Abstract

A radiometric model of the instrument line shape (ILS) of Fourier-transform spectrometers is presented. We show first that common line-shape models are based on distribution of the radiant intensity in the interferometer. The complete steps between the source and the ILS are exposed as the core of the model. Relationships between the ILS, the spectrum as measured by the instrument, and the spectrum as emitted by the scene are demonstrated from the ILS model. Then the formal radiometric modeling of the ILS is derived, including the contribution of the aperture of the optical system. The particular case of a centered circular aperture with a uniform Lambertian radiance in the field of view is discussed. Conditions are deduced to ensure that the only spectral variation of the ILS is a scaling with wave number, as is usually assumed in current line-shape models. The ILS dependence on the scene is also discussed, and the effect of taking into account the radiometry on the ILS is estimated for the case of an ideal thin lens used as a collimator.

© 2002 Optical Society of America

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References

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  1. T. Ogawa, ed., Proceedings of the Ninth International Workshop on Atmospheric Science from Space Using Fourier Transform Spectrometry, Kyoto, Japan, May 2000 (National Space Development Agency of Japan, Kyoto, 2000).
  2. C. L. Bennet, M. R. Carter, D. J. Fields, “Hyperspectral imaging in the infrared using LIFTIRS,” in Infrared Technology XXI, B. F. Andersen, M. S. Scholl, eds., Proc. SPIE2552, 274–283 (1995).
    [CrossRef]
  3. M. J. Persky, “A review of spaceborne infrared Fourier transform spectrometers,” Rev. Sci. Instrum. 66, 4763–4797 (1995).
    [CrossRef]
  4. J. Genest, P. Tremblay, “Instrument line shape of Fourier transform spectrometers: analytic solutions for nonuniformly illuminated off-axis detectors,” Appl. Opt. 38, 5438–5446 (1999).
    [CrossRef]
  5. K. W. Bowman, H. M. Worden, R. Beer, “Instrument line-shape modeling and correction for off-axis detectors in Fourier-transform spectrometry,” Appl. Opt. 39, 3765–3773 (2000).
    [CrossRef]
  6. J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, R. S. Booth, J. W. Brault, A. Labeyrie, eds., in Proceedings of the 15th Advanced Course of the Swiss Society of Astronomy and Astrophysics(Swiss Society of Astronomy and Astrophysics, Saas-Fee, Switzerland, 1985),pp. 1–61.
  7. E. Niple, A. Pires, K. Poultney, “Exact modeling of lineshape and wave-number 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. SPIE364, 11–20 (1982).
    [CrossRef]
  8. J. Connes, “Domaine d’utilisation de la méthode par transformée de Fourier,” J. Phys. Radium 19, 197–208 (1958).
    [CrossRef]
  9. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, UK, 1995).
    [CrossRef]
  10. B. K. Yap, W. A. M. Blumberg, R. E. Murphy, “Off-axis effect in a mosaic Michelson interferometer,” Appl. Opt. 21, 4176–4182 (1982).
    [CrossRef] [PubMed]
  11. P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, New York, 1986).
  12. J. Kauppinen, P. Saarinen, “Line-shape distortions in misaligned cube corners interferometers,” Appl. Opt. 31, 69–74 (1992).
    [CrossRef] [PubMed]
  13. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, the Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, UK, 1997).
  14. J. Genest, “Théorie de la cohérence optique et modélisation de la forme de raie des spectromètres par transformation de Fourier,”Ph.D. dissertation (Université Laval, Québec, Canada, 2001).
  15. M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, London, 1993).
  16. D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985).

2000

1999

1995

M. J. Persky, “A review of spaceborne infrared Fourier transform spectrometers,” Rev. Sci. Instrum. 66, 4763–4797 (1995).
[CrossRef]

1992

1982

1958

J. Connes, “Domaine d’utilisation de la méthode par transformée de Fourier,” J. Phys. Radium 19, 197–208 (1958).
[CrossRef]

Beer, R.

Bennet, C. L.

C. L. Bennet, M. R. Carter, D. J. Fields, “Hyperspectral imaging in the infrared using LIFTIRS,” in Infrared Technology XXI, B. F. Andersen, M. S. Scholl, eds., Proc. SPIE2552, 274–283 (1995).
[CrossRef]

Blumberg, W. A. M.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, London, 1993).

Bowman, K. W.

Brault, J. W.

J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, R. S. Booth, J. W. Brault, A. Labeyrie, eds., in Proceedings of the 15th Advanced Course of the Swiss Society of Astronomy and Astrophysics(Swiss Society of Astronomy and Astrophysics, Saas-Fee, Switzerland, 1985),pp. 1–61.

Carter, M. R.

C. L. Bennet, M. R. Carter, D. J. Fields, “Hyperspectral imaging in the infrared using LIFTIRS,” in Infrared Technology XXI, B. F. Andersen, M. S. Scholl, eds., Proc. SPIE2552, 274–283 (1995).
[CrossRef]

Connes, J.

J. Connes, “Domaine d’utilisation de la méthode par transformée de Fourier,” J. Phys. Radium 19, 197–208 (1958).
[CrossRef]

de Haseth, J. A.

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

Fields, D. J.

C. L. Bennet, M. R. Carter, D. J. Fields, “Hyperspectral imaging in the infrared using LIFTIRS,” in Infrared Technology XXI, B. F. Andersen, M. S. Scholl, eds., Proc. SPIE2552, 274–283 (1995).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, the Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, UK, 1997).

Genest, J.

J. Genest, 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, “Théorie de la cohérence optique et modélisation de la forme de raie des spectromètres par transformation de Fourier,”Ph.D. dissertation (Université Laval, Québec, Canada, 2001).

Griffiths, P. R.

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

Kauppinen, J.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, UK, 1995).
[CrossRef]

Murphy, R. E.

Niple, E.

E. Niple, A. Pires, K. Poultney, “Exact modeling of lineshape and wave-number 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. SPIE364, 11–20 (1982).
[CrossRef]

O’Shea, D. C.

D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985).

Persky, M. J.

M. J. Persky, “A review of spaceborne infrared Fourier transform spectrometers,” Rev. Sci. Instrum. 66, 4763–4797 (1995).
[CrossRef]

Pires, A.

E. Niple, A. Pires, K. Poultney, “Exact modeling of lineshape and wave-number 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. SPIE364, 11–20 (1982).
[CrossRef]

Poultney, K.

E. Niple, A. Pires, K. Poultney, “Exact modeling of lineshape and wave-number 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. SPIE364, 11–20 (1982).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, the Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, UK, 1997).

Saarinen, P.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, the Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, UK, 1997).

Tremblay, P.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, the Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, UK, 1997).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, London, 1993).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, UK, 1995).
[CrossRef]

Worden, H. M.

Yap, B. K.

Appl. Opt.

J. Phys. Radium

J. Connes, “Domaine d’utilisation de la méthode par transformée de Fourier,” J. Phys. Radium 19, 197–208 (1958).
[CrossRef]

Rev. Sci. Instrum.

M. J. Persky, “A review of spaceborne infrared Fourier transform spectrometers,” Rev. Sci. Instrum. 66, 4763–4797 (1995).
[CrossRef]

Other

J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, R. S. Booth, J. W. Brault, A. Labeyrie, eds., in Proceedings of the 15th Advanced Course of the Swiss Society of Astronomy and Astrophysics(Swiss Society of Astronomy and Astrophysics, Saas-Fee, Switzerland, 1985),pp. 1–61.

E. Niple, A. Pires, K. Poultney, “Exact modeling of lineshape and wave-number 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. SPIE364, 11–20 (1982).
[CrossRef]

T. Ogawa, ed., Proceedings of the Ninth International Workshop on Atmospheric Science from Space Using Fourier Transform Spectrometry, Kyoto, Japan, May 2000 (National Space Development Agency of Japan, Kyoto, 2000).

C. L. Bennet, M. R. Carter, D. J. Fields, “Hyperspectral imaging in the infrared using LIFTIRS,” in Infrared Technology XXI, B. F. Andersen, M. S. Scholl, eds., Proc. SPIE2552, 274–283 (1995).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, Cambridge, UK, 1995).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, the Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, UK, 1997).

J. Genest, “Théorie de la cohérence optique et modélisation de la forme de raie des spectromètres par transformation de Fourier,”Ph.D. dissertation (Université Laval, Québec, Canada, 2001).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, London, 1993).

D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985).

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

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Figures (6)

Fig. 1
Fig. 1

Relation between the solid angle element dΩ and the angles of the beam (θ, φ). The integration range Ω D can be split into a range over φ (φ D ) and a range over θ (θ D ).

Fig. 2
Fig. 2

(a) True spectrum S 00) of a uniform Lambertian source of 11 lines from 990 to 1000 cm-1. (b) Instrument line shapes ILS(σ, σ0) corresponding to the lines of the source for a square off-axis detector of 150 µm × 150 µm centered in the FOV at x = 250 µm and y = 200 µm and a circular-centered aperture of parameter ρ = 0.4. (c) Measured spectrum S m (σ) given by the spectrometer. (d) Estimated spectrum Ŝ m (σ) obtained by a Fourier transform of the finite-length interferogram (ΔX max = 4 cm).

Fig. 3
Fig. 3

Relationship between propagation angle θ and the position in the FOV (r, φ) with a collimator focal length f.

Fig. 4
Fig. 4

FOV (r, φ) and aperture (r A , φ A ) coordinates for computation of the radiant intensity I from the radiance of the source B after collimation of the FOV.

Fig. 5
Fig. 5

Circular centered aperture factor F A (σ̅), normalized to F A (1) for aperture radius to focal length ratios ρ = 0.01, 0.25, 0.4, 0.5, 1, 2, and 5. This factor represents the function that multiplies the ILS due to FOV only. The factor F A (σ̅) is well describe by a first-order relation with a very small slope.

Fig. 6
Fig. 6

Correction (parts per million) added by a circular aperture to the first moment of the ILS for a circular centered FOV uniformly illuminated (boxcar ILS) for different aperture radius to focal-length ratios ρ = 0.01, 0.25, 0.4, 0.5, 1, 2, and 5 as a function of the minimal normalized wave number detected σ̅min. The divergence of the beam is given by θmax = arccos σ̅min. We can see that the difference between the first moment of an ILS due to FOV only and an ILS taking into account the aperture increases for a wider boxcar ILS.

Equations (47)

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PΔX=σ0 P0ΔX, σ0dσ0,
P0ΔX, σ0=ΩD Iθ, φ; σ0×1+cos2πσ0ΔX cos θdΩ,
Sσ, σ0=cosΩD Iθ, φ; σ0cos2πσ0ΔX cos θdΩ,=ΩD Iθ, φ; σ0coscos2πσ0ΔX cos θdΩ,
=ΩD Iθ, φ; σ0δσ-σ0 cos θdΩ,
cosgΔX=2 0 gΔXcos2πσΔXdΔX
Sσ, σ0=θminθmaxφDθ Iθ, φ; σ0×δσ-σ0 cos θsin θdφdθ.
Sσ, σ0=1σ0σ0 cos θmaxσ0 cos θminφDarccosξ/σ0Iarccosξ/σ0, φ; σ0δσ-ξdφdξ.
Sσ, σ0=1σ0φDθ Iθ, φ; σ0dφθ=arccosσ/σ0σ0 cos θmaxσσ0 cos θmin0elsewhere,
S0σ0=σ Sσ, σ0dσ.
ILSσ, σ0=Sσ, σ0S0σ0, S0σ0>0.
S0σ0=σ Sσ, σ0dσ.
Smσ=σ0 Sσ, σ0dσ0.
Smσ=σ0 S0σ0ILSσ, σ0dσ0.
Ŝmσ=sincσΔXmax * σ0 S0σ0ILSσ, σ0dσ0.
Ŝmσ=σ0 S0σ0sincσΔXmax * ILSσ, σ0dσ0,
Ŝmσ=σ0 S0σ0ILSˆσ, σ0dσ0.
Er, φ; σ0=A Br, φ; α; σ0cos2 αdAR2,
cos α=ff2+r2+rA2-2rrA cosφ-φA1/2.
Er, φ; σ0=1f2A Br, φ; α; σ0cos4 αdA.
dP=EdS=IdΩ.
dS=rdrdφ=f2 tan θ sec2 θdθdφ=f2cos3 θsin θdθdφ=f2cos3 θdΩ.
I=f2cos3 θ E.
Iθ, φ; σ0=1cos3 θA Br, φ; α; σ0cos4 αdAr=f tan θ.
Sσ, σ0=1σ0φDθA Br, φ; α; σ0cos4 αcos3 θdAdφr=f tan θθ=arccosσ/σ0σ0 cos θmaxσσ0 cos θmin0elsewhere.
Br, φ; α; σ0Bσ0,
Iθ, φ; σ0=Bσ0cos3 θAcos4 αdAr=f tan θ.
Acos4 αdAr=f tan θ=πf221+RA2f2-sec2 θRA2f2-sec2 θ2+4 RA2f21/2.
Iθ, φ; σ0Iθ; σ0.
Sσ, σ0=Iθ; σ0σ0φDθdφθ=arccos(σ/σ0)σ0 cos θmaxσσ0 cos θmin0elsewhere.
FAσ¯=1cos3 θAcos4 αdAr=f tan θθ=arccos σ¯,
Fφσ¯=φDθdφθ=arccos σ¯cos θmaxσ¯cos θmin0,elsewhere,
FN=01 FA(σ¯)Fφ(σ¯)dσ¯.
Sσ, σ0=1σ0 Bσ0FAσ¯Fφσ¯,
S0σ0=Bσ0FN.
ILSσ, σ0=1σ0FAσ¯Fφσ¯FN,
=1σ0×aperture factor×FOV factor÷normalization factor.
Br, φ; α; σ0gr, φ; αhσ0.
M=01 σ¯ILSσ, σ0dσ¯01ILSσ, σ0dσ¯,
FAσ¯=πf22σ¯31+ρ2-1/σ¯2ρ2-1/σ¯22+4ρ21/2.
ILSσ, σ0=1σ01FN FAσ¯Fφσ¯1σ01FNc1+c2σ¯Fφσ¯,
c1=FAσ¯|σ¯=cos θmin-c2 cos θmin,
c2=FAσ¯|σ¯=cos θmin-FAσ¯|σ¯=cos θmaxcos θmin-cos θmax.
Mc1FN01 σF¯φσ¯dσ¯+c2FN01 σ¯2Fφσ¯dσ¯,
FNc101 Fφσ¯dσ¯+c201 σF¯φσ¯dσ¯.
M=01 σF¯φσ¯dσ¯01 Fφσ¯dσ¯,
V=01 σ¯2Fφσ¯dσ¯01 Fφσ¯dσ¯-M2.
ΔM=M-M=c2Vc1+c2M.

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