1 / Z 2 + tanh ( q 2 d 2 ) | d x p s × 100 .

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