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

Kazuro Kikuchi

doi:10.1364/OE.19.009868

1. Introduction

In the coherent optical receiver employing polarization diversity, the incoming signal having an arbitrary state of polarization (SOP) is separated into linear 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, 2].

In the polarization-multiplexed optical transmission system, on the other hand, two polarization tributaries travel through a fiber link having birefringence. Therefore, the output from each polarization port of the polarization-diversity coherent receiver includes both polarization tributaries, which must be separated in the digital domain. Polarization demultiplexing can be done by using a two-by-two matrix, and the constant-modulus algorithm (CMA) has widely been applied to adaptive control of matrix elements [3]. However, by using this algorithm, it is likely that each output converges with the same polarization tributary [4–6]. Such improper polarization demultiplexing occurs especially when polarization-dependent loss (PDL) cannot be ignored. In this paper, elucidating the physics behind CMA for polarization demultiplexing, we discuss the performance limit of CMA-based polarization demultiplexing and propose the method of improving its performance.

The organization of this paper is as follows: In Section 2, we describe the principle of operation of CMA-based polarization demultiplexing and discuss the condition where polarization demultiplexing is done properly without convergence with the same tributary. Section 3 presents simulation results on polarization-demultiplexing stability against SOP of the incoming signal. The method of improving the stability is also discussed. Section 4 concludes this paper.

2. Principle of CMA-based polarization demultiplexing

In this section, we describe how CMA enables polarization demultiplexing. First, we show that polarization demultiplexing is achieved when we suppress intensity fluctuations in each output from the polarization-diversity receiver, and that CMA is an efficient algorithm to suppress the intensity fluctuations. Next, we introduce the concept of the equivalent polarization vector in the digital domain. Using such concept, we discuss the stability of polarization demultiplexing in detail.

2.1. Fundamentals of CMA-based polarization demultiplexing

Figure 1 shows the model of our analyses. The polarization-multiplexed signal 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

| E_{in, x} (t) |^{2} = | E_{in, y} (t) |^{2} = 1.

The output complex amplitude from the fiber is then given as

[\begin{array}{l} E_{x} (t) \\ E_{y} (t) \end{array}] = T [\begin{array}{l} E_{i n, x} (t) \\ E_{i n, y} (t) \end{array}],

where T is the transfer matrix of the fiber expressed as

T = [\begin{matrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{matrix}] .

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

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.

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Using a digital coherent receiver, we measure E _x (t) and E _y (t), which appear at the x port and the y port of the polarization-diversity receiver, respectively [1]. With a two-by-two post-processing circuit, we multiply the vector [E _x (t), E _y (t)]^T by a matrix J given as

J = [\begin{matrix} J_{11} & J_{12} \\ J_{21} & J_{22} \end{matrix}] .

Then we obtain the following complex amplitudes at X and Y output ports of the two-by-two post-processing circuit:

[\begin{array}{l} E_{X} (t) \\ E_{Y} (t) \end{array}] = JT [\begin{array}{l} E_{i n, x} (t) \\ E_{i n, y} (t) \end{array}] .

Let the matrix C be defined as

C = JT,

where each element is given as

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} .

The intensity |E _X (t)|² measured at the X port is then given as

| 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 [\frac{C_{12} E_{i n, y} (t)}{C_{11} E_{i n, x} (t)}],

and |E _Y (t)|² at the Y port is written as

| 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 [\frac{C_{22} E_{i n, y} (t)}{C_{21} E_{i n, x} (t)}] .

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, 8]. When we control the matrix 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:

Case (X - I) {\begin{array}{l} C_{12} = J_{11} T_{12} + J_{12} T_{22} = 0 \\ | C_{11} | = | J_{11} T_{11} + J_{12} T_{21} | = 1 \end{array},

Case (X - II) {\begin{array}{l} C_{11} = J_{11} T_{11} + J_{12} T_{21} = 0 \\ | C_{12} | = | J_{11} T_{12} + J_{12} T_{22} | = 1 \end{array} .

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:

Case (Y - I) {\begin{array}{l} C_{21} = J_{21} T_{11} + J_{22} T_{21} = 0 \\ | C_{22} | = | J_{21} T_{12} + J_{22} T_{22} | = 1 \end{array},

Case (Y - II) {\begin{array}{l} C_{22} = J_{21} T_{12} + J_{22} T_{22} = 0 \\ | C_{21} | = | J_{21} T_{11} + J_{22} T_{21} | = 1 \end{array} .

When conditions for Case(X-I) and Case(Y-I) are satisfied simultaneously, the matrix C is written as

C = JT = [\begin{matrix} exp (j φ) & 0 \\ 0 & exp (j ψ) \end{matrix}],

where φ and ψ are real constants; therefore, we have

[\begin{array}{l} E_{X} (t) \\ E_{Y} (t) \end{array}] = [\begin{array}{l} E_{i n, x} (t) exp (j φ) \\ E_{i n, y} (t) exp (j ψ) \end{array}] .

Equation (20) shows that E _in,x and E _in,y appear at the 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.

On the other hand, when conditions for Case(X-II) and Case(Y-II) are satisfied, they yield

C = JT = [\begin{matrix} 0 & exp (j φ) \\ exp (j ψ) & 0 \end{matrix}] .

Therefore, noting that

[\begin{array}{l} E_{X} (t) \\ E_{Y} (t) \end{array}] = [\begin{array}{l} E_{i n, y} (t) exp (j φ) \\ E_{i n, x} (t) exp (j δ) \end{array}],

we find that E _in,x appears at the Y port, whereas E _in,y does at the X port. Although the x-and y-polarization tributaries are exchanged at the output ports, we can achieve polarization demultiplexing.

On the contrary, when Case(X-I) and Case(Y-II) are satisfied, we have

C = JT = [\begin{matrix} exp (j φ) & 0 \\ exp (j ψ) & 0 \end{matrix}] .

Therefore, outputs from the X and Y ports are given as

[\begin{array}{l} E_{X} (t) \\ E_{Y} (t) \end{array}] = [\begin{array}{l} E_{i n, x} (t) exp (j φ) \\ E_{i n, x} (t) exp (j ψ) \end{array}] .

In this case, the two polarization tributaries are not demultiplexed, because E _in,x appears at both ports. In addition, when conditions for Case(X-II) and Case(Y-I) are satisfied, we have

C = JT = [\begin{matrix} 0 & exp (j φ) \\ 0 & exp (j ψ) \end{matrix}] .

Since

[\begin{array}{l} E_{X} (t) \\ E_{Y} (t) \end{array}] = [\begin{array}{l} E_{i n, y} (t) exp (j φ) \\ E_{i n, y} (t) exp (j ψ) \end{array}],

E _in,y appears at both ports, which prohibits proper polarization demultiplexing.

2.2. Convergence property of CMA-based polarization demultiplexing

The discussion in 2.1 shows that to achieve polarization demultiplexing, we need to control the matrix J so that intensities at the X and Y ports become stationary. We can achieve this process using CMA.

We assume that measured complex amplitudes are sampled at the symbol rate and 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

[\begin{array}{l} E_{X} (n) \\ E_{Y} (n) \end{array}] = [\begin{matrix} p_{x x} (n) & p_{x y} (n) \\ p_{y x} (n) & p_{y y} (n) \end{matrix}] [\begin{array}{l} E_{x} (n) \\ E_{y} (n) \end{array}] .

These matrix elements are updated in a symbol-by-symbol manner as follows:

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),

where error signals are given as

ɛ_{X} (n) = 1 - | E_{X} (n) |^{2},

ɛ_{Y} (n) = 1 - | E_{Y} (n) |^{2},

and μ is a step-size parameter [3, 9].

Fig. 2 DSP circuit for controlling states of polarization.

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Through the above procedure, 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.

We assume that the matrix T is unitary ignoring PDL:

T = [\begin{matrix} T_{11} & T_{12} \\ - T_{12}^{*} & T_{11}^{*} \end{matrix}],

where

| T_{11} |^{2} + | T_{12} |^{2} = 1.

Let p(0) be also a unitary matrix given as

p
(0) = [\begin{matrix} p_{x x} (0) & p_{x y} (0) \\ - p_{x y} {(0)}^{*} & p_{x x} {(0)}^{*} \end{matrix}],

where

| p_{x x} (0) |^{2} + | p_{x y} (0) |^{2} = 1.

Initial-step outputs are given as

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),

where we define

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}^{*} .

Then, we find that the equivalent polarization vector E _X (0) for the complex amplitude from the X port is given as

E_{A_{0}} = [\begin{array}{l} X \\ Y^{*} \end{array}],

and the polarization vector for E _Y (0) is given as

E_{B_{0}} = [\begin{array}{l} - Y \\ X^{*} \end{array}] .

Note that 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].

Fig. 3 Trajectories of polarization vectors E _A₀ and E _B₀ on the Poincarè sphere. (a): S ₁ > 0 for A ₀ and (b): S ₁ < 0 for A ₀.

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In Fig. 3(a), the 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.

When 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

In this section, we study stability of CMA-based polarization demultiplexing through numerical analyses. First, in 3.1, theoretical results given in 2.2 are confirmed by numerical simulations. Next, in 3.2, we analyze stability of CMA-based polarization demultiplexing, which is dependent on SOP of the incoming signal, and propose a method of improving the stability.

3.1. Trajectory of equivalent polarization vectors on Poincarè sphere

We use the QPSK modulation format and the signal is sampled at the symbol rate. The step size parameter 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

B = [\begin{matrix} \sqrt{α} e^{j δ} & - \sqrt{1 - α} \\ \sqrt{1 - α} & \sqrt{α} e^{- j δ} \end{matrix}],

where α denotes the power-splitting ratio and δ the phase difference between x and y polarizations. We can express PDL as

D = R^{- 1} [\begin{matrix} 1 & 0 \\ 0 & γ \end{matrix}] R,

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

PDL [dB] = {20log}_{10} γ .

The transfer matrix T is given as a concatenation of B and D as T = BD [11].

The outputs from X and Y ports are given as

[\begin{array}{l} E_{X} (n) \\ E_{Y} (n) \end{array}] = [\begin{matrix} p_{x x} (n) & p_{x y} (n) \\ p_{y x} (n) & p_{y y} (n) \end{matrix}] T [\begin{array}{l} E_{i n, x} (n) \\ E_{i n, y} (n) \end{array}] .

If we define

[\begin{matrix} a (n) & b (n) \\ c (n) & d (n) \end{matrix}] = [\begin{matrix} p_{x x} (n) & p_{x y} (n) \\ p_{y x} (n) & p_{y y} (n) \end{matrix}] T,

equivalent polarization vectors for E _X (n) and E _Y (n) are respectively given as

E_{A_{n}} = [\begin{array}{l} a (n) / \sqrt{| a (n) |^{2} + | b (n) |^{2}} \\ b (n) / \sqrt{| a (n) |^{2} + b (n) |^{2}} \end{array}]

and

E_{B_{n}} = [\begin{array}{l} c (n) / \sqrt{| c (n) |^{2} + | d (n) |^{2}} \\ d (n) / \sqrt{| c (n) |^{2} + d (n) |^{2}} \end{array}] .

Trajectories of these vectors during the tap updating process can be plotted on the Poicarè sphere.

First, we assume that PDL= 0 dB, and R and p(0) are given as

R (0) = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}],

p (0) = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] .

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 _{A_n} and E _{B_n}, 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).

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°.

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Next, we introduce PDL into our calculation. Figure 5(a) shows trajectories of E _{A_n} and E _{B_n} 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

p (0) = [\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}] / \sqrt{2} .

we have trajectories of E _{A_n} and E _{B_n} 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.

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).

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3.2. Improvement of stability of CMA-based polarization demultiplexing

To discuss how properly polarization demultiplexing can be done with CMA, we change SOP of the incoming signal by sweeping 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.

We show polarization demultiplexing stability by a colored map on an 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.

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

View Table

First, we assume that 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.

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.

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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.

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To improve the stability, we can modify CMA following [4] and [5]. From the initial step (n = 0) through the n _s-th step (n = n _s), only matrix elements 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

p_{y y} (n) = p_{x x} {(n)}^{*},

p_{y x} (n) = - p_{x y} {(n)}^{*},

assuming that p is unitary. At the n _s-th step, the 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 _s + 1)-th step, we may achieve proper polarization demultiplexing.

The critical PDL value PDL_c to maintain the stability is estimated as follows: Instability occurs when the polarization vector of one polarization tributary converges with 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 $cos (π / 4) = 1 / \sqrt{2}$ , and the minimum amount of PDL giving such an inner product is tan(π/8), which means that PDL_c = −7.66 dB.

Using the modified CMA, we calculate maps of polarization-demultiplexing stability shown in Fig. 8. Figures 8(a)–8(e) are obtained when PDL= −1 dB, −3 dB, −5 dB, −7, and −8 dB, respectively. In this calculation, the initial matrix 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 _s 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_c = −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 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|.

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.

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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.

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4. Conclusion

We have elucidated the physics behind CMA-based polarization demultiplexing, using the concept of equivalent SOP in the digital domain. 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 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.

We have also analyzed the stability of CMA-based polarization demultiplexing through computer simulations. Sweeping SOP of the incoming signal as well as eigen states of PDL, we study the stability region on the 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.

The author thanks Y. Mori and Md. S. Faruk of The University of Tokyo for helpful discussions. This work was supported in part by Strategic Information and Communications R&D Promotion Programme (SCOPE) (081503001), the Ministry of Internal Affairs and Communications, Japan; and Grant-in-Aid for Scientific Research (A) (22246046), the Ministry of Education, Science, Sports and Culture, Japan.

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Color	X port	Y port	Matrix C
Red	x	y	Eq. (19)
Blue	y	x	Eq. (21)
White	x	y	Eq. (19)
	y	x	Eq. (21)
Yellow	x	x	Eq. (23)
Green	y	y	Eq. (25)
Black	undecided	undecided	undetermined

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

Abstract

1. Introduction

2. Principle of CMA-based polarization demultiplexing

2.1. Fundamentals of CMA-based polarization demultiplexing

2.2. Convergence property of CMA-based polarization demultiplexing

3. Stability analyses of CMA-based polarization demultiplexing

3.1. Trajectory of equivalent polarization vectors on Poincarè sphere

3.2. Improvement of stability of CMA-based polarization demultiplexing

4. Conclusion

References and links

Cited By

Figures (9)

Tables (1)

Equations (55)

Optics Express