Abstract

In this paper, dispersion analysis of optical components and systems is presented using a formalism based on the elementary matrices and the $N$-matrix, first described by Jones. This approach readily incorporates both phase and amplitude dispersion in a generalized dispersion framework. The method simplifies the analysis of the combined effects of group delay, differential group delay, amplitude slope, and differential amplitude slope as compared to traditional Jones matrix methods. Higher order polarization-mode dispersion and the effects of concatenation are presented along with a discussion of measurement principles. The application of the elementary matrix concept to Mueller matrix methods in Stokes space is also discussed.

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  1. C. D. Poole, T. E. Darcie, "Distortion related to polarization-mode dispersion in analog lightwave systems," J. Lightw. Technol. 11, 1749-1759 (1993).
  2. N. Gisin, J. P. Pellaux, "Polarization mode dispersion: Time versus frequency domain," Opt. Commun. 89, 316-323 (1992).
  3. E. Iannone, F. Matera, A. Galtarossa, G. Gianello, M. Schiano, "Effect of polarization dispersion on the performance in IM-DD communication systems," Photon. Technol. Lett. 5, 1247-1249 (1993).
  4. H. Kogelnik, L. E. Nelson, R. M. Jopson, "Polarization mode dispersion impairment in lightwave transmission systems," Proc. Opt. Fiber Commun. Conf. (OFC 2002) pp. 194.
  5. P. K. Kondamuri, C. Allen, D. L. Richards, "Variation of PMD-induced outage rates and durations with link length on buried standard single-mode fibers," Proc. Opt. Fiber Commun. Conf. (2005) pp. 1-3.
  6. F. Bruyère, "Impact of first- and second-order PMD in optical digital transmission systems," Opt. Fiber Technol. 2, 269-280 (1996).
  7. D. Wang, C. R. Menyuk, "Calculation of penalties due to polarization effects in a long-haul WDM system using a stokes parameter model," J. Lightw. Technol. 19, 487-494 (2001).
  8. N. Gisin, B. Huttner, "Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers," Opt. Commun. 142, 119-125 (1997).
  9. B. Huttner, C. Geiser, N. Gisin, "Polarization-induced distortions in optical networks with polarization-mode dispersion and polarization-dependent losses," IEEE J. Sel. Topics Quantum Electron. 6, 317-329 (2000).
  10. V. Chernyak, M. Chertkov, I. Gabitov, I. Kolokolov, V. Lebedev, "PMD-induced fluctuations of bit-error rate in optical fiber systems," J. Lightw. Technol. 22, 1155-1168 (2004).
  11. J. P. Gordon, H. Kogelnik, "PMD fundamentals: Polarization mode dispersion in optical fibers," Proc. Nat. Acad. Sci. USA (2000) pp. 4541-4550.
  12. P. Williams, "PMD measurement techniques and how to avoid the pitfalls," J. Opt. Fiber Commun. 1, 84-105 (2004) Rep. 1.
  13. H. Kogelnik, L. E. Nelson, J. P. Gordon, "The inverse PMD problem," J. Opt. Fiber Commun. 1, 107-122 (2004) Rep. 1.
  14. R. C. Jones, "A new calculus for the treatment of optical systems I. Description and discussion of the calculus," J. Opt. Soc. Amer. 31, 488-493 (1941).
  15. H. Hurwitz Jr., R. C. Jones, "A new calculus for the treatment of optical systems II. Proof of the three general equivalence theorems," J. Opt. Soc. Amer. 31, 493-499 (1941).
  16. R. C. Jones, "A new calculus for the treatment of optical systems III. The sohncke theory of optical activity," J. Opt. Soc. Amer. 31, 500-503 (1941).
  17. R. C. Jones, "A new calculus for the treatment of optical systems, IV," J. Opt. Soc. Amer. 32, 486-493 (1942).
  18. R. C. Jones, "A new calculus for the treatment of optical systems, V. A more general formulation and description of another calculus," J. Opt. Soc. Amer. 37, 107-110 (1947).
  19. R. C. Jones, "A new calculus for the treatment of optical systems, VI. Experimental determination or the matrix," J. Opt. Soc. Amer. 37, 110-112 (1947).
  20. R. C. Jones, "A new calculus for the treatment of optical systems. VII properties of the N-matrices," J. Opt. Soc. Amer. 38, 671-685 (1948).
  21. B. L. Heffner, "Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis," IEEE Photon. Technol. Lett. 4, 1066-1069 (1992).
  22. A. E. Willner, S. M. R. M. Nezam, L. Yan, Z. Pan, M. C. Hauer, "Monitoring and control of polarization-related impairments in optical fiber systems," J. Lightw. Technol. 22, 106-125 (2004).
  23. D. Penninckx, V. Morénas, "Jones matrix of polarization mode dispersion," Opt. Lett. 24, 875-877 (1999).
  24. E. Forestieri, L. Vincetti, "Exact evaluation of the jones matrix of a fiber in the presence of polarization mode dispersion of any order," J. Lightw. Technol. 19, 1898-1909 (2001).
  25. H. Kogelnik, L. E. Nelson, J. P. Gordon, R. M. Jopson, "Jones matrix for second order polarization mode dispersion," Opt. Lett. 25, 19-21 (2000).
  26. G. Arfken, Mathematical Methods for Physicists (Academic, 1985).
  27. C. Whitney, "Pauli-algebraic operators in polarization optics," J. Opt. Soc. Amer. 61, 1207-1213 (1971).
  28. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
  29. G. N. Ramachandran, S. Ramaseshan, Crystal Optics in Handbuch der Physik (Springer-Verlag, 1961).
  30. T. Do, H. Dong, P. Shum, "Measurement of the complex polarization mode dispersion vector using Jones matrix analysis," Appl. Opt. 46, 916-921 (2007).
  31. M. Reimer, D. Yevick, D. Dumas, "The accuracy of the magnus expansion for polarization-mode dispersion and polarization-dependent loss," J. Lightw. Technol. 16, 3337-3344 (2008).
  32. P. A. Williams, J. D. Kofler, "Narrow-band measurement of differential group delay by a six-state RF phase-shift technique: 40 fs single-measurement uncertainty," J. Lightw. Technol. 22, 448-456 (2004).
  33. P. A. Williams, "Modulation phase-shift measurements of PMD using only four launched polarization states: A new algorithm," Electron. Lett. 34, 1578-1579 (1999).
  34. H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, C. Q. Wu, "Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain," Opt. Exp. 14, 5067-5072 (2006).
  35. H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, C. Q. Wu, "Virtual generalized mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers," Photon. Technol. Lett. 19, 27-29 (2007).
  36. L. Chen, O. Chen, S. Hadjifaradji, X. Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," Photon. Technol. Lett. 16, 206-208 (2004).
  37. B. Szafraniec, D. M. Baney, "Elementary matrix-based vector optical network analysis," J. Lightw. Technol. 25, 1061-1069 (2007).
  38. B. Szafraniec, R. Mästle, D. M. Baney, "Polarization mode dispersion measurement based on continuous polarization modulation," Appl. Opt. 47, 1-9 (2008).
  39. J. W. Hovenier, "Structure of a general pure Mueller matrix," Appl. Opt. 33, 8318-8324 (1994).
  40. R. M. A. Azzam, "Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 44 matrix calculus," J. Opt. Soc. Amer. B, Opt. Phys. 68, 1756-1767 (1978).
  41. D. M. Baney, "Complex component characterization based on elementary matrices," Proc. Optical Fiber Communications Conf. (OFC 2009) (2009).

2008 (2)

M. Reimer, D. Yevick, D. Dumas, "The accuracy of the magnus expansion for polarization-mode dispersion and polarization-dependent loss," J. Lightw. Technol. 16, 3337-3344 (2008).

B. Szafraniec, R. Mästle, D. M. Baney, "Polarization mode dispersion measurement based on continuous polarization modulation," Appl. Opt. 47, 1-9 (2008).

2007 (3)

H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, C. Q. Wu, "Virtual generalized mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers," Photon. Technol. Lett. 19, 27-29 (2007).

B. Szafraniec, D. M. Baney, "Elementary matrix-based vector optical network analysis," J. Lightw. Technol. 25, 1061-1069 (2007).

T. Do, H. Dong, P. Shum, "Measurement of the complex polarization mode dispersion vector using Jones matrix analysis," Appl. Opt. 46, 916-921 (2007).

2006 (1)

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, C. Q. Wu, "Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain," Opt. Exp. 14, 5067-5072 (2006).

2004 (6)

P. A. Williams, J. D. Kofler, "Narrow-band measurement of differential group delay by a six-state RF phase-shift technique: 40 fs single-measurement uncertainty," J. Lightw. Technol. 22, 448-456 (2004).

A. E. Willner, S. M. R. M. Nezam, L. Yan, Z. Pan, M. C. Hauer, "Monitoring and control of polarization-related impairments in optical fiber systems," J. Lightw. Technol. 22, 106-125 (2004).

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

V. Chernyak, M. Chertkov, I. Gabitov, I. Kolokolov, V. Lebedev, "PMD-induced fluctuations of bit-error rate in optical fiber systems," J. Lightw. Technol. 22, 1155-1168 (2004).

P. Williams, "PMD measurement techniques and how to avoid the pitfalls," J. Opt. Fiber Commun. 1, 84-105 (2004) Rep. 1.

H. Kogelnik, L. E. Nelson, J. P. Gordon, "The inverse PMD problem," J. Opt. Fiber Commun. 1, 107-122 (2004) Rep. 1.

2001 (2)

D. Wang, C. R. Menyuk, "Calculation of penalties due to polarization effects in a long-haul WDM system using a stokes parameter model," J. Lightw. Technol. 19, 487-494 (2001).

E. Forestieri, L. Vincetti, "Exact evaluation of the jones matrix of a fiber in the presence of polarization mode dispersion of any order," J. Lightw. Technol. 19, 1898-1909 (2001).

2000 (2)

H. Kogelnik, L. E. Nelson, J. P. Gordon, R. M. Jopson, "Jones matrix for second order polarization mode dispersion," Opt. Lett. 25, 19-21 (2000).

B. Huttner, C. Geiser, N. Gisin, "Polarization-induced distortions in optical networks with polarization-mode dispersion and polarization-dependent losses," IEEE J. Sel. Topics Quantum Electron. 6, 317-329 (2000).

1999 (2)

D. Penninckx, V. Morénas, "Jones matrix of polarization mode dispersion," Opt. Lett. 24, 875-877 (1999).

P. A. Williams, "Modulation phase-shift measurements of PMD using only four launched polarization states: A new algorithm," Electron. Lett. 34, 1578-1579 (1999).

1997 (1)

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

1996 (1)

F. Bruyère, "Impact of first- and second-order PMD in optical digital transmission systems," Opt. Fiber Technol. 2, 269-280 (1996).

1994 (1)

1993 (2)

E. Iannone, F. Matera, A. Galtarossa, G. Gianello, M. Schiano, "Effect of polarization dispersion on the performance in IM-DD communication systems," Photon. Technol. Lett. 5, 1247-1249 (1993).

C. D. Poole, T. E. Darcie, "Distortion related to polarization-mode dispersion in analog lightwave systems," J. Lightw. Technol. 11, 1749-1759 (1993).

1992 (2)

N. Gisin, J. P. Pellaux, "Polarization mode dispersion: Time versus frequency domain," Opt. Commun. 89, 316-323 (1992).

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

1978 (1)

R. M. A. Azzam, "Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 44 matrix calculus," J. Opt. Soc. Amer. B, Opt. Phys. 68, 1756-1767 (1978).

1971 (1)

C. Whitney, "Pauli-algebraic operators in polarization optics," J. Opt. Soc. Amer. 61, 1207-1213 (1971).

1948 (1)

R. C. Jones, "A new calculus for the treatment of optical systems. VII properties of the N-matrices," J. Opt. Soc. Amer. 38, 671-685 (1948).

1947 (2)

R. C. Jones, "A new calculus for the treatment of optical systems, V. A more general formulation and description of another calculus," J. Opt. Soc. Amer. 37, 107-110 (1947).

R. C. Jones, "A new calculus for the treatment of optical systems, VI. Experimental determination or the matrix," J. Opt. Soc. Amer. 37, 110-112 (1947).

1942 (1)

R. C. Jones, "A new calculus for the treatment of optical systems, IV," J. Opt. Soc. Amer. 32, 486-493 (1942).

1941 (3)

R. C. Jones, "A new calculus for the treatment of optical systems I. Description and discussion of the calculus," J. Opt. Soc. Amer. 31, 488-493 (1941).

H. Hurwitz Jr., R. C. Jones, "A new calculus for the treatment of optical systems II. Proof of the three general equivalence theorems," J. Opt. Soc. Amer. 31, 493-499 (1941).

R. C. Jones, "A new calculus for the treatment of optical systems III. The sohncke theory of optical activity," J. Opt. Soc. Amer. 31, 500-503 (1941).

Appl. Opt. (3)

Electron. Lett. (1)

P. A. Williams, "Modulation phase-shift measurements of PMD using only four launched polarization states: A new algorithm," Electron. Lett. 34, 1578-1579 (1999).

IEEE J. Sel. Topics Quantum Electron. (1)

B. Huttner, C. Geiser, N. Gisin, "Polarization-induced distortions in optical networks with polarization-mode dispersion and polarization-dependent losses," IEEE J. Sel. Topics Quantum Electron. 6, 317-329 (2000).

IEEE Photon. Technol. Lett. (1)

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

J. Lightw. Technol. (8)

A. E. Willner, S. M. R. M. Nezam, L. Yan, Z. Pan, M. C. Hauer, "Monitoring and control of polarization-related impairments in optical fiber systems," J. Lightw. Technol. 22, 106-125 (2004).

M. Reimer, D. Yevick, D. Dumas, "The accuracy of the magnus expansion for polarization-mode dispersion and polarization-dependent loss," J. Lightw. Technol. 16, 3337-3344 (2008).

P. A. Williams, J. D. Kofler, "Narrow-band measurement of differential group delay by a six-state RF phase-shift technique: 40 fs single-measurement uncertainty," J. Lightw. Technol. 22, 448-456 (2004).

B. Szafraniec, D. M. Baney, "Elementary matrix-based vector optical network analysis," J. Lightw. Technol. 25, 1061-1069 (2007).

E. Forestieri, L. Vincetti, "Exact evaluation of the jones matrix of a fiber in the presence of polarization mode dispersion of any order," J. Lightw. Technol. 19, 1898-1909 (2001).

V. Chernyak, M. Chertkov, I. Gabitov, I. Kolokolov, V. Lebedev, "PMD-induced fluctuations of bit-error rate in optical fiber systems," J. Lightw. Technol. 22, 1155-1168 (2004).

D. Wang, C. R. Menyuk, "Calculation of penalties due to polarization effects in a long-haul WDM system using a stokes parameter model," J. Lightw. Technol. 19, 487-494 (2001).

C. D. Poole, T. E. Darcie, "Distortion related to polarization-mode dispersion in analog lightwave systems," J. Lightw. Technol. 11, 1749-1759 (1993).

J. Opt. Fiber Commun. (2)

P. Williams, "PMD measurement techniques and how to avoid the pitfalls," J. Opt. Fiber Commun. 1, 84-105 (2004) Rep. 1.

H. Kogelnik, L. E. Nelson, J. P. Gordon, "The inverse PMD problem," J. Opt. Fiber Commun. 1, 107-122 (2004) Rep. 1.

J. Opt. Soc. Amer. (8)

R. C. Jones, "A new calculus for the treatment of optical systems I. Description and discussion of the calculus," J. Opt. Soc. Amer. 31, 488-493 (1941).

H. Hurwitz Jr., R. C. Jones, "A new calculus for the treatment of optical systems II. Proof of the three general equivalence theorems," J. Opt. Soc. Amer. 31, 493-499 (1941).

R. C. Jones, "A new calculus for the treatment of optical systems III. The sohncke theory of optical activity," J. Opt. Soc. Amer. 31, 500-503 (1941).

R. C. Jones, "A new calculus for the treatment of optical systems, IV," J. Opt. Soc. Amer. 32, 486-493 (1942).

R. C. Jones, "A new calculus for the treatment of optical systems, V. A more general formulation and description of another calculus," J. Opt. Soc. Amer. 37, 107-110 (1947).

R. C. Jones, "A new calculus for the treatment of optical systems, VI. Experimental determination or the matrix," J. Opt. Soc. Amer. 37, 110-112 (1947).

R. C. Jones, "A new calculus for the treatment of optical systems. VII properties of the N-matrices," J. Opt. Soc. Amer. 38, 671-685 (1948).

C. Whitney, "Pauli-algebraic operators in polarization optics," J. Opt. Soc. Amer. 61, 1207-1213 (1971).

J. Opt. Soc. Amer. B, Opt. Phys. (1)

R. M. A. Azzam, "Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 44 matrix calculus," J. Opt. Soc. Amer. B, Opt. Phys. 68, 1756-1767 (1978).

Opt. Commun. (2)

N. Gisin, J. P. Pellaux, "Polarization mode dispersion: Time versus frequency domain," Opt. Commun. 89, 316-323 (1992).

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

Opt. Exp. (1)

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, C. Q. Wu, "Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain," Opt. Exp. 14, 5067-5072 (2006).

Opt. Fiber Technol. (1)

F. Bruyère, "Impact of first- and second-order PMD in optical digital transmission systems," Opt. Fiber Technol. 2, 269-280 (1996).

Opt. Lett. (2)

Photon. Technol. Lett. (3)

H. Dong, P. Shum, Y. D. Gong, M. Yan, J. Q. Zhou, C. Q. Wu, "Virtual generalized mueller matrix method for measurement of complex polarization-mode dispersion vector in optical fibers," Photon. Technol. Lett. 19, 27-29 (2007).

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

E. Iannone, F. Matera, A. Galtarossa, G. Gianello, M. Schiano, "Effect of polarization dispersion on the performance in IM-DD communication systems," Photon. Technol. Lett. 5, 1247-1249 (1993).

Other (7)

H. Kogelnik, L. E. Nelson, R. M. Jopson, "Polarization mode dispersion impairment in lightwave transmission systems," Proc. Opt. Fiber Commun. Conf. (OFC 2002) pp. 194.

P. K. Kondamuri, C. Allen, D. L. Richards, "Variation of PMD-induced outage rates and durations with link length on buried standard single-mode fibers," Proc. Opt. Fiber Commun. Conf. (2005) pp. 1-3.

J. P. Gordon, H. Kogelnik, "PMD fundamentals: Polarization mode dispersion in optical fibers," Proc. Nat. Acad. Sci. USA (2000) pp. 4541-4550.

D. M. Baney, "Complex component characterization based on elementary matrices," Proc. Optical Fiber Communications Conf. (OFC 2009) (2009).

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

G. N. Ramachandran, S. Ramaseshan, Crystal Optics in Handbuch der Physik (Springer-Verlag, 1961).

G. Arfken, Mathematical Methods for Physicists (Academic, 1985).

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