Abstract

We develop an analytical method for optimizing phase sensitive amplifiers for regeneration in multilevel phase encoded transmission systems. The model accurately predicts the optimum transfer function characteristics and identifies operating tolerances for different signal constellations and transmission scenarios. The results demonstrate the scalability of the scheme and show the significance of having simultaneous optimization of the transfer function and the signal alphabet. The model is general and can be applied to any regenerative system.

© 2013 Optical Society of America

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  1. P. J. Winzer and R.-J. Essiambre, J. Lightwave Technol. 24, 4711 (2006).
    [CrossRef]
  2. R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, J. Lightwave Technol. 28, 662 (2010).
    [CrossRef]
  3. R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
    [CrossRef]
  4. J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, Nat. Photonics 5, 748 (2011).
    [CrossRef]
  5. J. Kakande, R. Slavik, F. Parmigiani, P. Petropoulos, and D. Richardson, in Optical Fiber Communication Conference (Optical Society of America, 2012), p. OW1I.3.
  6. M. A. Sorokina and S. K. Turitsyn, http://arxiv.org/abs/1305.1537 .
  7. R. M. May, Nature 261, 459 (1976).
    [CrossRef]
  8. K. S. Turitsyn and S. K. Turitsyn, Opt. Lett. 37, 3600 (2012).
    [CrossRef]

2012

2011

J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, Nat. Photonics 5, 748 (2011).
[CrossRef]

2010

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, J. Lightwave Technol. 28, 662 (2010).
[CrossRef]

2006

1976

R. M. May, Nature 261, 459 (1976).
[CrossRef]

Andrekson, P. A.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Bogris, A.

J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, Nat. Photonics 5, 748 (2011).
[CrossRef]

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Dasgupta, S.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Ellis, A. D.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Essiambre, R. J.

Essiambre, R.-J.

Foschini, G. J.

Goebel, B.

Grüner-Nielsen, L.

J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, Nat. Photonics 5, 748 (2011).
[CrossRef]

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Herstrøm, S.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Jakobsen, D.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Kakande, J.

J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, Nat. Photonics 5, 748 (2011).
[CrossRef]

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

J. Kakande, R. Slavik, F. Parmigiani, P. Petropoulos, and D. Richardson, in Optical Fiber Communication Conference (Optical Society of America, 2012), p. OW1I.3.

Kramer, G.

Lundström, C.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

May, R. M.

R. M. May, Nature 261, 459 (1976).
[CrossRef]

O’Gorman, J.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Parmigiani, F.

J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, Nat. Photonics 5, 748 (2011).
[CrossRef]

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

J. Kakande, R. Slavik, F. Parmigiani, P. Petropoulos, and D. Richardson, in Optical Fiber Communication Conference (Optical Society of America, 2012), p. OW1I.3.

Petropoulos, P.

J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, Nat. Photonics 5, 748 (2011).
[CrossRef]

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

J. Kakande, R. Slavik, F. Parmigiani, P. Petropoulos, and D. Richardson, in Optical Fiber Communication Conference (Optical Society of America, 2012), p. OW1I.3.

Phelan, R.

J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, Nat. Photonics 5, 748 (2011).
[CrossRef]

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Richardson, D.

J. Kakande, R. Slavik, F. Parmigiani, P. Petropoulos, and D. Richardson, in Optical Fiber Communication Conference (Optical Society of America, 2012), p. OW1I.3.

Richardson, D. J.

J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, Nat. Photonics 5, 748 (2011).
[CrossRef]

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Sjödin, M.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Slavik, R.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

J. Kakande, R. Slavik, F. Parmigiani, P. Petropoulos, and D. Richardson, in Optical Fiber Communication Conference (Optical Society of America, 2012), p. OW1I.3.

Slavík, R.

J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, Nat. Photonics 5, 748 (2011).
[CrossRef]

Sorokina, M. A.

M. A. Sorokina and S. K. Turitsyn, http://arxiv.org/abs/1305.1537 .

Sygletos, S.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Syvridis, D.

J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, Nat. Photonics 5, 748 (2011).
[CrossRef]

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Turitsyn, K. S.

Turitsyn, S. K.

K. S. Turitsyn and S. K. Turitsyn, Opt. Lett. 37, 3600 (2012).
[CrossRef]

M. A. Sorokina and S. K. Turitsyn, http://arxiv.org/abs/1305.1537 .

Weerasuriya, R.

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

Winzer, P. J.

J. Lightwave Technol.

Nat. Photonics

R. Slavik, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, Nat. Photonics 4, 690 (2010).
[CrossRef]

J. Kakande, R. Slavík, F. Parmigiani, A. Bogris, D. Syvridis, L. Grüner-Nielsen, R. Phelan, P. Petropoulos, and D. J. Richardson, Nat. Photonics 5, 748 (2011).
[CrossRef]

Nature

R. M. May, Nature 261, 459 (1976).
[CrossRef]

Opt. Lett.

Other

J. Kakande, R. Slavik, F. Parmigiani, P. Petropoulos, and D. Richardson, in Optical Fiber Communication Conference (Optical Society of America, 2012), p. OW1I.3.

M. A. Sorokina and S. K. Turitsyn, http://arxiv.org/abs/1305.1537 .

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Figures (6)

Fig. 1.
Fig. 1.

Periodicity and slope characteristics of the PSA phase transfer function with varying input phase φ and regenerative parameter m.

Fig. 2.
Fig. 2.

(a) PSA phase transfer function for 8-PSK format. The optimal value of the parameter mopt=1/7 (shown by green solid line) demonstrates a plateau centered at the alphabet points, whereas a critical choice mcr=0.33 of the parameter (shown by red dashed line) is expected to lead to poorer regenerative performance. (b) Dependence of the optimal and critical values of m and the attraction region’s maximum half-width Δmax on the order M of the PSK modulation format.

Fig. 3.
Fig. 3.

Constellation diagram of 8-PSK after transmission with SNR=12dB (a) without regeneration and after 10 PSAs in cascade with (b) optimal m=1/7 and (c) nonoptimal m=0.4. One can observe dramatic gain in system performance due to the PSA optimization for the cascaded scenario.

Fig. 4.
Fig. 4.

SER as a function of SNR with cascaded regenerative PSA elements for (a) 4-PSK, (b) 8-PSK, and (c) 16-PSK formats. The SER of the linear channel is shown for comparison with a black dashed–dotted line. The SER of the system with the optimal and critical values of the parameter m is shown with green and red curves, respectively. The arrows show the SER evolution for increasing number of inline PSAs.

Fig. 5.
Fig. 5.

Dependence of the SNR improvement due to PSA application for the fixed value SER=103 as a function of the PSA parameter m for 8-PSK modulation format for the different numbers of PSAs (denoted by n). Faster saturation in performance improvement occurs for nonoptimal m values. The convergence of the numerical values mnum to the analytical mopt is shown by a black dotted curve.

Fig. 6.
Fig. 6.

Dependence of the SNR improvement due to PSA application for the fixed value SER=103 as a function of the PSA parameter m for 4 (blue), 8 (green), and 16 (red) PSK modulation formats. The cascaded scheme of 10 (dashed) and 20 (dotted) PSA elements gives significant improvement for high-order PSK. The sensitivity to the optimization parameter m increases rapidly for dense formats. The optimal value of m is shown by vertical lines of the corresponding color for each constellation size.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

Reiψ=reiφ(1+meiφM).
ψ=F(φ)=tan1(sin[φ]+msin[φ(1M)]cos[φ]+mcos[φ(1M)]).
|m|<mcr,mcr=2M2.
|mopt|=1M1.
Δ=(1|1m+mM1+m|)/(mM3(1m)2(1+m)3).

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