Abstract

A generalized Mueller matrix method (GMMM) is proposed to measure the polarization mode dispersion (PMD) in an optical fiber system with polarization-dependent loss or gain (PDL/G). This algorithm is based on the polar decomposition of a 4×4 matrix which corresponds to a Lorentz transformation. Compared to the generalized Poincaré sphere method, the GMMM can measure PMD accurately with a relatively larger frequency step, and the obtained PMD data has very low noise level.

© 2006 Optical Society of America

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References

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  1. N. Gisin, and B. Huttner, "Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers," Opt. Comm. 142, 119-125 (1997).
    [CrossRef]
  2. Yi Li, and A . Yariv, "Solutions to the dynamical equation of polarization-mode dispersion and polarization-dependent losses," J. Opt. Soc. Am. B 17,1821-1827 (2000).
    [CrossRef]
  3. L. Chen, O. Chen, S. Hadjifaradji, and X. Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16,206-208 (2004).
    [CrossRef]
  4. B. L. Heffner, "Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis," IEEE Photon. Technol. Lett. 4, 1066-1069 (1992).
    [CrossRef]
  5. A. Eyal, and M. Tur, "Measurement of polarization mode dispersion in systems having polarization dependent loss or gain," IEEE Photon. Technol. Lett. 9, 1256-1258 (1997).
    [CrossRef]
  6. R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999).
    [CrossRef]
  7. H. Dong, P. Shum, M. Yan, G. Ning, Y. Gong, and C. Wu, "Generalized frequency dependence of output Stokes parameters in an optical fiber system with PMD and PDL/PDG," Opt. Express 13,8875-8881 (2005).
    [CrossRef] [PubMed]
  8. R. Barakat, "Theory of the coherency matrix for light of arbitrary spectral bandwidth," J.Opt.Soc.Am. 53, 317-323 (1963).
    [CrossRef]
  9. R. Barakat, "Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory," Opt. Comm. 38, 159-161 (1981).
    [CrossRef]
  10. S.-Y. Lu and R. A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106-1113 (1996).
    [CrossRef]
  11. A. Bessa dos Santos, and J. P. von der weid, "PDL effects in PMD emulators made out with HiBi fibers: Building PMD/PDL emulators," IEEE Photon. Technol. Lett. 16,452-454 (2004).
    [CrossRef]
  12. R. M. Craig, S. L. Gilbert, and P. D. Hale, "High-resolution, nonmechanical approach to polarization-dependent transmission measurements," IEEE J. Lightwave Technol. 7, 1285-1294 (1998).
    [CrossRef]

2005 (1)

2004 (1)

L. Chen, O. Chen, S. Hadjifaradji, and X. Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16,206-208 (2004).
[CrossRef]

1999 (1)

R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999).
[CrossRef]

1998 (1)

R. M. Craig, S. L. Gilbert, and P. D. Hale, "High-resolution, nonmechanical approach to polarization-dependent transmission measurements," IEEE J. Lightwave Technol. 7, 1285-1294 (1998).
[CrossRef]

1997 (2)

N. Gisin, and B. Huttner, "Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers," Opt. Comm. 142, 119-125 (1997).
[CrossRef]

A. Eyal, and M. Tur, "Measurement of polarization mode dispersion in systems having polarization dependent loss or gain," IEEE Photon. Technol. Lett. 9, 1256-1258 (1997).
[CrossRef]

1996 (1)

1992 (1)

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

1981 (1)

R. Barakat, "Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory," Opt. Comm. 38, 159-161 (1981).
[CrossRef]

1963 (1)

R. Barakat, "Theory of the coherency matrix for light of arbitrary spectral bandwidth," J.Opt.Soc.Am. 53, 317-323 (1963).
[CrossRef]

Bao, X.

L. Chen, O. Chen, S. Hadjifaradji, and X. Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16,206-208 (2004).
[CrossRef]

Barakat, R.

R. Barakat, "Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory," Opt. Comm. 38, 159-161 (1981).
[CrossRef]

R. Barakat, "Theory of the coherency matrix for light of arbitrary spectral bandwidth," J.Opt.Soc.Am. 53, 317-323 (1963).
[CrossRef]

Chen, L.

L. Chen, O. Chen, S. Hadjifaradji, and X. Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16,206-208 (2004).
[CrossRef]

Chen, O.

L. Chen, O. Chen, S. Hadjifaradji, and X. Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16,206-208 (2004).
[CrossRef]

Chipman, R. A.

Craig, R. M.

R. M. Craig, S. L. Gilbert, and P. D. Hale, "High-resolution, nonmechanical approach to polarization-dependent transmission measurements," IEEE J. Lightwave Technol. 7, 1285-1294 (1998).
[CrossRef]

Dong, H.

Eyal, A.

A. Eyal, and M. Tur, "Measurement of polarization mode dispersion in systems having polarization dependent loss or gain," IEEE Photon. Technol. Lett. 9, 1256-1258 (1997).
[CrossRef]

Gilbert, S. L.

R. M. Craig, S. L. Gilbert, and P. D. Hale, "High-resolution, nonmechanical approach to polarization-dependent transmission measurements," IEEE J. Lightwave Technol. 7, 1285-1294 (1998).
[CrossRef]

Gisin, N.

N. Gisin, and B. Huttner, "Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers," Opt. Comm. 142, 119-125 (1997).
[CrossRef]

Gong, Y.

Hadjifaradji, S.

L. Chen, O. Chen, S. Hadjifaradji, and X. Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16,206-208 (2004).
[CrossRef]

Hale, P. D.

R. M. Craig, S. L. Gilbert, and P. D. Hale, "High-resolution, nonmechanical approach to polarization-dependent transmission measurements," IEEE J. Lightwave Technol. 7, 1285-1294 (1998).
[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]

Huttner, B.

N. Gisin, and B. Huttner, "Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers," Opt. Comm. 142, 119-125 (1997).
[CrossRef]

Jopson, R. M.

R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999).
[CrossRef]

Kogelnik, H.

R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999).
[CrossRef]

Lu, S.-Y.

Nelson, L. E.

R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999).
[CrossRef]

Ning, G.

Shum, P.

Tur, M.

A. Eyal, and M. Tur, "Measurement of polarization mode dispersion in systems having polarization dependent loss or gain," IEEE Photon. Technol. Lett. 9, 1256-1258 (1997).
[CrossRef]

Wu, C.

Yan, M.

IEEE J. Lightwave Technol. (1)

R. M. Craig, S. L. Gilbert, and P. D. Hale, "High-resolution, nonmechanical approach to polarization-dependent transmission measurements," IEEE J. Lightwave Technol. 7, 1285-1294 (1998).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

L. Chen, O. Chen, S. Hadjifaradji, and X. Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16,206-208 (2004).
[CrossRef]

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

A. Eyal, and M. Tur, "Measurement of polarization mode dispersion in systems having polarization dependent loss or gain," IEEE Photon. Technol. Lett. 9, 1256-1258 (1997).
[CrossRef]

R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999).
[CrossRef]

J. Opt. Soc. Am. A (1)

J.Opt.Soc.Am. (1)

R. Barakat, "Theory of the coherency matrix for light of arbitrary spectral bandwidth," J.Opt.Soc.Am. 53, 317-323 (1963).
[CrossRef]

Opt. Comm. (2)

R. Barakat, "Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory," Opt. Comm. 38, 159-161 (1981).
[CrossRef]

N. Gisin, and B. Huttner, "Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers," Opt. Comm. 142, 119-125 (1997).
[CrossRef]

Opt. Express (1)

Other (2)

Yi Li, and A . Yariv, "Solutions to the dynamical equation of polarization-mode dispersion and polarization-dependent losses," J. Opt. Soc. Am. B 17,1821-1827 (2000).
[CrossRef]

A. Bessa dos Santos, and J. P. von der weid, "PDL effects in PMD emulators made out with HiBi fibers: Building PMD/PDL emulators," IEEE Photon. Technol. Lett. 16,452-454 (2004).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Experimental configuration for PMD measurement in an optical fiber system.

Fig. 2.
Fig. 2.

DGD (a-d) and DAS (e-h) measurement results calculated by GMMM (blue lines) and the generalized Poincaré sphere method (red lines) with different wavelength step sizes.

Fig. 3.
Fig. 3.

Measured PDL at different wavelength in the fiber system under test.

Equations (13)

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( s out 0 s out 1 s out 2 s out 3 ) = M ( s in 0 s in 1 s in 2 s in 3 ) = ( m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 ) ( s in 0 s in 1 s in 2 s in 3 )
M 1 = 1 det M GM T G = ( m 11 m 21 m 31 m 41 m 12 m 22 m 32 m 42 m 13 m 23 m 33 m 43 m 14 m 24 m 34 m 44 ) det M
M 1 = M * det M = ( M 11 M 21 M 31 M 41 M 12 M 22 M 32 M 42 M 13 M 23 M 33 M 43 M 14 M 24 M 34 M 44 ) det M
m 14 = det ( m 21 m 22 m 23 m 31 m 32 m 33 m 41 m 42 m 43 ) det M , m 24 = det ( m 11 m 12 m 13 m 31 m 32 m 33 m 41 m 42 m 43 ) det M
m 34 = det ( m 11 m 12 m 13 m 21 m 22 m 23 m 41 m 42 m 43 ) det M , m 44 = det ( m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 ) det M
S ( ω + Δ ω ) = M ( ω + Δ ω ) M 1 ( ω ) S ( ω ) = M Δ S ( ω )
M Δ = t 0 ( 1 0 T 0 m R ) ( 1 D T D m D ) = t 0 ( 1 D T m R D m R m D )
m R = ( cos ϕ + r 1 2 ( 1 cos ϕ ) r 1 r 2 ( 1 cos ϕ ) + r 3 sin ϕ r 1 r 3 ( 1 cos ϕ ) r 2 sin ϕ r 1 r 2 ( 1 cos ϕ ) r 3 sin ϕ cos ϕ + r 2 2 ( 1 cos ϕ ) r 2 r 3 ( 1 cos ϕ ) + r 1 sin ϕ r 1 r 3 ( 1 cos ϕ ) + r 2 sin ϕ r 2 r 3 ( 1 cos ϕ ) r 1 sin ϕ cos ϕ + r 3 2 ( 1 cos ϕ ) )
m D = ( 1 D 2 + ( 1 1 D 2 ) d 1 2 ( 1 1 D 2 ) d 1 d 2 ( 1 1 D 2 ) d 1 d 3 ( 1 1 D 2 ) d 1 d 2 1 D 2 + ( 1 1 D 2 ) d 2 2 ( 1 1 D 2 ) d 2 d 3 ( 1 1 D 2 ) d 1 d 3 ( 1 1 D 2 ) d 2 d 3 1 D 2 + ( 1 1 D 2 ) d 3 2 )
d M d ω M 1 = lim Δ ω 0 M Δ I Δ ω = ( η ω Λ 1 Λ 2 Λ 3 Λ 1 η ω Ω 3 Ω 2 Λ 2 Ω 3 η ω Ω 1 Λ 3 Ω 2 Ω 1 η ω )
Λ = t 0 D Δ ω , Ω = ϕ r ̂ Δ ω
DGD = Re ( W · W ) , DAS = Im ( W · W )
PDL = 10 log m 11 + m 12 2 + m 13 2 + m 14 2 m 11 m 12 2 + m 13 2 + m 14 2

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