Abstract

A self-coherent receiver capable of demultiplexing PolMUX-signals without an external polarization controller is presented. Training sequences are introduced to estimate the polarization rotation, and a decision feedback recursive algorithm mitigates the random walk of the recovered field. The concept is tested for a PolMUX-DQPSK modulation format where one polarization carries a normal DQPSK signal while the other polarization is encoded as a progressive phase-shift DQPSK signal. An experimental demonstration of the scheme for a 112 Gbit/s PolMUX-DQPSK signal is presented.

© 2012 OSA

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2012 (3)

2011 (2)

2010 (2)

B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad/s polarization tracking in 200-Gb/s PDM-RZ-DQPSK transmission over 430 km,” IEEE Photon. Technol. Lett.22(9), 613–615 (2010).
[CrossRef]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett.22(21), 1601–1603 (2010).
[CrossRef]

2009 (1)

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, “C-band optical 90° hybrids based on Silicon-on insulator 4×4 waveguide coupler,” IEEE Photon. Technol. Lett.21(3), 143–145 (2009).
[CrossRef]

2008 (7)

2007 (1)

2006 (1)

2004 (1)

J. M. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron.10(2), 259–272 (2004).
[CrossRef]

2003 (1)

X. Wei, A. H. Gnauck, D. M. Gill, X. Liu, U.-V. Koc, S. Chandrasekhar, G. Raybon, and J. Leuthold, “Optical pi/2-DPSK and its tolerance to filtering and polarization-mode dispersion,” IEEE Photon. Technol. Lett.15(11), 1639–1641 (2003).
[CrossRef]

1988 (1)

M. Oerder and H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun.36(5), 605–612 (1988).
[CrossRef]

1983 (1)

A. J. Viterbi and A. N. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory29(4), 543–551 (1983).
[CrossRef]

Becker, J.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photon. Technol. Lett.24(1), 61–63 (2012).
[CrossRef]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett.22(21), 1601–1603 (2010).
[CrossRef]

Buhl, L. L.

Burrows, E.

Butrie, T.

Cappuzzo, M. A.

Centanni, J.

Chandrasekhar, S.

S. Chandrasekhar, X. Liu, A. Konczykowska, F. Jorge, J. Dupuy, and J. Godin, “Direct detection of 107-Gb/s polarization-multiplexed RZ-DQPSK without optical polarization demultiplexing,” IEEE Photon. Technol. Lett.20(22), 1878–1880 (2008).
[CrossRef]

X. Liu, S. Chandrasekhar, and A. Leven, “Digital self-coherent detection,” Opt. Express16(2), 792–803 (2008).
[CrossRef] [PubMed]

X. Wei, A. H. Gnauck, D. M. Gill, X. Liu, U.-V. Koc, S. Chandrasekhar, G. Raybon, and J. Leuthold, “Optical pi/2-DPSK and its tolerance to filtering and polarization-mode dispersion,” IEEE Photon. Technol. Lett.15(11), 1639–1641 (2003).
[CrossRef]

Charlet, G.

Chen, E. Y.

Chen, W.

De Man, E.

de Waardt, H.

Dentai, A.

Doerr, C.

Doerr, C. R.

Dominic, V.

Dreschmann, M.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photon. Technol. Lett.24(1), 61–63 (2012).
[CrossRef]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett.22(21), 1601–1603 (2010).
[CrossRef]

Dupuy, J.

S. Chandrasekhar, X. Liu, A. Konczykowska, F. Jorge, J. Dupuy, and J. Godin, “Direct detection of 107-Gb/s polarization-multiplexed RZ-DQPSK without optical polarization demultiplexing,” IEEE Photon. Technol. Lett.20(22), 1878–1880 (2008).
[CrossRef]

Duthel, T.

Ellis, A. D.

Ezra, S.-B.

Filsinger, V.

B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad/s polarization tracking in 200-Gb/s PDM-RZ-DQPSK transmission over 430 km,” IEEE Photon. Technol. Lett.22(9), 613–615 (2010).
[CrossRef]

Fludger, C. R. S.

Freude, W.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photon. Technol. Lett.24(1), 61–63 (2012).
[CrossRef]

J. Li, K. Worms, R. Maestle, D. Hillerkuss, W. Freude, and J. Leuthold, “Free-space optical delay interferometer with tunable delay and phase,” Opt. Express19(12), 11654–11666 (2011).
[CrossRef] [PubMed]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett.22(21), 1601–1603 (2010).
[CrossRef]

Geyer, J.

Gill, D. M.

C. R. Doerr, D. M. Gill, A. H. Gnauck, L. L. Buhl, P. J. Winzer, M. A. Cappuzzo, A. Wong-Foy, E. Y. Chen, and L. T. Gomez, “Monolithic demodulator for 40-Gbs DQPSK using a star coupler,” J. Lightwave Technol.24(1), 171–174 (2006).
[CrossRef]

X. Wei, A. H. Gnauck, D. M. Gill, X. Liu, U.-V. Koc, S. Chandrasekhar, G. Raybon, and J. Leuthold, “Optical pi/2-DPSK and its tolerance to filtering and polarization-mode dispersion,” IEEE Photon. Technol. Lett.15(11), 1639–1641 (2003).
[CrossRef]

Gnauck, A.

Gnauck, A. H.

C. R. Doerr, D. M. Gill, A. H. Gnauck, L. L. Buhl, P. J. Winzer, M. A. Cappuzzo, A. Wong-Foy, E. Y. Chen, and L. T. Gomez, “Monolithic demodulator for 40-Gbs DQPSK using a star coupler,” J. Lightwave Technol.24(1), 171–174 (2006).
[CrossRef]

X. Wei, A. H. Gnauck, D. M. Gill, X. Liu, U.-V. Koc, S. Chandrasekhar, G. Raybon, and J. Leuthold, “Optical pi/2-DPSK and its tolerance to filtering and polarization-mode dispersion,” IEEE Photon. Technol. Lett.15(11), 1639–1641 (2003).
[CrossRef]

Godin, J.

S. Chandrasekhar, X. Liu, A. Konczykowska, F. Jorge, J. Dupuy, and J. Godin, “Direct detection of 107-Gb/s polarization-multiplexed RZ-DQPSK without optical polarization demultiplexing,” IEEE Photon. Technol. Lett.20(22), 1878–1880 (2008).
[CrossRef]

Goldfarb, G.

Gomez, L. T.

Gottwald, E.

Higuma, K.

Hillerkuss, D.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photon. Technol. Lett.24(1), 61–63 (2012).
[CrossRef]

J. Li, K. Worms, R. Maestle, D. Hillerkuss, W. Freude, and J. Leuthold, “Free-space optical delay interferometer with tunable delay and phase,” Opt. Express19(12), 11654–11666 (2011).
[CrossRef] [PubMed]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett.22(21), 1601–1603 (2010).
[CrossRef]

Ho, K.-P.

J. M. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron.10(2), 259–272 (2004).
[CrossRef]

Huebner, M.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photon. Technol. Lett.24(1), 61–63 (2012).
[CrossRef]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett.22(21), 1601–1603 (2010).
[CrossRef]

Jansen, S. L.

Jorge, F.

S. Chandrasekhar, X. Liu, A. Konczykowska, F. Jorge, J. Dupuy, and J. Godin, “Direct detection of 107-Gb/s polarization-multiplexed RZ-DQPSK without optical polarization demultiplexing,” IEEE Photon. Technol. Lett.20(22), 1878–1880 (2008).
[CrossRef]

Josten, A.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photon. Technol. Lett.24(1), 61–63 (2012).
[CrossRef]

Kahn, J. M.

J. M. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron.10(2), 259–272 (2004).
[CrossRef]

Kato, M.

Kawanishi, T.

Khoe, G. D.

Khoe Giok-Djan,

Kikuchi, N.

Kish, F.

Koc, U.-V.

X. Wei, A. H. Gnauck, D. M. Gill, X. Liu, U.-V. Koc, S. Chandrasekhar, G. Raybon, and J. Leuthold, “Optical pi/2-DPSK and its tolerance to filtering and polarization-mode dispersion,” IEEE Photon. Technol. Lett.15(11), 1639–1641 (2003).
[CrossRef]

Koch, B.

B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad/s polarization tracking in 200-Gb/s PDM-RZ-DQPSK transmission over 430 km,” IEEE Photon. Technol. Lett.22(9), 613–615 (2010).
[CrossRef]

Koenig, S.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photon. Technol. Lett.24(1), 61–63 (2012).
[CrossRef]

Konczykowska, A.

S. Chandrasekhar, X. Liu, A. Konczykowska, F. Jorge, J. Dupuy, and J. Godin, “Direct detection of 107-Gb/s polarization-multiplexed RZ-DQPSK without optical polarization demultiplexing,” IEEE Photon. Technol. Lett.20(22), 1878–1880 (2008).
[CrossRef]

Koos, C.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photon. Technol. Lett.24(1), 61–63 (2012).
[CrossRef]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett.22(21), 1601–1603 (2010).
[CrossRef]

Krummrich, P. M.

Kuntz, M.

Lal, V.

Lambert, D.

Leuthold, J.

I. Tselniker, M. Nazarathy, S.-B. Ezra, J. Li, and J. Leuthold, “Self-coherent complex field reconstruction with in-phase and quadrature delay detection without a direct-detection branch,” Opt. Express20(14), 15452–15473 (2012).
[CrossRef] [PubMed]

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photon. Technol. Lett.24(1), 61–63 (2012).
[CrossRef]

J. Li, K. Worms, R. Maestle, D. Hillerkuss, W. Freude, and J. Leuthold, “Free-space optical delay interferometer with tunable delay and phase,” Opt. Express19(12), 11654–11666 (2011).
[CrossRef] [PubMed]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett.22(21), 1601–1603 (2010).
[CrossRef]

X. Wei, A. H. Gnauck, D. M. Gill, X. Liu, U.-V. Koc, S. Chandrasekhar, G. Raybon, and J. Leuthold, “Optical pi/2-DPSK and its tolerance to filtering and polarization-mode dispersion,” IEEE Photon. Technol. Lett.15(11), 1639–1641 (2003).
[CrossRef]

Leven, A.

Li, J.

Little, B.

Liu, X.

X. Liu, S. Chandrasekhar, and A. Leven, “Digital self-coherent detection,” Opt. Express16(2), 792–803 (2008).
[CrossRef] [PubMed]

S. Chandrasekhar, X. Liu, A. Konczykowska, F. Jorge, J. Dupuy, and J. Godin, “Direct detection of 107-Gb/s polarization-multiplexed RZ-DQPSK without optical polarization demultiplexing,” IEEE Photon. Technol. Lett.20(22), 1878–1880 (2008).
[CrossRef]

X. Wei, A. H. Gnauck, D. M. Gill, X. Liu, U.-V. Koc, S. Chandrasekhar, G. Raybon, and J. Leuthold, “Optical pi/2-DPSK and its tolerance to filtering and polarization-mode dispersion,” IEEE Photon. Technol. Lett.15(11), 1639–1641 (2003).
[CrossRef]

Maestle, R.

Malendevich, R.

Mandai, K.

McCarthy, M. E.

Meyer, J.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photon. Technol. Lett.24(1), 61–63 (2012).
[CrossRef]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett.22(21), 1601–1603 (2010).
[CrossRef]

Meyr, H.

M. Oerder and H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun.36(5), 605–612 (1988).
[CrossRef]

Mirvoda, V.

B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad/s polarization tracking in 200-Gb/s PDM-RZ-DQPSK transmission over 430 km,” IEEE Photon. Technol. Lett.22(9), 613–615 (2010).
[CrossRef]

Nagarajan, R.

Nazarathy, M.

Nebendahl, B.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error vector magnitude as a performance measure for advanced modulation formats,” IEEE Photon. Technol. Lett.24(1), 61–63 (2012).
[CrossRef]

R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett.22(21), 1601–1603 (2010).
[CrossRef]

Nilsson, A.

Noé, R.

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X. Wei, A. H. Gnauck, D. M. Gill, X. Liu, U.-V. Koc, S. Chandrasekhar, G. Raybon, and J. Leuthold, “Optical pi/2-DPSK and its tolerance to filtering and polarization-mode dispersion,” IEEE Photon. Technol. Lett.15(11), 1639–1641 (2003).
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R. Schmogrow, D. Hillerkuss, M. Dreschmann, M. Huebner, M. Winter, J. Meyer, B. Nebendahl, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Real-time software-defined multiformat transmitter generating 64QAM at 28 GBd,” IEEE Photon. Technol. Lett.22(21), 1601–1603 (2010).
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Figures (12)

Fig. 1
Fig. 1

Polarization multiplexed transmission system with self-coherent receiver (SCD Rx). (a) System schematic with SCD Rx and its DSP modules. Transmitters Txx and Txy generate two signals Ex and Ey carrying data Sx and Sy, which are then combined in a PBC to form a PolMUX signal. In the fiber the signal experiences a random change of the state of polarization described by a matrix (C) which consists of arbitrary polarization rotations (R)n, arbitrary order of PMDn, and arbitrary phase offset (P). The signal is then detected in a SCD Rx. There are two options for realizing the optical front-end (OFE) for detecting the in-phase (I) and quadrature phase (Q) signal: (b) OFE with two quadrature phase-offset DIs (IQ-DI) (c) OFE with optical hybrid.

Fig. 2
Fig. 2

(a) Illustration of Tx and Rx polarization states. (b) Constellation diagram as sent off by the Txx (× ) and Txy (× ) and constellation diagrams as received by the Rx in the two polarizations x’ (•) and y’ (•) when being sent over the channel in Eq. (2). The two QPSK (× , × ) are mixed and form new constellations E x = ( E x E y ) / 2 (•) and E y = ( E x + E y ) / 2 (•). Zeros in the center of the constellations result from destructive interference.

Fig. 3
Fig. 3

Schematic drawing of the transmitter. S(n) is differentially encoded into a A(n) by adding A (n1)= A(n1) / | A(n1) | . The symbol z−1 (representing the z-transform) stands for a time delay by one bit. Then the A(n) is modulated on an optical carrier with angular frequency ω. The mixer output is the time sequence E(n).

Fig. 4
Fig. 4

Equivalent digital representation of the self-coherent detection receiver (SCD Rx). The received signal E x,y (n) is combined with its conjugated delayed copy that generates u x,y (n) . The symbol z−1 (representing the z-transform) stands for a time delay by one bit.

Fig. 5
Fig. 5

Equivalent digital representation of the PolMUX signal transmission system. A PolMUX signal E x and E y carries encoded symbols of Sx and Sy. After the channel transmission, the receiver projects the signals on new polarization axes which results in E x and E y . In the SCD Rx, the signal interferes with its delayed copy generating quantities u x and u y . After polarization and field recovery the transmitted symbols Sx and Sy are recovered and sent for further evaluation.

Fig. 6
Fig. 6

Polarization and field recovery algorithm to derive the symbol Sx and Sy from complex signal u x,y (n) . (a) The conventional field recovery algorithm using solely Eq. (8). The recovered fields E ^ x,y (n) are then sent to polarization demultiplexing (PolDEMUX) algorithm and differential decoder to retrieve Sx and Sy. This algorithm fails when there is noise accumulation. (b) New polarization and field recovery algorithms. It first performs the PolDEMUX, then uses decision circuits to remove noise. The ‘clean’ signal E ˜ x,y (n) is then multiplied with SOP change to generate the reference E ˜ x,y (n1) for the next field recovery.

Fig. 7
Fig. 7

Examples of PolMUX-DQPSK signal constellation diagrams wherein there is polarization mixing between the two signals. (a), the constellations of E x / C 11 (•) or E y / C 22 (•) have more weight from E x (×) or E y (×). (b), E x / C 11 or E y / C 22 have more weight from sub-constellation ( C 12 / C 11 ) E y or ( C 21 / C 22 ) E x . Symbol combinations close to zero are detected that might cause outages in the field reconstruction.

Fig. 8
Fig. 8

Example of a PolMUX-DQPSK signal constellations when the linear transformation (C) in the fiber consists of an exact −45° rotation. In the constellation of E x (•), the signal transits from 1x to 2x to the ambiguous point 3x. In the constellation of E y (•), the signal transits from 1y to 2y to 3y or 3′y and the yellow points 1 1 2 y , 2 1 2 y , and 2 1 2 y are the halfway samples.

Fig. 9
Fig. 9

PolMUX DQPSK signal constellation diagrams. Top row shows the phasor of the signals (Ex and Ey) and their transition lines, the bottom row shows the phasors of the encoded symbols (Sx and Sy) which are also the transitions of the signals (Ex and Ey).

Fig. 10
Fig. 10

Experimental setup, the signal is generated in a software defined Tx as a series of frames. In each frame, there is a training sequence (TSx,y) for one polarization followed with a DQPSK signal or a progressive-phase DQPSK signal. Then the signal is split and combined in a PBC to form a PolMUX signal. The signal is then experienced with an arbitrary polarization rotation and sent to the self-coherent receiver. An ASE source is used to emulate different OSNR levels.

Fig. 11
Fig. 11

Constellation diagrams of detected signals u x,y in x and y polarizations together with the recovered symbols E ^ x,y for 6 different polarization states. All measurements were performed at low ASE noise, and 3072 symbols were evaluated.

Fig. 12
Fig. 12

BER versus OSNR plots of self-coherent receiver for 6 different polarization states with polarization and field recovery (PFR) algorithm (solid symbols) and without PFR algorithm (empty symbols). The black curve shows the OSNR versus BER for a coherent receiver.

Equations (25)

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u x,y (t)= E x,y (t) E x,y *(tτ).
[ E x E y ]=[ C 11 C 12 C 21 C 22 ][ E x E y ],C=P R n PM D n R n1 R 2 PM D 1 R 1 , R n =[ cos( θ n ) sin( θ n ) sin( θ n ) cos( θ n ) ],PM D n (Δω)=[ 1 0 0 e j(Δω τ DGDn +δ ϕ PMDn ) ],P=[ 1 0 0 e jδ ϕ PBS ].
[ E x E y ]= 1 2 [ I x +j Q x I y +j Q y ]exp(jωt)= 1 2 [ 1 1 1 1 ][ E x E y ]= 1 2 [ E x E y E x + E y ].
A(n)=S(n) A (n1),
A x (1...8):011 1100 A x (8) A y (1...8):011 0011 A y (8) .
u x,y (n)= u x,y (n)= E x,y *(n1) E x,y (n)= A x,y *(n1) A x,y (n) = A x,y *(n1) S x,y (n) A x,y (n1)=| A x,y (n1) | S x,y (n),
u x (n)= E x *(n1) E x (n), u y (n)= E y *(n1) E y (n),
E ^ x (n)= u x (n) / E ^ x *(n1) , E ^ y (n)= u y (n) / E ^ y *(n1) .
S ^ x (n)= E ^ x *(n1) E ^ x (n)= A x *(n1) S x (n) A x (n1)= S x (n), S ^ y (n)= E ^ y *(n1) E ^ y (n)= A y *(n1) S y (n) A y (n1)= S y (n).
E out1 (t)=E(t)+E(tτ), E out2 (t)=E(t)E(tτ).
I out1 (t) | E(t)+E(tτ) | 2 | E(t) | 2 + | E(tτ) | 2 +2| E(t) || E(tτ) |cos(E(t)E(tτ)), I out2 (t) | E(t)E(tτ) | 2 | E(t) | 2 + | E(tτ) | 2 2| E(t) || E(tτ) |cos(E(t)E(tτ)),
I BR (t)= I out1 (t) I out2 (t)| E(t) || E(tτ) |cos(E(t)E(tτ)).
I BR (I) (t)| E(t) || E(tτ) |cos(E(t)E(tτ)), I BR (Q) (t)| E(t) || E(tτ) |cos(E(t)E(tτ)+ π 2 ) | E(t) || E(tτ) |sin(E(t)E(tτ)).
u(t)= I BR (I) (t)j I BR (Q) (t) | E(t) || E(tτ) |cos(E(t)E(tτ))+j| E(t) || E(tτ) |sin(E(t)E(tτ)) | E(t) || E(tτ) |exp[ j(E(t)E(tτ)) ]=E(t)E*(tτ).
A x (1...8):011 1100 A x (8) A y (1...8):011 0011 A y (8)
[ E x (n) E y (n) ] n=1...8 =[ C 11 C 12 C 21 C 22 ] [ E x (n) E y (n) ] n=1...8 ={ [ 0 0 ],[ C 11 + C 12 C 21 + C 22 ],[ C 11 + C 12 C 21 + C 22 ], [ C 11 C 21 ],[ C 11 C 21 ],[ C 12 C 22 ],[ C 12 C 22 ],[ C 11 E x (8)+ C 12 E y (8) C 21 E x (8)+ C 22 E y (8) ] }.
[ u x (n) u y (n) ] n=1...8 = [ E x *(n1) E x (n) E y *(n1) E y (n) ] n=1...8 ={ [ 0 0 ],[ 0 0 ],[ ( C 11 + C 12 )*( C 11 + C 12 ) ( C 21 + C 22 )*( C 21 + C 22 ) ],[ ( C 11 + C 12 )* C 11 ( C 21 + C 22 )* C 21 ], [ C 11 * C 11 C 21 * C 21 ],[ C 11 * C 12 C 21 * C 22 ],[ C 12 * C 12 C 22 * C 22 ],[ C 12 *( C 11 E x (8)+ C 12 E y (8)) C 22 *( C 21 E x (8)+ C 22 E y (8)) ] }.
u x (5)= C 11 * C 11 = | C 11 | 2 , u x (6)= C 11 * C 12 , u x (7)= C 12 * C 12 = | C 12 | 2 , u y (5)= C 21 * C 21 = | C 21 | 2 , u y (6)= C 21 * C 22 , u y (7)= C 22 * C 22 = | C 22 | 2 .
C ^ 11 = | u x (5) | 1 2 exp( j u x (6) )= C 11 exp[ j( C 12 ) ]= C 11 C 12 *, C ^ 12 = | u x (7) | 1 2 = C 12 exp[ j( C 12 ) ]= C 12 C 12 *, C ^ 21 = | u y (5) | 1 2 exp( j u y (6) )= C 21 exp[ j( C 22 ) ]= C 21 C 22 *, C ^ 22 = | u y (7) | 1 2 = C 22 exp[ j( C 22 ) ]= C 22 C 22 *.
[ E ^ x (8) E ^ y (8) ]=[ C ^ 11 E x (8)+ C ^ 12 E y (8) C ^ 21 E x (8)+ C ^ 22 E y (8) ]=[ E x (8) C 12 * E y (8) C 22 * ].
[ E ^ x (9) E ^ y (9) ]=[ u x (9) / E ^ x *(8) u y (9) / E ^ y *(8) ]=[ E x (9) C 12 * E y (9) C 22 * ].
[ E ^ x ( 9 ) E ^ y ( 9 ) ]= [ C ^ 11 C ^ 12 C ^ 21 C ^ 22 ] 1 [ E ^ x ( 9 ) E ^ y ( 9 ) ]= 1 C ^ 11 C ^ 22 C ^ 12 C ^ 21 [ C ^ 22 C ^ 12 C ^ 21 C ^ 11 ][ E x (9) C 12 * E y (9) C 22 * ] = 1 C 11 C 22 C 12 * C 22 * C ^ 12 C ^ 21 C 12 * C 22 * [ C 22 C 12 * C 22 * C 12 C 12 * C 22 * C 21 C 12 * C 22 * C 11 C 12 * C 22 * ][ E x (9) E y (9) ].
[ E ^ x (9) E ^ y (9) ]= 1 C 11 C 22 C 12 C 21 [ C 22 C 12 C 21 C 11 ][ E x (9) E y (9) ] = [ C 11 C 12 C 21 C 22 ] 1 [ C 11 C 12 C 21 C 22 ][ E x (9) E y (9) ]=[ E x (9) E y (9) ].
[ E ^ x ( 9 ) E ^ y ( 9 ) ]=[ C ^ 11 C ^ 12 C ^ 21 C ^ 22 ][ E ˜ x (9) E ˜ y (9) ]=[ C ^ 11 E x (9)+ C ^ 12 E y (9) C ^ 21 E x (9)+ C ^ 22 E y (9) ] =[ C 12 *( C 11 E x (9)+ C 12 E y (9)) C 22 *( C 21 E x (9)+ C 22 E y (9)) ]=[ E x (9) C 12 * E y (9) C 22 * ].
[ C 11 C 12 C 21 C 22 ]=[ 1 0 0 1 ]or[ C 11 C 12 C 21 C 22 ]=[ 0 1 1 0 ].

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