Abstract

A new high-accuracy method has been developed to transform asymmetric single-sided interferograms into spectra. We used a fraction (short, double-sided) of the recorded interferogram and applied an iterative correction to the complete recorded interferogram for the linear part of the phase induced by the various optical elements. Iterative phase correction enhanced the symmetry in the recorded interferogram. We constructed a symmetric double-sided interferogram and followed the Mertz procedure [Infrared Phys. 7, 17 (1967)] but with symmetric apodization windows and with a nonlinear phase correction deduced from this double-sided interferogram. In comparing the solution spectrum with the source spectrum we applied the Rayleigh resolution criterion with a Gaussian instrument line shape. The accuracy of the solution is excellent, ranging from better than 0.1% for a blackbody spectrum to a few percent for a complicated atmospheric radiance spectrum.

© 2002 Optical Society of America

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References

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  1. P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, New York, 1986).
  2. D. B. Chase, “Phase correction in FT-IR,” Appl. Spectrosc. 36, 240–244 (1982).
    [CrossRef]
  3. H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, 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]
  4. R. C. M. Learner, A. P. Thorne, I. Wynne-Jones, J. W. Brault, M. C. Abrams, “Phase correction of emission line Fourier transform spectra,” J. Opt. Soc. Am. A 12, 2165–2171 (1995).
    [CrossRef]
  5. K. Rahmelow, W. Hubner, “Phase correction in Fourier transform spectroscopy: subsequent displacement correction and error limit,” Appl. Opt. 36, 6678–6686 (1997).
    [CrossRef]
  6. C. D. Barnet, J. M. Blaisdell, J. Susskind, “Practical method for rapid and accurate computation of interferometric spectra for remote sensing application,” IEEE Trans. Geosci. Remote Sens. 38, 169–183 (2000).
    [CrossRef]
  7. L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7, 17–23 (1967).
    [CrossRef]
  8. M. L. Forman, W. H. Steel, G. A. Vanasse, “Correction of asymmetric interferograms obtained in Fourier spectroscopy,” J. Opt. Soc. Am. 56, 59–63 (1966).
    [CrossRef]
  9. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]
  10. Based on probability theory, the probability of measuring a signal with a Gaussian probability function g(ν, ν1 = Δ) in the presence of zero-mean noise with probability-density function g(ν, ν1 = 0) is ∫Δ/2∞g(ν, Δ)dν = 90% when a threshold of detection ν ≥ Δ/2 is chosen.
  11. A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
    [CrossRef]

2000 (1)

C. D. Barnet, J. M. Blaisdell, J. Susskind, “Practical method for rapid and accurate computation of interferometric spectra for remote sensing application,” IEEE Trans. Geosci. Remote Sens. 38, 169–183 (2000).
[CrossRef]

1997 (1)

1995 (1)

1988 (1)

1982 (1)

1978 (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

1967 (1)

L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7, 17–23 (1967).
[CrossRef]

1966 (1)

Abrams, M. C.

Acharya, P. K.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Adler-Golden, S. M.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Allred, C. L.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Anderson, G. P.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Barnet, C. D.

C. D. Barnet, J. M. Blaisdell, J. Susskind, “Practical method for rapid and accurate computation of interferometric spectra for remote sensing application,” IEEE Trans. Geosci. Remote Sens. 38, 169–183 (2000).
[CrossRef]

Berk, A.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Bernstein, L. S.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Blaisdell, J. M.

C. D. Barnet, J. M. Blaisdell, J. Susskind, “Practical method for rapid and accurate computation of interferometric spectra for remote sensing application,” IEEE Trans. Geosci. Remote Sens. 38, 169–183 (2000).
[CrossRef]

Brault, J. W.

Buijs, H.

Chase, D. B.

Chetwynd, J. H.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

de Haseth, J. A.

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, New York, 1986).

Dothe, H.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Forman, M. L.

Griffiths, P. R.

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, New York, 1986).

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Hoke, M. L.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Howell, H. B.

Hubner, W.

Jeong, L. S.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

LaPorte, D. D.

Learner, R. C. M.

Matthew, M. W.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Mertz, L.

L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7, 17–23 (1967).
[CrossRef]

Pukall, B.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Rahmelow, K.

Revercomb, H. E.

Richtsmeier, S. C.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Smith, W. L.

Sromovsky, L. A.

Steel, W. H.

Susskind, J.

C. D. Barnet, J. M. Blaisdell, J. Susskind, “Practical method for rapid and accurate computation of interferometric spectra for remote sensing application,” IEEE Trans. Geosci. Remote Sens. 38, 169–183 (2000).
[CrossRef]

Thorne, A. P.

Vanasse, G. A.

Wynne-Jones, I.

Appl. Opt. (2)

Appl. Spectrosc. (1)

IEEE Trans. Geosci. Remote Sens. (1)

C. D. Barnet, J. M. Blaisdell, J. Susskind, “Practical method for rapid and accurate computation of interferometric spectra for remote sensing application,” IEEE Trans. Geosci. Remote Sens. 38, 169–183 (2000).
[CrossRef]

Infrared Phys. (1)

L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7, 17–23 (1967).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Proc. IEEE (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Other (3)

Based on probability theory, the probability of measuring a signal with a Gaussian probability function g(ν, ν1 = Δ) in the presence of zero-mean noise with probability-density function g(ν, ν1 = 0) is ∫Δ/2∞g(ν, Δ)dν = 90% when a threshold of detection ν ≥ Δ/2 is chosen.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, New York, 1986).

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Figures (10)

Fig. 1
Fig. 1

Single-sided asymmetric interferogram I(-x minxx max) computed for blackbody source B(500 < ν < 1500 cm-1) at a temperature of 300 K. Short, double-sided interferogram I s (x) = I(-x minxx min) and the location of center burst x 0 are shown. Only the modulated part (ac) of the interferogram that contains information on the spectrum is shown, and the constant (dc) component is omitted.

Fig. 2
Fig. 2

Sinc(100x) with -x minxx max, exact shifted function sinc[100(x - Δ)], and approximations I(x - Δ) from the shift theorem, where I(x - Δ) computed with Eq. (5) using single-sided and reconstructed double-sided I(-xminxx max ) [Eq. (7)] is used. The residual ∊(x) = I(x - Δ) - sinc(x - Δ) shows the improved accuracy when Eq. (7) is used. The centerburst at x = 0 is not sampled on this x-axis grid and is located between two adjacent values of x. Δ = 0.075 + δ/4, where δ = 10-4 is the interval between adjacent points.

Fig. 3
Fig. 3

Asymmetrical hybrid apodization windows. x min = 5% of the window width (100 points). (a) Time domain where a ramp function is substituted for the range -x minxx min. Triangle, Blackman, Kaiser, and Chebyshev apodization windows for the range x minxx max. (b) Power spectral density magnitude 10 log10{|F[ w(x)]|2} in the frequency domain where the Nyquist frequency is 1. Attenuation R = 100 dB for Chebyshev and Kaiser windows.

Fig. 4
Fig. 4

Symmetrical apodization windows (x min = x max) for which the peak location of the window (in the time domain) is at the middle (a) in the time domain and (b) in the frequency domain. Attenuation R = 100 dB for Chebyshev and Kaiser windows.

Fig. 5
Fig. 5

Rayleigh resolution criterion implemented with Gaussian functions. g(ν) = g(ν, ν 1) + g(ν, ν 1 + Δ) is the combined spectrum of two monochromatic sources, g1 and g2, at frequencies ν 1 = 1000 cm-1 and ν 2 = ν 1 + Δ separated by Δ = 4 cm-1.

Fig. 6
Fig. 6

Spectrum B(ν) W/(cm2 sr cm-1). (a) Planck function at a temperature of 300 K; (b) Planck function at a temperature of 300 K with five Lorentzian absorption lines of 10-cm-1 FWHM and transmissions 0.1, 0.2, 0.3, 0.4, and 0.5 at frequencies 600, 800, 1000, 1200 and 1400 cm-1, respectively; (c) atmospheric radiance (1-cm-1 resolution) computed with the modtran program for a 1976 U.S. Standard Atmosphere for an observer on the ground looking up. Wavelength range, 510–1500 cm-1.

Fig. 7
Fig. 7

Nonlinear phase error Φ1(ν) rad (obtained from a FTIR instrument) representing the dispersive optical elements in a spectrometer.

Fig. 8
Fig. 8

Radiance spectrum L(ν) W/(cm2 sr cm-1) [Eq. (11)], with Δ = 4 cm-1 for B(ν), and solution spectrum S(ν) W/(cm2 sr cm-1) (top). Residual percent difference 100|L(ν) - S(ν)|/L(ν) (middle). Absolute radiance difference |L(ν) - S(ν)| W/(cm2 sr cm-1) (bottom). Solution spectrum S(ν) is derived for Planck function B(ν) [Fig. 6(a)] for a Chebyshev apodizing window with R = 100 dB.

Fig. 9
Fig. 9

Same as Fig. 8 but here source spectrum B(ν) [Fig. 6(b)] is a Planck function with five Lorentzian absorption peaks (10-cm-1 FWHM) with transmissions of 0.1, 0.2, 0.3, and 0.5 at frequencies 600, 800, 1000, 1200, and 1400 cm-1. Two spectra, S1(ν) and S2(ν) (computed with a Chebyshev window with R = 100 dB), are shown. For S2(ν) the double-sided phase-corrected interferogram was used, and for S1(ν) zeros were substituted for the missing data -x maxx ≤ -x min in phase-corrected interferogram Id(x) [Eq. (7)].

Fig. 10
Fig. 10

Same as Fig. 8 but here source spectrum B(ν) [Fig. 6(c)] is atmospheric radiance (1-cm-1 resolution) computed with modtran and S(ν) is computed with a Kaiser window with R = 50 dB.

Equations (14)

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Ix=ν1ν2 Kν0.5Bνcos2πνx+Φ(ν)dν,
Φν=Φ1ν+Φ2ν,
Φ2ν=-2πνx0,
Ix= KνBνcos2πxνdν=F-1KνBν,Bν=Kν-1 Ixcos2πxνdx=Kν-1FIx,
Δδ= i=12n0-1 i|Isxi|i=12n0-1 |Isxi| -n0,
Ix-Δ= Iαδα-x-Δdα=  Iαδx-Δ-αdα
Ix-Δ=F-1FIxFδx-Δ= F-1FIxexp-j2πνΔ,
ν=νννNyquist-2νNyquist-νν>νNyquist,
Idx= I-x-xmaxx-xminIx-xmin<xxmax.
nν= 0.5n+m0+1n+m0even0.5n+m0+1n+m0odd,
Bˆν=2 FIdw/Kνndnd2νNyquist =2δFIdw/Kν FIdw/Kνnd2xmax-1/2W/ cm2 sr cm-1,
Sν=ReBν =ReBˆνcosΦˆ1ν+ImBˆνsinΦˆ1ν.
gν, ν1=exp-0.5ν-ν12/Δ/2.63822π0.5Δ/2.638,
Lν=0 Btgt, νdt =0 B(t) exp-0.5t-ν2/Δ/2.6382)2π0.5Δ/2.638dt.

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