Abstract

A novel matrix inversion approach is proposed to correct several contributions to the instrument line shape (ILS) of a Fourier transform spectrometer. The matrix formalism for the ILS is first quickly reviewed. Formal inversion of the ILS matrix is next discussed, along with its limitations. The stability of the inversion process for large field-of view- (FOV-) limited and highly off-axis line shapes is investigated. The effect of inversion on the noise that is present in the spectrum is also presented. Use of classical iterative inversion methods, coupled with efficient synthesis algorithms, is proposed as a way to drastically speed up the inversion process. The method is applied to correct HBr spectra obtained from a laboratory spectrometer that has an adjustable field of view. ILSS from six FOVs are brought to the same spectral axis and to the same ideal sinc shape.

© 2006 Optical Society of America

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References

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  1. J. Connes, "Domaine d'utilisation de la méthode par transformée de Fourier," J. Phys. Radium 19, 197-208 (1958).
    [CrossRef]
  2. E. Niple, A. Pires, and K. Poultney, "Exact modeling of lineshape and wavenumber variations for off-axis detectors in Fourier transform spectrometer (FTS) sensor systems," in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R.J.Huppi, ed., Proc. SPIE 364, 11-20 (1982).
  3. J. Kauppinen and P. Saarinen, "Line-shape distortions in misaligned cube corner interferometers," Appl. Opt. 31, 69-74 (1992).
    [CrossRef] [PubMed]
  4. J. Genest and P. Tremblay, "Impact of the optical aberrations on the line shape of Fourier-transform spectrometers," Vibr. Spectrosc. 29, 3-13 (2002).
    [CrossRef]
  5. R. Desbiens, J. Genest, and P. Tremblay, "Radiometry in line-shape modeling of Fourier-transform spectrometers," Appl. Opt. 41, 1424-1432 (2002).
    [CrossRef] [PubMed]
  6. R. Desbiens, P. Tremblay, and J. Genest, "Matrix algorithm for integration and inversion of instrument line shape," in Fourier Transform Spectroscopy (Optical Society of America, 2003), pp. 42-44.
  7. R. Desbiens, P. Tremblay, J. Genest, and J. P. Bouchard, "A matrix form for the instrument line shape of Fourier transform spectrometers yielding a fast integration algorithm to theoretical spectra," Appl. Opt. 45,546-557 (2006).
    [CrossRef] [PubMed]
  8. C. L. Bennett, M. R. Carter, and D. J. Fields, "Hyperspectral imaging in the infrared using LIFTIRS," in Infrared Technology XXI, B.F.Andresen and M.Strojnik, eds., Proc. SPIE 2552,274-283 (1995).
  9. L. M. Moreau and F. Grandmont, "Review of imaging spectrometers at ABB Bomem," in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery IX, S.S.Shen and P.E.Lewis, eds., Proc. SPIE 5093, 82-93 (2003).
  10. F. Grandmont, L. Drissen, and G. Joncas, "Development of an imaging Fourier transform spectrometer for astronomy," in Specialized Optical Developments in Astronomy, E.Atad-Ettedgui and S.D'Odorico, eds., Proc. SPIE 4842,392-401 (2003).
  11. J. Genest, A. Villemaire, and P. Tremblay, "Making the most of the through-put advantage in imaging Fourier transform spectrometers," in Fourier Transform Spectroscopy: 11th International Conference, J.A.de Haseth, ed. (American Institute of Physics, 1998).
  12. J. Genest and P. Tremblay, "Instrument line shape of Fourier-transform spectrometers: analytic solutions for nonuniformly illuminated off-axis detectors," Appl. Opt. 38, 5438-5446 (1999).
    [CrossRef]
  13. K. W. Bowman, H. M. Worden, and R. Beer, "Instrument line-shape modeling and correction for off-axis detectors in Fourier-transform spectrometry," Appl. Opt. 39, 3765-3773 (2000).
    [CrossRef]
  14. C. Cohen-Tannoudji, F. Laloe, and B. Diu, Mécanique Quantique (Hermann, 1996), Vol. 1.
  15. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).
  16. Y. Viniotis, Probability and Random Processes for Electrical Engineers (McGraw-Hill, 1998).
  17. H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, and L. A. Sromovsky, "Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the high-resolution interferometer sounder," Appl. Opt. 27, 3210-3218 (1988).
    [CrossRef] [PubMed]
  18. L. A. Sromovsky, "Radiometric errors in complex Fourier transform," Appl. Opt. 42, 1779-1787 (2003).
    [CrossRef] [PubMed]
  19. V. Pan and J. Reif, "Efficient parallel solution of linear systems," in Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, 1985), pp. 143-153.
    [CrossRef]
  20. R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, "Templates for the Solution of Linear Systems Building Blocks for Iterative Methods," SIAM, Philadelphia, 1994. anon@www.netlib.org/templates/templates.ps.
  21. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).
  22. S. Barnett, Matrices--Methods and Applications, Oxford Applied Mathematics and Computing Sciences Series (Oxford U. Press, 1990).
  23. J. P. Bouchard, "Étude expérimentale de la forme de raie des spectromètres par transformation de Fourier," Ph.D. dissertation (Université Laval, 2004).
  24. J.-P. Bouchard and P. Tremblay, "Experimental study of the instrument line-shape of Fourier-transform spectrometers using a high divergence, high resolution interferometer," in Fourier Transform Spectroscopy (Optical Society of America, 2005).

2006 (1)

R. Desbiens, P. Tremblay, J. Genest, and J. P. Bouchard, "A matrix form for the instrument line shape of Fourier transform spectrometers yielding a fast integration algorithm to theoretical spectra," Appl. Opt. 45,546-557 (2006).
[CrossRef] [PubMed]

2003 (1)

2002 (2)

J. Genest and P. Tremblay, "Impact of the optical aberrations on the line shape of Fourier-transform spectrometers," Vibr. Spectrosc. 29, 3-13 (2002).
[CrossRef]

R. Desbiens, J. Genest, and P. Tremblay, "Radiometry in line-shape modeling of Fourier-transform spectrometers," Appl. Opt. 41, 1424-1432 (2002).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

1992 (1)

1988 (1)

1958 (1)

J. Connes, "Domaine d'utilisation de la méthode par transformée de Fourier," J. Phys. Radium 19, 197-208 (1958).
[CrossRef]

Barnett, S.

S. Barnett, Matrices--Methods and Applications, Oxford Applied Mathematics and Computing Sciences Series (Oxford U. Press, 1990).

Barrett, R.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, "Templates for the Solution of Linear Systems Building Blocks for Iterative Methods," SIAM, Philadelphia, 1994. anon@www.netlib.org/templates/templates.ps.

Beer, R.

Bennett, C. L.

C. L. Bennett, M. R. Carter, and D. J. Fields, "Hyperspectral imaging in the infrared using LIFTIRS," in Infrared Technology XXI, B.F.Andresen and M.Strojnik, eds., Proc. SPIE 2552,274-283 (1995).

Berry, M.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, "Templates for the Solution of Linear Systems Building Blocks for Iterative Methods," SIAM, Philadelphia, 1994. anon@www.netlib.org/templates/templates.ps.

Bouchard, J. P.

R. Desbiens, P. Tremblay, J. Genest, and J. P. Bouchard, "A matrix form for the instrument line shape of Fourier transform spectrometers yielding a fast integration algorithm to theoretical spectra," Appl. Opt. 45,546-557 (2006).
[CrossRef] [PubMed]

J. P. Bouchard, "Étude expérimentale de la forme de raie des spectromètres par transformation de Fourier," Ph.D. dissertation (Université Laval, 2004).

Bouchard, J.-P.

J.-P. Bouchard and P. Tremblay, "Experimental study of the instrument line-shape of Fourier-transform spectrometers using a high divergence, high resolution interferometer," in Fourier Transform Spectroscopy (Optical Society of America, 2005).

Bowman, K. W.

Buijs, H.

Carter, M. R.

C. L. Bennett, M. R. Carter, and D. J. Fields, "Hyperspectral imaging in the infrared using LIFTIRS," in Infrared Technology XXI, B.F.Andresen and M.Strojnik, eds., Proc. SPIE 2552,274-283 (1995).

Chan, T. F.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, "Templates for the Solution of Linear Systems Building Blocks for Iterative Methods," SIAM, Philadelphia, 1994. anon@www.netlib.org/templates/templates.ps.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, F. Laloe, and B. Diu, Mécanique Quantique (Hermann, 1996), Vol. 1.

Connes, J.

J. Connes, "Domaine d'utilisation de la méthode par transformée de Fourier," J. Phys. Radium 19, 197-208 (1958).
[CrossRef]

Demmel, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, "Templates for the Solution of Linear Systems Building Blocks for Iterative Methods," SIAM, Philadelphia, 1994. anon@www.netlib.org/templates/templates.ps.

Desbiens, R.

R. Desbiens, P. Tremblay, J. Genest, and J. P. Bouchard, "A matrix form for the instrument line shape of Fourier transform spectrometers yielding a fast integration algorithm to theoretical spectra," Appl. Opt. 45,546-557 (2006).
[CrossRef] [PubMed]

R. Desbiens, J. Genest, and P. Tremblay, "Radiometry in line-shape modeling of Fourier-transform spectrometers," Appl. Opt. 41, 1424-1432 (2002).
[CrossRef] [PubMed]

R. Desbiens, P. Tremblay, and J. Genest, "Matrix algorithm for integration and inversion of instrument line shape," in Fourier Transform Spectroscopy (Optical Society of America, 2003), pp. 42-44.

Diu, B.

C. Cohen-Tannoudji, F. Laloe, and B. Diu, Mécanique Quantique (Hermann, 1996), Vol. 1.

Donato, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, "Templates for the Solution of Linear Systems Building Blocks for Iterative Methods," SIAM, Philadelphia, 1994. anon@www.netlib.org/templates/templates.ps.

Dongarra, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, "Templates for the Solution of Linear Systems Building Blocks for Iterative Methods," SIAM, Philadelphia, 1994. anon@www.netlib.org/templates/templates.ps.

Drissen, L.

F. Grandmont, L. Drissen, and G. Joncas, "Development of an imaging Fourier transform spectrometer for astronomy," in Specialized Optical Developments in Astronomy, E.Atad-Ettedgui and S.D'Odorico, eds., Proc. SPIE 4842,392-401 (2003).

Eijkhout, V.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, "Templates for the Solution of Linear Systems Building Blocks for Iterative Methods," SIAM, Philadelphia, 1994. anon@www.netlib.org/templates/templates.ps.

Fields, D. J.

C. L. Bennett, M. R. Carter, and D. J. Fields, "Hyperspectral imaging in the infrared using LIFTIRS," in Infrared Technology XXI, B.F.Andresen and M.Strojnik, eds., Proc. SPIE 2552,274-283 (1995).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Genest, J.

R. Desbiens, P. Tremblay, J. Genest, and J. P. Bouchard, "A matrix form for the instrument line shape of Fourier transform spectrometers yielding a fast integration algorithm to theoretical spectra," Appl. Opt. 45,546-557 (2006).
[CrossRef] [PubMed]

J. Genest and P. Tremblay, "Impact of the optical aberrations on the line shape of Fourier-transform spectrometers," Vibr. Spectrosc. 29, 3-13 (2002).
[CrossRef]

R. Desbiens, J. Genest, and P. Tremblay, "Radiometry in line-shape modeling of Fourier-transform spectrometers," Appl. Opt. 41, 1424-1432 (2002).
[CrossRef] [PubMed]

J. Genest and P. Tremblay, "Instrument line shape of Fourier-transform spectrometers: analytic solutions for nonuniformly illuminated off-axis detectors," Appl. Opt. 38, 5438-5446 (1999).
[CrossRef]

J. Genest, A. Villemaire, and P. Tremblay, "Making the most of the through-put advantage in imaging Fourier transform spectrometers," in Fourier Transform Spectroscopy: 11th International Conference, J.A.de Haseth, ed. (American Institute of Physics, 1998).

R. Desbiens, P. Tremblay, and J. Genest, "Matrix algorithm for integration and inversion of instrument line shape," in Fourier Transform Spectroscopy (Optical Society of America, 2003), pp. 42-44.

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Grandmont, F.

F. Grandmont, L. Drissen, and G. Joncas, "Development of an imaging Fourier transform spectrometer for astronomy," in Specialized Optical Developments in Astronomy, E.Atad-Ettedgui and S.D'Odorico, eds., Proc. SPIE 4842,392-401 (2003).

L. M. Moreau and F. Grandmont, "Review of imaging spectrometers at ABB Bomem," in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery IX, S.S.Shen and P.E.Lewis, eds., Proc. SPIE 5093, 82-93 (2003).

Howell, H. B.

Joncas, G.

F. Grandmont, L. Drissen, and G. Joncas, "Development of an imaging Fourier transform spectrometer for astronomy," in Specialized Optical Developments in Astronomy, E.Atad-Ettedgui and S.D'Odorico, eds., Proc. SPIE 4842,392-401 (2003).

Kauppinen, J.

Laloe, F.

C. Cohen-Tannoudji, F. Laloe, and B. Diu, Mécanique Quantique (Hermann, 1996), Vol. 1.

LaPorte, D. D.

Moreau, L. M.

L. M. Moreau and F. Grandmont, "Review of imaging spectrometers at ABB Bomem," in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery IX, S.S.Shen and P.E.Lewis, eds., Proc. SPIE 5093, 82-93 (2003).

Niple, E.

E. Niple, A. Pires, and K. Poultney, "Exact modeling of lineshape and wavenumber variations for off-axis detectors in Fourier transform spectrometer (FTS) sensor systems," in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R.J.Huppi, ed., Proc. SPIE 364, 11-20 (1982).

Pan, V.

V. Pan and J. Reif, "Efficient parallel solution of linear systems," in Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, 1985), pp. 143-153.
[CrossRef]

Pires, A.

E. Niple, A. Pires, and K. Poultney, "Exact modeling of lineshape and wavenumber variations for off-axis detectors in Fourier transform spectrometer (FTS) sensor systems," in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R.J.Huppi, ed., Proc. SPIE 364, 11-20 (1982).

Poultney, K.

E. Niple, A. Pires, and K. Poultney, "Exact modeling of lineshape and wavenumber variations for off-axis detectors in Fourier transform spectrometer (FTS) sensor systems," in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R.J.Huppi, ed., Proc. SPIE 364, 11-20 (1982).

Pozo, R.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, "Templates for the Solution of Linear Systems Building Blocks for Iterative Methods," SIAM, Philadelphia, 1994. anon@www.netlib.org/templates/templates.ps.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Reif, J.

V. Pan and J. Reif, "Efficient parallel solution of linear systems," in Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, 1985), pp. 143-153.
[CrossRef]

Revercomb, H. E.

Romine, C.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, "Templates for the Solution of Linear Systems Building Blocks for Iterative Methods," SIAM, Philadelphia, 1994. anon@www.netlib.org/templates/templates.ps.

Saarinen, P.

Smith, W. L.

Sromovsky, L. A.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Tremblay, P.

R. Desbiens, P. Tremblay, J. Genest, and J. P. Bouchard, "A matrix form for the instrument line shape of Fourier transform spectrometers yielding a fast integration algorithm to theoretical spectra," Appl. Opt. 45,546-557 (2006).
[CrossRef] [PubMed]

J. Genest and P. Tremblay, "Impact of the optical aberrations on the line shape of Fourier-transform spectrometers," Vibr. Spectrosc. 29, 3-13 (2002).
[CrossRef]

R. Desbiens, J. Genest, and P. Tremblay, "Radiometry in line-shape modeling of Fourier-transform spectrometers," Appl. Opt. 41, 1424-1432 (2002).
[CrossRef] [PubMed]

J. Genest and P. Tremblay, "Instrument line shape of Fourier-transform spectrometers: analytic solutions for nonuniformly illuminated off-axis detectors," Appl. Opt. 38, 5438-5446 (1999).
[CrossRef]

J. Genest, A. Villemaire, and P. Tremblay, "Making the most of the through-put advantage in imaging Fourier transform spectrometers," in Fourier Transform Spectroscopy: 11th International Conference, J.A.de Haseth, ed. (American Institute of Physics, 1998).

J.-P. Bouchard and P. Tremblay, "Experimental study of the instrument line-shape of Fourier-transform spectrometers using a high divergence, high resolution interferometer," in Fourier Transform Spectroscopy (Optical Society of America, 2005).

R. Desbiens, P. Tremblay, and J. Genest, "Matrix algorithm for integration and inversion of instrument line shape," in Fourier Transform Spectroscopy (Optical Society of America, 2003), pp. 42-44.

Van der Vorst, H.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, "Templates for the Solution of Linear Systems Building Blocks for Iterative Methods," SIAM, Philadelphia, 1994. anon@www.netlib.org/templates/templates.ps.

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Villemaire, A.

J. Genest, A. Villemaire, and P. Tremblay, "Making the most of the through-put advantage in imaging Fourier transform spectrometers," in Fourier Transform Spectroscopy: 11th International Conference, J.A.de Haseth, ed. (American Institute of Physics, 1998).

Viniotis, Y.

Y. Viniotis, Probability and Random Processes for Electrical Engineers (McGraw-Hill, 1998).

Worden, H. M.

Appl. Opt. (7)

J. Phys. Radium (1)

J. Connes, "Domaine d'utilisation de la méthode par transformée de Fourier," J. Phys. Radium 19, 197-208 (1958).
[CrossRef]

Vibr. Spectrosc. (1)

J. Genest and P. Tremblay, "Impact of the optical aberrations on the line shape of Fourier-transform spectrometers," Vibr. Spectrosc. 29, 3-13 (2002).
[CrossRef]

Other (15)

C. Cohen-Tannoudji, F. Laloe, and B. Diu, Mécanique Quantique (Hermann, 1996), Vol. 1.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Y. Viniotis, Probability and Random Processes for Electrical Engineers (McGraw-Hill, 1998).

V. Pan and J. Reif, "Efficient parallel solution of linear systems," in Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, 1985), pp. 143-153.
[CrossRef]

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, "Templates for the Solution of Linear Systems Building Blocks for Iterative Methods," SIAM, Philadelphia, 1994. anon@www.netlib.org/templates/templates.ps.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

S. Barnett, Matrices--Methods and Applications, Oxford Applied Mathematics and Computing Sciences Series (Oxford U. Press, 1990).

J. P. Bouchard, "Étude expérimentale de la forme de raie des spectromètres par transformation de Fourier," Ph.D. dissertation (Université Laval, 2004).

J.-P. Bouchard and P. Tremblay, "Experimental study of the instrument line-shape of Fourier-transform spectrometers using a high divergence, high resolution interferometer," in Fourier Transform Spectroscopy (Optical Society of America, 2005).

C. L. Bennett, M. R. Carter, and D. J. Fields, "Hyperspectral imaging in the infrared using LIFTIRS," in Infrared Technology XXI, B.F.Andresen and M.Strojnik, eds., Proc. SPIE 2552,274-283 (1995).

L. M. Moreau and F. Grandmont, "Review of imaging spectrometers at ABB Bomem," in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery IX, S.S.Shen and P.E.Lewis, eds., Proc. SPIE 5093, 82-93 (2003).

F. Grandmont, L. Drissen, and G. Joncas, "Development of an imaging Fourier transform spectrometer for astronomy," in Specialized Optical Developments in Astronomy, E.Atad-Ettedgui and S.D'Odorico, eds., Proc. SPIE 4842,392-401 (2003).

J. Genest, A. Villemaire, and P. Tremblay, "Making the most of the through-put advantage in imaging Fourier transform spectrometers," in Fourier Transform Spectroscopy: 11th International Conference, J.A.de Haseth, ed. (American Institute of Physics, 1998).

E. Niple, A. Pires, and K. Poultney, "Exact modeling of lineshape and wavenumber variations for off-axis detectors in Fourier transform spectrometer (FTS) sensor systems," in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R.J.Huppi, ed., Proc. SPIE 364, 11-20 (1982).

R. Desbiens, P. Tremblay, and J. Genest, "Matrix algorithm for integration and inversion of instrument line shape," in Fourier Transform Spectroscopy (Optical Society of America, 2003), pp. 42-44.

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Figures (15)

Fig. 1
Fig. 1

(Color online) Self-apodization envelopes for various wavenumbers. ILS for a centered circular FOV such that its radius is over focal length R∕f = 0.2, the interferogram has 513 points, and the ILS is integrated on 64 samples. Wavenumbers in this figure assume a 632.8 nm sampling interval.

Fig. 2
Fig. 2

(Color online) Interferogram restoration envelopes for the self-apodization envelopes of Fig. 1. ILS for a centered circular detector such that R∕f = 0.2, the interferogram has 513 points, and the ILS is integrated on 64 samples. Wavenumbers in this figure assume a 632.8 nm sampling interval.

Fig. 3
Fig. 3

Inverse of the example ILS matrix in the interferogram domain. The amplitude of the coefficients is represented on a logarithmic intensity scale. ILS for a centered circular detector such that R∕f = 0.2, the interferogram has 513 points, and the ILS is integrated on 64 samples.

Fig. 4
Fig. 4

(Color online) Self-apodization envelopes for various wavenumbers. ILS for an off-axis circular detector such that the off-axis position over focal length x∕f = 0.15 and R∕f = 0.05, the interferogram has 513 points, and the ILS is integrated on 64 samples. Wavenumbers in this figure assume a 632.8 nm sampling interval.

Fig. 5
Fig. 5

(Color online) Interferogram restoration envelopes for the self-apodization envelopes of Fig. 4. ILS for an off-axis circular detector such that x∕f = 0.15 and R∕f = 0.05, the interferogram has 513 points, and the ILS is integrated on 64 samples. Wavenumbers in this figure assume a 632.8 nm sampling interval.

Fig. 6
Fig. 6

Inverse of an off-axis ILS matrix in the interferogram domain. The amplitude of the coefficients is represented on a logarithmic intensity scale. ILS for an off-axis circular detector such that x∕f = 0.15 and R∕f = 0.05 the interferogram has 513 points, and the ILS is integrated on 64 samples.

Fig. 7
Fig. 7

(Color online) Diagonal of the covariance matrix of the real and imaginary parts of the spectral noise after ILS inversion (σ b 2 = 1). The line shape of a centered circular detector, R∕f = 0.08, has 513 points.

Fig. 8
Fig. 8

(Color online) Diagonal of the covariance matrix of the real and imaginary parts of the spectral noise after ILS inversion (σ b 2 = 1). The line shape of a centered circular detector, R∕f = 0.2, has 513 points.

Fig. 9
Fig. 9

Example of the noise covariance matrix in the interferogram domain. The amplitude of the coefficients is represented on a logarithmic intensity scale. The line shape of a centered circular detector has R∕f = 0.2, 513 points.

Fig. 10
Fig. 10

Example of the spectral noise covariance matrix after ILS inversion: top, real part and bottom, imaginary part. The amplitude of the coefficients is represented on a logarithmic intensity scale. The line shape of a centered circular detector, R∕f = 0.2, has 513 points.

Fig. 11
Fig. 11

(Color online) Corrected absorption HBr spectra.

Fig. 12
Fig. 12

(Color online) Difference between line positions and average line position after correction as a function of the FOV. Each trace corresponds to one HBr line. Corrected values are near zero.

Fig. 13
Fig. 13

Difference between positions of uncorrected lines and the average value of the corrected line for all FOVs as a function of the line's wavenumber. Each numbered curve represents a distinct FOV, as described in Table 1.

Fig. 14
Fig. 14

Difference between the positions of corrected lines and the average value of the corrected line for all FOVs as a function of the line's wavenumber. Each numbered curve represents a distinct FOV, as described in Table 1.

Fig. 15
Fig. 15

Zoom of the HBr P1 line, illustrating the difference between left, the uncorrected and right, the corrected line shape. All corrected ILSs have the same sinc-limited width and are located at the same position.

Tables (1)

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Table 1 Instrument Circular FOV Parameters for ILS Inversion (550 mm Focal Length)

Equations (35)

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S m ( σ ) = S o ( σ o ) I L S ( σ , σ o ) d σ o .
s m  =  ILS s o .
s o = F i o , i o = F 1 s o ,
s m = F i m , i m = F 1 s m ,
i m = F 1 s m ,        
= F 1 ILS s o ,        
= F 1 ILS FF 1 s o ,        
= ILS ˜ i o ,        
I m ( x ) = I o ( x o ) ILS ˜ ( x , x o ) d x o .
ILS ( σ , σ o ) = 1 | σ o | H ( σ σ o ) .
1 | σ o | H ( σ σ o ) 1 σ x f ( x σ o ) σ o x o 1 | x | H ( x o x ) ,
ILS ˜ ( x , x o ) = 1 | x | H ( x o x ) = ILS ( x o , x ) .
s o = ILS d     1 s m ,
i o = ILS ˜ d     1 i m ,
ILS ˜ d     1 = ( ILS d     1 ) T .
i m = ILS ˜ d F 1 s o .
i o = ILS ˜ d     1 F 1 s m .
C i = σ b     2 ILS ˜ d     1 ( ILS ˜ d     1 ) T .
P = [ 1 2 0 0 0 1 2 0 1 2 0 1 2 0 0 0 1 0 0 0 1 2 0 1 2 0 1 2 0 0 0 1 2 ] , P ¯ = [ 1 2 0 0 0 1 2 0 1 2 0 1 2 0 0 0 0 0 0 0 1 2 0 1 2 0 1 2 0 0 0 1 2 ] .
P P T = P , P ¯ P ¯ T = P ¯ ,
FPF * = P , F P ¯ F * = P ¯ ,
C s   Re = σ b     2 ILS d     1 FPP T ( F * ) T ( ILS d     1 ) T , = σ b     2 ILS d     1 P ( ILS d     1 ) T ,
C s   Im = σ b     2 ILS d     1 F P ¯ P ¯ T F ( ILS d     1 ) T ; = σ b     2 ILS d     1 P ¯ ( ILS d     1 ) T .
i ^ o ( k + 1 ) = i ^ o ( k ) + B [ i m ILS ˜ d i ^ o ( k ) ] ,
R = I B ILS ˜ d ,
B = τ ILS ˜ d     T ,
R = I τ ILS ˜ d     T ILS ˜ d .
R = diag ( 1 τ λ 1 , 1 τ λ 2 , . . . , 1 τ λ N ) ,
0 < τ < 2 max ( λ i ) .
i ^ o ( k + 1 ) = i ^ o ( k ) + τ ILS ˜ d     T [ i m ILS ˜ d i ^ o ( k ) ] .
i ^ o ( k + 1 ) = i ^ o ( k ) + τ F ILS ˜ d F 1 [ i m ILS ˜ d i ^ o ( k ) ] .
τ = 2 / [ min ( λ i ) + max ( λ i ) ] .
I L S ˜ T I L S ˜ = ILS T ( x 1 , x o ) ILS ( x o , x 2 ) d x = 1 | x 1 | H ( x o x 1 ) 1 | x 2 | H ( x o x 2 ) d x o ,
SR ( x 2 ) = 1 | x 1 | H ( x o x 1 ) 1 | x 2 | H ( x o x 2 ) d x o d x 1 .
SR ( x 2 ) = [ 1 | x 1 | H ( x o x 1 ) d x 1 ] 1 | x 2 | H ( x o x 2 ) d x o = 1 | x 1 | H ( x o x 1 ) d x 1 1 | x 2 | H ( x o x 2 ) d x o .

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