OSA's Digital Library

Journal of Lightwave Technology

Journal of Lightwave Technology

| A JOINT IEEE/OSA PUBLICATION

  • Vol. 32, Iss. 6 — Mar. 15, 2014
  • pp: 1246–1257

Four-dimensional Rotations in Coherent Optical Communications

Magnus Karlsson

Journal of Lightwave Technology, Vol. 32, Issue 6, pp. 1246-1257 (2014)


View Full Text Article

Acrobat PDF (1367 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations
  • Export Citation/Save Click for help

Abstract

To model electromagnetic wave propagation for coherent communications without polarization dependent losses, the unitary $2 \times 2$ Jones transfer matrix formalism is typically used. In this study, we propose an alternative formalism to describe such transformations based on rotations in four-dimensional (4d) Euclidean space. This formalism is usually more attractive from a communication theoretical perspective, since decisions and symbol errors can be related to geometric concepts such as Euclidean distances between points and decision boundaries. Since 4d rotations is a richer description than the conventional Jones calculus, having six rather than four degrees of freedom (DOF), we propose an extension of the Jones calculus to handle all six DOF. In addition, we show that the two extra DOF in the 4d description represents transformations that are nonphysical for propagating photons, since they does not obey the fundamental quantum mechanical boson commutation relations. Finally, we exemplify on how the nonphysical rotations can change the polarization-phase degeneracy of well-known constellations such as single-polarization QPSK, polarization-multiplexed (PM-)QPSK and polarization-switched (PS-) QPSK. For example, we show how PM-QPSK, which is well known to consist of four polarization states each having four-fold phase degeneracy, can be represented as eight states of polarizations, each with binary phase degeneracy.

© 2014 IEEE

Citation
Magnus Karlsson, "Four-dimensional Rotations in Coherent Optical Communications," J. Lightwave Technol. 32, 1246-1257 (2014)
http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-32-6-1246


Sort:  Year  |  Journal  |  Reset

References

  1. 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).
  2. H. Hurwitz, Jr.R. C. Jones, "A new calculus for the treatment of optical systems—II: Proof of three equivalence theorems," J. Opt. Soc. Amer. 31, 493-495 (1941).
  3. 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).
  4. R. C. Jones, "A new calculus for the treatment of optical systems—IV," J. Opt. Soc. Amer. 32, 486-493 (1942).
  5. 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).
  6. R. C. Jones, "A new calculus for the treatment of optical systems—VI. Experimental determination of the matrix," J. Opt. Soc. Amer. 37, 110-112 (1947).
  7. R. C. Jones, "A new calculus for the treatment of optical systems—VII. Properties of the N-matrices," J. Opt. Soc. Amer. 38, 671 -683 (1948).
  8. R. C. Jones, "New calculus for the treatment of optical systems—VIII. Electromagnetic theory," J. Opt. Soc. Amer. 46 , 126-131 (1956 ).
  9. H. Mueller, “Memorandumon the polarization optics of the photo-elastic shutter, NDRC project OEMsr-576,” National Defence Research Committee, Tech. Rep. no. 2, Nov. 15, 1943..
  10. N. G. Parke III, “Matrix optics,” Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, NA, USA, 1948..
  11. G. G. Stokes, "On the composition and resolution of streams of polarized light from different sources," Trans. Cambridge Phil. Soc. 9, 399-416 (1852).
  12. P. Johannisson, M. Sjödin, M. Karlsson, H. Wymeersch, E. Agrell, P. A. Andrekson, " Modified constant modulus algorithm for polarization-switched QPSK," Opt. Exp. 19, 7734-7741 (2011).
  13. H. Takenaka, "A unified formalism for polarization optics by using group theory," Nouvelle revue d’optique 4, (1973).
  14. S. R. Cloude, "Group theory and polarisation algebra ," Optik (Stuttgart) 75, 26-36 (1986).
  15. S. Betti, F. Curti, G. De Marchis, E. Iannone, "Multilevel coherent optical system based on Stokes parameters modulation ," J. Lightw. Tech. 8, 1127-1136 (1990).
  16. S. Betti, F. Curti, G. De Marchis, E. Iannone, "A novel multilevel coherent optical system: 4-quadrature signaling ," J. Lightw. Tech. 9, 514-523 (1991).
  17. R. Cusani, E. Iannone, A. Salonico, M. Todaro, "An efficient multilevel coherent optical system: M-4Q-QAM ," J. Lightw. Tech. 10, 777-786 (1992).
  18. (2013). “Rotations in 4-dimensional euclidean space,” [Online]. Available: http://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space.
  19. M. Karlsson and E. Agrell, “Four-dimensional optimized constellations for coherent optical transmission systems,” in Proc. Europ. Conf. Opt. Comm., 2010, Paper WeC3..
  20. M. Karlsson. (2013). “The connection between polarization calculus and four-dimensional rotations,” [Online]. Available: http://arxiv.org/abs/1303.1836.
  21. J. N. Damask, Polarization Optics in Telecommunications (Springer Verlag , 2005).
  22. N. J. Frigo, "A generalized geometrical representation of coupled mode theory," IEEE J. Quantum Electron. 22, 2131-2140 (1986).
  23. J. P. Gordon, H. Kogelnik, "PMD fundamentals: Polarization mode dispersion in optical fibers," Proc. Nat. Acad. Sci. USA 97, 4541-4550 (2000).
  24. E. Agrell, M. Karlsson, "Power-Efficient modulation formats in coherent transmission systems," J. Lightw. Tech. 27, 5115-5126 (2009).
  25. W. H. Louisell, A. Yariv, A. E. Siegman, "Quantum fluctuations and noise in parametric processes—I," Phys. Rev. 124, 1646 (1961).
  26. N. J. Frigo, F. Bucholtz, "Geometrical representation of optical propagation phase," J. Lightw. Tech. 27, 3283 -3293 (2009).
  27. N. J. Frigo, F. Bucholtz, and C. V. McLaughlin, “Polarization in phase modulated optical links: Jones- and generalized stokes-space analysis,” J. Lightw. Technol., vol. 31, no. 9, pp. 1503–1511, May 2013. [Online]. Available: http://jlt.osa.org/abstract.cfm?URI=jlt-31-9-1503.
  28. M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Exp., vol. 17, no. 13, pp. 10 814–10 819, 2009..
  29. M. Sjödin, P. Johannisson, H. Wymeersch, P. Andrekson, M. Karlsson, "Comparison of polarization-switched QPSK and polarization-multiplexed QPSK at 30 Gbit/s," Opt. Exp. 19, 7839-7846 (2011).
  30. J. Renaudier, P. Serena, A. Bononi, M. Salsi, O. Bertran-Pardo, H. Mardoyan, P. Tran, E. Dutisseuil, G. Charlet, S. Bigo, "Generation and detection of 28 Gbaud polarization switched-QPSK in WDM long-haul transmission systems," J. Lightw. Tech. 30, 1312-1318 (2012).
  31. G. Arfken, Mathematical Methods for Physicists, Third Edition ( Academic, 1985).
  32. M. Born, E. Wolf, Principles of optics, 7th ed (Cambridge Univ. Press, 1999).
  33. U. Fano, "A Stokes-parameter technique for the treatment of polarization in quantum mechanics," Phys. Rev. 93, 121-123 (1954).
  34. D. L. Falkoff, J. E. MacDonald, "On the Stokes parameters for polarized radiation," J. Opt. Soc. Amer. 41, 861- 862 (1951).
  35. W. P. Bowen, N. Treps, R. Schnabel, P. K. Lam, "Experimental demonstration of continuous variable polarization entanglement," Phys. Rev. Lett. 89, 253601 (2002).
  36. C. McKinstrie, M. Raymer, S. Radic, M. Vasilyev, "Quantum mechanics of phase-sensitive amplification in a fiber," Opt. Comm. 257, 146 -163 (2006).
  37. N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A, vol. 65, p. 052306, Apr. 2002. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.65.052306.
  38. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Cited By

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article

OSA is a member of CrossRef.

CrossCheck Deposited