Polarization demultiplexing in Stokes space

Bogdan Szafraniec; Bernd Nebendahl; Todd Marshall

doi:10.1364/OE.18.017928

Polarization demultiplexing in Stokes space

Bogdan Szafraniec, Bernd Nebendahl, and Todd Marshall

Optics Express
Vol. 18,
Issue 17,
pp. 17928-17939
(2010)
•https://doi.org/10.1364/OE.18.017928

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Bogdan Szafraniec, Bernd Nebendahl, and Todd Marshall, "Polarization demultiplexing in Stokes space," Opt. Express 18, 17928-17939 (2010)

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Related Topics
Table of Contents Category
- Fiber Optics and Optical Communications
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Coherent communications (060.1660)
- Modulation (060.4080)
- Multiplexing (060.4230)

About this Article
History
- Original Manuscript: June 13, 2010
- Revised Manuscript: July 22, 2010
- Manuscript Accepted: July 28, 2010
- Published: August 5, 2010

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Open Access

Abstract

A technique is demonstrated for polarization demultiplexing of arbitrary complex-modulated signals. The technique is based entirely on the observation of samples in Stokes space, does not involve demodulation and is modulation format independent. The data in Stokes space is used to find the best fit plane and the normal to it which contains the origin. This normal identifies the two orthogonal polarization states of transmission and the desired polarization alignment transformation matrix. The technique is verified experimentally and is compared with the constant modulus algorithm.

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References

View by:
Article Order
|
Year
|
Author
|
Publication

J. Treichler and B. Agee, “A new approach to multipath correction of constant modulus signals,” IEEE Trans. Acoust. Speech Signal Process. 31(2), 459–472 (1983).
[Crossref]
A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[Crossref]
H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing, (John Wiley & Sons, 1998).
R. Noe, “Phase noise-tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with feedforward carrier recovery,” IEEE J. Lightwave Technol. 23(2), 802–808 (2005).
[Crossref]
J. Renaudier, G. Charlet, M. Salsi, O. B. Pardo, H. Mardoyan, P. Tran, and S. Bigo, “Linear fiber impairments mitigation of 40-Gbit/s polarization-multiplexed QPSK by digital processing in a coherent receiver,” IEEE J. Lightwave Technol. 26(1), 36–42 (2008).
[Crossref]
S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
[Crossref] [PubMed]
P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Measurement of PMD tolerance in 40-Gb/s polarization-multiplexed RZ-DQPSK,” Opt. Express 16(17), 13398–13404 (2008).
[Crossref] [PubMed]
M. Martinelli, P. Martelli, and S. M. Pietralunga, “Polarization stabilization in optical communications systems,” IEEE J. Lightwave Technol. 24(11), 4172–4183 (2006).
[Crossref]
M. Tseytlin, O. Ritterbush, and A. Salamon, “Digital, endless polarization control for polarization multiplexed fiber-optic communications,” in Optical Fiber Communication Conference, Technical Digest (Optical Society of America, 2003), paper MF83.
N. Leven, Kaneda, and Y. Chen, “A real-time CMA-Based 10 Gb/s polarization demultiplexing coherent receiver implemented in an FPGA,” in Optical Fiber Communication Conference and National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuO2.
D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. Commun. Radar and Signal Process. F 130(1), 11–16 (1983).
[Crossref]
H. Zhang, Z. Tao, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, “Polarization demultiplexing based on independent component analysis in optical coherent receivers,” in 34th European Conference on Optical Communication, ECOC 2008, (Brussels, 2008), pp. 1–2.
G. Arfken, Mathematical Methods for Physicists, (Academic Press, 1985).
C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach, (John Wiley & Sons, 1998), Chap. 4.
C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal states in a 147-km undersea lightwave cable,” IEEE J. Lightwave Technol. 6(7), 1185–1190 (1988).
[Crossref]

2008 (3)

J. Renaudier, G. Charlet, M. Salsi, O. B. Pardo, H. Mardoyan, P. Tran, and S. Bigo, “Linear fiber impairments mitigation of 40-Gbit/s polarization-multiplexed QPSK by digital processing in a coherent receiver,” IEEE J. Lightwave Technol. 26(1), 36–42 (2008).
[Crossref]

S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
[Crossref] [PubMed]

P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Measurement of PMD tolerance in 40-Gb/s polarization-multiplexed RZ-DQPSK,” Opt. Express 16(17), 13398–13404 (2008).
[Crossref] [PubMed]

2006 (1)

M. Martinelli, P. Martelli, and S. M. Pietralunga, “Polarization stabilization in optical communications systems,” IEEE J. Lightwave Technol. 24(11), 4172–4183 (2006).
[Crossref]

2005 (1)

R. Noe, “Phase noise-tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with feedforward carrier recovery,” IEEE J. Lightwave Technol. 23(2), 802–808 (2005).
[Crossref]

1988 (1)

C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal states in a 147-km undersea lightwave cable,” IEEE J. Lightwave Technol. 6(7), 1185–1190 (1988).
[Crossref]

1983 (3)

J. Treichler and B. Agee, “A new approach to multipath correction of constant modulus signals,” IEEE Trans. Acoust. Speech Signal Process. 31(2), 459–472 (1983).
[Crossref]

A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[Crossref]

D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. Commun. Radar and Signal Process. F 130(1), 11–16 (1983).
[Crossref]

Agee, B.

J. Treichler and B. Agee, “A new approach to multipath correction of constant modulus signals,” IEEE Trans. Acoust. Speech Signal Process. 31(2), 459–472 (1983).
[Crossref]

Bergano, N. S.

Bigo, S.

Boffi, P.

Brandwood, D. H.

D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. Commun. Radar and Signal Process. F 130(1), 11–16 (1983).
[Crossref]

Charlet, G.

Ferrario, M.

Marazzi, L.

Mardoyan, H.

Martelli, P.

M. Martinelli, P. Martelli, and S. M. Pietralunga, “Polarization stabilization in optical communications systems,” IEEE J. Lightwave Technol. 24(11), 4172–4183 (2006).
[Crossref]

Martinelli, M.

M. Martinelli, P. Martelli, and S. M. Pietralunga, “Polarization stabilization in optical communications systems,” IEEE J. Lightwave Technol. 24(11), 4172–4183 (2006).
[Crossref]

Noe, R.

R. Noe, “Phase noise-tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with feedforward carrier recovery,” IEEE J. Lightwave Technol. 23(2), 802–808 (2005).
[Crossref]

Pardo, O. B.

Parolari, P.

Pietralunga, S. M.

M. Martinelli, P. Martelli, and S. M. Pietralunga, “Polarization stabilization in optical communications systems,” IEEE J. Lightwave Technol. 24(11), 4172–4183 (2006).
[Crossref]

Poole, C. D.

Renaudier, J.

Righetti, A.

Salsi, M.

Savory, S. J.

S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
[Crossref] [PubMed]

Schulte, H. J.

Siano, R.

Tran, P.

Treichler, J.

J. Treichler and B. Agee, “A new approach to multipath correction of constant modulus signals,” IEEE Trans. Acoust. Speech Signal Process. 31(2), 459–472 (1983).
[Crossref]

Viterbi, A. J.

A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[Crossref]

Viterbi, A. M.

A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[Crossref]

Wagner, R. E.

IEE Proc. Commun. Radar and Signal Process. F (1)

D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. Commun. Radar and Signal Process. F 130(1), 11–16 (1983).
[Crossref]

IEEE J. Lightwave Technol. (4)

R. Noe, “Phase noise-tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with feedforward carrier recovery,” IEEE J. Lightwave Technol. 23(2), 802–808 (2005).
[Crossref]

M. Martinelli, P. Martelli, and S. M. Pietralunga, “Polarization stabilization in optical communications systems,” IEEE J. Lightwave Technol. 24(11), 4172–4183 (2006).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process. (1)

J. Treichler and B. Agee, “A new approach to multipath correction of constant modulus signals,” IEEE Trans. Acoust. Speech Signal Process. 31(2), 459–472 (1983).
[Crossref]

IEEE Trans. Inf. Theory (1)

A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[Crossref]

Opt. Express (2)

S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
[Crossref] [PubMed]

Other (6)

H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing, (John Wiley & Sons, 1998).

M. Tseytlin, O. Ritterbush, and A. Salamon, “Digital, endless polarization control for polarization multiplexed fiber-optic communications,” in Optical Fiber Communication Conference, Technical Digest (Optical Society of America, 2003), paper MF83.

N. Leven, Kaneda, and Y. Chen, “A real-time CMA-Based 10 Gb/s polarization demultiplexing coherent receiver implemented in an FPGA,” in Optical Fiber Communication Conference and National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuO2.

H. Zhang, Z. Tao, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, “Polarization demultiplexing based on independent component analysis in optical coherent receivers,” in 34th European Conference on Optical Communication, ECOC 2008, (Brussels, 2008), pp. 1–2.

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

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach, (John Wiley & Sons, 1998), Chap. 4.

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Fig. 1

Ideal polarization-multiplexed QPSK data in Stokes space. Inset shows QPSK data in the complex plane.

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Fig. 2

Boundaries of the polarization-multiplexed data in Stokes space. Inset shows hypothetical modulation format covering all points within the unit circle of the complex plane.

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Fig. 3

Measured 40Gb/s polarization-multiplexed QPSK data in Stokes space. The data set includes transitions that traverse the complex plane. The upper-left inset shows the same set of data after polarization alignment.

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Fig. 4

Demodulated 40Gb/s polarization-multiplexed QPSK data from two polarization channels. Each polarization channel shows about 10000 symbols.

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Fig. 5

Speed of convergence of the CMA technique for different values of step size μ.

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Fig. 6

Speed of convergence of Stokes space polarization alignment.

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Equations (12)

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W_{i + 1} = W_{i} - μ ε_{i} X_{i}^{H},

M = (\begin{matrix} a & b \\ - b^{*} & a^{*} \end{matrix}) .

M^{- 1} = (\begin{matrix} a^{*} & - b \\ b^{*} & a \end{matrix}) .

E = \frac{1}{\sqrt{2}} (\begin{matrix} e_{x} \\ e_{y} \end{matrix}) = \frac{1}{\sqrt{2}} (\begin{matrix} a_{x} \exp (j ω t + j ϕ_{x}) \\ a_{y} \exp (j ω t + j ϕ_{y}) \end{matrix}),

S = (\begin{matrix} s_{0} \\ s_{1} \\ s_{2} \\ s_{3} \end{matrix}) = \frac{1}{2} (\begin{matrix} e_{x} e_{x}^{*} + e_{y} e_{y}^{*} \\ e_{x} e_{x}^{*} - e_{y} e_{y}^{*} \\ e_{x}^{*} e_{y} + e_{x} e_{y}^{*} \\ - j e_{x}^{*} e_{y} + j e_{x} e_{y}^{*} \end{matrix}) = \frac{1}{2} (\begin{matrix} a_{x}^{2} + a_{y}^{2} \\ a_{x}^{2} - a_{y}^{2} \\ 2 a_{x} a_{y} \cos Δ ϕ \\ 2 a_{x} a_{y} \sin Δ ϕ \end{matrix}),

E = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ r \exp (j θ) \end{matrix}),

S = (\begin{matrix} s_{0} \\ s_{1} \\ s_{2} \\ s_{3} \end{matrix}) = \frac{1}{2} (\begin{matrix} 1 + r^{2} \\ 1 - r^{2} \\ 2 r \cos θ \\ 2 r \sin θ \end{matrix}) .

J_{1} = (\begin{matrix} \cos (α) \\ \sin (α) \exp (j Δ ϕ) \end{matrix}), J_{2} = (\begin{matrix} \sin (α) \\ - \cos (α) \exp (j Δ ϕ) \end{matrix}),

M^{- 1} = (\begin{matrix} \cos (α) \exp (j Δ ϕ / 2) & \sin (α) \exp (- j Δ ϕ / 2) \\ - \sin (α) \exp (j Δ ϕ / 2) & \cos (α) \exp (- j Δ ϕ / 2) \end{matrix}) .

(\begin{matrix} e_{h} \\ e_{v} \end{matrix}) = M^{- 1} (\begin{matrix} e_{x} \\ e_{y} \end{matrix}) .

ε_{i} = x_{i} - s .

ε_{i} = (\begin{matrix} (y_{h} y_{h}^{*} - 1) y_{h} \\ (y_{v} y_{v}^{*} - 1) y_{v} \end{matrix}) .

Abstract

References

2008 (3)

2006 (1)

2005 (1)

1988 (1)

1983 (3)

Agee, B.

Bergano, N. S.

Bigo, S.

Boffi, P.

Brandwood, D. H.

Charlet, G.

Ferrario, M.

Marazzi, L.

Mardoyan, H.

Martelli, P.

Martinelli, M.

Noe, R.

Pardo, O. B.

Parolari, P.

Pietralunga, S. M.

Poole, C. D.

Renaudier, J.

Righetti, A.

Salsi, M.

Savory, S. J.

Schulte, H. J.

Siano, R.

Tran, P.

Treichler, J.

Viterbi, A. J.

Viterbi, A. M.

Wagner, R. E.

IEE Proc. Commun. Radar and Signal Process. F (1)

IEEE J. Lightwave Technol. (4)

IEEE Trans. Acoust. Speech Signal Process. (1)

IEEE Trans. Inf. Theory (1)

Opt. Express (2)

Other (6)

Cited By

Figures (6)

Equations (12)

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