Abstract

The instrument line shape (ILS) of a Fourier-transform spectrometer is expressed in a matrix form. For all line shape effects that scale with wavenumber, the ILS matrix is shown to be transposed in the spectral and interferogram domains. The novel representation of the ILS matrix in the interferogram domain yields an insightful physical interpretation of the underlying process producing self-apodization. Working in the interferogram domain circumvents the problem of taking into account the effects of finite optical path difference and permits a proper discretization of the equations. A fast algorithm in O(Nlog2N), based on the fractional Fourier transform, is introduced that permits the application of a constant resolving power line shape to theoretical spectra or forward models. The ILS integration formalism is validated with experimental data.

© 2006 Optical Society of America

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  1. R. Desbiens, J. Genest, and P. Tremblay, "Radiometry in line-shape modeling of Fourier-transform spectrometers," Appl. Opt. 41, 1424-1432 (2002).
    [CrossRef] [PubMed]
  2. 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]
  3. 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]
  4. F. Bouffard, P. Tremblay, R. Desbiens, and J.-P. Bouchard, "Synthetic spectra of Fourier transform spectrometers using analytically modeled instrument line shape inclusion," in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR) (Science and Technology Corporation, 2001), pp. 326-335.
  5. E. Sarkissian and K. W. Bowman, "Application of a nonuniform spectral resampling transform in Fourier-transform spectrometry," Appl. Opt. 42, 1122-1131 (2003).
    [CrossRef] [PubMed]
  6. A. Vance, W. Bell, C. Paton-Walsh, and B. Jolliffe, "Fourier transform infra-red instrument line shape determination using an infra-red helium:neon laser and a low pressure gas cell," inPolar Stratospheric Ozone 1997, European Commission Air Pollution Research Rep. 66, Proceedings of the Fourth European Symposium on Polar Stratospheric Ozone, European Community (European Commission on Air Pollution, 1998), pp. 735-738.
  7. F. Hase, T. Blumenstock, and C. Paton-Walsh, "Analysis of the instrumental line shape of high-resolution Fourier transform IR spectrometers with gas cell measurements and new retrieval software," Appl. Opt. 38, 3417-3422 (1999).
    [CrossRef]
  8. R. D. Kaiser, "Wavelength calibration and instrument line shape estimation of a LWIR hyperspectral sensor from in-scene data [4049-28]," in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Shen and M. R. Descour, eds., Proc. SPIE. 4049, 284-287 (2000).
    [CrossRef]
  9. S. Ichizawa, K. Kondo, M. Suzuki, H. Shimoda, and T. Ogawa, "Improvement of instrument line shape of IMG," in Infrared Spaceborne Remote Sensing IX, M. Strojnik and B. F. Andresen, eds., Proc. SPIE 4486, 326-333 (2002).
    [CrossRef]
  10. P. Raspollini, P. Ade, B. Carli, and M. Ridolfi, "Correction of instrument line-shape distortions in Fourier transform spectroscopy," Appl. Opt. 37, 3697-3704 (1998).
    [CrossRef]
  11. M. Ahro, J. Kauppinen, and I. Salomaa, "Detection and correction of instrumental line-shape distortions in Fourier spectroscopy," Appl. Opt. 39, 6230-6237 (2000).
    [CrossRef]
  12. F. Bouffard, "Une approche matricielle pour la modélisation et la correction des effets instrumentaux en spectrométrie par transformation de Fourier," master's degree dissertation (Université Laval, 2002).
  13. R. Desbiens, P. Tremblay, and J. Genest, "Matrix algorithm for integration and inversion of instrument line shape," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 42-44.
  14. C. Cohen-Tannoudji, F. Laloë, and B. Diu, Mécanique Quantique (Hermann, 1996), Vol. 1.
  15. J. Connes, "Domaine d'utilisation de la méthode par transformée de Fourier," J. Phys. Radium 19, 197-208 (1958).
    [CrossRef]
  16. E. Niple, A. Pires, and K. Poultney, "Exact modeling of lineshape and wavenumber variations for off-axis detectors in Fourier transform spectrometers (FTS) sensor systems," in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R. J. Huppi, ed., Proc. SPIE 0364, 11-20 (1982).
  17. J. Kauppinen and P. Saarinen, "Line-shape distortions in misaligned cube corners interferometers," Appl. Opt. 31, 69-74 (1992).
    [CrossRef] [PubMed]
  18. J. Genest and P. Tremblay, "Impact of the optical aberrations on the line shape of Fourier-transform spectrometers," Vibrational Spectrosc. 29(1-2), 1-13 (2002).
  19. 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).
  20. D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," SIAM Rev. 33, 389-404 (1991).
    [CrossRef]
  21. D. H. Bailey and P. N. Swarztrauber, "Errata for: The fractional Fourier transform and applications," http://crd.1b1.gov/∼dhbailey/dhbpapers/.
  22. D. H. Bailey and P. N. Swarztrauber, "A fast method for the numerical evaluation of continuous Fourier transforms," SIAM J. Sci. Comput. 15, 1105-1110 (1994).
    [CrossRef]
  23. L. I. Bluestein, "A linear filtering approach to the computation of discrete Fourier transform," IEEE Trans. Audio Electroacoust. 18, 451-455 (1970).
    [CrossRef]
  24. J.-P. Bouchard, "Étude expérimentale de la forme de raie des spectromètres par transformation de Fourier," Ph.D. dissertation (Université Laval, 2004).
  25. J.-P. Bouchard, P. Tremblay, R. Desbiens, and F. Bouffard, "Detailed line-shape measurements using a high resolution, high divergence Fourier transform spectrometer," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics series (Optical Society of America, 2003), pp. 25-27.
  26. M. Chamberland, V. Farley, L. Belhumeur, F. Williams, J. Lawrence, P. Tremblay, and R. Desbiens, "The Instrument LineShape, an imperative parameter for the absolute spectral calibration of an FTS," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 160-166.
  27. L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
    [CrossRef]
  28. L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
    [CrossRef]

2003 (2)

E. Sarkissian and K. W. Bowman, "Application of a nonuniform spectral resampling transform in Fourier-transform spectrometry," Appl. Opt. 42, 1122-1131 (2003).
[CrossRef] [PubMed]

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

2002 (2)

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

S. Ichizawa, K. Kondo, M. Suzuki, H. Shimoda, and T. Ogawa, "Improvement of instrument line shape of IMG," in Infrared Spaceborne Remote Sensing IX, M. Strojnik and B. F. Andresen, eds., Proc. SPIE 4486, 326-333 (2002).
[CrossRef]

2000 (3)

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]

R. D. Kaiser, "Wavelength calibration and instrument line shape estimation of a LWIR hyperspectral sensor from in-scene data [4049-28]," in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Shen and M. R. Descour, eds., Proc. SPIE. 4049, 284-287 (2000).
[CrossRef]

M. Ahro, J. Kauppinen, and I. Salomaa, "Detection and correction of instrumental line-shape distortions in Fourier spectroscopy," Appl. Opt. 39, 6230-6237 (2000).
[CrossRef]

1999 (2)

1998 (2)

P. Raspollini, P. Ade, B. Carli, and M. Ridolfi, "Correction of instrument line-shape distortions in Fourier transform spectroscopy," Appl. Opt. 37, 3697-3704 (1998).
[CrossRef]

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

1994 (1)

D. H. Bailey and P. N. Swarztrauber, "A fast method for the numerical evaluation of continuous Fourier transforms," SIAM J. Sci. Comput. 15, 1105-1110 (1994).
[CrossRef]

1992 (1)

1991 (1)

D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," SIAM Rev. 33, 389-404 (1991).
[CrossRef]

1982 (1)

E. Niple, A. Pires, and K. Poultney, "Exact modeling of lineshape and wavenumber variations for off-axis detectors in Fourier transform spectrometers (FTS) sensor systems," in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R. J. Huppi, ed., Proc. SPIE 0364, 11-20 (1982).

1970 (1)

L. I. Bluestein, "A linear filtering approach to the computation of discrete Fourier transform," IEEE Trans. Audio Electroacoust. 18, 451-455 (1970).
[CrossRef]

1958 (1)

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

Ade, P.

Ahro, M.

Bailey, D. H.

D. H. Bailey and P. N. Swarztrauber, "A fast method for the numerical evaluation of continuous Fourier transforms," SIAM J. Sci. Comput. 15, 1105-1110 (1994).
[CrossRef]

D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," SIAM Rev. 33, 389-404 (1991).
[CrossRef]

D. H. Bailey and P. N. Swarztrauber, "Errata for: The fractional Fourier transform and applications," http://crd.1b1.gov/∼dhbailey/dhbpapers/.

Barbe, A.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Beer, R.

Belhumeur, L.

M. Chamberland, V. Farley, L. Belhumeur, F. Williams, J. Lawrence, P. Tremblay, and R. Desbiens, "The Instrument LineShape, an imperative parameter for the absolute spectral calibration of an FTS," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 160-166.

Bell, W.

A. Vance, W. Bell, C. Paton-Walsh, and B. Jolliffe, "Fourier transform infra-red instrument line shape determination using an infra-red helium:neon laser and a low pressure gas cell," inPolar Stratospheric Ozone 1997, European Commission Air Pollution Research Rep. 66, Proceedings of the Fourth European Symposium on Polar Stratospheric Ozone, European Community (European Commission on Air Pollution, 1998), pp. 735-738.

Benner, D. Chris

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Bluestein, L. I.

L. I. Bluestein, "A linear filtering approach to the computation of discrete Fourier transform," IEEE Trans. Audio Electroacoust. 18, 451-455 (1970).
[CrossRef]

Blumenstock, T.

Bouchard, J.-P.

F. Bouffard, P. Tremblay, R. Desbiens, and J.-P. Bouchard, "Synthetic spectra of Fourier transform spectrometers using analytically modeled instrument line shape inclusion," in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR) (Science and Technology Corporation, 2001), pp. 326-335.

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, P. Tremblay, R. Desbiens, and F. Bouffard, "Detailed line-shape measurements using a high resolution, high divergence Fourier transform spectrometer," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics series (Optical Society of America, 2003), pp. 25-27.

Bouffard, F.

J.-P. Bouchard, P. Tremblay, R. Desbiens, and F. Bouffard, "Detailed line-shape measurements using a high resolution, high divergence Fourier transform spectrometer," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics series (Optical Society of America, 2003), pp. 25-27.

F. Bouffard, P. Tremblay, R. Desbiens, and J.-P. Bouchard, "Synthetic spectra of Fourier transform spectrometers using analytically modeled instrument line shape inclusion," in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR) (Science and Technology Corporation, 2001), pp. 326-335.

F. Bouffard, "Une approche matricielle pour la modélisation et la correction des effets instrumentaux en spectrométrie par transformation de Fourier," master's degree dissertation (Université Laval, 2002).

Bowman, K. W.

Brown, L. R.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Camy-Peyret, C.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Carleer, M. R.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Carli, B.

Chamberland, M.

M. Chamberland, V. Farley, L. Belhumeur, F. Williams, J. Lawrence, P. Tremblay, and R. Desbiens, "The Instrument LineShape, an imperative parameter for the absolute spectral calibration of an FTS," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 160-166.

Chance, K.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Clerbaux, C.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, F. Laloë, 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]

Dana, V.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Desbiens, R.

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

F. Bouffard, P. Tremblay, R. Desbiens, and J.-P. Bouchard, "Synthetic spectra of Fourier transform spectrometers using analytically modeled instrument line shape inclusion," in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR) (Science and Technology Corporation, 2001), pp. 326-335.

R. Desbiens, P. Tremblay, and J. Genest, "Matrix algorithm for integration and inversion of instrument line shape," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 42-44.

M. Chamberland, V. Farley, L. Belhumeur, F. Williams, J. Lawrence, P. Tremblay, and R. Desbiens, "The Instrument LineShape, an imperative parameter for the absolute spectral calibration of an FTS," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 160-166.

J.-P. Bouchard, P. Tremblay, R. Desbiens, and F. Bouffard, "Detailed line-shape measurements using a high resolution, high divergence Fourier transform spectrometer," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics series (Optical Society of America, 2003), pp. 25-27.

Devi, V. M.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Diu, B.

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

Edwards, D. P.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Farley, V.

M. Chamberland, V. Farley, L. Belhumeur, F. Williams, J. Lawrence, P. Tremblay, and R. Desbiens, "The Instrument LineShape, an imperative parameter for the absolute spectral calibration of an FTS," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 160-166.

Fayt, A.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

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).

Flaud, J.-M.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Gamache, R. R.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Genest, J.

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]

R. Desbiens, P. Tremblay, and J. Genest, "Matrix algorithm for integration and inversion of instrument line shape," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 42-44.

J. Genest and P. Tremblay, "Impact of the optical aberrations on the line shape of Fourier-transform spectrometers," Vibrational Spectrosc. 29(1-2), 1-13 (2002).

Goldman, A.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Hase, F.

Ichizawa, S.

S. Ichizawa, K. Kondo, M. Suzuki, H. Shimoda, and T. Ogawa, "Improvement of instrument line shape of IMG," in Infrared Spaceborne Remote Sensing IX, M. Strojnik and B. F. Andresen, eds., Proc. SPIE 4486, 326-333 (2002).
[CrossRef]

Jacquemart, D.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Jolliffe, B.

A. Vance, W. Bell, C. Paton-Walsh, and B. Jolliffe, "Fourier transform infra-red instrument line shape determination using an infra-red helium:neon laser and a low pressure gas cell," inPolar Stratospheric Ozone 1997, European Commission Air Pollution Research Rep. 66, Proceedings of the Fourth European Symposium on Polar Stratospheric Ozone, European Community (European Commission on Air Pollution, 1998), pp. 735-738.

Jucks, K. W.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Kaiser, R. D.

R. D. Kaiser, "Wavelength calibration and instrument line shape estimation of a LWIR hyperspectral sensor from in-scene data [4049-28]," in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Shen and M. R. Descour, eds., Proc. SPIE. 4049, 284-287 (2000).
[CrossRef]

Kauppinen, J.

Kondo, K.

S. Ichizawa, K. Kondo, M. Suzuki, H. Shimoda, and T. Ogawa, "Improvement of instrument line shape of IMG," in Infrared Spaceborne Remote Sensing IX, M. Strojnik and B. F. Andresen, eds., Proc. SPIE 4486, 326-333 (2002).
[CrossRef]

Lafferty, W. J.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Laloë, F.

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

Lawrence, J.

M. Chamberland, V. Farley, L. Belhumeur, F. Williams, J. Lawrence, P. Tremblay, and R. Desbiens, "The Instrument LineShape, an imperative parameter for the absolute spectral calibration of an FTS," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 160-166.

Mandin, J.-M.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Mandin, J.-Y.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Massie, S. T.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

McCann, A.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Nemtchinov, V.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Newnham, D. A.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Niple, E.

E. Niple, A. Pires, and K. Poultney, "Exact modeling of lineshape and wavenumber variations for off-axis detectors in Fourier transform spectrometers (FTS) sensor systems," in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R. J. Huppi, ed., Proc. SPIE 0364, 11-20 (1982).

Ogawa, T.

S. Ichizawa, K. Kondo, M. Suzuki, H. Shimoda, and T. Ogawa, "Improvement of instrument line shape of IMG," in Infrared Spaceborne Remote Sensing IX, M. Strojnik and B. F. Andresen, eds., Proc. SPIE 4486, 326-333 (2002).
[CrossRef]

Paton-Walsh, C.

F. Hase, T. Blumenstock, and C. Paton-Walsh, "Analysis of the instrumental line shape of high-resolution Fourier transform IR spectrometers with gas cell measurements and new retrieval software," Appl. Opt. 38, 3417-3422 (1999).
[CrossRef]

A. Vance, W. Bell, C. Paton-Walsh, and B. Jolliffe, "Fourier transform infra-red instrument line shape determination using an infra-red helium:neon laser and a low pressure gas cell," inPolar Stratospheric Ozone 1997, European Commission Air Pollution Research Rep. 66, Proceedings of the Fourth European Symposium on Polar Stratospheric Ozone, European Community (European Commission on Air Pollution, 1998), pp. 735-738.

Perrin, A.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Pires, A.

E. Niple, A. Pires, and K. Poultney, "Exact modeling of lineshape and wavenumber variations for off-axis detectors in Fourier transform spectrometers (FTS) sensor systems," in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R. J. Huppi, ed., Proc. SPIE 0364, 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 spectrometers (FTS) sensor systems," in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R. J. Huppi, ed., Proc. SPIE 0364, 11-20 (1982).

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).

Raspollini, P.

Ridolfi, M.

Rinsland, C. P.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Rothman, L. S.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Saarinen, P.

Salomaa, I.

Sarkissian, E.

Schroeder, J.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Shimoda, H.

S. Ichizawa, K. Kondo, M. Suzuki, H. Shimoda, and T. Ogawa, "Improvement of instrument line shape of IMG," in Infrared Spaceborne Remote Sensing IX, M. Strojnik and B. F. Andresen, eds., Proc. SPIE 4486, 326-333 (2002).
[CrossRef]

Smith, K. M.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Smith, M. A. H.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Suzuki, M.

S. Ichizawa, K. Kondo, M. Suzuki, H. Shimoda, and T. Ogawa, "Improvement of instrument line shape of IMG," in Infrared Spaceborne Remote Sensing IX, M. Strojnik and B. F. Andresen, eds., Proc. SPIE 4486, 326-333 (2002).
[CrossRef]

Swarztrauber, P. N.

D. H. Bailey and P. N. Swarztrauber, "A fast method for the numerical evaluation of continuous Fourier transforms," SIAM J. Sci. Comput. 15, 1105-1110 (1994).
[CrossRef]

D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," SIAM Rev. 33, 389-404 (1991).
[CrossRef]

D. H. Bailey and P. N. Swarztrauber, "Errata for: The fractional Fourier transform and applications," http://crd.1b1.gov/∼dhbailey/dhbpapers/.

Tang, K.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

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).

Toth, R. A.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Tremblay, P.

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]

F. Bouffard, P. Tremblay, R. Desbiens, and J.-P. Bouchard, "Synthetic spectra of Fourier transform spectrometers using analytically modeled instrument line shape inclusion," in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR) (Science and Technology Corporation, 2001), pp. 326-335.

R. Desbiens, P. Tremblay, and J. Genest, "Matrix algorithm for integration and inversion of instrument line shape," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 42-44.

J. Genest and P. Tremblay, "Impact of the optical aberrations on the line shape of Fourier-transform spectrometers," Vibrational Spectrosc. 29(1-2), 1-13 (2002).

J.-P. Bouchard, P. Tremblay, R. Desbiens, and F. Bouffard, "Detailed line-shape measurements using a high resolution, high divergence Fourier transform spectrometer," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics series (Optical Society of America, 2003), pp. 25-27.

M. Chamberland, V. Farley, L. Belhumeur, F. Williams, J. Lawrence, P. Tremblay, and R. Desbiens, "The Instrument LineShape, an imperative parameter for the absolute spectral calibration of an FTS," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 160-166.

Vance, A.

A. Vance, W. Bell, C. Paton-Walsh, and B. Jolliffe, "Fourier transform infra-red instrument line shape determination using an infra-red helium:neon laser and a low pressure gas cell," inPolar Stratospheric Ozone 1997, European Commission Air Pollution Research Rep. 66, Proceedings of the Fourth European Symposium on Polar Stratospheric Ozone, European Community (European Commission on Air Pollution, 1998), pp. 735-738.

Vander Auwera, J.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Varanasi, P.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

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).

Wattson, R. B.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Williams, F.

M. Chamberland, V. Farley, L. Belhumeur, F. Williams, J. Lawrence, P. Tremblay, and R. Desbiens, "The Instrument LineShape, an imperative parameter for the absolute spectral calibration of an FTS," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 160-166.

Worden, H. M.

Yoshino, K.

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

Appl. Opt. (8)

IEEE Trans. Audio Electroacoust. (1)

L. I. Bluestein, "A linear filtering approach to the computation of discrete Fourier transform," IEEE Trans. Audio Electroacoust. 18, 451-455 (1970).
[CrossRef]

J. Phys. Radium (1)

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

J. Quant. Spectrosc. Radiat. Transfer (2)

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J.-M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J.-M. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, and K. Yoshino, "HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition," J. Quant. Spectrosc. Radiat. Transfer 60, 665-710 (1998).
[CrossRef]

L. S. Rothman, A. Barbe, D. Chris Benner, L. R. Brown, C. Camy-Peyret, M. R. Carleer, K. Chance, C. Clerbaux, V. Dana, V. M. Devi, A. Fayt, J.-M. Flaud, R. R. Gamache, A. Goldman, D. Jacquemart, K. W. Jucks, W. J. Lafferty, J.-Y. Mandin, S. T. Massie, V. Nemtchinov, D. A. Newnham, A. Perrin, C. P. Rinsland, J. Schroeder, K. M. Smith, M. A. H. Smith, K. Tang, R. A. Toth, J. Vander Auwera, P. Varanasi, and K. Yoshino, "The HITRAN molecular spectroscopic database: edition of 2000 including updates through 2001," J. Quant. Spectrosc. Radiat. Transfer 82, 5-44 (2003).
[CrossRef]

Proc. SPIE (2)

S. Ichizawa, K. Kondo, M. Suzuki, H. Shimoda, and T. Ogawa, "Improvement of instrument line shape of IMG," in Infrared Spaceborne Remote Sensing IX, M. Strojnik and B. F. Andresen, eds., Proc. SPIE 4486, 326-333 (2002).
[CrossRef]

E. Niple, A. Pires, and K. Poultney, "Exact modeling of lineshape and wavenumber variations for off-axis detectors in Fourier transform spectrometers (FTS) sensor systems," in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R. J. Huppi, ed., Proc. SPIE 0364, 11-20 (1982).

Proc. SPIE. (1)

R. D. Kaiser, "Wavelength calibration and instrument line shape estimation of a LWIR hyperspectral sensor from in-scene data [4049-28]," in Algorithms for Multispectral, Hyperspectral, and Ultraspectral Imagery VI, S. S. Shen and M. R. Descour, eds., Proc. SPIE. 4049, 284-287 (2000).
[CrossRef]

SIAM J. Sci. Comput. (1)

D. H. Bailey and P. N. Swarztrauber, "A fast method for the numerical evaluation of continuous Fourier transforms," SIAM J. Sci. Comput. 15, 1105-1110 (1994).
[CrossRef]

SIAM Rev. (1)

D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," SIAM Rev. 33, 389-404 (1991).
[CrossRef]

Other (11)

D. H. Bailey and P. N. Swarztrauber, "Errata for: The fractional Fourier transform and applications," http://crd.1b1.gov/∼dhbailey/dhbpapers/.

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, P. Tremblay, R. Desbiens, and F. Bouffard, "Detailed line-shape measurements using a high resolution, high divergence Fourier transform spectrometer," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics series (Optical Society of America, 2003), pp. 25-27.

M. Chamberland, V. Farley, L. Belhumeur, F. Williams, J. Lawrence, P. Tremblay, and R. Desbiens, "The Instrument LineShape, an imperative parameter for the absolute spectral calibration of an FTS," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 160-166.

A. Vance, W. Bell, C. Paton-Walsh, and B. Jolliffe, "Fourier transform infra-red instrument line shape determination using an infra-red helium:neon laser and a low pressure gas cell," inPolar Stratospheric Ozone 1997, European Commission Air Pollution Research Rep. 66, Proceedings of the Fourth European Symposium on Polar Stratospheric Ozone, European Community (European Commission on Air Pollution, 1998), pp. 735-738.

F. Bouffard, P. Tremblay, R. Desbiens, and J.-P. Bouchard, "Synthetic spectra of Fourier transform spectrometers using analytically modeled instrument line shape inclusion," in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR) (Science and Technology Corporation, 2001), pp. 326-335.

J. Genest and P. Tremblay, "Impact of the optical aberrations on the line shape of Fourier-transform spectrometers," Vibrational Spectrosc. 29(1-2), 1-13 (2002).

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).

F. Bouffard, "Une approche matricielle pour la modélisation et la correction des effets instrumentaux en spectrométrie par transformation de Fourier," master's degree dissertation (Université Laval, 2002).

R. Desbiens, P. Tremblay, and J. Genest, "Matrix algorithm for integration and inversion of instrument line shape," in Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 42-44.

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

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Figures (9)

Fig. 1
Fig. 1

In the spectral domain, the ILS kernel spreads radiance toward lower wavenumbers. In the interferogram domain, it spreads the radiance toward higher OPDs.

Fig. 2
Fig. 2

The ILS stretches the true interferogram. On the optical axis, the apparent interferogram has its maximal average frequency. Off-axis rays see less OPD and therefore yield slower interferograms.

Fig. 3
Fig. 3

Piecewise integration of the instrument's line shape permits the use of higher-order numerical integration. Each piece of the ILS is a smooth function without discontinuity.

Fig. 4
Fig. 4

Duality of the interpolation in the spectral and interferogram domains. Interpolation artifacts show up near the maximal OPD and at high frequencies. Error toward the end of the vector in one domain gives rise to an oscillatory error in the other domain.

Fig. 5
Fig. 5

Examples of the interpolation error in the interferogram and in the spectrum. An ILS for a centered circular FOV with R∕f = 0.2 is integrated to a 513 point interferogram.

Fig. 6
Fig. 6

Interpolation error is reduced significantly after the application of the nonlinear filter sets a few points (16) to zero in the spectrum and the interferogram simultaneously. Centered circular FOV with R∕f = 0.2; 513 point interferogram.

Fig. 7
Fig. 7

Instrument line shapes measured with a laser for various highly divergent off-axis circular FOVs. Filled circles represent experimental measurements; solid curves are used for the integrated ILS model and for the difference between the model and the measurements (below on the same scale).

Fig. 8
Fig. 8

Measured and integrated HBr spectra compared. A 4.20 mm radius field stop is placed in the focal plane of a 550 mm parabola such that its center is 4.63 mm off axis. An offset of 20 is added to the difference between measured and integrated spectra.

Fig. 9
Fig. 9

Comparison of the measured and integrated HBr spectra for the P4 line. The field stop with a 4 mm radius is placed at six different locations in the FOV. An offset of 8 is added to the difference between measured and integrated spectra.

Tables (2)

Tables Icon

Table 1 Instrument FOV Parameters for Characterization of Laser Line Shapes a

Tables Icon

Table 2 Gas and Instrument FOV Parameters for Characterization of HBr Line Shapes a

Equations (44)

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

S m ( σ ) =     S 0 ( σ 0 ) ILS ( σ , σ 0 ) d σ 0 .
[ σ ] s m = [ σ 0 σ ILS ( σ , σ 0 ) ] ILS [ σ 0 ] s 0 ,
s m = ILS   s 0 .
ILS ( σ , σ 0 ) = 1 | σ 0 | H ( σ σ 0 ) .
ILS = [         ] .
S a ( σ ) =     I a ( x a ) exp ( i 2 π σ a x a ) d x a ,
[ σ ] s = [ x σ exp ( i 2 π σ x ) ] [ x ] i .
s 0 = i 0 ,     i 0 = 1 s 0 ,
s m = i m ,     i m = 1 s m .
i m = 1 s m       = 1 ILS   s 0
      = 1 ILS   1 s 0       = ILS ˜     i 0 ,
I m ( x ) =     I 0 ( x 0 ) ILS ˜ ( x , x 0 ) d x 0 ,
[ x ] i m = [ x 0 x ILS ˜ ( x , x 0 ) ] ILS ˜ [ x 0 ] i 0 .
1 | σ 0 | H ( σ σ 0 ) - 1 σ x f ( x σ 0 ) σ 0 x 0 1 | x | H ( x 0 x ) .
ILS ˜    ( x , x 0 ) = 1 | x | H ( x 0 x ) = ILS ( x 0 , x ) .
I m ( x ) =     I 0 ( x 0 ) ILS ( x 0 , x ) d x 0 ,
ILS ˜ = [ ] .
I 0 ( x 0 ) = n = I 0 ( n Δ x 0 ) sinc ( x 0 n Δ x 0 Δ x 0 ) .
I 0 ( x 0 ) n = N N I 0 ( n Δ x 0 ) D N ( x 0 n Δ x 0 Δ x 0 ) .
D N ( ξ ) 1 2 N + 1    n = N N exp [ i 2 π n ξ / ( 2 N + 1 ) ] = sinc ( ξ ) sinc ( ξ / 2 N + 1 ) .
I m ( x ) 1 | x | n = I 0 ( n Δ x 0 ) × [ H ( x 0 x ) D N ( x 0 n Δ x 0 Δ x 0 ) d x 0 ] .
x 0 / x = α .
I m ( x ) n = I 0 ( n Δ x 0 ) [ 0 1 H ( α ) D N ( α x n Δ x 0 Δ x 0 ) d α ] .
I m ( m Δ x 0 ) n = I 0 ( n Δ x 0 ) [ 0 1 H ( α ) D N ( α m n ) d α ] .
i m [ m ] = n = i 0 [ n ] ILS ˜ d [ m , n ] ,
ILS ˜ d [ m , n ] 0 1 H ( α ) D N ( α m n ) d α .
s [ k ] = n = N N i [ n ] exp [ i 2 π n k / ( 2 N + 1 ) ]
i [ n ] = 1 2 N + 1 k = N N s [ k ] exp [ i 2 π n k / ( 2 N + 1 ) ]
1 2 N + 1 m = N N n = N N exp [ i 2 π m j / ( 2 N + 1 ) ]     exp [ i 2 π n k / ( 2 N + 1 ) ] D N ( α m n ) = D N ( α k j ) ,
ILS d [ j , k ] 0 1 H ( α ) D N ( α k j ) d α ,
s m [ j ] k = N N s 0 [ k ] ILS d [ j , k ] .
   ILS ˜ d [ m , n ] i c i Δ α H ( α i ) D N ( α i m n ) .
i m [ n ] = a [ n ] + 1 2 N + 1 k b [ k ] exp [ i 2 π n k / ( 2 N + 1 ) ] ,
s m [ k ] = b [ k ] + n a [ n ] exp [ i 2 π n k / ( 2 N + 1 ) ] ,
i m [ n ] = i m [ n ] a [ n ] 1 2 N + 1 k b [ k ]    exp [ i 2 π n k / ( 2 N + 1 ) ] ,
s m [ k ] = s m [ k ] b [ k ] n a [ n ]     exp [ i 2 π n k / ( 2 N + 1 ) ] .
s m [ j ] = k = N N s 0 [ k ] ILS d [ j , k ]
            = k = N N s 0 [ k ] [ Δ α i c i H ( α i ) D N ( α i k j ) ] .
i m [ m ] = k = N N s 0 [ k ] [ Δ α 2 N + 1 i c i H ( α i ) exp ( i 2 π k m β i ) ] ,
            = Δ α 2 N + 1 i c i H ( α i ) [ k = N N s 0 [ k ] exp ( i 2 π k m β i ) ] ,
k = N N s 0 [ k ] exp ( i 2 π k m β i ) ,
2 k m = m 2 + k 2 ( m k ) 2 ,
i m [ m ] = Δ α 2 N + 1 i c i H ( α i ) ( k = N N s 0 [ k ] { exp i π [ m 2 + k 2 ( m - k ) 2 ] β i } )
      = Δ α 2 N + 1 i c i H ( α i ) exp ( i π m 2 β i ) × ( k = N N s 0 [ k ] exp ( i π k 2 β i ) exp [ i π ( m k ) 2 β i ] ) .

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