Abstract

The emerging astronomical technique known as wide-field spatiospectral interferometry can provide hyperspectral images with spatial resolutions that are unattainable with a single monolithic-aperture observatory. The theoretical groundwork for operation and data measurement is presented in full detail, including relevant coherence theory. We also discuss a data processing technique for recovering a hyperspectral image from an interferometric data set as well as the unusual effective transfer function of the system.

© 2017 Optical Society of America

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References

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  1. R. G. Lyon, S. A. Rinehart, D. T. Leisawitz, and N. Memarsadeghi, “Wide-field imaging interferometry testbed (WIIT): image construction algorithms,” Proc. SPIE 7013, 70131M (2008).
    [Crossref]
  2. R. G. Lyon, D. T. Leisawitz, S. A. Rinehart, N. Memarsadeghi, and E. J. Sinukoff, “Wide-field imaging interferometry spatial-spectral image synthesis algorithms,” Proc. SPIE 8445, 84450B (2012).
    [Crossref]
  3. N. Elias, M. Harwit, D. T. Leisawitz, and S. A. Rinehart, “The mathematics of double-Fourier interferometers,” Astrophys. J. 657, 1178–1200 (2007).
    [Crossref]
  4. S. T. Thurman and J. R. Fienup, “Multi-aperture Fourier transform imaging spectroscopy: theory and imaging properties,” Opt. Express 13, 2160–2175 (2005).
    [Crossref]
  5. S. T. Thurman and J. R. Fienup, “Fizeau Fourier transform imaging spectroscopy: missing data reconstruction,” Opt. Express 16, 6631–6645 (2008).
    [Crossref]
  6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  7. A. Zeilinger, “General properties of lossless beam splitters in interferometry,” Am. J. Phys. 49, 882–883 (1981).
    [Crossref]
  8. J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2000).
  9. J.-M. Mariotti and S. T. Ridgway, “Double Fourier spatio-spectral interferometry–combining high spectral and high spatial resolution in the near infrared,” Astron. Astrophys. 195, 350–363 (1988).
  10. S. T. Thurman and J. R. Fienup, “Dealiased spectral images from aliased Fizeau Fourier transform spectroscopy measurements,” J. Opt. Soc. Am. A 24, 68–73 (2007).
    [Crossref]
  11. D. B. Leviton, B. J. Frey, D. T. Leisawitz, A. J. Martino, W. L. Maynard, L. G. Mundy, S. A. Rinehart, S. H. Teng, and X. Zhang, “Wide-field imaging interferometry testbed 3: metrology subsystem,” Proc. SPIE 4852, 827–838 (2003).
    [Crossref]
  12. S. T. Thurman and J. R. Fienup, “Signal-to-noise ratio trade-offs associated with coarsely sampled Fourier transform spectroscopy,” J. Opt. Soc. Am. A 24, 2817–2821 (2007).
    [Crossref]
  13. J. D. Monnier, “Optical interferometry in astronomy,” Rep. Prog. Phys. 66, 789–857 (2003).
    [Crossref]
  14. R. Juanola-Parramon, D. M. Fenech, and G. Savini, “Architecture and performance of the space-based far-infrared interferometer instrument simulator,” Mon. Not. R. Astron. Soc. 457, 3457–3469 (2016).
    [Crossref]
  15. J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 417–426 (1974).
  16. R. Narayan and R. Nityananda, “Maximum entropy image restoration in astronomy,” Annu. Rev. Astron. Astrophys. 24, 127–170 (1986).
    [Crossref]
  17. E. Thiébaut and J. Young, “Principles of Image reconstruction in optical interferometry: tutorial,” J. Opt. Soc. Am. A 34, 904–923 (2017).
    [Crossref]
  18. J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
    [Crossref]
  19. L. Mandel, “Concepts of cross-spectral purity in coherence theory,” J. Opt. Soc. Am. 51, 1342–1350 (1961).
    [Crossref]

2017 (1)

2016 (2)

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

R. Juanola-Parramon, D. M. Fenech, and G. Savini, “Architecture and performance of the space-based far-infrared interferometer instrument simulator,” Mon. Not. R. Astron. Soc. 457, 3457–3469 (2016).
[Crossref]

2012 (1)

R. G. Lyon, D. T. Leisawitz, S. A. Rinehart, N. Memarsadeghi, and E. J. Sinukoff, “Wide-field imaging interferometry spatial-spectral image synthesis algorithms,” Proc. SPIE 8445, 84450B (2012).
[Crossref]

2008 (2)

S. T. Thurman and J. R. Fienup, “Fizeau Fourier transform imaging spectroscopy: missing data reconstruction,” Opt. Express 16, 6631–6645 (2008).
[Crossref]

R. G. Lyon, S. A. Rinehart, D. T. Leisawitz, and N. Memarsadeghi, “Wide-field imaging interferometry testbed (WIIT): image construction algorithms,” Proc. SPIE 7013, 70131M (2008).
[Crossref]

2007 (3)

2005 (1)

2003 (2)

D. B. Leviton, B. J. Frey, D. T. Leisawitz, A. J. Martino, W. L. Maynard, L. G. Mundy, S. A. Rinehart, S. H. Teng, and X. Zhang, “Wide-field imaging interferometry testbed 3: metrology subsystem,” Proc. SPIE 4852, 827–838 (2003).
[Crossref]

J. D. Monnier, “Optical interferometry in astronomy,” Rep. Prog. Phys. 66, 789–857 (2003).
[Crossref]

1988 (1)

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

1986 (1)

R. Narayan and R. Nityananda, “Maximum entropy image restoration in astronomy,” Annu. Rev. Astron. Astrophys. 24, 127–170 (1986).
[Crossref]

1981 (1)

A. Zeilinger, “General properties of lossless beam splitters in interferometry,” Am. J. Phys. 49, 882–883 (1981).
[Crossref]

1974 (1)

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 417–426 (1974).

1961 (1)

Elias, N.

N. Elias, M. Harwit, D. T. Leisawitz, and S. A. Rinehart, “The mathematics of double-Fourier interferometers,” Astrophys. J. 657, 1178–1200 (2007).
[Crossref]

Fenech, D. M.

R. Juanola-Parramon, D. M. Fenech, and G. Savini, “Architecture and performance of the space-based far-infrared interferometer instrument simulator,” Mon. Not. R. Astron. Soc. 457, 3457–3469 (2016).
[Crossref]

Ferrari, A.

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

Fienup, J. R.

Frey, B. J.

D. B. Leviton, B. J. Frey, D. T. Leisawitz, A. J. Martino, W. L. Maynard, L. G. Mundy, S. A. Rinehart, S. H. Teng, and X. Zhang, “Wide-field imaging interferometry testbed 3: metrology subsystem,” Proc. SPIE 4852, 827–838 (2003).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2000).

Harwit, M.

N. Elias, M. Harwit, D. T. Leisawitz, and S. A. Rinehart, “The mathematics of double-Fourier interferometers,” Astrophys. J. 657, 1178–1200 (2007).
[Crossref]

Heininger, M.

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

Hofmann, K.-H.

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

Högbom, J. A.

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 417–426 (1974).

Juanola-Parramon, R.

R. Juanola-Parramon, D. M. Fenech, and G. Savini, “Architecture and performance of the space-based far-infrared interferometer instrument simulator,” Mon. Not. R. Astron. Soc. 457, 3457–3469 (2016).
[Crossref]

Leisawitz, D. T.

R. G. Lyon, D. T. Leisawitz, S. A. Rinehart, N. Memarsadeghi, and E. J. Sinukoff, “Wide-field imaging interferometry spatial-spectral image synthesis algorithms,” Proc. SPIE 8445, 84450B (2012).
[Crossref]

R. G. Lyon, S. A. Rinehart, D. T. Leisawitz, and N. Memarsadeghi, “Wide-field imaging interferometry testbed (WIIT): image construction algorithms,” Proc. SPIE 7013, 70131M (2008).
[Crossref]

N. Elias, M. Harwit, D. T. Leisawitz, and S. A. Rinehart, “The mathematics of double-Fourier interferometers,” Astrophys. J. 657, 1178–1200 (2007).
[Crossref]

D. B. Leviton, B. J. Frey, D. T. Leisawitz, A. J. Martino, W. L. Maynard, L. G. Mundy, S. A. Rinehart, S. H. Teng, and X. Zhang, “Wide-field imaging interferometry testbed 3: metrology subsystem,” Proc. SPIE 4852, 827–838 (2003).
[Crossref]

Leviton, D. B.

D. B. Leviton, B. J. Frey, D. T. Leisawitz, A. J. Martino, W. L. Maynard, L. G. Mundy, S. A. Rinehart, S. H. Teng, and X. Zhang, “Wide-field imaging interferometry testbed 3: metrology subsystem,” Proc. SPIE 4852, 827–838 (2003).
[Crossref]

Lyon, R. G.

R. G. Lyon, D. T. Leisawitz, S. A. Rinehart, N. Memarsadeghi, and E. J. Sinukoff, “Wide-field imaging interferometry spatial-spectral image synthesis algorithms,” Proc. SPIE 8445, 84450B (2012).
[Crossref]

R. G. Lyon, S. A. Rinehart, D. T. Leisawitz, and N. Memarsadeghi, “Wide-field imaging interferometry testbed (WIIT): image construction algorithms,” Proc. SPIE 7013, 70131M (2008).
[Crossref]

Mandel, L.

L. Mandel, “Concepts of cross-spectral purity in coherence theory,” J. Opt. Soc. Am. 51, 1342–1350 (1961).
[Crossref]

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

Mariotti, J.-M.

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

Martino, A. J.

D. B. Leviton, B. J. Frey, D. T. Leisawitz, A. J. Martino, W. L. Maynard, L. G. Mundy, S. A. Rinehart, S. H. Teng, and X. Zhang, “Wide-field imaging interferometry testbed 3: metrology subsystem,” Proc. SPIE 4852, 827–838 (2003).
[Crossref]

Mary, D.

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

Maynard, W. L.

D. B. Leviton, B. J. Frey, D. T. Leisawitz, A. J. Martino, W. L. Maynard, L. G. Mundy, S. A. Rinehart, S. H. Teng, and X. Zhang, “Wide-field imaging interferometry testbed 3: metrology subsystem,” Proc. SPIE 4852, 827–838 (2003).
[Crossref]

Memarsadeghi, N.

R. G. Lyon, D. T. Leisawitz, S. A. Rinehart, N. Memarsadeghi, and E. J. Sinukoff, “Wide-field imaging interferometry spatial-spectral image synthesis algorithms,” Proc. SPIE 8445, 84450B (2012).
[Crossref]

R. G. Lyon, S. A. Rinehart, D. T. Leisawitz, and N. Memarsadeghi, “Wide-field imaging interferometry testbed (WIIT): image construction algorithms,” Proc. SPIE 7013, 70131M (2008).
[Crossref]

Millour, F.

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

Monnier, J. D.

J. D. Monnier, “Optical interferometry in astronomy,” Rep. Prog. Phys. 66, 789–857 (2003).
[Crossref]

Mundy, L. G.

D. B. Leviton, B. J. Frey, D. T. Leisawitz, A. J. Martino, W. L. Maynard, L. G. Mundy, S. A. Rinehart, S. H. Teng, and X. Zhang, “Wide-field imaging interferometry testbed 3: metrology subsystem,” Proc. SPIE 4852, 827–838 (2003).
[Crossref]

Narayan, R.

R. Narayan and R. Nityananda, “Maximum entropy image restoration in astronomy,” Annu. Rev. Astron. Astrophys. 24, 127–170 (1986).
[Crossref]

Nityananda, R.

R. Narayan and R. Nityananda, “Maximum entropy image restoration in astronomy,” Annu. Rev. Astron. Astrophys. 24, 127–170 (1986).
[Crossref]

Ridgway, S. T.

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

Rinehart, S. A.

R. G. Lyon, D. T. Leisawitz, S. A. Rinehart, N. Memarsadeghi, and E. J. Sinukoff, “Wide-field imaging interferometry spatial-spectral image synthesis algorithms,” Proc. SPIE 8445, 84450B (2012).
[Crossref]

R. G. Lyon, S. A. Rinehart, D. T. Leisawitz, and N. Memarsadeghi, “Wide-field imaging interferometry testbed (WIIT): image construction algorithms,” Proc. SPIE 7013, 70131M (2008).
[Crossref]

N. Elias, M. Harwit, D. T. Leisawitz, and S. A. Rinehart, “The mathematics of double-Fourier interferometers,” Astrophys. J. 657, 1178–1200 (2007).
[Crossref]

D. B. Leviton, B. J. Frey, D. T. Leisawitz, A. J. Martino, W. L. Maynard, L. G. Mundy, S. A. Rinehart, S. H. Teng, and X. Zhang, “Wide-field imaging interferometry testbed 3: metrology subsystem,” Proc. SPIE 4852, 827–838 (2003).
[Crossref]

Sanchez-Bermudez, J.

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

Savini, G.

R. Juanola-Parramon, D. M. Fenech, and G. Savini, “Architecture and performance of the space-based far-infrared interferometer instrument simulator,” Mon. Not. R. Astron. Soc. 457, 3457–3469 (2016).
[Crossref]

Schertl, D.

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

Schutz, A.

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

Sinukoff, E. J.

R. G. Lyon, D. T. Leisawitz, S. A. Rinehart, N. Memarsadeghi, and E. J. Sinukoff, “Wide-field imaging interferometry spatial-spectral image synthesis algorithms,” Proc. SPIE 8445, 84450B (2012).
[Crossref]

Teng, S. H.

D. B. Leviton, B. J. Frey, D. T. Leisawitz, A. J. Martino, W. L. Maynard, L. G. Mundy, S. A. Rinehart, S. H. Teng, and X. Zhang, “Wide-field imaging interferometry testbed 3: metrology subsystem,” Proc. SPIE 4852, 827–838 (2003).
[Crossref]

Thiébaut, E.

E. Thiébaut and J. Young, “Principles of Image reconstruction in optical interferometry: tutorial,” J. Opt. Soc. Am. A 34, 904–923 (2017).
[Crossref]

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

Thurman, S. T.

Vannier, M.

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

Weigelt, G.

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

Wolf, E.

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

Young, J.

E. Thiébaut and J. Young, “Principles of Image reconstruction in optical interferometry: tutorial,” J. Opt. Soc. Am. A 34, 904–923 (2017).
[Crossref]

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

Zeilinger, A.

A. Zeilinger, “General properties of lossless beam splitters in interferometry,” Am. J. Phys. 49, 882–883 (1981).
[Crossref]

Zhang, X.

D. B. Leviton, B. J. Frey, D. T. Leisawitz, A. J. Martino, W. L. Maynard, L. G. Mundy, S. A. Rinehart, S. H. Teng, and X. Zhang, “Wide-field imaging interferometry testbed 3: metrology subsystem,” Proc. SPIE 4852, 827–838 (2003).
[Crossref]

Am. J. Phys. (1)

A. Zeilinger, “General properties of lossless beam splitters in interferometry,” Am. J. Phys. 49, 882–883 (1981).
[Crossref]

Annu. Rev. Astron. Astrophys. (1)

R. Narayan and R. Nityananda, “Maximum entropy image restoration in astronomy,” Annu. Rev. Astron. Astrophys. 24, 127–170 (1986).
[Crossref]

Astron. Astrophys. (2)

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 417–426 (1974).

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

Astrophys. J. (1)

N. Elias, M. Harwit, D. T. Leisawitz, and S. A. Rinehart, “The mathematics of double-Fourier interferometers,” Astrophys. J. 657, 1178–1200 (2007).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Mon. Not. R. Astron. Soc. (1)

R. Juanola-Parramon, D. M. Fenech, and G. Savini, “Architecture and performance of the space-based far-infrared interferometer instrument simulator,” Mon. Not. R. Astron. Soc. 457, 3457–3469 (2016).
[Crossref]

Opt. Express (2)

Proc. SPIE (4)

R. G. Lyon, S. A. Rinehart, D. T. Leisawitz, and N. Memarsadeghi, “Wide-field imaging interferometry testbed (WIIT): image construction algorithms,” Proc. SPIE 7013, 70131M (2008).
[Crossref]

R. G. Lyon, D. T. Leisawitz, S. A. Rinehart, N. Memarsadeghi, and E. J. Sinukoff, “Wide-field imaging interferometry spatial-spectral image synthesis algorithms,” Proc. SPIE 8445, 84450B (2012).
[Crossref]

J. Sanchez-Bermudez, E. Thiébaut, K.-H. Hofmann, M. Heininger, D. Schertl, G. Weigelt, F. Millour, A. Schutz, A. Ferrari, M. Vannier, D. Mary, and J. Young, “The 2016 interferometric imaging beauty contest,” Proc. SPIE 9907, 99071D (2016).
[Crossref]

D. B. Leviton, B. J. Frey, D. T. Leisawitz, A. J. Martino, W. L. Maynard, L. G. Mundy, S. A. Rinehart, S. H. Teng, and X. Zhang, “Wide-field imaging interferometry testbed 3: metrology subsystem,” Proc. SPIE 4852, 827–838 (2003).
[Crossref]

Rep. Prog. Phys. (1)

J. D. Monnier, “Optical interferometry in astronomy,” Rep. Prog. Phys. 66, 789–857 (2003).
[Crossref]

Other (2)

J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2000).

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

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Figures (3)

Fig. 1.
Fig. 1. Simplified diagram of a wide-field spatiospectral interferometer for a fixed baseline, showing coordinate relationships and basic system configuration.
Fig. 2.
Fig. 2. Illustration of the information content probed by the datacube associated with each baseline vector. Each datacube maps out a truncated oblique cone in the spectral-spatial frequency domain whose location depends on the particular value of the baseline vector.
Fig. 3.
Fig. 3. Simplified diagram of a wide-field spatiospectral interferometer an additional beam splitter and mirror to demonstrate that the same system can be used for conventional Fourier transform imaging spectroscopy.

Equations (73)

Equations on this page are rendered with MathJax. Learn more.

α=(α,β)=(ξ,η)z=ξz,
Ein(xin,α;κ)=γ1ei2πκ(zct)exp(iπκzxin2)Es(α;κ)×exp(iπκzα2)exp(i2πκxin·α),
γ1=izκ
Eap(xin,α;κ)=A(xin;κ)Ein(xin,α;κ),
E1,ap(xin,α;κ)=A1(xinB2;κ)Ein(xin,α;κ),
E2,ap(xin,α;κ)=A2(xin+B2;κ)Ein(xin,α;κ),
An(xin;κ)={an(xn,ap;κ)eiϕn(xn,ap;κ)0,|xin|D/2,|xin|>D/2,
E1,pup(xpup,α;κ)=E1,ap(xpup+B2,α;κ)=A1(xpup;κ)Ein(xpup+B2,α;κ)=γ1eiϕ1(κ)ei2πκ[zc(t+τ1)]exp[iπκz(xpup+12B)2]Es(α;κ)×A1(xpup;κ)exp(iπκzα2)exp[i2πκα·(xpup+12B)],
E2,pup(xpup,α;κ)=E2,ap(xpupB2,α;κ)=γ1eiϕ2(κ)ei2πκ[zc(t+τ2)]exp[iπκz(xpup12B)2]Es(α;κ)×A2(xpup;κ)exp(iπκzα2)exp[i2πκα·(xpup12B)],
E3=rbsE1+tbsE2,
E4=tbsE1+rbsE2,
Es*(α;κ)Es(α;κ)=Ws(α,α;κ)δ(κκ)σSs(α;κ)δ(αα,κκ),
|Es(α;κ)|2σSs(α;κ),
=1tbtatatbdt.
θ=(θx,θy)=(ximg,yimg)f=ximgf,
E1,im(θ,α;κ)=κifE1,pup(xpup,α;κ)ei2πκθ·xpupd2xpup=γ2eiϕ1(κ)exp(iπκ4zB2)ei2πκ[zc(t+τ1)]Es(α;κ)×exp(iπκzα2)eiπκα·BA1(xpup;κ)exp(iπκzxpup2)×exp[i2πκ(θαB2z)·xpup]d2xpup=γ2eiϕ1(κ)exp(iπκ4zB2)ei2πκ[zc(t+τ1)]exp(iπκzα2),×h1[(θB2z)α;κ]Es(α;κ)eiπκα·B,
E2,im(θ,α;κ)=κifE2,pup(xpup,α;κ)ei2πκθ·xpupd2xpup=γ2eiϕ2(κ)exp(iπκ4zB2)ei2πκ[zc(t+τ2)]exp(iπκzα2)×h2[(θ+B2z)α;κ]Es(α;κ)eiπκα·B,
hn(θ;κ)=An(xpup;κ)exp(iπκzxpup2)×exp(i2πκθ·xpup)d2xpup
γ2=γ1κif=izκκif=zf.
I1(θ)=|011E1,im(θ,α;κ)d2αdκ|2=|γ2|2001111h1*[(θB2z)α;κ]×h1[(θB2z)α;κ]Es*(α;κ)Es(α;κ)×d2αd2αdκdκ=|γ3|2011p1,1[(θB2z)α;κ]Ss(α;κ)d2αdκ,
I2(θ)=|011E2,im(θ,α;κ)d2αdκ|2=|γ3|2011p2,2[(θ+B2z)α;κ]Ss(α;κ)d2αdκ,
γ3=σ1/2γ2κif=1π1/2κzfκif=ziπ1/2f2,
pn,n(θ,κ)=|hn(θ,κ)|2.
I3(θ)=|011E3,im(θ,α;κ)d2αdκ|2=|011[rbsE1,im(θ,α;κ)+tbsE2,im(θ,α;κ)]d2αdκ|2=RI1(θ)+TI2(θ)+2Re[rbs*tbs001111E1,im*(θ,α;κ)×E2,im(θ,α;κ)d2αd2αdκdκ]=12[I1(θ)+I2(θ)]+Re[eiπ200W12im(θ;κ)δ(κκ)dκdκ]=12[I1(θ)+I2(θ)]Im[0W12im(θ;κ)dκ],
I4(θ)=|011E4,im(θ,α;κ)d2αdκ|2=12[I1(θ)+I2(θ)]+Im[0W12im(θ;κ)dκ],
W12im(θ;κ)=1111E1,im*(θ,α;κ)E2,im(θ,α;κ)d2αd2α=|γ2|21111eiΔϕ(κ)exp(iπκ4zB2)eiπκzα2ei2πκ[zc(t+τ1)]×eiπκα·Bexp(iπκ4zB2)eiπκzα2ei2πκ[zc(t+τ2)]eiπκα·B×h1*[(θB2z)α;κ]h2[(θ+B2z)α;κ]×Es*(α;κ)Es(α;κ)d2αd2α=|γ3|2eiΔϕ(κ)ei2πκcΔτ×11h1*[(θB2z)α;κ]h2[(θ+B2z)α;κ]×ei2πκα·BSs(α;κ)d2α
I3,4(θ)=12[I1(θ)+I2(θ)]Im[0W12im(θ;κ)dκ].
|B|max2z1(κmax|B|max)11,z12κmax|B|max2,
I1(θ)=|γ3|2011p1,1(θα;κ)Ss(α;κ)d2αdκ,
I2(θ)=|γ3|2011p2,2(θα;κ)Ss(α;κ)d2αdκ,
W12im(θ;κ)=|γ3|2eiΔϕ(κ)ei2πκΔL×11p1,2(θα;κ)ei2πκα·BSs(α;κ)d2α,
pm,n(θ;κ)=hm*(θ;κ)hn(θ;κ),
I3,4(θ)=12[I1(θ)+I2(θ)]|γ3|2011Ss(α;κ)×Im[eiΔϕ(κ)p1,2(θα;κ)ei2πκ(α·B+ΔL)]d2αdκ.
p1,1(θ;κ)=p2,2(θ;κ)=p1,2(θ;κ),
I1(θ)=I2(θ),
I3,4(θ)=I1(θ)|γ3|2011p1,1(θα;κ)Ss(α;κ)×Im[eiΔϕ(κ)ei2πκ(α·B+ΔL)]d2αdκ=I1(θ)±|γ3|2011p1,1(θα;κ)Ss(α;κ)×sin[2πκ(α·B+ΔL)Δϕ(κ)]d2αdκ.
resolution:Δκmin1ΔLmaxΔLmin=12ΔLmax,
range:κmaxNL2Δκmin=NL4ΔLmax,
Δαmin=(κmax|B|max)1=λmin|B|max.
ΔLZPD=α·B.
FOV=|α|max|α|minΔLmin|B|max+ΔLmax|B|max=2ΔLmax|B|max.
I3,4(θ,B;ΔL)=Ibias(θ)|γ3|2011Ss(α;κ)×Im[eiΔϕ(κ)p1,2(θα;κ)ei2πκ(α·B+ΔL)]d2αdκ,
Ibias(θ)=12[I1(θ)+I2(θ)]limΔLmax12ΔLmaxΔLmaxΔLmaxI3,4(θ,B;ΔL)dΔL,
Ibsr(θ,B;ΔL)=I3,4(θ,B;ΔL)Ibias(θ)=|γ3|2011Ss(α;κ)×Im[eiΔϕ(κ)p1,2(θα;κ)ei2πκ(α·B+ΔL)]d2αdκ.
Ibsi(θ,B;ΔL)=H[Ibsr(θ,B;ΔL)]=±|γ3|2011Ss(α;κ)×Re[eiΔϕ(κ)p1,2(θα;κ)ei2πκ(α·B+ΔL)]d2αdκ,
Ibs(θ,B;ΔL)=Ibsr(θ,B;ΔL)+iIbsi(θ,B;ΔL)=±i|γ3|2011p1,2(θα;κ)Ss(α;κ)×exp[iΔϕ(κ)i2πκ(α·B+ΔL)]d2αdκ.
Si(θ,B;κ)=ΔLmaxΔLmaxIbs(θ,B;ΔL)ei2πκΔLdΔL=±i|γ3|20(ΔLmaxΔLmaxei2πκΔLei2πκΔLdΔL)11p1,2(θα;κ)×Ss(α;κ)exp[iΔϕ(κ)i2πκ(α·B+ΔL)]d2αdκ=±i|γ3|202ΔLmaxsinc[2ΔLmax(κκ)]11p1,2(θα;κ)×Ss(α;κ)eiΔϕ(κ)exp(i2πκα·B)d2αdκ=±i2ΔLmax|γ3|2sinc(2ΔLmaxκ)*κ[11p1,2(θα;κ)×Ss(α;κ)eiΔϕ(κ)exp(i2πκα·B)d2α]=±i2ΔLmax|γ3|2sinc(2ΔLmaxκ)*κ{[Ss(θ;κ)exp(i2πκθ·B)]*θp1,2(θ;κ)eiΔϕ(κ)},
S˜i(fθ,B;κ)=11Si(θ,B;κ)exp(i2πθ·fθ)d2θ=±i2ΔLmax|γ3|2110sinc[2ΔLmax(κκ)]×{[Ss(θ;κ)exp(i2πκθ·B)]*θp1,2(θ;κ)×eiΔϕ(κ)}exp(i2πθ·fθ)dκd2θ=±i2ΔLmax|γ3|2110sinc[2ΔLmax(κκ)]×[S˜s(fθκB;κ)p˜1,2(fθ;κ)eiΔϕ(κ)]dκ=±i2ΔLmax|γ3|2sinc(2ΔLmaxκ)*κ[S˜s(fθκB;κ)p˜1,2(fθ;κ)eiΔϕ(κ)],
p˜1,2(fθ;κ)=p1,2(θ;κ)exp(i2πθ·fθ)d2θ
Sihr(θ;κ)=n=1NbSi(θ,Bn;κ)ei2πκθ·Bn=±i2ΔLmax|γ3|2n=1Nb{0sinc[2ΔLmax(κκ)]×11Ss(α;κ)p1,2(θα;κ)eiΔϕ(κ)×exp[i2π(κθκα)·Bn]d2αdκ}.
Sihr(θ;κ)=±i2ΔLmax|γ3|2n=1Nb{0sinc[2ΔLmax(κκ)]×exp[i2π(κκ)θ·Bn]11Ss(α;κ)eiΔϕ(κ)×p1,2(θα;κ)exp[i2πκ(θα)·Bn]d2αdκ}=±i2ΔLmax|γ3|2n=1Nb[sinc(2ΔLmaxκ)×exp(i2πκθ·Bn)]*κ{Ss(θ;κ)*θ[p1,2(θ;κ)eiΔϕ(κ)×exp(i2πκθ·Bn)]}.
S˜ihr(fθ;κ)=11Sihr(θ;κ)exp(i2πθ·fθ)d2θ=±i2ΔLmax|γ3|211n=1Nb0sinc[2ΔLmax(κκ)]×exp[i2π(κκ)θ·Bn]{Ss(θ;κ)*θ[p1,2(θ;κ)eiΔϕ(κ)exp(i2πκθ·Bn)]}×exp(i2πθ·fθ)dκd2θ=±i2ΔLmax|γ3|20sinc[2ΔLmax(κκ)]×n=1Nb{S˜s[fθ+(κκ)Bn;κ]p˜1,2(fθ+κBn;κ)}dκ.
OTFeff(fθ;κ)n=1Nbp˜1,2(fθ+κBn;κ).
S˜ihr(fθ;κ)=n=1NbS˜i(fθ+κBn,Bn;κ),
I1(xpup)=|011E1,pup(xpup,α;κ)d2αdκ|2=|γ1|20|A1(xpup;κ)|211|Es(α;κ)|2d2αdκ,
I2(xpup)=|γ1|20|A2(xpup;κ)|211|Es(α;κ)|2d2αdκ.
I3(xpup)=|011E3,pup(xpup,α;κ)d2αdκ|2=|011[rbsE1,pup(xpup,α;κ)+tbsE2,pup(xpup,α;κ)]d2αdκ|2=RI1(xpup)+TI2(xpup)+2Re[rbs*tbs001111E1,pup*(xpup,α;κ)×E2,pup(xpup,α;κ)d2αd2αdκdκ]=12[I1(xpup)+I2(xpup)]Im[001111E1,pup*(xpup,α;κ)×E2,pup(xpup,α;κ)d2αd2αdκdκ],
I4(xpup)=12[I1(xpup)+I2(xpup)]+Im[001111E1,pup*(xpup,α;κ)×E2,pup(xpup,α;κ)d2αd2αdκdκ],
I3,4(xpup)=12[I1(xpup)+I2(xpup)]Im[001111E1,pup*(xpup,α;κ)×E2,pup(xpup,α;κ)d2αd2αdκdκ].
E1*(κ)E2(κ)=W12(κ)δ(κκ).
I3,4(xpup)=12[I1(xpup)+I2(xpup)]Im[0W12pup(xpup,κ)dκ],
W12pup(xpup,κ)=1111E1,pup*(xpup,α;κ)×E2,pup(xpup,α;κ)d2αd2α
W12pup(xpup,κ)=|γ1|21111eiΔϕ(κ)ei2πκ[zc(t+τ1)]exp[iπκz(xpup+12B)2]×Es*(α;κ)A1*(xpup;κ)eiπκzα2exp[i2πκα·(xpup+12B)]×ei2πκ[zc(t+τ2)]exp[iπκz(xpup12B)2]Es(α;κ)×A2(xpup;κ)eiπκzα2exp[i2πκα·(xpup12B)]d2αd2α=|γ1|21111eiΔϕ(κ)ei2πκcΔτexp(i2πκzxpup·B)×Es*(α;κ)Es(α;κ)A1*(xpup;κ)A2(xpup;κ)×exp[iπκz(α2α2)]×exp{iπκ[(α+α)·B2(αα)·xpup]}d2αd2α.
Es*(α;κ)Es(α;κ)=Ws(α,α;κ)δ(κκ)σSs(α;κ)δ(αα,κκ).
|Es(α;κ)|2σSs(α;κ).
I1(xpup)=|γ4|20|A1(xpup;κ)|211Ss(α;κ)d2αdκ,
I2(xpup)=|γ4|20|A2(xpup;κ)|211Ss(α;κ)d2αdκ,
W12pup(xpup,κ)=|γ4|211eiΔϕ(κ)ei2πκcΔτA1*(xpup;κ)A2(xpup;κ)×exp(i2πκzxpup·B)W12in(α;κ)d2α,
W12in(α;κ)=Ss(α;κ)ei2πκα·B,
γ4=γ1σ1/2=izπ1/2κ2.
Γ12(τ)=0W12(ν)ei2πντdν=c0W12(κ)ei2πκcτdκ.
I3,4(xpup)=12[I1(xpup)+I2(xpup)]c1|γ4|2Im{eiΔϕA1*(xpup)A2(xpup)×11c0exp[i2πcκ(Δτ+xpup·Bcz)]W12in(α,κ)dκd2α}=12[I1(xpup)+I2(xpup)]c1|γ4|2Im[eiΔϕA1*(xpup)A2(xpup)×11Γ12in(α,Δτ+xpup·Bcz)d2α],
I3,4(xpup)=12[I1(xpup)+I2(xpup)]|γ4|2Im{110eiΔϕ(κ)A1*(xpup,κ)A2(xpup,κ)×exp[i2πcκ(Δτ+xpup·Bcz+α·Bc)]×Ss(α;κ)dκd2α}.

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