Abstract

By combining two well-established techniques—multispectral imaging with a tunable filter and regularized data inversion—a fast and accurate method for measuring, at each pixel of a CCD array, the spectrum emitted by a radiating object is obtained. The method is demonstrated with ruby R1- and R2-line fluorescence, using a narrowband tilt-tunable Fabry–Perot interference filter for data acquisition. Since the data inversion problem is badly ill conditioned, regularization is essential to obtain meaningful results. The reconstruction is capable of a peak wavelength accuracy of 0.01nm, sufficient to measure the small wavelength shifts encountered in rubylike materials due to changes in the local crystal lattice stress. Thus, the technique presents a novel approach to piezospectroscopic imaging and offers 2 to 3 orders of magnitude faster mapping of local stress than current techniques.

© 2007 Optical Society of America

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  1. J. D. Barnett, S. Block, and G. J. Piermarini, Rev. Sci. Instrum. 44, 1 (1973).
    [CrossRef]
  2. L. Grabner, J. Appl. Phys. 49, 580 (1978).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. S. L. Wunder and P. E. Schoen, J. Appl. Phys. 52, 3772 (1981).
    [CrossRef]

2001 (1)

J. A. Nychka and D. R. Clarke, Surf. Coat. Technol. 146, 110 (2001).
[CrossRef]

1995 (1)

J. He and D. R. Clarke, J. Am. Ceram. Soc. 78, 1347 (1995).
[CrossRef]

1993 (1)

Q. Ma and D. R. Clarke, J. Am. Ceram. Soc. 76, 1433 (1993).
[CrossRef]

1992 (1)

P. C. Hansen, SIAM (Soc. Ind. Appl. Math.) Rev. 34, 561 (1992).

1983 (1)

1981 (1)

S. L. Wunder and P. E. Schoen, J. Appl. Phys. 52, 3772 (1981).
[CrossRef]

1978 (1)

L. Grabner, J. Appl. Phys. 49, 580 (1978).
[CrossRef]

1973 (1)

J. D. Barnett, S. Block, and G. J. Piermarini, Rev. Sci. Instrum. 44, 1 (1973).
[CrossRef]

Abbiss, J. B.

J. B. Abbiss, M. Defrise, C. de Mol, and H. S. Dhadwal, J. Opt. Soc. Am. 73, 1470 (1983).
[CrossRef]

J. B. Abbiss, J. C. Allen, R. P. Bocker, and H. J. Whitehouse, "Fast image reconstruction based on the regularized pseudoinverse of the imaging operator," Proc. SPIE 1767, 93 (1992).

Allen, J. C.

J. B. Abbiss, J. C. Allen, R. P. Bocker, and H. J. Whitehouse, "Fast image reconstruction based on the regularized pseudoinverse of the imaging operator," Proc. SPIE 1767, 93 (1992).

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (V.H. Winston and Sons, 1977).

Barnett, J. D.

J. D. Barnett, S. Block, and G. J. Piermarini, Rev. Sci. Instrum. 44, 1 (1973).
[CrossRef]

Block, S.

J. D. Barnett, S. Block, and G. J. Piermarini, Rev. Sci. Instrum. 44, 1 (1973).
[CrossRef]

Bocker, R. P.

J. B. Abbiss, J. C. Allen, R. P. Bocker, and H. J. Whitehouse, "Fast image reconstruction based on the regularized pseudoinverse of the imaging operator," Proc. SPIE 1767, 93 (1992).

Clarke, D. R.

J. A. Nychka and D. R. Clarke, Surf. Coat. Technol. 146, 110 (2001).
[CrossRef]

J. He and D. R. Clarke, J. Am. Ceram. Soc. 78, 1347 (1995).
[CrossRef]

Q. Ma and D. R. Clarke, J. Am. Ceram. Soc. 76, 1433 (1993).
[CrossRef]

de Mol, C.

Defrise, M.

Dhadwal, H. S.

Grabner, L.

L. Grabner, J. Appl. Phys. 49, 580 (1978).
[CrossRef]

Hansen, P. C.

P. C. Hansen, SIAM (Soc. Ind. Appl. Math.) Rev. 34, 561 (1992).

P. C. Hansen, in Computational Inverse Problems in Electrocardiology, P.Johnston, ed. (WIT Press, 2001), pp. 119-142.

He, J.

J. He and D. R. Clarke, J. Am. Ceram. Soc. 78, 1347 (1995).
[CrossRef]

Ma, Q.

Q. Ma and D. R. Clarke, J. Am. Ceram. Soc. 76, 1433 (1993).
[CrossRef]

Nychka, J. A.

J. A. Nychka and D. R. Clarke, Surf. Coat. Technol. 146, 110 (2001).
[CrossRef]

Piermarini, G. J.

J. D. Barnett, S. Block, and G. J. Piermarini, Rev. Sci. Instrum. 44, 1 (1973).
[CrossRef]

Schoen, P. E.

S. L. Wunder and P. E. Schoen, J. Appl. Phys. 52, 3772 (1981).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (V.H. Winston and Sons, 1977).

Whitehouse, H. J.

J. B. Abbiss, J. C. Allen, R. P. Bocker, and H. J. Whitehouse, "Fast image reconstruction based on the regularized pseudoinverse of the imaging operator," Proc. SPIE 1767, 93 (1992).

Wunder, S. L.

S. L. Wunder and P. E. Schoen, J. Appl. Phys. 52, 3772 (1981).
[CrossRef]

J. Am. Ceram. Soc. (2)

Q. Ma and D. R. Clarke, J. Am. Ceram. Soc. 76, 1433 (1993).
[CrossRef]

J. He and D. R. Clarke, J. Am. Ceram. Soc. 78, 1347 (1995).
[CrossRef]

J. Appl. Phys. (2)

S. L. Wunder and P. E. Schoen, J. Appl. Phys. 52, 3772 (1981).
[CrossRef]

L. Grabner, J. Appl. Phys. 49, 580 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Rev. Sci. Instrum. (1)

J. D. Barnett, S. Block, and G. J. Piermarini, Rev. Sci. Instrum. 44, 1 (1973).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) Rev. (1)

P. C. Hansen, SIAM (Soc. Ind. Appl. Math.) Rev. 34, 561 (1992).

Surf. Coat. Technol. (1)

J. A. Nychka and D. R. Clarke, Surf. Coat. Technol. 146, 110 (2001).
[CrossRef]

Other (3)

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (V.H. Winston and Sons, 1977).

P. C. Hansen, in Computational Inverse Problems in Electrocardiology, P.Johnston, ed. (WIT Press, 2001), pp. 119-142.

J. B. Abbiss, J. C. Allen, R. P. Bocker, and H. J. Whitehouse, "Fast image reconstruction based on the regularized pseudoinverse of the imaging operator," Proc. SPIE 1767, 93 (1992).

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Figures (3)

Fig. 1
Fig. 1

Experimental transmission curves with a commercial (solid-etalon) Fabry–Perot filter, for four values of the angle of incidence, together with a fitted transmission curve, using Eq. (2).

Fig. 2
Fig. 2

Directly measured spectrum (using a high-resolution spectrometer) of ruby, and the reconstructed spectrum, using Tikhonov regularization. Differences in peak wavelength values based on a seven-point parabolic fit are given for both peaks.

Fig. 3
Fig. 3

Image of differences between reconstructed and true peak wavelength values (in nm), as obtained for each pixel.

Equations (5)

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g ( θ ) = T ( θ , λ ) f ( λ ) d λ ,
T = T 0 [ 1 + F sin 2 ( 2 π λ L n s cos θ i ) ] 1 ,
g = T f ,
f = ( T T ) 1 T g ,
f = ( T T + α I ) 1 T g ,

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