Abstract

We propose a space-demultiplexing algorithm based on signal analysis in higher-order Poincaré spheres for optical transmission systems supported by space-division multiplexing. This algorithm is modulation format agnostic and does not require training sequences. We show that any arbitrary pair of tributaries signals can be represented in a higher-order Poincaré sphere. In such sphere, the crosstalk between any two tributary signals can be reversed by computing and realigning the best fit plane. Using this procedure for all possible combinations of tributaries the transmitted signal is successfully recovered, with negligible signal-to-noise ratio (SNR) penalties for quadrature phase-shift keying (QPSK) and 16-quadrature amplitude modulation (QAM) constellations, and with a SNR penalty as lower as 0.5 dB for the 64-QAM.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
  7. S. Ziaie, N. J. Muga, F. P. Guiomar, G. M. Fernandes, R. M. Ferreira, A. Shahpari, A. L. Teixeira, and A. N. Pinto, “Experimental assessment of the adaptive stokes space-based polarization demultiplexing for optical metro and access networks,” J. Lightwave Technol. 33(23), 4968–4974 (2015).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  22. A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19(10), 9714–9736 (2011).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  28. R. Shafik, M. Rahman, and A. Islam, “On the Extended Relationships Among EVM, BER and SNR as Performance Metrics”. In International Conference on Electrical and Computer Engineering (ICECE), 408–411, (2006).

2016 (1)

2015 (6)

G. Milione, M. Lavery, H. Huang, Y. Ren, G. Xie, T.A. Nguyen, E. Karimi, L. Marrucci, D. Nolan, R. Alfano, and others, “4×20 Gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de) multiplexer,” Opt. Lett. 40(9), 1980–1983 (2015).
[Crossref] [PubMed]

S. Arik, K.-P. Ho, and J. Kahn, “Delay spread reduction in mode-division multiplexing: Mode coupling versus delay compensation,” J. Lightwave Technol. 33(21), 4504–4512 (2015).
[Crossref]

N. Muga and A. Pinto, “Extended Kalman filter vs. geometrical approach for Stokes space-based polarization demultiplexing,” J. Lightwave Technol. 33(23), 4826–4833 (2015).
[Crossref]

S. Ziaie, N. J. Muga, F. P. Guiomar, G. M. Fernandes, R. M. Ferreira, A. Shahpari, A. L. Teixeira, and A. N. Pinto, “Experimental assessment of the adaptive stokes space-based polarization demultiplexing for optical metro and access networks,” J. Lightwave Technol. 33(23), 4968–4974 (2015).
[Crossref]

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

G. Milione, D. A. Nolan, and R. R. Alfano, “Determining principal modes in a multimode optical fiber using the mode dependent signal delay method,” J. Opt. Soc. Am. B 32(1), 143–149 (2015).
[Crossref]

2014 (5)

2013 (2)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nature Photonics 7(5), 354–362 (2013).
[Crossref]

N. J. Muga and A. N. Pinto, “Digital PDL compensation in 3D Stokes space,” J. Lightwave Technol. 31(13), 2122–2130 (2013).
[Crossref]

2012 (1)

2011 (2)

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19(10), 9714–9736 (2011).
[Crossref] [PubMed]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Physical Review letters 107(5), 053601 (2011).
[Crossref]

2010 (3)

2007 (1)

2000 (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97(9), 4541–4550 (2000).
[Crossref] [PubMed]

1998 (1)

Ahmed, N.

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

Aiello, A.

Alfano, R.

Alfano, R. R.

G. Milione, D. A. Nolan, and R. R. Alfano, “Determining principal modes in a multimode optical fiber using the mode dependent signal delay method,” J. Opt. Soc. Am. B 32(1), 143–149 (2015).
[Crossref]

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Physical Review letters 107(5), 053601 (2011).
[Crossref]

Antonelli, C.

Arik, S.

Askarov, D.

Bai, N.

Bosco, G.

M. Visintin, G. Bosco, P. Poggiolini, and F. Forghieri, “Adaptive digital equalization in optical coherent receivers with Stokes-space update algorithm”, J. Lightwave Technol. 32(24), 4759–4767 (2014).
[Crossref]

G. Bosco, M. Visintin, P. Poggiolini, and F. Forghieri, “A novel update algorithm in stokes space for adaptive equalization in coherent receivers,” in “Optical Fiber Communication Conference (OFC),” (2014), pp. Th3E-6.

Buchali, F.

F. Buchali, H. Buelow, K. Schuh, and W. Idler, “4D-CMA: Enabling Separation of Channel Compensation and Polarization Demultiplex,” in “Optical Fiber Communication Conference (OFC),” (2015), p. Th2A.15.

Buelow, H.

F. Buchali, H. Buelow, K. Schuh, and W. Idler, “4D-CMA: Enabling Separation of Channel Compensation and Polarization Demultiplex,” in “Optical Fiber Communication Conference (OFC),” (2015), p. Th2A.15.

Caballero, F. J. V.

F. J. V. Caballero, A. Zanaty, F. Pittalá, G. Goeger, Y. Ye, I. T. Monroy, and W. Rosenkranz, “Frequency-domain 2×2 mimo equalizer with stokes space updating algorithm,” in “Signal Processing in Photonic Communications,”(2016), pp. SpW2G-5.

F. J. V. Caballero, A. Zanaty, F. Pittalá, G. Goeger, and Y. Ye, “Efficient SDM-MIMO Stokes-space equalization,” in “European Conference on Optical Communications (ECOC), (2016)

Cao, Y.

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

da Silva, H. J. A.

Fernandes, G. M.

Ferreira, F. M.

Ferreira, R. M.

Fini, J. M.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nature Photonics 7(5), 354–362 (2013).
[Crossref]

Fonseca, D.

Forghieri, F.

M. Visintin, G. Bosco, P. Poggiolini, and F. Forghieri, “Adaptive digital equalization in optical coherent receivers with Stokes-space update algorithm”, J. Lightwave Technol. 32(24), 4759–4767 (2014).
[Crossref]

G. Bosco, M. Visintin, P. Poggiolini, and F. Forghieri, “A novel update algorithm in stokes space for adaptive equalization in coherent receivers,” in “Optical Fiber Communication Conference (OFC),” (2014), pp. Th3E-6.

Gabriel, C.

Goeger, G.

F. J. V. Caballero, A. Zanaty, F. Pittalá, G. Goeger, Y. Ye, I. T. Monroy, and W. Rosenkranz, “Frequency-domain 2×2 mimo equalizer with stokes space updating algorithm,” in “Signal Processing in Photonic Communications,”(2016), pp. SpW2G-5.

F. J. V. Caballero, A. Zanaty, F. Pittalá, G. Goeger, and Y. Ye, “Efficient SDM-MIMO Stokes-space equalization,” in “European Conference on Optical Communications (ECOC), (2016)

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97(9), 4541–4550 (2000).
[Crossref] [PubMed]

Guiomar, F. P.

Ho, K.

Ho, K.-P.

Holleczek, A.

Huang, H.

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

G. Milione, M. Lavery, H. Huang, Y. Ren, G. Xie, T.A. Nguyen, E. Karimi, L. Marrucci, D. Nolan, R. Alfano, and others, “4×20 Gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de) multiplexer,” Opt. Lett. 40(9), 1980–1983 (2015).
[Crossref] [PubMed]

Idler, W.

F. Buchali, H. Buelow, K. Schuh, and W. Idler, “4D-CMA: Enabling Separation of Channel Compensation and Polarization Demultiplex,” in “Optical Fiber Communication Conference (OFC),” (2015), p. Th2A.15.

Islam, A.

R. Shafik, M. Rahman, and A. Islam, “On the Extended Relationships Among EVM, BER and SNR as Performance Metrics”. In International Conference on Electrical and Computer Engineering (ICECE), 408–411, (2006).

Jiang, J.

Kahn, J.

Kahn, J. M.

Karimi, E.

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97(9), 4541–4550 (2000).
[Crossref] [PubMed]

Lavery, M.

Lavery, M. PJ

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

Leuchs, G.

Li, G.

Li, M. J.

Li, M.-J.

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

Marquardt, C.

Marrucci, L.

Marshall, T.

Mecozzi, A.

Milione, G.

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

G. Milione, M. Lavery, H. Huang, Y. Ren, G. Xie, T.A. Nguyen, E. Karimi, L. Marrucci, D. Nolan, R. Alfano, and others, “4×20 Gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de) multiplexer,” Opt. Lett. 40(9), 1980–1983 (2015).
[Crossref] [PubMed]

G. Milione, D. A. Nolan, and R. R. Alfano, “Determining principal modes in a multimode optical fiber using the mode dependent signal delay method,” J. Opt. Soc. Am. B 32(1), 143–149 (2015).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Physical Review letters 107(5), 053601 (2011).
[Crossref]

Monroy, I. T.

F. J. V. Caballero, A. Zanaty, F. Pittalá, G. Goeger, Y. Ye, I. T. Monroy, and W. Rosenkranz, “Frequency-domain 2×2 mimo equalizer with stokes space updating algorithm,” in “Signal Processing in Photonic Communications,”(2016), pp. SpW2G-5.

Muga, N.

Muga, N. J.

Nebendahl, B.

Nelson, L. E.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nature Photonics 7(5), 354–362 (2013).
[Crossref]

Nguyen, T. A.

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

Nguyen, T.A.

Nolan, D.

Nolan, D. A.

G. Milione, D. A. Nolan, and R. R. Alfano, “Determining principal modes in a multimode optical fiber using the mode dependent signal delay method,” J. Opt. Soc. Am. B 32(1), 143–149 (2015).
[Crossref]

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Physical Review letters 107(5), 053601 (2011).
[Crossref]

M. J. Li and D. A. Nolan, “Fiber spin-profile designs for producing fibers with low polarization mode dispersion,” Opt. Lett. 23(21), 1659–1661 (1998).
[Crossref]

Pinto, A.

Pinto, A. N.

Pittalá, F.

F. J. V. Caballero, A. Zanaty, F. Pittalá, G. Goeger, and Y. Ye, “Efficient SDM-MIMO Stokes-space equalization,” in “European Conference on Optical Communications (ECOC), (2016)

F. J. V. Caballero, A. Zanaty, F. Pittalá, G. Goeger, Y. Ye, I. T. Monroy, and W. Rosenkranz, “Frequency-domain 2×2 mimo equalizer with stokes space updating algorithm,” in “Signal Processing in Photonic Communications,”(2016), pp. SpW2G-5.

Poggiolini, P.

M. Visintin, G. Bosco, P. Poggiolini, and F. Forghieri, “Adaptive digital equalization in optical coherent receivers with Stokes-space update algorithm”, J. Lightwave Technol. 32(24), 4759–4767 (2014).
[Crossref]

G. Bosco, M. Visintin, P. Poggiolini, and F. Forghieri, “A novel update algorithm in stokes space for adaptive equalization in coherent receivers,” in “Optical Fiber Communication Conference (OFC),” (2014), pp. Th3E-6.

Rahman, M.

R. Shafik, M. Rahman, and A. Islam, “On the Extended Relationships Among EVM, BER and SNR as Performance Metrics”. In International Conference on Electrical and Computer Engineering (ICECE), 408–411, (2006).

Ren, Y.

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

G. Milione, M. Lavery, H. Huang, Y. Ren, G. Xie, T.A. Nguyen, E. Karimi, L. Marrucci, D. Nolan, R. Alfano, and others, “4×20 Gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de) multiplexer,” Opt. Lett. 40(9), 1980–1983 (2015).
[Crossref] [PubMed]

Richardson, D. J.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nature Photonics 7(5), 354–362 (2013).
[Crossref]

Rosenkranz, W.

F. J. V. Caballero, A. Zanaty, F. Pittalá, G. Goeger, Y. Ye, I. T. Monroy, and W. Rosenkranz, “Frequency-domain 2×2 mimo equalizer with stokes space updating algorithm,” in “Signal Processing in Photonic Communications,”(2016), pp. SpW2G-5.

Savory, S. J.

S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

Schuh, K.

F. Buchali, H. Buelow, K. Schuh, and W. Idler, “4D-CMA: Enabling Separation of Channel Compensation and Polarization Demultiplex,” in “Optical Fiber Communication Conference (OFC),” (2015), p. Th2A.15.

Shafik, R.

R. Shafik, M. Rahman, and A. Islam, “On the Extended Relationships Among EVM, BER and SNR as Performance Metrics”. In International Conference on Electrical and Computer Engineering (ICECE), 408–411, (2006).

Shahpari, A.

Shemirani, M. B.

Shtaif, M.

Szafraniec, B.

Sztul, H. I.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Physical Review letters 107(5), 053601 (2011).
[Crossref]

Teixeira, A. L.

Tur, M.

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

Visintin, M.

M. Visintin, G. Bosco, P. Poggiolini, and F. Forghieri, “Adaptive digital equalization in optical coherent receivers with Stokes-space update algorithm”, J. Lightwave Technol. 32(24), 4759–4767 (2014).
[Crossref]

G. Bosco, M. Visintin, P. Poggiolini, and F. Forghieri, “A novel update algorithm in stokes space for adaptive equalization in coherent receivers,” in “Optical Fiber Communication Conference (OFC),” (2014), pp. Th3E-6.

Wilde, J. P.

Willner, A. E.

H. Huang, G. Milione, M. PJ Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 21 (2015).

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F. J. V. Caballero, A. Zanaty, F. Pittalá, G. Goeger, and Y. Ye, “Efficient SDM-MIMO Stokes-space equalization,” in “European Conference on Optical Communications (ECOC), (2016)

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F. J. V. Caballero, A. Zanaty, F. Pittalá, G. Goeger, and Y. Ye, “Efficient SDM-MIMO Stokes-space equalization,” in “European Conference on Optical Communications (ECOC), (2016)

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Figures (8)

Fig. 1
Fig. 1 Schematic representation of the decomposition of the Jones space in gs SO(3) subspaces, i.e., higher-order Poincaré spheres. The tributary signals from the same spatial channel are represented in a sphere (i.e., intramode case), whereas the tributary signals from distinct spatial channels are represented in a ellipsoid. Note that, the subindexes f and g are defined in subsection (2.1.1) and (2.1.2) for the intra and intermode cases, respectively.
Fig. 2
Fig. 2 Boundaries of the signals in a higher-order Poincaré sphere produced by the set of matrices Λ(g,2n), with g denoting an arbitrary tributary signal comprised in (11). Insets (a) and (b) show the unit circle which contains an arbitrary modulation format.
Fig. 3
Fig. 3 Four tributary QPSK signals, i.e. two modes with two polarization-multiplexed tributaries, are represented in the higher-order Poincaré spheres with the symmetric plane (i.e. the best fit plane) to the samples. The sub-captions indicate tributary signals, (f, g), represented in each subspace.
Fig. 4
Fig. 4 a) Schematic of the proposed space-demultiplexing step algorithm. The samples are represented in a higher-order Poincaré sphere in order to estimate the normal to the best fit plane. Then, the matrix F(f,g) is calculated and applied to the signal. b) Schematic of the proposed space-demultiplexing algorithm. The received samples are sequentially launched in gs space-demultiplexing steps. Then, the suitable sequence of these steps is chosen by the minimization the sum of the absolute value of the residuals, i.e. the parameter ζ.
Fig. 5
Fig. 5 Representation in the higher-order Poincaré spheres of the QPSK signal, shown in Fig. 3, after transmission through a FMF. In each figure, the deeper red color disks represent the higher order Poincaré sphere.
Fig. 6
Fig. 6 Representation in the higher-order Poincaré spheres of the four tributary QPSK signals after mode-demultiplexing. These signals are obtained from the signal represented in Fig. 5 and the output constellations are well compared with input constellations present in Fig. 3.
Fig. 7
Fig. 7 Convergence of the coefficients of the demultiplexing matrix as function of the number of samples for a QPSK signal. The dash line repressents the coefficients of the channel matrix, Mtot.
Fig. 8
Fig. 8 (a) SNR penalty induced by the space-demultiplexing algorithm as function of the number of samples considered in the calculations of the inverse channel matrix. Inset show in log scale the SNR penalty as function of the number of samples. The QPSK signals are assumed with an optical SNR of 17 dB. The 16-QAM and the 64-QAM signals are assumed with an optical SNR of 23 and 30 dB, respectively. Figure 8(b), (d) and (f) show the constellation diagram before demultiplexing for the QPSK, the 16-QAM and the 64-QAM, respectively. Figure 8(c), (e) and (g) show the constellation diagram after demultiplexing for the QPSK, the 16-QAM and the 64-QAM, respectively. Note that, the number of samples used to calculate the inverse channel matrix are pointed out in Fig. 8(a).

Tables (1)

Tables Icon

Table 1 Penalty induced by the space-demultiplexing algorithm as function of the SNR for a QPSK signal. We assume 10000 samples in the calculation of the inverse channel matrix. The results are presented in decibels.

Equations (52)

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| ψ = ( υ x 1 , υ y 1 , υ x n , υ y n ) T ,
| ψ = ( υ 1 , υ 2 , υ h , υ 2 n ) T ,
υ ( z , t ) = a ( z , t ) e [ i ( ω t + ϕ ( z , t ) ) ] ,
Ψ = ( Ψ 1 , Ψ 2 Ψ D ) T ,
Ψ i = ψ | Λ i | ψ .
Ψ ( f , g ) = ( Ψ 1 ( f , g ) , Ψ 2 ( f , g ) , Ψ 3 ( f , g ) ) T ,
Ψ 1 ( f , g ) = | e f | ψ | 2 | e g | ψ | 2 ,
Ψ 2 ( f , g ) = 2 Re ( e f | ψ * e g | ψ ) ,
Ψ 3 ( f , g ) = 2 Im ( e f | ψ * e g | ψ ) ,
Λ 1 ( f , g ) ( k , l ) = { n if k = g , l = g n if k = f , l = f , 0 otherwise
Λ 2 ( f , g ) ( k , l ) = { n if k = f , l = g n if k = g , l = j , 0 otherwise
Λ 3 ( f , g ) ( k , l ) = { i n if k = f , l = g i n if k = g , l = f , 0 otherwise
Ψ 1 ( f , g ) = ψ | Λ 2 ( f , g ) | ψ = n ( a f 2 a g 2 ) ,
Ψ 2 ( f , g ) = ψ | Λ 2 ( f , g ) | ψ = 2 n a f a g cos δ f g ,
Ψ 3 ( f , g ) = ψ | Λ 3 ( f , g ) | ψ = 2 n a f a g sin δ f g ,
Λ 1 ( f , g ) ( k , l ) = { n n l 2 + n l k if k = g , l = g n n l 2 + n l k if k = f , l = f , 0 otherwise
Λ 2 ( f , g ) ( k , l ) = { n if k = f , l = g n if k = g , l = f , 0 otherwise
Λ 3 ( f , g ) ( k , l ) = { i n if k = f , l = g i n if k = g , l = f , 0 otherwise
Ψ 1 ( f , g ) = ψ | Λ 1 ( f , g ) | ψ = n n l 2 + n l κ ( a f 2 a g 2 ) ,
Ψ 2 ( f , g ) = ψ | Λ 2 ( f , g ) | ψ = 2 n a f a g cos δ f g ,
Ψ 3 ( f , g ) = ψ | Λ 3 ( f , g ) | ψ = 2 n a f a g sin δ f g .
| D = 1 2 ( | e f + | e g ) ,
| A = 1 2 ( | e f | e g ) ,
| R = 1 2 ( | e f + i | e g ) ,
| L = 1 2 ( | e f i | e g ) ,
| ϒ = 1 2 n ( 1 , , 1 , r e i φ ) T ,
Ψ 1 ( f , g ) = κ 2 1 n ( n l 2 + n l ) ( 1 r 2 ) ,
Ψ 2 ( f , g ) = r n cos φ ,
Ψ 3 ( f , g ) = r 2 sin φ .
| ϒ = 1 2 n ( 1 , , r e i φ , , 1 ) T
Ψ 1 ( f , g ) = 1 2 n ( 1 r 2 ) ,
Ψ 2 ( f , g ) = r n cos φ ,
Ψ 3 ( f , g ) = r n sin φ .
| e 1 = ( 1 , 0 , 0 , 0 ) T ,
| e 2 = ( 0 , 1 , 0 , 0 ) T ,
| e 3 = ( 0 , 0 , 1 , 0 ) T ,
| e 4 = ( 0 , 0 , 0 , 1 ) T ,
( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) = 1 2 ( 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ) + 1 2 ( 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ) + 1 2 ( 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 ) + 1 2 ( 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 ) ,
Λ 1 ( 1 , 3 ) = 1 2 ( 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ) ,
Λ 2 ( 1 , 3 ) = 2 ( 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 ) ,
Λ 3 ( 1 , 3 ) = 2 ( 0 0 i 0 0 0 0 0 i 0 0 0 0 0 0 0 ) .
| ψ o u t = M tot ( ω ) | ψ i n ,
M tot ( ω ) = M MD ( ω ) e i 2 ω 2 β ¯ 2 L ,
M MD ( ω ) = k = 1 n s M MD k ( ω ) ,
M MD k ( Ω ) = diag ( e g 1 k 2 e g 2 n k 2 ) l = 1 n step V k l Θ ( Ω ) U k l H ,
Θ ( Ω ) = diag ( e i ω τ 1 e i ω τ i e i ω τ 2 n ) ,
M tot = g = 1 2 n 1 f = g + 1 2 n ( f , g ) ,
a Ψ 1 ( f , g ) + b Ψ 2 ( f , g ) + c Ψ 3 ( f , g ) = 0 ,
| φ o u t = F ( f , g ) | φ i n ,
F ( f , g ) ( k , l ) = { cos ( p ) e i q 2 if k = g , l = g sin ( p ) e i q 2 if k = g , l = f sin ( p ) e i q 2 if k = f , l = g cos ( p ) e i q 2 if k = f , l = f 1 if k = l and k , l f , g 0 otherwise ,
Δ = SNR i n SNR o u t ,
S N R = 1 E V M 2 .

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