Abstract

The instrument line shape (ILS) of Fourier-transform spectrometers is modeled within a framework that enables us to take into account the partial coherence of optical fields. The cross spectral density and the angular coherence functions are used to develop a global ILS model including all possible geometric defects that can be introduced by a realistic two-beam interferometer. Tilt and shear no longer only reduce the modulation efficiency but are presented as contributors to the ILS. The case of an incoherent secondary planar source is covered and agrees with previously known results. However, it shows a coupling among tilt, shear, and optical path difference (OPD). A quasi-coherent source is also studied. Differences between the incoherent and the quasi-coherent cases are highlighted. The relative localization of the reference laser beam in the interferometer is shown to be of significance to provide a sampling scale that minimizes the OPD, or phase, induced by angular misalignment.

© 2005 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2002 (2)

2001 (1)

ABB Bomem Inc., The DA8 series FT-IR spectrometers.Commercial brochure and instrument specification (2001).

2000 (1)

1998 (1)

1995 (1)

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

1974 (1)

1966 (1)

1965 (1)

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

1958 (1)

J. Connes, “Domaine d’utilisation de la méthode par transformée de Fourier,” J. Phys. Radium 19, 197–208 (1958).
[CrossRef]

Beer, R.

Bennet, C. L.

C. L. Bennet, M. R. Carter, D. J. Fields, “Hyperspectral imaging in the infrared using LIFTIRS,” in Infrared Technology XXI, B. F. Andresen, M. Strojnik, eds., Proc. SPIE2552, 274–283 (1995).
[CrossRef]

Bernath, P. F.

Boone, C. D.

Bouchard, J.-P.

F. Bouffard, P. Tremblay, R. Desbiens, J.-P. Bouchard, “Synthetic spectra of Fourier transform spectrometers using analytically modeled instrument line shape inclusion,” in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR), J. Ferriter, D. Faubert, eds.(Science and Technology Corp., 2001), pp. 326–335.

Bouffard, F.

F. Bouffard, P. Tremblay, R. Desbiens, J.-P. Bouchard, “Synthetic spectra of Fourier transform spectrometers using analytically modeled instrument line shape inclusion,” in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR), J. Ferriter, D. Faubert, eds.(Science and Technology Corp., 2001), pp. 326–335.

Bowman, K. W.

Brault, J. W.

J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, Proceedings of the 15th Advanced Course of the Swiss Society of Astronomy and Astrophysics, A. O. Benz, M. C. E. Huber, M. Mayor, eds.(Swiss Society of Astronomy and Astrophysics, 1985), pp. 1–61.

Carter, M. R.

C. L. Bennet, M. R. Carter, D. J. Fields, “Hyperspectral imaging in the infrared using LIFTIRS,” in Infrared Technology XXI, B. F. Andresen, M. Strojnik, eds., Proc. SPIE2552, 274–283 (1995).
[CrossRef]

Châteauneuf, F.

M.-A. Soucy, F. Châteauneuf, C. Deutsch, J. Giroux, “Status of the ACE-FTS instrument development,” in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR), J. Ferriter, D. Faubert, eds. (Science and Technology Corp., 2001).

Connes, J.

J. Connes, “Domaine d’utilisation de la méthode par transformée de Fourier,” J. Phys. Radium 19, 197–208 (1958).
[CrossRef]

Desbiens, R.

R. Desbiens, J. Genest, P. Tremblay, “Radiometry in line-shape modeling of Fourier-transform spectrometers,” Appl. Opt. 41, 1424–1432 (2002).
[CrossRef] [PubMed]

R. Desbiens, P. Tremblay, “Families of optimal parametric windows having arbitrary secondary lobe profile,” in Fourier Transform Spectroscopy, Vol. 51 of 2001 OSA Technical Digest Series (Optical Society of America, 2001), pp. 41–43.

R. Desbiens, P. Tremblay, J. Genest, “Matrix algorithm for integration and inversion of instrument line shape,” in Fourier Transform Spectroscopy, D. Hausamann, R. McKellar, eds., Vol. 84 of 2003 OSA Trends in Optics and Photonics (Optical Society of America, 2003), pp. 42–44.

F. Bouffard, P. Tremblay, R. Desbiens, J.-P. Bouchard, “Synthetic spectra of Fourier transform spectrometers using analytically modeled instrument line shape inclusion,” in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR), J. Ferriter, D. Faubert, eds.(Science and Technology Corp., 2001), pp. 326–335.

Deutsch, C.

M.-A. Soucy, F. Châteauneuf, C. Deutsch, J. Giroux, “Status of the ACE-FTS instrument development,” in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR), J. Ferriter, D. Faubert, eds. (Science and Technology Corp., 2001).

Fields, D. J.

C. L. Bennet, M. R. Carter, D. J. Fields, “Hyperspectral imaging in the infrared using LIFTIRS,” in Infrared Technology XXI, B. F. Andresen, M. Strojnik, eds., Proc. SPIE2552, 274–283 (1995).
[CrossRef]

Fortin, S.

A. J. Villemaire, S. Fortin, J. Giroux, T. Smithson, R. J. Oermann, “Imaging Fourier transform spectrometer,” in Aerosense ’95 Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, L. R. Illing, eds., Proc. SPIE2480, 387–397 (1995).
[CrossRef]

Genest, J.

R. Desbiens, J. Genest, P. Tremblay, “Radiometry in line-shape modeling of Fourier-transform spectrometers,” Appl. Opt. 41, 1424–1432 (2002).
[CrossRef] [PubMed]

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

J. Genest, P. Tremblay, “General analytic solutions for the ILS of nonuniformly illuminated off-axis detectors,” in Fourier Transform Spectroscopy: New Methods and Applications (Optical Society of America, 1999), pp. 58–60.

R. Desbiens, P. Tremblay, J. Genest, “Matrix algorithm for integration and inversion of instrument line shape,” in Fourier Transform Spectroscopy, D. Hausamann, R. McKellar, eds., Vol. 84 of 2003 OSA Trends in Optics and Photonics (Optical Society of America, 2003), pp. 42–44.

Giroux, J.

A. J. Villemaire, S. Fortin, J. Giroux, T. Smithson, R. J. Oermann, “Imaging Fourier transform spectrometer,” in Aerosense ’95 Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, L. R. Illing, eds., Proc. SPIE2480, 387–397 (1995).
[CrossRef]

M.-A. Soucy, F. Châteauneuf, C. Deutsch, J. Giroux, “Status of the ACE-FTS instrument development,” in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR), J. Ferriter, D. Faubert, eds. (Science and Technology Corp., 2001).

Goorvitch, D.

Kunz, L. W.

Maillard, J.-P.

W. Posselt, J.-P. Maillard, G. Wright, “NIRCAM-IFTS: imaging Fourier transform spectrometer for NGST,” in Next Generation Space Telescope Science and Technology, Vol. 27 of Astronomical Society of the Pacific Conference Series (Astronomical Society of the Pacific, 1999), pp. 303–310.

Mandel, L.

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
[CrossRef]

McLeod, S. D.

Oermann, R. J.

A. J. Villemaire, S. Fortin, J. Giroux, T. Smithson, R. J. Oermann, “Imaging Fourier transform spectrometer,” in Aerosense ’95 Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, L. R. Illing, eds., Proc. SPIE2480, 387–397 (1995).
[CrossRef]

Overbeck, J. A.

K. D. Stumpf, J. A. Overbeck, “CrIS optical system design,” in Infrared Spaceborne Remote Sensing IX, M. Strojnik, B. F. Andresen, eds., Proc. SPIE4486, 437–444 (2002).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).

Persky, M. J.

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

Posselt, W.

W. Posselt, J.-P. Maillard, G. Wright, “NIRCAM-IFTS: imaging Fourier transform spectrometer for NGST,” in Next Generation Space Telescope Science and Technology, Vol. 27 of Astronomical Society of the Pacific Conference Series (Astronomical Society of the Pacific, 1999), pp. 303–310.

Smithson, T.

A. J. Villemaire, S. Fortin, J. Giroux, T. Smithson, R. J. Oermann, “Imaging Fourier transform spectrometer,” in Aerosense ’95 Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, L. R. Illing, eds., Proc. SPIE2480, 387–397 (1995).
[CrossRef]

Soucy, M.-A.

M.-A. Soucy, F. Châteauneuf, C. Deutsch, J. Giroux, “Status of the ACE-FTS instrument development,” in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR), J. Ferriter, D. Faubert, eds. (Science and Technology Corp., 2001).

Stumpf, K. D.

K. D. Stumpf, J. A. Overbeck, “CrIS optical system design,” in Infrared Spaceborne Remote Sensing IX, M. Strojnik, B. F. Andresen, eds., Proc. SPIE4486, 437–444 (2002).
[CrossRef]

Tenjimbayashi, K.

K. Tenjimbayashi, “Technique of recording and judging the sign of tilt in one interferogram,” in International Conference on Optical Fabrication and Testing, T. Kasia, ed., Proc. SPIE2576, 326–334 (1995).
[CrossRef]

Tremblay, P.

R. Desbiens, J. Genest, P. Tremblay, “Radiometry in line-shape modeling of Fourier-transform spectrometers,” Appl. Opt. 41, 1424–1432 (2002).
[CrossRef] [PubMed]

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

R. Desbiens, P. Tremblay, “Families of optimal parametric windows having arbitrary secondary lobe profile,” in Fourier Transform Spectroscopy, Vol. 51 of 2001 OSA Technical Digest Series (Optical Society of America, 2001), pp. 41–43.

J. Genest, P. Tremblay, “General analytic solutions for the ILS of nonuniformly illuminated off-axis detectors,” in Fourier Transform Spectroscopy: New Methods and Applications (Optical Society of America, 1999), pp. 58–60.

R. Desbiens, P. Tremblay, J. Genest, “Matrix algorithm for integration and inversion of instrument line shape,” in Fourier Transform Spectroscopy, D. Hausamann, R. McKellar, eds., Vol. 84 of 2003 OSA Trends in Optics and Photonics (Optical Society of America, 2003), pp. 42–44.

F. Bouffard, P. Tremblay, R. Desbiens, J.-P. Bouchard, “Synthetic spectra of Fourier transform spectrometers using analytically modeled instrument line shape inclusion,” in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR), J. Ferriter, D. Faubert, eds.(Science and Technology Corp., 2001), pp. 326–335.

Villemaire, A.

Villemaire, A. J.

A. J. Villemaire, S. Fortin, J. Giroux, T. Smithson, R. J. Oermann, “Imaging Fourier transform spectrometer,” in Aerosense ’95 Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, L. R. Illing, eds., Proc. SPIE2480, 387–397 (1995).
[CrossRef]

Williams, C. S.

Wolf, E.

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
[CrossRef]

Worden, H. M.

Wright, G.

W. Posselt, J.-P. Maillard, G. Wright, “NIRCAM-IFTS: imaging Fourier transform spectrometer for NGST,” in Next Generation Space Telescope Science and Technology, Vol. 27 of Astronomical Society of the Pacific Conference Series (Astronomical Society of the Pacific, 1999), pp. 303–310.

Appl. Opt. (6)

Commercial brochure and instrument specification (1)

ABB Bomem Inc., The DA8 series FT-IR spectrometers.Commercial brochure and instrument specification (2001).

J. Phys. Radium (1)

J. Connes, “Domaine d’utilisation de la méthode par transformée de Fourier,” J. Phys. Radium 19, 197–208 (1958).
[CrossRef]

Rev. Mod. Phys. (1)

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Rev. Sci. Instrum. (1)

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

Other (13)

J. Genest, P. Tremblay, “General analytic solutions for the ILS of nonuniformly illuminated off-axis detectors,” in Fourier Transform Spectroscopy: New Methods and Applications (Optical Society of America, 1999), pp. 58–60.

R. Desbiens, P. Tremblay, “Families of optimal parametric windows having arbitrary secondary lobe profile,” in Fourier Transform Spectroscopy, Vol. 51 of 2001 OSA Technical Digest Series (Optical Society of America, 2001), pp. 41–43.

K. Tenjimbayashi, “Technique of recording and judging the sign of tilt in one interferogram,” in International Conference on Optical Fabrication and Testing, T. Kasia, ed., Proc. SPIE2576, 326–334 (1995).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
[CrossRef]

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).

F. Bouffard, P. Tremblay, R. Desbiens, J.-P. Bouchard, “Synthetic spectra of Fourier transform spectrometers using analytically modeled instrument line shape inclusion,” in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR), J. Ferriter, D. Faubert, eds.(Science and Technology Corp., 2001), pp. 326–335.

R. Desbiens, P. Tremblay, J. Genest, “Matrix algorithm for integration and inversion of instrument line shape,” in Fourier Transform Spectroscopy, D. Hausamann, R. McKellar, eds., Vol. 84 of 2003 OSA Trends in Optics and Photonics (Optical Society of America, 2003), pp. 42–44.

K. D. Stumpf, J. A. Overbeck, “CrIS optical system design,” in Infrared Spaceborne Remote Sensing IX, M. Strojnik, B. F. Andresen, eds., Proc. SPIE4486, 437–444 (2002).
[CrossRef]

M.-A. Soucy, F. Châteauneuf, C. Deutsch, J. Giroux, “Status of the ACE-FTS instrument development,” in Proceedings of the International Symposium on Spectral Sensing Research (ISSSR), J. Ferriter, D. Faubert, eds. (Science and Technology Corp., 2001).

A. J. Villemaire, S. Fortin, J. Giroux, T. Smithson, R. J. Oermann, “Imaging Fourier transform spectrometer,” in Aerosense ’95 Imaging Spectrometry, M. R. Descour, J. M. Mooney, D. L. Perry, L. R. Illing, eds., Proc. SPIE2480, 387–397 (1995).
[CrossRef]

C. L. Bennet, M. R. Carter, D. J. Fields, “Hyperspectral imaging in the infrared using LIFTIRS,” in Infrared Technology XXI, B. F. Andresen, M. Strojnik, eds., Proc. SPIE2552, 274–283 (1995).
[CrossRef]

W. Posselt, J.-P. Maillard, G. Wright, “NIRCAM-IFTS: imaging Fourier transform spectrometer for NGST,” in Next Generation Space Telescope Science and Technology, Vol. 27 of Astronomical Society of the Pacific Conference Series (Astronomical Society of the Pacific, 1999), pp. 303–310.

J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, Proceedings of the 15th Advanced Course of the Swiss Society of Astronomy and Astrophysics, A. O. Benz, M. C. E. Huber, M. Mayor, eds.(Swiss Society of Astronomy and Astrophysics, 1985), pp. 1–61.

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Figures (9)

Fig. 1
Fig. 1

Relations among cross spectral densities, angular coherence, and PSD. The cross PSD W(x1, y1, 0, x2, y2, σ0) is assumed known in a secondary source plane (a physical stop at the entrance of the instrument). The spectral density S0) is integrated in the detector plane at the output of the interferometer.

Fig. 2
Fig. 2

Director cosines and propagation angles in the half-space. z is the interferometer optical axis so that θ is the off-axis angle of a plane wave in the instrument.

Fig. 3
Fig. 3

Michelson interferometer creating two virtual images of the secondary plane source.

Fig. 4
Fig. 4

Coordinate transformation performed by the interferometer. There are two paths r1 and r2 to the same points on the detector. The alternative view is that the interferometer creates interference in the field at two distinct coordinates.

Fig. 5
Fig. 5

OPD and shear induced by a tilt between two coordinate systems. The shear is not the same at the coordinate systems (y0) and on the detector y0 + (z1z0)α because of the tilt angle α.

Fig. 6
Fig. 6

OPD and shear induced by a tilt between two coordinate systems. Rotation of a mirror about an off-axis pivot point induces a coordinate transformation between the images of the planar secondary source. Shear accumulates and can be partially compensated, as tilted beams propagate toward the detector.

Fig. 7
Fig. 7

Interference of rays with an angularly incoherent source. Only rays at the same angle before the tilt can interfere.

Fig. 8
Fig. 8

Line shape for a centered circular detector having a width of 1 cm−1 at 10,000 wave numbers. Top, line shape at σ0 = 10,000 cm−1; middle, line shape at σ0 = 5000 cm−1; bottom, line shape at σ0 = 1000 cm−1. Solid lines are the unperturbed ILS; dashed curves represent the multiplicative effect of shear; and dashed–dotted curves are the tilt contribution. Tilt angle is 40 µm and the shear is 3 mm. The instrument aperture has a 1 cm radius.

Fig. 9
Fig. 9

Interference of rays with an angularly quasicoherent source. Because of the aperture angular selectivity, only the rays at the same angle after the tilt will produce appreciable interference.

Equations (52)

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

Γ ( r 1 , r 2 , τ ) = E * ( r 1 , t ) E ( r 2 , t + τ ) ,
P = D Γ ( r , r , 0 ) d x d y .
W ( r 1 , r 2 , σ 0 ) = τ σ 0 [ Γ ( r 1 , r 2 , τ ) ] ,
W ( r 1 , r 2 , σ 0 ) = U * ( r 1 , σ 0 ) U ( r 2 , σ 0 ) ,
S ( σ 0 ) = D W ( r , r , σ 0 ) d x d y .
U ( r , σ 0 ) = a ( p , q , σ 0 ) exp [ i 2 π σ 0 ( p x + q y + m z ) ] d p d q .
p = sin θ cos ϕ ,
q = sin θ sin ϕ ,
m = cos θ ,
p 2 + q 2 + m 2 = 1 .
W ( r 1 , r 2 , σ 0 ) = A ( p 1 , q 1 , p 2 , q 2 , σ 0 ) × exp [ i 2 π σ 0 ( p 2 x 2 + q 2 y 2 + m 2 z 2 p 1 x 1 q 1 y 1 m 1 * z 1 ) ] d p 1 d q 1 d p 2 d q 2 .
A ( p 1 , q 1 , p 2 , q 2 , σ 0 ) = σ 0 4 W ( x 1 , y 1 , 0 , x 2 , y 2 , 0 , σ 0 ) × exp [ i 2 π σ 0 ( p 2 x 2 + q 2 y 2 p 1 x 1 q 1 y 1 ) ] × d x 1 d y 1 d x 2 d y 2 .
S ( σ 0 ) = D A ( p 1 , q 1 , p 2 , q 2 , 0 , σ 0 ) × exp [ i 2 π σ 0 ( p 2 x + q 2 y + m 2 z p 1 x q 1 y m 1 * z ) ] d p 1 d q 1 d p 2 d q 2 d x d y ,
U ( r , σ 0 ) = U ( r 1 , σ 0 ) + U ( r 2 , σ 0 ) ,
W ( r , r , σ 0 ) = [ U ( r 1 , σ 0 ) + U ( r 2 , σ 0 ) ] [ U ( r 1 , σ 0 ) + U ( r 2 , σ 0 ) ] * = U * ( r 1 , σ 0 ) U ( r 1 , σ 0 ) + U * ( r 1 , σ 0 ) U ( r 2 , σ 0 ) + U ( r 1 , σ 0 ) U * ( r 2 , σ 0 ) + U * ( r 2 , σ 0 ) U ( r 2 , σ 0 ) = W ( r 1 , r 1 , σ 0 ) + W ( r 1 , r 2 , σ 0 ) + W * ( r 1 , r 2 , σ 0 ) + W ( r 2 , r 2 , σ 0 ) .
S ( σ 0 ) = D W ( r 1 , r 2 , σ 0 ) d x 1 d y 1 ,
x 2 = x 1 x 0 , y 2 = cos α ( y 1 y 0 ) sin α ( z 1 z 0 ) , z 2 = sin α ( y 1 y 0 ) + cos α ( z 1 z 0 ) ,
S ( σ 0 ) = D p 1 2 + q 1 2 1 p 2 2 + p 2 2 1 A ( p 1 , q 1 , p 2 , q 2 , σ 0 ) × exp ( i 2 π σ 0 { p 2 ( x 1 x 0 ) + q 2 [ cos α ( y 1 y 0 ) sin α ( z 1 z 0 ) ] + m 2 [ sin α ( y 1 y 0 ) + cos α ( z 1 z 0 ) } ] ) × exp [ i 2 π σ 0 ( p 1 x 1 q 1 y 1 m 1 z 1 ) ] d p 1 d q 1 d p 2 d q 2 d x 1 d y 1 ,
S ( σ 0 ) = π R 2 p 1 2 + q 1 2 1 p 2 2 + p 2 2 1 A ( p 1 , q 1 , p 2 , q 2 , σ 0 ) 2 J 1 { 2 π σ 0 R [ ( q 2 cos α + m 2 sin α q 1 ) 2 + ( p 2 p 1 ) 2 ] 1 / 2 } 2 π σ 0 R [ ( q 2 cos α + m 2 sin α q 1 ) 2 + ( p 2 p 1 ) 2 ] 1 / 2 × exp { i 2 π σ 0 [ p 2 x 0 ( q 2 cos α + m 2 sin α ) y 0 + ( q 2 sin α m 2 cos α ) z 0 ] } × exp [ i 2 π σ 0 ( m 2 cos α q 2 sin α m 1 ) z 1 ] d p 1 d q 1 d p 2 d q 2 ,
2 J 1 ( 2 π σ 0 R α ) 2 π σ 0 R α .
S ( σ 0 ) π R 2 p 1 2 + q 1 2 1 p 2 2 + q 2 2 1 A ( p 1 , q 1 , p 2 , q 2 , σ 0 ) 2 J 1 { 2 π σ 0 R [ ( q 2 q 1 + m 2 α ) 2 + ( p 2 p 1 ) 2 ] 1 / 2 } 2 π σ 0 R [ ( q 2 q 1 + m 2 α ) 2 + ( p 2 p 1 ) 2 ] 1 / 2 × exp { i 2 π σ 0 [ p 2 x 0 ( q 2 + m 2 α ) y 0 + ( q 2 α m 2 ) z 0 ] } × exp [ i 2 π σ 0 ( m 2 m 1 q 2 α ) z 1 ] d p 1 d q 1 d p 2 d q 2 .
A ( p 1 , q 1 , p 2 , q 2 , σ 0 ) = ( σ 0 ) exp { [ ( p 1 p 2 ) 2 + ( q 1 q 2 ) 2 ] / s 2 ,
S ( σ 0 ) π R 2 p 1 2 + q 1 2 1 p 2 2 + q 2 2 1 ( σ 0 ) exp { [ ( p 1 p 2 ) 2 + ( q 1 q 2 ) 2 ] / s 2 } × 2 J 1 { 2 π σ 0 R [ ( q 2 q 1 + m 2 α ) 2 + ( p 2 p 1 ) 2 ] 1 / 2 } 2 π σ 0 R [ ( q 2 q 1 + m 2 α ) 2 + ( p 2 p 1 ) 2 ] 1 / 2 × exp { i 2 π σ 0 [ p 2 x 0 ( q 2 + m 2 α ) y 0 + ( q 2 α m 2 ) z 0 ] } × exp [ i 2 π σ 0 ( m 2 m 1 q 2 α ) z 1 ] d p 1 d q 1 d p 2 d q 2 .
S ( σ 0 ) π R 2 p 2 2 + q 2 2 1 ( σ 0 ) exp [ ( u 2 + υ 2 ) / s 2 ] × 2 J 1 { 2 π σ 0 R [ ( υ + m 2 α ) 2 + u 2 ] 1 / 2 } 2 π σ 0 R [ ( υ + m 2 α ) 2 + u 2 ] 1 / 2 × exp { i 2 π σ 0 [ p 2 x 0 ( q 2 + m 2 α ) y 0 + ( q 2 α m 2 ) z 0 ] } exp [ i 2 π σ 0 ( m 2 m 1 q 2 α ) z 1 ] d u d υ d p 2 d q 2 .
S ( σ 0 ) π R 2 2 J 1 { 2 π σ 0 R [ ( υ + m 2 α ) 2 + u 2 ] 1 / 2 } 2 π σ 0 R [ ( υ + m 2 α ) 2 + u 2 ] 1 / 2 d u d υ × p 2 2 + q 2 2 1 ( σ 0 ) exp [ ( m 2 α / s ) 2 ] exp { i 2 π σ 0 × [ p 2 x 0 ( q 2 + m 2 α ) y 0 + ( q 2 α m 2 ) z 0 ] } d p 2 d q 2 .
S ( σ 0 ) π R 2 β ( 2 π σ 0 R ) φ ϑ L ( σ 0 ) exp [ ( cos θ α / s ) 2 ] × exp { i 2 π σ 0 [ x 0 sin θ cos ϕ ( y 0 α z 0 ) sin θ sin ϕ + ( z 0 + α y 0 ) cos θ ] } cos θ sin θ d θ d ϕ , 1 σ 0 2 φ ϑ L ( σ 0 ) exp [ ( cos θ α / s ) 2 ] × exp { i 2 π σ 0 [ x 0 sin θ cos ϕ ( y 0 + α z 0 ) sin θ sin ϕ + ( z 0 + α y 0 ) cos θ ] } cos θ sin θ d θ d ϕ .
S ( σ 0 ) 2 π 2 R 2 β ( 2 π σ 0 R ) ϑ L ( σ 0 ) × exp [ ( cos θ α / s ) 2 ] J 0 { 2 π σ 0 sin θ [ x 0 2 + ( y 0 α z 0 ) 2 ] 1 / 2 } × exp [ i 2 π σ 0 ( z 0 + α y 0 ) cos θ ] cos θ sin θ d ϕ ,
A ( p 1 , q 1 , p 2 , q 2 , σ 0 ) = ( p 2 , q 2 , σ 0 ) δ ( p 1 p 2 , q 1 q 2 ) ,
S ( σ 0 ) = π R 2 p 2 2 + q 2 2 1 ( p 2 , q 2 , σ 0 ) 2 J 1 ( 2 π σ 0 R α m 2 ) 2 π σ 0 R α m 2 × exp { i 2 π σ 0 [ x 0 p 2 + ( y 0 α z 0 + α z 1 ) q 2 + ( y 0 α + z 0 ) m 2 ] d p 2 d q 2
S ( σ 0 ) = π R 2 φ ϑ L ( θ , ϕ , σ 0 ) 2 J 1 ( 2 π σ 0 R α cos θ ) 2 π σ 0 R α cos θ × exp { i 2 π σ 0 [ x 0 sin θ cos ϕ + ( y 0 α z 0 + α z 1 ) sin θ sin ϕ + ( y 0 α + z 0 ) cos θ ] } cos θ sin θ d θ d ϕ ,
S ( σ 0 ) = 2 π 2 R 2 ϑ L ( θ , σ 0 ) 2 J 1 ( 2 π σ 0 R α cos θ ) 2 π σ 0 R α cos θ × J 0 ( 2 π σ 0 { x 0 2 + [ y 0 + α ( z 1 z 0 ) ] 2 } sin θ ) × exp [ i 2 π σ 0 ( y 0 α + z 0 ) cos θ sin θ d θ ,
Δ z t = ( 1 cos α ) z t + y t sin α y t α ,
Δ y t = ( 1 cos α ) y t z t sin α z t α ,
z 0 z s + y t α ,
y 0 y s z t α .
Δ Y y 0 + α ( z 1 z 0 ) , y s z t α + α ( z 1 z s y t α ) , y s + α ( z 1 z s z t ) ,
Δ Z z 0 + α y 0 , = z s + y t α + α ( y s z t α ) , z s + α ( y t + y s ) ,
Δ z r = z s + α ( y t r + y s ) ,
z s = Δ z r α ( y t r + y s ) .
S ( Δ z r , σ 0 ) = 2 π 2 R 2 ϑ L ( θ , σ 0 ) 2 J 1 ( 2 π σ 0 R α cos θ ) 2 π σ 0 R α cos θ × J 0 ( 2 π σ 0 { x 0 2 + [ y s + α ( z 1 Δ z r z t ) ] 2 } 1 / 2 sin θ ) × exp [ i 2 π σ 0 [ Δ z r + α ( y t y t r ) ] cos θ sin θ d θ ,
S ( σ , σ 0 ) = Δ z r σ [ S ( σ 0 , Δ z r ) ] = 2 π 2 R 2 ϑ L ( θ , σ 0 ) 2 J 1 ( 2 π σ 0 R α cos θ ) 2 π σ 0 R α cos θ × exp [ i 2 π σ 0 α ( y t y t r ) cos θ ] × Δ z r σ [ J 0 ( 2 π σ 0 { x 0 2 + [ y s + α ( z 1 Δ z r z t ) ] 2 } 1 / 2 sin θ ) × exp ( i 2 π σ 0 Δ z r cos θ ) ] cos θ sin θ d θ .
J 0 ( 2 π σ 0 { x 0 2 + [ y s + α ( z 1 Δ z r z t ) ] 2 } 1 / 2 sin θ ) = J 0 ( 2 π σ 0 { x 0 2 + [ y s + α ( z 1 z t ) ] 2 } 1 / 2 sin θ ) + α ( 2 π σ 0 ) 2 [ y s + α ( z 1 z t ) ] sin 2 θ Δ z r 2 2 J 1 ( 2 π σ 0 { x 0 2 + [ y s + α ( z 1 z t ) ] 2 } 1 / 2 sin θ 2 π σ 0 { x 0 2 + [ y s + α ( z 1 z t ) ] 2 } 1 / 2 sin θ + O ( Δ z r 2 ) .
α ( 2 π σ 0 ) 2 [ y s + α ( z 1 z t ) ] sin 2 θ Δ z r 2 1 .
2 π σ 0 α 2 π σ 0 [ y s + α ( z 1 z t ) ] sin θ sin θ Δ z r 2 .
Δ z r sin θ R .
S ( σ , σ 0 ) 2 π 2 R 2 ϑ L ( θ , σ 0 ) 2 J 1 ( 2 π σ 0 R α cos θ ) 2 π σ 0 R α cos θ J 0 × ( 2 π σ 0 { x 0 2 + [ y s + α ( z 1 z t ) ] 2 } 1 / 2 sin θ ) × exp [ i 2 π σ 0 α ( y t y t r ) cos θ ] × F Δ z r σ [ exp ( i 2 π σ 0 Δ z r × cos θ ) ] cos θ sin θ d θ , 2 π 2 R 2 θ max θ max L ( θ , σ 0 ) 2 J 1 ( 2 π σ 0 R α cos θ ) 2 π σ 0 R α cos θ × J 0 ( 2 π σ 0 { x 0 2 + [ y s + α ( z 1 z t ) ] 2 } 1 / 2 sin θ ) × exp [ i 2 π σ 0 α ( y t y t r ) cos θ ] δ ( σ σ 0 cos θ ) cos θ sin θ d θ .
S ( σ , σ 0 ) 2 π 2 R 2 L [ arccos ( σ / σ 0 ) , σ 0 ] 2 J 1 ( 2 π R α σ ) 2 π R α σ × J 0 ( 2 π { x 0 2 + [ y s + α ( z 1 z t ) ] 2 } 1 / 2 ( σ 0 2 σ 2 ) 1 / 2 ) × exp [ i 2 π α ( y t y t r ) σ ] σ / σ o 2 .
Δ Y y s α ( z t + z s ) .
S ( σ 0 ) = 2 π 2 R 2 β ( 2 π σ 0 R ) ϑ L ( θ , σ 0 ) × exp [ ( cos θ α / s ) 2 ] J 0 ( 2 π σ 0 sin θ { x 0 2 + [ y s α ( z t + z s ) ] 2 } 1 / 2 ) × exp { i 2 π σ 0 [ z s + α ( y t + y s ) ] cos θ } cos θ sin θ d θ .
S ( Δ z r , σ 0 ) = 2 π 2 R 2 β ( 2 π σ 0 R ) ϑ L ( θ , σ 0 ) exp [ ( cos θ α / s ) 2 ] J 0 ( 2 π σ 0 sin θ { x 0 2 + [ y s α ( z t + Δ z r ) ] 2 } 1 / 2 ) × exp { i 2 π σ 0 [ Δ z r + α ( y t y t r ) ] cos θ } cos θ sin θ d θ ,
J 0 ( 2 π σ 0 sin θ { x 0 2 + [ y s α ( z t + Δ z r ) ] 2 } 1 / 2 ) = J 0 { 2 π σ 0 sin θ [ x 0 2 + ( y s α z t ) 2 ] 1 / 2 } + α ( 2 π σ 0 ) 2 ( y s α z t ) sin 2 θ Δ z r 2 × 2 J 1 ( 2 π σ 0 sin θ { x 0 2 [ y s α z t ) ] 2 } 1 / 2 ) 2 π σ 0 sin θ [ x 0 2 ( y s α z t ) 2 ] 1 / 2 + O ( Δ z r 2 ) .
S ( σ , σ 0 ) 2 π 2 R 2 β ( 2 π σ 0 R ) L [ arccos ( σ / σ 0 ) , σ 0 ] × exp { [ ( σ α ) / ( s σ 0 ) ] 2 } J 0 { 2 π σ 0 [ x 0 2 + ( y s α z t ) 2 ] 1 / 2 } exp [ i 2 π α ( y t y t r ) σ ] σ / σ o 2 .

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