Abstract

In the digital coherent optical receiver, we can achieve polarization demultiplexing in the digital domain, using a two-by-two matrix controlled by the constant-modulus algorithm (CMA). In this paper, after elucidating the physics behind CMA for polarization demultiplexing, we discuss the performance limit of CMA-based polarization demultiplexing through computer simulations. The method of improving its performance is also demonstrated.

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  1. K. Kikuchi and S. Tsukamoto, “Evaluation of sensitivity of the digital coherent receiver,” J. Lightwave. Technol. 26, 1817–1822 (2008).
    [CrossRef]
  2. K. Kikuchi, “Coherent optical communications: Historical perspectives and future directions,” in High Spectral Density Optical Communication Technology , M. Nakazawa, K. Kikuchi, and T. Miyazaki, eds. (Springer, 2010), Chap. 2.
    [CrossRef]
  3. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16, 804–817 (2008).
    [CrossRef] [PubMed]
  4. K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” in 2008 Technical Digest of IEEE/LEOS Summer Topical Meeting (IEEE/LEOS, 2008), paper MC2.2.
  5. L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in 2009 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2009), paper OMT2.
  6. C. Xie and S. Chandrasekhar, “Two-stage constant modulus algorithm equalizer for singularity free operation and optical performance monitoring in optical coherent receiver,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), paper OMK3.
  7. S. Hinz, D. Sandel, F. Wüst, and R. Noé, “Interference detection enabling 2×20Gbit/s RZ polarisation division multiplex transmission,” Electron. Lett. 37, 511–512 (2001).
    [CrossRef]
  8. M. Yagi, S. Satomi, and S. Ryu, “Field trial of 160-Gbit/s, polarization-division multiplexed RZ-DQPSK transmission system using automatic polarization control,” in 2008 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2008), paper OThT7.
  9. D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun . COM-28, 1867–1875 (1980).
    [CrossRef]
  10. S. Ryu, Coherent Lightwave Communication Systems (Artech House, Inc., 1995), Chap. 6.
  11. Md. S. Faruk, Y. Mori, C. Zhang, K. Igarashi, and K. Kikuchi, “Multi-impairment monitoring from adaptive finite-impulse-response filters in a digital coherent receiver,” Opt. Express 18, 26929–26936 (2010).
    [CrossRef]

2010

2008

K. Kikuchi and S. Tsukamoto, “Evaluation of sensitivity of the digital coherent receiver,” J. Lightwave. Technol. 26, 1817–1822 (2008).
[CrossRef]

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

2001

S. Hinz, D. Sandel, F. Wüst, and R. Noé, “Interference detection enabling 2×20Gbit/s RZ polarisation division multiplex transmission,” Electron. Lett. 37, 511–512 (2001).
[CrossRef]

1980

D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun . COM-28, 1867–1875 (1980).
[CrossRef]

Chandrasekhar, S.

C. Xie and S. Chandrasekhar, “Two-stage constant modulus algorithm equalizer for singularity free operation and optical performance monitoring in optical coherent receiver,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), paper OMK3.

Faruk, Md. S.

Godard, D. N.

D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun . COM-28, 1867–1875 (1980).
[CrossRef]

Hinz, S.

S. Hinz, D. Sandel, F. Wüst, and R. Noé, “Interference detection enabling 2×20Gbit/s RZ polarisation division multiplex transmission,” Electron. Lett. 37, 511–512 (2001).
[CrossRef]

Hoshida, T.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in 2009 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2009), paper OMT2.

Igarashi, K.

Kikuchi, K.

Md. S. Faruk, Y. Mori, C. Zhang, K. Igarashi, and K. Kikuchi, “Multi-impairment monitoring from adaptive finite-impulse-response filters in a digital coherent receiver,” Opt. Express 18, 26929–26936 (2010).
[CrossRef]

K. Kikuchi and S. Tsukamoto, “Evaluation of sensitivity of the digital coherent receiver,” J. Lightwave. Technol. 26, 1817–1822 (2008).
[CrossRef]

K. Kikuchi, “Coherent optical communications: Historical perspectives and future directions,” in High Spectral Density Optical Communication Technology , M. Nakazawa, K. Kikuchi, and T. Miyazaki, eds. (Springer, 2010), Chap. 2.
[CrossRef]

K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” in 2008 Technical Digest of IEEE/LEOS Summer Topical Meeting (IEEE/LEOS, 2008), paper MC2.2.

Liu, L.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in 2009 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2009), paper OMT2.

Mori, Y.

Noé, R.

S. Hinz, D. Sandel, F. Wüst, and R. Noé, “Interference detection enabling 2×20Gbit/s RZ polarisation division multiplex transmission,” Electron. Lett. 37, 511–512 (2001).
[CrossRef]

Oda, S.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in 2009 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2009), paper OMT2.

Rasmussen, J. C.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in 2009 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2009), paper OMT2.

Ryu, S.

S. Ryu, Coherent Lightwave Communication Systems (Artech House, Inc., 1995), Chap. 6.

M. Yagi, S. Satomi, and S. Ryu, “Field trial of 160-Gbit/s, polarization-division multiplexed RZ-DQPSK transmission system using automatic polarization control,” in 2008 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2008), paper OThT7.

Sandel, D.

S. Hinz, D. Sandel, F. Wüst, and R. Noé, “Interference detection enabling 2×20Gbit/s RZ polarisation division multiplex transmission,” Electron. Lett. 37, 511–512 (2001).
[CrossRef]

Satomi, S.

M. Yagi, S. Satomi, and S. Ryu, “Field trial of 160-Gbit/s, polarization-division multiplexed RZ-DQPSK transmission system using automatic polarization control,” in 2008 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2008), paper OThT7.

Savory, S. J.

Tao, Z.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in 2009 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2009), paper OMT2.

Tsukamoto, S.

K. Kikuchi and S. Tsukamoto, “Evaluation of sensitivity of the digital coherent receiver,” J. Lightwave. Technol. 26, 1817–1822 (2008).
[CrossRef]

Wüst, F.

S. Hinz, D. Sandel, F. Wüst, and R. Noé, “Interference detection enabling 2×20Gbit/s RZ polarisation division multiplex transmission,” Electron. Lett. 37, 511–512 (2001).
[CrossRef]

Xie, C.

C. Xie and S. Chandrasekhar, “Two-stage constant modulus algorithm equalizer for singularity free operation and optical performance monitoring in optical coherent receiver,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), paper OMK3.

Yagi, M.

M. Yagi, S. Satomi, and S. Ryu, “Field trial of 160-Gbit/s, polarization-division multiplexed RZ-DQPSK transmission system using automatic polarization control,” in 2008 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2008), paper OThT7.

Yan, W.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in 2009 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2009), paper OMT2.

Zhang, C.

Electron. Lett.

S. Hinz, D. Sandel, F. Wüst, and R. Noé, “Interference detection enabling 2×20Gbit/s RZ polarisation division multiplex transmission,” Electron. Lett. 37, 511–512 (2001).
[CrossRef]

IEEE Trans. Commun

D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun . COM-28, 1867–1875 (1980).
[CrossRef]

J. Lightwave. Technol.

K. Kikuchi and S. Tsukamoto, “Evaluation of sensitivity of the digital coherent receiver,” J. Lightwave. Technol. 26, 1817–1822 (2008).
[CrossRef]

Opt. Express

Other

S. Ryu, Coherent Lightwave Communication Systems (Artech House, Inc., 1995), Chap. 6.

K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” in 2008 Technical Digest of IEEE/LEOS Summer Topical Meeting (IEEE/LEOS, 2008), paper MC2.2.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers,” in 2009 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2009), paper OMT2.

C. Xie and S. Chandrasekhar, “Two-stage constant modulus algorithm equalizer for singularity free operation and optical performance monitoring in optical coherent receiver,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), paper OMK3.

K. Kikuchi, “Coherent optical communications: Historical perspectives and future directions,” in High Spectral Density Optical Communication Technology , M. Nakazawa, K. Kikuchi, and T. Miyazaki, eds. (Springer, 2010), Chap. 2.
[CrossRef]

M. Yagi, S. Satomi, and S. Ryu, “Field trial of 160-Gbit/s, polarization-division multiplexed RZ-DQPSK transmission system using automatic polarization control,” in 2008 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2008), paper OThT7.

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Figures (9)

Fig. 1
Fig. 1

Model of the polarization demultiplexing circuit. The polarization multiplexed signal given as E in (t) travels through a fiber, whose transfer function is T. The coherent receiver Rx measures two polarization components E x and E y , which are transformed into E X and E Y by the matrix J.

Fig. 2
Fig. 2

DSP circuit for controlling states of polarization.

Fig. 3
Fig. 3

Trajectories of polarization vectors E A 0 and E B 0 on the Poincarè sphere. (a): S 1 > 0 for A 0 and (b): S 1 < 0 for A 0.

Fig. 4
Fig. 4

Calculated trajectories of polarization vectors on the Poincarè sphere when PDL= 0 dB. Red and blue curves are those of E A n and E B n , respectively. (a): α = 0.55 and δ = 30°, and (b): α = 0.45 and δ = 30°.

Fig. 5
Fig. 5

Calculated trajectories of polarization vectors on the Poincarè sphere when PDL= −3 dB. Red and blue curves are those of E A n and E B n , respectively. In (a), p(0) is given as Eq. (52), and in (b), p(0) is given as Eq. (53).

Fig. 6
Fig. 6

Map of polarization demultiplexing stability when p(0) is given by Eq. (52). (a): PDL= −1 dB, (b): PDL= −3 dB, (c): PDL= −5 dB, and (d): PDL= −7 dB.

Fig. 7
Fig. 7

Map of polarization demultiplexing stability when p(0) is given by Eq. (53). (a): PDL= −1 dB, (b): PDL= −3 dB, (c): PDL= −5 dB, and (d): PDL= −7 dB.

Fig. 8
Fig. 8

Map of polarization demultiplexing stability when the improved CMA is applied. p(0) is given by Eq. (52). (a): PDL= −1 dB, (b): PDL= −3 dB, (c): PDL= −5 dB, (d): PDL= −7 dB, and (e): PDL= −8 dB.

Fig. 9
Fig. 9

Map of polarization demultiplexing stability when the improved CMA is applied. p(0) is given by Eq. (53). (a): PDL= −1 dB, (b): PDL= −3 dB, (c): PDL= −5 dB, (d): PDL= −7 dB, and (e): PDL= −8 dB.

Tables (1)

Tables Icon

Table 1 Relation Among the Color on the Map, the Demultiplexed Tributary, and the Converged Matrix Form

Equations (55)

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| E in , x ( t ) | 2 = | E in , y ( t ) | 2 = 1.
[ E x ( t ) E y ( t ) ] = T  [ E i n , x ( t ) E i n , y ( t ) ] ,
T  = [ T 11 T 12 T 21 T 22 ] .
J  = [ J 11 J 12 J 21 J 22 ] .
[ E X ( t ) E Y ( t ) ] = JT  [ E i n , x ( t ) E i n , y ( t ) ] .
C  = JT ,
C 11 = J 11 T 11 + J 12 T 21 ,
C 12 = J 11 T 12 + J 12 T 22 ,
C 21 = J 21 T 11 + J 22 T 21 ,
C 22 = J 21 T 12 + J 22 T 22 .
| E X ( t ) | 2 = | C 11 E i n , x ( t ) | 2 + | C 12 E i n , y ( t ) | 2 + 2 | C 11 C 12 E i n , x ( t ) E i n , y ( t ) | cos θ X ( t ) ,
θ X ( t ) = arg [ C 12 E i n , y ( t ) C 11 E i n , x ( t ) ] ,
| E Y ( t ) | 2 = | C 21 E i n , x ( t ) | 2 + | C 22 E i n , y ( t ) | 2 + 2 | C 21 C 22 E i n , x ( t ) E i n , y ( t ) | cos θ Y ( t ) ,
θ Y ( t ) = arg [ C 22 E i n , y ( t ) C 21 E i n , x ( t ) ] .
Case (X - I)  { C 12 = J 11 T 12 + J 12 T 22 = 0 | C 11 | = | J 11 T 11 + J 12 T 21 | = 1      ,
Case (X - II)  { C 11 = J 11 T 11 + J 12 T 21 = 0 | C 12 | = | J 11 T 12 + J 12 T 22 | = 1       .
Case (Y - I)  { C 21 = J 21 T 11 + J 22 T 21 = 0 | C 22 | = | J 21 T 12 + J 22 T 22 | = 1      ,
Case (Y - II)  { C 22 = J 21 T 12 + J 22 T 22 = 0 | C 21 | = | J 21 T 11 + J 22 T 21 | = 1       .
C  =  JT  = [ exp ( j φ ) 0 0 exp ( j ψ ) ] ,
[ E X ( t ) E Y ( t ) ] = [ E i n , x ( t ) exp ( j φ ) E i n , y ( t ) exp ( j ψ ) ] .
C  =  JT  = [ 0 exp ( j φ ) exp ( j ψ ) 0 ] .
[ E X ( t ) E Y ( t ) ] = [ E i n , y ( t ) exp ( j φ ) E i n , x ( t ) exp ( j δ ) ] ,
C  =  JT  = [ exp ( j φ ) 0 exp ( j ψ ) 0 ] .
[ E X ( t ) E Y ( t ) ] = [ E i n , x ( t ) exp ( j φ ) E i n , x ( t ) exp ( j ψ ) ] .
C  =  JT  = [ 0 exp ( j φ ) 0 exp ( j ψ ) ] .
[ E X ( t ) E Y ( t ) ] = [ E i n , y ( t ) exp ( j φ ) E i n , y ( t ) exp ( j ψ ) ] ,
[ E X ( n ) E Y ( n ) ] = [ p x x ( n ) p x y ( n ) p y x ( n ) p y y ( n ) ] [ E x ( n ) E y ( n ) ] .
p x x ( n + 1 ) = p x x ( n ) + μ ɛ X ( n ) E X ( n ) E x * ( n ) ,
p x y ( n + 1 ) = p x y ( n ) + μ ɛ X ( n ) E X ( n ) E y * ( n ) ,
p y y ( n + 1 ) = p y y ( n ) + μ ɛ Y ( n ) E Y ( n ) E y * ( n ) ,
p y x ( n + 1 ) = p y x ( n ) + μ ɛ Y ( n ) E Y ( n ) E x * ( n ) ,
ɛ X ( n ) = 1 - | E X ( n ) | 2 ,
ɛ Y ( n ) = 1 - | E Y ( n ) | 2 ,
T  = [ T 11 T 12 - T 12 * T 11 * ] ,
| T 11 | 2 + | T 12 | 2 = 1.
p (0)  = [ p x x ( 0 ) p x y ( 0 ) - p x y ( 0 ) * p x x ( 0 ) * ] ,
| p x x ( 0 ) | 2 + | p x y ( 0 ) | 2 = 1.
E X ( 0 ) = X E i n , x ( 0 ) + Y * E i n , y ( 0 ) ,
E Y ( 0 ) = - Y E i n , x ( 0 ) + X * E i n , y ( 0 ) ,
X = p x x ( 0 ) T 11 - p x y ( 0 ) T 12 * ,
Y * = p x x ( 0 ) T 12 + p x y ( 0 ) T 11 * .
E A 0 = [ X Y * ] ,
E B 0 = [ - Y X * ] .
B  = [ α e j δ - 1 - α 1 - α α e - j δ ] ,
D  = R - 1 [ 1 0 0 γ ] R ,
PDL  [ dB ] = 20log 10 γ .
[ E X ( n ) E Y ( n ) ] = [ p x x ( n ) p x y ( n ) p y x ( n ) p y y ( n ) ] T  [ E i n , x ( n ) E i n , y ( n ) ] .
[ a ( n ) b ( n ) c ( n ) d ( n ) ] = [ p x x ( n ) p x y ( n ) p y x ( n ) p y y ( n ) ] T ,
E A n = [ a ( n ) / | a ( n ) | 2 + | b ( n ) | 2 b ( n ) / | a ( n ) | 2 + b ( n ) | 2 ]
E B n = [ c ( n ) / | c ( n ) | 2 + | d ( n ) | 2 d ( n ) / | c ( n ) | 2 + d ( n ) | 2 ] .
R (0)  = [ 1 0 0 1 ] ,
p (0)  = [ 1 0 0 1 ] .
p (0)  =    [ 1 1 - 1 1 ] / 2 .
p y y ( n ) = p x x ( n ) * ,
p y x ( n ) = - p x y ( n ) * ,

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