## Performance analyses of polarization demultiplexing based on constant-modulus algorithm in digital coherent optical receivers |

Optics Express, Vol. 19, Issue 10, pp. 9868-9880 (2011)

http://dx.doi.org/10.1364/OE.19.009868

Acrobat PDF (1100 KB)

### 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.

© 2011 OSA

## 1. Introduction

*x*- and

*y*-polarization components with a polarization beam splitter (PBS), whereas a linearly-polarized local oscillator (LO) is equally split into these polarizations. Such polarization-diversity coherent receiver can measure complex amplitudes of

*x*- and

*y*-polarization components simultaneously through phase-diversity homodyne detection [1

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]

## 2. Principle of CMA-based polarization demultiplexing

### 2.1. Fundamentals of CMA-based polarization demultiplexing

**E**

*(*

_{in}*t*) = [

*E*(

_{in,x}*t*),

*E*(

_{in,y}*t*)]

^{T}is launched on a fiber for transmission, where

*E*(

_{in,x}*t*) and

*E*(

_{in,y}*t*) denote complex amplitudes of the signal electric filed for the

*x*-polarization tributary and the

*y*-polarization tributary, respectively, and “T” means to transpose the vector. We assume the use of the

*M*-ary PSK modulation format and normalize the input signal amplitude such that The output complex amplitude from the fiber is then given as where

**T**is the transfer matrix of the fiber expressed as Note that

**T**is a unitary Jones matrix provided that polarization-dependent loss (PDL) is negligible.

*E*

*(*

_{x}*t*) and

*E*

*(*

_{y}*t*), which appear at the

*x*port and the

*y*port of the polarization-diversity receiver, respectively [1

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

*E*

*(*

_{x}*t*)

*, E*

*(*

_{y}*t*)]

^{T}by a matrix

**J**given as Then we obtain the following complex amplitudes at

*X*and

*Y*output ports of the two-by-two post-processing circuit: Let the matrix

**C**be defined as where each element is given as The intensity |

*E*

*(*

_{X}*t*)|

^{2}measured at the

*X*port is then given as and |

*E*

*(*

_{Y}*t*)|

^{2}at the

*Y*port is written as Since the phase difference between complex amplitudes of the

*x*-polarization tributary and the

*y*-polarization tributary varies randomly at the symbol rate of

*M*-ary PSK modulation, third terms in Eqs. (11) and (13) change at the symbol rate [7

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]

**J**so that time variation of |

*E*

*(*

_{X}*t*)|

^{2}is suppressed and

*|E*

*(*

_{X}*t*)

*|*

^{2}approaches to 1, as is actually done in CMA, the matrix

**J**satisfies either of the following two conditions: On the other hand, when we control the matrix

**J**so that time variation of

*|E*

*(*

_{Y}*t*)

*|*

^{2}is suppressed and

*|E*

*(*

_{Y}*t*)

*|*

^{2}approaches to 1, either of the following two conditions for the matrix

**J**should be satisfied:

**C**is written as where

*φ*and

*ψ*are real constants; therefore, we have Equation (20) shows that

*E*

*and*

_{in,x}*E*

*appear at the*

_{in,y}*X*port and

*Y*port, respectively, which means that polarization demultiplexing can be achieved. Phases of the complex amplitudes are not fixed yet but determined by the phase-tracking circuit. It should also be noted that PDL is compensated for simultaneously with polarization demultiplexing.

*E*

*appears at the*

_{in,x}*Y*port, whereas

*E*

*does at the*

_{in,y}*X*port. Although the

*x*-and

*y*-polarization tributaries are exchanged at the output ports, we can achieve polarization demultiplexing.

*X*and

*Y*ports are given as In this case, the two polarization tributaries are not demultiplexed, because

*E*

*appears at both ports. In addition, when conditions for Case(X-II) and Case(Y-I) are satisfied, we have Since*

_{in,x}*E*

*appears at both ports, which prohibits proper polarization demultiplexing.*

_{in,y}### 2.2. Convergence property of CMA-based polarization demultiplexing

**J**so that intensities at the

*X*and

*Y*ports become stationary. We can achieve this process using CMA.

*n*denotes the number of samples. CMA-based polarization demultiplexing can be done by the circuit shown in Fig. 2, and its function is expressed as These matrix elements are updated in a symbol-by-symbol manner as follows: where error signals are given as and

*μ*is a step-size parameter [3

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

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]

*ε*

*(*

_{X}*n*) and

*ε*

*(*

_{Y}*n*) converges with 0. In 2.1, we have shown that polarization tributaries can be demultiplexed in such a case; however, proper polarization demultiplexing is done only when either the combination of Case(X-I) and Case(Y-I) or the combination of Case(X-II) and Case(Y-II) is achieved. In the following, by drawing the SOP trajectory on the Poincarè sphere during the tap updating process, we discuss how these conditions are realized.

**T**is unitary ignoring PDL: where Let

**p**(0) be also a unitary matrix given as where Initial-step outputs are given as where we define Then, we find that the equivalent polarization vector

*E*

*(0) for the complex amplitude from the*

_{X}*X*port is given as and the polarization vector for

*E*

*(0) is given as Note that*

_{Y}**E**

_{A0}and

**E**

_{B0}are orthogonal to each other when

**T**and

**p**(0) are unitary. Therefore, if

**E**

_{A0}is located at

*A*

_{0}on the Poincarè sphere as shown in Fig. 3,

**E**

_{B0}is at the antipodal point

*B*

_{0}. The

*S*

_{1}coordinate of

*A*

_{0}in the Stokes space is given as |

*X*|

^{2}− |

*Y*|

^{2}, whereas that of

*B*

_{0}as |

*Y*|

^{2}− |

*X|*

^{2}[10].

*S*

_{1}value of

*A*

_{0}is positive, and in Fig. 3(b), that is negative. When CMA suppresses the change in |

*E*

*|*

_{X,Y}^{2}, |

*X*||

*Y*| should be reduced. Noting that |

*X|*

^{2}+ |

*Y*|

^{2}= 1, we find that when |

*X*| > |

*Y*| for

*A*

_{0}, the point

*A*

_{0}moves toward

*A*(linear

*x*polarization), whereas

*B*

_{0}toward

*B*(linear

*y*polarization), as shown in Fig. 3(a); thus, proper polarization demultiplexing can be done. On the other hand, when

*|X| < |Y|*for

*A*

_{0}, the point

*A*

_{0}moves toward

*B*(linear

*y*polarizaton), whereas

*B*

_{0}toward

*A*(linear

*x*polarization), as shown in Fig. 3(b). This means that the

*x*-polarization tributary is output from the

*Y*port, and the

*y*-polarization tributary from the

*X*port. It should also be noted that when

*S*

_{1}values of

*A*

_{0}and

*B*

_{0}are close to 0, the convergence property of CMA becomes worse, which results in unstable polarization demultiplexing characteristics.

**T**is not a unitary matrix, as is the case in the transmission system having PDL, we cannot exclude the possibility that

*S*

_{1}values for

*A*

_{0}and

*B*

_{0}have the same sign. In such a case, outputs from both ports converge with the same polarization tributary, which limits the performance of CMA-based polarization demultiplexing.

## 3. Stability analyses of CMA-based polarization demultiplexing

### 3.1. Trajectory of equivalent polarization vectors on Poincarè sphere

*μ*used in CMA is 1/16, and the number of symbols is 256. Let a unitary matrix expressing the fiber birefringence

**B**be written as where

*α*denotes the power-splitting ratio and

*δ*the phase difference between

*x*and

*y*polarizations. We can express PDL as where 0 ≤

*γ*≤ 1 and the unitary matrix

**R**converts eigen states of PDL to

*x*and

*y*polarizations. The PDL in dB is given as The transfer matrix

**T**is given as a concatenation of

**B**and

**D**as

**T**=

**BD**[11

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]

*X*and

*Y*ports are given as If we define equivalent polarization vectors for

*E*

*(*

_{X}*n*) and

*E*

*(*

_{Y}*n*) are respectively given as and Trajectories of these vectors during the tap updating process can be plotted on the Poicarè sphere.

**R**and

**p**(0) are given as When the Jones matrix of the fiber

**B**is defined by

*α*= 0.55 and

*δ*= 30° in Eq. (44), for example, red and blue curves in Fig. 4(a) are trajectories of

**E**

_{An}and

**E**

_{Bn}, respectively, showing that the

*x*-polarization tributary appears at the

*X*port, whereas the

*y*-polarization tributary appears at the

*Y*port. On the other hand, Fig. 4(b) shows those obtained when

*α*= 0.45 and

*δ*= 30° in

**B**. In this case, the

*x*-polarization tributary appears at the

*Y*port, whereas the

*y*-polarization tributary appears at the

*X*port. From these figures, we find that when

*S*

_{1}of the polarization vector at the initial step of CMA is positive, the vector converges with the

*x*polarization; on the contrary, when

*S*

_{1}is negative, the vector converges with the

*y*polarization. As far as PDL is ignored, we can achieve proper polarization demultiplexing, choosing an arbitrary unitary matrix for

**p**(0).

**E**

_{An}and

**E**

_{Bn}when PDL= −3 dB, and we assume that

*α*= 0.45 and

*δ*= 30° in

**B**. Both polarization vectors converge with the same polarization tributary, because signs of

*S*

_{1}of the both vectors are the same at the initial step. However, changing the matrix

**p**(0) so that we have trajectories of

**E**

_{An}and

**E**

_{Bn}shown in Fig. 5(b), where proper polarization demultiplexing is done. Thus, when PDL cannot be ignored, we need to adjust

**p**(0) so that proper polarization demultiplexing is achieved. However, as PDL increases, the requirement for

**p**(0) becomes more and more stringent.

### 3.2. Improvement of stability of CMA-based polarization demultiplexing

*α*and

*δ*in the range where 0 ≤

*α*≤ 1 and −

*π*<

*δ*≤

*π*. In this calculation, the initial unitary matrix

**p**(0) is fixed. The eigen states of PDL (

*i.e.*, eigen vectors of the matrix

**R**) is swept over the entire Poincarè sphere for each set of

*α*and

*δ*to estimate the worst case of the stability. The step size parameter

*μ*used in CMA is 1/16.

*α*-

*δ*plane consisting of 20 × 20 segments. Table 1 represents the relation between the color and the demultiplexed tributary. The converged matrix form for

**C**is also shown for each color, being used for identifying demultiplexed tributaries in our calculations. Only in red and blue regions, stable polarization demultiplexing is always done. In the white region, we can achieve proper polarization demultiplexing, although

*x*and

*y*tributaries are sometimes exchanged at output ports while the eigen vectors of

**R**are varied. In regions with other colors, proper polarization demultiplexing is prohibited.

**p**(0) is given as Eq. (52). The number of symbols is 256 in the calculation. Figures 6(a)–6(d) show maps of polarization-demultiplexing stability when PDL= −1 dB, −3 dB, −5 dB, and −7 dB, respectively. The yellow region, where both output ports have the same polarization tributary, becomes widespread as the amount of PDL increases. When we choose

**p**(0) given as Eq. (53), stability maps for PDL= −1 dB, −3 dB, −5 dB, and −7 dB are shown in Figs. 7(a)–7(d), respectively. Also in this case, the unstable region spreads with the increase in PDL. It should be stressed that sets of (

*α, δ*) in the yellow region transform SOP of the received signal close to the meridian of the Poincarè sphere with

*S*

_{1}= 0, bringing forth the unstable operation of polarization demultiplexing.

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.

*n*= 0) through the

*n*

*-th step (*

_{s}*n*=

*n*

*), only matrix elements*

_{s}*p*

*(*

_{xx}*n*) and

*p*

*(*

_{xy}*n*) are updated by using CMA. On the other hand, we determine matrix elements

*p*

*(*

_{yy}*n*) and

*p*

*(*

_{yx}*n*) from assuming that

**p**is unitary. At the

*n*

*-th step, the*

_{s}*X*-port output almost converges with either

*A*or

*B*on the Poincarè sphere as shown in Fig. 3; on the other hand, owing to the unitarity of

**p**, the distance between the transformed

*Y*-port output and

*X*-port output is maintained on the Poincarè sphere. Therefore, if the amount of PDL is small enough, the polarization vector of the transformed

*Y*-port output is located on the hemisphere opposite to the

*X*-port output. Thus, applying CMA to all of the matrix elements from the (

*n*

*+ 1)-th step, we may achieve proper polarization demultiplexing.*

_{s}*to maintain the stability is estimated as follows: Instability occurs when the polarization vector of one polarization tributary converges with*

_{c}*A*or

*B*and that of the other tributary is on the meridian with

*S*

_{1}= 0. In such a case, the inner product between the two polarization vectors is

*π/*8), which means that PDL

*= −7.66 dB.*

_{c}**p**(0) is given by Eq. (52). For PDL= −1 dB, −3 dB, −5 dB, −7, and −8 dB, numbers of symbols used in the calculation are 256, 512, 1024, 1024, and 1024, whereas

*n*

*are 112, 256, 512, 512, and 512, respectively. We find that the unstable region shrinks greatly with this method. Proper polarization demultiplexing can be done when the amount of PDL is smaller than 5 dB, whereas the small unstable region remains when PDL= −7 dB. Although PDL*

_{s}*= −7.66 dB, the convergence property of CMA becomes worth at around PDL= −7 dB, giving unstable polarization demultiplexing. When PDL= −8 dB, proper polarization demultiplexing is entirely prohibited. Figure 9 is the stability map obtained when the initial matrix*

_{c}**p**(0) is Eq. (53). Figures 8(a)–8(e) correspond to PDL=–1 dB, −3 dB, −5 dB, −7 dB, and −8 dB, respectively. Also in this case, the stability is improved by the modified CMA just the same as Fig. 8, when the amount of PDL is much smaller than |PDL

*|.*

_{c}## 4. Conclusion

*S*

_{1}of the polarization vector at the initial step of CMA is positive, the vector converges with the

*x*polarization tributary; in contrast, when

*S*

_{1}is negative, the vector converges with the

*y*polarization tributary. In the case that PDL is negligible, signs of

*S*

_{1}of the two polarization vectors are different from each other if we choose a unitary matrix for

**p**(0) at the initial step of CMA. In such a case, we can always achieve proper polarization demultiplexing. However, when the transmission system has PDL, we cannot exclude the possibility that

*S*

_{1}values for both polarization vectors have the same sign at the initial step of CMA. Two polarization vectors converge with the same polarization tributary in this case.

*α*-

*δ*plane. When the amount of PDL is small enough (PDL ≃ −5 dB), we can always achieve proper polarization demultiplexing, using the modified CMA, where updating of all taps starts after convergence of one of the polarization vectors.

## References and links

1. | K. Kikuchi and S. Tsukamoto, “Evaluation of sensitivity of the digital coherent receiver,” J. Lightwave. Technol. |

2. | K. Kikuchi, “Coherent optical communications: Historical perspectives and future directions,” in |

3. | S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express |

4. | K. Kikuchi, “Polarization-demultiplexing algorithm in the digital coherent receiver,” in 2008 Technical Digest of |

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 |

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 |

7. | S. Hinz, D. Sandel, F. Wüst, and R. Noé, “Interference detection enabling 2×20Gbit/s RZ polarisation division multiplex transmission,” Electron. Lett. |

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 |

9. | D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun . |

10. | S. Ryu, |

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 |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.2330) Fiber optics and optical communications : Fiber optics communications

(060.2920) Fiber optics and optical communications : Homodyning

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: March 1, 2011

Revised Manuscript: April 19, 2011

Manuscript Accepted: April 20, 2011

Published: May 5, 2011

**Citation**

Kazuro Kikuchi, "Performance analyses of polarization demultiplexing based on constant-modulus algorithm in digital coherent optical receivers," Opt. Express **19**, 9868-9880 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9868

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### References

- 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]
- S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16, 804–817 (2008). [CrossRef] [PubMed]
- 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.
- 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]
- 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.
- D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun . COM-28, 1867–1875 (1980). [CrossRef]
- S. Ryu, Coherent Lightwave Communication Systems (Artech House, Inc., 1995), Chap. 6.
- 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]

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