Abstract

We examine a series of experimentally realizable procedures for wide-bandwidth polarization mode dispersion compensation based on Taylor and Chebyshev approximations to the transfer matrix for light polarization in optical fibers. Our results demonstrate that a symmetric ordering of compensator elements in the Taylor procedure improves performance and that methods based on the Chebyshev approximation can significantly widen the compensation bandwidth.

© 2005 Optical Society of America

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References

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  1. M. Shtaif, A. Mecozzi, M. Tur, J. A. Nagel, “A compensator for the effects of high-order polarization mode dispersion in optical fibers,” IEEE Photonics Technol. Lett. 12, 434–436 (2000).
    [CrossRef]
  2. A. Yariv, Optical Fiber Communication, 5th ed. (Oxford U. Press, 1997).
  3. A. Eyal, W. K. Marshall, M. Tur, A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35, 1658–1659 (1999).
    [CrossRef]
  4. T. Kudou, M. Iguchi, M. Masuda, T. Ozeki, “Theoretical basis of polarization mode dispersion equalization up to the second order,” J. Lightwave Technol. 18, 614–617 (2000).
    [CrossRef]
  5. E. Hellstrom, H. Sunnerud, M. Westlund, M. Karlsson, “Third-order dispersion compensation using a phase modulator,” J. Lightwave Technol. 21, 1188–1197 (2003).
    [CrossRef]
  6. H. Sunnerud, M. Karlsson, C. Xie, P. A. Andrekson, “Polarization-mode dispersion in high-speed fiber-optic transmission systems,” J. Lightwave Technol. 20, 2204–2219 (2002).
    [CrossRef]
  7. L. Moller, “Filter synthesis for broad-band PMD compensation in WDM systems,” IEEE Photonics Technol. Lett. 12, 1258–1260 (2000).
    [CrossRef]
  8. A. Eyal, A. Yariv, “Design of broad-band PMD compensation filters,” IEEE Photonics Technol. Lett. 14, 1088–1090 (2002).
    [CrossRef]
  9. W. H. Press, B. P. Flannery, S. A. Teukolsky, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).
  10. F. Heismann, “Accurate Jones matrix expansion for all orders of polarization mode dispersion,” Opt. Lett. 28, 2013–2015 (2003).
    [CrossRef] [PubMed]
  11. I. T. Lima, R. Khosravani, P. Ebrahimi, E. Ibragimov, C. R. Menyuk, A. E. Willner, “Comparison of polarization mode dispersion emulators,” J. Lightwave Technol. 19, 1872–1881 (2001).
    [CrossRef]
  12. M. Karlsson, “Probability density functions of the differential group delay in optical fiber communication systems,” J. Lightwave Technol. 19, 324–331 (2001).
    [CrossRef]
  13. H. Kogelnik, L. E. Nelson, J. P. Gordon, “Emulation and inversion of polarization-mode dispersion,” J. Lightwave Technol. 21, 482–495 (2003).
    [CrossRef]
  14. T. J. Rivlin, The Chebyshev Polynomials (Wiley, 1974).
  15. M. Glasner, D. Yevick, B. Hermansson, “Generalized propagation formulas of arbitrarily high order,” J. Chem. Phys. 95, 8266–8272 (1991).
    [CrossRef]

2003 (3)

2002 (2)

2001 (2)

2000 (3)

L. Moller, “Filter synthesis for broad-band PMD compensation in WDM systems,” IEEE Photonics Technol. Lett. 12, 1258–1260 (2000).
[CrossRef]

T. Kudou, M. Iguchi, M. Masuda, T. Ozeki, “Theoretical basis of polarization mode dispersion equalization up to the second order,” J. Lightwave Technol. 18, 614–617 (2000).
[CrossRef]

M. Shtaif, A. Mecozzi, M. Tur, J. A. Nagel, “A compensator for the effects of high-order polarization mode dispersion in optical fibers,” IEEE Photonics Technol. Lett. 12, 434–436 (2000).
[CrossRef]

1999 (1)

A. Eyal, W. K. Marshall, M. Tur, A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35, 1658–1659 (1999).
[CrossRef]

1991 (1)

M. Glasner, D. Yevick, B. Hermansson, “Generalized propagation formulas of arbitrarily high order,” J. Chem. Phys. 95, 8266–8272 (1991).
[CrossRef]

Andrekson, P. A.

Ebrahimi, P.

Eyal, A.

A. Eyal, A. Yariv, “Design of broad-band PMD compensation filters,” IEEE Photonics Technol. Lett. 14, 1088–1090 (2002).
[CrossRef]

A. Eyal, W. K. Marshall, M. Tur, A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35, 1658–1659 (1999).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Glasner, M.

M. Glasner, D. Yevick, B. Hermansson, “Generalized propagation formulas of arbitrarily high order,” J. Chem. Phys. 95, 8266–8272 (1991).
[CrossRef]

Gordon, J. P.

Heismann, F.

Hellstrom, E.

Hermansson, B.

M. Glasner, D. Yevick, B. Hermansson, “Generalized propagation formulas of arbitrarily high order,” J. Chem. Phys. 95, 8266–8272 (1991).
[CrossRef]

Ibragimov, E.

Iguchi, M.

Karlsson, M.

Khosravani, R.

Kogelnik, H.

Kudou, T.

Lima, I. T.

Marshall, W. K.

A. Eyal, W. K. Marshall, M. Tur, A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35, 1658–1659 (1999).
[CrossRef]

Masuda, M.

Mecozzi, A.

M. Shtaif, A. Mecozzi, M. Tur, J. A. Nagel, “A compensator for the effects of high-order polarization mode dispersion in optical fibers,” IEEE Photonics Technol. Lett. 12, 434–436 (2000).
[CrossRef]

Menyuk, C. R.

Moller, L.

L. Moller, “Filter synthesis for broad-band PMD compensation in WDM systems,” IEEE Photonics Technol. Lett. 12, 1258–1260 (2000).
[CrossRef]

Nagel, J. A.

M. Shtaif, A. Mecozzi, M. Tur, J. A. Nagel, “A compensator for the effects of high-order polarization mode dispersion in optical fibers,” IEEE Photonics Technol. Lett. 12, 434–436 (2000).
[CrossRef]

Nelson, L. E.

Ozeki, T.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Rivlin, T. J.

T. J. Rivlin, The Chebyshev Polynomials (Wiley, 1974).

Shtaif, M.

M. Shtaif, A. Mecozzi, M. Tur, J. A. Nagel, “A compensator for the effects of high-order polarization mode dispersion in optical fibers,” IEEE Photonics Technol. Lett. 12, 434–436 (2000).
[CrossRef]

Sunnerud, H.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Tur, M.

M. Shtaif, A. Mecozzi, M. Tur, J. A. Nagel, “A compensator for the effects of high-order polarization mode dispersion in optical fibers,” IEEE Photonics Technol. Lett. 12, 434–436 (2000).
[CrossRef]

A. Eyal, W. K. Marshall, M. Tur, A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35, 1658–1659 (1999).
[CrossRef]

Westlund, M.

Willner, A. E.

Xie, C.

Yariv, A.

A. Eyal, A. Yariv, “Design of broad-band PMD compensation filters,” IEEE Photonics Technol. Lett. 14, 1088–1090 (2002).
[CrossRef]

A. Eyal, W. K. Marshall, M. Tur, A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35, 1658–1659 (1999).
[CrossRef]

A. Yariv, Optical Fiber Communication, 5th ed. (Oxford U. Press, 1997).

Yevick, D.

M. Glasner, D. Yevick, B. Hermansson, “Generalized propagation formulas of arbitrarily high order,” J. Chem. Phys. 95, 8266–8272 (1991).
[CrossRef]

Electron. Lett. (1)

A. Eyal, W. K. Marshall, M. Tur, A. Yariv, “Representation of second-order polarisation mode dispersion,” Electron. Lett. 35, 1658–1659 (1999).
[CrossRef]

IEEE Photonics Technol. Lett. (3)

L. Moller, “Filter synthesis for broad-band PMD compensation in WDM systems,” IEEE Photonics Technol. Lett. 12, 1258–1260 (2000).
[CrossRef]

A. Eyal, A. Yariv, “Design of broad-band PMD compensation filters,” IEEE Photonics Technol. Lett. 14, 1088–1090 (2002).
[CrossRef]

M. Shtaif, A. Mecozzi, M. Tur, J. A. Nagel, “A compensator for the effects of high-order polarization mode dispersion in optical fibers,” IEEE Photonics Technol. Lett. 12, 434–436 (2000).
[CrossRef]

J. Chem. Phys. (1)

M. Glasner, D. Yevick, B. Hermansson, “Generalized propagation formulas of arbitrarily high order,” J. Chem. Phys. 95, 8266–8272 (1991).
[CrossRef]

J. Lightwave Technol. (6)

Opt. Lett. (1)

Other (3)

A. Yariv, Optical Fiber Communication, 5th ed. (Oxford U. Press, 1997).

W. H. Press, B. P. Flannery, S. A. Teukolsky, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

T. J. Rivlin, The Chebyshev Polynomials (Wiley, 1974).

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Figures (9)

Fig. 1
Fig. 1

(a) Average correlation, calculated as a function of the normalized frequency parameter, Δ ν = Δ ω τ ¯ 2 π , with first- to fifth-order Taylor-series expansions and τ ¯ = 65 ps . The maximum Δ ω for both the Chebyshev and the Taylor approximations is 2 π × 10 GHz . (b) Comparison of the instantaneous DGD, Ω ( ω ) , for a single-fiber realization, obtained with the sixth-order Taylor and Chebyshev approximations to the PMD vector.

Fig. 2
Fig. 2

Minimum average correlation (solid curve), standard deviation about the minimum average correlation (dotted–dashed curve), and the minimum correlation for 99% confidence (dashed curve) as a function of the normalized frequency parameter calculated with the Chebyshev approximation for the PMD.

Fig. 3
Fig. 3

Average correlation (figure of merit) for the first- to fourth-order Chebyshev approximations to the PMD vector, plotted as a function of the normalized parameter Δ ν = Δ ω τ ¯ 2 π .

Fig. 4
Fig. 4

Variation of the residual with the normalized parameter Δ ν = Δ ω bd τ ¯ 2 π after the first- to third-order asymmetric compensation of Ref. [8].

Fig. 5
Fig. 5

As in Fig. 4, but after appropriate symmetrization of the operator products.

Fig. 6
Fig. 6

Variation with optical frequency of the residuals associated with first- to fourth-order compensation based on symmetric exponential products resulting from Chebyshev approximations of the designated orders to the inverse transmission matrix.

Fig. 7
Fig. 7

As in the previous figure, but without operator symmetrization.

Fig. 8
Fig. 8

Comparison of the residuals expressed as a function of optical frequency with C = 1.5 for symmetric (dotted curves) second- and third-order Chebyshev compensators and the corresponding second- and third-order standard compensators (solid curves).

Fig. 9
Fig. 9

Result of two-stage compensation in which (1) a second-order compensator processes the output of a first-order Chebyshev compensator and (2) a second-order compensator is followed by a third-order compensator.

Equations (33)

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Ω ( ω ) i 2 Ω ( ω ) σ = T ( 1 ) ( ω ) T 1 ( ω ) ,
Φ ( Δ ω ) = 2 Ω ( ω 0 + Δ ω ) Ω ( ω 0 + Δ ω ) Ω ( ω 0 + Δ ω ) 2 + Ω ( ω 0 + Δ ω ) 2 .
C j = 2 M k = 1 M Ω ( M Δ ω 1 cos [ π ( k 0.5 ) M ] + ω 0 ) cos ( π j ( k 0.5 ) M ) ,
Ω ( ω ) ( k = 0 M 1 C k P k ( ω ) ) 1 2 C 0 ,
P 0 ( ω ) = 1 ,
P 1 ( ω ) = ω ω 0 M Δ ω 1 ,
P 2 ( ω ) = 2 ( ω ω 0 M Δ ω 1 ) 2 1 ,
P 3 ( ω ) = 4 ( ω ω 0 M Δ ω 1 ) 3 3 ( ω ω 0 M Δ ω 1 ) .
T ( ω 0 + Δ ω ) = T ( ω 0 ) exp ( Δ ω N 1 ) exp ( 1 2 ! Δ ω 2 N 2 ) exp ( 1 3 ! Δ ω 3 N 3 ) exp ( 1 4 ! Δ ω 4 N 4 ) .
N 1 = T 1 ( ω 0 ) T ( 1 ) ( ω 0 ) ,
N 2 = T 1 ( ω 0 ) T ( 2 ) ( ω 0 ) N 1 2 ,
N 3 = T 1 ( ω 0 ) T ( 3 ) ( ω 0 ) N 1 3 3 N 1 N 2 ,
N 4 = T 1 ( ω 0 ) T ( 4 ) ( ω 0 ) N 1 4 3 N 2 2 4 N 1 N 3 6 N 1 2 N 2 .
T ( 1 ) ( ω ) = [ T ( 1 ) ( ω ) T 1 ( ω ) ] T ( ω ) Ω ( ω ) T ( ω ) .
Ω ( ω ) = n = 0 Ω ( n ) n ! ( Δ ω ) n ,
T ( ω 0 + Δ ω ) = exp [ S 0 Δ ω + S 1 2 ( Δ ω ) 2 + S 2 6 ( Δ ω ) 3 + ] T ( ω 0 ) ,
T ( 1 ) ( ω 0 ) T ( ω 0 ) 1 = Ω ( 0 ) = S 0 ,
T ( 2 ) ( ω 0 ) T ( ω 0 ) 1 = Ω ( 1 ) + Ω ( 0 ) Ω ( 0 ) = S 1 + S 0 S 0 ,
T ( 3 ) ( ω 0 ) T ( ω 0 ) 1 = Ω ( 2 ) + 2 Ω ( 1 ) Ω ( 0 ) + Ω ( 0 ) Ω ( 1 ) + Ω ( 0 ) Ω ( 0 ) Ω ( 0 ) = S 2 + 1.5 ( S 1 S 0 + S 0 S 1 ) + S 0 S 0 S 0 ,
S 0 = Ω ( 0 ) ,
S 1 = Ω ( 1 ) ,
S 2 = Ω ( 2 ) + 1 2 [ Ω ( 1 ) , Ω ( 0 ) ] ,
S 0 = T ( 1 ) ( ω 0 ) T 1 ( ω 0 ) ,
S 1 = T ( 2 ) ( ω 0 ) T 1 ( ω 0 ) ( Ω ( 0 ) ) 2 ,
S 2 = T ( 3 ) ( ω 0 ) T 1 ( ω 0 ) 1.5 ( Ω ( 1 ) Ω ( 0 ) + Ω ( 0 ) Ω ( 1 ) ) ( Ω ( 0 ) ) 3 .
T ( ω 0 + Δ ω ) = exp ( 1 2 S 0 Δ ω ) exp ( 1 2 S 1 Δ ω 2 ) exp ( 1 2 S 0 Δ ω ) T ( ω 0 ) + O ( Δ ω 3 ) ,
T res ( ω ) = T 1 ( ω 0 ) exp ( 1 2 Δ ω Ω ( 0 ) ) exp ( 1 2 Δ ω 2 Ω ( 1 ) ) exp ( 1 2 Δ ω Ω ( 0 ) ) T ( ω ) .
T res ( ω ) = [ 1 2 C 0 P 0 ( Δ ω ) + C 1 P 1 ( Δ ω ) + C 2 P 2 ( Δ ω ) + + C M 1 P M 1 ( Δ ω ) ] T ( ω ) ,
T res ( ω ) = ( I + A 1 A 0 1 Δ ω + 1 2 ! A 2 A 0 1 Δ ω 2 + ) A 0 T ( ω ) = exp ( 1 2 Ω ̃ ( 0 ) Δ ω ) exp ( 1 2 Ω ̃ ( 1 ) Δ ω 2 ) exp ( 1 2 Ω ̃ ( 0 ) Δ ω ) A 0 T ( ω ) ,
Ω ̃ ( 0 ) = A 1 A 0 1 ,
Ω ̃ ( 1 ) = A 2 A 0 1 ( Ω ̃ ( 0 ) ) 2 ,
Ω ̃ ( 2 ) = A 3 A 0 1 2 Ω ̃ ( 1 ) Ω ̃ ( 0 ) Ω ̃ ( 0 ) Ω ̃ ( 1 ) ( Ω ̃ ( 0 ) ) 3 .
T res ( ω ) = exp ( Ω ̃ ( 0 ) Δ ω ) exp ( 1 2 Ω ̃ ( 1 ) Δ ω 2 ) exp ( 1 6 Ω ̃ ( 2 ) Δ ω 3 ) A 0 T ( ω ) ,

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