Abstract

The bandwidth limitations of Poole’s higher-order polarization-mode dispersion (PMD) interpretation are examined. Correlations and errors related to the truncation of the PMD Taylor series are determined by analysis and simulation. As the PMD order increases, the effective bandwidth of the Poole representation is found to grow slowly beyond the bandwidth of the principal state applicable to first-order PMD.

© 2003 Optical Society of America

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References

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  1. C. D. Poole and J. A. Nagel, �??Polarization effects in lightwave systems�?? in Optical Fiber Telecommunications IIIA, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997).
  2. H. Kogelnik, L. E. Nelson, and R. M. Jopson, �??Polarization mode dispersion�?? in Optical Fiber Telecommunications IVB, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).
  3. L. E. Nelson, R. M. Jopson, H. Kogelnik and J. P. Gordon, �??Measurement of polarization mode dispersion vectors using the polarization-dependent signal delay method,�?? Opt. Express 6, 158-167 (2000), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-6-8-158">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-6-8-158</a>
    [CrossRef] [PubMed]
  4. G. J. Foschini, and C. D. Poole, �??Statistical theory of polarization dispersion in single mode fibers,�?? J. Lightwave Technol. 9, 1439-1456 (1991).
    [CrossRef]
  5. H. Kogelnik, L. E. Nelson, and J. P. Gordon, �??Emulation and inversion of polarization mode dispersion,�?? J. Lightwave Technol. 21, 482-495 (2003).
    [CrossRef]
  6. F. Bruyere, �??Impact of first- and second-order PMD in optical digital transmission systems,�?? Opt. Fiber Technol. 2, 269-280 (1996).
    [CrossRef]
  7. D. Penninckx and V. Morénas, �??Jones matrix of polarization mode dispersion,�?? Opt. Lett. 24, 875-877 (1999).
    [CrossRef]
  8. H. Kogelnik, L. E. Nelson, J. P. Gordon and R. M. Jopson, �??Jones matrix for second-order polarization mode dispersion,�?? Opt. Lett. 25, 19-21 (2000).
    [CrossRef]
  9. C. Francia and D Penninckx, �??Polarization mode dispersion in single-mode optical fibers: Time impulse response,�?? in Proc. 1999 IEEE Internat. Conf. on Communications 3 (Institute of Electrical and Electronics Engineers, Piscataway, 1999), pp. 1731-1735.
  10. A. Eyal, W. Marshall, M. Tur and A. Yariv, �??Representation of second-order PMD,�?? Electron. Lett. 35, 1658- 1659 (1999).
    [CrossRef]
  11. Y. Li, A. Eyal, P.-O. Hedekvist and A. Yariv, �??Measurement of high-order polarization mode dispersion,�?? Photon. Technol. Lett. 12, 861-863 (2000).
    [CrossRef]
  12. A. Eyal, Y. Li, W. K. Marshall and A. Yariv, �??Statistical determination of the length dependence of highorder polarization mode dispersion,�?? Opt. Lett. 25, 875-877 (2000).
    [CrossRef]
  13. M. Karlsson and J. Brentel, �??Autocorrelation function of the polarization-mode dispersion vector,�?? Opt. Lett. 24, 939-941 (1999).
    [CrossRef]
  14. M. Shtaif, A. Mecozzi, and A. Nagel, �??Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal state,�?? Photon. Technol. Lett. 12, 53-55 (2000).
    [CrossRef]
  15. A. O. Lima, I. T. Lima Jr., C. R. Menyuk, and T. Adali, "Comparison of penalties resulting from first-order and all-order polarization mode dispersion distortions in optical fiber transmission systems," Opt. Lett. 28, 310-312 (2003).
    [CrossRef] [PubMed]

Electron Lett. (1)

A. Eyal, W. Marshall, M. Tur and A. Yariv, �??Representation of second-order PMD,�?? Electron. Lett. 35, 1658- 1659 (1999).
[CrossRef]

IEEE Internat. Conf. on Communications (1)

C. Francia and D Penninckx, �??Polarization mode dispersion in single-mode optical fibers: Time impulse response,�?? in Proc. 1999 IEEE Internat. Conf. on Communications 3 (Institute of Electrical and Electronics Engineers, Piscataway, 1999), pp. 1731-1735.

J. Lightwave Technol. (2)

G. J. Foschini, and C. D. Poole, �??Statistical theory of polarization dispersion in single mode fibers,�?? J. Lightwave Technol. 9, 1439-1456 (1991).
[CrossRef]

H. Kogelnik, L. E. Nelson, and J. P. Gordon, �??Emulation and inversion of polarization mode dispersion,�?? J. Lightwave Technol. 21, 482-495 (2003).
[CrossRef]

Opt. Express (1)

Opt. Fiber. Technol. (1)

F. Bruyere, �??Impact of first- and second-order PMD in optical digital transmission systems,�?? Opt. Fiber Technol. 2, 269-280 (1996).
[CrossRef]

Opt. Lett. (5)

Photon. Technol. Lett. (2)

M. Shtaif, A. Mecozzi, and A. Nagel, �??Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal state,�?? Photon. Technol. Lett. 12, 53-55 (2000).
[CrossRef]

Y. Li, A. Eyal, P.-O. Hedekvist and A. Yariv, �??Measurement of high-order polarization mode dispersion,�?? Photon. Technol. Lett. 12, 861-863 (2000).
[CrossRef]

Other (2)

C. D. Poole and J. A. Nagel, �??Polarization effects in lightwave systems�?? in Optical Fiber Telecommunications IIIA, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997).

H. Kogelnik, L. E. Nelson, and R. M. Jopson, �??Polarization mode dispersion�?? in Optical Fiber Telecommunications IVB, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).

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Figures (7)

Fig. 1.
Fig. 1.

Output PMD vector τ⃗(ω) of a fiber with a mean DGD of 35 ps as a function of frequency [3]. Measurements were performed with the PSD and the MMM methods. The figure shows the DGD and the three vector components τi .

Fig. 2.
Fig. 2.

The rms magnitude Mn of the Taylor series terms from theory as a function of the frequency deviation Δνω/2π. The mean DGD of the fiber is 1 ps. The lowest six PMD orders are shown.

Fig. 3.
Fig. 3.

Correlation Cn for the lowest six PMD orders as a function of the frequency deviation Δνω/2π. The mean DGD of the fiber is 1 ps.

Fig. 4.
Fig. 4.

Diagram for the second-order prediction τ⃗pred at ω 0ω. τ⃗(ω 0ω) is the actual PMD vector at ω 0ω for which the prediction is attempted.

Fig. 5.
Fig. 5.

Error functions En for the lowest four PMD orders as a function of the frequency deviation Δνω/2π. The mean DGD of the fiber is 1 ps. The results of theory (lines) and simulation (markers) are shown.

Fig. 6.
Fig. 6.

Error functions for the lowest 30 PMD orders. The mean DGD of the fiber is 1 ps.

Fig. 7.
Fig. 7.

Full bandwidth for 1-ps mean DGD using the three different measures of the error of a truncated Taylor series.

Tables (1)

Tables Icon

Table 1. Frequency deviation (1-ps mean DGD) and μ -values at the higher-order half-bandwidth limit where the E-functions reach a value of 0.06, for selected PMD orders. The bandwidth increase relative to the first-order bandwidth of the PSP is shown in the rightmost column.

Equations (17)

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τ = Δ τ p ,
τ ( ω ) = τ ( ω 0 ) + τ ω Δ ω + 1 2 τ ωω Δ ω 2 + 1 n ! τ ( n ) Δ ω n ,
Δ ω PSP Δ τ = π 4 .
μ Δ ω Δ τ 2 3 = π 8 Δ τ Δ ω .
C 1 ( μ ) = τ ( ω 0 + Δ ω ) · τ ( ω 0 ) τ 2 = ( 1 e μ 2 ) μ 2 .
M n + 1 ( 1 n ! τ ( n ) Δ ω n ) rms Δ τ rms = ( 2 n ) ! n ! ( n + 1 ) ! μ n ,
τ pred ( ω + Δ ω ) = τ ( ω 0 ) + τ ω Δ ω + . . + 1 ( n 1 ) ! τ ( n 1 ) Δ ω n 1 ,
C n = τ ( ω 0 + Δ ω ) · τ pred ( ω + Δ ω ) τ 2 ,
E n = ( τ ( ω 0 + Δ ω ) τ pred ) 2 τ 2 .
E 1 = 2 2 C 1 .
C n + 1 C n = ( 1 n ) μ n n ! d n d μ n C 1 ,
C n + 1 = 1 v = v min ( 1 ) n + v ( 2 v 3 ) ! n ! v ! ( 2 v n 3 ) ! μ 2 v 2 ,
E n + 1 E n = 2 ( 2 n 1 ) ! ( n 1 ) ! n ! ( n + 1 ) ! μ 2 n μ n d d μ E n .
E n + 1 = v = n + 2 ( 1 ) n + v 2 ( 2 v 3 ) ! n ! v ! ( 2 v n 3 ) ! μ 2 v 2 .
E n + 1 = 2 ( 2 n + 1 ) ! n ! ( n + 1 ) ! ( n + 2 ) ! μ 2 n + 2 + . . . .
E n = i = 1 m i M n + i 2
m i = ( 1 ) i 1 ( n + i 1 ) ! ( n + i 2 ) ! ( n 1 ) ! ( n + 2 i 2 ) !

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