Abstract

In this paper, we propose an in-service method to simultaneously monitor both nominal and effective values of differential group delay (DGD) in wavelength-division multiplexing (WDM) optical communication systems, in a per channel basis. The method is based on coherent heterodyne detection of the optical signal. We have demonstrated that the technique is capable to recover nominal DGD values from 0 ps to 90 ps while, at same time, to provide the effective DGD parameter, related to the impairment of optical channels. The relationship between the Q factor and effective DGD was also demonstrated, both numerically and experimentally, for distinct nominal values of DGD inserted on the system, by varying the state of polarization (SOP) of the optical signal at the input of the DGD element.

© 2013 OSA

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References

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    [CrossRef] [PubMed]
  7. B. J. Soller, “Second-order PMD in optical components,” in Luna Technologies, May 13rd, 2005, http://lunainc.com/wp-content/uploads/2012/08/2nd-Order-PMD.pdf , last access: 2012, October 29th.
  8. R. Hui and M. O'Sullivan, Fiber optic measurement techniques (Elsevier Academic Press, 2009).
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    [CrossRef]
  11. J.-P. Von der Weid, L. Thenenaz, and J.-P. Pelleaux, “Interferometer measurements of chromatic dispersion and polarization mode dispersion in highly birefringent single mode fibers,” Electron. Lett.23(4), 151–152 (1987), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4257386 .
    [CrossRef]
  12. P. Hernday, “Dispersion measurements,” in Fiber-optic test and measurement, Denis Derickson ed., (Prentice Hall, 1998).
  13. B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Tech. Lett. 4(9), 1066–1069 (1992), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=157151 .
  14. B. L. Heffner, “Accurate, automated measurement of differential group delay dispersion and principal state variation using Jones matrix eigenanalysis,” IEEE Photon. Tech. Lett. 5(7), 814–817 (1993), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=229816 .
  15. M. Sköld, B. E. Olsson, M. Karlsson, and P. A. Andrekson, “Low-cost multiparameter optical performance monitoring based on polarization modulation,” J. Lightwave Technol.27(2), 128–138 (2009), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-27-2-128 .
    [CrossRef]
  16. C. Floridia, G. C. C. P. Simões, E. W. Bezerra, M. M. Feres, and M. A. Romero, “Simplified approach to low-cost multiparameter monitoring based on low frequency polarization modulation,” Microw. Optic. Technol. Lett. 54(8), 1820–1824 (2012), http://onlinelibrary.wiley.com/doi/10.1002/mop.26956/full .
    [CrossRef]
  17. G.-W. Lu, M.-H. Cheung, L.-K. Chen, and C.-K. Chan, “Simple PMD-insensitive OSNR monitoring scheme assisted by transmitter-side polarization scrambling,” Opt. Express14(1), 58–62 (2006).
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    [CrossRef] [PubMed]
  19. B. Fu and R. Hui, “Fiber chromatic dispersion and polarization-mode dispersion monitoring using coherent detection,” IEEE Photon. Tech. Lett. 17(7), 1561–1563 (2005), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1453677 .
  20. R. Hui, R. Saunders, B. Heffner, D. Richards, B. Fu, and P. Adany, “Non-blocking PMD monitoring in live optical systems,” Electron. Lett.43(1), 53–54 (2007), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4068491 .
    [CrossRef]
  21. J. Jiang, S. Sundhararajan, D. Richards, S. Oliva, and R. Hui, “PMD monitoring in traffic-carrying optical systems and its statistical analysis,” Opt. Express16(18), 14057–14063 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14057 .
    [CrossRef] [PubMed]
  22. J. Jiang, D. Richards, S. Oliva, and R. Hui, “PMD and PDL monitoring of traffic-carrying transatlantic fibre-optic system,” Electron. Lett.45(2), 123–124 (2009), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4752656 .
    [CrossRef]
  23. J. Jiang, D. Richards, S. Oliva, P. Green, and R. Hui, “In-situ monitoring of PMD and PDL in a traffic-carrying transatlantic fiber-optic system,” in National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper NMB1, http://www.opticsinfobase.org/abstract.cfm?URI=NFOEC-2009-NMB1 .
  24. http://mathforum.org/library/drmath/view/63765.htm , last access: 2010, September 10th.
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    [CrossRef]
  26. S. Betti, F. Curti, B. Daino, G. De Marchis, E. Lannone, and F. Matera, “Evolution of the bandwidth of the principal states of polarization in single-mode fibers,” Opt. Lett.16(7), 467–469 (1991).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  28. M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett.24(14), 939–941 (1999).
    [CrossRef] [PubMed]

2012 (1)

2009 (2)

M. Sköld, B. E. Olsson, M. Karlsson, and P. A. Andrekson, “Low-cost multiparameter optical performance monitoring based on polarization modulation,” J. Lightwave Technol.27(2), 128–138 (2009), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-27-2-128 .
[CrossRef]

J. Jiang, D. Richards, S. Oliva, and R. Hui, “PMD and PDL monitoring of traffic-carrying transatlantic fibre-optic system,” Electron. Lett.45(2), 123–124 (2009), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4752656 .
[CrossRef]

2008 (1)

2007 (1)

R. Hui, R. Saunders, B. Heffner, D. Richards, B. Fu, and P. Adany, “Non-blocking PMD monitoring in live optical systems,” Electron. Lett.43(1), 53–54 (2007), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4068491 .
[CrossRef]

2006 (1)

2004 (1)

2000 (2)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A.97(9), 4541–4550 (2000), http://www.pnas.org/content/97/9/4541.full.pdf+html (PNAS).
[CrossRef] [PubMed]

H. Kogelnik, L. E. Nelson, J. P. Gordon, and R. M. Jopson, “Jones matrix for second-order polarization mode dispersion,” Opt. Lett.25(1), 19–21 (2000).
[CrossRef] [PubMed]

1999 (2)

R. M. Jopson, L. E. Nelson, and H. Kogelnik, “Measurement of second-order polarization-mode dispersion vectors in optical fibers,” IEEE Photon. Tech. Lett. 11(9), 1153–1155 (1999), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=784234 .

M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett.24(14), 939–941 (1999).
[CrossRef] [PubMed]

1998 (1)

1993 (1)

B. L. Heffner, “Accurate, automated measurement of differential group delay dispersion and principal state variation using Jones matrix eigenanalysis,” IEEE Photon. Tech. Lett. 5(7), 814–817 (1993), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=229816 .

1992 (1)

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Tech. Lett. 4(9), 1066–1069 (1992), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=157151 .

1991 (2)

N. Gisin, J.-P. Von der Weid, and J.-P. Pelleaux, “Polarization mode dispersion of short and long single-mode fibers,” J. Lightwave Technol.9(7), 821–827 (1991), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=85780 .
[CrossRef]

S. Betti, F. Curti, B. Daino, G. De Marchis, E. Lannone, and F. Matera, “Evolution of the bandwidth of the principal states of polarization in single-mode fibers,” Opt. Lett.16(7), 467–469 (1991).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

J.-P. Von der Weid, L. Thenenaz, and J.-P. Pelleaux, “Interferometer measurements of chromatic dispersion and polarization mode dispersion in highly birefringent single mode fibers,” Electron. Lett.23(4), 151–152 (1987), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4257386 .
[CrossRef]

Adany, P.

R. Hui, R. Saunders, B. Heffner, D. Richards, B. Fu, and P. Adany, “Non-blocking PMD monitoring in live optical systems,” Electron. Lett.43(1), 53–54 (2007), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4068491 .
[CrossRef]

Andrekson, P. A.

Aso, O.

Betti, S.

Brentel, J.

Chan, C.-K.

Chen, L.-K.

Cheung, M.-H.

Chowdhury, D. Q.

Curti, F.

Daino, B.

De Marchis, G.

Feres, M. M.

Floridia, C.

Fu, B.

R. Hui, R. Saunders, B. Heffner, D. Richards, B. Fu, and P. Adany, “Non-blocking PMD monitoring in live optical systems,” Electron. Lett.43(1), 53–54 (2007), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4068491 .
[CrossRef]

Giles, C. R.

Gisin, N.

N. Gisin, J.-P. Von der Weid, and J.-P. Pelleaux, “Polarization mode dispersion of short and long single-mode fibers,” J. Lightwave Technol.9(7), 821–827 (1991), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=85780 .
[CrossRef]

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A.97(9), 4541–4550 (2000), http://www.pnas.org/content/97/9/4541.full.pdf+html (PNAS).
[CrossRef] [PubMed]

H. Kogelnik, L. E. Nelson, J. P. Gordon, and R. M. Jopson, “Jones matrix for second-order polarization mode dispersion,” Opt. Lett.25(1), 19–21 (2000).
[CrossRef] [PubMed]

Heffner, B.

R. Hui, R. Saunders, B. Heffner, D. Richards, B. Fu, and P. Adany, “Non-blocking PMD monitoring in live optical systems,” Electron. Lett.43(1), 53–54 (2007), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4068491 .
[CrossRef]

Heffner, B. L.

B. L. Heffner, “Accurate, automated measurement of differential group delay dispersion and principal state variation using Jones matrix eigenanalysis,” IEEE Photon. Tech. Lett. 5(7), 814–817 (1993), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=229816 .

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Tech. Lett. 4(9), 1066–1069 (1992), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=157151 .

Hui, R.

J. Jiang, D. Richards, S. Oliva, and R. Hui, “PMD and PDL monitoring of traffic-carrying transatlantic fibre-optic system,” Electron. Lett.45(2), 123–124 (2009), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4752656 .
[CrossRef]

J. Jiang, S. Sundhararajan, D. Richards, S. Oliva, and R. Hui, “PMD monitoring in traffic-carrying optical systems and its statistical analysis,” Opt. Express16(18), 14057–14063 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14057 .
[CrossRef] [PubMed]

R. Hui, R. Saunders, B. Heffner, D. Richards, B. Fu, and P. Adany, “Non-blocking PMD monitoring in live optical systems,” Electron. Lett.43(1), 53–54 (2007), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4068491 .
[CrossRef]

Jiang, J.

J. Jiang, D. Richards, S. Oliva, and R. Hui, “PMD and PDL monitoring of traffic-carrying transatlantic fibre-optic system,” Electron. Lett.45(2), 123–124 (2009), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4752656 .
[CrossRef]

J. Jiang, S. Sundhararajan, D. Richards, S. Oliva, and R. Hui, “PMD monitoring in traffic-carrying optical systems and its statistical analysis,” Opt. Express16(18), 14057–14063 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14057 .
[CrossRef] [PubMed]

Jopson, R. M.

H. Kogelnik, L. E. Nelson, J. P. Gordon, and R. M. Jopson, “Jones matrix for second-order polarization mode dispersion,” Opt. Lett.25(1), 19–21 (2000).
[CrossRef] [PubMed]

R. M. Jopson, L. E. Nelson, and H. Kogelnik, “Measurement of second-order polarization-mode dispersion vectors in optical fibers,” IEEE Photon. Tech. Lett. 11(9), 1153–1155 (1999), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=784234 .

Karlsson, M.

Kogelnik, H.

H. Kogelnik, L. E. Nelson, J. P. Gordon, and R. M. Jopson, “Jones matrix for second-order polarization mode dispersion,” Opt. Lett.25(1), 19–21 (2000).
[CrossRef] [PubMed]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A.97(9), 4541–4550 (2000), http://www.pnas.org/content/97/9/4541.full.pdf+html (PNAS).
[CrossRef] [PubMed]

R. M. Jopson, L. E. Nelson, and H. Kogelnik, “Measurement of second-order polarization-mode dispersion vectors in optical fibers,” IEEE Photon. Tech. Lett. 11(9), 1153–1155 (1999), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=784234 .

Lannone, E.

Lu, G.-W.

Matera, F.

Mauro, Y.

Mlejnek, M.

Nelson, L. E.

H. Kogelnik, L. E. Nelson, J. P. Gordon, and R. M. Jopson, “Jones matrix for second-order polarization mode dispersion,” Opt. Lett.25(1), 19–21 (2000).
[CrossRef] [PubMed]

R. M. Jopson, L. E. Nelson, and H. Kogelnik, “Measurement of second-order polarization-mode dispersion vectors in optical fibers,” IEEE Photon. Tech. Lett. 11(9), 1153–1155 (1999), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=784234 .

Oliva, S.

J. Jiang, D. Richards, S. Oliva, and R. Hui, “PMD and PDL monitoring of traffic-carrying transatlantic fibre-optic system,” Electron. Lett.45(2), 123–124 (2009), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4752656 .
[CrossRef]

J. Jiang, S. Sundhararajan, D. Richards, S. Oliva, and R. Hui, “PMD monitoring in traffic-carrying optical systems and its statistical analysis,” Opt. Express16(18), 14057–14063 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14057 .
[CrossRef] [PubMed]

Olsson, B. E.

Pelleaux, J.-P.

N. Gisin, J.-P. Von der Weid, and J.-P. Pelleaux, “Polarization mode dispersion of short and long single-mode fibers,” J. Lightwave Technol.9(7), 821–827 (1991), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=85780 .
[CrossRef]

J.-P. Von der Weid, L. Thenenaz, and J.-P. Pelleaux, “Interferometer measurements of chromatic dispersion and polarization mode dispersion in highly birefringent single mode fibers,” Electron. Lett.23(4), 151–152 (1987), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4257386 .
[CrossRef]

Piech, G. A.

Poole, C. D.

Richards, D.

J. Jiang, D. Richards, S. Oliva, and R. Hui, “PMD and PDL monitoring of traffic-carrying transatlantic fibre-optic system,” Electron. Lett.45(2), 123–124 (2009), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4752656 .
[CrossRef]

J. Jiang, S. Sundhararajan, D. Richards, S. Oliva, and R. Hui, “PMD monitoring in traffic-carrying optical systems and its statistical analysis,” Opt. Express16(18), 14057–14063 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14057 .
[CrossRef] [PubMed]

R. Hui, R. Saunders, B. Heffner, D. Richards, B. Fu, and P. Adany, “Non-blocking PMD monitoring in live optical systems,” Electron. Lett.43(1), 53–54 (2007), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4068491 .
[CrossRef]

Romero, M. A.

Roudas, I.

Saunders, R.

R. Hui, R. Saunders, B. Heffner, D. Richards, B. Fu, and P. Adany, “Non-blocking PMD monitoring in live optical systems,” Electron. Lett.43(1), 53–54 (2007), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4068491 .
[CrossRef]

Simões, G. C. C. P.

Sköld, M.

Sundhararajan, S.

Thenenaz, L.

J.-P. Von der Weid, L. Thenenaz, and J.-P. Pelleaux, “Interferometer measurements of chromatic dispersion and polarization mode dispersion in highly birefringent single mode fibers,” Electron. Lett.23(4), 151–152 (1987), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4257386 .
[CrossRef]

Vasilyev, M.

Von der Weid, J.-P.

N. Gisin, J.-P. Von der Weid, and J.-P. Pelleaux, “Polarization mode dispersion of short and long single-mode fibers,” J. Lightwave Technol.9(7), 821–827 (1991), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=85780 .
[CrossRef]

J.-P. Von der Weid, L. Thenenaz, and J.-P. Pelleaux, “Interferometer measurements of chromatic dispersion and polarization mode dispersion in highly birefringent single mode fibers,” Electron. Lett.23(4), 151–152 (1987), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4257386 .
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (3)

R. Hui, R. Saunders, B. Heffner, D. Richards, B. Fu, and P. Adany, “Non-blocking PMD monitoring in live optical systems,” Electron. Lett.43(1), 53–54 (2007), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4068491 .
[CrossRef]

J. Jiang, D. Richards, S. Oliva, and R. Hui, “PMD and PDL monitoring of traffic-carrying transatlantic fibre-optic system,” Electron. Lett.45(2), 123–124 (2009), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4752656 .
[CrossRef]

J.-P. Von der Weid, L. Thenenaz, and J.-P. Pelleaux, “Interferometer measurements of chromatic dispersion and polarization mode dispersion in highly birefringent single mode fibers,” Electron. Lett.23(4), 151–152 (1987), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4257386 .
[CrossRef]

J. Lightwave Technol. (3)

Opt. Express (2)

Opt. Lett. (5)

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(9), 4541–4550 (2000), http://www.pnas.org/content/97/9/4541.full.pdf+html (PNAS).
[CrossRef] [PubMed]

Other (13)

C. R. Menyuk and A. Galtarossa, Polarization mode dispersion: Optical and fiber communications reports (Springer Science + Business Media, Inc., 2005).

C. Yu, “Polarization mode dispersion monitoring,” in Optical performance monitoring: advanced techniques for next-generation photonic networks, Calvin C. K. Chan ed., (Elsevier Academic Press, 2010), Chap. 4.

R. M. Jopson, L. E. Nelson, and H. Kogelnik, “Measurement of second-order polarization-mode dispersion vectors in optical fibers,” IEEE Photon. Tech. Lett. 11(9), 1153–1155 (1999), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=784234 .

F. Heismann, “Polarization mode dispersion: Fundamentals and impact on optical communications systems,” in Proceedings of IEEE 24th European Conference on Optical Communication (Institute of Electrical and Electronics Engineers, Madrid, 1998), pp. 51–79, http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=729518 .

B. J. Soller, “Second-order PMD in optical components,” in Luna Technologies, May 13rd, 2005, http://lunainc.com/wp-content/uploads/2012/08/2nd-Order-PMD.pdf , last access: 2012, October 29th.

R. Hui and M. O'Sullivan, Fiber optic measurement techniques (Elsevier Academic Press, 2009).

C. Floridia, G. C. C. P. Simões, E. W. Bezerra, M. M. Feres, and M. A. Romero, “Simplified approach to low-cost multiparameter monitoring based on low frequency polarization modulation,” Microw. Optic. Technol. Lett. 54(8), 1820–1824 (2012), http://onlinelibrary.wiley.com/doi/10.1002/mop.26956/full .
[CrossRef]

P. Hernday, “Dispersion measurements,” in Fiber-optic test and measurement, Denis Derickson ed., (Prentice Hall, 1998).

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Tech. Lett. 4(9), 1066–1069 (1992), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=157151 .

B. L. Heffner, “Accurate, automated measurement of differential group delay dispersion and principal state variation using Jones matrix eigenanalysis,” IEEE Photon. Tech. Lett. 5(7), 814–817 (1993), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=229816 .

B. Fu and R. Hui, “Fiber chromatic dispersion and polarization-mode dispersion monitoring using coherent detection,” IEEE Photon. Tech. Lett. 17(7), 1561–1563 (2005), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1453677 .

J. Jiang, D. Richards, S. Oliva, P. Green, and R. Hui, “In-situ monitoring of PMD and PDL in a traffic-carrying transatlantic fiber-optic system,” in National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper NMB1, http://www.opticsinfobase.org/abstract.cfm?URI=NFOEC-2009-NMB1 .

http://mathforum.org/library/drmath/view/63765.htm , last access: 2010, September 10th.

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Figures (7)

Fig. 1
Fig. 1

(a) Our proposed PMD monitor block diagram and (b) in the inset: the PMD monitor presented in [20,21]. PD: photodetector, RF: radiofrequency, A/D: analog-to-digital.

Fig. 2
Fig. 2

Signal spectrum after the coherent heterodyne detection. S(f) is the signal spectral power density; fIF is the intermediate RF frequency, given by the difference between the carrier frequency, fC, and the local oscillator frequency, fLO; f1, f2, f3 and f4 are the central frequencies of each narrow bandpass filter used to slice the signal spectrum after the coherent detection.

Fig. 3
Fig. 3

Poincarè sphere depicting the Stokes vectors for each narrowband spectrum slice centered in frequencies f1, f2, f3 and f4, and its respective projections on the least squares Π plane. The Stokes vectors are given by S1, S2, S3 and S4, represented by the dot dashed lines. Projections V1, V2, V3 and V4 of those Stokes vectors onto the least squares Π plane are represented by the continuous lines. Ω PSP represents the optical fiber principal state of polarization vector, which is collinear to the Π plane normal vector.

Fig. 4
Fig. 4

Experimental setup composed by a 10 Gb/s optical pattern generator, a manual polarization controller (PC), a DGD module to emulate different DGD values, a 90/10 splitter and an oscilloscope.

Fig. 5
Fig. 5

Poincarè spheres containing the set of experimentally obtained Stokes vectors for different values of DGD inserted in the system under test: (a) 15 ps, (b) 25 ps, (c) 45 ps and (d) 90 ps. In these figures, we present intermediate states of polarization (SOPs) at the input of DGD module, to allow a better view of the Stokes vectors and their projections on the least squares plan as Fig. 3. Stokes vectors for each narrowband spectrum slice, centered in frequencies f1 (blue), f2 (magenta), f3 (green) and f4 (red) are presented in dot-dashed lines and their respectively projections on the least squares Π plane are present in continuous lines. The PSP vector Ω PSP and the least square Π plan can be also seen in each plot.

Fig. 6
Fig. 6

Experimental results for different SOPs in the input of the DGD module device. (a) for 15 ps, (b) for 25 ps, (c) for 70 ps and (d) for 90 ps of DGD. Blue solid lines show the value set in the DGD module device; red dot dashed lines show the value of the effective DGD, evaluated through the technique proposed in [8,16,17]; green dashed lines show the nominal DGD measured through the technique proposed in this paper; and the violet solid lines show the electrical SNR behavior, measured through the oscilloscope eye diagram.

Fig. 7
Fig. 7

Comparison of simulation and the experimental results concerning the relationship between Q factor and effective DGD behavior.

Equations (29)

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P E ¯ ( f i )= P S P LO | J S J LO | 2 ,
P i = η i P S ( f i ) P LO | J S ( f i ) J LO | 2 ,
| J S ( f i ) J LO | 2 = 1 2 ( 1+ S S ( f i ) S LO ),
P i = η i P s ( f i ) P LO 1 2 ( 1+ S S ( f i ) S LO ),
P i = η i 2 P S ( f i ) P LO ( 1+ S S,x ( f i ) S LO,x + S S,y ( f i ) S LO,y + S S,z ( f i ) S LO,z ),
s S,x ( f i )= P i (0) P i (1) ( P i (0) P i (1) ) 2 + ( P i (2) P i (3) ) 2 + ( P i (4) P i (5) ) 2 s S,y ( f i )= P i (2) P i (3) ( P i (0) P i (1) ) 2 + ( P i (2) P i (3) ) 2 + ( P i (4) P i (5) ) 2 , s S,z ( f i )= P i (4) P i (5) ( P i (0) P i (1) ) 2 + ( P i (2) P i (3) ) 2 + ( P i (4) P i (5) ) 2
τ= Δθ 2πΔf ,
DG D eff =DG D nom sin(2φ),
Ω (ω)=τ(ω) q ^ (ω),
d Ω (ω) dω = dτ(ω) dω q ^ (ω)+τ(ω) d q ^ (ω) dω .
Δ f PSP τ ¯ =k,
SOPMD=τ(ω) d q ^ (ω) dω = τ 1 + τ 2 2 q ^ 2 q ^ 1 Δω = τ 1 + τ 2 2 q ^ 2 q ^ 1 2πΔ f 21 ,
P i (0) = η i 2 P S ( f i ) P LO ( 1+ S S,x ( f i ) S LO,x (0) + S S,y ( f i ) S LO,y (0) + S S,z ( f i ) S LO,z (0) ),
P i (1) = η i 2 P S ( f i ) P LO ( 1 S S,x ( f i ) S LO,x (0) S S,y ( f i ) S LO,y (0) S S,z ( f i ) S LO,z (0) ).
η i 2 P S ( f i ) P LO = P i (0) + P i (1) 2 .
η i P S ( f i ) P LO 2 = P i (0) + P i (1) 2 = P i (2) + P i (3) 2 = P i (4) + P i (5) 2 .
P i (0) = η i 2 P S ( f i ) P LO ( 1+ S S,x ( f i ) S LO,x (0) + S S,y ( f i ) S LO,y (0) + S S,z ( f i ) S LO,z (0) ),
P i (2) = η i 2 P S ( f i ) P LO ( 1+ S S,x ( f i ) S LO,x (2) + S S,y ( f i ) S LO,y (2) + S S,z ( f i ) S LO,z (2) ),
P i (4) = η i 2 P S ( f i ) P LO ( 1+ S S,x ( f i ) S LO,x (4) + S S,y ( f i ) S LO,y (4) + S S,z ( f i ) S LO,z (4) ).
( S LO,x (0) S LO,y (0) S LO,z (0) S LO,x (2) S LO,y (2) S LO,z (2) S LO,x (4) S LO,y (4) S LO,z (4) )( S S,x ( f i ) S S,y ( f i ) S S,z ( f i ) )= 2 η i P S ( f i ) P LO ( P i (0) P i (2) P i (4) )1( 1 1 1 ).
S LO (0) =( S LO,x (0) S LO,y (0) S LO,z (0) )=( 1 0 0 ),
S LO (2) =( S LO,x (2) S LO,y (2) S LO,z (2) )=( 0 1 0 ),
S LO (4) =( S LO,x (4) S LO,y (4) S LO,z (4) )=( 0 0 1 ).
( S S,x ( f i ) S S,y ( f i ) S S,z ( f i ) )= 2 η i P S ( f i ) P LO ( P i (0) P i (2) P i (4) )( 1 1 1 ).
S S,x ( f i )= 2 η i P S ( f i ) P LO P i (0) 1 S S,y ( f i )= 2 η i P S ( f i ) P LO P i (2) 1. S S,z ( f i )= 2 η i P S ( f i ) P LO P i (4) 1
S S,x ( f i )= 2 P i (0) + P i (1) P i (0) 1 S S,x ( f i )= P i (0) P i (1) P i (0) + P i (1) S S,y ( f i )= 2 P i (2) + P i (3) P i (2) 1 S S,y ( f i )= P i (2) P i (3) P i (2) + P i (3) . S S,z ( f i )= 2 P i (4) + P i (5) P i (4) 1 S S,z ( f i )= P i (4) P i (5) P i (4) + P i (5)
S S,0 ( f i )= ( S S,x ( f i ) ) 2 + ( S S,y ( f i ) ) 2 + ( S S,z ( f i ) ) 2 = S S,0 ( f i )= ( P i (0) P i (1) P i (0) + P i (1) ) 2 + ( P i (2) P i (3) P i (2) + P i (3) ) 2 + ( P i (4) P i (5) P i (4) + P i (5) ) 2 ,
S S,0 ( f i )= ( P i (0) P i (1) ) 2 + ( P i (2) P i (3) ) 2 + ( P i (4) P i (5) ) 2 ( P i (0) + P i (1) ) .
s S,x ( f i )= P i (0) P i (1) ( P i (0) P i (1) ) 2 + ( P i (2) P i (3) ) 2 + ( P i (4) P i (5) ) 2 s S,y ( f i )= P i (2) P i (3) ( P i (0) P i (1) ) 2 + ( P i (2) P i (3) ) 2 + ( P i (4) P i (5) ) 2 . s S,z ( f i )= P i (4) P i (5) ( P i (0) P i (1) ) 2 + ( P i (2) P i (3) ) 2 + ( P i (4) P i (5) ) 2

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