Polarization demultiplexing in stokes space for coherent optical PDM-OFDM

Zhenming Yu; Xingwen Yi; Qi Yang; Ming Luo; Jing Zhang; Lei Chen; Kun Qiu

doi:10.1364/OE.21.003885

1. Introduction

Coherent optical orthogonal frequency-division multiplexing (CO-OFDM) has emerged as a promising solution for next-generation high-capacity and long-haul optical communication systems [1–4]. Optical OFDM at 40-Gb/s or even 100-Gb/s data rate becomes feasible with recent advances in high-speed CMOS technologies and optical modulation and detection technologies [5–12]. Coherent optical OFDM (CO-OFDM) has similar benefits to coherent single-carrier transmissions, but with a lower computation complexity [13–15]. Polarization multiplexing is a simple approach to double the spectral efficiency and used widely. In single carrier system, constant modulus algorithm (CMA) and the decision-directed least-mean-square (DD-LMS) algorithm are usually applied to polarization demultiplexing, which shows a slow speed of convergence [16, 17]. In CO-OFDM systems, polarization-demultiplexing generally requires some periodic training symbols (TS’s) to recover the states of polarization by digital signal processing. However, these TSs increase the system redundancy, and reduces the spectral efficiency of CO-OFDM [18]. Inserting TSs with a longer period seems to lower the overhead, but it tends to fail if the optical channel changes within one period, especially for polarization-related effects that can change faster than millisecond.

Bogdan et al. proposed an efficient method of polarization demultiplexing in Stokes space (SS-PDM) [19]. This method does not require knowledge of the modulation format and its convergence speed is fast. The experimental demonstration of its feasibility has been reported in a single-carrier system. The principle of this method is to convert the received PDM signals into Stokes space, take advantages of the characteristics of their distribution in Stokes space to find an inverse Jones matrix, and then multiply it with the PDM signals in the time domain to re-rotate its polarization states digitally. Apparently, it is assumed that the inverse Jones matrix is frequency-independent. However, the polarization parameters in fibers are generally frequency-dependent and cannot always be ignored. Therefore it is not straightforward to apply SS-PDM to single-carrier signals with frequency-dependent polarization effects. Further, the OFDM signal in the time domain is noise-like and the distribution of OFDM signal in Stokes space cannot satisfy the conditions of SS-PDM accurately. In this paper, we first convert the OFDM signal to the frequency domain by FFT and then apply the SS-PDM for individual subcarriers. Because each subcarrier has a much lower speed and narrower bandwidth, the polarization effects that it experiences can be treated as flat, and the distribution in Stokes space of OFDM signals in the frequency domain also meets the requirements of SS-PDM. Consequently, SS-PDM can be applied to high-speed, wide-bandwidth OFDM signals, but on per subcarrier basis. The convergence speed of this method is very fast and it also can reduce overhead. The performance of this method is also comparable with that of the conventional method.

2. Principle of SS-PDM for CO-OFDM

To elucidate the principle of SS-PDM for CO-OFDM, we introduce a simple transmission model of CO-OFDM, following the similar terminology in [20]. Within one OFDM frame, the indices of OFDM symbol and subcarrier are i and k, respectively. We use Jones vector ${\overset{⇀}{s}}^{t} = {[s_{x} s_{y}]}^{T}$ to represent the PDM signal with two polarizations, i.e., $s_{x}$ and $s_{y}$ . After the FFT window synchronization, the channel model for individual subcarriers is,

{\vec{s}}_{_{k i}}^{r} = \exp (j ϕ_{i}) \cdot \exp (j Φ_{k}) \cdot M_{k} \cdot {\vec{s}}_{_{k i}}^{t} + n_{k i} .

where

Φ

is the phase dispersion owing to the fiber chromatic dispersion,

ϕ

is the laser phase noise, M is a unitary matrix for polarization rotation, n is the random noise. From the channel model, fiber dispersion and polarization rotation is frequency-dependent on the subcarrier index. In fact, the DGD in fiber depends on the optical frequency and the bandwidth of the principal states

Δ λ_{p s p} \approx 1 n m / 〈 Δ τ 〉 [p s]

is a wavelength range over which the DGD is reasonably constant [21].

Δ τ

is the differential group delay. After long-distance optical fiber transmissions, the DGD can easily reach 10~100ps and the bandwidth of the principle states becomes very narrow. The polarization rotation matrix has to be considered as frequency-dependent, and therefore, it is difficult to apply SS-PDM for single carrier signals with larger than 10-GHz bandwidth. Fortunately, for OFDM signals, its subcarriers have a much lower speed and narrower bandwidth, and therefore, the polarization effects that the individual subcarrier experiences can be treated as flat. The distribution of individual OFDM subcarriers in Stokes space also has the characteristics that can be used by SS-PDM to align the polarization states [19].These are the motivations that we convert the OFDM signal to the frequency domain by FFT to distinguish the different frequency subcarriers firstly and then calculate the polarization rotation matrix M for each OFDM subcarrier. The unitary matrix M of the k-th subcarrier can be expressed as:

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

Using Jones vector, the PDM signals of the k^th subcarrier before and after polarization rotation can be expressed as

{\vec{s}}_{o, k} = M_{k} \cdot {\vec{s}}_{i, k}

. The process of polarization demultiplexing is equivalent to find the inverse matrix

M^{- 1}

, and the polarization alignment is expressed as

{\hat{\vec{s}}}_{i, k} = M_{k}^{- 1} \cdot {\vec{s}}_{o, k}

. The thrust is then to find

M^{- 1}

by SS-PDM.

At first, we convert the Jones vector s(k) of the k-th subcarrier of received PDM-OFDM signals into the Stokes vectors [22]. The Jones vector s(k) can be expressed as $s (k) = \frac{1}{\sqrt{2}} {[s_{x}, s_{y}]}^{T}$ and it can be transformed into the Stokes vector S by:

S = (\begin{matrix} s_{0} \\ s_{1} \\ s_{2} \\ s_{3} \end{matrix}) = \frac{1}{2} (\begin{matrix} s_{x} s_{x}^{*} + s_{y} s_{y}^{*} \\ s_{x} s_{x}^{*} - s_{y} s_{y}^{*} \\ s_{x}^{*} s_{y} + s_{x} s_{y}^{*} \\ - j s_{x}^{*} s_{y} + j s_{x} s_{y}^{*} \end{matrix}) .

The first component of the Stokes vector,

s_{0}

, represents the total power. The other three components

s_{1}, s_{2}, s_{3}

represent 0° linear, 45° linear, and circularly polarized light, respectively. Figures 1(a) and 1(b) show the principle of SS-PDM by analyzing the experimental data. Figure 1(a) is the Stokes vectors in Poincare sphere of one subcarrier with 4QAM modulation before polarization demultiplexing. H and V represent the linear horizontal and linear vertical polarization states, respectively. LSP is the least squares plane of these Stokes vectors in Poincare sphere, and S is the normal of LSP. The theory of singular value decomposition (SVD) is applied to find the LSP [23]. Although there are only tens of data points, the normal of the LSP is successfully found and applied to calculate

M^{- 1}

[19]. Figure 1(b) shows that the polarization has been aligned to H-V direction properly, i.e., polarization demultiplexing.

Fig. 1 Stokes vectors in Poincare sphere of one OFDM subcarrier with 4QAM modulation (a) before SS-PDM and (b) after SS-PDM; Constellations of one OFDM subcarrier (c) before and (d) after SS-PDM, and(c) after channel compensation and phase noise compensation, respectively.

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If the normal of LSP is S: ( $s_{1}, s_{2}, s_{3}$ ), the $M^{- 1}$ can be express by SS-PDM as:

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

where

Δ ϕ = \arctan (s_{3}, s_{2})

, and

α = \arctan (s_{2}^{2} + s_{3}^{2}, s_{1})

. Once we have determined

M^{- 1}

for every subcarrier, we can continue other DSP to recovery the transmitted data, such as channel compensation and laser phase noise compensation. The final data after all the DSP is:

{\hat{\vec{s}}}_{_{k i}}^{t} = \exp (- j {\hat{ϕ}}_{i}) \cdot \exp (- j {\hat{Φ}}_{k}) \cdot M_{k}^{- 1} \cdot {\vec{s}}_{_{k i}}^{r} .

By applying the SS-PDM on individual subcarrier, SS-PDM can deal with the frequency-dependent polarization effect, but it is difficult to be applied in single-carrier systems. Note this channel compensation only includes simple constellation rotation. Figures 1(c) and 1(d) are the data of one OFDM subcarrier with 4QAM before and after SS-PDM, respectively. Figure 1(e) is after channel compensation and phase noise compensation, ready for data decoding.

In addition, there are a few unique features for our method worth further discussions,

i) The SS-PDM is carried out right after the FFT window synchronization or before the channel compensation and laser phase noise compensation. This is because our method does not specify a modulation format and it is based on the statistic feature in Stokes space.
ii) After the FFT window synchronization, the data on each subcarrier exclude the transition between symbols, unlike the single-carrier systems with oversampling [19]. This helps the successful finding of the normal without using many data points. As shown in Figs. 1(a) and 1(b), we can find the normal using only tens of symbol points in Stokes space. Further, using less data also means faster convergence speed.
iii) The SS-PDM may swap the data of the two orthogonal polarizations of received OFDM signals, because the normal of LSP can be represented as two directions: S or –S, as shown in Fig. 1. This can be readily corrected by a logical comparison. Or we can correct it using the joint information of neighboring subcarriers' normal, since the neighboring subcarriers should have similar normal.
iv) The computational complexity of this method is mainly the calculation of the normal vector S of the LSP, which is based on the mature algorithm. Our challenge is that this calculation is for each subcarrier. Therefore we will carry out further research to reduce the computational complexity.

3. Experimental Setup and results

The experimental step of SS-PDM-OFDM is shown in Fig. 2 . At the transmitter, one laser source is fed in to the optical IQ modulator to carry the OFDM signal. The transmitted signal is generated off-line by MATLAB program with a data sequence of 2¹⁵-1 pseudo-random binary sequence (PRBS) and mapped onto 4QAM or 16QAM constellation. An arbitrary waveform generator (AWG) is used to produce I/Q RF signals at 10 GS/s. An optical intensity modulator is used to further duplicate the signal to three copies. The frequency of the driving sine wave signal is at 6.71875 GHz, which is intentionally set to satisfy the condition of orthogonal band multiplexing [2]. All the clock resources are phase-locked to the AWG using a 10 MHz reference clock. Each polarization components was mapped onto 86 OFDM frequency subcarriers. Six out of total subcarriers are used to estimate the phase noise. We use 128-point IFFT to convert the frequency domain signal to the time domain. We also use 1/8 of OFDM symbol period for cyclic prefix to avoid the fiber dispersion. The optical OFDM signal is then polarization multiplexed by a pair of polarization beam splitter/combiner with one branch delayed by one OFDM symbol. The transmission link is a fiber recirculation loop, which contains four spans of 80km SSMF whose loss is compensated by Raman amplifiers. At the receiver side, a local oscillator is coupled into polarization diversity optical hybrid to mix with the signal. The signal is detected by typical coherent receivers. The four RF signals for the two IQ components are then fed into an oscillator scope and are acquired at 50 GS/s and processed off-line with a MATLAB program using 2x2 MIMO-OFDM models.

Fig. 2 Experimental setup. PBS/C: Polarization beam splitter/combiner; ECL: External cavity laser; SW: switch; OTF: optical tunable filter.

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To accurately assess the performance of PDM-OFDM with polarization demultiplexing in Stokes space (SS-PDM-OFDM), we compare the experimental results with those of using 20 training symbols in every 200 symbols for channel estimation without using intra-symbol frequency-domain averaging [3]. Figure 3(a) shows the back-to-back (B2B) performance and transmission over 5440km SSMF performance of PDM-OFDM with training symbols (TS-PDM-OFDM) and SS-PDM-OFDM when the subcarriers are modulated by 4QAM. Figure 3(d) shows the back-to-back (B2B) performance and transmission over 960km SSMF performance of TS-PDM-OFDM and SS-PDM-OFDM when the subcarriers are modulated by 16QAM. Figures 3(b), 3(c), 3(e) and 3(f) are the constellations for the two polarizations of the recovered SS-PDM-OFDM signals modulated by 4QAM and 16QAM, respectively. The BTB performance of SS-PDM-OFDM is almost the same as that of TS-PDM-OFDM. The performance after transmitting over fiber of SS-PDM-OFDM is slightly worse compared with that of TS-PDM-OFDM, 0.5 dB approximately.

Fig. 3 (a) BER results of 66.7-Gb/s TS-PDM-OFDM and SS-PDM-OFDM with 4QAM subcarrier modulation; (b) and (c) Constellations after transmission over 5440km with 4QAM subcarrier modulation vs. OSNR 18.42dB; (d) BER results of 133.3-Gb/s TS-PDM-OFDM and SS-PDM-OFDM with 16QAM subcarrier modulation; (e) and (f) Constellations after transmission over 960km with 16QAM subcarrier modulation vs. OSNR 23.82dB.

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4. Speed of convergence

One advantage of polarization demultiplexing in Stokes space used in PDM-OFDM is that its speed of convergence is very fast. The SS-PDM is conducted after the FFT window synchronization and FFT, and the resulted data points exclude the transition between symbols, unlike the single-carrier system [19]. This helps the successful finding of the normal without using many data points, as shown in Figs. 1(a) and 1(b). Figure 4 shows the absolute values of the first row elements of the matrix $M^{- 1}$ vs. the number of OFDM symbols to analyze the speed of convergence. The a and b plotted in Fig. 4 represent the two values in the first row of the unitary matrix $M^{- 1}$ . Figure 4(a) is for 4QAM subcarrier modulation and Fig. 4(b) is for 16QAM subcarrier modulation. 16QAM modulation needs slightly more symbols for convergence than 4QAM modulation. However, one hundred symbols are enough for both of them, corresponding to $1.28 μ s$ at most, which is enough to track the dynamic change of polarization in field fibers.

Fig. 4. (a) Speed of convergence of Stokes space polarization alignment for (a) 4QAM subcarrier modulation and (b) 16QAM subcarrier modulation.

5. Conclusion

In this paper we have proposed a method of polarization demultiplexing in Stokes space for coherent optical PDM-OFDM without inserting training symbols. We first convert the received OFDM signals to the frequency domain by FFT. Then polarization demultiplexing in Stokes space can be applied to high-speed, wide-bandwidth OFDM signals, but on per subcarrier basis. It works for different modulation formats on OFDM subcarriers and requires no training symbols, which means a reduced overhead. This method has been verified in the experiments of 66.6-Gb/s coherent optical PDM-OFDM with 4QAM subcarrier modulation and 133.3-Gb/s coherent optical PDM-OFDM with 16QAM subcarrier modulation. The experiments results show that the performance of SS-PDM is almost the same as that of the conventional method. At last, we have analyzed the speed of convergence of this method, which is on the order of microsecond. Since the computation complexity is important in optical fiber transmissions, our future work is to identify the optimal algorithm and compare it with the conventional methods for polarization demultiplexing.

Acknowledgement

This work was supported in part by 863 Program（No. 2012AA011302) and （No. 2013AA010503), NSFC (No. 61071097, and (No 61107060), and State Key Laboratory of Advanced Optical Communication Systems and Networks.

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Polarization demultiplexing in stokes space for coherent optical PDM-OFDM

Abstract

1. Introduction

2. Principle of SS-PDM for CO-OFDM

3. Experimental Setup and results

4. Speed of convergence

5. Conclusion

Acknowledgement

References and links

Cited By

Figures (3)

Equations (5)

Optics Express