Abstract

Spectra measured by off-axis detectors in a high-resolution Fourier-transform spectrometer are characterized by frequency scaling, asymmetry and broadening of their line shape, and self-apodization in the corresponding interferogram. For a narrow-band input spectrum and a specified detector geometry, a formalism is presented that accounts for these effects with separate terms. Some of the terms are used to correct the larger off-axis effects as part of the calibration. The remaining terms are used to model the residual effects with the on-axis instrument line shape. We extend this approach to the broadband case using filter banks. The technique is applied to simulated spectra for the Tropospheric Emissions Spectrometer. This approach is shown to maintain a radiometric accuracy to less than 0.1%.

© 2000 Optical Society of America

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References

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  1. J. Malliard, D. Simons, C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 185–193 (1994).
    [CrossRef]
  2. K. Silk, E. Schildkraut, “Imaging Fourier transform spectroscopy for remote chemical sensing,” in Electro-Optic Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 169–177 (1996).
    [CrossRef]
  3. R. Beer, T. Glavich, “Remote sensing of the troposphere by infrared emissions spectroscopy,” in Advanced Optical Instrumentation for Remote Sensing of the Earth from Space, G. Duchossois, F. L. Herr, R. J. Sander, eds., Proc. SPIE1129, 42–51 (1989).
    [CrossRef]
  4. D. Siméoni, C. Singer, G. Chalon, “Infrared Atmospheric Sounding Interferometer,” Acta Astron. 40, 113–118 (1997).
    [CrossRef]
  5. J. J. Puschell, P. Tompkins, “Imaging spectrometers for future earth observing systems,” in Earth Observing Systems II, W. L. Barnes, ed., Proc. SPIE3117, 36–48 (1997).
    [CrossRef]
  6. R. Beer, T. Glavich, D. Rider, “Tropospheric Emission Spectrometer (TES) for the Earth Observing System (EOS) Chemistry I Satellite: I. Objectives, requirements and instrument overview,” Appl. Opt. MS. 16825.
  7. R. Beer, “Tropospheric Emissions Spectrometer: scientific objectives and approach, goals and requirements,” (Jet Propulsion Laboratory, Pasadena, Calif., 1996).
  8. R. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).
  9. J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).
  10. 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]
  11. A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  12. G. Strang, T. Nyugen, Wavelets and Filter Banks (Wellesley-Cambridge, Wellesley, Mass., 1996).
  13. S. Mallat, A Wavelet Tour of Signal Processing (Academic, New York, 1998).

1999 (1)

1997 (1)

D. Siméoni, C. Singer, G. Chalon, “Infrared Atmospheric Sounding Interferometer,” Acta Astron. 40, 113–118 (1997).
[CrossRef]

Beer, R.

R. Beer, T. Glavich, “Remote sensing of the troposphere by infrared emissions spectroscopy,” in Advanced Optical Instrumentation for Remote Sensing of the Earth from Space, G. Duchossois, F. L. Herr, R. J. Sander, eds., Proc. SPIE1129, 42–51 (1989).
[CrossRef]

R. Beer, “Tropospheric Emissions Spectrometer: scientific objectives and approach, goals and requirements,” (Jet Propulsion Laboratory, Pasadena, Calif., 1996).

Bell, R.

R. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).

Chalon, G.

D. Siméoni, C. Singer, G. Chalon, “Infrared Atmospheric Sounding Interferometer,” Acta Astron. 40, 113–118 (1997).
[CrossRef]

Chamberlain, J.

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).

Clark, C.

J. Malliard, D. Simons, C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

Genest, J.

Glavich, T.

R. Beer, T. Glavich, “Remote sensing of the troposphere by infrared emissions spectroscopy,” in Advanced Optical Instrumentation for Remote Sensing of the Earth from Space, G. Duchossois, F. L. Herr, R. J. Sander, eds., Proc. SPIE1129, 42–51 (1989).
[CrossRef]

Kerr, J.

J. Malliard, D. Simons, C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, New York, 1998).

Malliard, J.

J. Malliard, D. Simons, C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

Massey, S.

J. Malliard, D. Simons, C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

Nyugen, T.

G. Strang, T. Nyugen, Wavelets and Filter Banks (Wellesley-Cambridge, Wellesley, Mass., 1996).

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Puschell, J. J.

J. J. Puschell, P. Tompkins, “Imaging spectrometers for future earth observing systems,” in Earth Observing Systems II, W. L. Barnes, ed., Proc. SPIE3117, 36–48 (1997).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Schildkraut, E.

K. Silk, E. Schildkraut, “Imaging Fourier transform spectroscopy for remote chemical sensing,” in Electro-Optic Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 169–177 (1996).
[CrossRef]

Silk, K.

K. Silk, E. Schildkraut, “Imaging Fourier transform spectroscopy for remote chemical sensing,” in Electro-Optic Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 169–177 (1996).
[CrossRef]

Siméoni, D.

D. Siméoni, C. Singer, G. Chalon, “Infrared Atmospheric Sounding Interferometer,” Acta Astron. 40, 113–118 (1997).
[CrossRef]

Simons, D.

J. Malliard, D. Simons, C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

Singer, C.

D. Siméoni, C. Singer, G. Chalon, “Infrared Atmospheric Sounding Interferometer,” Acta Astron. 40, 113–118 (1997).
[CrossRef]

Smith, S.

J. Malliard, D. Simons, C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

Strang, G.

G. Strang, T. Nyugen, Wavelets and Filter Banks (Wellesley-Cambridge, Wellesley, Mass., 1996).

Tompkins, P.

J. J. Puschell, P. Tompkins, “Imaging spectrometers for future earth observing systems,” in Earth Observing Systems II, W. L. Barnes, ed., Proc. SPIE3117, 36–48 (1997).
[CrossRef]

Tremblay, P.

Acta Astron. (1)

D. Siméoni, C. Singer, G. Chalon, “Infrared Atmospheric Sounding Interferometer,” Acta Astron. 40, 113–118 (1997).
[CrossRef]

Appl. Opt. (1)

Other (11)

J. Malliard, D. Simons, C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. L. Crawford, E. R. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

K. Silk, E. Schildkraut, “Imaging Fourier transform spectroscopy for remote chemical sensing,” in Electro-Optic Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 169–177 (1996).
[CrossRef]

R. Beer, T. Glavich, “Remote sensing of the troposphere by infrared emissions spectroscopy,” in Advanced Optical Instrumentation for Remote Sensing of the Earth from Space, G. Duchossois, F. L. Herr, R. J. Sander, eds., Proc. SPIE1129, 42–51 (1989).
[CrossRef]

J. J. Puschell, P. Tompkins, “Imaging spectrometers for future earth observing systems,” in Earth Observing Systems II, W. L. Barnes, ed., Proc. SPIE3117, 36–48 (1997).
[CrossRef]

R. Beer, T. Glavich, D. Rider, “Tropospheric Emission Spectrometer (TES) for the Earth Observing System (EOS) Chemistry I Satellite: I. Objectives, requirements and instrument overview,” Appl. Opt. MS. 16825.

R. Beer, “Tropospheric Emissions Spectrometer: scientific objectives and approach, goals and requirements,” (Jet Propulsion Laboratory, Pasadena, Calif., 1996).

R. Bell, Introductory Fourier Transform Spectroscopy (Academic, New York, 1972).

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

G. Strang, T. Nyugen, Wavelets and Filter Banks (Wellesley-Cambridge, Wellesley, Mass., 1996).

S. Mallat, A Wavelet Tour of Signal Processing (Academic, New York, 1998).

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Figures (18)

Fig. 1
Fig. 1

Off-axis geometry and assumed pixel response function for the TES detectors.

Fig. 2
Fig. 2

(a) Simulated on-axis interferogram for a limb view at pixel 8. (b) Simulated off-axis interferogram measured at pixel 8. The off-axis interferogram decreases in amplitude with OPD. (c) The instrument line shape (ILS) for a spectrum measured on-axis and off-axis. The off-axis spectrum is attenuated in amplitude, asymmetric about the center frequency, line broadened, and shifted down in frequency with respect to the on-axis ILS for the same input frequency.

Fig. 3
Fig. 3

(a) Magnitude of the Fresnel kernel and (b) the phase of the Fresnel kernel for ν = 2537.5 cm-1.

Fig. 4
Fig. 4

(a) Residual Fresnel phase function and (b) the Fourier transform of the residual ILS function.

Fig. 5
Fig. 5

Validation steps for off-axis ILS correction and modeling algorithms. L1B, level 1B; L2, level 2.

Fig. 6
Fig. 6

Spectra and spectral differences at different stages in the off-axis ILS correction and residual modeling. (a) On-axis spectrum, (b) off-axis spectrum, (c) difference between on-axis and off-axis spectrum, (d) spectrum after level 1B correction, (e) spectrum with level 1B correction and level 2 modeling. Horizontal lines in (d) and (a) indicate a 0.1% error level.

Fig. 7
Fig. 7

Expected pixel response along array axis. Although the physical width of each pixel is 0.75 mrad, the response is wider than this because of diffraction and charge diffusion. Stars represent the points included in the interferogram model integration. (Angles corresponding to intensities lower than 2 dB below the maximum were not included.)

Fig. 8
Fig. 8

Two-channel filter bank.

Fig. 9
Fig. 9

Frequency response of the biorthogonal filters. The low-pass frequency response is H 0(ω) and the high-pass frequency response is G 0(ω).

Fig. 10
Fig. 10

Four-channel filter bank based on an iteration of the two-channel case. Only the analysis bank is shown. The synthesis bank is simply the reverse of the analysis bank in which the downsamplers are replaced by upsamplers.

Fig. 11
Fig. 11

Level 1B (L1B) processing: The functions h 0[n] and h 1[n] refer to low-pass filters and g 0[n] and g 1[n] refer to high-pass filters. The up arrows and down arrows refer to upsampling and downsampling, respectively.

Fig. 12
Fig. 12

Level 2 processing.

Fig. 13
Fig. 13

Magnitude of the Fresnel kernel, which is also called the self-apodization function, for 1045 and 1055 cm-1.

Fig. 14
Fig. 14

Residual Fresnel phase function for 1045 and 1055 cm-1. The complex exponent of these functions are the residual ILS functions.

Fig. 15
Fig. 15

Comparison between the on-axis spectrum and the off-axis spectrum after level 1B correction. The off-axis spectrum was split into two subbands by use of filter banks. The self-apodization functions were applied to the respective interferograms of the subbands. The subbands were reconstructed and then scale corrected. The error stated above the difference plot is defined as the maximum difference between the on-axis and level 1B (L1B) corrected off-axis divided by the mean value of the on-axis spectrum. MLS, mid-latitude summer.

Fig. 16
Fig. 16

Comparison between the on-axis spectrum with the residual ILS functions and the off-axis spectrum after level 1B (L1B) correction. The error increases for spectral points that are farther away from the correcting frequencies. L2, level 2.

Fig. 17
Fig. 17

Pixel response of the detector under uniform and gradient illumination.

Fig. 18
Fig. 18

Self-apodization functions and residual ILS phase functions under uniform and gradient illumination.

Tables (3)

Tables Icon

Table 1 Error Levels for ILS Correction and Modeling Processing Steps by use of Simulated Limb Views for Pixel 1 (eighth off-axis pixel)a

Tables Icon

Table 2 Filter Bank Coefficients

Tables Icon

Table 3 Error Levels for ILS Correction and Modeling Processing Steps by use of Simulated Limb Views for Pixel 1 (Eighth off-axis pixel) for Both Cases of Known (uniform) Illumination and Illumination Mismatch (gradient)

Equations (19)

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Ix=-βwβwαiαi+1- Bα, βPα, βLν×exp-i2π cosϕα, βνxdνdαdβ,
tan2 ϕ=tan2 α+tan2 β.
Ix=--βwβwαiαi+1Bα, βPα, βcosϕα, β×Fνcosϕα, βdαdβexp-i2πνxdν.
Rν=-βwβwαiαi+1Bα, βPα, βcosϕα, β Fνcosϕα, βdαdβ.
ϕ2α2+β2,
cos ϕα, β1-α2+β22.
Ix2 Re0 LνAνxexp-i2πνxdν,
Aνx=-βwβwαiαi+1 Bα, βPα, β×expi2πα2νxexpi2πβ2νxdαdβ.
Aνx=|Aνx|expiΦνx.
Φνx=2πρνx+ψνx.
Ix=2 Re0|Aνx|expiψνxLν×exp-i2π1-ρνxdν.
Ix2|Aν¯x|Reexpiψν¯x0 Lν¯×exp-i2π1-ρνxdν,
Ix2|Aν¯x|Reexpiψν¯xILx1-ρ,
Lν=L1ν+L2ν++LMν,
ν¯n=νn+νn+12,
 2In=I2n.
 2In=In2n=0, ±2, ±4,0 otherwise.
operations needed to generate outputof analysis bank <2MN.
=max|Lon axisω-LL1Bω|meanLon axisω.

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