Abstract

We derive an analytic expression for the evolution of the differential group delay (DGD) in a fiber-optic circulating loop. We distill a simple analytic solution for the average DGD and show that it accumulates approximately linearly with transmission distance. The scaling of this linear function is confirmed experimentally using a long-haul fiber-optic transmission test-bed. This result is contrasted with DGD accumulation in a long straight-line transmission system, where the average DGD accumulates as a square root of the system’s length.

© 2004 Optical Society of America

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  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]
  2. A. Caltarossa, A. Pizzinat, and F. Matera, “Statistical description of optical system performances due to random coupling on the principal states of polarization,” IEEE Photon. Technol. Lett. 13, 1307–1309 (2001).
    [CrossRef]
  3. M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999).
    [CrossRef]
  4. M. Shtaif and A. Mecozzi, “Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal states,” IEEE Photon. Technol. Lett. 12, 53–55 (2000).
    [CrossRef]
  5. J. Garnier, J. Fatome, and G. Le Meur, “Statistical analysis of pulse propagation driven by polarization-mode dispersion,” J. Opt. Soc. Am. B 19, 1968–1977 (2002).
    [CrossRef]
  6. Q. Lin and G. P. Agrawal, “Correlation theory of polarization mode dispersion in optical fibers,” J. Opt. Soc. Am. B 20, 292–301 (2003).
    [CrossRef]
  7. S. Lee, Q. Yu, L.-S. Yan, Y. Xie, O. H. Adamczyk, and A. E. Willner, “A short recirculating fiber loop testbed with accurate reproduction of Maxwellian PMD statistics,” Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper WT-2.
  8. H. Xu, J. Wen, J. Zweck, L. Yan, C. Menyuk, G. Carter, “The effects of distributed PMD, PDL, and loop scrambling on BER distributions in a recirculating loop used to emulate long-haul terrestrial transmission,” Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003), paper TuO2.
  9. H. Kogelnik, R. Jopson, and L. Nelson, “Polarization-mode dispersion,” Optical Fiber Telecommunications IV-B, I. Kaminow and T. Li, eds. (Academic, New York, 2002), pp. 725–861.
  10. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. 97, 4541–4550 (2000).
    [CrossRef] [PubMed]
  11. Y. Tan, J. Yang, W. L. Kath, and C. R. Menyuk, “Transient evolution of the polarization dispersion vector’s probability distribution,” J. Opt. Soc. Am. B 19, 992–1000 (2002).
    [CrossRef]
  12. S. V. Chernikov and J. R. Taylor, “Measurement of normalization factor of n2 for random polarization in optical fibers,” Opt. Lett. 21, 1559–1561 (1996).
    [CrossRef] [PubMed]
  13. A. Vannucci and A. Bononi, “Statistical characterization of the Jones matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol. 20, 811–821 (2002).
    [CrossRef]
  14. B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992).
    [CrossRef]
  15. 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]
  16. D. N. Maywar, D. F. Grosz, A. Kung, L. Altman, M. Movassaghi, A. Agarwal, S. Banerjee, and T. H. Wood, “Ultra-wideband transmission of 1.28 Tbit/s (128×10.7 Gbit/s) over 2000 km using 50% RZ data,” Electron. Lett. 38, 1573–1575 (2002).
    [CrossRef]

2003 (1)

2002 (4)

Y. Tan, J. Yang, W. L. Kath, and C. R. Menyuk, “Transient evolution of the polarization dispersion vector’s probability distribution,” J. Opt. Soc. Am. B 19, 992–1000 (2002).
[CrossRef]

A. Vannucci and A. Bononi, “Statistical characterization of the Jones matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol. 20, 811–821 (2002).
[CrossRef]

J. Garnier, J. Fatome, and G. Le Meur, “Statistical analysis of pulse propagation driven by polarization-mode dispersion,” J. Opt. Soc. Am. B 19, 1968–1977 (2002).
[CrossRef]

D. N. Maywar, D. F. Grosz, A. Kung, L. Altman, M. Movassaghi, A. Agarwal, S. Banerjee, and T. H. Wood, “Ultra-wideband transmission of 1.28 Tbit/s (128×10.7 Gbit/s) over 2000 km using 50% RZ data,” Electron. Lett. 38, 1573–1575 (2002).
[CrossRef]

2001 (1)

A. Caltarossa, A. Pizzinat, and F. Matera, “Statistical description of optical system performances due to random coupling on the principal states of polarization,” IEEE Photon. Technol. Lett. 13, 1307–1309 (2001).
[CrossRef]

2000 (2)

M. Shtaif and A. Mecozzi, “Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal states,” IEEE Photon. Technol. Lett. 12, 53–55 (2000).
[CrossRef]

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

1999 (1)

1996 (1)

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]

1992 (1)

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992).
[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]

Agarwal, A.

D. N. Maywar, D. F. Grosz, A. Kung, L. Altman, M. Movassaghi, A. Agarwal, S. Banerjee, and T. H. Wood, “Ultra-wideband transmission of 1.28 Tbit/s (128×10.7 Gbit/s) over 2000 km using 50% RZ data,” Electron. Lett. 38, 1573–1575 (2002).
[CrossRef]

Agrawal, G. P.

Altman, L.

D. N. Maywar, D. F. Grosz, A. Kung, L. Altman, M. Movassaghi, A. Agarwal, S. Banerjee, and T. H. Wood, “Ultra-wideband transmission of 1.28 Tbit/s (128×10.7 Gbit/s) over 2000 km using 50% RZ data,” Electron. Lett. 38, 1573–1575 (2002).
[CrossRef]

Banerjee, S.

D. N. Maywar, D. F. Grosz, A. Kung, L. Altman, M. Movassaghi, A. Agarwal, S. Banerjee, and T. H. Wood, “Ultra-wideband transmission of 1.28 Tbit/s (128×10.7 Gbit/s) over 2000 km using 50% RZ data,” Electron. Lett. 38, 1573–1575 (2002).
[CrossRef]

Bononi, A.

A. Vannucci and A. Bononi, “Statistical characterization of the Jones matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol. 20, 811–821 (2002).
[CrossRef]

Brentel, J.

Caltarossa, A.

A. Caltarossa, A. Pizzinat, and F. Matera, “Statistical description of optical system performances due to random coupling on the principal states of polarization,” IEEE Photon. Technol. Lett. 13, 1307–1309 (2001).
[CrossRef]

Chernikov, S. V.

Fatome, J.

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]

Garnier, J.

Gordon, J. P.

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

Grosz, D. F.

D. N. Maywar, D. F. Grosz, A. Kung, L. Altman, M. Movassaghi, A. Agarwal, S. Banerjee, and T. H. Wood, “Ultra-wideband transmission of 1.28 Tbit/s (128×10.7 Gbit/s) over 2000 km using 50% RZ data,” Electron. Lett. 38, 1573–1575 (2002).
[CrossRef]

Heffner, B. L.

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992).
[CrossRef]

Karlsson, M.

Kath, W. L.

Kogelnik, H.

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

Kung, A.

D. N. Maywar, D. F. Grosz, A. Kung, L. Altman, M. Movassaghi, A. Agarwal, S. Banerjee, and T. H. Wood, “Ultra-wideband transmission of 1.28 Tbit/s (128×10.7 Gbit/s) over 2000 km using 50% RZ data,” Electron. Lett. 38, 1573–1575 (2002).
[CrossRef]

Le Meur, G.

Lin, Q.

Matera, F.

A. Caltarossa, A. Pizzinat, and F. Matera, “Statistical description of optical system performances due to random coupling on the principal states of polarization,” IEEE Photon. Technol. Lett. 13, 1307–1309 (2001).
[CrossRef]

Maywar, D. N.

D. N. Maywar, D. F. Grosz, A. Kung, L. Altman, M. Movassaghi, A. Agarwal, S. Banerjee, and T. H. Wood, “Ultra-wideband transmission of 1.28 Tbit/s (128×10.7 Gbit/s) over 2000 km using 50% RZ data,” Electron. Lett. 38, 1573–1575 (2002).
[CrossRef]

Mecozzi, A.

M. Shtaif and A. Mecozzi, “Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal states,” IEEE Photon. Technol. Lett. 12, 53–55 (2000).
[CrossRef]

Menyuk, C. R.

Movassaghi, M.

D. N. Maywar, D. F. Grosz, A. Kung, L. Altman, M. Movassaghi, A. Agarwal, S. Banerjee, and T. H. Wood, “Ultra-wideband transmission of 1.28 Tbit/s (128×10.7 Gbit/s) over 2000 km using 50% RZ data,” Electron. Lett. 38, 1573–1575 (2002).
[CrossRef]

Pizzinat, A.

A. Caltarossa, A. Pizzinat, and F. Matera, “Statistical description of optical system performances due to random coupling on the principal states of polarization,” IEEE Photon. Technol. Lett. 13, 1307–1309 (2001).
[CrossRef]

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

Shtaif, M.

M. Shtaif and A. Mecozzi, “Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal states,” IEEE Photon. Technol. Lett. 12, 53–55 (2000).
[CrossRef]

Tan, Y.

Taylor, J. R.

Vannucci, A.

A. Vannucci and A. Bononi, “Statistical characterization of the Jones matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol. 20, 811–821 (2002).
[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]

Wood, T. H.

D. N. Maywar, D. F. Grosz, A. Kung, L. Altman, M. Movassaghi, A. Agarwal, S. Banerjee, and T. H. Wood, “Ultra-wideband transmission of 1.28 Tbit/s (128×10.7 Gbit/s) over 2000 km using 50% RZ data,” Electron. Lett. 38, 1573–1575 (2002).
[CrossRef]

Yang, J.

Electron. Lett. (2)

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

D. N. Maywar, D. F. Grosz, A. Kung, L. Altman, M. Movassaghi, A. Agarwal, S. Banerjee, and T. H. Wood, “Ultra-wideband transmission of 1.28 Tbit/s (128×10.7 Gbit/s) over 2000 km using 50% RZ data,” Electron. Lett. 38, 1573–1575 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992).
[CrossRef]

A. Caltarossa, A. Pizzinat, and F. Matera, “Statistical description of optical system performances due to random coupling on the principal states of polarization,” IEEE Photon. Technol. Lett. 13, 1307–1309 (2001).
[CrossRef]

M. Shtaif and A. Mecozzi, “Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal states,” IEEE Photon. Technol. Lett. 12, 53–55 (2000).
[CrossRef]

J. Lightwave Technol. (2)

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]

A. Vannucci and A. Bononi, “Statistical characterization of the Jones matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol. 20, 811–821 (2002).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (2)

Proc. Natl. Acad. Sci. (1)

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

Other (3)

S. Lee, Q. Yu, L.-S. Yan, Y. Xie, O. H. Adamczyk, and A. E. Willner, “A short recirculating fiber loop testbed with accurate reproduction of Maxwellian PMD statistics,” Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper WT-2.

H. Xu, J. Wen, J. Zweck, L. Yan, C. Menyuk, G. Carter, “The effects of distributed PMD, PDL, and loop scrambling on BER distributions in a recirculating loop used to emulate long-haul terrestrial transmission,” Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003), paper TuO2.

H. Kogelnik, R. Jopson, and L. Nelson, “Polarization-mode dispersion,” Optical Fiber Telecommunications IV-B, I. Kaminow and T. Li, eds. (Academic, New York, 2002), pp. 725–861.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup: (a) General setup; AOS, acousto-optic switch; PC, manual polarization controller; HP-PC, HP8169 polarization controller; Mux, multiplexer; Demux, demultiplexer; Filter, 0.6-nm-wide spectral filter; HP-Pol, HP8509B polarimeter; AtoD, National Instruments PCI 1610 AtoD converter. (b) Close-up on the loop: LS1, load acousto-optic switch; LS2, loop acousto-optic switch; PCB and PCC, manual polarization controllers; PCA, voltage-controlled Polarite polarization controller.

Fig. 2
Fig. 2

Example of a time-resolved measurement of one coordinate of the Stokes vector. The lower scale corresponds to samples taken by the analog-to-digital board at a sampling rate of 100 kHz. The upper scale corresponds to time.

Fig. 3
Fig. 3

Stokes vector on the Poincaré sphere at frequency measured every 20 seconds for (a) back-to-back, (b) one round trip, (c) five round trips, and (d) ten round trips. The measurement period was 1 hour and 20 minutes.

Fig. 4
Fig. 4

Average DGD as a function of the number of circulations in the loop (triangular markers) and a fit using the analytic derivation (continuous line) for the nonscrambled loop; average DGD as a function of the number of circulations in the loop (square markers) and a fit using the analytic derivation (dotted curve) for the randomized loop.

Equations (20)

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[T-1T-iτ1]e1=0
[(TN)-1(TN)-iτN]eN=0.
[(T-(N-1)(T-1T)TN-1+T-(N-2)(T-1T)TN-2
++T-1T)-iτN]eN=0.
T=cos φ exp(iθ1)-sin φ exp(iθ2)sin φ exp(-iθ2)cos φ exp(-iθ1),
(e1+, e1-)=(t+, t-)Φ,
(t+, t-)=(e1+, e1-)Φ+.
Φ=cos ξ exp(iψ1)-sin ξ exp(iψ2)sin ξ exp(-iψ2)cos ξ exp(-iψ1).
E+[(T-(N-1)(T-1T)λ+N-1+T-(N-2)(T-1T)λ+N-2++T-1T)]t++E-[(T-(N-1)(T-1T)λ-N-1+T-(N-2)(T-1T)λ-N-2++T-1T)]t-
=iτN(E+t++E-t-).
iτ1E+N cos 2ξ+E- 1-exp(iΛN)1-exp(iΛ)×exp(iψ)sin 2ξt+-iτ1E-N cos 2ξ-E+1-exp(iΛN)1-exp(iΛ) exp(iψ)* sin 2ξt-
=iτN(E+t++E-t-),
τN2=τ12N2 cos2 2ξ+sin2 2ξ sin2(NΛ/2)sin2(Λ/2).
f(2ξ, ψ)=14π 02πdψ2ξ=02ξ=πd(2ξ)sin 2ξf(2ξ, ψ).
sin2(NΛ/2)sin2(Λ/2)=12π 0πdθ10πdφ sin 2φ×sin2(N arccos[cos θ1 cos φ])sin2(arccos[cos θ1 cos φ]).
sin2(NΛ/2)sin2(Λ/2)=1
τN2=τ12(N2+2)/3.
dS/dω=τ×S.
S2-S12=(Δω)2τ212(S1+S2)2 sin2 α,
τ2=6Δω S2-S12S2+S12.

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