Abstract

An instrument line-shape correction method adapted to imaging Fourier- transform spectrometers is demonstrated. The method calibrates all pixels on the same spectral grid and permits a direct comparison of the spectral features between pixels such as emission or absorption lines. Computation speed is gained by using matrix line-shape integration formalism rather than properly inverting the line shape of each pixel. A monochromatic source is used to characterize the spectral shift of each pixel, and a line-shape correction scheme is then applied on measured interferograms. This work is motivated by the emergence of affordable infrared CCD cameras that are currently being integrated in imaging Fourier-transform spectrometers.

© 2007 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 line-shape 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).
  3. R. Desbiens, P. Tremblay, J. Genest, and J.-P. Bouchard, "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]
  4. R. Desbiens, J. Genest, P. Tremblay, and J.-P. Bouchard, "Correction of instrument line shape in Fourier transform spectrometry using matrix inversion," Appl. Opt. 45, 5270-5280 (2006).
    [CrossRef] [PubMed]
  5. 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]
  6. 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]
  7. R. C. M. Learner and A. P. Thorne, "Wavelength calibration of Fourier-transform emission spectra with applications to Fe I," J. Opt. Soc. Am. B 5, 2045-2049 (1988).
    [CrossRef]
  8. M. L. Salit, J. C. Travis, and M. R. Winchester, "Practical wavelength calibration considerations for UV-visible Fourier-transform spectroscopy," Appl. Opt. 35, 2960-2970 (1996).
    [CrossRef] [PubMed]
  9. L. I. Bluestein, "A linear filtering approach to the computation of discrete Fourier transform," Northeast Elec. Res. and Eng. Meeting Rec. 10, 218-219 (1968).
  10. L. R. Rabiner, R. W. Schafer, and C. M. Rader, "The chirp z-transform algorithm," IEEE Trans. Audio Electroacoust. 17, 8692-92 (1969).
    [CrossRef]
  11. A. Kuze, H. Nakajima, J. Tanii, and Y. Sasano, "Conceptual design of solar occultation FTS for inclined-orbit satellite (SOFIS) on GCOM-A1," in Infrared Spaceborne Remote Sensing VIII, Proc. SPIE 4131, 4541-4548 (2000).
  12. J.-P. Bouchard, P. Tremblay, R. Desbiens, and F. Bouffard, "Detailed line-shape measurements using a high resolution, high divergence Fourier transform spectrometer," Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 25-27.
  13. 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," Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 160-166.
  14. J. Genest and S. Potvin, "FFTmt," MEX-file available on MATLAB Central (http://www.mathworks.com/matlabcentral/fileexchange/) (2006).

2006 (2)

2003 (1)

2000 (1)

A. Kuze, H. Nakajima, J. Tanii, and Y. Sasano, "Conceptual design of solar occultation FTS for inclined-orbit satellite (SOFIS) on GCOM-A1," in Infrared Spaceborne Remote Sensing VIII, Proc. SPIE 4131, 4541-4548 (2000).

1999 (1)

1996 (1)

1988 (1)

1982 (1)

E. Niple, A. Pires, and K. Poultney, "Exact modeling of line-shape 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).

1969 (1)

L. R. Rabiner, R. W. Schafer, and C. M. Rader, "The chirp z-transform algorithm," IEEE Trans. Audio Electroacoust. 17, 8692-92 (1969).
[CrossRef]

1968 (1)

L. I. Bluestein, "A linear filtering approach to the computation of discrete Fourier transform," Northeast Elec. Res. and Eng. Meeting Rec. 10, 218-219 (1968).

1958 (1)

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

Appl. Opt. (5)

IEEE Trans. Audio Electroacoust. (1)

L. R. Rabiner, R. W. Schafer, and C. M. Rader, "The chirp z-transform algorithm," IEEE Trans. Audio Electroacoust. 17, 8692-92 (1969).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. Radium (1)

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

Northeast Elec. Res. and Eng. Meeting Rec. (1)

L. I. Bluestein, "A linear filtering approach to the computation of discrete Fourier transform," Northeast Elec. Res. and Eng. Meeting Rec. 10, 218-219 (1968).

Other (5)

E. Niple, A. Pires, and K. Poultney, "Exact modeling of line-shape 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).

A. Kuze, H. Nakajima, J. Tanii, and Y. Sasano, "Conceptual design of solar occultation FTS for inclined-orbit satellite (SOFIS) on GCOM-A1," in Infrared Spaceborne Remote Sensing VIII, Proc. SPIE 4131, 4541-4548 (2000).

J.-P. Bouchard, P. Tremblay, R. Desbiens, and F. Bouffard, "Detailed line-shape measurements using a high resolution, high divergence Fourier transform spectrometer," 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," Fourier Transform Spectroscopy, Vol. 84 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. 160-166.

J. Genest and S. Potvin, "FFTmt," MEX-file available on MATLAB Central (http://www.mathworks.com/matlabcentral/fileexchange/) (2006).

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Figures (11)

Fig. 1
Fig. 1

(Color online) Line shapes of rectangular pixels near and off axis. When pixels are very small, the FOV contribution to the ILS is narrower than the sinc from interferogram truncation shown here as dotted curves.

Fig. 2
Fig. 2

(Color online) Illustration of the proposed two-step ILS removal procedure. (a) First, a shifted Dirac function is added to the near-axis pixel interferogram. This steps aligns the near-axis line shape peak with off-axis pixel line shape. (b) Spectra of both pixels are then rescaled to realign the line shapes with unitary normalized wavenumber.

Fig. 3
Fig. 3

(Color online) Residuals at the interferogram extremity after shifting the ILS by integrating a Dirac delta function in an ideal interferogram. A positive shift implies shifting toward lower frequencies, and a negative shift is applied toward higher frequencies.

Fig. 4
Fig. 4

(Color online) Nominal wavenumber distribution for a monochromatic source as measured by an imaging FTS with 200 × 200 pixels for a laser line estimated at 6519.7581 cm 1 ( 1533.8 nm ) .

Fig. 5
Fig. 5

(Color online) Experimental setup for near-infrared line shape characterization and correction.

Fig. 6
Fig. 6

(Color online) Line shapes of three pixels before correction.

Fig. 7
Fig. 7

(Color online) Line shapes of the same three pixels after correction.

Fig. 8
Fig. 8

(Color online) (a) Nominal wavenumber distribution before correction. (b) Nominal wavenumber distribution obtained after intrameasurement calibration. The mean wavenumber ( 6519.7515 cm 1 ) has been subtracted, and a standard deviation of 7 × 10 6 cm 1 yields 1 ppb precision.

Fig. 9
Fig. 9

(Color online) (a) Same nominal wavenumber distribution before correction as in Fig. 8. (b) Wavenumber distribution at approximately mean value 6519.748 cm 1 obtained after intermeasurement calibration. A standard deviation of 3 × 10 3 cm 1 yields 0.5 ppm precision.

Fig. 10
Fig. 10

(Color online) Throughput of the correction algorithm as a function of interferogram length on a single processor.

Fig. 11
Fig. 11

(Color online) Overall postprocessing times obtained for a 200 × 200 × 18, 963 datacube (1.5 Gbytes) on four processors compared with the actual measurement time.

Equations (11)

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s m [ j ] = k = N N s 0 [ k ] ILS d [ j , k ] ,
ILS d [ j , k ] = 0 1 H ( σ ¯ ) D N ( σ ¯ k j ) δ σ ¯ ,
D N ( ξ ) = 1 2 N + 1 n = - N N exp [ i 2 π n ξ 2 N + 1 ] .
s γ ¯ [ j ] = k = - N N s m [ k ] ILS d [ j , k ] .
H ( σ ¯ ) = δ ( σ ¯ γ ¯ ) ,
s γ ¯ [ j ] = k = N N s m [ k ] 0 1 δ ( σ ¯ γ ¯ ) D N ( σ ¯ k j ) δ σ ¯ ,
= k = N N s m [ k ] D N ( γ ¯ k j ) ,
= k = N N s m [ k ] 1 2 N + 1 n = N N exp [ i 2 π m ( γ ¯ k j ) 2 N + 1 ] .
i γ ¯ [ m ] = 1 2 N + 1 k = N N s m [ k ] exp [ i 2 π k m β ] ,
i γ ¯ [ m ] = 1 2 N + 1 exp ( i π m 2 β ) × [ k = N N s m [ k ] exp ( i π k 2 β ) × exp ( i π ( m k ) 2 β ) ] .
γ ¯ ( x , y ) = 1 σ min σ ( x , y ) .

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