Abstract

We present a novel analytical expression relating the output state of polarization (SOP) and the polarization mode dispersion (PMD) vector, including polarization-dependent chromatic dispersion (PCD), in terms of the angle of precession of the output SOP around the PMD vector. We derive a number of new expressions incorporating for the first time this angle of precession. First, a general relation to study the effect of differential group delay, PCD, and chromatic dispersion on pulses of arbitrary shapes is given. From this general relation, we derive expressions for pulse broadening and power penalty for Gaussian pulses. Moreover, a new expression for PMD-induced power fading for single-sideband modulated radio frequency signals is also derived. Measured experimental results are presented to support the derived expressions.

© 2006 Optical Society of America

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  1. H. Kogelnik, R. M. Jopson, and L. E. Nelson, in Optical Fibre Telecommunications, I.P.Kaminow and TingyeLi, eds. (Academic, 2002), Vol. IV-B.
  2. J. P. Gorden and H. Kogelnik, "PMD fundamentals: polarization mode dispersion in optical fibers," Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
    [CrossRef]
  3. C. D. Poole, N. S. Bergano, R. E. Wanger, and H. J. Schulte, "Polarization dispersion and principal states in a 147-km undersea lightwave cable," J. Lightwave Technol. 6, 1185-1190 (1988).
    [CrossRef]
  4. F. Heismann, D. A. Fishman, and D. L. Wilson, "Automatic compensation of first-order polarization mode dispersion in a 10Gb/s transmission system," in Proceedings of the 24th European Conference on Optical Communication (ECOC) '98 (IEE, 1998), pp. 529-530.
  5. Ping Lu, Liang Chen, and Xiaoyi Bao, "Polarization mode dispersion and polarization dependent loss for a pulse in single-mode fibers," J. Lightwave Technol. 19, 856-860 (2001).
    [CrossRef]
  6. S. J. Savory and F. P. Payne, "Pulse propagation in fibers with polarization-mode dispersion," J. Lightwave Technol. 19, 350-357 (2001).
    [CrossRef]
  7. J. Sanchez Garcia, A. Galindo Gonzalez, and M. Larraz Iribas, "Polarization mode dispersion power penalty; influence of rise/fall times, receiver Q and amplifier noise," IEEE Photonics Technol. Lett. 8, 1719-1721 (1996).
    [CrossRef]
  8. C. D. Poole, R. W. Tkach, A. R. Chraplyvy, and D. A. Fishman, "Fading in lightwave systems due to polarization mode dispersion," IEEE Photonics Technol. Lett. 3, 68-70 (1991).
    [CrossRef]
  9. H. F. Haunstein, H. M. Kallert, and H. Kogelink, "Fast PMD penalty measurement using polarization scrambling," in Optical Fiber Communications Conference (OFC) 2002, Vol. 70 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper WQ6, pp. 305-306.
    [CrossRef]
  10. G. Ishikawa and H. Ooi, "Polarization mode dispersion sensitivity and monitoring in 40Gbit/s OTDM and 10Gbis/s NRZ transmission experiments," in Optical Fiber Communications Conference (OFC) 1998 (Optical Society of America, 1998), paper WC5, pp. 117-119.
  11. C. D. Poole and R. E. Wagner, "Phenomenological approach to polarization dispersion in long single mode fibers," Electron. Lett. 22, 1029-1030 (1986).
    [CrossRef]
  12. J. Wang, J. M. Joseph, and M. Kahn, "Impact of chromatic and polarization-mode dispersion on DPSK systems using interferometric demodulation and direct detection," J. Lightwave Technol. 22, 362-371 (2004).
    [CrossRef]
  13. C. D. Poole and D. L. Favin, "Polarization mode dispersion measurements based on transmission spectra through a polarizer," J. Lightwave Technol. 12, 917-929 (1994).
    [CrossRef]
  14. Chongqing Wu, Optical Waveguide Theory (Tinghua U. Press, 2000), pp. 133-138 (in Chinese).

2004 (1)

2001 (2)

2000 (1)

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

1996 (1)

J. Sanchez Garcia, A. Galindo Gonzalez, and M. Larraz Iribas, "Polarization mode dispersion power penalty; influence of rise/fall times, receiver Q and amplifier noise," IEEE Photonics Technol. Lett. 8, 1719-1721 (1996).
[CrossRef]

1994 (1)

C. D. Poole and D. L. Favin, "Polarization mode dispersion measurements based on transmission spectra through a polarizer," J. Lightwave Technol. 12, 917-929 (1994).
[CrossRef]

1991 (1)

C. D. Poole, R. W. Tkach, A. R. Chraplyvy, and D. A. Fishman, "Fading in lightwave systems due to polarization mode dispersion," IEEE Photonics Technol. Lett. 3, 68-70 (1991).
[CrossRef]

1988 (1)

C. D. Poole, N. S. Bergano, R. E. Wanger, and H. J. Schulte, "Polarization dispersion and principal states in a 147-km undersea lightwave cable," J. Lightwave Technol. 6, 1185-1190 (1988).
[CrossRef]

1986 (1)

C. D. Poole and R. E. Wagner, "Phenomenological approach to polarization dispersion in long single mode fibers," Electron. Lett. 22, 1029-1030 (1986).
[CrossRef]

Bao, Xiaoyi

Bergano, N. S.

C. D. Poole, N. S. Bergano, R. E. Wanger, and H. J. Schulte, "Polarization dispersion and principal states in a 147-km undersea lightwave cable," J. Lightwave Technol. 6, 1185-1190 (1988).
[CrossRef]

Chen, Liang

Chraplyvy, A. R.

C. D. Poole, R. W. Tkach, A. R. Chraplyvy, and D. A. Fishman, "Fading in lightwave systems due to polarization mode dispersion," IEEE Photonics Technol. Lett. 3, 68-70 (1991).
[CrossRef]

Favin, D. L.

C. D. Poole and D. L. Favin, "Polarization mode dispersion measurements based on transmission spectra through a polarizer," J. Lightwave Technol. 12, 917-929 (1994).
[CrossRef]

Fishman, D. A.

C. D. Poole, R. W. Tkach, A. R. Chraplyvy, and D. A. Fishman, "Fading in lightwave systems due to polarization mode dispersion," IEEE Photonics Technol. Lett. 3, 68-70 (1991).
[CrossRef]

F. Heismann, D. A. Fishman, and D. L. Wilson, "Automatic compensation of first-order polarization mode dispersion in a 10Gb/s transmission system," in Proceedings of the 24th European Conference on Optical Communication (ECOC) '98 (IEE, 1998), pp. 529-530.

Garcia, J. Sanchez

J. Sanchez Garcia, A. Galindo Gonzalez, and M. Larraz Iribas, "Polarization mode dispersion power penalty; influence of rise/fall times, receiver Q and amplifier noise," IEEE Photonics Technol. Lett. 8, 1719-1721 (1996).
[CrossRef]

Gonzalez, A. Galindo

J. Sanchez Garcia, A. Galindo Gonzalez, and M. Larraz Iribas, "Polarization mode dispersion power penalty; influence of rise/fall times, receiver Q and amplifier noise," IEEE Photonics Technol. Lett. 8, 1719-1721 (1996).
[CrossRef]

Gorden, J. P.

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

Haunstein, H. F.

H. F. Haunstein, H. M. Kallert, and H. Kogelink, "Fast PMD penalty measurement using polarization scrambling," in Optical Fiber Communications Conference (OFC) 2002, Vol. 70 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper WQ6, pp. 305-306.
[CrossRef]

Heismann, F.

F. Heismann, D. A. Fishman, and D. L. Wilson, "Automatic compensation of first-order polarization mode dispersion in a 10Gb/s transmission system," in Proceedings of the 24th European Conference on Optical Communication (ECOC) '98 (IEE, 1998), pp. 529-530.

Iribas, M. Larraz

J. Sanchez Garcia, A. Galindo Gonzalez, and M. Larraz Iribas, "Polarization mode dispersion power penalty; influence of rise/fall times, receiver Q and amplifier noise," IEEE Photonics Technol. Lett. 8, 1719-1721 (1996).
[CrossRef]

Ishikawa, G.

G. Ishikawa and H. Ooi, "Polarization mode dispersion sensitivity and monitoring in 40Gbit/s OTDM and 10Gbis/s NRZ transmission experiments," in Optical Fiber Communications Conference (OFC) 1998 (Optical Society of America, 1998), paper WC5, pp. 117-119.

Jopson, R. M.

H. Kogelnik, R. M. Jopson, and L. E. Nelson, in Optical Fibre Telecommunications, I.P.Kaminow and TingyeLi, eds. (Academic, 2002), Vol. IV-B.

Joseph, J. M.

Kahn, M.

Kallert, H. M.

H. F. Haunstein, H. M. Kallert, and H. Kogelink, "Fast PMD penalty measurement using polarization scrambling," in Optical Fiber Communications Conference (OFC) 2002, Vol. 70 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper WQ6, pp. 305-306.
[CrossRef]

Kogelink, H.

H. F. Haunstein, H. M. Kallert, and H. Kogelink, "Fast PMD penalty measurement using polarization scrambling," in Optical Fiber Communications Conference (OFC) 2002, Vol. 70 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper WQ6, pp. 305-306.
[CrossRef]

Kogelnik, H.

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

H. Kogelnik, R. M. Jopson, and L. E. Nelson, in Optical Fibre Telecommunications, I.P.Kaminow and TingyeLi, eds. (Academic, 2002), Vol. IV-B.

Lu, Ping

Nelson, L. E.

H. Kogelnik, R. M. Jopson, and L. E. Nelson, in Optical Fibre Telecommunications, I.P.Kaminow and TingyeLi, eds. (Academic, 2002), Vol. IV-B.

Ooi, H.

G. Ishikawa and H. Ooi, "Polarization mode dispersion sensitivity and monitoring in 40Gbit/s OTDM and 10Gbis/s NRZ transmission experiments," in Optical Fiber Communications Conference (OFC) 1998 (Optical Society of America, 1998), paper WC5, pp. 117-119.

Payne, F. P.

Poole, C. D.

C. D. Poole and D. L. Favin, "Polarization mode dispersion measurements based on transmission spectra through a polarizer," J. Lightwave Technol. 12, 917-929 (1994).
[CrossRef]

C. D. Poole, R. W. Tkach, A. R. Chraplyvy, and D. A. Fishman, "Fading in lightwave systems due to polarization mode dispersion," IEEE Photonics Technol. Lett. 3, 68-70 (1991).
[CrossRef]

C. D. Poole, N. S. Bergano, R. E. Wanger, and H. J. Schulte, "Polarization dispersion and principal states in a 147-km undersea lightwave cable," J. Lightwave Technol. 6, 1185-1190 (1988).
[CrossRef]

C. D. Poole and R. E. Wagner, "Phenomenological approach to polarization dispersion in long single mode fibers," Electron. Lett. 22, 1029-1030 (1986).
[CrossRef]

Savory, S. J.

Schulte, H. J.

C. D. Poole, N. S. Bergano, R. E. Wanger, and H. J. Schulte, "Polarization dispersion and principal states in a 147-km undersea lightwave cable," J. Lightwave Technol. 6, 1185-1190 (1988).
[CrossRef]

Tkach, R. W.

C. D. Poole, R. W. Tkach, A. R. Chraplyvy, and D. A. Fishman, "Fading in lightwave systems due to polarization mode dispersion," IEEE Photonics Technol. Lett. 3, 68-70 (1991).
[CrossRef]

Wagner, R. E.

C. D. Poole and R. E. Wagner, "Phenomenological approach to polarization dispersion in long single mode fibers," Electron. Lett. 22, 1029-1030 (1986).
[CrossRef]

Wang, J.

Wanger, R. E.

C. D. Poole, N. S. Bergano, R. E. Wanger, and H. J. Schulte, "Polarization dispersion and principal states in a 147-km undersea lightwave cable," J. Lightwave Technol. 6, 1185-1190 (1988).
[CrossRef]

Wilson, D. L.

F. Heismann, D. A. Fishman, and D. L. Wilson, "Automatic compensation of first-order polarization mode dispersion in a 10Gb/s transmission system," in Proceedings of the 24th European Conference on Optical Communication (ECOC) '98 (IEE, 1998), pp. 529-530.

Wu, Chongqing

Chongqing Wu, Optical Waveguide Theory (Tinghua U. Press, 2000), pp. 133-138 (in Chinese).

Electron. Lett. (1)

C. D. Poole and R. E. Wagner, "Phenomenological approach to polarization dispersion in long single mode fibers," Electron. Lett. 22, 1029-1030 (1986).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

J. Sanchez Garcia, A. Galindo Gonzalez, and M. Larraz Iribas, "Polarization mode dispersion power penalty; influence of rise/fall times, receiver Q and amplifier noise," IEEE Photonics Technol. Lett. 8, 1719-1721 (1996).
[CrossRef]

C. D. Poole, R. W. Tkach, A. R. Chraplyvy, and D. A. Fishman, "Fading in lightwave systems due to polarization mode dispersion," IEEE Photonics Technol. Lett. 3, 68-70 (1991).
[CrossRef]

J. Lightwave Technol. (5)

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

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

Other (5)

H. Kogelnik, R. M. Jopson, and L. E. Nelson, in Optical Fibre Telecommunications, I.P.Kaminow and TingyeLi, eds. (Academic, 2002), Vol. IV-B.

F. Heismann, D. A. Fishman, and D. L. Wilson, "Automatic compensation of first-order polarization mode dispersion in a 10Gb/s transmission system," in Proceedings of the 24th European Conference on Optical Communication (ECOC) '98 (IEE, 1998), pp. 529-530.

H. F. Haunstein, H. M. Kallert, and H. Kogelink, "Fast PMD penalty measurement using polarization scrambling," in Optical Fiber Communications Conference (OFC) 2002, Vol. 70 of Trends in Optics and Photonics Series (Optical Society of America, 2002), paper WQ6, pp. 305-306.
[CrossRef]

G. Ishikawa and H. Ooi, "Polarization mode dispersion sensitivity and monitoring in 40Gbit/s OTDM and 10Gbis/s NRZ transmission experiments," in Optical Fiber Communications Conference (OFC) 1998 (Optical Society of America, 1998), paper WC5, pp. 117-119.

Chongqing Wu, Optical Waveguide Theory (Tinghua U. Press, 2000), pp. 133-138 (in Chinese).

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Figures (4)

Fig. 1
Fig. 1

Experimental setup to observe power penalty due to PMD and CD, and power fading for PMD monitoring.

Fig. 2
Fig. 2

Power penalty versus angle between the output SOP and the PMD vector, with optimum decision threshold for PMD and CD.

Fig. 3
Fig. 3

RF power fading versus angle between the output SOP and the PMD vector for SSB modulation.

Fig. 4
Fig. 4

Power penalty versus DGD for the worst case and for the case with polarization scrambling, with optimum decision threshold for PMD and CD.

Equations (33)

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S o ( ω ) ω = τ × S o ( ω ) .
S o ( ω ) = Ω cos ( 2 α ) + [ S o ( ω 0 ) Ω cos ( 2 α ) ] cos ( Δ τ Δ ω ) + B sin ( 2 α ) sin ( Δ τ Δ ω ) .
S o ( ω ) = Ω cos ( 2 α ) + [ S o ( ω 0 ) Ω cos ( 2 α ) ] cos ( Δ τ Δ ω Δ τ 1 Δ ω 2 ) + B sin ( 2 α ) sin ( Δ τ Δ ω Δ τ 1 Δ ω 2 ) .
f Ω + ( t ) = cos α 2 π + F ( ω ω 0 ) exp { j [ ( Δ τ Δ ω 2 ) ( Δ τ 1 Δ ω 2 2 ) ( β 2 z Δ ω 2 2 ) ] } exp [ j ( ω ω 0 ) t ] d ω ,
f Ω ( t ) = sin α 2 π + F ( ω ω 0 ) exp { j [ ( Δ τ Δ ω 2 ) ( Δ τ 1 Δ ω 2 2 ) + ( β 2 z Δ ω 2 2 ) ] } exp [ j ( ω ω 0 ) t ] d ω ,
f Ω + ( t ) = T 0 cos α T 0 2 j ( β 2 z + Δ τ 1 ) exp { ( Δ τ 2 + t ) 2 2 [ T 0 2 j ( β 2 z + Δ τ 1 ) ] } ,
f Ω ( t ) = T 0 sin α T 0 2 j ( β 2 z Δ τ 1 ) exp { ( Δ τ 2 t ) 2 2 [ T 0 2 j ( β 2 z Δ τ 1 ) ] } .
P ( t ) = f Ω + ( t ) 2 + f Ω ( t ) 2 = T 0 2 cos 2 α T 0 4 + ( Δ τ 1 + β 2 z ) 2 exp [ ( Δ τ 2 + t ) 2 T 0 2 T 0 4 + ( Δ τ 1 + β 2 z ) 2 ] + T 0 2 sin 2 α T 0 4 + ( β 2 z Δ τ 1 ) 2 exp [ ( Δ τ 2 t ) 2 T 0 2 T 0 4 + ( β 2 z Δ τ 1 ) 2 ] .
σ 2 = + t 2 P ( t ) d t P 0 [ + t P ( t ) d t P 0 ] 2 = T 0 2 2 + ( Δ τ 1 ) 2 + ( β 2 z ) 2 2 T 0 2 + cos ( 2 α ) Δ τ 1 β 2 z T 0 2 + ( Δ τ 2 ) 2 sin 2 ( 2 α ) .
ε ( dB ) = B [ ( Δ τ 1 ) 2 + ( β 2 z ) 2 2 T 0 4 + cos ( 2 α ) Δ τ 1 β 2 z T 0 4 ] + A ( Δ τ 2 T 0 ) 2 sin 2 ( 2 α ) ,
ε ( dB ) = B ( Δ τ 1 + β 2 z ) 2 2 T 0 4 .
ε ( dB ) = B ( Δ τ 1 β 2 z ) 2 2 T 0 4 .
ε ¯ ( dB ) = B ( Δ τ 1 ) 2 + ( β 2 z ) 2 2 T 0 4 + 2 A 3 ( Δ τ 2 T 0 ) 2 .
f Ω + ( t ) = ( cos α ) M I i { cos ( ω 0 t ) + m 4 cos [ ( ω 0 + ω m ) t + ( Δ τ ω m 2 ) ( β 2 z ω m 2 2 ) ] } ,
f Ω ( t ) = ( sin α ) M I i { cos ( ω 0 t ) + m 4 cos [ ( ω 0 + ω m ) t ( Δ τ ω m 2 ) ( β 2 z ω m 2 2 ) ] } ,
I ( t ) = ( cos α ) 2 R M 2 I i { ( m 2 ) cos [ ω m ( t + Δ τ 2 ) ( β 2 z ω m 2 2 ) ] } + ( sin α ) 2 R M 2 I i { ( m 2 ) cos [ ω m ( t Δ τ 2 ) ( β 2 z ω m 2 2 ) ] } .
P ( ω m ) { 1 [ sin ( 2 α ) ] 2 [ sin ( Δ τ ω m 2 ) ] 2 } .
P ( f m ) { 1 2 3 [ sin ( π Δ τ f m ) ] 2 } .
P = τ × = [ 0 Ω 3 Ω 2 Ω 3 0 Ω 1 Ω 2 Ω 1 0 ] .
S o ( ω ) = exp [ P ( ω ω 0 ) ] S o ( ω 0 ) ,
T = [ Ω 3 Ω 1 Ω 3 j Ω 2 Δ τ Ω 1 Ω 3 + j Ω 2 Δ τ Ω 2 Ω 1 Ω 2 + j Ω 3 Δ τ Ω 1 Ω 2 j Ω 3 Δ τ Ω 1 Ω 1 2 Δ τ 2 Ω 1 2 Δ τ 2 ] .
exp [ P ( ω ω 0 ) ] = T [ 1 0 0 0 exp [ j Δ τ ( ω ω 0 ) ] 0 0 0 exp [ j Δ τ ( ω ω 0 ) ] ] T 1 .
S o ( ω ) = def ( Ω 1 Δ τ 2 ( Ω 1 s 01 + Ω 2 s 02 + Ω 3 s 03 ) + [ s 01 Ω 1 Δ τ 2 ( Ω 1 s 01 + Ω 2 s 02 + Ω 3 s 03 ) ] cos ( Δ τ Δ ω ) + 1 Δ τ ( Ω 2 s 03 Ω 3 s 02 ) sin ( Δ τ Δ ω ) Ω 2 Δ τ 2 ( Ω 1 s 01 + Ω 2 s 02 + Ω 3 s 03 ) + [ s 02 Ω 2 Δ τ 2 ( Ω 1 s 01 + Ω 2 s 02 + Ω 3 s 03 ) ] cos ( Δ τ Δ ω ) + 1 Δ τ ( Ω 3 s 01 Ω 1 s 03 ) sin ( Δ τ Δ ω ) Ω 3 Δ τ 2 ( Ω 1 s 01 + Ω 2 s 02 + Ω 3 s 03 ) + [ s 03 Ω 3 Δ τ 2 ( Ω 1 s 01 + Ω 2 s 02 + Ω 3 s 03 ) ] cos ( Δ τ Δ ω ) + 1 Δ τ ( Ω 1 s 02 Ω 2 s 01 ) sin ( Δ τ Δ ω ) ) .
S o ( ω ) = Ω [ Ω S o ( ω 0 ) ] + { S o ( ω 0 ) Ω [ Ω S o ( ω 0 ) ] } cos ( Δ τ Δ ω ) + Ω × S o ( ω 0 ) sin ( Δ τ Δ ω ) ,
S o ( ω ) = Ω cos ( 2 α ) + [ S o ( ω 0 ) Ω cos ( 2 α ) ] cos ( Δ τ Δ ω ) + B sin ( 2 α ) sin ( Δ τ Δ ω ) .
e ( ω ) = cos α exp ( j ϕ + ) e Ω + + sin α exp ( j ϕ ) e Ω .
e ( ω ) = cos π 4 exp ( j ϕ + ) e Ω + + sin π 4 exp ( j ϕ ) e Ω .
S o ( ω ) = S o ( ω 0 ) cos ( Δ τ Δ ω Δ τ 1 Δ ω 2 ) + B sin ( Δ τ Δ ω Δ τ 1 Δ ω 2 ) .
S o ( ω ) = ( cos ( Δ ω Δ τ Δ τ 1 Δ ω 2 ) sin ( Δ ω Δ τ Δ τ 1 Δ ω 2 ) 0 ) .
e ( ω ) = ( cos ( Δ τ Δ ω 2 Δ τ 1 Δ ω 2 2 ) exp ( j 0 ) sin ( Δ τ Δ ω 2 Δ τ 1 Δ ω 2 2 ) exp ( j 0 ) ) .
e Ω + = [ cos ( π 4 ) exp ( j π 4 ) sin ( π 4 ) exp ( j π 4 ) ] .
tan ϕ + = cos ( θ + θ p ) cos ( θ θ p ) tan ( δ δ p ) ,
tan ϕ = sin ( θ + θ p ) sin ( θ θ p ) tan ( δ δ p ) .

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