Abstract

Coherent optical transmission systems have a four-dimensional (4–D) signal space (two quadratures in two polarizations). These four dimensions can be used to create modulation formats that have a better power efficiency (higher sensitivity) than the conventional binary phase shift keying/quadrature phase shift keying (BPSK/QPSK) signals. Several examples are given, with some emphasis on a 24-level format and an 8-level format, including descriptions of how they can be realized and expressions for their symbol and bit error probabilities. These formats are, respectively, an extension and a subset of the commonly used 16-level dual-polarization QPSK format. Sphere packing simulations in 2, 3, and 4 dimensions, up to 32 levels, are used to verify their optimality. The numerical results, as the number of levels increases, are shown to agree with lattice-theoretical results. Finally, we point out that the use of these constellations will lead to improved fundamental sensitivity limits for optical communication systems, and they may also be relevant as a way of reducing power demands and/or nonlinear influence.

© 2009 IEEE

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  2. A. Leven, N. Kaneda, U. Koc, Y. Chen, "Coherent receivers for practical optical communication systems," Proc. Opt. Fiber Commun. Nat. Fiber Opt. Eng. Conf. (OFC/NFOEC) (2007).
  3. A. Leven, N. Kaneda, Y.-K. Chen, "A real-time CMA-based 10 Gb/s polarization demultiplexing coherent receiver implemented in an FPGA," Proc. Opt. Fiber Commun./Nat. Fiber Opt. Eng. Conf. (OFC/NFOEC) (2008).
  4. H. Sun, K. Wu, K. Roberts, "Real-time measurements of a 40 Gb/s coherent system," Opt. Exp. 16, 873-879 (2008).
  5. D. Ly-Gagnon, K. Katoh, K. Kikuchi, "Unrepeatered optical transmission of 20 Gbit/s quadrature phase-shift keying signals over 210 km using homodyne phase-diversity receiver and digital signal processing," Electron. Lett. 41, 206-207 (2005).
  6. G. Charlet, N. Maaref, J. Renaudier, H. Mardoyan, P. Tran, S. Bigo, "Transmission of 40 Gb/s QPSK with coherent detection over ultra-long distance improved by nonlinearity mitigation," Proc. Eur. Conf. Opt. Commun. (ECOC) (2006) pp. 35-36.
  7. S. Tsukamoto, D. Ly-Gagnon, K. Katoh, K. Kikuchi, "Coherent demodulation of 40-Gbit/s polarization-multiplexed QPSK signals with 16-GHz spacing after 200-km transmission," Proc. Opt. Fiber Commun. Nat. Fiber Opt. Eng. Conf. (OFC/NFOEC) (2005).
  8. G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, S. Bigo, "Performance comparison of singly-polarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43, 1109-1111 (2007).
  9. J. Renaudier, G. Charlet, M. Salsi, O. Pardo, H. Mardoyan, P. Tran, S. Bigo, "Linear fiber impairments mitigation of 40-Gbit/s polarization-multiplexed QPSK by digital processing in a coherent receiver," J. Lightw. Technol. 26, 36-42 (2008).
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  22. R. Cusani, E. Iannone, A. Salonico, M. Todaro, "An efficient multilevel coherent optical system: M-4Q-QAM," J. Lightw. Technol. 10, 777-786 (1992).
  23. M. Karlsson, E. Agrell, "Which is the most power-efficient modulation format in optical links?," Opt. Exp., 1713 pp. 10814-10819 (2009) http://www.opticsexpress.org/abstract.cfm?URI=oe-17-13-10814.
  24. S. Benedetto, P. Poggiolini, "Theory of polarization shift keying modulation," IEEE Trans. Commun. 40, 708-721 (1992).
  25. S. Benedetto, A. Djupsjöbacka, B. Lagerström, R. Paoletti, P. Poggiolini, G. Mijic, "Multilevel polarization modulation using a specifically designed LiNbO$_3$ device," IEEE Photon. Technol. Lett. 6, 949-951 (1994).
  26. H. Zhao, E. Agrell, M. Karlsson, "Unequal bit error probability in coherent QPSK fiber-optic systems using phase modulator based transmitters," Eur. Trans. Telecommun. 19, 895-906 (2007).
  27. H. Zhao, M. Karlsson, E. Agrell, "Transmitter comparison and unequal bit error probabilities in coherent QPSK systems," Proc. Opt. Fiber Commun. Nat. Fiber Opt. Eng. Conf. (OFC/NFOEC) (2007).
  28. H. Zhao, E. Agrell, M. Karlsson, "Intersymbol interference in DQPSK fiber-optic systems," Eur. Trans. Telecommun. .
  29. K. Kikuchi, S. Tsukamoto, "Evaluation of sensitivity of the digital coherent receiver," J. Lightw. Technol. 26, 1817-1822 (2008).
  30. S. Benedetto, E. Biglieri, Principles of Digital Transmission: With Wireless Applications (Kluwer, 1999).
  31. J. Proakis, Digital Communications (McGraw-Hill, 2001).
  32. H. Bülow, "Polarization QAM modulation (POL-QAM) for coherent detection schemes," Proc. Opt. Fiber Commun. Nat. Fiber Opt. Eng. Conf. (OFC/NFOEC) (2009).
  33. J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer-Verlag, 1999).
  34. O. Musin, "The kissing number in four dimensions," Ann. Math. 168, 1-32 (2008).
  35. N. J. A. Sloane, R. H. Hardin, T. S. Duff, J. H. Conway, (1997)"Minimal-energy clusters, library of 3-d clusters, library of 4-d clusters," http://www.research.att.com/~njas/cluster/.
  36. R. L. Graham, N. J. A. Sloane, "Penny-packing and two-dimensional codes," Discrete Comput. Geom. 5, 1-11 (1990).
  37. M. Simon, S. Hinedi, W. Lindsey, Digital Communication Techniques: Signal Design and Detection (Prentice-Hall, 1995).
  38. L. Xiao, X. Dong, "New analytical expressions for orthogonal, biorthogonal, and transorthogonal signaling in Nakagami fading channels with diversity reception," IEEE Trans. Wireless Commun. 4, 1418-1424 (2005).
  39. M. K. Simon, R. Annavajjala, "On the optimality of bit detection of certain digital modulations," IEEE Trans. Commun. 53, 299-307 (2005).
  40. G. Foschini, R. Gitlin, S. Weinstein, "Optimization of two-dimensional signal constellations in the presence of Gaussian noise," IEEE Trans. Commun. COM-22, 28-38 (1974).
  41. N. J. A. Sloane, R. H. Hardin, T. S. Duff, J. H. Conway, "Minimal-energy clusters of hard spheres," Discrete Comput. Geom. 14, 237-259 (1995).
  42. J.-E. Porath, T. Aulin, "Design of multidimensional signal constellations," Inst. Electr. Eng. Proc.-Commun. 150, 317-323 (2003).
  43. J. R. Pierce, "Comparison of three-phase modulation with two-phase and four-phase modulation," IEEE Trans. Commun. COM-28, 1098-1099 (1980).
  44. N. Ekanayake, T. Tjhung, "On ternary phase-shift keyed signaling," IEEE Trans. Inf. Theory IT-28, 658-660 (1982).
  45. J. P. Gordon, L. R. Walker, W. H. Louisell, "Quantum statistics of masers and attenuators," Phys. Rev. 130, 806-812 (1963).
  46. J. R. Pierce, "Optical channels: Practical limits with photon counting," IEEE Trans. Commun. 26, 1819-1821 (1978).

2008 (5)

H. Sun, K. Wu, K. Roberts, "Real-time measurements of a 40 Gb/s coherent system," Opt. Exp. 16, 873-879 (2008).

J. Renaudier, G. Charlet, M. Salsi, O. Pardo, H. Mardoyan, P. Tran, S. Bigo, "Linear fiber impairments mitigation of 40-Gbit/s polarization-multiplexed QPSK by digital processing in a coherent receiver," J. Lightw. Technol. 26, 36-42 (2008).

E. Ip, A. Lau, D. Barros, J. Kahn, "Coherent detection in optical fiber systems," Opt. Exp. 16, 753-791 (2008) (erratum, vol. 16, no. p. 26, p. 21943, 2008).

K. Kikuchi, S. Tsukamoto, "Evaluation of sensitivity of the digital coherent receiver," J. Lightw. Technol. 26, 1817-1822 (2008).

O. Musin, "The kissing number in four dimensions," Ann. Math. 168, 1-32 (2008).

2007 (2)

G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, S. Bigo, "Performance comparison of singly-polarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43, 1109-1111 (2007).

H. Zhao, E. Agrell, M. Karlsson, "Unequal bit error probability in coherent QPSK fiber-optic systems using phase modulator based transmitters," Eur. Trans. Telecommun. 19, 895-906 (2007).

2006 (1)

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, Y. Achiam, "Synchronous QPSK transmission at 1.6 Gbit/s with standard DFB lasers and real-time digital receiver," Electron. Lett. 42, 1175-1176 (2006).

2005 (3)

D. Ly-Gagnon, K. Katoh, K. Kikuchi, "Unrepeatered optical transmission of 20 Gbit/s quadrature phase-shift keying signals over 210 km using homodyne phase-diversity receiver and digital signal processing," Electron. Lett. 41, 206-207 (2005).

L. Xiao, X. Dong, "New analytical expressions for orthogonal, biorthogonal, and transorthogonal signaling in Nakagami fading channels with diversity reception," IEEE Trans. Wireless Commun. 4, 1418-1424 (2005).

M. K. Simon, R. Annavajjala, "On the optimality of bit detection of certain digital modulations," IEEE Trans. Commun. 53, 299-307 (2005).

2004 (1)

J. Kahn, K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Topics Quantum Electron. 10, 259-272 (2004).

2003 (1)

J.-E. Porath, T. Aulin, "Design of multidimensional signal constellations," Inst. Electr. Eng. Proc.-Commun. 150, 317-323 (2003).

1995 (1)

N. J. A. Sloane, R. H. Hardin, T. S. Duff, J. H. Conway, "Minimal-energy clusters of hard spheres," Discrete Comput. Geom. 14, 237-259 (1995).

1994 (1)

S. Benedetto, A. Djupsjöbacka, B. Lagerström, R. Paoletti, P. Poggiolini, G. Mijic, "Multilevel polarization modulation using a specifically designed LiNbO$_3$ device," IEEE Photon. Technol. Lett. 6, 949-951 (1994).

1992 (2)

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

S. Benedetto, P. Poggiolini, "Theory of polarization shift keying modulation," IEEE Trans. Commun. 40, 708-721 (1992).

1991 (2)

S. Betti, F. Curti, G. D. Marchis, and E. Iannone, "A novel multilevel coherent optical system: 4-quadrature signaling," J. Lightw. Technol. 9, 514-523 (1991).

S. Betti, G. De Marchis, E. Iannone, P. Lazzaro, "Homodyne optical coherent systems based on polarization modulation," J. Lightw. Technol. 9, 1314-1320 (1991).

1990 (2)

S. Betti, F. Curti, G. De Marchis, E. Iannone, "Exploiting fibre optics transmission capacity: 4-quadrature multilevel signalling," Electron. Lett. 26, 992-993 (1990).

R. L. Graham, N. J. A. Sloane, "Penny-packing and two-dimensional codes," Discrete Comput. Geom. 5, 1-11 (1990).

1989 (1)

D. Saha, T. Birdsall, "Quadrature-quadrature phase-shift keying," IEEE Trans. Commun. 37, 437-448 (1989).

1982 (1)

N. Ekanayake, T. Tjhung, "On ternary phase-shift keyed signaling," IEEE Trans. Inf. Theory IT-28, 658-660 (1982).

1980 (1)

J. R. Pierce, "Comparison of three-phase modulation with two-phase and four-phase modulation," IEEE Trans. Commun. COM-28, 1098-1099 (1980).

1978 (1)

J. R. Pierce, "Optical channels: Practical limits with photon counting," IEEE Trans. Commun. 26, 1819-1821 (1978).

1977 (1)

L. Zetterberg, H. Brändström, "Codes for combined phase and amplitude modulated signals in a four-dimensional space," IEEE Trans. Commun. COM-25, 943-950 (1977).

1974 (2)

G. Welti, J. Lee, "Digital transmission with coherent four-dimensional modulation," IEEE Trans. Inf. Theory IT-20, 497-502 (1974).

G. Foschini, R. Gitlin, S. Weinstein, "Optimization of two-dimensional signal constellations in the presence of Gaussian noise," IEEE Trans. Commun. COM-22, 28-38 (1974).

1963 (1)

J. P. Gordon, L. R. Walker, W. H. Louisell, "Quantum statistics of masers and attenuators," Phys. Rev. 130, 806-812 (1963).

Ann. Math. (1)

O. Musin, "The kissing number in four dimensions," Ann. Math. 168, 1-32 (2008).

Discrete Comput. Geom. (2)

R. L. Graham, N. J. A. Sloane, "Penny-packing and two-dimensional codes," Discrete Comput. Geom. 5, 1-11 (1990).

N. J. A. Sloane, R. H. Hardin, T. S. Duff, J. H. Conway, "Minimal-energy clusters of hard spheres," Discrete Comput. Geom. 14, 237-259 (1995).

Electron. Lett. (4)

S. Betti, F. Curti, G. De Marchis, E. Iannone, "Exploiting fibre optics transmission capacity: 4-quadrature multilevel signalling," Electron. Lett. 26, 992-993 (1990).

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, Y. Achiam, "Synchronous QPSK transmission at 1.6 Gbit/s with standard DFB lasers and real-time digital receiver," Electron. Lett. 42, 1175-1176 (2006).

D. Ly-Gagnon, K. Katoh, K. Kikuchi, "Unrepeatered optical transmission of 20 Gbit/s quadrature phase-shift keying signals over 210 km using homodyne phase-diversity receiver and digital signal processing," Electron. Lett. 41, 206-207 (2005).

G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, S. Bigo, "Performance comparison of singly-polarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43, 1109-1111 (2007).

Eur. Trans. Telecommun. (2)

H. Zhao, E. Agrell, M. Karlsson, "Unequal bit error probability in coherent QPSK fiber-optic systems using phase modulator based transmitters," Eur. Trans. Telecommun. 19, 895-906 (2007).

H. Zhao, E. Agrell, M. Karlsson, "Intersymbol interference in DQPSK fiber-optic systems," Eur. Trans. Telecommun. .

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

J. Kahn, K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Topics Quantum Electron. 10, 259-272 (2004).

IEEE Photon. Technol. Lett. (1)

S. Benedetto, A. Djupsjöbacka, B. Lagerström, R. Paoletti, P. Poggiolini, G. Mijic, "Multilevel polarization modulation using a specifically designed LiNbO$_3$ device," IEEE Photon. Technol. Lett. 6, 949-951 (1994).

IEEE Trans. Commun. (7)

S. Benedetto, P. Poggiolini, "Theory of polarization shift keying modulation," IEEE Trans. Commun. 40, 708-721 (1992).

D. Saha, T. Birdsall, "Quadrature-quadrature phase-shift keying," IEEE Trans. Commun. 37, 437-448 (1989).

L. Zetterberg, H. Brändström, "Codes for combined phase and amplitude modulated signals in a four-dimensional space," IEEE Trans. Commun. COM-25, 943-950 (1977).

M. K. Simon, R. Annavajjala, "On the optimality of bit detection of certain digital modulations," IEEE Trans. Commun. 53, 299-307 (2005).

G. Foschini, R. Gitlin, S. Weinstein, "Optimization of two-dimensional signal constellations in the presence of Gaussian noise," IEEE Trans. Commun. COM-22, 28-38 (1974).

J. R. Pierce, "Comparison of three-phase modulation with two-phase and four-phase modulation," IEEE Trans. Commun. COM-28, 1098-1099 (1980).

J. R. Pierce, "Optical channels: Practical limits with photon counting," IEEE Trans. Commun. 26, 1819-1821 (1978).

IEEE Trans. Inf. Theory (2)

N. Ekanayake, T. Tjhung, "On ternary phase-shift keyed signaling," IEEE Trans. Inf. Theory IT-28, 658-660 (1982).

G. Welti, J. Lee, "Digital transmission with coherent four-dimensional modulation," IEEE Trans. Inf. Theory IT-20, 497-502 (1974).

IEEE Trans. Wireless Commun. (1)

L. Xiao, X. Dong, "New analytical expressions for orthogonal, biorthogonal, and transorthogonal signaling in Nakagami fading channels with diversity reception," IEEE Trans. Wireless Commun. 4, 1418-1424 (2005).

Inst. Electr. Eng. Proc.-Commun. (1)

J.-E. Porath, T. Aulin, "Design of multidimensional signal constellations," Inst. Electr. Eng. Proc.-Commun. 150, 317-323 (2003).

J. Lightw. Technol. (5)

J. Renaudier, G. Charlet, M. Salsi, O. Pardo, H. Mardoyan, P. Tran, S. Bigo, "Linear fiber impairments mitigation of 40-Gbit/s polarization-multiplexed QPSK by digital processing in a coherent receiver," J. Lightw. Technol. 26, 36-42 (2008).

S. Betti, F. Curti, G. D. Marchis, and E. Iannone, "A novel multilevel coherent optical system: 4-quadrature signaling," J. Lightw. Technol. 9, 514-523 (1991).

S. Betti, G. De Marchis, E. Iannone, P. Lazzaro, "Homodyne optical coherent systems based on polarization modulation," J. Lightw. Technol. 9, 1314-1320 (1991).

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

K. Kikuchi, S. Tsukamoto, "Evaluation of sensitivity of the digital coherent receiver," J. Lightw. Technol. 26, 1817-1822 (2008).

Opt. Exp. (2)

H. Sun, K. Wu, K. Roberts, "Real-time measurements of a 40 Gb/s coherent system," Opt. Exp. 16, 873-879 (2008).

E. Ip, A. Lau, D. Barros, J. Kahn, "Coherent detection in optical fiber systems," Opt. Exp. 16, 753-791 (2008) (erratum, vol. 16, no. p. 26, p. 21943, 2008).

Phys. Rev. (1)

J. P. Gordon, L. R. Walker, W. H. Louisell, "Quantum statistics of masers and attenuators," Phys. Rev. 130, 806-812 (1963).

Other (16)

K.-P. Ho, Phase-Modulated Optical Communication Systems (Springer-Verlag, 2005).

G. Taricco, E. Biglieri, V. Castellani, "Applicability of four-dimensional modulations to digital satellites: A simulation study," Proc. IEEE Global Telecommun. Conf. (GLOBECOM) (1993) pp. 28-34.

G. Jacobsen, Noise in Digital Optical Transmission Systems (Artech House, 1994).

L. Kazovsky, S. Benedetto, A. Willner, Optical Fiber Communication Systems (Artech House, 1996).

G. Charlet, N. Maaref, J. Renaudier, H. Mardoyan, P. Tran, S. Bigo, "Transmission of 40 Gb/s QPSK with coherent detection over ultra-long distance improved by nonlinearity mitigation," Proc. Eur. Conf. Opt. Commun. (ECOC) (2006) pp. 35-36.

S. Tsukamoto, D. Ly-Gagnon, K. Katoh, K. Kikuchi, "Coherent demodulation of 40-Gbit/s polarization-multiplexed QPSK signals with 16-GHz spacing after 200-km transmission," Proc. Opt. Fiber Commun. Nat. Fiber Opt. Eng. Conf. (OFC/NFOEC) (2005).

A. Leven, N. Kaneda, U. Koc, Y. Chen, "Coherent receivers for practical optical communication systems," Proc. Opt. Fiber Commun. Nat. Fiber Opt. Eng. Conf. (OFC/NFOEC) (2007).

A. Leven, N. Kaneda, Y.-K. Chen, "A real-time CMA-based 10 Gb/s polarization demultiplexing coherent receiver implemented in an FPGA," Proc. Opt. Fiber Commun./Nat. Fiber Opt. Eng. Conf. (OFC/NFOEC) (2008).

S. Benedetto, E. Biglieri, Principles of Digital Transmission: With Wireless Applications (Kluwer, 1999).

J. Proakis, Digital Communications (McGraw-Hill, 2001).

H. Bülow, "Polarization QAM modulation (POL-QAM) for coherent detection schemes," Proc. Opt. Fiber Commun. Nat. Fiber Opt. Eng. Conf. (OFC/NFOEC) (2009).

J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer-Verlag, 1999).

M. Simon, S. Hinedi, W. Lindsey, Digital Communication Techniques: Signal Design and Detection (Prentice-Hall, 1995).

N. J. A. Sloane, R. H. Hardin, T. S. Duff, J. H. Conway, (1997)"Minimal-energy clusters, library of 3-d clusters, library of 4-d clusters," http://www.research.att.com/~njas/cluster/.

M. Karlsson, E. Agrell, "Which is the most power-efficient modulation format in optical links?," Opt. Exp., 1713 pp. 10814-10819 (2009) http://www.opticsexpress.org/abstract.cfm?URI=oe-17-13-10814.

H. Zhao, M. Karlsson, E. Agrell, "Transmitter comparison and unequal bit error probabilities in coherent QPSK systems," Proc. Opt. Fiber Commun. Nat. Fiber Opt. Eng. Conf. (OFC/NFOEC) (2007).

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