Abstract

Fourier transform imaging spectroscopy (FTIS) can be performed with a multi-aperture optical system by making a series of intensity measurements, while introducing optical path differences (OPD’s) between various subapertures, and recovering spectral data by the standard Fourier post-processing technique. The imaging properties for multi-aperture FTIS are investigated by examining the imaging transfer functions for the recovered spectral images. For systems with physically separated subapertures, the imaging transfer functions are shown to vanish necessarily at the DC spatial frequency. Also, it is shown that the spatial frequency coverage of particular systems may be improved substantially by simultaneously introducing multiple OPD’s during the measurements, at the expense of limiting spectral coverage and causing the spectral resolution to vary with spatial frequency.

© 2005 Optical Society of America

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References

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  1. J. S. Fender, “Synthetic apertures: an overview,” in Synthetic Aperture Systems, J. S. Fender, ed., Proc. SPIE440, 2–7 (1983).
  2. S.-J. Chung, D. W. Miller, and O. L. de Weck, “Design and implementation of sparse aperture imaging systems,” in Highly Innovative Space Telescope Concepts, H. A. MacEwen, ed., Proc. SPIE4849, 181–191 (2002).
  3. D. Redding, S. Basinger, A. E. Lowman, A. Kissil, P. Bely, R. Burg, and R. Lyon, “Wavefront sensing for a next generation space telescope,” in Space Telescopes and Instruments V, P. Y. Bely and J. B. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).
  4. R. L. Kendrick, A. L. Duncan, and R. Sigler, “Imaging Fizeau interferometer: experimental results,” presented at Frontiers in Optics, Tucson, Arizona, 5–9 Oct. 2003 (post-deadline paper 15).
  5. R. G. Paxman, T. J. Schultz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [Crossref]
  6. J. R. Fienup, “MTF and integration time versus fill factor for sparse-aperture imaging systems,” in Imaging Technologies and Telescopes, J. W. Bilbro, et al., eds., Proc. SPIE4091, 43–47 (2000).
  7. J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” in Image Reconstruction from Incomplete Data II, P. J. Bones, et al., eds., Proc. SPIE4792, 1–8 (2002).
  8. J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy, (Wiley-VCH, Berlin, 2001).
    [Crossref]
  9. N. J. E. Johnson, “Spectral imaging with the Michelson interferometer,” in Infrared Imaging Systems Technology, Proc. SPIE226, 2–9 (1980).
  10. C. L. Bennett, M. Carter, D. Fields, and J. Hernandez, “Imaging Fourier transform spectrometer,” in Imaging Spectrometry of the Terrestrial Environment, G. Vane, ed., Proc. SPIE1937, 191–200 (1993).
  11. M. R. Carter, C. L. Bennett, D. J. Fields, and F. D. Lee, “Livermore imaging Fourier transform infrared spectrometer,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, and L. R. Illing, eds., Proc. SPIE2480, 380–386 (1995).
  12. K. Itoh and Y. Ohtsuka, “Fourier transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A 3, 94–100 (1986).
    [Crossref]
  13. J.-M. Mariotti and S. T. Ridgeway, “Double Fourier spatio-spectral interferometry: combining high spectral and high spatial resolution in the near infrared,” Astron. Astrophys. 195, 350–363 (1988).
  14. M. Frayman and J. A. Jamieson, “Scene imaging and spectroscopy using a spatial spectral interferometer,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckingridge, ed., Proc. SPIE1237, 585–603 (1990).
  15. R. L. Kendrick, E. H. Smith, and A. L. Duncan, “Imaging Fourier transform spectrometry with a Fizeau interferometer,” in Interferometry in Space, M. Shao, ed., Proc. SPIE4852, 657–662 (2003).
  16. Provided through the courtesy of Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California (http://aviris.jpl.nasa.gov/).
  17. C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. 57, 297–303 (1967).
    [Crossref]
  18. B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” International Journal of Imaging Systems and Technology 6, 297–304 (1995).
    [Crossref]
  19. S. T. Thurman and J. R. Fienup, “Fourier transform imaging spectroscopy with a multiple-aperture telescope: band-by-band image reconstruction,” in Optical, Infrared, and Millimeter Space Telescopes, J. C. Mather, ed., Proc. SPIE5487-68 (2004).
  20. S. T. Thurman and J. R. Fienup, “Reconstruction of multispectral image cubes from multiple-telescope array Fourier transform imaging spectrometer,” presented at Frontiers in Optics, Rochester, New York, 10–14 Oct. 2004, paper FTuB3.
  21. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, Cambridge, 1995).
  22. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed., (Cambridge University Press, Cambridge, 2002) Sec. 10.2.
  23. J. W. Goodman, Statistical Optics, (Wiley, New York, 2000) Sec. 3.5.
  24. J. Goodman, Introduction to Fourier Optics2nd ed., (McGraw-Hill, New York, 1996).
  25. M. J. Beran and G. B. Parrent, “The mutual coherence of incoherent radiation,” Nuovo Cimento 27, 1049–1065 (1963).
    [Crossref]

1995 (1)

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” International Journal of Imaging Systems and Technology 6, 297–304 (1995).
[Crossref]

1992 (1)

1988 (1)

J.-M. Mariotti and S. T. Ridgeway, “Double Fourier spatio-spectral interferometry: combining high spectral and high spatial resolution in the near infrared,” Astron. Astrophys. 195, 350–363 (1988).

1986 (1)

1967 (1)

1963 (1)

M. J. Beran and G. B. Parrent, “The mutual coherence of incoherent radiation,” Nuovo Cimento 27, 1049–1065 (1963).
[Crossref]

Basinger, S.

D. Redding, S. Basinger, A. E. Lowman, A. Kissil, P. Bely, R. Burg, and R. Lyon, “Wavefront sensing for a next generation space telescope,” in Space Telescopes and Instruments V, P. Y. Bely and J. B. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).

Bely, P.

D. Redding, S. Basinger, A. E. Lowman, A. Kissil, P. Bely, R. Burg, and R. Lyon, “Wavefront sensing for a next generation space telescope,” in Space Telescopes and Instruments V, P. Y. Bely and J. B. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).

Bennett, C. L.

C. L. Bennett, M. Carter, D. Fields, and J. Hernandez, “Imaging Fourier transform spectrometer,” in Imaging Spectrometry of the Terrestrial Environment, G. Vane, ed., Proc. SPIE1937, 191–200 (1993).

M. R. Carter, C. L. Bennett, D. J. Fields, and F. D. Lee, “Livermore imaging Fourier transform infrared spectrometer,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, and L. R. Illing, eds., Proc. SPIE2480, 380–386 (1995).

Beran, M. J.

M. J. Beran and G. B. Parrent, “The mutual coherence of incoherent radiation,” Nuovo Cimento 27, 1049–1065 (1963).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed., (Cambridge University Press, Cambridge, 2002) Sec. 10.2.

Burg, R.

D. Redding, S. Basinger, A. E. Lowman, A. Kissil, P. Bely, R. Burg, and R. Lyon, “Wavefront sensing for a next generation space telescope,” in Space Telescopes and Instruments V, P. Y. Bely and J. B. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).

Carter, M.

C. L. Bennett, M. Carter, D. Fields, and J. Hernandez, “Imaging Fourier transform spectrometer,” in Imaging Spectrometry of the Terrestrial Environment, G. Vane, ed., Proc. SPIE1937, 191–200 (1993).

Carter, M. R.

M. R. Carter, C. L. Bennett, D. J. Fields, and F. D. Lee, “Livermore imaging Fourier transform infrared spectrometer,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, and L. R. Illing, eds., Proc. SPIE2480, 380–386 (1995).

Chung, S.-J.

S.-J. Chung, D. W. Miller, and O. L. de Weck, “Design and implementation of sparse aperture imaging systems,” in Highly Innovative Space Telescope Concepts, H. A. MacEwen, ed., Proc. SPIE4849, 181–191 (2002).

de Weck, O. L.

S.-J. Chung, D. W. Miller, and O. L. de Weck, “Design and implementation of sparse aperture imaging systems,” in Highly Innovative Space Telescope Concepts, H. A. MacEwen, ed., Proc. SPIE4849, 181–191 (2002).

Duncan, A. L.

R. L. Kendrick, A. L. Duncan, and R. Sigler, “Imaging Fizeau interferometer: experimental results,” presented at Frontiers in Optics, Tucson, Arizona, 5–9 Oct. 2003 (post-deadline paper 15).

R. L. Kendrick, E. H. Smith, and A. L. Duncan, “Imaging Fourier transform spectrometry with a Fizeau interferometer,” in Interferometry in Space, M. Shao, ed., Proc. SPIE4852, 657–662 (2003).

Fender, J. S.

J. S. Fender, “Synthetic apertures: an overview,” in Synthetic Aperture Systems, J. S. Fender, ed., Proc. SPIE440, 2–7 (1983).

Fields, D.

C. L. Bennett, M. Carter, D. Fields, and J. Hernandez, “Imaging Fourier transform spectrometer,” in Imaging Spectrometry of the Terrestrial Environment, G. Vane, ed., Proc. SPIE1937, 191–200 (1993).

Fields, D. J.

M. R. Carter, C. L. Bennett, D. J. Fields, and F. D. Lee, “Livermore imaging Fourier transform infrared spectrometer,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, and L. R. Illing, eds., Proc. SPIE2480, 380–386 (1995).

Fienup, J. R.

R. G. Paxman, T. J. Schultz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[Crossref]

S. T. Thurman and J. R. Fienup, “Fourier transform imaging spectroscopy with a multiple-aperture telescope: band-by-band image reconstruction,” in Optical, Infrared, and Millimeter Space Telescopes, J. C. Mather, ed., Proc. SPIE5487-68 (2004).

S. T. Thurman and J. R. Fienup, “Reconstruction of multispectral image cubes from multiple-telescope array Fourier transform imaging spectrometer,” presented at Frontiers in Optics, Rochester, New York, 10–14 Oct. 2004, paper FTuB3.

J. R. Fienup, “MTF and integration time versus fill factor for sparse-aperture imaging systems,” in Imaging Technologies and Telescopes, J. W. Bilbro, et al., eds., Proc. SPIE4091, 43–47 (2000).

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” in Image Reconstruction from Incomplete Data II, P. J. Bones, et al., eds., Proc. SPIE4792, 1–8 (2002).

Frayman, M.

M. Frayman and J. A. Jamieson, “Scene imaging and spectroscopy using a spatial spectral interferometer,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckingridge, ed., Proc. SPIE1237, 585–603 (1990).

Goodman, J.

J. Goodman, Introduction to Fourier Optics2nd ed., (McGraw-Hill, New York, 1996).

Goodman, J. W.

J. W. Goodman, Statistical Optics, (Wiley, New York, 2000) Sec. 3.5.

Griffith, D.

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” in Image Reconstruction from Incomplete Data II, P. J. Bones, et al., eds., Proc. SPIE4792, 1–8 (2002).

Harrington, L.

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” in Image Reconstruction from Incomplete Data II, P. J. Bones, et al., eds., Proc. SPIE4792, 1–8 (2002).

Helstrom, C. W.

Hernandez, J.

C. L. Bennett, M. Carter, D. Fields, and J. Hernandez, “Imaging Fourier transform spectrometer,” in Imaging Spectrometry of the Terrestrial Environment, G. Vane, ed., Proc. SPIE1937, 191–200 (1993).

Hunt, B. R.

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” International Journal of Imaging Systems and Technology 6, 297–304 (1995).
[Crossref]

Itoh, K.

Jamieson, J. A.

M. Frayman and J. A. Jamieson, “Scene imaging and spectroscopy using a spatial spectral interferometer,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckingridge, ed., Proc. SPIE1237, 585–603 (1990).

Johnson, N. J. E.

N. J. E. Johnson, “Spectral imaging with the Michelson interferometer,” in Infrared Imaging Systems Technology, Proc. SPIE226, 2–9 (1980).

Kauppinen, J.

J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy, (Wiley-VCH, Berlin, 2001).
[Crossref]

Kendrick, R. L.

R. L. Kendrick, A. L. Duncan, and R. Sigler, “Imaging Fizeau interferometer: experimental results,” presented at Frontiers in Optics, Tucson, Arizona, 5–9 Oct. 2003 (post-deadline paper 15).

R. L. Kendrick, E. H. Smith, and A. L. Duncan, “Imaging Fourier transform spectrometry with a Fizeau interferometer,” in Interferometry in Space, M. Shao, ed., Proc. SPIE4852, 657–662 (2003).

Kissil, A.

D. Redding, S. Basinger, A. E. Lowman, A. Kissil, P. Bely, R. Burg, and R. Lyon, “Wavefront sensing for a next generation space telescope,” in Space Telescopes and Instruments V, P. Y. Bely and J. B. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).

Kowalczyk, A. M.

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” in Image Reconstruction from Incomplete Data II, P. J. Bones, et al., eds., Proc. SPIE4792, 1–8 (2002).

Lee, F. D.

M. R. Carter, C. L. Bennett, D. J. Fields, and F. D. Lee, “Livermore imaging Fourier transform infrared spectrometer,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, and L. R. Illing, eds., Proc. SPIE2480, 380–386 (1995).

Lowman, A. E.

D. Redding, S. Basinger, A. E. Lowman, A. Kissil, P. Bely, R. Burg, and R. Lyon, “Wavefront sensing for a next generation space telescope,” in Space Telescopes and Instruments V, P. Y. Bely and J. B. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).

Lyon, R.

D. Redding, S. Basinger, A. E. Lowman, A. Kissil, P. Bely, R. Burg, and R. Lyon, “Wavefront sensing for a next generation space telescope,” in Space Telescopes and Instruments V, P. Y. Bely and J. B. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, Cambridge, 1995).

Mariotti, J.-M.

J.-M. Mariotti and S. T. Ridgeway, “Double Fourier spatio-spectral interferometry: combining high spectral and high spatial resolution in the near infrared,” Astron. Astrophys. 195, 350–363 (1988).

Miller, D. W.

S.-J. Chung, D. W. Miller, and O. L. de Weck, “Design and implementation of sparse aperture imaging systems,” in Highly Innovative Space Telescope Concepts, H. A. MacEwen, ed., Proc. SPIE4849, 181–191 (2002).

Miller, J. J.

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” in Image Reconstruction from Incomplete Data II, P. J. Bones, et al., eds., Proc. SPIE4792, 1–8 (2002).

Mooney, J. A.

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” in Image Reconstruction from Incomplete Data II, P. J. Bones, et al., eds., Proc. SPIE4792, 1–8 (2002).

Ohtsuka, Y.

Parrent, G. B.

M. J. Beran and G. B. Parrent, “The mutual coherence of incoherent radiation,” Nuovo Cimento 27, 1049–1065 (1963).
[Crossref]

Partanen, J.

J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy, (Wiley-VCH, Berlin, 2001).
[Crossref]

Paxman, R. G.

Redding, D.

D. Redding, S. Basinger, A. E. Lowman, A. Kissil, P. Bely, R. Burg, and R. Lyon, “Wavefront sensing for a next generation space telescope,” in Space Telescopes and Instruments V, P. Y. Bely and J. B. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).

Ridgeway, S. T.

J.-M. Mariotti and S. T. Ridgeway, “Double Fourier spatio-spectral interferometry: combining high spectral and high spatial resolution in the near infrared,” Astron. Astrophys. 195, 350–363 (1988).

Schultz, T. J.

Sigler, R.

R. L. Kendrick, A. L. Duncan, and R. Sigler, “Imaging Fizeau interferometer: experimental results,” presented at Frontiers in Optics, Tucson, Arizona, 5–9 Oct. 2003 (post-deadline paper 15).

Smith, E. H.

R. L. Kendrick, E. H. Smith, and A. L. Duncan, “Imaging Fourier transform spectrometry with a Fizeau interferometer,” in Interferometry in Space, M. Shao, ed., Proc. SPIE4852, 657–662 (2003).

Thurman, S. T.

S. T. Thurman and J. R. Fienup, “Fourier transform imaging spectroscopy with a multiple-aperture telescope: band-by-band image reconstruction,” in Optical, Infrared, and Millimeter Space Telescopes, J. C. Mather, ed., Proc. SPIE5487-68 (2004).

S. T. Thurman and J. R. Fienup, “Reconstruction of multispectral image cubes from multiple-telescope array Fourier transform imaging spectrometer,” presented at Frontiers in Optics, Rochester, New York, 10–14 Oct. 2004, paper FTuB3.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, Cambridge, 1995).

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed., (Cambridge University Press, Cambridge, 2002) Sec. 10.2.

Astron. Astrophys. (1)

J.-M. Mariotti and S. T. Ridgeway, “Double Fourier spatio-spectral interferometry: combining high spectral and high spatial resolution in the near infrared,” Astron. Astrophys. 195, 350–363 (1988).

International Journal of Imaging Systems and Technology (1)

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” International Journal of Imaging Systems and Technology 6, 297–304 (1995).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nuovo Cimento (1)

M. J. Beran and G. B. Parrent, “The mutual coherence of incoherent radiation,” Nuovo Cimento 27, 1049–1065 (1963).
[Crossref]

Other (19)

J. R. Fienup, “MTF and integration time versus fill factor for sparse-aperture imaging systems,” in Imaging Technologies and Telescopes, J. W. Bilbro, et al., eds., Proc. SPIE4091, 43–47 (2000).

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” in Image Reconstruction from Incomplete Data II, P. J. Bones, et al., eds., Proc. SPIE4792, 1–8 (2002).

J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy, (Wiley-VCH, Berlin, 2001).
[Crossref]

N. J. E. Johnson, “Spectral imaging with the Michelson interferometer,” in Infrared Imaging Systems Technology, Proc. SPIE226, 2–9 (1980).

C. L. Bennett, M. Carter, D. Fields, and J. Hernandez, “Imaging Fourier transform spectrometer,” in Imaging Spectrometry of the Terrestrial Environment, G. Vane, ed., Proc. SPIE1937, 191–200 (1993).

M. R. Carter, C. L. Bennett, D. J. Fields, and F. D. Lee, “Livermore imaging Fourier transform infrared spectrometer,” in Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, and L. R. Illing, eds., Proc. SPIE2480, 380–386 (1995).

J. S. Fender, “Synthetic apertures: an overview,” in Synthetic Aperture Systems, J. S. Fender, ed., Proc. SPIE440, 2–7 (1983).

S.-J. Chung, D. W. Miller, and O. L. de Weck, “Design and implementation of sparse aperture imaging systems,” in Highly Innovative Space Telescope Concepts, H. A. MacEwen, ed., Proc. SPIE4849, 181–191 (2002).

D. Redding, S. Basinger, A. E. Lowman, A. Kissil, P. Bely, R. Burg, and R. Lyon, “Wavefront sensing for a next generation space telescope,” in Space Telescopes and Instruments V, P. Y. Bely and J. B. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).

R. L. Kendrick, A. L. Duncan, and R. Sigler, “Imaging Fizeau interferometer: experimental results,” presented at Frontiers in Optics, Tucson, Arizona, 5–9 Oct. 2003 (post-deadline paper 15).

S. T. Thurman and J. R. Fienup, “Fourier transform imaging spectroscopy with a multiple-aperture telescope: band-by-band image reconstruction,” in Optical, Infrared, and Millimeter Space Telescopes, J. C. Mather, ed., Proc. SPIE5487-68 (2004).

S. T. Thurman and J. R. Fienup, “Reconstruction of multispectral image cubes from multiple-telescope array Fourier transform imaging spectrometer,” presented at Frontiers in Optics, Rochester, New York, 10–14 Oct. 2004, paper FTuB3.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, Cambridge, 1995).

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed., (Cambridge University Press, Cambridge, 2002) Sec. 10.2.

J. W. Goodman, Statistical Optics, (Wiley, New York, 2000) Sec. 3.5.

J. Goodman, Introduction to Fourier Optics2nd ed., (McGraw-Hill, New York, 1996).

M. Frayman and J. A. Jamieson, “Scene imaging and spectroscopy using a spatial spectral interferometer,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckingridge, ed., Proc. SPIE1237, 585–603 (1990).

R. L. Kendrick, E. H. Smith, and A. L. Duncan, “Imaging Fourier transform spectrometry with a Fizeau interferometer,” in Interferometry in Space, M. Shao, ed., Proc. SPIE4852, 657–662 (2003).

Provided through the courtesy of Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California (http://aviris.jpl.nasa.gov/).

Supplementary Material (3)

» Media 1: AVI (455 KB)     
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» Media 3: AVI (746 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Illustration of multiple-telescope array with four subaperture telescopes.

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Fig. 2.
Fig. 2.

Simplified refractive model for a multi-aperture optical system.

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Fig. 3.
Fig. 3.

Pupil of optical system used in simulations.

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Fig. 4.
Fig. 4.

Movie (455KB) showing the effect of the OPD’s on the optical system at ν=ν 0 as the time-delay variable is changed from τ=0 to τ=3/ν 0: (a) the magnitude of the relative phase delay of each subaperture, where white represents 0 and black represents ±π, (b) the PSF, and (c) the magnitude of the real part of the OTF.

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Fig. 5.
Fig. 5.

Localization of FTIS signal in: (a) the raw intensity data cube, (b) the spectral image cube, and (c) the spectral-spatial transform cube. In each cube, the FTIS signal is localized to the darkly shaded regions.

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Fig. 6.
Fig. 6.

Image intensity versus τ for the point object simulation: (a) at Point A, (b) at Point B, and contributions to the intensity at Point B due to the interference between subapertures: (c) 1 and 2, (d) 2 and 3, and (e) 1 and 3.

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Fig. 7.
Fig. 7.

Spectral data from point object simulation at positive temporal frequencies in the ν′-domain: (a) at Point A (real-valued) and (b) at Point B (real and imaginary parts).

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Fig. 8.
Fig. 8.

The extended object simulation: (a) movie (582KB) of the object data versus ν(the still frame shows the data at ν=1.03ν 0), (b) size of the pupil in spatial frequencies corresponding to ν=1.03ν 0, and (c) movie (746KB) of the image intensity versus τ(the still frame shows the image intensity at τ=0).

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Fig. 9.
Fig. 9.

Spectral image data from the extended-object simulation. The top row shows the real part of spectral images at: (a) ν′=0.34ν 0, (c) ν′=0.68ν 0, and (e) ν′=1.03ν0. The bottom row shows the Fourier magnitude of each image. For the spectral images, note that dark grays represent negative values and light grays represent positive values.

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Fig. 10.
Fig. 10.

Composite spectral image data from extended object simulation: (a) the real part of the spectral image at ν=1.03ν 0, (b) the corresponding Fourier magnitude, (c) the imaginary part of the same spectral image, and (d) the corresponding Fourier magnitude. For the spectral images, note that dark grays represent negative values, middle gray represents zero, and light grays represent positive values.

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Equations (44)

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T pup ( ξ , η , ν , τ ) = ∑ q = 1 Q T q ( ξ , η , ν ) exp ( i 2 π ν γ q τ ) ,
I ( x , y , τ ) = κ ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ 1 M 2 S o ( x ′ M , y ′ M , ν ) h ( x − x ′ , y − y ′ , ν , τ ) dx ′ dy ′ d ν ,
h ( x , y , ν , τ ) = ∑ q = 1 Q h q , q ( x , y , ν ) + ∑ p = 1 Q ∑ q = 1 q ≠ p Q h p , q ( x , y , ν ) exp [ i 2 π ν ( γ p − γ q ) τ ] .
h p , q ( x , y , ν ) = t p ( x , y , ν ) t q * ( x , y , ν ) ,
t q ( x , y , ν ) = 1 λ 2 f i 2 ∫ − ∞ ∞ ∫ − ∞ ∞ T q ( ξ , η , ν ) exp [ − i 2 π ( x λ f i ξ + y λ f i η ) ] d ξ d η .
H ( f x , f y , ν , τ ) = ∫ − ∞ ∞ ∫ − ∞ ∞ h ( x , y , ν , τ ) exp [ − i 2 π ( f x x + f y y ) ] d x d y ∫ − ∞ ∞ ∫ − ∞ ∞ h ( x , y , ν , τ ) d x d y
= T pup ( − λ f i f x , − λ f i f y , ν , τ ) ★ T pup ( − λ f i f x , − λ f i f y , ν , τ ) ∫ − ∞ ∞ ∫ − ∞ ∞ ∣ T pup ( ξ , η , ν , τ ) ∣ 2 d ξ d η
= ∑ q = 1 Q H q , q ( f x , f y , ν ) + ∑ p = 1 Q ∑ q = 1 q ≠ p Q H p , q ( f x , f y , ν ) exp [ i 2 π ν ( γ p − γ q ) τ ] ,
H p , q ( f x , f y , ν ) = ∫ − ∞ ∞ ∫ − ∞ ∞ h p , q ( x , y , ν ) exp [ − i 2 π ( f x x + f y y ) ] d x d y ∫ − ∞ ∞ ∫ − ∞ ∞ h ( x , y , ν , τ ) d x d y
= T p ( − λ f i f x , − λ f i f y , ν ) ★ T q ( − λ f i f x , − λ f i f y , ν ) ∫ − ∞ ∞ ∫ − ∞ ∞ ∣ T pup ( ξ , η , ν , τ ) ∣ 2 d ξ d η .
S i ( x , y , ν ′ ) = κ ∑ p = 1 Q ∑ q = 1 q ≠ p Q ∫ − ∞ ∞ ∫ − ∞ ∞ 1 ∣ γ p − γ q ∣ M 2 S o ( x ′ M , y ′ M , ν ′ γ p − γ q )
× h p , q ( x − x ′ , y − y ′ , ν ′ γ p − γ q ) d x ′ d y ′ .
G i ( f x , f y , ν ′ ) = κ ∑ p = 1 Q ∑ q = 1 q ≠ p Q 1 ∣ γ p − γ q ∣ G o ( M f x , M f y , ν ′ γ p − γ q ) H p , q ( f x , f y , ν ′ γ p − γ q ) ,
S comp ( x , y , ν ) = ∑ Δ γ > 0 Δ γ S i ( x , y , Δ γ ν ) for ν 1 ≤ ν ≤ ν 2 ,
S comp ( x , y , ν ) = κ ∑ p = 1 Q ∑ q = 1 Δ γ > 0 Q ∫ − ∞ ∞ ∫ − ∞ ∞ 1 M 2 S o ( x ′ M , y ′ M , ν )
× h p , q ( x − x ′ , y − y ′ , ν ) d x ′ d y ′ for ν 1 ≤ ν ≤ ν 2 .
S i ( x , y , ν ′ ) = S i * ( x , y , − ν ′ ) ,
G i ( f x , f y , ν ′ ) = G i * ( − f x , − f y , − ν ′ ) .
S i ( Re ) ( x , y , ν ′ ) = 1 2 [ S i ( x , y , ν ′ ) + S i * ( x , y , ν ′ ) ] ,
G i ( Re ) ( f x , f y , ν ′ ) = 1 2 [ G i ( f x , f y , ν ′ ) + G i * ( − f x , − f y , ν ′ ) ] ,
G i ( Re ) ( f x , f y , ν ′ ) = κ ∑ p = 1 Q ∑ q = 1 q ≠ p Q 1 ∣ γ p − γ q ∣ G o ( M f x , M f y , ν ′ γ p − γ q )
× 1 2 [ H p , q ( f x , f y , ν ′ γ p − γ q ) + H p , q * ( − f x , − f y , ν ′ γ p − γ q ) ] .
G i ( Im ) ( f x , f y , ν ′ ) = κ ∑ p = 1 Q ∑ q = 1 q ≠ p Q 1 ∣ γ p − γ q ∣ G o ( M f x , M f y , ν ′ γ p − γ q )
× 1 2 i [ H p , q ( f x , f y , ν ′ γ p − γ q ) − H p , q * ( − f x , − f y , ν ′ γ p − γ q ) ] .
S i ( x , y , ν ′ ) = κ ∑ p = 1 Q ∑ q = 1 q ≠ p Q ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ 1 M 2 S o ( x ′ M , y ′ M , ν ) h p , q ( x − x ′ , y − y ′ , ν )
× 2 τ max sin⁡ c [ 2 τ max ( γ p − γ q ) ( ν ′ γ p − γ q − ν ) ] d x ′ d y ′ d ν ,
S comp ( x , y , ν ) ≈ κ ∑ p = 1 Q ∑ q = 1 Δ γ > 0 Q ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ 1 M 2 S o ( x ′ M , y ′ M , ν ′ ) h p , q ( x − x ′ , y − y ′ , ν ′ )
× 2 τ max ( γ p − γ q ) sin⁡ c [ 2 τ max ( γ p − γ q ) ( ν − ν ′ ) ] d x ′ d y ′ d ν ′ for ν 1 ≤ ν ≤ ν 2 .
S o ( x ′ , y ′ , ν ) = E rect [ ( ν − ν 0 ) ⁄ ( ν 2 − ν 1 ) ] δ ( x ′ , y ′ ) ,
I ( x , y , τ ) = κ E ∫ ν 1 ν 2 h ( x , y , ν , τ ) d ν .
t q ( x , y , ν ) = π R 2 λ 2 f i 2 jinc ( 2 R λ f i x 2 + y 2 ) exp [ − i 2 π λ f i ( x ξ q + y η q ) ] ,
τ p , q = x B ( ξ p − ξ q ) + y B ( η p − η q ) c f i ( γ p − γ q ) .
W ( z ) ( x 1 , y 1 , x 2 , y 2 , ν ) δ ( ν − ν ' ) = 〈 V ( x 1 , y 1 , ν ) V * ( x 2 , y 2 , ν ′ ) 〉 ,
W ( d ) ( x 1 , y 1 , x 2 , y 2 , ν ) = 1 λ 2 d 2 ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ d x 1 ′ d y 1 ′ d x 1 ′ d y 2 ′ W ( 0 ) ( x 1 ′ , y 1 ′ , x 2 ′ , y 2 ′ , ν )
× exp { i π λ d [ ( x 1 − x 1 ′ ) 2 + ( y 1 − y 1 ′ ) 2 − ( x 2 − x 2 ′ ) 2 − ( y 2 − y 2 ′ ) 2 ] } ,
V trans ( x , y , ν ) = T ( x , y , ν ) V inc ( x , y , ν ) ,
W trans ( 0 ) ( x 1 , y 1 , x 2 , y 2 , ν ) = T ( x 1 , y 1 , ν ) T * ( x 2 , y 2 , ν ) W inc ( 0 ) ( x 1 , y 1 , x 2 , y 2 , ν ) ,
W ( i ) ( x 1 , y 1 , x 2 , y 2 , ν , τ ) = ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ d x 1 ′ d y 1 ′ d x 2 ′ d y 2 ′ 1 M 2 W ( o ) ( x 1 ′ M , y 1 ′ M , x 2 ′ M , y 2 ′ M , ν )
× exp [ i π λ f o ( 1 − d 1 f o ) ( x ′ 1 2 M 2 + y ′ 1 2 M 2 − x ′ 2 2 M 2 − y ′ 2 2 M 2 ) ]
× exp [ i π λ f i ( 1 − d 2 f i ) ( x 1 2 + y 1 2 − x 2 2 − y 2 2 ) ]
× { ∑ q = 1 Q ∑ p = 1 Q t q ( x 1 − x 1 ′ , y 1 − y 1 ′ , ν ) t p * ( x 2 − x 2 ′ , y 2 − y 2 ′ , ν )
× exp [ i 2 π ν ( γ q − γ p ) τ ] ,
I ( x , y , τ ) = ∫ − ∞ ∞ W ( i ) ( x , y , x , y , ν , τ ) d ν .
W ( o ) ( x 1 ′ , y 1 ′ , x 2 ′ , y 2 ′ , ν ) = κ S o ( x ′ 1 , y 1 ′ , ν ) δ ( x 1 ′ − x 2 ′ , y 1 ′ − y 2 ′ )

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