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.

©2010 Optical Society of America

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2008 (1)

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)

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]

1992 (1)

1991 (1)

1984 (1)

1977 (1)

1972 (1)

1968 (1)

1967 (1)

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

1966 (2)

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)

Bajpai, O. P.

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]

Barducci, A.

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]

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]

Basheerullah, B. G.

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]

Bliss, M.

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

Boreman, G. D.

Buckwald, R. A.

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]

Cabib, D.

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]

Casini, A.

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]

Castagnoli, F.

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]

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.

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]

Ciprian, D.

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]

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]

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]

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.

Fronterhouse, D. S.

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]

Garin, Y.

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]

Genest, J.

Griffiths, P. R.

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.

Harnisch, B.

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]

Hilliard, R. L.

Hlubina, P.

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]

Holota, K.

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]

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.

Jacquinot, P.

Jennings, R. E.

Jones, B. A.

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]

Junttila, M.-L.

Kahn, F. D.

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

Kauppinen, J.

Kawata, S.

Lucey, P. G.

Lunacek, J.

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]

Marcoionni, P.

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]

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]

Meigs, A. D.

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]

Mertz, L.

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

Minami, S.

Morandi, M.

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]

Okamoto, T.

Otten, L. J.

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]

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.

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]

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]

Poggesi, M.

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]

Posselt, W.

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]

Prinzing, P.

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]

Rabindranath, B.

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]

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.

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]

Rost, M.

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]

Sakai, H.

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.

Sloane, H. J.

Soenksen, D. G.

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]

Steel, W. H.

Subramaniam, S.

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]

Taboury, J.

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Figures (5)

Fig. 1
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
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
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
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
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)

Tables Icon

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 ∫ 0 + ∞ 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 .
t r ( κ ) = ρ ( κ ) τ i ( κ ) ρ ( κ ) τ i ( κ ) exp ( j π ) = − ρ ( κ ) τ i ( κ ) t t ( κ ) = 1 − ρ ( κ ) τ i ( κ ) 1 − ρ ( κ ) τ i ( κ ) = [ 1 − ρ ( κ ) ] τ i ( κ ) .
I ( x ) = ∫ 0 + ∞ 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 ) = 2 ρ ( κ ) ( 1 − ρ ( κ ) ) ρ 2 ( κ ) + [ 1 − ρ ( κ ) ] 2 .
OPD ( x , κ ) = ϖ ( κ ) x ,
I S ( x ) = [ I ( OPD ( x , κ ) ) ∗ 1 d rect ( x d ) ] comb ( x p ) 1 D rect ( x − x 0 D ) D = M p ,
OPD max = ϖ ( κ ) ( x 0 + D / 2 ) = M b δ OPD b = ( x 0 + D / 2 ) D .
I T { I S ( x ) } − 1 = ϖ ( κ ) C T { I S ( x ) } − 1 | κ ϖ ( κ )
I S S ( x ) = [ I ( x ) ∗ 1 d rect ( x d ) ] comb ( x p ) 1 2 x 0 + D rect ( x 2 x 0 + D ) OPD max = M b κ S = ( x 0 + D 2 ) ϖ ( κ ) κ S = 1 δ OPD ,
I T − 1 { I S S ( x ) } = { ∑ m = − ∞ m = + ∞ i ˜ ( κ − m κ S ) sinc [ ( κ − m κ S )   d OPD ] } ∗ sinc ( 2 κ   M b κ S ) i ˜ ( κ ) = I T − 1 { I T [ i ( κ ) ] } | κ .
κ s = 1 δ OPD ≥ 2 κ max S ( κ ) H ( κ ) i ( κ ) = 0     ∀ κ ≥ κ max ,
F { I T − 1 { I S S ( x ) } } = { i ˜ ( κ ) sinc ( κ d OPD ) } ∗ sinc ( 2 κ   M b κ S ) .
δ κ = 1 OPD max δ λ = λ 2 OPD max .
λ min ≥ 2 δ OPD δ λ ( λ ) = λ 2 b M δ OPD } ⇒ δ λ ( λ ) λ min ≥ 2 λ 2 b M .
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 ,
I T − 1 { I T { i ( κ ) } } = Re { ∫ − ∞ + ∞ ∫ − ∞ + ∞ 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 .
Re { ϖ ( κ ) ∫ − ∞ + ∞ exp { − 2 π j x [ ϖ ( ξ ) ξ − ϖ ( κ ) κ ] } d x } = ϖ ( κ ) δ [ ϖ ( ξ ) ξ − ϖ ( κ ) κ ] ,
ϖ ( ξ ) ξ − ϖ ( κ ) κ = ϖ A i r ( ξ − κ ) + ϖ B S [ ξ n ( ξ ) − κ n ( κ ) ] Re { ϖ ( κ ) ∫ − ∞ + ∞ 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 ) ] } .
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 ) } } ≅ ∫ − ∞ + ∞ i ( ξ ) ϖ ( κ ) δ { ϖ A i r ( ξ − κ ) + ϖ B S ( ξ − κ ) [ n ( κ ) + κ n ′ ( κ ) ] } d ξ .
I T − 1 { I T { i ( κ ) } } = i ( κ ) ϖ ( κ ) ϖ ( κ ) + ϖ ′ ( κ ) κ .
F { I T − 1 { I S S ( x ) } } = { i ( κ ) ϖ ( κ ) ϖ ( κ ) + ϖ ′ ( κ ) κ sinc ( κ d OPD ) } ∗ sinc ( 2 κ OPD max ) .
OPD ( x , κ ) = ϖ ( κ ) x = [ ϖ A i r + ϖ B S n ( κ ) ] x OPD g ( x , κ ) = d [ ϖ ( κ ) κ ] d κ x = [ ϖ ( κ ) + ϖ ′ ( κ ) κ ] x = [ ϖ A i r + ϖ B S N ( κ ) ] x ϖ ( κ ) ϖ ( κ ) + ϖ ′ ( κ ) κ = OPD ( x , κ ) OPD g ( x , κ ) .
ζ ( κ ) = I F T − 1 { z ( x ) } σ ζ = σ z ,
F { ‖ I F T − 1 { I S S ( x ) + z ( x ) } ‖ } =      = ‖ [ i ( κ ) exp ( − 2 π j κ Δ OPD ) 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 ) ϖ ( κ ) ϖ ( κ ) + ϖ ′ ( κ ) κ sinc ( κ d OPD ) ] ∗ sinc ( 2 κ OPD max ) } 2 + σ z 2 ,
E { F { ‖ I F T − 1 { I S S ( x ) + z ( x ) } ‖ 2 } } =                     + { [ i ( κ ) ϖ ( κ ) ϖ ( κ ) + ϖ ′ ( κ ) κ sinc ( κ d OPD ) ] ∗ sinc ( 2 κ OPD max ) } 2 +                     + [ σ z ( p h o t ) ϖ ( κ ) ϖ ( κ ) + ϖ ′ ( κ ) κ sinc ( κ d OPD ) ] 2 ∗ sinc 2 ( 2 κ OPD max ) + σ z 2 .
z ( x ) = z E ( x ) + z O ( x ) 2 z E ( x ) = z ( x ) + z ( − x ) 2 z O ( x ) = z ( x ) − z ( − x ) 2 σ z E 2 = σ z O 2 = σ z 2 2 .
E { F { ‖ I F T − 1 { I S S ( x ) + z ( x ) } ‖ 2 } − σ z 2 } 1 2 =                   = { [ i ( κ ) + I 0 δ ( κ ) ] ϖ ( κ ) ϖ ( κ ) + ϖ ′ ( κ ) κ sinc ( κ d OPD ) } ∗ sinc ( 2 κ OPD max ) ;
λ max = 2 OPD max = 2 δ OPD M b λ min = 2 δ OPD .
I ( x ) = { F T { i ( κ ) } | x = 0 + F T { i ( κ ) } 2 + z ( x )        0 ≤ x ≤ OPD max 0                                                           e l s e w h e r e .
〈 σ z 2 〉 = 1 OPD max ∫ − ∞ + ∞ ‖ z ( x ) ‖ 2 d x ,
〈 ‖ I ( x ) ‖ 2 〉 = 1 OPD max ∫ − ∞ + ∞ ‖ I ( x ) ‖ 2 d x .
〈 S N R I ( x ) 2 〉 = ∫ − ∞ + ∞ ‖ I ( x ) ‖ 2 d x ∫ − ∞ + ∞ ‖ z ( x ) ‖ 2 d x .
〈 S N R F T − 1 { I ( x ) } 2 〉 = ∫ − ∞ + ∞ ‖ F T − 1 { I ( x ) } ‖ 2 d κ ∫ − ∞ + ∞ ‖ ζ ( κ ) ‖ 2 d κ ,
∫ − ∞ + ∞ ‖ I i n f o r m a t i v e ( x ) ‖ 2 d x = 1 4 ∫ − ∞ + ∞ ‖ i ( κ ) ‖ 2 d κ .
i ( κ ) = I 0 rect ( κ − κ 0 B ) I I n f o r m a t i v e ( OPD ( x ) ) = B I 0 2 sinc ( B  OPD ( x ) ) exp ( − 2 π j κ 0 OPD ( x ) ) .
δ i F T S ≅ ‖ B I 0 2 sinc ( B  OPD max ) exp ( − 2 π j κ 0 OPD max ) ‖ ≤ B I 0 2 1 π B  OPD max .
δ i F T S ≈ B I 0 2 δ κ π B = I 0 2 δ κ π .
δ i F T S = 1 2 F T { i ( κ ) } ≈ A F T S F T { i ( κ ) } | OPD = 0 2 | B OPD max | l + 1          l ≥ 0.
δ i F T S ≈ A F T S 2 F T { i ( κ ) } | OPD = 0 ( δ κ B ) l + 1       l ≥ 0 δ i D I S P ≈ i ( κ ) δ κ .
δ i F T S ( δ κ ) < δ i D I S P ( δ κ )       ∀ δ κ < δ 0 .
S N R min ≅ F T { i ( κ ) } | OPD = 0 2 δ i F T S ≈ 1 A F T S ( K max ) l + 1 K max = B δ κ ,
Q min ≅ log 2 ( F T { i ( κ ) } | OPD = 0 δ i F T S ) ≈ 1 + ( l + 1 ) log 2 ( K max ) − log 2 ( A F T S ) .
OPD max < OPD n .
d σ Φ ( OPD , κ ) = S ( κ ) H ( κ ) i ( κ ) c h κ { 1 + V ( κ ) cos [ 2 π κ OPD ( x , κ ) + π ] } d κ .
σ I ( OPD ) = ∫ 0 + ∞ c h κ S ( κ ) H ( κ ) i ( κ ) { 1 + V ( κ ) cos [ 2 π κ OPD ( x , κ ) + π ] } d κ .
σ I ( OPD ) = c h κ a v { F T { i ( κ ) } | OPD = 0 + F T { i ( κ ) } } 2 κ a v = ∫ 0 + ∞ κ S ( κ ) H ( κ ) i ( κ ) { 1 + V ( κ ) cos [ 2 π κ OPD ( x , κ ) + π ] } d κ ∫ 0 + ∞ S ( κ ) H ( κ ) i ( κ ) { 1 + V ( κ ) cos [ 2 π κ OPD ( x , κ ) + π ] } d κ ,
S N R max e f f ( F T S ) = ‖ F T { i ( κ ) } / 2 ‖ c h κ a v { 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 F T { i ( κ ) } | OPD = 0 2 c h κ a v ( δ κ B ) l + 1 .
S N R max e f f ( D I S P ) ≅ A D I S P ( δ κ B ) 1 2 .

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