Transmission of multi-polarization-multiplexed signals: another freedom to explore?

Zhiyu Chen; Lianshan Yan; Wei Pan; Bin Luo; Yinghui Guo; Hengyun Jiang; Anlin Yi; Yafei Sun; Xiaoxia Wu

doi:10.1364/OE.21.011590

1. Introduction

In recent years, both the system capacity and the spectral efficiency (SE) have been significantly improved to meet the demands of the ever-growing data traffic [1, 2]. Thanks to the advanced modulation formats (i.e. orthogonal frequency-division multiplexing, phase-shift-keying (PSK) and quadrature amplitude modulation (QAM)) and coherent detection techniques, the spectral efficiency with close to 10 bit/s/Hz or even higher has been achieved [3–5]. However, the growth of the data traffic does not appear to be leveling off any time soon, and most likely it will continue to grow in an exponential trend. Thus, many researchers focus on how to further increase the SE and overall capacity [5–18]. Generally there are five major candidates, including higher-order modulation formats (i. e. 256-QAM, 512-QAM and so on) [5, 6], polarization-division multiplexing (PDM) [7–10], frequency-division multiplexing (FDM) [11–13], space-division multiplexing (SDM) [14–16] and orbital-angular momentum [17]. Among these technologies, advanced higher-order modulation formats combined with PDM scheme, in which information is encoded on amplitude, phase and polarization of the light wave, seem to be more practical in the near future [19–22].

Up to now, only two orthogonal states of polarization (SOPs) are available in traditional PDM systems due to the relatively simple demultiplexing method and manageable crosstalk between two SOPs. It would be highly desirable if we can further utilize the multidimensional polarization as it can provide infinite multiplexing freedoms in theory. However, due to the difficulty of polarization management, the demonstrated PDM systems are limited to two polarization states.

Previously, we have performed the first-step investigation about the feasibility of signal multiplexing with four SOPs for on-off-keying (OOK) signals [23]. Compared with traditional OOK system, the proposed scheme could quadruple the system capacity and SE directly. In this paper, a thorough analysis of the 4PM system is presented. In our scheme, the signal is generated by combining two conventional PDM signals, and demodulated using coherent detection technology combined with post digital signal processing (DSP). The angle between any two neighbor SOPs is set to be 45° (i.e. 0, 45, 90 and 135° at the transmitter). Then, the details of phase synchronization scheme are presented. Furthermore, we also analyze the impact of the crosstalk from polarization-mode dispersion (PMD) on four-polarization-multiplexing (4PM) systems. Simulation results show that the transmission distance of a 4 × 10-Gbit/s 4PM non-return-to-zero OOK (NRZ-OOK) could be extended from ~50 km to more than ~80 km using feedback-decision equalizer (FDE). The potential of 4PM-OOK system is also shown comparable performance with 40-Gbit/s PDM-QPSK systems under the same SE, which indicates that utilizing the freedom of polarization may also be another potential solution for future networks.

2. Principle and theoretical model for 4PM-OOK signal

Figure 1 shows the generation and demultiplexing scheme for the 4PM-OOK system. The NRZ-OOK signal E₁, E₂, E₃ and E₄ (CH1-CH4) are generated from the laser1, which can be written as $E_{i} = | A_{i} | \exp (j ω_{c} t + ϕ)$ (i = 1 ~4), where A_i, $ω_{c}$ and $ϕ$ are the amplitudes, carrier angular frequency and phase of laser, respectively. Thus, after optical coupler2 (OC2), the multiplexed signal is expressed as follows. Note that there needs to be an additional $π / 2$ phase shift in the lower arm of the OC2.

\begin{array}{l} E_{m} = E_{x} + E_{y} \\ = [\hat{x} (\frac{\sqrt{2}}{2} A_{1} + \frac{1}{2} j A_{3} - \frac{1}{2} j A_{4}) + \hat{y} (\frac{\sqrt{2}}{2} A_{2} + \frac{1}{2} j A_{3} + \frac{1}{2} j A_{4})] \exp (j ω_{c} t + ϕ), \end{array}

where E_x and E_y are the horizontal and vertical polarization modes, respectively.

Fig. 1 (a) Setup for four SOPs multiplexing, transmission and demodulation. (b) Receiver configuration for 4PM-OOK signal. OC: optical coupler; PC: polarization controller; VOA: variable optical attenuator; SMF: single-mode fiber; PBS: polarization beam splitter; LO: local oscillator laser; PD: photodetector; ADC: analog to digital converter; DSP: digital signal processing; CHn: channel n, (n = 1-4).

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In order to demodulate the signal, firstly the SOP of E₁ should be adjusted to be aligned with the x-axis of PBS3 by a polarization control (PC7). Assuming that the angular frequency and initial phase of the local oscillator (LO) laser are $ω_{c}$ and $θ$ , the LO laser can be written as $E_{L O} = | A_{L O} | \exp [j (ω_{c} t + θ)]$ . Next, the signals combined with a LO laser are fed into two optical hybrids, whose outputs are $E_{x} \pm j E_{L O}$ , $E_{x} \pm E_{L O}$ , $E_{y} \pm j E_{L O}$ and $E_{y} \pm E_{L O}$ , respectively. Here, we only use the four outputs ( $E_{x} \pm j E_{L O}$ and $E_{y} \pm j E_{L O}$ ) and obtain the expressions as follows:

{\begin{cases} {E^{'}}_{1} = \frac{\sqrt{2}}{2} \cdot {\frac{\sqrt{2}}{2} A_{1} \exp [j (ω_{c} t + ϕ)] + \frac{1}{2} j A_{3} \exp [j (ω_{c} t + ϕ)] - \frac{1}{2} j A_{4} \exp [j (ω_{c} t + ϕ)] + j A_{L O} \exp [j (ω_{c} t + θ)]} \\ = \frac{\sqrt{2}}{2} \cdot {\frac{\sqrt{2}}{2} A_{1} \exp [j (ω_{c} t + ϕ)] + j B_{1} \exp j (ω_{c} t)}, \\ {E^{'}}_{2} = \frac{\sqrt{2}}{2} \cdot {\frac{\sqrt{2}}{2} A_{1} \exp [j (ω_{c} t + ϕ)] + \frac{1}{2} j A_{3} \exp [j (ω_{c} t + ϕ)] - \frac{1}{2} j A_{4} \exp [j (ω_{c} t + ϕ)] - j A_{L O} \exp [j (ω_{c} t + θ)]} \\ = \frac{\sqrt{2}}{2} \cdot {\frac{\sqrt{2}}{2} A_{1} \exp [j (ω_{c} t + ϕ)] + j B_{2} \exp j (ω_{c} t)}, \\ {E^{'}}_{3} = \frac{\sqrt{2}}{2} \cdot {\frac{\sqrt{2}}{2} A_{2} \exp [j (ω_{c} t + ϕ)] + \frac{1}{2} j A_{3} \exp [j (ω_{c} t + ϕ)] + \frac{1}{2} j A_{4} \exp [j (ω_{c} t + ϕ)] + j A_{L O} \exp [j (ω_{c} t + θ)]} \\ = \frac{\sqrt{2}}{2} \cdot {\frac{\sqrt{2}}{2} A_{2} \exp [j (ω_{c} t + ϕ)] + j B_{3} \exp j (ω_{c} t)}, \\ {E^{'}}_{4} = \frac{\sqrt{2}}{2} \cdot {\frac{\sqrt{2}}{2} A_{2} \exp [j (ω_{c} t + ϕ)] + \frac{1}{2} j A_{3} \exp [j (ω_{c} t + ϕ)] + \frac{1}{2} j A_{4} \exp [j (ω_{c} t + ϕ)] - j A_{L O} \exp [j (ω_{c} t + θ)]} \\ = \frac{\sqrt{2}}{2} \cdot {\frac{\sqrt{2}}{2} A_{2} \exp [j (ω_{c} t + ϕ)] + j B_{4} \exp j (ω_{c} t)}, \end{cases}

where E’_n (n = 1~4) are the outputs of the optical hybrids.

[\begin{array}{l} B_{1} \\ B_{2} \\ B_{3} \\ B_{4} \end{array}] = [\begin{array}{l} \frac{1}{2} A_{3} \exp (j ϕ) - \frac{1}{2} A_{4} \exp (j ϕ) + A_{L O} \exp (j θ) \\ \frac{1}{2} A_{3} \exp (j ϕ) - \frac{1}{2} A_{4} \exp (j ϕ) - A_{L O} \exp (j θ) \\ \frac{1}{2} A_{3} \exp (j ϕ) + \frac{1}{2} A_{4} \exp (j ϕ) + A_{L O} \exp (j θ) \\ \frac{1}{2} A_{3} \exp (j ϕ) + \frac{1}{2} A_{4} \exp (j ϕ) - A_{L O} \exp (j θ) \end{array}] .

Afterwards, the optical fields are detected by photodetectors (PDs) with responsivity $ℜ$ to produce the photocurrents i_n, which are given by

\begin{array}{l} i_{1} = \frac{ℜ}{2} [\frac{\sqrt{2}}{2} A_{1} \exp j (ω_{c} t + ϕ) + j B_{1} \exp j (ω_{c} t)] \cdot [\frac{\sqrt{2}}{2} A_{1} \exp j (- ω_{c} t - ϕ) - j B_{1}^{*} \exp j (- ω_{c} t)] \\ = \frac{ℜ}{2} [\frac{1}{2} A_{1}^{2} + B_{1} \cdot B_{1}^{*} - \frac{\sqrt{2}}{2} j A_{1} B_{1}^{*} \exp (j ϕ) + \frac{\sqrt{2}}{2} j A_{1} B_{1} \exp (- j ϕ)], \end{array}

i_{2} = \frac{ℜ}{2} [\frac{1}{2} A_{1}^{2} + B_{2} \cdot B_{2}^{*} - \frac{\sqrt{2}}{2} j A_{1} B_{2}^{*} \exp (j ϕ) + \frac{\sqrt{2}}{2} j A_{1} B_{2} \exp (- j ϕ)],

i_{7} = \frac{ℜ}{2} [\frac{1}{2} A_{2}^{2} + B_{3} \cdot B_{3}^{*} - \frac{\sqrt{2}}{2} j A_{2} B_{3}^{*} \exp (j ϕ) + \frac{\sqrt{2}}{2} j A_{2} B_{3} \exp (- j ϕ)],

i_{8} = \frac{ℜ}{2} [\frac{1}{2} A_{2}^{2} + B_{4} \cdot B_{4}^{*} - \frac{\sqrt{2}}{2} j A_{2} B_{4}^{*} \exp (j ϕ) + \frac{\sqrt{2}}{2} j A_{2} B_{4} \exp (- j ϕ)] .

For simplicity, assuming that $ϕ = θ$ , the sum of the third and fourth terms in Eqs. (4)–(7) becomes zero. Thus, i_n can be simplified as follows:

i_{1} = ℜ [\frac{1}{4} A_{1}^{2} + {(\frac{\sqrt{2}}{4} A_{3} - \frac{\sqrt{2}}{4} A_{4} + \frac{\sqrt{2}}{2} A_{L O})}^{2}],

i_{2} = ℜ [\frac{1}{4} A_{1}^{2} + {(\frac{\sqrt{2}}{4} A_{3} - \frac{\sqrt{2}}{4} A_{4} - \frac{\sqrt{2}}{2} A_{L O})}^{2}],

i_{7} = ℜ [\frac{1}{4} A_{2}^{2} + {(\frac{\sqrt{2}}{4} A_{3} + \frac{\sqrt{2}}{4} A_{4} + \frac{\sqrt{2}}{2} A_{L O})}^{2}],

i_{8} = ℜ [\frac{1}{4} A_{2}^{2} + {(\frac{\sqrt{2}}{4} A_{3} + \frac{\sqrt{2}}{4} A_{4} - \frac{\sqrt{2}}{2} A_{L O})}^{2}],

and

i_{1} - i_{2} = ℜ (A_{3} A_{L O} - A_{4} A_{L O}),

i_{7} - i_{8} = ℜ (A_{3} A_{L O} + A_{4} A_{L O}) .

Combining Eqs. (8)–(13), the four SOPs multiplexing signal can be demodulated and the four output channels are respectively described as follows, which can be used for digital signal processing (DSP).

A_{1} = \frac{1}{ℜ} \sqrt{2 (i_{1} + i_{2}) - \frac{{(i_{1} - i_{2})}^{2}}{2 A_{L O}^{2}} - 2 A_{L O}^{2}},

A_{2} = \frac{1}{ℜ} \sqrt{2 (i_{7} + i_{8}) - \frac{{(i_{7} - i_{8})}^{2}}{2 A_{L O}^{2}} - 2 A_{L O}^{2}},

A_{3} = \frac{i_{1} - i_{2} + i_{7} - i_{8}}{2 ℜ A_{L O}},

A_{4} = \frac{i_{1} - i_{2} - (i_{7} - i_{8})}{2 ℜ A_{L O}} .

3. Phase synchronization

In the practical systems, the assumption of $ϕ = θ$ is difficult to be implemented. Thus, when $ϕ \neq θ$ , the photocurrents i₁ and i₂ become

\begin{array}{l} i_{1} = \frac{ℜ}{2} [\frac{1}{2} A_{1}^{2} + \frac{1}{4} A_{3}^{2} + \frac{1}{4} A_{4}^{2} + A_{L O}^{2} - \frac{1}{2} A_{3} A_{4} + A_{3} A_{L O} \cos (ϕ - θ) \\ - A_{4} A_{L O} \cos (ϕ - θ) + \sqrt{2} A_{1} A_{L O} \sin (ϕ - θ)], \end{array}

\begin{array}{l} i_{2} = \frac{ℜ}{2} [\frac{1}{2} A_{1}^{2} + \frac{1}{4} A_{3}^{2} + \frac{1}{4} A_{4}^{2} + A_{L O}^{2} - \frac{1}{2} A_{3} A_{4} - A_{3} A_{L O} \cos (ϕ - θ) \\ + A_{4} A_{L O} \cos (ϕ - θ) - \sqrt{2} A_{1} A_{L O} \sin (ϕ - θ)] . \end{array}

In this case, the difference of i₁ and i₂ becomes

i_{1} - i_{2} = ℜ [A_{3} A_{L O} \cos (ϕ - θ) - A_{4} A_{L O} \cos (ϕ - θ) + \sqrt{2} A_{1} A_{L O} \sin (ϕ - θ)] .

It is different from the Eq. (12), which leads to the large error when demodulating the channels. Thus, an algorithm is required to synchronize the phase of the local oscillator with the transmitter laser. If we assume that the phases of carrier and LO drift slowly, one solution is shown in Fig. 2, where i₃ and i₄ are given by (E_x + E_LO) × (E_x-E_LO)* and (E_x + E_LO) × (E_x-E_LO)*, respectively.

Fig. 2 Phase synchronization scheme for 4PM-OOK signal. Re{X}: real part of X.

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\begin{array}{l} i_{3} = \frac{ℜ}{2} [\frac{1}{2} A_{1}^{2} + \frac{1}{4} A_{3}^{2} + \frac{1}{4} A_{4}^{2} + A_{L O}^{2} - \frac{1}{2} A_{3} A_{4} - A_{3} A_{L O} \sin (ϕ - θ) \\ + A_{4} A_{L O} \sin (ϕ - θ) + \sqrt{2} A_{1} A_{L O} \cos (ϕ - θ)], \end{array}

\begin{array}{l} i_{4} = \frac{ℜ}{2} [\frac{1}{2} A_{1}^{2} + \frac{1}{4} A_{3}^{2} + \frac{1}{4} A_{4}^{2} + A_{L O}^{2} - \frac{1}{2} A_{3} A_{4} + A_{3} A_{L O} \sin (ϕ - θ) \\ - A_{4} A_{L O} \sin (ϕ - θ) - \sqrt{2} A_{1} A_{L O} \cos (ϕ - θ)] . \end{array}

By subtracting i₃ from i₄, we can obtain the expression as follow

i_{3} - i_{4} = - ℜ [A_{3} A_{L O} \sin (ϕ - θ) - A_{4} A_{L O} \sin (ϕ - θ) - \sqrt{2} A_{1} A_{L O} \cos (ϕ - θ)] .

Afterwards, the expression of point A (i.e. Figure 2) is given by

E_{A} = ℜ [A_{3} A_{L O} \exp j (ϕ - θ) - A_{4} A_{L O} \exp j (ϕ - θ) - \sqrt{2} A_{1} A_{L O} \exp j (ϕ - θ + \frac{π}{2})] .

The relative phase difference between LO and transmitter laser can be estimated according to the Eq. (24). To cancelled the phase asynchronization, the E_A should be shifted by $ϕ - θ$ , and the output of B can be written as

\begin{array}{l} E_{B} = E_{A} \cdot \exp [- j (ϕ - θ)] \\ = ℜ [A_{3} A_{L O} - A_{4} A_{L O} - j \sqrt{2} A_{1} A_{L O}] . \end{array}

Finally, by retrieving the real part of the E_B, we can obtain the output of phase synchronization as

i_{o u t} = ℜ (A_{3} A_{L O} - A_{4} A_{L O}) .

Compared the Eq. (26) with Eq. (12), the phase asynchronization between LO and transmitter laser is successfully compensated. In addition, the frequency offset between LO laser and input signals also need to be compensated in the DSP unit.

4. Demodulation for 4PM-PSK signal

The PSK modulation format has recently attracted increasing interests due to the 3-dB receiver sensitivity enhancement and better tolerance to nonlinear effects [24, 25]. Thus, we also investigate the demodulation scheme for 4PM-PSK signal. In this case, the inputs are given by $E_{i} = | A | \exp (j ω_{c} t + ϕ_{i})$ , where $ϕ_{i}$ is the phase information of the i-th channel. After OC2, the multiplexing signal can be expressed as

E_{m} = A \exp (j ω_{c} t) {\begin{cases} \hat{x} [\frac{\sqrt{2}}{2} \exp (j ϕ_{1}) + \frac{1}{2} j \exp (j ϕ_{3}) - \frac{1}{2} j \exp (j ϕ_{4})] \\ + \hat{y} [\frac{\sqrt{2}}{2} \exp (j ϕ_{2}) + \frac{1}{2} j \exp (j ϕ_{3}) + \frac{1}{2} j \exp (j ϕ_{4})] \end{cases}} .

As shown in Fig. 3, similar to the demodulation scheme for 4PM-OOK signal, the divided signals (E_x & E_y) are combined with a LO laser into the corresponding optical hybrid, whose the outputs are E_x ± jE_LO, E_x ± E_LO, E_y ± jE_LO and E_y ± E_LO, respectively. Here, E_LO is given by $E_{L O} = | A | \exp (j ω_{c} t + ϕ_{L O})$ , where $ϕ_{L O}$ is the initial phase of the LO laser. After photoelectric conversion, eight signals can be expressed as

Fig. 3 Demodulation scheme for PSK signal. BPD: balance photodetector.

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{\begin{cases} i_{1}^{(1)} = ℜ \cdot (E_{x} + j E_{L O}) \cdot (^{E_{x} + j E_{L O}) *} \\ = ℜ | A |^{2} [\frac{\sqrt{2}}{2} \sin (ϕ_{1} - ϕ_{3}) + \frac{\sqrt{2}}{2} \sin (ϕ_{4} - ϕ_{1}) + \sqrt{2} \sin (ϕ_{1} - ϕ_{L O}) - \frac{1}{2} \cos (ϕ_{3} - ϕ_{4}) \\ + \cos (ϕ_{3} - ϕ_{L O}) - \cos (ϕ_{L O} - ϕ_{4}) + 2] \\ i_{1}^{(2)} = ℜ \cdot (E_{x} - j E_{L O}) \cdot (^{E_{x} - j E_{L O}) *} \\ = ℜ | A |^{2} [\frac{\sqrt{2}}{2} \sin (ϕ_{1} - ϕ_{3}) + \frac{\sqrt{2}}{2} \sin (ϕ_{4} - ϕ_{1}) - \sqrt{2} \sin (ϕ_{1} - ϕ_{L O}) - \frac{1}{2} \cos (ϕ_{3} - ϕ_{4}) \\ - \cos (ϕ_{3} - ϕ_{L O}) + \cos (ϕ_{L O} - ϕ_{4}) + 2] \end{cases},

{\begin{cases} i_{2}^{(1)} = ℜ \cdot (E_{x} + E_{L O}) \cdot (^{E_{x} + E_{L O}) *} \\ = ℜ | A |^{2} [\frac{\sqrt{2}}{2} \sin (ϕ_{1} - ϕ_{3}) + \frac{\sqrt{2}}{2} \sin (ϕ_{4} - ϕ_{1}) + \sqrt{2} \cos (ϕ_{1} - ϕ_{L O}) - \frac{1}{2} \cos (ϕ_{3} - ϕ_{4}) \\ - \sin (ϕ_{3} - ϕ_{L O}) + \sin (ϕ_{4} - ϕ_{L O}) + 2] \\ i_{2}^{(2)} = ℜ \cdot (E_{x} - E_{L O}) \cdot (^{E_{x} - E_{L O}) *} \\ = ℜ | A |^{2} [\frac{\sqrt{2}}{2} \sin (ϕ_{1} - ϕ_{3}) + \frac{\sqrt{2}}{2} \sin (ϕ_{4} - ϕ_{1}) - \sqrt{2} \cos (ϕ_{1} - ϕ_{L O}) - \frac{1}{2} \cos (ϕ_{3} - ϕ_{4}) \\ + \sin (ϕ_{3} - ϕ_{L O}) - \sin (ϕ_{4} - ϕ_{L O}) + 2] \end{cases},

{\begin{cases} i_{3}^{(1)} = ℜ \cdot (E_{y} + E_{L O}) \cdot (^{E_{y} + E_{L O}) *} \\ = ℜ | A |^{2} [\frac{\sqrt{2}}{2} \sin (ϕ_{2} - ϕ_{3}) + \frac{\sqrt{2}}{2} \sin (ϕ_{2} - ϕ_{4}) + \sqrt{2} \cos (ϕ_{2} - ϕ_{L O}) + \frac{1}{2} \cos (ϕ_{3} - ϕ_{4}) \\ - \sin (ϕ_{3} - ϕ_{L O}) - \sin (ϕ_{4} - ϕ_{L O}) + 2] \\ i_{3}^{(2)} = ℜ \cdot (E_{y} - E_{L O}) \cdot (^{E_{y} - E_{L O}) *} \\ = ℜ | A |^{2} [\frac{\sqrt{2}}{2} \sin (ϕ_{2} - ϕ_{3}) + \frac{\sqrt{2}}{2} \sin (ϕ_{2} - ϕ_{4}) - \sqrt{2} \cos (ϕ_{2} - ϕ_{L O}) + \frac{1}{2} \cos (ϕ_{3} - ϕ_{4}) \\ + \sin (ϕ_{3} - ϕ_{L O}) + \sin (ϕ_{4} - ϕ_{L O}) + 2] \end{cases},

{\begin{cases} i_{4}^{(1)} = ℜ \cdot (E_{y} + j E_{L O}) \cdot (^{E_{y} + j E_{L O}) *} \\ = ℜ | A |^{2} [\frac{\sqrt{2}}{2} \sin (ϕ_{2} - ϕ_{3}) + \frac{\sqrt{2}}{2} \sin (ϕ_{2} - ϕ_{4}) + \sqrt{2} \sin (ϕ_{2} - ϕ_{L O}) + \frac{1}{2} \cos (ϕ_{3} - ϕ_{4}) \\ + \cos (ϕ_{3} - ϕ_{L O}) + \cos (ϕ_{4} - ϕ_{L O}) + 2] \\ i_{4}^{(2)} = ℜ \cdot (E_{y} - j E_{L O}) \cdot (^{E_{y} - j E_{L O}) *} \\ = ℜ | A |^{2} [\frac{\sqrt{2}}{2} \sin (ϕ_{2} - ϕ_{3}) + \frac{\sqrt{2}}{2} \sin (ϕ_{2} - ϕ_{4}) - \sqrt{2} \sin (ϕ_{2} - ϕ_{L O}) + \frac{1}{2} \cos (ϕ_{3} - ϕ_{4}) \\ - \cos (ϕ_{3} - ϕ_{L O}) - \cos (ϕ_{4} - ϕ_{L O}) + 2] \end{cases} .

Afterwards, the outputs of four balance detectors are

[\begin{array}{l} i_{1} \\ i_{2} \\ i_{3} \\ i_{4} \end{array}] = [\begin{array}{l} i_{1}^{(1)} - i_{1}^{(2)} \\ i_{2}^{(1)} - i_{2}^{(2)} \\ i_{3}^{(1)} - i_{3}^{(2)} \\ i_{4}^{(1)} - i_{4}^{(2)} \end{array}] = [\begin{array}{l} 2 ℜ | A |^{2} [\sqrt{2} \sin (ϕ_{1} - ϕ_{L O}) + \cos (ϕ_{3} - ϕ_{L O}) - \cos (ϕ_{4} - ϕ_{L O})] \\ 2 ℜ | A |^{2} [\sqrt{2} \cos (ϕ_{1} - ϕ_{L O}) - \sin (ϕ_{3} - ϕ_{L O}) + \sin (ϕ_{4} - ϕ_{L O})] \\ 2 ℜ | A |^{2} [\sqrt{2} \cos (ϕ_{2} - ϕ_{L O}) - \sin (ϕ_{3} - ϕ_{L O}) - \sin (ϕ_{4} - ϕ_{L O})] \\ 2 ℜ | A |^{2} [\sqrt{2} \sin (ϕ_{2} - ϕ_{L O}) + \cos (ϕ_{3} - ϕ_{L O}) + \cos (ϕ_{4} - ϕ_{L O})] \end{array}] .

By solving these equations, the phase information of four channels can be recovered, which could be used as the channel demodulation algorithm of the DSP. However, for 4PM-PSK system, the operations of DSP at the receiver are more complex, which also significantly increases the complexity of the receiver. Thus, in a practical application, there are still some issues to be solved including simplifying the demodulation schemes, etc. These issues will be further investigated in our next step.

5. Crosstalk due to PMD

Generally, PMD is considered to be one of the major impairments in PDM systems [26, 27]. It causes the output SOP of a fully polarized input signal to vary with frequency. Due to the limited bandwidth of the signal, the orthogonal SOPs cannot be completely separated by using a PC and a PBS, which results in coherent crosstalk because of PMD. It is obvious that PMD is still one of the major obstacles in MPM systems. Thus, to analyze the impact of crosstalk due to the PMD on our 4PM system, the demodulated model is simplified as shown in Fig. 4.

Fig. 4 Demodulated model of a 4PM system. PC1 and PC2 control the launch angle into fiber and PBS, respectively. CHx: channel x; CHy: channel y.

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The 4PM-OOK signal is described by Eq. (1). The demodulated model in this paper is similar to the conventional PDM system (PC & PBS). Thus, the spectral power density coupled from E_y channel to E_x can be expressed as [26]

I_{y x} (ω) = {\tilde{A}}_{y} {\tilde{A}}_{y}^{*} {(\frac{ω \cdot \vec{τ} \times {\hat{s}}_{E y} (0)}{2})}^{2},

where the output spectral amplitude

{\tilde{A}}_{y}

is the Fourier transform of

A_{y}

(A_y = |E_y|).

ω

,

\vec{τ}

and

{\hat{s}}_{E y}

are the angular frequency deviation from the carrier, PMD vector, and the Stokes vector of channel y (CHy).

Considering the case of launching the signal at 45° to the principal state of polarization-maintaining fiber (PMF), we can obtain ${[\vec{τ} \times {\hat{s}}_{E y} (0)]}^{2} = {(Δ τ)}^{2}$ , where $Δ τ$ is the differential group delay (DGD). According to the definition of $I_{y x} (ω) = {\tilde{A}}_{y x} (ω) \cdot {\tilde{A}}_{y x}^{*} (ω)$ , one can recover the spectral amplitude of crosstalk from CHy as ${\tilde{A}}_{y x} = j (ω \cdot Δ τ / 2) {\tilde{A}}_{y}$ . Thus, by neglecting the depletion of the channel x (CHx), the output spectrum of CHx becomes ${\tilde{A}}^{'}_{x} (ω) = {\tilde{A}}_{x} (ω) + j (ω \cdot Δ τ / 2) {\tilde{A}}_{y} (ω)$ . Afterwards, the Fourier transform of the spectral amplitude ${\tilde{A}}^{'}_{x} (ω)$ can be written as [26] ${A^{'}}_{x} (t) = A_{x} (t) + \frac{Δ τ}{2} {\dot{A}}_{y} (t)$ , where ${A^{'}}_{x} (t)$ ${E^{'}}_{x}$ is the degraded signal induced by PMD, and ${\dot{A}}_{y} = \frac{d A_{y}}{d t}$ is the time derivative. Thus, we can obtain the expressions after PBS:

{\begin{cases} {E^{'}}_{x} = [(\frac{\sqrt{2}}{2} A_{1} + \frac{1}{2} j A_{3} - \frac{1}{2} j A_{4}) + j \frac{Δ τ ω_{c}}{2} (\frac{\sqrt{2}}{2} {\dot{A}}_{2} + \frac{1}{2} j {\dot{A}}_{3} + \frac{1}{2} j {\dot{A}}_{4})] \exp [j (ω_{c} + ϕ)] \\ {E^{'}}_{y} = [(\frac{\sqrt{2}}{2} A_{2} + \frac{1}{2} j A_{3} + \frac{1}{2} j A_{4}) + j \frac{Δ τ ω_{c}}{2} (\frac{\sqrt{2}}{2} {\dot{A}}_{1} + \frac{1}{2} j {\dot{A}}_{3} - \frac{1}{2} j {\dot{A}}_{4})] \exp [j (ω_{c} + ϕ)] \end{cases} .

Substituting Eq. (34) into Eqs. (1)–(17) and ignoring the term of $Δ τ^{2}$ , the new expressions for the four outputs are given as

{A^{'}}_{1} = \sqrt{A_{1}^{2} - \frac{\sqrt{2}}{2} Δ τ ω_{c} (A_{1} {\dot{A}}_{3} + A_{1} {\dot{A}}_{4})},

{A^{'}}_{2} = \sqrt{A_{2}^{2} - \frac{\sqrt{2}}{2} Δ τ ω_{c} (A_{2} {\dot{A}}_{3} + A_{2} {\dot{A}}_{4})},

A_{3}^{'} = A_{3} + \frac{\sqrt{2}}{4} Δ τ ω_{c} ({\dot{A}}_{1} + {\dot{A}}_{2}),

A_{4}^{'} = A_{4} + \frac{\sqrt{2}}{4} Δ τ ω_{c} (- {\dot{A}}_{1} + {\dot{A}}_{2}) .

In these equations, the second term is the crosstalk induced by the PMD impairment. Taking channel 4 as an example, the time derivative of the interfering terms ${\dot{A}}_{1}$ and ${\dot{A}}_{2}$ imply the crosstalk only occurs at the edges of the pulse instead of during the whole pulse period. In addition, the crosstalk mainly comes from the neighboring polarization channels, while the impact from the orthogonal polarization channel can be mitigated by using signal processing algorithms.

On the other hand, when PMD-induced impairments are well compensated, polarization dependent loss (PDL) becomes a primary source of the system degradation, which has been carefully studied in PDM systems [28, 29]. It is obvious that PDL degrades the signals more seriously in MPM systems. It serves to cause the power imbalance during the four polarization states, which leads to the different optical signal-to-noise (OSNR) to the tributaries. Without PMD compensation, the combining effects of PMD and PDL may lead more severe performance fluctuations or degradations [30].

6. Setup and simulation results

The simulation setup of 4PM-OOK system as shown in Fig. 1 is performed by using OptiSim simulation platform, whose credibility has been validated in many published literatures [31, 32]. The 10-Gbit/s NRZ-OOK signal with the wavelength of 1550-nm is generated by a Mach-Zehnder modulator with a 2⁷-1 pseudorandom bit sequence. The signal is then split into four streams by a coupler and polarization multiplexed to generate two traditional PDM signals using PBS1 and PBS2, respectively. Here, three spools of 1-km SMF (SMF1, SMF2 and SMF3) are inserted in corresponding branches to decorrelate the data stream; and variable optical attenuators (VOAs) are applied in four channels to balance the optical power among them. In order to obtain the four SOPs multiplexing signal, two PDM signals are combined through two PCs (PC5 & PC6) and a coupler (OC2). The multiplexed signal is then launched into the SMF. Two issues should be mentioned about such multiplexing configuration: (i) the principal axis of the PBS1 is adjusted to be 45° with respect of that of PBS2; (ii) there needs to be an additional $π / 2$ phase shift in the lower arm of OC2, which can be realized by using a phase modulator (PM) or a phase shifter.

After the transmission, the multiplexed signal is divided into two orthogonal streams by PBS3 as shown in Fig. 1(b). Here, PC7 is used to align the SOP of E₁ with the one of the principal axis of PBS3. Then, these two streams are respectively mixed in the 90°optical hybrid with a local oscillator laser. The outputs are detected by eight photodetectors, and digitized by analog to digital converters (ADCs), respectively. Afterwards, the phase of LO is synchronized with the transmitter laser (Fig. 2), and four polarization-channels are demodulated through a DSP unit incorporating our algorithms. In addition, the performance of demodulated signal would be degraded due to the dispersion. Thus, a feedback decision equalizer (FDE) is also required to compensate the dispersion as shown in Fig. 1(b).

Figure 5 shows the back-to-back performance of the proposed demodulator. According to the Eqs. (14)–(17), all the four channels are successfully demultiplexed assisted by the digital signal processing. The eye-diagrams for all outputs are clearly open, but they exhibit different performance. It is because the expressions of Eqs. (14) and (16) (or Eqs. (15) and (17)) are different. Actually, due to the terms of (i₁-i₂) and (i₅-i₆) in Eqs. (16) and (17), four corresponding PDs (i.e. Fig. 1) can be replaced by two balance detectors (BPDs), which could significantly improve the signal performance. Thus, as expected, the signal qualities of CH3 and CH4 are better than other two channels as shown in Fig. 5.

Fig. 5 Back-to-back eye-diagrams for one typical input and four demodulated outputs.

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In order to investigate the PMD crosstalk, a section of polarization-maintaining fiber with the DGD of 10.2-ps is inserted before the demodulator. As an illustration, the crosstalks coupled from CHn (n = 1~3) to CH4 are shown in Fig. 6(a)–6(c), respectively. If there is only one input (E₁ or E₂), the crosstalk from CH1 (or CH2) is serious (solid line), while that from CH3 is slight (Fig. 6(c)). Furthermore, the interference terms ${\dot{A}}_{1}$ and ${\dot{A}}_{2}$ have opposite sign at the leading (or trailing) edge of the corresponding channel, which agrees well with the Eq. (38). In addition, the spectral power density coupled from CH1 to CH4 is also obtained when the DGD value is 10.2-ps, as shown in Fig. 6(d). The BER performance are shown in Fig. 7, there is ~3-dB power difference between CH1 (or CH2) and CH3 (or CH4) at the BER of 10⁻⁹. It is the major disadvantage of this demodulation technology.

Fig. 6 Crosstalk coupled form CH1 (a), CH2 (b) or CH3(c) to CH4 in the absence of CH4. Dash line in (a)-(c) is the bit patterns of the channel 1, 2 and 3; solid line is the bit patterns of channel 4 when there is only one input (CH1 (a), CH2 (b) or CH3 (c)). (d) Coupled spectral power density from CH1 to CH4 in absence of CH2, CH3 and CH4.

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Fig. 7 Back-to-back BER versus received power before PBS3.

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In order to overcome this shortage, the second demodulated scheme (named as 4PM-OOK-2) has been proposed as shown in Fig. 8. The OC3 is inserted before the demodulator, and split the multiplexed signal into two streams. PC7 is used to adjust the input polarization to be 45° with respect to the principal state of polarization (PSP) of PBS3, while PC8 is used to align with the input SOP and the PSP of PBS4. Following two PBSs, four streams $E_{x x}$ , $E_{y y}$ , $E_{x}$ and $E_{y}$ with different polarization states are obtained, where

Fig. 8 Second demodulated scheme for 4PM-OOK systems (4PM-OOK-2).

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E_{x x} = \hat{x} \hat{x} (\frac{\sqrt{2}}{2} j A_{3} + \frac{1}{2} A_{1} + \frac{1}{2} A_{2}) \exp (j ω_{c} t + ϕ),

E_{y y} = \hat{y} \hat{y} (\frac{\sqrt{2}}{2} j A_{4} - \frac{1}{2} A_{1} + \frac{1}{2} A_{2}) \exp (j ω_{c} t + ϕ) .

Next, the signals combined with a local oscillator laser are fed into four optical hybrids, whose outputs are $E_{x x} \pm E_{L O}$ , $E_{y y} \pm E_{L O}$ , $E_{x} \pm j E_{L O}$ and $E_{y} \pm j E_{L O}$ , respectively. Then, four BPDs are used to produce the photocurrents i_n, which are given by

i_{1} = \frac{ℜ}{2} [(E_{x x} + E_{L O}) \cdot {(E_{x x} + E_{L O})}^{*} - (E_{x x} - E_{L O}) \cdot {(E_{x x} - E_{L O})}^{*}] = ℜ (A_{1} + A_{2}) A_{L O},

i_{2} = \frac{ℜ}{2} [(E_{y y} + E_{L O}) \cdot {(E_{y y} + E_{L O})}^{*} - (E_{y y} - E_{L O}) \cdot {(E_{y y} - E_{L O})}^{*}] = ℜ (A_{2} - A_{1}) A_{L O},

i_{3} = \frac{ℜ}{2} [(E_{x} + j E_{L O}) \cdot {(E_{x} + j E_{L O})}^{*} - (E_{x} - j E_{L O}) \cdot {(E_{x} - j E_{L O})}^{*}] = ℜ (A_{3} - A_{4}) A_{L O},

i_{4} = \frac{ℜ}{2} [(E_{y} + j E_{L O}) \cdot {(E_{y} + E_{L O})}^{*} - (E_{y} - E_{L O}) \cdot {(E_{y} - E_{L O})}^{*}] = ℜ (A_{3} + A_{4}) A_{L O} .

Therefore, the expressions of the demodulation outputs are

A_{1} = \frac{i_{1} - i_{2}}{2 ℜ A_{L O}},

A_{2} = \frac{i_{1} + i_{2}}{2 ℜ A_{L O}},

A_{3} = \frac{i_{3} + i_{4}}{2 ℜ A_{L O}},

A_{4} = \frac{i_{3} - i_{4}}{2 ℜ A_{L O}} .

Figure 9 shows the back-to-back eye-diagrams for four demodulated outputs, which indicates that the performance of CH1 and CH2 are greatly enhanced by using the second demodulation scheme (4PM-OOK-2). The eye-diagrams for all outputs are clearly open and the demodulated performances for all channels are quite similar (i.e. the difference is negligible).

Fig. 9 Back-to-back eye-diagrams for the second demodulation scheme for OOK signal (4PM-OOK-2).

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Figure 10 shows the eye closure performance versus the DGD value. Here, the eye closure is defined by $C l o s u r e = 10 \log_{10} (\frac{a v e r a g e o p e n i n g}{o p e n i n g})$ , where the eye opening is the difference between the minimum value of the samples related to a logical “1” and the maximum value of samples related to a logical “0”; and the average opening is the difference between the mean values of the samples related to a logic “1” and “0”. The left and right inserts of Fig. 10 show the typical eye diagrams of the degraded signals when DGD is 10.2-ps. As expected, the eye closure of single polarization signal and PDM signal is smaller than that of 4PM signals under the same DGD value. Furthermore, the performance of 4PM system decreases more rapidly than the other two systems with the DGD becoming larger.

Fig. 10 Eye closure performance versus different DGD values. The inserts are the eye-diagrams for different multiplexing signals in the presence of 10.2-ps DGD.

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To further study the transmission performance, the polarization multiplexed signal is fed into a 30-km SMF. In conventional PDM systems, FDE is widely used to compensate for the dispersion, which is also applied here to the 4PM system. Similar to the previous schemes [33, 34], our FDE also consists of a feedforward filter, a decision block and a feedback filter as shown in Fig. 1(b). Figure 11(a) shows the BER with and without dispersion compensation versus the received power before PBS3. The power penalty improvements for all the outputs compared to the degraded signals at the reference BER of 10⁻³ are 1.6, 1.7, 1.2 and 1.6 dB, respectively. Figure 11(b) shows the Q-factor versus transmission distance with and without FDE. Taking the CH1 as an example, the performance of signal significantly decreases during the 50 km transmission (square line). It is because the inter-symbol and inter-channel interferences induced by chromatic dispersion (CD) and PMD limit the transmission distance in 4PM system more seriously than conventional PDM systems. But thanks to the FDE, the transmission distance is extended from ~50 km to ~80 km in our simulation (circle line in Fig. 11(b)) with the same Q-factor of 8.5-dB. Furthermore, the transmission distance would be extended to 500-km by using matched dispersion compensation fiber (DCF) as shown in the insert of Fig. 11(b). Thus, if improved dispersion compensation algorithms are employed [35, 36], the multiplexed signal could also be transmitted to nearly 500-km.

Fig. 11 (a) BER versus received power before PBS3 (with and without compensation); (b) Q-factor versus transmission distance with and without the FDE. W/: with; W/O: without; Comp.: compensation; FDE: feedback decision equalizer; DCF: dispersion compensation fiber. Insert: dispersion compensation by using DCF.

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Next, to investigate the practical usage of the MPM system, we compare the back to back performance of 4 × 10Gb/s 4PM-OOK system with that of 2 × 20Gb/s PDM-QPSK system under the same spectral efficiency. Here the QPSK signal is demodulated by differential detection. As shown in Fig. 12, the power penalty difference is less than 0.5-dB at the BER of 10⁻³ under the same bit rate. It indicates that the proposed scheme can be used as an alternative to PDM-QPSK in a flexible-rate coherent system, which demonstrates the potential of utilizing the freedom of polarization for future optical networks to further increase the system capacity and spectral efficiency. On the other hand, compared to the well-established modulation schemes, there are still lots of problems (not limitations), such as increasing transmission distance, feasibility of advanced modulation formats, and so on. However, all of these remaining issues are worth being investigated further.

Fig. 12 Back to back BER of the 4PM-OOK-2 systems and the PDM-QPSK systems.

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Finally, we also investigate the demodulation performance when the inputs are 4 × 40Gb/s (160-Gib/s) 4PM-OOK signals. Figure 13(a) shows the back-to-back eye-diagrams for demodulated outputs. As expected, the performance of system is degraded as the bit rate increasing. The BER performance is shown in Fig. 13(b). Compared with the 40-Gb/s 4PM-OOK system, ~6-dB power penalty at the BER of 10⁻³ has been obtained.

Fig. 13 (a) Back-to-back eye-diagrams (5-ps/div) and (b) BER performance for 4 × 40Gbit/s 4PM-OOK signals when using the second demodulation scheme.

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7. Discussion and conclusion

We have proposed a novel configuration of signal multiplexing with four polarization states, and investigated its transmission performance over SMF. Based on coherent detection and DSP, the demodulated schemes for 4PM-OOK and 4PM-PSK were presented. We further investigated the impact of the crosstalk induced by PMD on the 4PM-OOK system. Thanks to the FDE, the dispersion could be compensated and the transmission distance was extended from ~50 km to 80 km. The performance of proposed systems was comparable to that of PDM-QPSK systems with the same spectral efficiency by comparing the back-to-back transmission. Such scheme may stimulate the explore of new freedom in high-capacity optical communication system in addition to the wavelength (WDM), the time (TDM), the space or mode (SDM) as well as multi-level modulation. While we have to admit that there are too many issues worthwhile pursuing about the new freedom, including simplifying the MUX/DEMUX configurations, accommodating advanced modulation formats, utilizing cost-effective signal processing algorithms, exploring the effects of degrading effects, etc..

Acknowledgments

This research is supported by the National Basic Research Program of China (2012CB315704), the Natural Science Foundation of China (No. 61275068, 61111140390), the Key Grant Project of Chinese Ministry of Education under Grant 313049, the 2013 Doctoral Innovation funds of Southwest Jiaotong University and the Fundamental Research Funds for the Central Universities. The authors would like thank for valuable discussions with Dr. Xiang Liu from Bell Labs, Alcatel-Lucent, and Prof. William Shieh from University of Melbourne.

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Transmission of multi-polarization-multiplexed signals: another freedom to explore?

Abstract

1. Introduction

2. Principle and theoretical model for 4PM-OOK signal

3. Phase synchronization

4. Demodulation for 4PM-PSK signal

5. Crosstalk due to PMD

6. Setup and simulation results

7. Discussion and conclusion

Acknowledgments

References and links

Cited By

Figures (13)

Equations (48)

Optics Express