Abstract

We present a novel, to our knowledge, fiber-based single-channel polarization-sensitive spectral interferometry system that provides depth-resolved measurement of polarization transformations of light reflected from a sample. Algebraic expressions for the Stokes parameters at the output of the interferometer are derived for light reflected from a birefringent sample by using the cross-spectral density function. By insertion of a fiber-optic spectral polarimetry instrument into the detection path of a common-path spectral interferometer, the full set of Stokes parameters of light reflected from a sample can be obtained with a single optical frequency scan. The methodology requires neither polarization-control components nor prior knowledge of the polarization state of light incident on the sample. The fiber-based single-channel polarization-sensitive spectral interferometer and analysis are demonstrated by measurement of phase retardation and fast-axis angle of a birefringent mica plate.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2005 (2)

2004 (3)

2003 (4)

2002 (2)

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagia, "Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography," Opt. Lett. 27, 1803-1805 (2002).
[CrossRef]

2000 (1)

1999 (1)

1998 (1)

G. Häusler and M. W. Lindner, "'Coherence radar' and 'spectral radar'—new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

1997 (5)

1996 (1)

J. P. Hamaker, J. D. Bregman, and R. J. Sault, "Understanding radio polarimetry. I. Mathematical foundation," Astron. Astrophys., Suppl. Ser. 117, 137-147 (1996).
[CrossRef]

1995 (4)

A. Dutt and V. Rokhlin, "Fast Fourier transforms for nonequispaced data, II," Appl. Comput. Harmon. Anal. 2, 85-100 (1995).
[CrossRef]

A. F. Ferchher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117, 43-48 (1995).
[CrossRef]

U. Schnell, E. Zimmermann, and R. Dändliker, "Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry," Pure Appl. Opt. 4, 643-651 (1995).
[CrossRef]

V. N. Kumar and D. N. Rao, "Using interference in the frequency domain for precise determination of thickness and refractive indices of normal dispersive materials," J. Opt. Soc. Am. B 12, 1559-1563 (1995).
[CrossRef]

1994 (1)

1976 (1)

1971 (1)

1969 (1)

1961 (1)

Bajraszewski, T.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Bouma, B. E.

Bregman, J. D.

J. P. Hamaker, J. D. Bregman, and R. J. Sault, "Understanding radio polarimetry. I. Mathematical foundation," Astron. Astrophys., Suppl. Ser. 117, 137-147 (1996).
[CrossRef]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Cao, X. D.

Cense, B.

Chandrasekharan, V.

Chen, T. C.

Chen, Z.

Chinn, S. R.

Choma, M. A.

Damany, H.

Dändliker, R.

U. Schnell, E. Zimmermann, and R. Dändliker, "Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry," Pure Appl. Opt. 4, 643-651 (1995).
[CrossRef]

Dave, D. P.

E. Kim, D. P. Dave, and T. E. Milner, "Fiber optic spectral polarimeter using a broadband swept laser source," Opt. Commun. 249, 351-356 (2005).
[CrossRef]

de Boer, J.

de Boer, J. F.

Dutt, A.

A. Dutt and V. Rokhlin, "Fast Fourier transforms for nonequispaced data, II," Appl. Comput. Harmon. Anal. 2, 85-100 (1995).
[CrossRef]

El-Zaiat, S. Y.

M. Medhat and S. Y. El-Zaiat, "Interferometric determination of the birefringence dispersion of anisotropic materials," Opt. Commun. 141, 145-149 (1997).
[CrossRef]

A. F. Ferchher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117, 43-48 (1995).
[CrossRef]

Endo, T.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagia, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Fercher, A. F.

Ferchher, A. F.

A. F. Ferchher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117, 43-48 (1995).
[CrossRef]

Fittinghoff, D. N.

Fujimoto, J. G.

Hamaker, J. P.

J. P. Hamaker, J. D. Bregman, and R. J. Sault, "Understanding radio polarimetry. I. Mathematical foundation," Astron. Astrophys., Suppl. Ser. 117, 137-147 (1996).
[CrossRef]

Häusler, G.

G. Häusler and M. W. Lindner, "'Coherence radar' and 'spectral radar'—new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

Hitzenberger, C. K.

Itoh, M.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagia, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagia, "Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography," Opt. Lett. 27, 1803-1805 (2002).
[CrossRef]

Izatt, J. A.

Jung, W.

Kamp, G.

A. F. Ferchher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117, 43-48 (1995).
[CrossRef]

Kane, D. J.

Katada, C.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagia, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Kato, T.

Kemp, N. J.

Kim, E.

E. Kim, D. P. Dave, and T. E. Milner, "Fiber optic spectral polarimeter using a broadband swept laser source," Opt. Commun. 249, 351-356 (2005).
[CrossRef]

Konoplev, O. A.

Kowalczyk, A.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Kulhavy, M.

Kumar, V. N.

Leitgeb, R.

R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, "Performance of Fourier domain vs. time domain optical coherence tomography," Opt. Express 11, 889-894 (2003).
[CrossRef] [PubMed]

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Lexer, F.

Lindner, M. W.

G. Häusler and M. W. Lindner, "'Coherence radar' and 'spectral radar'—new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

Makita, S.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagia, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagia, "Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography," Opt. Lett. 27, 1803-1805 (2002).
[CrossRef]

Mandel, L.

Medhat, M.

M. Medhat and S. Y. El-Zaiat, "Interferometric determination of the birefringence dispersion of anisotropic materials," Opt. Commun. 141, 145-149 (1997).
[CrossRef]

Meyerhofer, D. D.

Milner, T. E.

Mutoh, M.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagia, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Nassif, N.

Nelson, J. S.

Oka, K.

Park, B. H.

Park, J.

Peterson, K. A.

Pierce, M. C.

Rao, D. N.

Rokhlin, V.

A. Dutt and V. Rokhlin, "Fast Fourier transforms for nonequispaced data, II," Appl. Comput. Harmon. Anal. 2, 85-100 (1995).
[CrossRef]

Rylander, H. G.

Sandeman, R. J.

Sarunic, M. V.

Sault, R. J.

J. P. Hamaker, J. D. Bregman, and R. J. Sault, "Understanding radio polarimetry. I. Mathematical foundation," Astron. Astrophys., Suppl. Ser. 117, 137-147 (1996).
[CrossRef]

Saxer, C. E.

Schnell, U.

U. Schnell, E. Zimmermann, and R. Dändliker, "Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry," Pure Appl. Opt. 4, 643-651 (1995).
[CrossRef]

Smirl, A. L.

Sutoh, Y.

Swanson, E. A.

Takahashi, M.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagia, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Tearney, G. J.

Trebino, R.

Vakhtin, A. B.

Walecki, W. J.

Wojtkowski, M.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Wolf, E.

L. Mandel and E. Wolf, "Spectral coherence and the concept of cross-spectral purity," J. Opt. Soc. Am. 66, 529-535 (1976).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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

Wood, W. R.

Yang, C.

Yasuno, Y.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagia, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagia, "Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography," Opt. Lett. 27, 1803-1805 (2002).
[CrossRef]

Yatagia, T.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagia, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagia, "Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography," Opt. Lett. 27, 1803-1805 (2002).
[CrossRef]

Yun, S. H.

Zaatari, H. N.

Zhang, J.

Zhao, Y.

Zheng, I.

Zimmermann, E.

U. Schnell, E. Zimmermann, and R. Dändliker, "Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry," Pure Appl. Opt. 4, 643-651 (1995).
[CrossRef]

Appl. Comput. Harmon. Anal. (1)

A. Dutt and V. Rokhlin, "Fast Fourier transforms for nonequispaced data, II," Appl. Comput. Harmon. Anal. 2, 85-100 (1995).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagia, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Astron. Astrophys., Suppl. Ser. (1)

J. P. Hamaker, J. D. Bregman, and R. J. Sault, "Understanding radio polarimetry. I. Mathematical foundation," Astron. Astrophys., Suppl. Ser. 117, 137-147 (1996).
[CrossRef]

J. Biomed. Opt. (2)

G. Häusler and M. W. Lindner, "'Coherence radar' and 'spectral radar'—new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

A. F. Ferchher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117, 43-48 (1995).
[CrossRef]

M. Medhat and S. Y. El-Zaiat, "Interferometric determination of the birefringence dispersion of anisotropic materials," Opt. Commun. 141, 145-149 (1997).
[CrossRef]

E. Kim, D. P. Dave, and T. E. Milner, "Fiber optic spectral polarimeter using a broadband swept laser source," Opt. Commun. 249, 351-356 (2005).
[CrossRef]

Opt. Express (4)

Opt. Lett. (9)

N. Nassif, B. Cense, B. H. Park, S. H. Yun, T. C. Chen, B. E. Bouma, G. J. Tearney, and J. F. de Boer, "In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography," Opt. Lett. 29, 480-482 (2004).
[CrossRef] [PubMed]

J. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, "Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography," Opt. Lett. 28, 2067-2069 (2003).
[CrossRef] [PubMed]

C. E. Saxer, J. F. de Boer, B. H. Park, Y. Zhao, Z. Chen, and J. S. Nelson, "High-speed fiber-based polarization-sensitive optical coherence tomography of in vivo human skin," Opt. Lett. 25, 1355-1357 (2000).
[CrossRef]

Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagia, "Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography," Opt. Lett. 27, 1803-1805 (2002).
[CrossRef]

W. J. Walecki, D. N. Fittinghoff, A. L. Smirl, and R. Trebino, "Characterization of the polarization state of weak ultrashort coherent signals by dual-channel spectral interferometry," Opt. Lett. 22, 81-83 (1997).
[CrossRef] [PubMed]

S. R. Chinn, E. A. Swanson, and J. G. Fujimoto, "Optical coherence tomography using a frequency-tunable optical source," Opt. Lett. 22, 340-342 (1997).
[CrossRef] [PubMed]

I. Zheng, O. A. Konoplev, and D. D. Meyerhofer, "Determination of the optical-axis orientation of a uniaxial crystal by frequency-domain interferometry," Opt. Lett. 22, 931-933 (1997).
[CrossRef] [PubMed]

K. Oka and T. Kato, "Spectroscopic polarimetry with a channeled spectrum," Opt. Lett. 24, 1475-1477 (1999).
[CrossRef]

X. D. Cao and D. D. Meyerhofer, "Frequency-domain interferometer for measurement of the polarization mode dispersion in single-mode optical fibers," Opt. Lett. 19, 1837-1839 (1994).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

U. Schnell, E. Zimmermann, and R. Dändliker, "Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry," Pure Appl. Opt. 4, 643-651 (1995).
[CrossRef]

Other (3)

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of a fiber-based single-channel polarization-sensitive spectral interferometer with a FOSPI.

Fig. 2
Fig. 2

Output intensity from a fiber-based single-channel polarization-sensitive spectral interferometer with FOSPI versus optical frequency.

Fig. 3
Fig. 3

Fourier transform magnitude of Fig. 2. (a) Interference between the front and the back surfaces of the glass window. (b) Interference between the back surfaces of the glass window and the sample retarder.

Fig. 4
Fig. 4

(a) Estimated phase retardation ( δ ) and (b) fast-axis angle ( α ) over the set orientation of a mica retarder.

Equations (44)

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

Γ i j ( r 1 , r 2 , t 1 , t 2 ) E i ( r 1 , t 1 ) E j * ( r 2 , t 2 ) ,
Γ i j ( r 1 , r 2 , τ ) = E i ( r 1 , t + τ ) E j * ( r 2 , t ) ,
E ( r , t ) = E ( r , ν ) e i 2 π ν t d ν ,
Φ i j ( r 1 , r 2 , ν 1 , ν 2 ) = E i ( r 1 , ν 1 ) E j * ( r 2 , ν 2 ) = E i ( r 1 , t 1 ) E j * ( r 2 , t 2 ) e i 2 π ν 1 t 1 e i 2 π ν 2 t 2 d t 1 d t 2 = Γ i j ( r 1 , r 2 , t 1 , t 2 ) e i 2 π ν 1 t 1 e i 2 π ν 2 t 2 d t 1 d t 2 .
Φ i j ( r 1 , r 2 , ν 1 , ν 2 ) = δ ( ν 1 ν 2 ) W i j ( r 1 , r 2 , ν 2 ) ,
W i j ( r 1 , r 2 , ν ) = Γ i j ( r 1 , r 2 , τ ) e i 2 π ν τ d τ
I ( r ) = Γ i i ( r , r , 0 ) = W i i ( r , r , ν ) d ν .
J = E E * = ( E x ( ν ) E x * ( ν ) E x ( ν ) E y * ( ν ) E y ( ν ) E x * ( ν ) E y ( ν ) E y * ( ν ) ) ,
S = ( S 0 ( ν ) S 1 ( ν ) S 2 ( ν ) S 3 ( ν ) ) = A J , A = [ 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ] .
J = ( W x x ( ν ) W x y ( ν ) W y x ( ν ) W y y ( ν ) ) ,
S = ( W x x ( ν ) + W y y ( ν ) W x x ( ν ) W y y ( ν ) 2 R ( W x y ( ν ) ) 2 I ( W x y ( ν ) ) ) ,
J out = ( T E in ) ( T * E in * ) = ( T T * ) J in .
S out = A ( T T * ) J in
= [ A ( T T * ) A 1 ] S in
= M S in ,
A 1 = 1 2 [ 1 1 0 0 0 0 1 i 0 0 1 i 1 1 0 0 ] .
E i ( r , ν ) = E i ( r 1 , ν ) e i 2 π ν t 1 + E i ( r 2 , ν ) e i 2 π ν t 2 ,
W i j ( r , r , ν ) = W i j ( 1 ) ( r , r , ν ) + W i j ( 2 ) ( r , r , ν ) + 2 R { W i j ( r 1 , r 2 , ν ) e i 2 π ν τ } , i = j ,
W i j ( r , r , ν ) = W i j ( 1 ) ( r , r , ν ) + W i j ( 2 ) ( r , r , ν ) + W i j ( r 1 , r 2 , ν ) e i 2 π ν τ + W i j ( r 2 , r 1 , ν ) e i 2 π ν τ , i j ,
I ( r ) = I ( 1 ) ( r ) + I ( 2 ) ( r ) + 2 R { Γ i i ( r 1 , r 2 , τ ) } ,
F 1 { W i i ( r , r , ν ) } = F 1 { W i i ( 1 ) ( r , r , ν ) } + F 1 { W i i ( 2 ) ( r , r , ν ) } + Γ i i ( r 1 , r 2 , t τ ) + Γ i i * ( r 1 , r 2 , t + τ ) .
S = S ( 1 ) + S ( 2 ) + S ( i ) ,
S ( i ) = ( 2 R { ( W x x ( r 1 , r 2 , ν ) + W y y ( r 1 , r 2 , ν ) ) e i 2 π ν τ } 2 R { ( W x x ( r 1 , r 2 , ν ) W y y ( r 1 , r 2 , ν ) ) e i 2 π ν τ } 2 R { W x y ( r 1 , r 2 , ν ) e i 2 π ν τ + W x y ( r 2 , r 1 , ν ) e i 2 π ν τ } 2 I { W x y ( r 1 , r 2 , ν ) e i 2 π ν τ + W x y ( r 2 , r 1 , ν ) e i 2 π ν τ } ) .
T = r s [ cos ( δ ( ν ) 2 ) + i sin ( δ ( ν ) 2 ) cos 2 α i sin ( δ ( ν ) 2 ) sin 2 α i sin ( δ ( ν ) 2 ) sin 2 α cos ( ( ν ) 2 ) i sin ( δ ( ν ) 2 ) cos 2 α ] ,
S 0 ( i ) ( ν ) = 2 r s cos Δ ( ν ) cos δ ( ν ) 2 S 0 ( 1 ) ( ν ) + 2 r s sin Δ ( ν ) sin δ ( ν ) 2 ( cos 2 α S 1 ( 1 ) ( ν ) + sin 2 α S 2 ( 1 ) ( ν ) ) ,
S 1 ( i ) ( ν ) = 2 r s cos ( ν ) ( cos δ ( ν ) 2 S 1 ( 1 ) ( ν ) sin δ ( ν ) 2 sin 2 α S 3 ( 1 ) ( ν ) ) + 2 r s sin Δ ( ν ) sin δ ( ν ) 2 cos 2 α s 0 ( 1 ) ( ν ) ,
S 2 ( i ) ( ν ) = 2 r s cos Δ ( ν ) ( cos δ ( ν ) 2 S 2 ( 1 ) ( ν ) + sin δ ( ν ) 2 cos 2 α S 3 ( 1 ) ( ν ) ) + 2 r s sin Δ ( ν ) sin δ ( ν ) 2 sin 2 α S 0 ( 1 ) ( ν ) ,
S 3 ( i ) ( ν ) = 2 r s cos Δ ( ν ) ( sin δ ( ν ) 2 sin 2 α S 1 ( 1 ) ( ν ) sin δ ( ν ) 2 cos 2 α S 2 ( 1 ) ( ν ) + cos δ ( ν ) 2 S 3 ( 1 ) ( ν ) ) ,
Δ ( ν ) = 2 π ν τ = 4 π c ( L + n ¯ d ) ν ,
δ ( ν ) = 4 π c Δ n d ν .
I out ( ν ) = 1 2 S 0 , in ( ν ) + 1 2 cos ϕ 2 ( ν ) S 1 , in ( ν ) + 1 2 sin ϕ 1 ( ν ) sin ϕ 2 ( ν ) S 2 , in ( ν ) 1 2 cos ϕ 1 ( ν ) sin ϕ 2 ( ν ) S 3 , in ( ν ) = 1 2 S 0 , in ( ν ) + 1 2 cos ϕ 2 ( ν ) S 1 , in ( ν ) + 1 4 S 23 , in ( ν ) cos ( ϕ 2 ( ν ) ϕ 1 ( ν ) + arg ( S 23 , in ( ν ) ) ) 1 4 S 23 , in ( ν ) cos ( ϕ 2 ( ν ) + ϕ 1 ( ν ) arg ( S 23 , in ( ν ) ) ) ,
ϕ 1 ( 2 ) ( ν ) = 2 π ν Δ n ( ν ) c L 1 ( 12 ) ,
I out ( i ) ( ν ) = r s cos Δ ( ν ) cos δ ( ν ) 2 S 0 ( 1 ) ( ν ) + r s sin Δ ( ν ) sin δ ( ν ) 2 ( cos 2 α S 1 ( 1 ) ( ν ) + sin 2 α S 2 ( 1 ) ( ν ) ) + 1 2 r s [ ( cos δ ( ν ) 2 S 1 ( 1 ) ( ν ) sin δ ( ν ) 2 sin 2 α S 3 ( 1 ) ( ν ) ) cos ( Δ ( ν ) ϕ 2 ( ν ) ) + sin δ ( ν ) 2 cos 2 α S 0 ( 1 ) ( ν ) sin ( Δ ( ν ) ϕ 2 ( ν ) ) ] + 1 2 r s [ ( cos δ ( ν ) 2 S 1 ( 1 ) ( ν ) sin δ ( ν ) 2 sin 2 α S 3 ( 1 ) ( ν ) ) cos ( Δ ( ν ) + ϕ 2 ( ν ) ) + sin δ ( ν ) 2 cos 2 α S 0 ( 1 ) ( ν ) sin ( Δ ( ν ) + ϕ 2 ( ν ) ) ] + 1 4 r s [ ( cos δ ( ν ) 2 S 2 ( 1 ) ( ν ) + sin δ ( ν ) 2 cos 2 α S 3 ( 1 ) ( ν ) ) cos ( Δ ( ν ) ϕ 2 ( ν ) + ϕ 1 ( ν ) ) + { sin δ ( ν ) 2 sin 2 α ( S 0 ( 1 ) ( ν ) + S 1 ( 1 ) ( ν ) ) sin δ ( ν ) 2 cos 2 α S 2 ( 1 ) ( ν ) + cos δ ( ν ) 2 S 3 ( 1 ) ( ν ) } sin ( Δ ( ν ) ϕ 2 ( ν ) + ϕ 1 ( ν ) ) ] + 1 4 r s [ ( cos δ ( ν ) 2 S 2 ( 1 ) ( ν ) + sin δ ( ν ) 2 cos 2 α S 3 ( 1 ) ( ν ) ) cos ( Δ ( ν ) + ϕ 2 ( ν ) ϕ 1 ( ν ) ) + { sin δ ( ν ) 2 sin 2 α ( S 0 ( 1 ) ( ν ) S 1 ( 1 ) ( ν ) ) + sin δ ( ν ) 2 cos 2 α S 2 ( 1 ) ( ν ) cos δ ( ν ) 2 S 3 ( 1 ) ( ν ) } sin ( Δ ( ν ) + ϕ 2 ( ν ) ϕ 1 ( ν ) ) ] 1 4 r s [ ( cos δ ( ν ) 2 S 2 ( 1 ) ( ν ) + sin δ ( ν ) 2 cos 2 α S 3 ( 1 ) ( ν ) ) cos ( Δ ( ν ) ϕ 2 ( ν ) ϕ 1 ( ν ) ) + { sin δ ( ν ) 2 sin 2 α ( S 0 ( 1 ) ( ν ) S 1 ( 1 ) ( ν ) ) + sin δ ( ν ) 2 cos 2 α S 2 ( 1 ) ( ν ) cos δ ( ν ) 2 S 3 ( 1 ) ( ν ) } sin ( Δ ( ν ) ϕ 2 ( ν ) ϕ 1 ( ν ) ) ] 1 4 r s [ ( cos δ ( ν ) 2 S 2 ( 1 ) ( ν ) + sin δ ( ν ) 2 cos 2 α S 3 ( 1 ) ( ν ) ) cos ( Δ ( ν ) + ϕ 2 ( ν ) + ϕ 1 ( ν ) ) + { sin δ ( ν ) 2 sin 2 α ( S 0 ( 1 ) ( ν ) + S 1 ( 1 ) ( ν ) ) sin δ ( ν ) 2 cos 2 α S 2 ( 1 ) ( ν ) + cos δ ( ν ) 2 S 3 ( 1 ) ( ν ) } sin ( Δ ( ν ) + ϕ 2 ( ν ) + ϕ 1 ( ν ) ) ] .
Δ ( ν ) = 2 π L o ν + 2 π L 1 ( ν ) ,
ϕ i ( ν ) = 2 π l i , o ν + 2 π l i , 1 ( ν ) .
L o : 1 2 r s e i Δ ( ν ) { cos δ ( ν ) 2 S 0 ( 1 ) ( ν ) i sin δ ( ν ) 2 ( cos 2 α S 1 ( 1 ) ( ν ) + sin 2 α S 2 ( 1 ) ( ν ) ) } ,
L o + l 2 , o : 1 4 r s e i ϕ 2 e i Δ ( ν ) { ( cos δ ( ν ) 2 S 1 ( 1 ) sin δ ( ν ) 2 sin 2 α S 3 ( 1 ) ( ν ) ) i ( sin δ ( ν ) 2 cos 2 α S 0 ( 1 ) ( ν ) ) } ,
L o + l 2 , o l 1 , o : 1 8 r s e i ( ϕ 2 ( ν ) ϕ 1 ( ν ) ) e i Δ ( ν ) [ ( cos δ ( ν ) 2 S 2 ( 1 ) ( ν ) + sin δ ( ν ) 2 cos 2 α S 3 ( 1 ) ( ν ) ) i { sin δ ( ν ) 2 sin 2 α ( S 0 ( 1 ) ( ν ) S 1 ( 1 ) ( ν ) ) + sin δ ( ν ) 2 cos 2 α S 2 ( 1 ) ( ν ) cos δ ( ν ) 2 S 3 ( 1 ) ( ν ) } ] ,
L o + l 2 , o + l 1 , o : 1 8 r s e i ( ϕ 2 ( ν ) + ϕ 1 ( ν ) ) e i Δ ( ν ) [ ( cos δ ( ν ) 2 S 2 ( 1 ) ( ν ) + sin δ ( ν ) 2 cos 2 α S 3 ( 1 ) ( ν ) ) i { sin δ ( ν ) 2 sin 2 α ( S 0 ( 1 ) ( ν ) + S 1 ( 1 ) ( ν ) ) sin δ ( ν ) 2 cos 2 α S 2 ( 1 ) ( ν ) + cos δ ( ν ) 2 S 3 ( 1 ) ( ν ) } ] .
1 2 r s cos δ ( ν ) 2 S 0 ( 1 ) ( ν ) ,
1 4 r s sin δ ( ν ) 2 cos 2 α S 0 ( 1 ) ( ν ) ,
1 4 r s sin δ ( ν ) 2 sin 2 α S 0 ( 1 ) ( ν ) ,
tan δ ( ν ) 2 = 2 expression ( 36 ) 2 + expression ( 37 ) 2 expression ( 35 ) ,
tan α = expression ( 37 ) expression ( 36 ) .

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