Theoretical aspects of Fourier Transform Spectrometry and common path triangular interferometers

Alessandro Barducci; Donatella Guzzi; Cinzia Lastri; Paolo Marcoionni; Vanni Nardino; Ivan Pippi

doi:10.1364/OE.18.011622

Theoretical aspects of Fourier Transform Spectrometry and common path triangular interferometers

Alessandro Barducci, Donatella Guzzi, Cinzia Lastri, Paolo Marcoionni, Vanni Nardino, and Ivan Pippi

Optics Express
Vol. 18,
Issue 11,
pp. 11622-11649
(2010)
•https://doi.org/10.1364/OE.18.011622

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Alessandro Barducci, Donatella Guzzi, Cinzia Lastri, Paolo Marcoionni, Vanni Nardino, and Ivan Pippi, "Theoretical aspects of Fourier Transform Spectrometry and common path triangular interferometers," Opt. Express 18, 11622-11649 (2010)

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Related Topics
Table of Contents Category
- Spectroscopy
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Remote sensing and sensors (120.0280)
- Interferometry (120.3180)
- Passive remote sensing (280.4991)
- Spectrometers (300.6190)
- Spectroscopy, Fourier transforms (300.6300)

About this Article
History
- Original Manuscript: February 24, 2010
- Revised Manuscript: March 17, 2010
- Manuscript Accepted: March 19, 2010
- Published: May 18, 2010

Accessible

Open Access

Abstract

Recent investigations have induced relevant advancements of imaging interferometry, which is becoming a viable option for Earth remote sensing. Various research programs have chosen the Sagnac configuration for new imaging interferometers. Due to the growing diffusion of this technique, we have developed a self-contained theory for describing the signal produced by triangular FTSs and its optimal processing. We investigate the relevant disadvantages of multiplexing, and compare dispersive with FTS instruments. The paper addresses some methods for correcting the phase error, and the non-unitary transformation performed by a Sagnac interferometer. The effect of noise on spectral estimations is discussed.

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References

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Year
|
Author
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Publication

M. J. Persky, “A review of spaceborne infrared Fourier transform spectrometers for remote sensing,” Rev. Sci. Instrum. 66(10), 4763–4797 (1995).
[Crossref]
B. Harnisch, W. Posselt, K. Holota, H. O. Tittel, and M. Rost, “Compact Fourier-transform imaging spectrometer for small satellite missions,” Acta Astronaut. 52(9-12), 803–811 (2003).
[Crossref]
M.-L. Junttila, “Stationary Fourier transform spectrometer,” Appl. Opt. 31(21), 4106–4112 (1992).
[Crossref] [PubMed]
L. J. Otten, A. D. Meigs, B. A. Jones, P. Prinzing, and D. S. Fronterhouse, “Payload Qualification and Optical Performance Test Results for the MightySat II.1 Hyperspectral Imager,” Proc. SPIE 3498, 231–238 (1998).
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[Crossref]
L. J. Otten, R. G. Sellar, and J. B. Rafert, “MightySatII.1 Fourier transform hyperspectral imager payload performance,” Proc. SPIE 2583, 566–575 (1995).
[Crossref]
P. G. Lucey, K. A. Horton, and T. Williams, “Performance of a long-wave infrared hyperspectral imager using a Sagnac interferometer and an uncooled microbolometer array,” Appl. Opt. 47(28), F107–F113 (2008).
[Crossref] [PubMed]
A. Barducci, P. Marcoionni, I. Pippi, and M. Poggesi, “Simulation of the Performance of a Stationary Imaging Interferometer for High Resolution Monitoring of the Earth,” Proc. SPIE 4540, 112–121 (2001).
[Crossref]
A. Barducci, A. Casini, F. Castagnoli, P. Marcoionni, M. Morandi, and I. Pippi, “Performance assessment of a Stationary Interferometer for High-Resolution Remote Sensing,” Proc. SPIE 4725, 547–555 (2002).
[Crossref]
M. Bliss, “Demonstration of a static Fourier transform spectrometer,” Proc. SPIE 3541, 103–109 (1999).
[Crossref]
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[Crossref]
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[Crossref]
R. F. Horton, “Optical Design for High Ètendue Imaging Fourier Transform Spectrometer,” Proc. SPIE 2819, 300–315 (1996).
[Crossref]
D. Cabib, R. A. Buckwald, Y. Garin, and D. G. Soenksen, “Spatially resolved Fourier transform spectroscopy (spectral imaging): a powerful tool for quantitative analytical microscopy”, in Optical diagnostics of living cells on biofluids,” Proc. SPIE 2678, 278–291 (1996).
[Crossref]
J. Genest, P. Tremblay, and A. Villemaire, “Throughput of tilted interferometers,” Appl. Opt. 37(21), 4819–4822 (1998).
[Crossref]
P. Jacquinot, “The luminosity of spectrometers with Prisms, Grating, or Fabry-Perot Etalons,” J. Opt. Soc. Am. 44(10), 761–765 (1954).
[Crossref]
M. R. Descour, “The Throughput Advantage In Imaging Fourier-Transform Spectrometers,” Proc. SPIE 2819, 285–290 (1997).
[Crossref]
R. G. Sellar and G. D. Boreman, “Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer,” Appl. Opt. 44(9), 1614–1624 (2005).
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[Crossref] [PubMed]
M.-L. Junttila, J. Kauppinen, and E. Ikonen, “Performance limits of stationary Fourier spectrometers,” J. Opt. Soc. Am. A 8(9), 1457–1462 (1991).
[Crossref]
R. L. Hilliard and G. G. Shepherd, “Wide-Angle Michelson Interferometer for Measuring Doppler Line Widths,” J. Opt. Soc. Am. 56(3), 362–369 (1966).
[Crossref]
P. Hlubina, D. Ciprian, J. Lunacek, and R. Chlebus, “Phase retrieval from the spectral interference signal used to measure thickness of SiO2 thin film on silicon wafer,” Appl. Phys. B 88(3), 397–403 (2007), doi:.
[Crossref]
L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7(1), 17–23 (1967).
[Crossref]
H. Sakai, G. A. Vanasse, and M. L. Forman, “Spectral Recovery in Fourier Spectroscopy,” J. Opt. Soc. Am. 58(1), 84–90 (1968).
[Crossref]
M. L. Forman, W. H. Steel, and G. A. Vanasse, “Correction of Asymmetric Interferograms Obtained in Fourier Spectroscopy,” J. Opt. Soc. Am. 56(1), 59–63 (1966).
[Crossref]
J. Connes, “Recherches sur la spectroscopie par transformation de Fourier,” Revue d’Optique 40, 45–265 (1961).
T. Okamoto, S. Kawata, and S. Minami, “Fourier transform spectrometer with a self-scanning photodiode array,” Appl. Opt. 23(2), 269–273 (1984).
[Crossref] [PubMed]
A. Papoulis, Probability Random Variables, and Stochastic Processes, (McGraw-Hill International Editions, Third Edition, 1991).
P. R. Griffiths, H. J. Sloane, and R. W. Hannah, “Interferometers vs monochromators: separating the optical and digital advantages,” Appl. Spectrosc. 31(6), 485–495 (1977).
[Crossref]
R. J. Bell, Introductory Fourier Transform Spectroscopy, (Academic Press, New York and London, 1972).
P. B. Fellgett, “Conclusions on multiplex methods,” Journal de Physique, Colloque C2, Supplément au n. 3–4 Tome 28, mars-avril 1967, pp.: 165–171, (1967).
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[Crossref]
P. B. Fellgett, “The nature and origin of multiplex Fourier spectrometry,” Notes Rec. R. Soc. 60(1), 91–93 (2006).
[Crossref]
F. D. Kahn, “The signal: noise ratio of a suggested spectral analyzer,” Astrophys. J. 129, 518–520 (1959).
[Crossref]
W. Schumann, and T. S. Lomheim, “Infrared hyperspectral imaging Fourier transform and dispersive spectrometers: comparison of signal-to-noise-based performance,” Imaging Spectrometry VII, San Diego, CA, USA, SPIE vol. 4480, (2001).

2008 (1)

P. G. Lucey, K. A. Horton, and T. Williams, “Performance of a long-wave infrared hyperspectral imager using a Sagnac interferometer and an uncooled microbolometer array,” Appl. Opt. 47(28), F107–F113 (2008).
[Crossref] [PubMed]

2007 (1)

P. Hlubina, D. Ciprian, J. Lunacek, and R. Chlebus, “Phase retrieval from the spectral interference signal used to measure thickness of SiO2 thin film on silicon wafer,” Appl. Phys. B 88(3), 397–403 (2007), doi:.
[Crossref]

2006 (2)

P. B. Fellgett, “The nature and origin of multiplex Fourier spectrometry,” Notes Rec. R. Soc. 60(1), 91–93 (2006).
[Crossref]

Y. Ferrec, J. Taboury, H. Sauer, and P. Chavel, “Optimal geometry for Sagnac and Michelson interferometers used as spectral imagers,” Opt. Eng. 45(11), 115601-115606 (2006).
[Crossref]

2005 (1)

R. G. Sellar and G. D. Boreman, “Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer,” Appl. Opt. 44(9), 1614–1624 (2005).
[Crossref] [PubMed]

2003 (2)

S. Subramaniam, B. Y. Ravindra, B. Rabindranath, B. G. Basheerullah, P. V. Viswanath, and O. P. Bajpai, “Stationary spatially modulated fourier transform spectro-radiometer,” J. Indian Soc. Remote Sens. 31(3), 187–196 (2003).
[Crossref]

B. Harnisch, W. Posselt, K. Holota, H. O. Tittel, and M. Rost, “Compact Fourier-transform imaging spectrometer for small satellite missions,” Acta Astronaut. 52(9-12), 803–811 (2003).
[Crossref]

2002 (1)

A. Barducci, A. Casini, F. Castagnoli, P. Marcoionni, M. Morandi, and I. Pippi, “Performance assessment of a Stationary Interferometer for High-Resolution Remote Sensing,” Proc. SPIE 4725, 547–555 (2002).
[Crossref]

2001 (1)

A. Barducci, P. Marcoionni, I. Pippi, and M. Poggesi, “Simulation of the Performance of a Stationary Imaging Interferometer for High Resolution Monitoring of the Earth,” Proc. SPIE 4540, 112–121 (2001).
[Crossref]

1999 (1)

M. Bliss, “Demonstration of a static Fourier transform spectrometer,” Proc. SPIE 3541, 103–109 (1999).
[Crossref]

1998 (2)

L. J. Otten, A. D. Meigs, B. A. Jones, P. Prinzing, and D. S. Fronterhouse, “Payload Qualification and Optical Performance Test Results for the MightySat II.1 Hyperspectral Imager,” Proc. SPIE 3498, 231–238 (1998).
[Crossref]

J. Genest, P. Tremblay, and A. Villemaire, “Throughput of tilted interferometers,” Appl. Opt. 37(21), 4819–4822 (1998).
[Crossref]

1997 (1)

M. R. Descour, “The Throughput Advantage In Imaging Fourier-Transform Spectrometers,” Proc. SPIE 2819, 285–290 (1997).
[Crossref]

1996 (2)

R. F. Horton, “Optical Design for High Ètendue Imaging Fourier Transform Spectrometer,” Proc. SPIE 2819, 300–315 (1996).
[Crossref]

D. Cabib, R. A. Buckwald, Y. Garin, and D. G. Soenksen, “Spatially resolved Fourier transform spectroscopy (spectral imaging): a powerful tool for quantitative analytical microscopy”, in Optical diagnostics of living cells on biofluids,” Proc. SPIE 2678, 278–291 (1996).
[Crossref]

1995 (2)

L. J. Otten, R. G. Sellar, and J. B. Rafert, “MightySatII.1 Fourier transform hyperspectral imager payload performance,” Proc. SPIE 2583, 566–575 (1995).
[Crossref]

M. J. Persky, “A review of spaceborne infrared Fourier transform spectrometers for remote sensing,” Rev. Sci. Instrum. 66(10), 4763–4797 (1995).
[Crossref]

1993 (1)

P. D. Hammer, F. P. J. Valero, and D. L. Peterson, “An imaging interferometer for terrestrial remote sensing,” Proc. SPIE 1937, 244–255 (1993).
[Crossref]

1967 (1)

L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7(1), 17–23 (1967).
[Crossref]

1966 (2)

M. L. Forman, W. H. Steel, and G. A. Vanasse, “Correction of Asymmetric Interferograms Obtained in Fourier Spectroscopy,” J. Opt. Soc. Am. 56(1), 59–63 (1966).
[Crossref]

R. L. Hilliard and G. G. Shepherd, “Wide-Angle Michelson Interferometer for Measuring Doppler Line Widths,” J. Opt. Soc. Am. 56(3), 362–369 (1966).
[Crossref]

1961 (1)

J. Connes, “Recherches sur la spectroscopie par transformation de Fourier,” Revue d’Optique 40, 45–265 (1961).

1959 (1)

F. D. Kahn, “The signal: noise ratio of a suggested spectral analyzer,” Astrophys. J. 129, 518–520 (1959).
[Crossref]

1958 (1)

P. B. Fellgett, “I. — les principes généraux des méthodes nouvelles en spectroscopie interférentielle A propos de la théorie du spectromètre interférentiel multiplex,” J. Phys. Radium 19(3), 187–191 (1958).
[Crossref]

1954 (1)

P. Jacquinot, “The luminosity of spectrometers with Prisms, Grating, or Fabry-Perot Etalons,” J. Opt. Soc. Am. 44(10), 761–765 (1954).
[Crossref]

Bajpai, O. P.

Barducci, A.

Basheerullah, B. G.

Bliss, M.

M. Bliss, “Demonstration of a static Fourier transform spectrometer,” Proc. SPIE 3541, 103–109 (1999).
[Crossref]

Boreman, G. D.

R. G. Sellar and G. D. Boreman, “Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer,” Appl. Opt. 44(9), 1614–1624 (2005).
[Crossref] [PubMed]

Buckwald, R. A.

Cabib, D.

Casini, A.

Castagnoli, F.

Chavel, P.

Y. Ferrec, J. Taboury, H. Sauer, and P. Chavel, “Optimal geometry for Sagnac and Michelson interferometers used as spectral imagers,” Opt. Eng. 45(11), 115601-115606 (2006).
[Crossref]

Chlebus, R.

Ciprian, D.

Clark, T. A.

Connes, J.

J. Connes, “Recherches sur la spectroscopie par transformation de Fourier,” Revue d’Optique 40, 45–265 (1961).

Descour, M. R.

M. R. Descour, “The Throughput Advantage In Imaging Fourier-Transform Spectrometers,” Proc. SPIE 2819, 285–290 (1997).
[Crossref]

Fellgett, P. B.

P. B. Fellgett, “The nature and origin of multiplex Fourier spectrometry,” Notes Rec. R. Soc. 60(1), 91–93 (2006).
[Crossref]

Ferrec, Y.

Y. Ferrec, J. Taboury, H. Sauer, and P. Chavel, “Optimal geometry for Sagnac and Michelson interferometers used as spectral imagers,” Opt. Eng. 45(11), 115601-115606 (2006).
[Crossref]

Forman, M. L.

H. Sakai, G. A. Vanasse, and M. L. Forman, “Spectral Recovery in Fourier Spectroscopy,” J. Opt. Soc. Am. 58(1), 84–90 (1968).
[Crossref]

M. L. Forman, W. H. Steel, and G. A. Vanasse, “Correction of Asymmetric Interferograms Obtained in Fourier Spectroscopy,” J. Opt. Soc. Am. 56(1), 59–63 (1966).
[Crossref]

Fronterhouse, D. S.

Garin, Y.

Genest, J.

J. Genest, P. Tremblay, and A. Villemaire, “Throughput of tilted interferometers,” Appl. Opt. 37(21), 4819–4822 (1998).
[Crossref]

Griffiths, P. R.

P. R. Griffiths, H. J. Sloane, and R. W. Hannah, “Interferometers vs monochromators: separating the optical and digital advantages,” Appl. Spectrosc. 31(6), 485–495 (1977).
[Crossref]

Hammer, P. D.

P. D. Hammer, F. P. J. Valero, and D. L. Peterson, “An imaging interferometer for terrestrial remote sensing,” Proc. SPIE 1937, 244–255 (1993).
[Crossref]

Hannah, R. W.

P. R. Griffiths, H. J. Sloane, and R. W. Hannah, “Interferometers vs monochromators: separating the optical and digital advantages,” Appl. Spectrosc. 31(6), 485–495 (1977).
[Crossref]

Harnisch, B.

Hilliard, R. L.

R. L. Hilliard and G. G. Shepherd, “Wide-Angle Michelson Interferometer for Measuring Doppler Line Widths,” J. Opt. Soc. Am. 56(3), 362–369 (1966).
[Crossref]

Hlubina, P.

Holota, K.

Horton, K. A.

Horton, R. F.

R. F. Horton, “Optical Design for High Ètendue Imaging Fourier Transform Spectrometer,” Proc. SPIE 2819, 300–315 (1996).
[Crossref]

Ikonen, E.

M.-L. Junttila, J. Kauppinen, and E. Ikonen, “Performance limits of stationary Fourier spectrometers,” J. Opt. Soc. Am. A 8(9), 1457–1462 (1991).
[Crossref]

Jacquinot, P.

P. Jacquinot, “The luminosity of spectrometers with Prisms, Grating, or Fabry-Perot Etalons,” J. Opt. Soc. Am. 44(10), 761–765 (1954).
[Crossref]

Jennings, R. E.

Jones, B. A.

Junttila, M.-L.

M.-L. Junttila, “Stationary Fourier transform spectrometer,” Appl. Opt. 31(21), 4106–4112 (1992).
[Crossref] [PubMed]

M.-L. Junttila, J. Kauppinen, and E. Ikonen, “Performance limits of stationary Fourier spectrometers,” J. Opt. Soc. Am. A 8(9), 1457–1462 (1991).
[Crossref]

Kahn, F. D.

F. D. Kahn, “The signal: noise ratio of a suggested spectral analyzer,” Astrophys. J. 129, 518–520 (1959).
[Crossref]

Kauppinen, J.

M.-L. Junttila, J. Kauppinen, and E. Ikonen, “Performance limits of stationary Fourier spectrometers,” J. Opt. Soc. Am. A 8(9), 1457–1462 (1991).
[Crossref]

Kawata, S.

T. Okamoto, S. Kawata, and S. Minami, “Fourier transform spectrometer with a self-scanning photodiode array,” Appl. Opt. 23(2), 269–273 (1984).
[Crossref] [PubMed]

Lucey, P. G.

Lunacek, J.

Marcoionni, P.

Meigs, A. D.

Mertz, L.

L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7(1), 17–23 (1967).
[Crossref]

Minami, S.

T. Okamoto, S. Kawata, and S. Minami, “Fourier transform spectrometer with a self-scanning photodiode array,” Appl. Opt. 23(2), 269–273 (1984).
[Crossref] [PubMed]

Morandi, M.

Okamoto, T.

T. Okamoto, S. Kawata, and S. Minami, “Fourier transform spectrometer with a self-scanning photodiode array,” Appl. Opt. 23(2), 269–273 (1984).
[Crossref] [PubMed]

Otten, L. J.

L. J. Otten, R. G. Sellar, and J. B. Rafert, “MightySatII.1 Fourier transform hyperspectral imager payload performance,” Proc. SPIE 2583, 566–575 (1995).
[Crossref]

Persky, M. J.

M. J. Persky, “A review of spaceborne infrared Fourier transform spectrometers for remote sensing,” Rev. Sci. Instrum. 66(10), 4763–4797 (1995).
[Crossref]

Peterson, D. L.

P. D. Hammer, F. P. J. Valero, and D. L. Peterson, “An imaging interferometer for terrestrial remote sensing,” Proc. SPIE 1937, 244–255 (1993).
[Crossref]

Pippi, I.

Poggesi, M.

Posselt, W.

Prinzing, P.

Rabindranath, B.

Rafert, J. B.

L. J. Otten, R. G. Sellar, and J. B. Rafert, “MightySatII.1 Fourier transform hyperspectral imager payload performance,” Proc. SPIE 2583, 566–575 (1995).
[Crossref]

Ravindra, B. Y.

Rost, M.

Sakai, H.

H. Sakai, G. A. Vanasse, and M. L. Forman, “Spectral Recovery in Fourier Spectroscopy,” J. Opt. Soc. Am. 58(1), 84–90 (1968).
[Crossref]

Sauer, H.

Y. Ferrec, J. Taboury, H. Sauer, and P. Chavel, “Optimal geometry for Sagnac and Michelson interferometers used as spectral imagers,” Opt. Eng. 45(11), 115601-115606 (2006).
[Crossref]

Sellar, R. G.

R. G. Sellar and G. D. Boreman, “Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer,” Appl. Opt. 44(9), 1614–1624 (2005).
[Crossref] [PubMed]

L. J. Otten, R. G. Sellar, and J. B. Rafert, “MightySatII.1 Fourier transform hyperspectral imager payload performance,” Proc. SPIE 2583, 566–575 (1995).
[Crossref]

Shepherd, G. G.

R. L. Hilliard and G. G. Shepherd, “Wide-Angle Michelson Interferometer for Measuring Doppler Line Widths,” J. Opt. Soc. Am. 56(3), 362–369 (1966).
[Crossref]

Sloane, H. J.

P. R. Griffiths, H. J. Sloane, and R. W. Hannah, “Interferometers vs monochromators: separating the optical and digital advantages,” Appl. Spectrosc. 31(6), 485–495 (1977).
[Crossref]

Soenksen, D. G.

Steel, W. H.

M. L. Forman, W. H. Steel, and G. A. Vanasse, “Correction of Asymmetric Interferograms Obtained in Fourier Spectroscopy,” J. Opt. Soc. Am. 56(1), 59–63 (1966).
[Crossref]

Subramaniam, S.

Taboury, J.

Y. Ferrec, J. Taboury, H. Sauer, and P. Chavel, “Optimal geometry for Sagnac and Michelson interferometers used as spectral imagers,” Opt. Eng. 45(11), 115601-115606 (2006).
[Crossref]

Tittel, H. O.

Tremblay, P.

J. Genest, P. Tremblay, and A. Villemaire, “Throughput of tilted interferometers,” Appl. Opt. 37(21), 4819–4822 (1998).
[Crossref]

Valero, F. P. J.

P. D. Hammer, F. P. J. Valero, and D. L. Peterson, “An imaging interferometer for terrestrial remote sensing,” Proc. SPIE 1937, 244–255 (1993).
[Crossref]

Vanasse, G. A.

H. Sakai, G. A. Vanasse, and M. L. Forman, “Spectral Recovery in Fourier Spectroscopy,” J. Opt. Soc. Am. 58(1), 84–90 (1968).
[Crossref]

M. L. Forman, W. H. Steel, and G. A. Vanasse, “Correction of Asymmetric Interferograms Obtained in Fourier Spectroscopy,” J. Opt. Soc. Am. 56(1), 59–63 (1966).
[Crossref]

Villemaire, A.

J. Genest, P. Tremblay, and A. Villemaire, “Throughput of tilted interferometers,” Appl. Opt. 37(21), 4819–4822 (1998).
[Crossref]

Viswanath, P. V.

Walmsley, D. A.

Williams, T.

Acta Astronaut. (1)

Appl. Opt. (6)

T. Okamoto, S. Kawata, and S. Minami, “Fourier transform spectrometer with a self-scanning photodiode array,” Appl. Opt. 23(2), 269–273 (1984).
[Crossref] [PubMed]

M.-L. Junttila, “Stationary Fourier transform spectrometer,” Appl. Opt. 31(21), 4106–4112 (1992).
[Crossref] [PubMed]

J. Genest, P. Tremblay, and A. Villemaire, “Throughput of tilted interferometers,” Appl. Opt. 37(21), 4819–4822 (1998).
[Crossref]

R. G. Sellar and G. D. Boreman, “Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer,” Appl. Opt. 44(9), 1614–1624 (2005).
[Crossref] [PubMed]

Appl. Phys. B (1)

Appl. Spectrosc. (1)

P. R. Griffiths, H. J. Sloane, and R. W. Hannah, “Interferometers vs monochromators: separating the optical and digital advantages,” Appl. Spectrosc. 31(6), 485–495 (1977).
[Crossref]

Astrophys. J. (1)

F. D. Kahn, “The signal: noise ratio of a suggested spectral analyzer,” Astrophys. J. 129, 518–520 (1959).
[Crossref]

Infrared Phys. (1)

L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7(1), 17–23 (1967).
[Crossref]

J. Indian Soc. Remote Sens. (1)

J. Opt. Soc. Am. (4)

P. Jacquinot, “The luminosity of spectrometers with Prisms, Grating, or Fabry-Perot Etalons,” J. Opt. Soc. Am. 44(10), 761–765 (1954).
[Crossref]

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Fig. 1

Layout of a stationary interferometer in the Sagnac (triangular common path) configuration. The light from the objective is first collimated by the input lens L1, and the semi-reflecting surface of the beam-splitter BS originates two rays (reflected and transmitted beam), which travel the instrument on the same triangular path limited by the two folding mirrors M1 and M2. On the exit port, light is focused onto a CCD plane by the lens L2.

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Fig. 2

Theoretical representation of the link between spectral resolution and minimum reconstructed wavelength computed at 500 nm (see Eq. (13)). The red curve represents the “standard” interferometer configuration sampled in a symmetric way with respect to the fringe pattern centre $b = 0.5$ , while blue curve is characteristic for an optimal configuration where only one side of the interferogram is sampled $b = 1$ . Let us note that this last case ( $b = 1$ ) is unrealistic because symmetrisation requires a rough estimate of the interferogram centre, that can be achieved gathering two or three samples around the interferogram centre. This makes b only slightly less than unit, thus the blue curve still is a realistic representation of the best interferometer performance.

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Fig. 3

Schematic of signals composing the measured interferogram (black curve). The interferogram is made up of two components, a dc term (green line) that holds the source half-energy and the Fourier (cosine) Transform of the source spectrum (orange curve) that contains the desired spectral information. While the measured signal (interferogram) always is high and does not decreases with increasing the spectral resolution, the useful signal (orange curve) is tiny for high OPDs. The interferogram model here described arises from Eq. (3) and constitutes a basic property obeyed by any interferometers. This behavior is the main drawback of Fourier Transform Spectrometers.

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Fig. 4

Schematic of signals composing the spectral estimate obtained stemming from a generic interferogram measurement. The interferogram is made up of two components, a constant dc term that inverse transforms in a pulse at zero wavenumber (green line), and the true source signal that inverse transforms in the desired spectrum (orange curve). The meaning of these two interferogram component in the conjugate OPD domain is discussed in the text and illustrated in Fig. 3. This picture undoubtedly shows that the dc term, which is at the core of the multuplexing advantage, does not bring source information apart its energy.

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Fig. 5

Schematic of signals related to the Fourier Transform of the source spectrum (orange curve). The point where the envelope of the source Fourier Transform (blue curve) reaches the level of noise (black curve) defines the noise OPD limit ${OPD}_{n}$ . Interferogram samples gathered at OPDs greater than ${OPD}_{n}$ does not bring source information.

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Tables (1)

Table 1 Main differences between dispersive and interferometric technique about throughput, multiplexing, radiometric accuracy and SNR issues.

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Equations (48)

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I (x) = A \int_{0}^{+ \infty} S (κ) {‖ t_{r} (κ) E (κ) \exp (j φ) + t_{t} (κ) E (κ) \exp (j φ + 2 π j κ OPD (x, κ)) ‖}^{2} d κ .

\begin{array}{l} t_{r} (κ) = \sqrt{ρ (κ)} \sqrt{τ_{i} (κ)} \sqrt{ρ (κ)} \sqrt{τ_{i} (κ)} \exp (j π) = - ρ (κ) τ_{i} (κ) \\ t_{t} (κ) = \sqrt{1 - ρ (κ)} \sqrt{τ_{i} (κ)} \sqrt{1 - ρ (κ)} \sqrt{τ_{i} (κ)} = [1 - ρ (κ)] τ_{i} (κ) . \end{array}

\begin{array}{l} I (x) = \int_{0}^{+ \infty} S (κ) H (κ) i (κ) {1 + V (κ) \cos [2 π κ OPD (x, κ) + π]} d κ \\ H (κ) = τ_{i}^{2} (κ) [ρ^{2} (κ) + {[1 - ρ (κ)]}^{2}] \\ V (κ) = \frac{2 ρ (κ) (1 - ρ (κ))}{ρ^{2} (κ) + {[1 - ρ (κ)]}^{2}} . \end{array}

OPD (x, κ) = ϖ (κ) x,

\begin{array}{l} I_{S} (x) = [I (OPD (x, κ)) * \frac{1}{d} rect (\frac{x}{d})] comb (\frac{x}{p}) \frac{1}{D} rect (\frac{x - x_{0}}{D}) \\ D = M p, \end{array}

\begin{array}{l} {OPD}_{\max} = ϖ (κ) (x_{0} + D / 2) = M b δ_{OPD} \\ b = \frac{(x_{0} + D / 2)}{D} . \end{array}

I T {}^{- 1}{I_{S} (x)} = ϖ (κ) {C T {}^{- 1}{I_{S} (x)} |}_{κ ϖ (κ)}

\begin{array}{l} I_{S S} (x) = [I (x) * \frac{1}{d} rect (\frac{x}{d})] comb (\frac{x}{p}) \frac{1}{2 x_{0} + D} rect (\frac{x}{2 x_{0} + D}) \\ {OPD}_{\max} = \frac{M b}{κ_{S}} = (x_{0} + \frac{D}{2}) ϖ (κ) \\ κ_{S} = \frac{1}{δ_{OPD}}, \end{array}

\begin{array}{l} I T^{- 1} {I_{S S} (x)} = {\sum_{m = - \infty}^{m = + \infty} \tilde{i} (κ - m κ_{S}) sinc [(κ - m κ_{S}) d_{OPD}]} * sinc (\frac{2 κ M b}{κ_{S}}) \\ \tilde{i} (κ) = {I T^{- 1} {I T [i (κ)]} |}_{κ} . \end{array}

\begin{array}{l} κ_{s} = \frac{1}{δ_{OPD}} \geq 2 κ_{\max} \\ S (κ) H (κ) i (κ) = 0 \forall κ \geq κ_{\max}, \end{array}

F {I T^{- 1} {I_{S S} (x)}} = {\tilde{i} (κ) sinc (κ d_{OPD})} * sinc (\frac{2 κ M b}{κ_{S}}) .

\begin{array}{l} δ_{κ} = \frac{1}{{OPD}_{\max}} \\ δ_{λ} = \frac{λ^{2}}{{OPD}_{\max}} . \end{array}

\begin{array}{l} λ_{\min} \geq 2 δ_{OPD} \\ δ_{λ} (λ) = \frac{λ^{2}}{b M δ_{OPD}} \end{array}} \Rightarrow δ_{λ} (λ) λ_{\min} \geq 2 \frac{λ^{2}}{b M} .

\begin{array}{l} OPD (x, κ) = n (κ) {GPD}_{B S} (x) + {GPD}_{A i r} (x) = n (κ) ϖ_{B S} x + ϖ_{A i r} x \\ ϖ (κ) = n (κ) ϖ_{B S} + ϖ_{A i r}, \end{array}

\begin{array}{l} I T^{- 1} {I T {i (κ)}} = Re {\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} i (ξ) \exp {- 2 π j ϖ (ξ) x ξ + 2 π j ϖ (κ) x κ} d ξ ϖ (κ) d x} \\ d OPD = ϖ (κ) d x . \end{array}

Re {ϖ (κ) \int_{- \infty}^{+ \infty} \exp {- 2 π j x [ϖ (ξ) ξ - ϖ (κ) κ]} d x} = ϖ (κ) δ [ϖ (ξ) ξ - ϖ (κ) κ],

\begin{array}{l} ϖ (ξ) ξ - ϖ (κ) κ = ϖ_{A i r} (ξ - κ) + ϖ_{B S} [ξ n (ξ) - κ n (κ)] \\ Re {ϖ (κ) \int_{- \infty}^{+ \infty} \exp {- 2 π j x [ϖ (ξ) ξ - ϖ (κ) κ]} d x} = \\ = ϖ (κ) δ {ϖ_{A i r} (ξ - κ) + ϖ_{B S} [ξ n (ξ) - κ n (κ)]} . \end{array}

\begin{array}{l} n (ξ) ≅ n (κ) + n^{'} (κ) (ξ - κ) \\ ξ n (ξ) - κ n (κ) ≅ (ξ - κ) [n (κ) + ξ n^{'} (κ)] + o [{(ξ - κ)}^{2}] ≅ (ξ - κ) [n (κ) + κ n^{'} (κ)] \\ I T^{- 1} {I T {i (κ)}} ≅ \int_{- \infty}^{+ \infty} i (ξ) ϖ (κ) δ {ϖ_{A i r} (ξ - κ) + ϖ_{B S} (ξ - κ) [n (κ) + κ n^{'} (κ)]} d ξ . \end{array}

I T^{- 1} {I T {i (κ)}} = i (κ) \frac{ϖ (κ)}{ϖ (κ) + ϖ^{'} (κ) κ} .

F {I T^{- 1} {I_{S S} (x)}} = {i (κ) \frac{ϖ (κ)}{ϖ (κ) + ϖ^{'} (κ) κ} sinc (κ d_{OPD})} * sinc (2 κ {OPD}_{\max}) .

\begin{array}{l} OPD (x, κ) = ϖ (κ) x = {[ϖ_{A i r} + ϖ_{B S} n (κ)]}^{} x \\ {OPD}_{g} (x, κ) = \frac{d [ϖ (κ) κ]}{d κ} x = [ϖ (κ) + ϖ^{'} (κ) κ] x = [ϖ_{A i r} + ϖ_{B S} N (κ)] x \\ \frac{ϖ (κ)}{ϖ (κ) + ϖ^{'} (κ) κ} = \frac{OPD (x, κ)}{{OPD}_{g} (x, κ)} . \end{array}

\begin{array}{l} ζ (κ) = I F T^{- 1} {z (x)} \\ σ_{ζ} = σ_{z}, \end{array}

\begin{array}{l} F {‖ I F T^{- 1} {I_{S S} (x) + z (x)} ‖} = \\ = ‖ [i (κ) \exp (- 2 π j κ Δ_{OPD}) \frac{ϖ (κ)}{ϖ (κ) + ϖ^{'} (κ) κ} sinc (κ d_{OPD})] * sinc (2 κ {OPD}_{\max}) + ζ (κ) ‖ \\ E {F {{‖ I F T^{- 1} {I_{S S} (x) + z (x)} ‖}^{2}}} = \\ = {[i (κ) \frac{ϖ (κ)}{ϖ (κ) + ϖ^{'} (κ) κ} sinc (κ d_{OPD})] * sinc (2 κ {OPD}_{\max})}^{2} + σ_{z}^{2}, \end{array}

\begin{array}{l} E {F {{‖ I F T^{- 1} {I_{S S} (x) + z (x)} ‖}^{2}}} = \\ + {[i (κ) \frac{ϖ (κ)}{ϖ (κ) + ϖ^{'} (κ) κ} sinc (κ d_{OPD})] * sinc (2 κ {OPD}_{\max})}^{2} + \\ + {[σ_{z} (p h o t) \frac{ϖ (κ)}{ϖ (κ) + ϖ^{'} (κ) κ} sinc (κ d_{OPD})]}^{2} * {sinc}^{2} (2 κ {OPD}_{\max}) + σ_{z}^{2} . \end{array}

\begin{array}{l} z (x) = \frac{z_{E} (x) + z_{O} (x)}{2} \\ z_{E} (x) = \frac{z (x) + z (- x)}{2} \\ z_{O} (x) = \frac{z (x) - z (- x)}{2} \\ σ_{z_{E}}^{2} = σ_{z_{O}}^{2} = \frac{σ_{z}^{2}}{2} . \end{array}

\begin{array}{l} E {F {{‖ I F T^{- 1} {I_{S S} (x) + z (x)} ‖}^{2}} - σ_{z}^{2}}^{\frac{1}{2}} = \\ = {[i (κ) + I_{0} δ (κ)] \frac{ϖ (κ)}{ϖ (κ) + ϖ^{'} (κ) κ} sinc (κ d_{OPD})} * sinc (2 κ {OPD}_{\max}); \end{array}

\begin{array}{l} λ_{\max} = 2 {OPD}_{\max} = 2 δ_{OPD} M b \\ λ_{\min} = 2 δ_{OPD} . \end{array}

I (x) = {\begin{cases} \frac{{F T {i (κ)} |}_{x = 0} + F T {i (κ)}}{2} + z (x) 0 \leq x \leq {OPD}_{\max} \\ 0 e l s e w h e r e . \end{cases}

〈 σ_{z}^{2} 〉 = \frac{1}{{OPD}_{\max}} \int_{- \infty}^{+ \infty} {‖ z (x) ‖}^{2} d x,

〈 {‖ I (x) ‖}^{2} 〉 = \frac{1}{{OPD}_{\max}} \int_{- \infty}^{+ \infty} {‖ I (x) ‖}^{2} d x .

〈 S N R_{I (x)}^{2} 〉 = \frac{\int_{- \infty}^{+ \infty} {‖ I (x) ‖}^{2} d x}{\int_{- \infty}^{+ \infty} {‖ z (x) ‖}^{2} d x} .

〈 S N R_{F T^{- 1} {I (x)}}^{2} 〉 = \frac{\int_{- \infty}^{+ \infty} {‖ F T^{- 1} {I (x)} ‖}^{2} d κ}{\int_{- \infty}^{+ \infty} {‖ ζ (κ) ‖}^{2} d κ},

\int_{- \infty}^{+ \infty} {‖ I_{i n f o r m a t i v e} (x) ‖}^{2} d x = \frac{1}{4} \int_{- \infty}^{+ \infty} {‖ i (κ) ‖}^{2} d κ .

\begin{array}{l} i (κ) = I_{0} rect (\frac{κ - κ_{0}}{B}) \\ I_{I n f o r m a t i v e} (OPD (x)) = \frac{B I_{0}}{2} sinc (B OPD (x)) \exp (- 2 π j κ_{0} OPD (x)) . \end{array}

δ_{i}^{F T S} ≅ ‖ \frac{B I_{0}}{2} sinc (B {OPD}_{\max}) \exp (- 2 π j κ_{0} {OPD}_{\max}) ‖ \leq \frac{B I_{0}}{2} \frac{1}{π B {OPD}_{\max}} .

δ_{i}^{F T S} \approx \frac{B I_{0}}{2} \frac{δ_{κ}}{π B} = \frac{I_{0}}{2} \frac{δ_{κ}}{π} .

δ_{i}^{F T S} = \frac{1}{2} F T {i (κ)} \approx \frac{A_{F T S} {F T {i (κ)} |}_{OPD = 0}}{2 {| B {OPD}_{\max} |}^{l + 1}} l \geq 0.

\begin{array}{l} δ_{i}^{F T S} \approx \frac{A_{F T S}}{2} {F T {i (κ)} |}_{OPD = 0} {(\frac{δ_{κ}}{B})}^{l + 1} l \geq 0 \\ δ_{i}^{D I S P} \approx i (κ) δ_{κ} . \end{array}

δ_{i}^{F T S} (δ_{κ}) < δ_{i}^{D I S P} (δ_{κ}) \forall δ_{κ} < δ_{0} .

\begin{array}{l} S N R_{\min} ≅ \frac{{F T {i (κ)} |}_{OPD = 0}}{2 δ_{i}^{F T S}} \approx \frac{1}{A_{F T S}} {(K_{\max})}^{l + 1} \\ K_{\max} = \frac{B}{δ_{κ}}, \end{array}

Q_{\min} ≅ \log_{2} (\frac{{F T {i (κ)} |}_{OPD = 0}}{δ_{i}^{F T S}}) \approx 1 + (l + 1) \log_{2} (K_{\max}) - \log_{2} (A_{F T S}) .

{OPD}_{\max} < {OPD}_{n} .

d σ_{Φ (OPD, κ)} = \sqrt{S (κ) H (κ) \frac{i (κ)}{c h κ} {1 + V (κ) \cos [2 π κ OPD (x, κ) + π]} d κ} .

σ_{I (OPD)} = \sqrt{\int_{0}^{+ \infty} c h κ S (κ) H (κ) i (κ) {1 + V (κ) \cos [2 π κ OPD (x, κ) + π]} d κ} .

\begin{array}{l} σ_{I (OPD)} = \sqrt{c h κ_{a v} \frac{{{F T {i (κ)} |}_{OPD = 0} + F T {i (κ)}}}{2}} \\ κ_{a v} = \frac{\int_{0}^{+ \infty} κ S (κ) H (κ) i (κ) {1 + V (κ) \cos [2 π κ OPD (x, κ) + π]} d κ}{\int_{0}^{+ \infty} S (κ) H (κ) i (κ) {1 + V (κ) \cos [2 π κ OPD (x, κ) + π]} d κ}, \end{array}

S N R_{\max}^{e f f} (F T S) = \frac{‖ F T {i (κ)} / 2 ‖}{\sqrt{c h κ_{a v} \frac{{{F T {i (κ)} |}_{OPD = 0} + F T {i (κ)}}}{2}}} .

{S N R_{\max}^{e f f} (F T S) |}_{{OPD}_{\max}} ≅ A_{F T S} \sqrt{\frac{{F T {i (κ)} |}_{OPD = 0}}{2 c h κ_{a v}}} {(\frac{δ_{κ}}{B})}^{l + 1} .

S N R_{\max}^{e f f} (D I S P) ≅ A_{D I S P} {(\frac{δ_{κ}}{B})}^{\frac{1}{2}} .

Issue	Dispersive spectrometer	FTS device
Jacquinot advantage	NO	YES (some imaging FTSs adopt an input slit losing this advantage)
Fellgett’s advantage	YES (for those devices that employ array detectors)	YES (for those devices that employ array detectors)
Multiplexing effect	-	YES
SNR improvement by the $F T^{- 1} {}$ operator	-	NO
Physical signal	$i (κ)$	$[{F T {i (κ)} \|}_{OPD = 0} + F T {i (κ)}] / 2$
Effective signal	$i (κ)$	$F T {i (κ)} / 2$
Minimal radiometric Accuracy	$\approx i (κ) δ_{κ}$	$\approx {\frac{A_{F T S}}{2} F T {i (κ)} \|}_{OPD = 0} {(\frac{δ_{κ}}{B})}^{l + 1}$
Effective SNR (proportional to)	${(δ_{κ})}^{\frac{1}{2}}$	${(\frac{δ_{κ}}{B})}^{l + 1}$
Lost samples	Lack of a single spectral channel	No spectral channel is lost – the lost information gives rise to a small error in any spectral channels

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