Abstract

A simple and powerful method for obtaining analytic instrument line shapes (ILS’s) for Fourier transform spectrometers is explained. ILS’s for off-axis circular and rectangular detectors are calculated to illustrate the method. Results match earlier ILS simulations. The contribution of the nonuniformity of light intensity across the detector surface is also taken into account.

© 1999 Optical Society of America

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References

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  1. G. A. Vanasse, H. Sakai, “Fourier spectroscopy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. 6, Chap. 7, pp. 259–330.
    [CrossRef]
  2. E. Niple, A. Pires, K. Poultney, “Exact modeling of line-shape and wave-number variations for off-axis detectors in Fourier transform spectrometer (FTS) sensor systems,” in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R. J. Huppi, ed., Proc. SPIE364, 11–20 (1982).
    [CrossRef]
  3. 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]
  4. J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, Proceedings of the 15th Advanced Course of the Swiss Society of Astronomy and Astrophysics, A. O. Benz, M. C. E. Huber, M. Mayor, eds. (Swiss Society of Astronomy and Astrophysics, Saas-Fee, 1985), pp. 1–61.
  5. J. Connes, “Domaine d’utilisation de la Méthode par Transforme de Fourier,” J. Phys. Rad. 19, 197–208 (1958).
    [CrossRef]
  6. P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley, New York, 1986).
  7. J. Giroux, A. Villemaire, “Off-axis effects in image formation,” in Proceedings of the Fifth International Workshop on Athmospheric Science from Space by Using Fourier Transform Spectroscopy (Japan Resource Observation System Organization, Tokyo, 1994), pp. 331–352.
  8. J. Kauppinen, P. Saarinen, “Line-shape distortions in misaligned cube corner interferometers,” Appl. Opt. 31, 69–74 (1992).
    [CrossRef] [PubMed]
  9. P. Saarinen, J. Kauppinen, “Spectral line-shape distortions in Michelson interferometers due to off-focus radiation source,” Appl. Opt. 31, 2353–2359 (1992).
    [CrossRef] [PubMed]

1992 (2)

1982 (1)

1958 (1)

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

Blumberg, W. A. M.

Brault, J. W.

J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, Proceedings of the 15th Advanced Course of the Swiss Society of Astronomy and Astrophysics, A. O. Benz, M. C. E. Huber, M. Mayor, eds. (Swiss Society of Astronomy and Astrophysics, Saas-Fee, 1985), pp. 1–61.

Connes, J.

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

de Haseth, J. A.

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

Giroux, J.

J. Giroux, A. Villemaire, “Off-axis effects in image formation,” in Proceedings of the Fifth International Workshop on Athmospheric Science from Space by Using Fourier Transform Spectroscopy (Japan Resource Observation System Organization, Tokyo, 1994), pp. 331–352.

Griffiths, P. R.

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

Kauppinen, J.

Murphy, R. E.

Niple, E.

E. Niple, A. Pires, K. Poultney, “Exact modeling of line-shape and wave-number variations for off-axis detectors in Fourier transform spectrometer (FTS) sensor systems,” in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R. J. Huppi, ed., Proc. SPIE364, 11–20 (1982).
[CrossRef]

Pires, A.

E. Niple, A. Pires, K. Poultney, “Exact modeling of line-shape and wave-number variations for off-axis detectors in Fourier transform spectrometer (FTS) sensor systems,” in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R. J. Huppi, ed., Proc. SPIE364, 11–20 (1982).
[CrossRef]

Poultney, K.

E. Niple, A. Pires, K. Poultney, “Exact modeling of line-shape and wave-number variations for off-axis detectors in Fourier transform spectrometer (FTS) sensor systems,” in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R. J. Huppi, ed., Proc. SPIE364, 11–20 (1982).
[CrossRef]

Saarinen, P.

Sakai, H.

G. A. Vanasse, H. Sakai, “Fourier spectroscopy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. 6, Chap. 7, pp. 259–330.
[CrossRef]

Vanasse, G. A.

G. A. Vanasse, H. Sakai, “Fourier spectroscopy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. 6, Chap. 7, pp. 259–330.
[CrossRef]

Villemaire, A.

J. Giroux, A. Villemaire, “Off-axis effects in image formation,” in Proceedings of the Fifth International Workshop on Athmospheric Science from Space by Using Fourier Transform Spectroscopy (Japan Resource Observation System Organization, Tokyo, 1994), pp. 331–352.

Yap, B. K.

Appl. Opt. (3)

J. Phys. Rad. (1)

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

Other (5)

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

J. Giroux, A. Villemaire, “Off-axis effects in image formation,” in Proceedings of the Fifth International Workshop on Athmospheric Science from Space by Using Fourier Transform Spectroscopy (Japan Resource Observation System Organization, Tokyo, 1994), pp. 331–352.

J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, Proceedings of the 15th Advanced Course of the Swiss Society of Astronomy and Astrophysics, A. O. Benz, M. C. E. Huber, M. Mayor, eds. (Swiss Society of Astronomy and Astrophysics, Saas-Fee, 1985), pp. 1–61.

G. A. Vanasse, H. Sakai, “Fourier spectroscopy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. 6, Chap. 7, pp. 259–330.
[CrossRef]

E. Niple, A. Pires, K. Poultney, “Exact modeling of line-shape and wave-number variations for off-axis detectors in Fourier transform spectrometer (FTS) sensor systems,” in Technologies of Cryogenically Cooled Sensors and Fourier Transform Spectrometers II, R. J. Huppi, ed., Proc. SPIE364, 11–20 (1982).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Off-axis plane wave in a Michelson interferometer.

Fig. 2
Fig. 2

Zoom of Fig. 1.

Fig. 3
Fig. 3

Boxcar ILS obtained for a circular centered detector.

Fig. 4
Fig. 4

Collimating lens.

Fig. 5
Fig. 5

Schematic of the proposed method for calculating the ILS.

Fig. 6
Fig. 6

Geometry for an off-axis circular detector.

Fig. 7
Fig. 7

ILS’s for various positions of an off-axis circular detector such that R = 0.15f. The frequency scale is normalized to ν/ν 0.

Fig. 8
Fig. 8

Two off-axis rectangular detectors.

Fig. 9
Fig. 9

Calculation of the intercepted angle ϕ from r min to r c1 for a rectangular detector.

Fig. 10
Fig. 10

ILS’s for various positions of an off-axis square pixel such that A = B = 0.15f. The frequency scale is normalized to ν/ν 0.

Fig. 11
Fig. 11

ILS for an on-axis square pixel such that A = B = 0.15f. The frequency scale is normalized to ν/ν 0.

Fig. 12
Fig. 12

ILS’s for a Gaussian nonuniform source with an on-axis circular detector such that R = 0.15f. The frequency scale is normalized to ν/ν 0.

Tables (3)

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Table 1 Instrument Line Shape for an Off-axis Circular Detector

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Table 2 Instrument Line Shape for an Off-Axis Rectangular Detector

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Table 3 Instrument Line Shapes for an On-Axis Square Detector

Equations (32)

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OPD=AB+C=ΔXmcos θ+ΔXm cos 2θcos θ=ΔXm+ΔXm2 cos2 θ - 1cos θ=2ΔXm cos θ=ΔX0 cos θ.
Iθ, ΔX0=L1+cos2πν0ΔX0 cos θ,
IΔX0=Ω L01+cos2πν0ΔX0 cos θdΩ.
IΔX0=02π0θm L01+cos2πν0ΔX0 cos θsin θdθdϕ.
IΔX0=-2πL0sin2πν0ΔX0 cos θ2πν0ΔX00θm=2πL0-sin2πν0ΔX0 cos θm2πν0ΔX0+sin2πν0ΔX02πν0ΔX0.
IΔX0=2πL02πν0ΔX0 2 cos2πν0ΔX01+cos θm2×sin2πν0ΔX01-cos θm2.
IΔX0=2πL01-cos θmsinc2πν0ΔX01-cos θm2×cos2πν0ΔX01+cos θm2.
ν=ν0 cos θ.
dν=-ν0 sin θdθ.
dΩ=dA/R2=sin θdθdϕ.
dΩ=dA cos θ/R2dA/f2.
r=f tan θ,
ν=ν0 cos θ,
r=fν02/ν2-11/2;
In=2ϕ2π=2 arccos y/r2π.
x2+y-rc2 =R2,  x2+y2=r2.
y=rc2+r2-R22rc,
In=1/π arccosrc2+r2-R22rcr.
ν=ν01+r2/f21/2,
rmin=xc-A2+yc-B21/2,
rc1=xc-A2+yc+B21/2,
rc2=xc+A2+yc-B21/2,
rmax=xc+A2+yc+B21/2,
ϕ=π/2-α1-α2=π/2-arcsinyc-Br-arcsinxc-Ar.
In=14-12π arcsinyc-Br-12π arcsinxc-Ar.
r=fν02/ν2-11/2,
ν=ν01+r2/f21/2,
Inr=12πϕintcpr IAr, ϕdϕ,
Inν=12πϕintcpfν02/ν2-11/2 IAfν02/ν2-11/2ϕdϕ,
Inν=ϕintcpfν02/ν2-11/22π IAfν02/ν2-11/2.
IAr=exp-r2/a2,
Inν=exp- f2a2ν02/ν2-1Mπ×arccosrc2+f2ν02/ν2-1-R22rcfν02/ν2-11/2.

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