Abstract

We report on an exact analytic method to evaluate the optical eye diagram due to polarization-mode dispersion (PMD), polarization-dependent loss (PDL), and chromatic dispersion (CD) in dynamic fiber links for any given optical pulse sequence. With this method we consider all orders of PMD and PDL, and we obtain the time-dependent average output light intensity as well as its corresponding variations. With this method we can obtain accurate impairment evaluations for communication systems due to given PMD, PDL, and CD. Time jitter caused by the PMD and PDL polarization direction correlation is also discussed.

© 2004 Optical Society of America

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References

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  1. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541–4550 (2000).
    [CrossRef] [PubMed]
  2. N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119–125 (1997).
    [CrossRef]
  3. B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
    [CrossRef]
  4. P. Lu, L. Chen, and X. Bao, “Polarization mode dispersion and polarization dependent loss for a pulse in single-mode fibers,” J. Lightwave Technol. 19, 856–860 (2001).
    [CrossRef]
  5. Y. Li and A. Yariv, “Solution to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,” J. Opt. Soc. Am. B 17, 1821–1827 (2000).
    [CrossRef]
  6. F. Bruyére and O. Audouin, “Penalties in long-haul optical amplifier systems due to polarization dependent loss and gain,” IEEE Photon. Technol. Lett. 6, 654–656 (1994).
    [CrossRef]
  7. E. Lichtman, “Limitations imposed by polarization dependent gain and loss on all-optical ultralong communication systems,” J. Lightwave Technol. 13, 906–913 (1995).
    [CrossRef]
  8. G. Biodini, W. L. Kath, and C. R. Menyuk, “Importance sampling for polarization mode dispersion,” IEEE Photonics Technol. Lett. 14, 310–312 (2002).
    [CrossRef]
  9. S. L. Fogal, G. Biondini, and W. L. Kath, “Multiple importance sampling for first- and second-order polarization-mode dispersion,” IEEE Photon. Technol. Lett. 14, 1273–1275 (2002), errata, 14, 1487 (2002).
    [CrossRef]
  10. D. Yevick, “Multicanonical communication system modeling: application to PMD statistics,” IEEE Photon. Technol. Lett. 14, 1512–1514 (2002).
    [CrossRef]
  11. D. Yevick, “The accuracy of multicanonical system models,” IEEE Photon. Technol. Lett. 15, 224–226 (2003).
    [CrossRef]
  12. M. Shtaif, “The Brownian-bridge method for simulating polarization mode dispersion in optical communication systems,” IEEE Photon. Technol. Lett. 15, 51–53 (2003).
    [CrossRef]
  13. M. Karlsson, “Polarization mode dispersion-induced pulse broadening in optical fibers,” Opt. Lett. 23, 688–690 (1998).
    [CrossRef]
  14. M. Wang, T. Li, and S. Jian, “Analytical theory of pulse broadening due to polarization mode dispersion and polarization-dependent loss,” Opt. Commun. 223, 75–80 (2003).
    [CrossRef]
  15. A. Eyal, D. Kuperman, O. Dimenstein, and M. Tur, “Polarization dependence of the intensity modulation transfer function of an optical system with PMD and PDL,” IEEE Photon. Technol. Lett. 14, 1515–1517 (2002).
    [CrossRef]
  16. R. Feced, S. J. Savory, and A. Hadjifotiou, “Interaction between polarization mode dispersion and polarization-dependent losses in optical communication links,” J. Opt. Soc. Am. B 20, 424–433 (2003).
    [CrossRef]
  17. L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Principal state vector autocorrelation in a fiber optic system having both polarization-mode dispersion and polarization dependent loss,” in Applications of Photonic Technology 6, R. A. Lessard and G. A. Lampropoulos, eds., Proc. SPIE 5260, 382–385 (2003).
    [CrossRef]
  18. L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Effect of local PMD and PDL directional correlation on the principal state of polarization vector autocorrelation,” Opt. Express 11, 3141–3146 (2003), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  19. M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999).
    [CrossRef]
  20. S. Hadjifaradji, L. Chen, and X. Bao, “Eye diagram evaluation in single mode fibers having PMD, PDL and CD,” in Holey Fibers and Photonic Crystals/Polarization Mode Dispersion/Photonics Time/Frequency Measurement and Control, 2003, Digest of the LEOS Summer Topical Meetings (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 53–54.
  21. A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14, 313–315 (2002).
    [CrossRef]
  22. A. Galtarossa and L. Palmieri, “The exact statistics of polarization-dependent loss in fiber-optic links,” IEEE Photon. Technol. Lett. 15, 57–59 (2003).
    [CrossRef]

2003 (7)

D. Yevick, “The accuracy of multicanonical system models,” IEEE Photon. Technol. Lett. 15, 224–226 (2003).
[CrossRef]

M. Shtaif, “The Brownian-bridge method for simulating polarization mode dispersion in optical communication systems,” IEEE Photon. Technol. Lett. 15, 51–53 (2003).
[CrossRef]

M. Wang, T. Li, and S. Jian, “Analytical theory of pulse broadening due to polarization mode dispersion and polarization-dependent loss,” Opt. Commun. 223, 75–80 (2003).
[CrossRef]

L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Principal state vector autocorrelation in a fiber optic system having both polarization-mode dispersion and polarization dependent loss,” in Applications of Photonic Technology 6, R. A. Lessard and G. A. Lampropoulos, eds., Proc. SPIE 5260, 382–385 (2003).
[CrossRef]

A. Galtarossa and L. Palmieri, “The exact statistics of polarization-dependent loss in fiber-optic links,” IEEE Photon. Technol. Lett. 15, 57–59 (2003).
[CrossRef]

R. Feced, S. J. Savory, and A. Hadjifotiou, “Interaction between polarization mode dispersion and polarization-dependent losses in optical communication links,” J. Opt. Soc. Am. B 20, 424–433 (2003).
[CrossRef]

L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Effect of local PMD and PDL directional correlation on the principal state of polarization vector autocorrelation,” Opt. Express 11, 3141–3146 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

2002 (4)

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14, 313–315 (2002).
[CrossRef]

G. Biodini, W. L. Kath, and C. R. Menyuk, “Importance sampling for polarization mode dispersion,” IEEE Photonics Technol. Lett. 14, 310–312 (2002).
[CrossRef]

D. Yevick, “Multicanonical communication system modeling: application to PMD statistics,” IEEE Photon. Technol. Lett. 14, 1512–1514 (2002).
[CrossRef]

A. Eyal, D. Kuperman, O. Dimenstein, and M. Tur, “Polarization dependence of the intensity modulation transfer function of an optical system with PMD and PDL,” IEEE Photon. Technol. Lett. 14, 1515–1517 (2002).
[CrossRef]

2001 (1)

2000 (3)

Y. Li and A. Yariv, “Solution to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,” J. Opt. Soc. Am. B 17, 1821–1827 (2000).
[CrossRef]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541–4550 (2000).
[CrossRef] [PubMed]

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

1999 (1)

1998 (1)

1997 (1)

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119–125 (1997).
[CrossRef]

1995 (1)

E. Lichtman, “Limitations imposed by polarization dependent gain and loss on all-optical ultralong communication systems,” J. Lightwave Technol. 13, 906–913 (1995).
[CrossRef]

1994 (1)

F. Bruyére and O. Audouin, “Penalties in long-haul optical amplifier systems due to polarization dependent loss and gain,” IEEE Photon. Technol. Lett. 6, 654–656 (1994).
[CrossRef]

Audouin, O.

F. Bruyére and O. Audouin, “Penalties in long-haul optical amplifier systems due to polarization dependent loss and gain,” IEEE Photon. Technol. Lett. 6, 654–656 (1994).
[CrossRef]

Bao, X.

L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Effect of local PMD and PDL directional correlation on the principal state of polarization vector autocorrelation,” Opt. Express 11, 3141–3146 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Principal state vector autocorrelation in a fiber optic system having both polarization-mode dispersion and polarization dependent loss,” in Applications of Photonic Technology 6, R. A. Lessard and G. A. Lampropoulos, eds., Proc. SPIE 5260, 382–385 (2003).
[CrossRef]

P. Lu, L. Chen, and X. Bao, “Polarization mode dispersion and polarization dependent loss for a pulse in single-mode fibers,” J. Lightwave Technol. 19, 856–860 (2001).
[CrossRef]

Biodini, G.

G. Biodini, W. L. Kath, and C. R. Menyuk, “Importance sampling for polarization mode dispersion,” IEEE Photonics Technol. Lett. 14, 310–312 (2002).
[CrossRef]

Brentel, J.

Bruyére, F.

F. Bruyére and O. Audouin, “Penalties in long-haul optical amplifier systems due to polarization dependent loss and gain,” IEEE Photon. Technol. Lett. 6, 654–656 (1994).
[CrossRef]

Chen, L.

L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Effect of local PMD and PDL directional correlation on the principal state of polarization vector autocorrelation,” Opt. Express 11, 3141–3146 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Principal state vector autocorrelation in a fiber optic system having both polarization-mode dispersion and polarization dependent loss,” in Applications of Photonic Technology 6, R. A. Lessard and G. A. Lampropoulos, eds., Proc. SPIE 5260, 382–385 (2003).
[CrossRef]

P. Lu, L. Chen, and X. Bao, “Polarization mode dispersion and polarization dependent loss for a pulse in single-mode fibers,” J. Lightwave Technol. 19, 856–860 (2001).
[CrossRef]

Dimenstein, O.

A. Eyal, D. Kuperman, O. Dimenstein, and M. Tur, “Polarization dependence of the intensity modulation transfer function of an optical system with PMD and PDL,” IEEE Photon. Technol. Lett. 14, 1515–1517 (2002).
[CrossRef]

Eyal, A.

A. Eyal, D. Kuperman, O. Dimenstein, and M. Tur, “Polarization dependence of the intensity modulation transfer function of an optical system with PMD and PDL,” IEEE Photon. Technol. Lett. 14, 1515–1517 (2002).
[CrossRef]

Feced, R.

Galtarossa, A.

A. Galtarossa and L. Palmieri, “The exact statistics of polarization-dependent loss in fiber-optic links,” IEEE Photon. Technol. Lett. 15, 57–59 (2003).
[CrossRef]

Geiser, C.

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

Gisin, N.

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119–125 (1997).
[CrossRef]

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541–4550 (2000).
[CrossRef] [PubMed]

Hadjifaradji, S.

L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Effect of local PMD and PDL directional correlation on the principal state of polarization vector autocorrelation,” Opt. Express 11, 3141–3146 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Principal state vector autocorrelation in a fiber optic system having both polarization-mode dispersion and polarization dependent loss,” in Applications of Photonic Technology 6, R. A. Lessard and G. A. Lampropoulos, eds., Proc. SPIE 5260, 382–385 (2003).
[CrossRef]

Hadjifotiou, A.

Huttner, B.

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119–125 (1997).
[CrossRef]

Jian, S.

M. Wang, T. Li, and S. Jian, “Analytical theory of pulse broadening due to polarization mode dispersion and polarization-dependent loss,” Opt. Commun. 223, 75–80 (2003).
[CrossRef]

Karlsson, M.

Kath, W. L.

G. Biodini, W. L. Kath, and C. R. Menyuk, “Importance sampling for polarization mode dispersion,” IEEE Photonics Technol. Lett. 14, 310–312 (2002).
[CrossRef]

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541–4550 (2000).
[CrossRef] [PubMed]

Kuperman, D.

A. Eyal, D. Kuperman, O. Dimenstein, and M. Tur, “Polarization dependence of the intensity modulation transfer function of an optical system with PMD and PDL,” IEEE Photon. Technol. Lett. 14, 1515–1517 (2002).
[CrossRef]

Li, T.

M. Wang, T. Li, and S. Jian, “Analytical theory of pulse broadening due to polarization mode dispersion and polarization-dependent loss,” Opt. Commun. 223, 75–80 (2003).
[CrossRef]

Li, Y.

Lichtman, E.

E. Lichtman, “Limitations imposed by polarization dependent gain and loss on all-optical ultralong communication systems,” J. Lightwave Technol. 13, 906–913 (1995).
[CrossRef]

Lu, P.

Mecozzi, A.

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14, 313–315 (2002).
[CrossRef]

Menyuk, C. R.

G. Biodini, W. L. Kath, and C. R. Menyuk, “Importance sampling for polarization mode dispersion,” IEEE Photonics Technol. Lett. 14, 310–312 (2002).
[CrossRef]

Palmieri, L.

A. Galtarossa and L. Palmieri, “The exact statistics of polarization-dependent loss in fiber-optic links,” IEEE Photon. Technol. Lett. 15, 57–59 (2003).
[CrossRef]

Savory, S. J.

Shtaif, M.

M. Shtaif, “The Brownian-bridge method for simulating polarization mode dispersion in optical communication systems,” IEEE Photon. Technol. Lett. 15, 51–53 (2003).
[CrossRef]

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14, 313–315 (2002).
[CrossRef]

Tur, M.

A. Eyal, D. Kuperman, O. Dimenstein, and M. Tur, “Polarization dependence of the intensity modulation transfer function of an optical system with PMD and PDL,” IEEE Photon. Technol. Lett. 14, 1515–1517 (2002).
[CrossRef]

Waddy, D. S.

L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Principal state vector autocorrelation in a fiber optic system having both polarization-mode dispersion and polarization dependent loss,” in Applications of Photonic Technology 6, R. A. Lessard and G. A. Lampropoulos, eds., Proc. SPIE 5260, 382–385 (2003).
[CrossRef]

L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Effect of local PMD and PDL directional correlation on the principal state of polarization vector autocorrelation,” Opt. Express 11, 3141–3146 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

Wang, M.

M. Wang, T. Li, and S. Jian, “Analytical theory of pulse broadening due to polarization mode dispersion and polarization-dependent loss,” Opt. Commun. 223, 75–80 (2003).
[CrossRef]

Yariv, A.

Yevick, D.

D. Yevick, “The accuracy of multicanonical system models,” IEEE Photon. Technol. Lett. 15, 224–226 (2003).
[CrossRef]

D. Yevick, “Multicanonical communication system modeling: application to PMD statistics,” IEEE Photon. Technol. Lett. 14, 1512–1514 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (7)

A. Eyal, D. Kuperman, O. Dimenstein, and M. Tur, “Polarization dependence of the intensity modulation transfer function of an optical system with PMD and PDL,” IEEE Photon. Technol. Lett. 14, 1515–1517 (2002).
[CrossRef]

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14, 313–315 (2002).
[CrossRef]

A. Galtarossa and L. Palmieri, “The exact statistics of polarization-dependent loss in fiber-optic links,” IEEE Photon. Technol. Lett. 15, 57–59 (2003).
[CrossRef]

F. Bruyére and O. Audouin, “Penalties in long-haul optical amplifier systems due to polarization dependent loss and gain,” IEEE Photon. Technol. Lett. 6, 654–656 (1994).
[CrossRef]

D. Yevick, “Multicanonical communication system modeling: application to PMD statistics,” IEEE Photon. Technol. Lett. 14, 1512–1514 (2002).
[CrossRef]

D. Yevick, “The accuracy of multicanonical system models,” IEEE Photon. Technol. Lett. 15, 224–226 (2003).
[CrossRef]

M. Shtaif, “The Brownian-bridge method for simulating polarization mode dispersion in optical communication systems,” IEEE Photon. Technol. Lett. 15, 51–53 (2003).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

G. Biodini, W. L. Kath, and C. R. Menyuk, “Importance sampling for polarization mode dispersion,” IEEE Photonics Technol. Lett. 14, 310–312 (2002).
[CrossRef]

J. Lightwave Technol. (2)

E. Lichtman, “Limitations imposed by polarization dependent gain and loss on all-optical ultralong communication systems,” J. Lightwave Technol. 13, 906–913 (1995).
[CrossRef]

P. Lu, L. Chen, and X. Bao, “Polarization mode dispersion and polarization dependent loss for a pulse in single-mode fibers,” J. Lightwave Technol. 19, 856–860 (2001).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119–125 (1997).
[CrossRef]

M. Wang, T. Li, and S. Jian, “Analytical theory of pulse broadening due to polarization mode dispersion and polarization-dependent loss,” Opt. Commun. 223, 75–80 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Proc. Natl. Acad. Sci. U.S.A. (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541–4550 (2000).
[CrossRef] [PubMed]

Proc. SPIE (1)

L. Chen, S. Hadjifaradji, D. S. Waddy, and X. Bao, “Principal state vector autocorrelation in a fiber optic system having both polarization-mode dispersion and polarization dependent loss,” in Applications of Photonic Technology 6, R. A. Lessard and G. A. Lampropoulos, eds., Proc. SPIE 5260, 382–385 (2003).
[CrossRef]

Other (2)

S. Hadjifaradji, L. Chen, and X. Bao, “Eye diagram evaluation in single mode fibers having PMD, PDL and CD,” in Holey Fibers and Photonic Crystals/Polarization Mode Dispersion/Photonics Time/Frequency Measurement and Control, 2003, Digest of the LEOS Summer Topical Meetings (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 53–54.

S. L. Fogal, G. Biondini, and W. L. Kath, “Multiple importance sampling for first- and second-order polarization-mode dispersion,” IEEE Photon. Technol. Lett. 14, 1273–1275 (2002), errata, 14, 1487 (2002).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Sample optical eye diagram for average DGD of 10 ps with (a) and (b) a PDL of 5 dB, (c) decibels zero. Note that the asymmetry in (b) is caused by the high value of PDL and the PMD and PDL polarization directional correlation. Other parameters are T=100 ps, D=50%, and the Q factor is obtained from Eq. (19).

Fig. 2
Fig. 2

Optical eye diagrams for (a) wavelengths at 1550 nm; and the Q factor plotted (b) as a function of duty cycle and (c) as a function of the chirp factor.

Equations (32)

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

Eout(ω)=AN exp[iϕCD(ω)]TN(ω)Ein(ω)=AN exp[iϕCD(ω)]exp(-iωβ1·σ/2) × exp(α1·σ/2)exp(-iωβN·σ/2) × exp(αN·σ/2)Ein(ω),
Ein(ω)=fin(ω)xy.
fin(ω)=12π - exp(iωt)Ein(t)dt.
I(t)=Eout*(t)·Eout(t)=|AN|22π -dω-dω exp{i[ϕCD(ω)-ϕCD(ω)]} × [x*y*]TN(ω)TN(ω)xy×exp[i(ω-ω)t]fin*(ω)fin(ω).
TN(ω)TN(ω)=Γ0,N(ω, ω)I+ΓN(ω, ω)·σ=exp(αN·σ/2)exp(iωβN·σ/2) × [Γ0,N-1(ω, ω)I+ΓN-1(ω, ω)·σ]exp(-iωβN·σ/2)exp(αN·σ/2),
Γ0,N(ω, ω)=cosh αN cos(βNΔω)Γ0,N-1+i cosh αN sin(βNΔω)(βˆN·ΓN-1)+sinh αN cos(βNω¯)(αˆN·ΓN-1)+sinh αN sin(βNω¯)×[αˆN·(ΓN-1×βˆN)]+(αˆN·βˆN) × {i sinh αN sin(βNΔω)Γ0,N-1+(βˆN·ΓN-1)sinh αN[cos(βNΔω)-cos(βNω¯)]},
ΓN(ω, ω)=βˆN{i sin(βNΔω)Γ0,N-1+[cos(βNΔω)-cos(βNω¯)](βˆN·ΓN-1)}+cos(βNω¯)ΓN-1+sin(βNω¯)(ΓN-1×βˆN)+αˆN(sinh αN cos(βNΔω)Γ0,N-1+i sinh αN sin(βNΔω)(βˆN·ΓN-1)+(cosh αN-1){i(αˆN·βˆN)sin(βNΔω)(Γ0,N-1)+(αˆN·βˆN)[cos(βNΔω)-cos(βNω¯)] × (βˆN·ΓN-1)+cos(βNω¯)(αˆN·ΓN-1)+sin(βNω¯) × [αˆN·(ΓN-1×βˆN)]}),
[x*y*]TN(ω)TN(ω)xy
=Γ0,N(ω, ω)+ΓN(ω, ω)·mˆ,
nˆ·A0nˆ·B,
(nˆ·A)(nˆ·B)=13 A·B.
(αˆj·A)(βˆj·B)=13αˆ·βˆ(A·B),
I(t)=|AN|22π -dω-dω exp{i[ϕCD(ω)-ϕCD(ω)]} × Γ0,N+ΓN·mˆexp[i(ω-ω)t] × fin*(ω)fin(ω)=|AN|22π -dω-dω exp{i[ϕCD(ω)-ϕCD(ω)]}
×Γ0,Nexp[i(ω-ω)t]fin*(ω)fin(ω).
[I(t)]2=|AN|44π2 -dω1-dω1-dω2-dω2 × exp{i[ϕCD(ω1)-ϕCD(ω1)+ϕCD(ω2)-ϕCD(ω2)]}exp[i(ω1-ω1+ω2-ω2)t] × [Γ0,N(ω1, ω1)+ΓN(ω1, ω1)·mˆ]×[Γ0,N(ω2, ω2)+ΓN(ω2, ω2)·mˆ]fin*(ω1)fin(ω1)fin*(ω2)fin(ω2)=|AN|44π2 -dω1-dω1-dω2-dω2 × exp{i[ϕCD(ω1)-ϕCD(ω1)+ϕCD(ω2)-ϕCD(ω2)]}exp[i(ω1-ω1+ω2-ω2)t] × [Γ0,N(ω1, ω1)Γ0,N(ω2, ω2)]+13ΓN(ω1, ω1)·ΓN(ω2, ω2) × fin*(ω1)fin(ω1)fin*(ω2)fin(ω2),
Γ0,N(ω, ω)=[cosh α cos(βΔω)+iαˆ·βˆsinh α sin(βΔω)]Γ0,N-1(ω, ω)==[cosh α cos(βΔω)+iαˆ·βˆsinh α sin(βΔω)]NΓ0,0(ω, ω).
limNΓ0,N(ω, ω)=exp12η2-18Δτ2(ω-ω)2+12iαˆ·βˆ(η2Δτ2)1/2 × (ω-ω).
Γ0,N(ω1, ω1)Γ0,N(ω2, ω2)ΓN(ω1, ω1)·ΓN(ω2, ω2)
=a11a12a21a22Γ0,N-1(ω1, ω1)Γ0,N-1(ω2, ω2)ΓN-1(ω1, ω1)·ΓN-1(ω2, ω2)==a11a12a21a22N10.
a11=cosh2 α cos(βΔω1)cos(βΔω2)+i2 sinh 2α sin[β(Δω1+Δω2)]αˆ·βˆ-sinh2 α sin(βΔω1)sin(βΔω2)(αˆ·βˆ)2,
a12=13 sinh2 α[cos(βΔω1)cos(βΔω2)+sin(βω¯1)sin(βω¯2)]-cosh2 α sin(βΔω1)×sin(βΔω2)+i2αˆ·βˆsinh 2α{sin[β(Δω1+Δω2)]+sin(βΔω1)×[cos(βΔω2)-cos(βω¯2)]+sin(βΔω2) × [cos(βΔω1)-cos(βω¯1)]}+(αˆ·βˆ)2sinh2 α{cos(βΔω1)[cos(βΔω2)-cos(βω¯2)]+cos(βΔω2)[cos(βΔω1)-cos(βω¯1)]-sin(βω¯1)sin(βω¯2)},
a21=sinh2 α cos(βΔω1)cos(βΔω2)-sin(βΔω1)sin(βΔω2)+αˆ·βˆ×{i sinh α cosh α sin[β(Δω1+Δω2)]-sinh2 α sin(βΔω1)sin(βΔω2)},
a22=13 2 cos[β(ω¯1-ω¯2)]+cos(βΔω1)cos(βΔω2)+sinh2 α{cos[β(ω¯1-ω¯2)]-sin(βΔω1)sin(βΔω2)}+i2αˆ·βˆsinh 2α sin[β(Δω1+Δω2)]+(αˆ·βˆ)2sinh2 α{cos(βΔω1)cos(βΔω2)-cos[β(ω¯1-ω¯2)]}.
limNΓ0,N(ω1, ω1)Γ0,N(ω2, ω2)
=egcoshf2+h231/2+ff2+h231/2 × sinhf2+h231/2,
limNΓN(ω1, ω1)·ΓN(ω2, ω2)
=hegf2+h231/2 sinhf2+h231/2,
f=13η2-16Δτ2[(Δω1)2+(Δω2)2-(ω¯1-ω¯2)2]+13iαˆ·βˆ(η2Δτ2)1/2(Δω1+Δω2),
g=23η2-13Δτ2(Δω1)2+(Δω2)2+12(ω¯1-ω¯2)2+23iαˆ·βˆ(η2Δτ2)1/2(Δω1+Δω2),
h=η2-Δτ2Δω1Δω2+iαˆ·βˆ(η2Δτ2)1/2 × (Δω1+Δω2).
E(t)=A2 1+cos2πtDTfor -DT2tDT20otherwise.
Q=I(t)max-I(t)minσ(t)max+σ(t)min,

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