Abstract

We present analytic third- and fourth-order expansions of the Jones matrix as products of exponentials of individual matrices. In our first formalism, these are polarization mode dispersion (PMD) matrices of definite orders. We then discuss an alternative procedure that instead employs exponentials of general skew-Hermitian matrices with a low-order dependence on the deviation of the optical frequency from a central reference frequency. Our expressions correspond to PMD compensators formed from a succession of relatively simple optical components each of which has the frequency response of a single operator in the product.

© 2006 Optical Society of America

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References

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  1. A. Eyal and A. Yariv, "Design of broad-band PMD compensation filters," IEEE Photon. Technol. Lett. 14, 1088-1090 (2002).
    [CrossRef]
  2. 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]
  3. H. Kogelnik, L. E. Nelson, and J. P. Gordon, "Emulation and inversion of polarization-mode dispersion," J. Lightwave Technol. 21, 482-495 (2003).
    [CrossRef]
  4. E. Forestieri and G. Prati, "Exact analytical evaluation of second-order PMD impact on the outage probability for a compensated system," J. Lightwave Technol. 22, 988-996 (2004).
    [CrossRef]
  5. F. Heismann, "Accurate Jones matrix expansion for all orders of polarization mode dispersion," Opt. Lett. 28, 2013-2015 (2003).
    [CrossRef] [PubMed]
  6. M. Glasner, D. Yevick, and B. Hermansson, "Generalized propagation formulas of arbitrarily high order," J. Chem. Phys. 95, 8266-8272 (1991).
    [CrossRef]
  7. M. Suzuki, "General theory of fractal path integrals with applications to many-body theories and statistical physics," J. Math. Phys. 32, 400-407 (1991).
    [CrossRef]
  8. T. Kudou, M. Iguchi, M. Masuda, and T. Ozeki, "Theoretical basis of polarization mode dispersion equalization up to the second order," J. Lightwave Technol. 18, 614-617 (2000).
    [CrossRef]
  9. T. Kudou, K. Shimizu, K. Harada, and T. Ozeki, "Synthesis of grating lattice circuits," J. Lightwave Technol. 17, 347-353 (1999).
    [CrossRef]
  10. D. O. Yevick, M. Chanachowicz, M. Reimer, M. O'Sullivan, W. Huang, and T. Lu, "Chebyshev and Taylor approximations of polarization mode dispersion for improved compensation bandwidth," J. Opt. Soc. Am. A 22, 1662-1667 (2005).
    [CrossRef]
  11. T. Lu, W. Huang, D. Yevick, M. O'Sullivan, and M. Reimer, "A multicanonical comparison of PMD compensator performance," J. Opt. Soc. Am. A 22, 2804-2809 (2005).
    [CrossRef]
  12. A. Eyal, W. K. Marshall, M. Tur, and A. Yariv, "Representation of second-order polarisation mode dispersion," Electron. Lett. 35, 1658-1659 (1999).
    [CrossRef]
  13. D. Yevick, "Multicanonical evaluation of joint probability density functions in communication system modeling," IEEE Photonics Technol. Lett. 15, 1540-1542 (2003).
    [CrossRef]
  14. J. N. Damask, P. R. Myers, G. J. Simer, and A. Boschi, "Methods to construct programmable PMD sources--Part II: instrument demonstrations," J. Lightwave Technol. 22, 1006-1013 (2004).
    [CrossRef]

2005 (2)

2004 (2)

2003 (3)

2002 (1)

A. Eyal and A. Yariv, "Design of broad-band PMD compensation filters," IEEE Photon. Technol. Lett. 14, 1088-1090 (2002).
[CrossRef]

2000 (2)

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]

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

1999 (2)

T. Kudou, K. Shimizu, K. Harada, and T. Ozeki, "Synthesis of grating lattice circuits," J. Lightwave Technol. 17, 347-353 (1999).
[CrossRef]

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

1991 (2)

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

M. Suzuki, "General theory of fractal path integrals with applications to many-body theories and statistical physics," J. Math. Phys. 32, 400-407 (1991).
[CrossRef]

Boschi, A.

Chanachowicz, M.

Damask, J. N.

Eyal, A.

A. Eyal and A. Yariv, "Design of broad-band PMD compensation filters," IEEE Photon. Technol. Lett. 14, 1088-1090 (2002).
[CrossRef]

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

Forestieri, E.

Glasner, M.

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

Gordon, J. P.

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

Harada, K.

Heismann, F.

Hermansson, B.

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

Huang, W.

Iguchi, M.

Kogelnik, H.

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

Kudou, T.

Lu, T.

Marshall, W. K.

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

Masuda, M.

Myers, P. R.

Nelson, L. E.

O'Sullivan, M.

Ozeki, T.

Prati, G.

Reimer, M.

Shimizu, K.

Simer, G. J.

Suzuki, M.

M. Suzuki, "General theory of fractal path integrals with applications to many-body theories and statistical physics," J. Math. Phys. 32, 400-407 (1991).
[CrossRef]

Tur, M.

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

Yariv, A.

A. Eyal and A. Yariv, "Design of broad-band PMD compensation filters," IEEE Photon. Technol. Lett. 14, 1088-1090 (2002).
[CrossRef]

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

Yevick, D.

T. Lu, W. Huang, D. Yevick, M. O'Sullivan, and M. Reimer, "A multicanonical comparison of PMD compensator performance," J. Opt. Soc. Am. A 22, 2804-2809 (2005).
[CrossRef]

D. Yevick, "Multicanonical evaluation of joint probability density functions in communication system modeling," IEEE Photonics Technol. Lett. 15, 1540-1542 (2003).
[CrossRef]

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

Yevick, D. O.

Electron. Lett. (1)

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

IEEE Photon. Technol. Lett. (1)

A. Eyal and A. Yariv, "Design of broad-band PMD compensation filters," IEEE Photon. Technol. Lett. 14, 1088-1090 (2002).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

D. Yevick, "Multicanonical evaluation of joint probability density functions in communication system modeling," IEEE Photonics Technol. Lett. 15, 1540-1542 (2003).
[CrossRef]

J. Chem. Phys. (1)

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

J. Lightwave Technol. (5)

J. Math. Phys. (1)

M. Suzuki, "General theory of fractal path integrals with applications to many-body theories and statistical physics," J. Math. Phys. 32, 400-407 (1991).
[CrossRef]

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

Opt. Lett. (1)

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]

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Figures (4)

Fig. 1
Fig. 1

Residual DGD of various Taylor expansion techniques as a function of normalized frequency, ω τ ave 2 π , where τ ave is the average DGD.

Fig. 2
Fig. 2

Contour plots of the average bandwidth for each histogram bin as a joint function of the uncompensated first- and second-order normalized PMD for the output signals generated by a 30-section PMD emulator with compensators based on the expansions of (A) Ref. [12], (B) Eq. (9), and (C) Eq. (25) and (D) the three-operator method.

Fig. 3
Fig. 3

As in Fig. 2 but for the conditional 1 dB outage probability.

Fig. 4
Fig. 4

As in Fig. 3 but for the absolute 1 dB outage probability density.

Equations (39)

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E out ( ω ) = T ( ω ) E in ( ω ) .
U ( 1 ) ( ω ) = j 1 2 τ ( ω ) σ U ( ω ) D ( ω ) U ( ω ) ,
U ( ω ) = U ( 0 ) + 0 ω d ω 1 D ( ω 1 ) U ( ω 1 ) U ( 0 ) + p ̂ ( ω ) U ( ω ) .
U ( ω ) = [ I + p ̂ ( ω ) + p ̂ 2 ( ω ) + p ̂ 3 ( ω ) + ] U ( 0 ) = n = 0 p ̂ n ( ω ) U ( 0 )
D ( ω ) = n = 0 D n n ! ω n ,
p ̂ ( ω ) U ( 0 ) = 0 ω d ω 1 [ D 0 + ω 1 D 1 + ω 1 2 2 D 2 + ω 1 3 6 D 3 + O ( ω 1 4 ) ] U ( 0 ) = [ ω D 0 + ω 2 2 D 1 + ω 3 6 D 2 + ω 4 24 D 3 + O ( ω 5 ) ] U ( 0 ) ,
p ̂ 2 ( ω ) U ( 0 ) = 0 ω d ω 1 [ D 0 + ω 1 D 1 + ω 1 2 2 D 2 + O ( ω 1 3 ) ] [ ω 1 D 0 + ω 1 2 2 D 1 + ω 1 3 6 D 2 + O ( ω 1 4 ) ] U ( 0 ) = [ ω 2 2 D 0 2 + ω 3 6 ( 2 D 1 D 0 + D 0 D 1 ) + ω 4 24 ( D 0 D 2 + 3 D 1 2 + 3 D 2 D 0 ) + O ( ω 5 ) ] U ( 0 )
p ̂ 3 ( ω ) U ( 0 ) = 0 ω d ω 1 [ D 0 + ω 1 D 1 + O ( ω 1 2 ) ] [ ω 1 2 2 D 0 2 + ω 1 3 6 ( 2 D 1 D 0 + D 0 D 1 ) + O ( ω 1 4 ) ] U ( 0 ) = [ ω 3 6 D 0 3 + ω 4 24 ( 3 D 1 D 0 2 + 2 D 0 D 1 D 0 + D 0 2 D 1 ) + O ( ω 5 ) ] U ( 0 ) ,
p ̂ 4 ( ω ) U ( 0 ) = 0 ω d ω 1 [ D 0 + O ( ω 1 ) ] [ ω 1 3 6 D 0 3 + O ( ω 1 4 ) ] U ( 0 ) = [ ω 4 24 D 0 4 + O ( ω 5 ) ] U ( 0 ) .
U ( ω ) U 1 ( 0 ) = I + ω D 0 + ω 2 2 ( D 1 + D 0 2 ) + ω 3 6 ( D 2 + 2 D 1 D 0 + D 0 D 1 + D 0 3 ) + ω 4 24 ( D 3 + D 0 D 2 + 3 D 1 2 + 3 D 2 D 0 + 3 D 1 D 0 2 + 2 D 0 D 1 D 0 + D 0 2 D 1 + D 0 4 ) + O ( ω 5 ) ,
D 0 = U ( 1 ) ( 0 ) U 1 ( 0 ) ,
D 1 = U ( 2 ) ( 0 ) U 1 ( 0 ) D 0 2 ,
D 2 = U ( 3 ) ( 0 ) U 1 ( 0 ) 2 D 1 D 0 D 0 D 1 D 0 3 ,
D 3 = U ( 4 ) ( 0 ) U 1 ( 0 ) D 0 D 2 3 D 1 2 3 D 2 D 0 3 D 1 D 0 2 2 D 0 D 1 D 0 D 0 2 D 1 D 0 4 .
U ( ω ) U 1 ( 0 ) = exp ( D 0 ω 3 ) exp ( D 1 ω 2 2 ) exp ( 2 D 0 ω 3 ) exp ( D 2 ω 3 6 ) + O ( ω 4 ) ,
U ( ω ) U 1 ( 0 ) = exp ( D 1 ω 2 6 ) exp ( D 0 ω 4 ) exp ( D 2 ω 3 6 ) exp ( D 0 ω 4 ) exp ( D 1 ω 2 3 ) exp ( D 0 ω 2 ) exp ( D 3 ω 4 24 ) + O ( ω 5 ) .
exp [ D 0 ω + D 1 ω 2 2 + D 2 ω 3 6 + + O ( ω N + 1 ) ]
exp [ D 0 ω + D 1 ω 2 2 + O ( ω 3 ) ] = exp ( A 1 ω ) exp ( A 2 ω ) + O ( ω 3 ) ,
A 1 = D 0 A 2 ,
D 1 = D 0 A 2 A 2 D 0 [ D 0 , A 2 ] .
D l = j d l σ ,
A l = j a l σ ,
d 0 × a 2 = 1 2 d 1 ,
a 2 = d 1 2 d 0 d 0 × d 1 d 0 × d 1 + λ d 0 a 2 + λ d 0
A 2 = ( a 2 + 1 2 d 0 ) σ = A 2 + 1 2 D 0 .
exp [ D 0 ω + D 1 ω 2 2 + O ( ω 3 ) ]
= exp ( A 1 ω ) exp ( A 2 ω ) exp ( A 3 ω ) + O ( ω 3 ) ,
A 2 = D 0 A 1 A 3
D 1 = [ D 0 , A 3 ] + [ A 1 , D 0 ] + [ A 3 , A 1 ] .
U ( ω ) U 1 ( 0 ) = n = 0 3 D n ( 0 ) n ! ω n = exp ( A 1 ω 2 ) exp ( A 2 ω ) exp ( A 3 ω 2 ) + O ( ω 4 ) .
A 2 = D 0 ,
A 1 = D 1 3 δ A 3 6 ,
D 0 δ A 3 δ A 3 D 0 = D 2 .
2 d 0 × a 3 = d 2
a 3 = d 2 2 d 0 d 2 × d 0 d 2 × d 0 + λ d 0 a 3 + λ d 0 .
C = 1 24 ( 2 D 0 2 δ A 3 δ A 3 D 1 + 2 D 0 2 D 1 + D 1 δ A 3 + 4 D 1 D 0 2 2 δ A 3 D 0 2 + 3 D 1 2 ) 1 24 λ ( D 0 D 1 D 1 D 0 ) 1 24 ( D 3 + D 0 D 2 + 3 D 1 2 + 3 D 2 D 0 + 3 D 1 D 0 2 + 2 D 0 D 1 D 0 + D 0 2 D 1 + D 0 4 ) ,
= 1 24 ( 2 D 0 2 δ A 3 δ A 3 D 1 + D 1 δ A 3 2 δ A 3 D 0 2 D 3 D 0 D 2 3 D 2 D 0 + D 1 D 0 2 2 D 0 D 1 D 0 + D 0 2 D 1 D 0 4 ) 1 24 λ ( D 0 D 1 D 1 D 0 ) ,
1 24 λ [ D 0 , D 1 ] + 1 24 C .
U ( ω ) U 1 ( 0 ) = exp ( D 1 ω 2 3 ) exp ( D 0 ω ) exp ( D 1 ω 2 6 ) exp ( D 2 ω 3 ) + O ( ω 4 ) ,

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