Abstract

A contact-free, nondestructive laser photothermal radiometric instrumentation technique was developed to meet industrial demand for on-line steel hardness inspection and quality control. A series of industrial steel samples, flat or curvilinear, with different effective hardness case depths ranging between 0.21 and 1.78mm were measured. The results demonstrated that three measurement parameters (metrics) extracted from fast swept-sine photothermal excitation and measurements, namely, the phase minimum frequency fmin, the peak or trough frequency width W, and the area S, are complementary for evaluating widely different ranges of hardness case depth: fmin is most suitable for large case depths, and W and S for small case depths. It was also found that laser beam angular inclination with respect to the surface plane of the sample strongly affects hardness measurement resolution and that the phase frequency maximum is more reliable than the amplitude maximum for laser beam focusing on the sample surface.

© 2008 Optical Society of America

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References

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  1. A. Rosencwaig and A. Gersho, “Theory of the photoacoustic effect with solids,” J. Appl. Phys. 47, 64-69 (1976).
    [CrossRef]
  2. P. Nordal and S. O. Kanstad, “Photothermal radiometry,” Phys. Scr. 20, 659-662 (1979).
    [CrossRef]
  3. J. Shen and A. Mandelis, “Thermal-wave resonator cavity,” Rev. Sci. Instrum. 66, 4999-5005 (1995).
    [CrossRef]
  4. D. Fournier, A. C. Boccara, and J. Badoz, “Thermo-optical spectroscopy: detection by the “mirage effect,” Appl. Phys. Lett. 36, 130-132 (1980).
    [CrossRef]
  5. A. Salazar, A. Sanchez-Lavega, and J. M. Terron, “Effective thermal diffusivity of layered materials measured by modulated photothermal techniques,” J. Appl. Phys. 84, 3031-3041(1998).
    [CrossRef]
  6. T. D. Bennett and F. Yu, “A nondestructive technique for determining thermal properties of thermal barrier coatings,” J. Appl. Phys. 97, 013520 (2005).
    [CrossRef]
  7. P. Li and G. Zhou, “Photothermal radiometry probing of scars in the internal surface of a thin metal tube,” Appl. Opt. 31, 3781-3783 (1992).
    [CrossRef] [PubMed]
  8. M. Depriester, P. Hus, S. Delenclos, and A. Sahraoui, “New methodology for thermal parameter measurements in solids using photothermal radiometry,” Rev. Sci. Instrum. 76, 074902 (2005).
    [CrossRef]
  9. M. Reichling and H. Gronbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914-1922(1994).
    [CrossRef]
  10. J. Jaarinøn and M. Luukkala, “Numerical analysis of thermal waves in stratified media for non-destructive testing purposes,” J. Phys. (Paris) 44, C6-503 (1983).
    [CrossRef]
  11. T. T. N. Lan, H. G. Walther, G. Goch, and B. Schmitz, “Experimental results of photothermal microstructural depth profiling,” J. Appl. Phys. 78, 4108-4111 (1995).
    [CrossRef]
  12. H. G. Walther, D. Fournier, J. C. Krapez, M. Luukkala, B. Schmitz, C. Sibilia, H. Stamm, and J. Thoen, “Photothermal steel hardness measurements-results and perspectives,” Anal. Sci. 17, s165-s168 (2001).
  13. D. Fournier, J. P. Roger, A. Bellouati, C. Boué, H. Stamm, and F. Lakestani, “Correlation between hardness and thermal diffusivity,” Anal. Sci. 17, s158-s160 (2001).
  14. M. Munidasa, F. Funak, and A. Mandelis, “Application of a generalized methodology for quantitative thermal diffusivity depth profile reconstruction in manufactured inhomogeneous steel-based materials,” J. Appl. Phys. 83, 3495-3498(1998).
    [CrossRef]
  15. A. Mandelis, Diffusion-Wave Fields: Mathematical Methods and Green Functions (Springer, 2001), Chap. 3.
  16. H. Qu, C. Wang, X. Guo, and A. Mandelis, “Reconstruction of depth profiles of thermal conductivity of case-hardened steels using a three-dimensional photothermal technique,” J. Appl. Phys. , to be published.
  17. A. Mandelis, F. Funak, and M. Munidasa, “Generalized methodology for thermal diffusivity depth profile reconstruction in semi-infinite and finitely thick inhomogeneous solids,” J. Appl. Phys. 80, 5570-5578 (1996).
    [CrossRef]
  18. Standard SAE 9310, “Data on world wide metals and alloys,” (SAE International, 1990), SA-444.
  19. C. Wang, A. Mandelis, H. Qu, and Z. Chen, “Influence of laser beam size on measurement sensitivity of thermophysical property gradients in layered structures using thermal-wave techniques,” J. Appl. Phys. 103, 043510(2008).
    [CrossRef]
  20. L. Nicolaides and A. Mandelis, “Methods for surface roughness elimination from thermal-wave frequency scans in thermally inhomogeneous solids,” J. Appl. Phys. 90, 1255-1265(2001).
    [CrossRef]
  21. A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627-1639 (1964).
    [CrossRef]

2008 (1)

C. Wang, A. Mandelis, H. Qu, and Z. Chen, “Influence of laser beam size on measurement sensitivity of thermophysical property gradients in layered structures using thermal-wave techniques,” J. Appl. Phys. 103, 043510(2008).
[CrossRef]

2005 (2)

M. Depriester, P. Hus, S. Delenclos, and A. Sahraoui, “New methodology for thermal parameter measurements in solids using photothermal radiometry,” Rev. Sci. Instrum. 76, 074902 (2005).
[CrossRef]

T. D. Bennett and F. Yu, “A nondestructive technique for determining thermal properties of thermal barrier coatings,” J. Appl. Phys. 97, 013520 (2005).
[CrossRef]

2001 (3)

L. Nicolaides and A. Mandelis, “Methods for surface roughness elimination from thermal-wave frequency scans in thermally inhomogeneous solids,” J. Appl. Phys. 90, 1255-1265(2001).
[CrossRef]

H. G. Walther, D. Fournier, J. C. Krapez, M. Luukkala, B. Schmitz, C. Sibilia, H. Stamm, and J. Thoen, “Photothermal steel hardness measurements-results and perspectives,” Anal. Sci. 17, s165-s168 (2001).

D. Fournier, J. P. Roger, A. Bellouati, C. Boué, H. Stamm, and F. Lakestani, “Correlation between hardness and thermal diffusivity,” Anal. Sci. 17, s158-s160 (2001).

1998 (2)

M. Munidasa, F. Funak, and A. Mandelis, “Application of a generalized methodology for quantitative thermal diffusivity depth profile reconstruction in manufactured inhomogeneous steel-based materials,” J. Appl. Phys. 83, 3495-3498(1998).
[CrossRef]

A. Salazar, A. Sanchez-Lavega, and J. M. Terron, “Effective thermal diffusivity of layered materials measured by modulated photothermal techniques,” J. Appl. Phys. 84, 3031-3041(1998).
[CrossRef]

1996 (1)

A. Mandelis, F. Funak, and M. Munidasa, “Generalized methodology for thermal diffusivity depth profile reconstruction in semi-infinite and finitely thick inhomogeneous solids,” J. Appl. Phys. 80, 5570-5578 (1996).
[CrossRef]

1995 (2)

T. T. N. Lan, H. G. Walther, G. Goch, and B. Schmitz, “Experimental results of photothermal microstructural depth profiling,” J. Appl. Phys. 78, 4108-4111 (1995).
[CrossRef]

J. Shen and A. Mandelis, “Thermal-wave resonator cavity,” Rev. Sci. Instrum. 66, 4999-5005 (1995).
[CrossRef]

1994 (1)

M. Reichling and H. Gronbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914-1922(1994).
[CrossRef]

1992 (1)

1983 (1)

J. Jaarinøn and M. Luukkala, “Numerical analysis of thermal waves in stratified media for non-destructive testing purposes,” J. Phys. (Paris) 44, C6-503 (1983).
[CrossRef]

1980 (1)

D. Fournier, A. C. Boccara, and J. Badoz, “Thermo-optical spectroscopy: detection by the “mirage effect,” Appl. Phys. Lett. 36, 130-132 (1980).
[CrossRef]

1979 (1)

P. Nordal and S. O. Kanstad, “Photothermal radiometry,” Phys. Scr. 20, 659-662 (1979).
[CrossRef]

1976 (1)

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

1964 (1)

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627-1639 (1964).
[CrossRef]

Badoz, J.

D. Fournier, A. C. Boccara, and J. Badoz, “Thermo-optical spectroscopy: detection by the “mirage effect,” Appl. Phys. Lett. 36, 130-132 (1980).
[CrossRef]

Bellouati, A.

D. Fournier, J. P. Roger, A. Bellouati, C. Boué, H. Stamm, and F. Lakestani, “Correlation between hardness and thermal diffusivity,” Anal. Sci. 17, s158-s160 (2001).

Bennett, T. D.

T. D. Bennett and F. Yu, “A nondestructive technique for determining thermal properties of thermal barrier coatings,” J. Appl. Phys. 97, 013520 (2005).
[CrossRef]

Boccara, A. C.

D. Fournier, A. C. Boccara, and J. Badoz, “Thermo-optical spectroscopy: detection by the “mirage effect,” Appl. Phys. Lett. 36, 130-132 (1980).
[CrossRef]

Boué, C.

D. Fournier, J. P. Roger, A. Bellouati, C. Boué, H. Stamm, and F. Lakestani, “Correlation between hardness and thermal diffusivity,” Anal. Sci. 17, s158-s160 (2001).

Chen, Z.

C. Wang, A. Mandelis, H. Qu, and Z. Chen, “Influence of laser beam size on measurement sensitivity of thermophysical property gradients in layered structures using thermal-wave techniques,” J. Appl. Phys. 103, 043510(2008).
[CrossRef]

Delenclos, S.

M. Depriester, P. Hus, S. Delenclos, and A. Sahraoui, “New methodology for thermal parameter measurements in solids using photothermal radiometry,” Rev. Sci. Instrum. 76, 074902 (2005).
[CrossRef]

Depriester, M.

M. Depriester, P. Hus, S. Delenclos, and A. Sahraoui, “New methodology for thermal parameter measurements in solids using photothermal radiometry,” Rev. Sci. Instrum. 76, 074902 (2005).
[CrossRef]

Fournier, D.

H. G. Walther, D. Fournier, J. C. Krapez, M. Luukkala, B. Schmitz, C. Sibilia, H. Stamm, and J. Thoen, “Photothermal steel hardness measurements-results and perspectives,” Anal. Sci. 17, s165-s168 (2001).

D. Fournier, J. P. Roger, A. Bellouati, C. Boué, H. Stamm, and F. Lakestani, “Correlation between hardness and thermal diffusivity,” Anal. Sci. 17, s158-s160 (2001).

D. Fournier, A. C. Boccara, and J. Badoz, “Thermo-optical spectroscopy: detection by the “mirage effect,” Appl. Phys. Lett. 36, 130-132 (1980).
[CrossRef]

Funak, F.

M. Munidasa, F. Funak, and A. Mandelis, “Application of a generalized methodology for quantitative thermal diffusivity depth profile reconstruction in manufactured inhomogeneous steel-based materials,” J. Appl. Phys. 83, 3495-3498(1998).
[CrossRef]

A. Mandelis, F. Funak, and M. Munidasa, “Generalized methodology for thermal diffusivity depth profile reconstruction in semi-infinite and finitely thick inhomogeneous solids,” J. Appl. Phys. 80, 5570-5578 (1996).
[CrossRef]

Gersho, A.

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

Goch, G.

T. T. N. Lan, H. G. Walther, G. Goch, and B. Schmitz, “Experimental results of photothermal microstructural depth profiling,” J. Appl. Phys. 78, 4108-4111 (1995).
[CrossRef]

Golay, M. J. E.

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627-1639 (1964).
[CrossRef]

Gronbeck, H.

M. Reichling and H. Gronbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914-1922(1994).
[CrossRef]

Guo, X.

H. Qu, C. Wang, X. Guo, and A. Mandelis, “Reconstruction of depth profiles of thermal conductivity of case-hardened steels using a three-dimensional photothermal technique,” J. Appl. Phys. , to be published.

Hus, P.

M. Depriester, P. Hus, S. Delenclos, and A. Sahraoui, “New methodology for thermal parameter measurements in solids using photothermal radiometry,” Rev. Sci. Instrum. 76, 074902 (2005).
[CrossRef]

Jaarinøn, J.

J. Jaarinøn and M. Luukkala, “Numerical analysis of thermal waves in stratified media for non-destructive testing purposes,” J. Phys. (Paris) 44, C6-503 (1983).
[CrossRef]

Kanstad, S. O.

P. Nordal and S. O. Kanstad, “Photothermal radiometry,” Phys. Scr. 20, 659-662 (1979).
[CrossRef]

Krapez, J. C.

H. G. Walther, D. Fournier, J. C. Krapez, M. Luukkala, B. Schmitz, C. Sibilia, H. Stamm, and J. Thoen, “Photothermal steel hardness measurements-results and perspectives,” Anal. Sci. 17, s165-s168 (2001).

Lakestani, F.

D. Fournier, J. P. Roger, A. Bellouati, C. Boué, H. Stamm, and F. Lakestani, “Correlation between hardness and thermal diffusivity,” Anal. Sci. 17, s158-s160 (2001).

Lan, T. T. N.

T. T. N. Lan, H. G. Walther, G. Goch, and B. Schmitz, “Experimental results of photothermal microstructural depth profiling,” J. Appl. Phys. 78, 4108-4111 (1995).
[CrossRef]

Li, P.

Luukkala, M.

H. G. Walther, D. Fournier, J. C. Krapez, M. Luukkala, B. Schmitz, C. Sibilia, H. Stamm, and J. Thoen, “Photothermal steel hardness measurements-results and perspectives,” Anal. Sci. 17, s165-s168 (2001).

J. Jaarinøn and M. Luukkala, “Numerical analysis of thermal waves in stratified media for non-destructive testing purposes,” J. Phys. (Paris) 44, C6-503 (1983).
[CrossRef]

Mandelis, A.

C. Wang, A. Mandelis, H. Qu, and Z. Chen, “Influence of laser beam size on measurement sensitivity of thermophysical property gradients in layered structures using thermal-wave techniques,” J. Appl. Phys. 103, 043510(2008).
[CrossRef]

L. Nicolaides and A. Mandelis, “Methods for surface roughness elimination from thermal-wave frequency scans in thermally inhomogeneous solids,” J. Appl. Phys. 90, 1255-1265(2001).
[CrossRef]

M. Munidasa, F. Funak, and A. Mandelis, “Application of a generalized methodology for quantitative thermal diffusivity depth profile reconstruction in manufactured inhomogeneous steel-based materials,” J. Appl. Phys. 83, 3495-3498(1998).
[CrossRef]

A. Mandelis, F. Funak, and M. Munidasa, “Generalized methodology for thermal diffusivity depth profile reconstruction in semi-infinite and finitely thick inhomogeneous solids,” J. Appl. Phys. 80, 5570-5578 (1996).
[CrossRef]

J. Shen and A. Mandelis, “Thermal-wave resonator cavity,” Rev. Sci. Instrum. 66, 4999-5005 (1995).
[CrossRef]

H. Qu, C. Wang, X. Guo, and A. Mandelis, “Reconstruction of depth profiles of thermal conductivity of case-hardened steels using a three-dimensional photothermal technique,” J. Appl. Phys. , to be published.

A. Mandelis, Diffusion-Wave Fields: Mathematical Methods and Green Functions (Springer, 2001), Chap. 3.

Munidasa, M.

M. Munidasa, F. Funak, and A. Mandelis, “Application of a generalized methodology for quantitative thermal diffusivity depth profile reconstruction in manufactured inhomogeneous steel-based materials,” J. Appl. Phys. 83, 3495-3498(1998).
[CrossRef]

A. Mandelis, F. Funak, and M. Munidasa, “Generalized methodology for thermal diffusivity depth profile reconstruction in semi-infinite and finitely thick inhomogeneous solids,” J. Appl. Phys. 80, 5570-5578 (1996).
[CrossRef]

Nicolaides, L.

L. Nicolaides and A. Mandelis, “Methods for surface roughness elimination from thermal-wave frequency scans in thermally inhomogeneous solids,” J. Appl. Phys. 90, 1255-1265(2001).
[CrossRef]

Nordal, P.

P. Nordal and S. O. Kanstad, “Photothermal radiometry,” Phys. Scr. 20, 659-662 (1979).
[CrossRef]

Qu, H.

C. Wang, A. Mandelis, H. Qu, and Z. Chen, “Influence of laser beam size on measurement sensitivity of thermophysical property gradients in layered structures using thermal-wave techniques,” J. Appl. Phys. 103, 043510(2008).
[CrossRef]

H. Qu, C. Wang, X. Guo, and A. Mandelis, “Reconstruction of depth profiles of thermal conductivity of case-hardened steels using a three-dimensional photothermal technique,” J. Appl. Phys. , to be published.

Reichling, M.

M. Reichling and H. Gronbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914-1922(1994).
[CrossRef]

Roger, J. P.

D. Fournier, J. P. Roger, A. Bellouati, C. Boué, H. Stamm, and F. Lakestani, “Correlation between hardness and thermal diffusivity,” Anal. Sci. 17, s158-s160 (2001).

Rosencwaig, A.

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

Sahraoui, A.

M. Depriester, P. Hus, S. Delenclos, and A. Sahraoui, “New methodology for thermal parameter measurements in solids using photothermal radiometry,” Rev. Sci. Instrum. 76, 074902 (2005).
[CrossRef]

Salazar, A.

A. Salazar, A. Sanchez-Lavega, and J. M. Terron, “Effective thermal diffusivity of layered materials measured by modulated photothermal techniques,” J. Appl. Phys. 84, 3031-3041(1998).
[CrossRef]

Sanchez-Lavega, A.

A. Salazar, A. Sanchez-Lavega, and J. M. Terron, “Effective thermal diffusivity of layered materials measured by modulated photothermal techniques,” J. Appl. Phys. 84, 3031-3041(1998).
[CrossRef]

Savitzky, A.

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627-1639 (1964).
[CrossRef]

Schmitz, B.

H. G. Walther, D. Fournier, J. C. Krapez, M. Luukkala, B. Schmitz, C. Sibilia, H. Stamm, and J. Thoen, “Photothermal steel hardness measurements-results and perspectives,” Anal. Sci. 17, s165-s168 (2001).

T. T. N. Lan, H. G. Walther, G. Goch, and B. Schmitz, “Experimental results of photothermal microstructural depth profiling,” J. Appl. Phys. 78, 4108-4111 (1995).
[CrossRef]

Shen, J.

J. Shen and A. Mandelis, “Thermal-wave resonator cavity,” Rev. Sci. Instrum. 66, 4999-5005 (1995).
[CrossRef]

Sibilia, C.

H. G. Walther, D. Fournier, J. C. Krapez, M. Luukkala, B. Schmitz, C. Sibilia, H. Stamm, and J. Thoen, “Photothermal steel hardness measurements-results and perspectives,” Anal. Sci. 17, s165-s168 (2001).

Stamm, H.

H. G. Walther, D. Fournier, J. C. Krapez, M. Luukkala, B. Schmitz, C. Sibilia, H. Stamm, and J. Thoen, “Photothermal steel hardness measurements-results and perspectives,” Anal. Sci. 17, s165-s168 (2001).

D. Fournier, J. P. Roger, A. Bellouati, C. Boué, H. Stamm, and F. Lakestani, “Correlation between hardness and thermal diffusivity,” Anal. Sci. 17, s158-s160 (2001).

Terron, J. M.

A. Salazar, A. Sanchez-Lavega, and J. M. Terron, “Effective thermal diffusivity of layered materials measured by modulated photothermal techniques,” J. Appl. Phys. 84, 3031-3041(1998).
[CrossRef]

Thoen, J.

H. G. Walther, D. Fournier, J. C. Krapez, M. Luukkala, B. Schmitz, C. Sibilia, H. Stamm, and J. Thoen, “Photothermal steel hardness measurements-results and perspectives,” Anal. Sci. 17, s165-s168 (2001).

Walther, H. G.

H. G. Walther, D. Fournier, J. C. Krapez, M. Luukkala, B. Schmitz, C. Sibilia, H. Stamm, and J. Thoen, “Photothermal steel hardness measurements-results and perspectives,” Anal. Sci. 17, s165-s168 (2001).

T. T. N. Lan, H. G. Walther, G. Goch, and B. Schmitz, “Experimental results of photothermal microstructural depth profiling,” J. Appl. Phys. 78, 4108-4111 (1995).
[CrossRef]

Wang, C.

C. Wang, A. Mandelis, H. Qu, and Z. Chen, “Influence of laser beam size on measurement sensitivity of thermophysical property gradients in layered structures using thermal-wave techniques,” J. Appl. Phys. 103, 043510(2008).
[CrossRef]

H. Qu, C. Wang, X. Guo, and A. Mandelis, “Reconstruction of depth profiles of thermal conductivity of case-hardened steels using a three-dimensional photothermal technique,” J. Appl. Phys. , to be published.

Yu, F.

T. D. Bennett and F. Yu, “A nondestructive technique for determining thermal properties of thermal barrier coatings,” J. Appl. Phys. 97, 013520 (2005).
[CrossRef]

Zhou, G.

Anal. Chem. (1)

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627-1639 (1964).
[CrossRef]

Anal. Sci. (2)

H. G. Walther, D. Fournier, J. C. Krapez, M. Luukkala, B. Schmitz, C. Sibilia, H. Stamm, and J. Thoen, “Photothermal steel hardness measurements-results and perspectives,” Anal. Sci. 17, s165-s168 (2001).

D. Fournier, J. P. Roger, A. Bellouati, C. Boué, H. Stamm, and F. Lakestani, “Correlation between hardness and thermal diffusivity,” Anal. Sci. 17, s158-s160 (2001).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

D. Fournier, A. C. Boccara, and J. Badoz, “Thermo-optical spectroscopy: detection by the “mirage effect,” Appl. Phys. Lett. 36, 130-132 (1980).
[CrossRef]

J. Appl. Phys. (10)

A. Salazar, A. Sanchez-Lavega, and J. M. Terron, “Effective thermal diffusivity of layered materials measured by modulated photothermal techniques,” J. Appl. Phys. 84, 3031-3041(1998).
[CrossRef]

T. D. Bennett and F. Yu, “A nondestructive technique for determining thermal properties of thermal barrier coatings,” J. Appl. Phys. 97, 013520 (2005).
[CrossRef]

M. Munidasa, F. Funak, and A. Mandelis, “Application of a generalized methodology for quantitative thermal diffusivity depth profile reconstruction in manufactured inhomogeneous steel-based materials,” J. Appl. Phys. 83, 3495-3498(1998).
[CrossRef]

M. Reichling and H. Gronbeck, “Harmonic heat flow in isotropic layered systems and its use for thin film thermal conductivity measurements,” J. Appl. Phys. 75, 1914-1922(1994).
[CrossRef]

T. T. N. Lan, H. G. Walther, G. Goch, and B. Schmitz, “Experimental results of photothermal microstructural depth profiling,” J. Appl. Phys. 78, 4108-4111 (1995).
[CrossRef]

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

H. Qu, C. Wang, X. Guo, and A. Mandelis, “Reconstruction of depth profiles of thermal conductivity of case-hardened steels using a three-dimensional photothermal technique,” J. Appl. Phys. , to be published.

A. Mandelis, F. Funak, and M. Munidasa, “Generalized methodology for thermal diffusivity depth profile reconstruction in semi-infinite and finitely thick inhomogeneous solids,” J. Appl. Phys. 80, 5570-5578 (1996).
[CrossRef]

C. Wang, A. Mandelis, H. Qu, and Z. Chen, “Influence of laser beam size on measurement sensitivity of thermophysical property gradients in layered structures using thermal-wave techniques,” J. Appl. Phys. 103, 043510(2008).
[CrossRef]

L. Nicolaides and A. Mandelis, “Methods for surface roughness elimination from thermal-wave frequency scans in thermally inhomogeneous solids,” J. Appl. Phys. 90, 1255-1265(2001).
[CrossRef]

J. Phys. (Paris) (1)

J. Jaarinøn and M. Luukkala, “Numerical analysis of thermal waves in stratified media for non-destructive testing purposes,” J. Phys. (Paris) 44, C6-503 (1983).
[CrossRef]

Phys. Scr. (1)

P. Nordal and S. O. Kanstad, “Photothermal radiometry,” Phys. Scr. 20, 659-662 (1979).
[CrossRef]

Rev. Sci. Instrum. (2)

J. Shen and A. Mandelis, “Thermal-wave resonator cavity,” Rev. Sci. Instrum. 66, 4999-5005 (1995).
[CrossRef]

M. Depriester, P. Hus, S. Delenclos, and A. Sahraoui, “New methodology for thermal parameter measurements in solids using photothermal radiometry,” Rev. Sci. Instrum. 76, 074902 (2005).
[CrossRef]

Other (2)

A. Mandelis, Diffusion-Wave Fields: Mathematical Methods and Green Functions (Springer, 2001), Chap. 3.

Standard SAE 9310, “Data on world wide metals and alloys,” (SAE International, 1990), SA-444.

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Figures (17)

Fig. 1
Fig. 1

(a) Typical microhardness profile of a hardened steel sample measured with the mechanical indentation method HV0.5. Hardness decreases with depth. The effective case depth E is the depth where the hardness drops to 513 HV , i.e., 0.86 mm in this case. (b) Schematic diagram of a continuously inhomogeneous system including an inhomogeneous layer and a substrate m. The inhomogeneous layer is divided into n layers for theoretical treatment.

Fig. 2
Fig. 2

Various thermal conductivity depth profiles obtained using the k ( z ) ansatz, Eq. (5). The parameters used are as follows: curve K1, k 0 = 36 W / mK , k L 0 = 51.9 W / mK , q = 2 × 10 3 mm 1 , L 0 = 5 mm ; curve K2, k 0 = 35 W / mK , k L 0 = 51.9 W / mK , q = 0.5 × 10 3 mm 1 , L 0 = 5 mm ; curve K3, k 0 = 36 W / mK , k L 0 = 51.9 W / mK , q = 2 × 10 3 mm 1 , L 0 = 5 mm 1 ; curve K4, k 0 = 51.9 W / mK , k L 0 = 36 W / mK , q = 1 × 10 3 mm 1 , L 0 = 5 mm ; and curve K5, k 0 = 51.9 W / mK , k L 0 = W / mK , q = 1 × 10 3 mm 1 , L 0 = 5 mm .

Fig. 3
Fig. 3

Thermal conductivity depth profile of the hardened layer. Parameters used are k 0 = 20 W / mK , k L 0 = 36.0489 W / mK , q = 2529 mm 1 , and L 0 = 2.45 mm .

Fig. 4
Fig. 4

Amplitude and phase of a steel with inhomogeneous thermal conductivity simulating a case-hardened AISI 9310 normalized by the corresponding homogeneous AISI 9310 semi-infinite steel sample using several beam sizes a ( mm ) : (1) 0.01, (2) 0.02, (3) 0.05, (4) 0.5, (5) 1.0, (6) 2.0, (7) 5.0, (8)10, (9) 20, (10) 40, (11) 100. Other parameters of the hardened layer used are k 0 = 20 W / mK , k L 0 = 36.0489 W / mK , q = 2529 mm 1 , and L 0 = 2.45 mm .

Fig. 5
Fig. 5

Diagram of the HD-PTR system. The system consists of three parts. Part I: source and controlling system, including a diode laser, a laser controller, a detector TEC controller, and a computer for data acquisition and laser modulation. Part II: optical box, dimensions 19 × 19 × 10 cm 3 . F, optical fiber; C, collimator; D, TEC-cooled HgCdZnTe (MCZT) detector; M 1 and M 2 , steering mirrors; L, lens; P 1 and P 2 , gold-coated off-axis parabolic mirrors; W, Ca F 2 window. Part III, S, sample compartment.

Fig. 6
Fig. 6

Schematic cross section of cylindrical and gear-tooth samples, indicating the various measurement sites.

Fig. 7
Fig. 7

PTR SNR versus frequency from the end face measurement of an A-type nonhardened gear-tooth sample (Table 1). The instrumental settling and integration time is 1 s , and the number of cycles for settling and integration is 5. 30 measured points cover a frequency span of 1 500 Hz . (a) SNR of PTR amplitude and (b) SNR of PTR phase.

Fig. 8
Fig. 8

Time scan ( 200   s ) of a PTR signal at 10 Hz from a cylindrical hardened sample, A3. The measurements were taken after the laser had thermalized for 5 min . (a) PTR amplitude and (b) PTR phase.

Fig. 9
Fig. 9

Initial transient ( 0 8   s ) of the PTR signal in Fig. 8: (a) amplitude and (b) phase.

Fig. 10
Fig. 10

Long-term transient ( 0 200   s ) of the PTR signal in Fig. 8: (a) amplitude and (b) phase.

Fig. 11
Fig. 11

Sample positioning effect on the PTR signal at 10 Hz from the cross section of the cylindrical hardened sample, A3. The sample was translated 1000 μm backward and forward, away from the laser focal position ( z = 0 ) after PTR amplitude stabilization: (a) amplitude and (b) phase.

Fig. 12
Fig. 12

Thin oil-film effect on PTR signal. The end face of a nonhardened gear-tooth sample, A R , was probed before and after the protective oil layer was removed with methanol: (a) amplitude, (b) phase, (c) absolute phase difference between the measurements with and without the oil film.

Fig. 13
Fig. 13

Three parameters extracted from normalized PTR phase frequency scans: minimum frequency f min , trough/peak width W, and area S, and their correlation with sample hardness E: smaller f min , W, and S indicate deeper effective case depth. (a) Typical trough shape from the cylindrical hardened sample A1; (b) troughs from two root measurements of N-type gear-tooth samples, N2 ( E = 0.89 mm ) and N3 ( E = 0.97 mm ).

Fig. 14
Fig. 14

Correlation between PTR phase trough parameters and hardness of cylindrical sample set A: A 1 ( E = 1.78 mm ) , A 2 ( E = 1.37 mm ) , and A 3 ( E = 0.61 ) . (a) E versus f min , (b) E versus W, (c) E versus S, (d) hardness profiles measured by the indentation method.

Fig. 15
Fig. 15

Correlation between PTR phase trough parameters and flank hardness of gear-tooth sample set N: N 3 ( E = 1.16 mm ) , N 2 ( E = 1.03 ) . (a) E versus f min , (b) E versus W, (c) E versus S.

Fig. 16
Fig. 16

Correlation between PTR phase peak parameters and hardness of cylindrical sample set C: C 1 ( E = 0.41 mm ) , C 2 ( E = 0.31 mm ) , and C 3 ( E = 0.21 ) . (a) E versus S, (c) E versus W.

Fig. 17
Fig. 17

Hardness measurement resolution dependence on angle between sample-surface normal and laser-beam axis. A-type cylindrical samples A 1 ( E = 1.78 mm ) , A 2 ( E = 1.37 mm ) , and A 3 ( E = 0.61 ) were measured at different angles from 0 ° to 15 ° (almost perpendicular to the laser beam). θ = 5 ° was found to be the optimal measurement angle. (a) Laser-beam and sample surface geometry and angle definition, (b) E vesus f min , (c) E versus W, (d) E versus S at various angles.

Tables (4)

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Table 1 Industrial Steel Sample Matrix for PTR Measurements

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Table 2 Repeatability and Reproducibility Measurements on the End Face of the N3 Sample

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Table 3 Unknown Sample SA1 Evaluation

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Table 4 Oxidization Effects on the Sample Hardness

Equations (7)

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2 T i ( r , z , ω ) σ i 2 T i ( r , z , ω ) = 0 ,
k 0 T 0 ( r , 0 , ω ) z k 1 T 1 ( r , 0 , ω ) z = Q s ( 0 ) ,
Q s ( 0 ) = A s ( 1 R 1 ) P π a 2 e r 2 / a 2 .
T i ( r , z = 0 , ω ) = 0 T ˜ ( λ , z ) = 0 , ω ) J 0 ( λ r ) λ d λ ,
k = k 0 ( 1 + Δ e q z 1 + Δ ) ,
Δ = 1 k L 0 / k 0 k L 0 / k 0 e q L 0 .
μ H = ( α H / π f ) 1 / 2 ,

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