Abstract

Thermal-wave Slice Diffraction Tomography (TSDT) is a photothermal imaging technique for non-destructive detection of subsurface cross-sectional defects in opaque solids in the very-near-surface region (µm-mm). Conventional reconstructions of the well-posed propagating wave-field tomographies cannot be applied to the ill-posed thermal wave problem. Photothermal tomographic microscopy is used to collect experimental data that are numerically inverted with the Tikhonov regularization method to produce thermal diffusivity cross-sectional images in materials. Multiplicity of solutions, which is inherent to ill-posed problems, is resolved by adopting the L-curve method for optimization. For tomographic imaging of sub-surface defects, a new high-resolution radiometric setup is constructed, which reduces the broadening of images associated with previous low-resolution setups.

© 2000 Optical Society of America

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  1. G. Busse and K.F. Rank, “Stereoscopic Depth Analysis by Thermal Wave Transmission for Non-Destructive Evaluation,” Appl. Phys. Let. 42, 366 (1983).
    [CrossRef]
  2. A. Mandelis, “Theory of Photothermal Wave Diffraction Tomography via Spatial Laplace Spectral Decomposition,” J. Phys. A: Math. General 24, 2485 (1991).
    [CrossRef]
  3. A. Mandelis, “Theory of Photothermal Wave Diffraction and Interference in Condensed Phases,” J. Opt. Soc. Am. A 6, 298 (1989).
    [CrossRef]
  4. M. Munidasa and A. Mandelis, “Photopyroelectric Thermal-Wave Tomography of Aluminum with Ray-Optic Reconstruction,” J. Opt. Soc. Am. A 8,1851 (1991).
    [CrossRef]
  5. O. Pade and A. Mandelis, “Computational Thermal-Wave Slice Tomography with Backpropagation and Transmission Reconstructions,” Rev. Sci. Instrum. 64, 3548 (1993).
    [CrossRef]
  6. A. Mandelis, “Green’s Functions in Thermal Wave Physics: Cartesian Coordinate Representations,” J. Appl. Phys.,  78 (2), 647 (1995).
    [CrossRef]
  7. L. Nicolaides and A. Mandelis, “Image-Enhanced Thermal-Wave Slice Diffraction Tomography with Numerically Simulated Reconstructions,” Inv. Prob. 13, 1393 (1997).
    [CrossRef]
  8. L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997).
    [CrossRef]
  9. A. Mandelis, Diffusion Wave Fields: Green Functions and Mathematical Methods (Springer, New York, in press).
  10. A.N. Tikhonov, “On Stability Of Inverse Problems,” Dokl. Acad. Nauuk USSR,  39(5), 195 (1943).
  11. B. Hofmann, Regularization for Applied and Ill-Posed Problems, (Teubner, 1986).
  12. C. Hansen, “Analysis of Disrete Ill-Posed Problems by Means of the L-Curve,” SIAM Review,  34, 561 (1992).
    [CrossRef]
  13. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487 (1993).
    [CrossRef]
  14. A. Mandelis and M. Mieszkowski, “Thermal Wave Sub-Surface Defect Imaging and Tomography Apparatus,” U.S. Patent Number 4, 950, 897; Date: August 21, 1990.
  15. L. Qian and P. Li, “Photothermal radiometry measurement of thermal diffusivity,” Appl. Opt. 29, 4241 (1990).
    [CrossRef] [PubMed]
  16. G. Busse, D. Wu, and W. Karpen, “Thermal Wave Imaging with Phase Sensitive Modulated Thermography,” J. Appl. Phys. 71, 3962 (1992).
    [CrossRef]
  17. A.C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (IEEE Press, New York, 1988).
  18. A. Mandelis, “Diffusion waves and their uses,” Physics Today, 29, August 2000.
  19. E. Miller, L. Nicolaides, and A. Mandelis, “Nonlinear Inverse Scattering Methods for Thermal-Wave Slice Tomography: AWavelet Domain Approach,” J. Opt. Soc. Am. A,  15, 1545 (1998).
    [CrossRef]

1998 (1)

1997 (2)

L. Nicolaides and A. Mandelis, “Image-Enhanced Thermal-Wave Slice Diffraction Tomography with Numerically Simulated Reconstructions,” Inv. Prob. 13, 1393 (1997).
[CrossRef]

L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997).
[CrossRef]

1995 (1)

A. Mandelis, “Green’s Functions in Thermal Wave Physics: Cartesian Coordinate Representations,” J. Appl. Phys.,  78 (2), 647 (1995).
[CrossRef]

1993 (2)

O. Pade and A. Mandelis, “Computational Thermal-Wave Slice Tomography with Backpropagation and Transmission Reconstructions,” Rev. Sci. Instrum. 64, 3548 (1993).
[CrossRef]

C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487 (1993).
[CrossRef]

1992 (2)

C. Hansen, “Analysis of Disrete Ill-Posed Problems by Means of the L-Curve,” SIAM Review,  34, 561 (1992).
[CrossRef]

G. Busse, D. Wu, and W. Karpen, “Thermal Wave Imaging with Phase Sensitive Modulated Thermography,” J. Appl. Phys. 71, 3962 (1992).
[CrossRef]

1991 (2)

A. Mandelis, “Theory of Photothermal Wave Diffraction Tomography via Spatial Laplace Spectral Decomposition,” J. Phys. A: Math. General 24, 2485 (1991).
[CrossRef]

M. Munidasa and A. Mandelis, “Photopyroelectric Thermal-Wave Tomography of Aluminum with Ray-Optic Reconstruction,” J. Opt. Soc. Am. A 8,1851 (1991).
[CrossRef]

1990 (2)

A. Mandelis and M. Mieszkowski, “Thermal Wave Sub-Surface Defect Imaging and Tomography Apparatus,” U.S. Patent Number 4, 950, 897; Date: August 21, 1990.

L. Qian and P. Li, “Photothermal radiometry measurement of thermal diffusivity,” Appl. Opt. 29, 4241 (1990).
[CrossRef] [PubMed]

1989 (1)

1983 (1)

G. Busse and K.F. Rank, “Stereoscopic Depth Analysis by Thermal Wave Transmission for Non-Destructive Evaluation,” Appl. Phys. Let. 42, 366 (1983).
[CrossRef]

1943 (1)

A.N. Tikhonov, “On Stability Of Inverse Problems,” Dokl. Acad. Nauuk USSR,  39(5), 195 (1943).

Busse, G.

G. Busse, D. Wu, and W. Karpen, “Thermal Wave Imaging with Phase Sensitive Modulated Thermography,” J. Appl. Phys. 71, 3962 (1992).
[CrossRef]

G. Busse and K.F. Rank, “Stereoscopic Depth Analysis by Thermal Wave Transmission for Non-Destructive Evaluation,” Appl. Phys. Let. 42, 366 (1983).
[CrossRef]

Hansen, C.

C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487 (1993).
[CrossRef]

C. Hansen, “Analysis of Disrete Ill-Posed Problems by Means of the L-Curve,” SIAM Review,  34, 561 (1992).
[CrossRef]

Hofmann, B.

B. Hofmann, Regularization for Applied and Ill-Posed Problems, (Teubner, 1986).

Kak, A.C.

A.C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (IEEE Press, New York, 1988).

Karpen, W.

G. Busse, D. Wu, and W. Karpen, “Thermal Wave Imaging with Phase Sensitive Modulated Thermography,” J. Appl. Phys. 71, 3962 (1992).
[CrossRef]

Li, P.

Mandelis, A.

E. Miller, L. Nicolaides, and A. Mandelis, “Nonlinear Inverse Scattering Methods for Thermal-Wave Slice Tomography: AWavelet Domain Approach,” J. Opt. Soc. Am. A,  15, 1545 (1998).
[CrossRef]

L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997).
[CrossRef]

L. Nicolaides and A. Mandelis, “Image-Enhanced Thermal-Wave Slice Diffraction Tomography with Numerically Simulated Reconstructions,” Inv. Prob. 13, 1393 (1997).
[CrossRef]

A. Mandelis, “Green’s Functions in Thermal Wave Physics: Cartesian Coordinate Representations,” J. Appl. Phys.,  78 (2), 647 (1995).
[CrossRef]

O. Pade and A. Mandelis, “Computational Thermal-Wave Slice Tomography with Backpropagation and Transmission Reconstructions,” Rev. Sci. Instrum. 64, 3548 (1993).
[CrossRef]

M. Munidasa and A. Mandelis, “Photopyroelectric Thermal-Wave Tomography of Aluminum with Ray-Optic Reconstruction,” J. Opt. Soc. Am. A 8,1851 (1991).
[CrossRef]

A. Mandelis, “Theory of Photothermal Wave Diffraction Tomography via Spatial Laplace Spectral Decomposition,” J. Phys. A: Math. General 24, 2485 (1991).
[CrossRef]

A. Mandelis and M. Mieszkowski, “Thermal Wave Sub-Surface Defect Imaging and Tomography Apparatus,” U.S. Patent Number 4, 950, 897; Date: August 21, 1990.

A. Mandelis, “Theory of Photothermal Wave Diffraction and Interference in Condensed Phases,” J. Opt. Soc. Am. A 6, 298 (1989).
[CrossRef]

A. Mandelis, Diffusion Wave Fields: Green Functions and Mathematical Methods (Springer, New York, in press).

A. Mandelis, “Diffusion waves and their uses,” Physics Today, 29, August 2000.

Mieszkowski, M.

A. Mandelis and M. Mieszkowski, “Thermal Wave Sub-Surface Defect Imaging and Tomography Apparatus,” U.S. Patent Number 4, 950, 897; Date: August 21, 1990.

Miller, E.

Munidasa, M.

L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997).
[CrossRef]

M. Munidasa and A. Mandelis, “Photopyroelectric Thermal-Wave Tomography of Aluminum with Ray-Optic Reconstruction,” J. Opt. Soc. Am. A 8,1851 (1991).
[CrossRef]

Nicolaides, L.

E. Miller, L. Nicolaides, and A. Mandelis, “Nonlinear Inverse Scattering Methods for Thermal-Wave Slice Tomography: AWavelet Domain Approach,” J. Opt. Soc. Am. A,  15, 1545 (1998).
[CrossRef]

L. Nicolaides and A. Mandelis, “Image-Enhanced Thermal-Wave Slice Diffraction Tomography with Numerically Simulated Reconstructions,” Inv. Prob. 13, 1393 (1997).
[CrossRef]

L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997).
[CrossRef]

O’Leary, D. P.

C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487 (1993).
[CrossRef]

Pade, O.

O. Pade and A. Mandelis, “Computational Thermal-Wave Slice Tomography with Backpropagation and Transmission Reconstructions,” Rev. Sci. Instrum. 64, 3548 (1993).
[CrossRef]

Qian, L.

Rank, K.F.

G. Busse and K.F. Rank, “Stereoscopic Depth Analysis by Thermal Wave Transmission for Non-Destructive Evaluation,” Appl. Phys. Let. 42, 366 (1983).
[CrossRef]

Slaney, M.

A.C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (IEEE Press, New York, 1988).

Tikhonov, A.N.

A.N. Tikhonov, “On Stability Of Inverse Problems,” Dokl. Acad. Nauuk USSR,  39(5), 195 (1943).

Wu, D.

G. Busse, D. Wu, and W. Karpen, “Thermal Wave Imaging with Phase Sensitive Modulated Thermography,” J. Appl. Phys. 71, 3962 (1992).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Let. (1)

G. Busse and K.F. Rank, “Stereoscopic Depth Analysis by Thermal Wave Transmission for Non-Destructive Evaluation,” Appl. Phys. Let. 42, 366 (1983).
[CrossRef]

Dokl. Acad. Nauuk USSR (1)

A.N. Tikhonov, “On Stability Of Inverse Problems,” Dokl. Acad. Nauuk USSR,  39(5), 195 (1943).

Inv. Prob. (2)

L. Nicolaides and A. Mandelis, “Image-Enhanced Thermal-Wave Slice Diffraction Tomography with Numerically Simulated Reconstructions,” Inv. Prob. 13, 1393 (1997).
[CrossRef]

L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-Wave Infrared Radiometric Slice Diffraction Tomography with Back-Scattering and Transmission Reconstructions: Experimental,” Inv. Prob. 13, 1413 (1997).
[CrossRef]

J. Appl. Phys. (2)

G. Busse, D. Wu, and W. Karpen, “Thermal Wave Imaging with Phase Sensitive Modulated Thermography,” J. Appl. Phys. 71, 3962 (1992).
[CrossRef]

A. Mandelis, “Green’s Functions in Thermal Wave Physics: Cartesian Coordinate Representations,” J. Appl. Phys.,  78 (2), 647 (1995).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Phys. A: Math. General (1)

A. Mandelis, “Theory of Photothermal Wave Diffraction Tomography via Spatial Laplace Spectral Decomposition,” J. Phys. A: Math. General 24, 2485 (1991).
[CrossRef]

Rev. Sci. Instrum. (1)

O. Pade and A. Mandelis, “Computational Thermal-Wave Slice Tomography with Backpropagation and Transmission Reconstructions,” Rev. Sci. Instrum. 64, 3548 (1993).
[CrossRef]

SIAM J. Sci. Comput. (1)

C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487 (1993).
[CrossRef]

SIAM Review (1)

C. Hansen, “Analysis of Disrete Ill-Posed Problems by Means of the L-Curve,” SIAM Review,  34, 561 (1992).
[CrossRef]

U.S. Patent Number (1)

A. Mandelis and M. Mieszkowski, “Thermal Wave Sub-Surface Defect Imaging and Tomography Apparatus,” U.S. Patent Number 4, 950, 897; Date: August 21, 1990.

Other (4)

A.C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (IEEE Press, New York, 1988).

A. Mandelis, “Diffusion waves and their uses,” Physics Today, 29, August 2000.

A. Mandelis, Diffusion Wave Fields: Green Functions and Mathematical Methods (Springer, New York, in press).

B. Hofmann, Regularization for Applied and Ill-Posed Problems, (Teubner, 1986).

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Figures (12)

Fig. 1.
Fig. 1.

L-curve method: the corner of the curve corresponds to optimum regularization.

Fig. 2.
Fig. 2.

High-resolution tomographic radiometric microscope experimental set-up.

Fig. 3.
Fig. 3.

Line scan for cross-sectional imaging, of sample thickness d with a round subsurface defect.

Fig. 4.
Fig. 4.

Schematic of experimental method of thermal-wave tomographic scans for 3 laser and 3 detection positions.

Fig. 5.
Fig. 5.

Amplitude and phase in transmission of the line scan, f=11 Hz.

Fig. 6.
Fig. 6.

Transmission tomographic spatial-frequency scan of 0.1-mm deep defect at f=11 Hz. Amplitude and phase.

Fig. 7.
Fig. 7.

Homogeneous field of sample #2 at f=80 Hz. Experiment (square); Theory (eq. (20a) in Ref.[6]) (solid).

Fig. 8.
Fig. 8.

TSDT transmission reconstruction of Figure 6, averaged over 5 laser positions. True defect shown by solid line; average regularization parameter σ~1×10-6.

Fig. 9.
Fig. 9.

L-curve of x=1.5mm laser position reconstruction of Fig. 8; optimal regularization parameter σ~1×10-6.

Fig. 10.
Fig. 10.

Reconstruction from the vertical part of the L-curve (Fig. 9). Regularization σ=1×10-8.

Fig. 11.
Fig. 11.

Reconstruction from the flat part of the L-curve (Fig. 9). Regularization σ=0.1.

Fig. 12.
Fig. 12.

Average reconstruction of 0.1-mm deep defect with 0.15-mm diameter (front is the bottom and back is the top of image): a) f=80 Hz, b) f=300 Hz. Average reconstruction of 0.2-mm deep defect with 0.15-mm diameter: c) f=80 Hz, d) f=300 Hz.

Tables (1)

Tables Icon

Table 1. Born approximation validity for all thermal-wave reconstructions.

Equations (6)

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T s ( r ) = V 0 G 0 ( r r 0 ) T i ( r 0 ) F ( r 0 ) d V 0
F ( r ) = { q 0 2 [ n 2 ( r ) 1 ] ; r R 0 ; r R
n ( r ) = α 0 α ( r )
q 0 = ( 1 + i ) ( ω 2 α 0 ) 1 2
x σ = min { Ax b 2 + σ L ( x x 0 ) 2 }
an δ < λ 4

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