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.

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  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).

2012

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).

2011

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).

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).

2009

E. Agrell, M. Karlsson, "Power-Efficient modulation formats in coherent transmission systems," J. Lightw. Tech. 27, 5115-5126 (2009).

N. J. Frigo, F. Bucholtz, "Geometrical representation of optical propagation phase," J. Lightw. Tech. 27, 3283 -3293 (2009).

2006

C. McKinstrie, M. Raymer, S. Radic, M. Vasilyev, "Quantum mechanics of phase-sensitive amplification in a fiber," Opt. Comm. 257, 146 -163 (2006).

2002

W. P. Bowen, N. Treps, R. Schnabel, P. K. Lam, "Experimental demonstration of continuous variable polarization entanglement," Phys. Rev. Lett. 89, 253601 (2002).

2000

J. P. Gordon, H. Kogelnik, "PMD fundamentals: Polarization mode dispersion in optical fibers," Proc. Nat. Acad. Sci. USA 97, 4541-4550 (2000).

1992

R. Cusani, E. Iannone, A. Salonico, M. Todaro, "An efficient multilevel coherent optical system: M-4Q-QAM ," J. Lightw. Tech. 10, 777-786 (1992).

1991

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).

1990

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).

1986

S. R. Cloude, "Group theory and polarisation algebra ," Optik (Stuttgart) 75, 26-36 (1986).

N. J. Frigo, "A generalized geometrical representation of coupled mode theory," IEEE J. Quantum Electron. 22, 2131-2140 (1986).

1973

H. Takenaka, "A unified formalism for polarization optics by using group theory," Nouvelle revue d’optique 4, (1973).

1961

W. H. Louisell, A. Yariv, A. E. Siegman, "Quantum fluctuations and noise in parametric processes—I," Phys. Rev. 124, 1646 (1961).

1954

U. Fano, "A Stokes-parameter technique for the treatment of polarization in quantum mechanics," Phys. Rev. 93, 121-123 (1954).

1951

D. L. Falkoff, J. E. MacDonald, "On the Stokes parameters for polarized radiation," J. Opt. Soc. Amer. 41, 861- 862 (1951).

1948

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).

1947

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 of the matrix," J. Opt. Soc. Amer. 37, 110-112 (1947).

1942

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

1941

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 three equivalence theorems," J. Opt. Soc. Amer. 31, 493-495 (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).

1852

G. G. Stokes, "On the composition and resolution of streams of polarized light from different sources," Trans. Cambridge Phil. Soc. 9, 399-416 (1852).

J. Lightw. Tech.

E. Agrell, M. Karlsson, "Power-Efficient modulation formats in coherent transmission systems," J. Lightw. Tech. 27, 5115-5126 (2009).

IEEE J. Quantum Electron.

N. J. Frigo, "A generalized geometrical representation of coupled mode theory," IEEE J. Quantum Electron. 22, 2131-2140 (1986).

J. Opt. Soc. Amer.

D. L. Falkoff, J. E. MacDonald, "On the Stokes parameters for polarized radiation," J. Opt. Soc. Amer. 41, 861- 862 (1951).

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).

J. Lightw. Tech.

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).

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).

R. Cusani, E. Iannone, A. Salonico, M. Todaro, "An efficient multilevel coherent optical system: M-4Q-QAM ," J. Lightw. Tech. 10, 777-786 (1992).

N. J. Frigo, F. Bucholtz, "Geometrical representation of optical propagation phase," J. Lightw. Tech. 27, 3283 -3293 (2009).

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).

J. Opt. Soc. Amer.

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).

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).

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).

J. Opt. Soc. Amer.

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

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).

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).

R. C. Jones, "New calculus for the treatment of optical systems—VIII. Electromagnetic theory," J. Opt. Soc. Amer. 46 , 126-131 (1956 ).

Nouvelle revue d’optique

H. Takenaka, "A unified formalism for polarization optics by using group theory," Nouvelle revue d’optique 4, (1973).

Opt. Comm.

C. McKinstrie, M. Raymer, S. Radic, M. Vasilyev, "Quantum mechanics of phase-sensitive amplification in a fiber," Opt. Comm. 257, 146 -163 (2006).

Opt. Exp.

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).

Opt. Exp.

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).

Optik (Stuttgart)

S. R. Cloude, "Group theory and polarisation algebra ," Optik (Stuttgart) 75, 26-36 (1986).

Phys. Rev.

U. Fano, "A Stokes-parameter technique for the treatment of polarization in quantum mechanics," Phys. Rev. 93, 121-123 (1954).

W. H. Louisell, A. Yariv, A. E. Siegman, "Quantum fluctuations and noise in parametric processes—I," Phys. Rev. 124, 1646 (1961).

Phys. Rev. Lett.

W. P. Bowen, N. Treps, R. Schnabel, P. K. Lam, "Experimental demonstration of continuous variable polarization entanglement," Phys. Rev. Lett. 89, 253601 (2002).

Proc. Nat. Acad. Sci. USA

J. P. Gordon, H. Kogelnik, "PMD fundamentals: Polarization mode dispersion in optical fibers," Proc. Nat. Acad. Sci. USA 97, 4541-4550 (2000).

Trans. Cambridge Phil. Soc.

G. G. Stokes, "On the composition and resolution of streams of polarized light from different sources," Trans. Cambridge Phil. Soc. 9, 399-416 (1852).

Other

(2013). “Rotations in 4-dimensional euclidean space,” [Online]. Available: http://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space.

M. Karlsson and E. Agrell, “Four-dimensional optimized constellations for coherent optical transmission systems,” in Proc. Europ. Conf. Opt. Comm., 2010, Paper WeC3..

M. Karlsson. (2013). “The connection between polarization calculus and four-dimensional rotations,” [Online]. Available: http://arxiv.org/abs/1303.1836.

J. N. Damask, Polarization Optics in Telecommunications (Springer Verlag , 2005).

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..

N. G. Parke III, “Matrix optics,” Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, NA, USA, 1948..

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

M. Born, E. Wolf, Principles of optics, 7th ed (Cambridge Univ. Press, 1999).

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.

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..

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.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

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