Long-haul quasi-single-mode transmissions using few-mode fiber in presence of multi-path interference

Qi Sui; HongYu Zhang; John D. Downie; William A. Wood; Jason Hurley; Snigdharaj Mishra; Alan Pak Tao Lau; Chao Lu; Hwa-Yaw Tam; P. K. A. Wai

doi:10.1364/OE.23.003156

1. Introduction

Fiber nonlinearity is a fundamental performance limiting factor in long-haul transmissions. Nonlinearity compensation algorithms such as digital back-propagation [1] are new tools to combat fiber nonlinearities but they are too complex and only bring moderate transmission improvements at best. This partly motivates the current wave of research in FMF MIMO transmissions [2] as alternatives to increase transmission capacities but those are highly impractical at present as they require few-mode amplifiers, multiplexers, de-multiplexers in addition to highly complex digital signal processing (DSP). In this regard, Yaman et al. [3,4] proposed to use the FMF as a large effective-area single-mode fiber and launch signals in the fundamental LP₀₁ mode while keeping other single-mode components such as erbium-doped fiber amplifiers (EDFA) and transceivers in the link, otherwise known as QSM transmissions. The large-effective-area nature of the LP₀₁ mode considerably extended the transmission distance. However, this may be a specific and overly-optimistic scenario. In general, mode coupling between the LP₀₁ and LP₁₁ modes are expected to be present in practical QSM systems which will induce multi-path interference (MPI) and potentially limit transmission performance. In other significant QSM transmission work, 16 × 100 Gb/s PM-QPSK signals were successfully transmitted over 635 km of standard 50 μm core diameter multimode fiber, but the reach was ultimately limited by MPI effects [5]. Bai et al. [6] briefly studied DSP techniques to mitigate MPI but long-haul QSM transmission experiments with MPI compensation has yet to be demonstrated.

Recently, we reported 256 Gb/s PM-16-QAM QSM transmission with 100 km per span and EDFA amplification [7,8]. The MPI is compensated by an additional DD-LMS algorithm after carrier phase estimation (CPE) in a standard DSP platform. The DD-LMS increases the optimal signal launch power by around 3 dB and the transmission distance for 256 Gb/s PM-16-QAM signals with 100 km per span can be extended up to 2600 km, which is a new record for 256 Gb/s PM-16-QAM transmissions with 100-km spans. In this paper, we extend our investigations and develop the theoretical foundations for polarization-multiplexed QSM transmission systems in the presence of mode coupling and differential mode loss between different mode groups in the FMF. We show that in the weak coupling regime, the multi-span QSM channel is a Gaussian random process in frequency in the weak coupling regime. MPI compensation filters are derived and mode-coupling-induced performance penalties are characterized. In particular, we show that MPI impairments can be effectively compensated by appropriate DSP techniques while signal loss in higher-order modes at the end of each span plays a significant role in the overall performance gain of QSM transmission systems.

2. Mode-coupling dynamics and charactering the QSM channel

2.1 QSM channel model

Consider a span of few-mode fiber of length L modeled as a concatenation of K sections of length ΔL between which mode coupling may occur. For simplicity, we will initially only study the coupling between the fundamental mode and a higher-order mode with differential mode delay (DMD) ∆τ and differential mode loss (DML) Δα as shown in Fig. 1. The input E_in(ω) is the transmitted signal launched into the fundamental mode with unit power and E_out(ω) is the signal output at the fundamental mode at the fiber end. Because of mode-coupling along the fiber, some signals will be present at the higher-order mode at the fiber end and will be stripped off as a consequence of butt-coupling with single-mode EDFAs or coherent receiver.

Fig. 1 The 2-mode coupling model for one span of QSM transmission.

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In this case, the output signal is given by

E_{o u t} (ω) = [\begin{matrix} 1 & 0 \end{matrix}] \prod_{k = 1}^{K} (G (ω) R {}_{k}) \cdot [\begin{matrix} E_{i n} (ω) \\ 0 \end{matrix}]

where

G (ω) = [\begin{matrix} 1 & 0 \\ 0 & e^{- (j ω Δ τ + \frac{1}{2} Δ α) Δ L} \end{matrix}]

takes into account the effects of DMD and DML. The matrix

R_{k} = [\begin{matrix} \sqrt{1 - η_{k}} & \sqrt{η_{k}} e^{- j θ_{k}} \\ - \sqrt{η_{k}} e^{j θ_{k}} & \sqrt{1 - η_{k}} \end{matrix}]

models random mode coupling at the k^th section through independent identically distributed (i.i.d.) uniform random variables η_k and θ_k with distributions U[0, 2κΔL] and U[0, 2π] respectively. The coupling coefficient κ can be viewed as a measure of coupling strength. We note that there are other 2-mode models proposed in the literature [9,10] but they are fundamentally very similar to ours and give identical statistical predictions. Furthermore, the dynamics of statistical mode coupling in linear fibers have been extensively studied in the ‘90s in which a continuous-fiber model is adopted [11,12]. We hereby choose to use the discrete concatenating-fiber model to better align with theoretical analysis in recent spatial-division multiplexing literature and we again emphasize that the conclusions drawn will be identical regardless of the fiber model chosen.

The transfer function of one span of QSM system is then given by

H_{1 - s p a n} (ω) = [\begin{matrix} 1 & 0 \end{matrix}] \prod_{k = 1}^{K} (G (ω) R {}_{k}) \cdot [\begin{matrix} 1 \\ 0 \end{matrix}]

and let h_1-span(t) denote the corresponding impulse response. We are interested in the statistics of h_1-span(t) as it characterizes the amount of MPI and the length of the MPI compensation filter. In particular, starting from the Power Spectral Density relation

E [| h_{1 - s p a n} (t) |^{2}] \overset{F}{\leftrightarrow} E [H_{1 - s p a n}^{*} (ω') H_{1 - s p a n} (ω' + ω)] .

Appendix I shows that in the weak coupling regime where κL is small,

E [{| h_{1 - s p a n} (t) |}^{2}] = {\begin{matrix} e^{- κ L} δ (t) + e^{- κ L} \cdot \underset{E [| h_{M P I, 1 - s p a n} (t) |^{2}]}{\underset{︸}{\frac{κ^{2} L}{Δ τ} e^{- \frac{Δ α t}{Δ τ}} (1 - \frac{t}{Δ τ L})}}, & 0 \leq t \leq Δ τ L \\ 0 & otherwise \end{matrix},

where E[·] denotes expectation and

E [| h_{M P I, 1 - s p a n} (t) |^{2}]

are the MPI component of h_1-span(t). For a multi-span QSM system with per-span amplification, let

\otimes

denotes convolution and h_MPI,n(t) be the MPI component of the n^th span of fiber. In this case, the overall impulse response h(t) is given by

\begin{matrix} h (t) = (δ (t) + h_{M P I, 1} (t)) \otimes (δ (t) + h_{M P I, 2} (t)) \otimes ... \otimes (δ (t) + h_{M P I, N} (t)) \\ \approx (δ (t) + \sum_{n = 1}^{N} h_{M P I, n} (t)) \end{matrix}

in the weak coupling regime where h_MPI,n(t) is small. Now, since h_MPI,n(t) are statistically independent for different spans,

\begin{matrix} E [{| h (t) |}^{2}] \approx δ (t) + \sum_{n = 1}^{N} E [{| h_{M P I, n} (t) |}^{2}] \\ = {\begin{matrix} δ (t) + \underset{E [{| h_{M P I} (t) |}^{2}]}{\underset{︸}{\frac{N κ^{2} L}{Δ τ} e^{- \frac{Δ α t}{Δ τ}} (1 - \frac{t}{Δ τ L})}} & 0 \leq t \leq Δ τ L \\ 0 & otherwise \end{matrix} \end{matrix}

and the corresponding MPI power is

P_{M P I} = \int_{0}^{Δ τ L} E [{| h_{M P I} (t) |}^{2}] d t = N κ^{2} \cdot \frac{Δ α L + e^{- Δ α L} - 1}{Δ α^{2}} .

Note that the MPI power is independent of DMD.

The 2-mode coupling model can be readily generalized to 6 spatial-polarization modes for realistic polarization-multiplexed QSM transmissions. In this case, the input-output relation is given by

[\begin{matrix} E_{o u t, x} (ω) \\ E_{o u t, y} (ω) \end{matrix}] = H (ω) E_{i n} (ω) + V (ω) = [\begin{matrix} H_{x x} (ω) & H_{x y} (ω) \\ H_{y x} (ω) & H_{x x} (ω) \end{matrix}] [\begin{matrix} E_{i n, x} (ω) \\ E_{i n, y} (ω) \end{matrix}] + [\begin{matrix} V_{x} (ω) \\ V_{y} (ω) \end{matrix}],

where V_x₍_y₎(ω) is the Fourier Transform of the collective amplifier noise of the link, which is modelled as a circularly symmetric white Gaussian random process. Signals in the two fundamental modes (and any two modes of the 4 higher-order modes) are assumed to couple freely between each other i.e. the corresponding η_k will be uniformly distributed between 0 and 1. For coupling between any fundamental mode and any higher-order mode, the coupling coefficient is changed from κ to κ/4. The derivations for the expected impulse response powers for polarization-multiplexed QSM systems are largely similar and will be omitted here. The results are

E [{| h_{x x} (t) |}^{2}] = E [{| h_{y y} (t) |}^{2}] = δ (t) + \frac{1}{4} E [{| h_{M P I} (t) |}^{2}]

and

E [{| h_{x y} (t) |}^{2}] = E [{| h_{y x} (t) |}^{2}] = \frac{1}{4} E [{| h_{M P I} (t) |}^{2}] .

In addition, we are also interested in the probability density function (pdf) of the QSM channel H(ω). In Appendix II, we show that in the weak coupling regime, the pdf of H(ω) is independent of ω and approximately complex circularly symmetric Gaussian with

H_{x x (y y)} (ω) \sim N (1, \frac{1}{8} P_{M P I} I) and H_{x y (y x)} (ω) \sim N (0, \frac{1}{8} P_{M P I} I),

where I is the identity matrix. Altogether, in the weak coupling regime, the transfer function of a N-span QSM system H(ω) is a stationary Gaussian random process in frequency with auto-correlation function

E [| h_{x x (y y)} (t) |^{2}]

and

E [| h_{x y (y x)} (t) |^{2}]

.

Simulations are conducted to verify the theoretical predictions and results shown in Fig. 2, which indicates good agreements between them. Further simulations reveal that $E [| h_{x x (y y)} (t) |^{2}]$ is insensitive to typical differential group delays (DGD) between the two fundamental modes and thus PMD effects. This should be expected as typical FMFs have DMDs much larger than the DGD between the fundamental modes.

Fig. 2 Theoretical and simulation results for (a) Expected impulse response power $E [| h_{x x (y y)} (t) |^{2}]$ and (b) pdf of transfer function H_xx₍_yy)(ω) for a 32 Gbaud polarization-multiplexed QSM system. The link consists of 20 spans of 100-km fiber with κL = 0.1, DMD Δτ = 1.28 ns/km and ∆α = 0.1 dB/km.

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2.2 Transmission penalties and DSP mitigation strategies for QSM systems

Mode coupling effects and DMD induce MPI and inter-symbol interference (ISI) in QSM transmissions. In principle, the ISI or MPI length should scale with increasing DMD and decreasing DML. The MPI length dictates the number of taps and hence the computational complexity of the MPI compensation filter in a digital coherent receiver. If we define τ_MPI as the length of time that contains 99% of the MPI power, we have

\int_{0}^{τ_{M P I}} {| h_{M P I} (t) |}^{2} d t = \int_{0}^{τ_{M P I}} (1 - \frac{t}{Δ τ L}) \cdot e^{- \frac{Δ α t}{Δ τ}} d t = 0.99 \int_{0}^{Δ τ L} (1 - \frac{t}{Δ τ L}) \cdot e^{- \frac{Δ α t}{Δ τ}} d t

and the solution is given by

τ_{M P I} = \frac{Δ τ}{Δ α} (Δ α L - W ((0.01 (Δ α L - 1) - 0.99 e^{- Δ α L}) e^{Δ α L - 1}) - 1) \overset{Δ α L large}{\to} \frac{Δ τ}{Δ α} \cdot \ln 100

where W(∙) is the Lambert W function [13]. Figure 3 shows the MPI length τ_MPI for various DMD and DML values for a 32Gbaud QSM transmission system.

Fig. 3 The MPI length τ_MPI for various DMD Δτ and DML Δα for a 32 Gbaud polarization-multiplexed QSM transmission system.

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In the frequency domain, MPI manifests itself as frequency-selective fading, very much like multi-path fading in wireless communications. A sample realization of H_xx₍_yy)(ω) and the pdf of its amplitude is shown in Fig. 4(a). Wiener filter W(ω) (that minimize the expected mean-square error) can be used to compensate MPI and is given by

W (ω) = {(H^{†} (ω) H (ω) S_{S} (ω) + N_{0} I)}^{- 1} H^{†} (ω) S_{S} (ω),

where H^†(ω) is the conjugate transpose of H(ω). S_S(ω) is the power spectral density of the input signal. N₀ is the power spectral density of the amplifier noise V_x₍_y₎(ω) and the filter output is given by

Fig. 4 (a) A sample realization of H_xx_/_yy(ω) and (b) pdf of the amplitude |H_xx_/_yy(ω)| for a 20-span QSM system with 100-km span length, κL = 0.1, Δα = 1 dB/km and ∆τ = 1.28 ns/km.

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[\begin{matrix} R_{o u t, x} (ω) \\ R_{o u t, y} (ω) \end{matrix}] = W (ω) H (ω) E_{i n} (ω) + W (ω) V (ω) = U (ω) E_{i n} (ω) + W (ω) V (ω)

Figure 5 shows the Q vs. OSNR for 32Gbaud-PM-16-QAM transmissions with and without MPI compensation for a 20-span QSM system with a random realization of the transfer function H(ω). The Q value is derived from $Q = \sqrt{2} e r f c^{- 1} (2 \cdot B E R)$ . It is clear that MPI significantly degrades system performance if left uncompensated. Fortunately, Wiener filtering can almost eliminate the effects of MPI, but not completely. This is because Wiener filtering is based on the minimum mean-squared error-criterion which introduces noise enhancement, residual ISI and hence a performance penalty. Furthermore, the penalty is in general random due to the random fading nature of H_xx₍_yy)(ω). If we also consider the residual ISI as an additional source of noise, this MPI-induced penalty Λ_MPI will be given by

Λ_{M P I} = \frac{\int ({| u_{12} (ω) |}^{2} + {| u_{21} (ω) |}^{2}) S_{S} (ω) d ω + \int ({| w_{11} (ω) |}^{2} + {| w_{12} (ω) |}^{2} + {| w_{21} (ω) |}^{2} + {| w_{22} (ω) |}^{2}) N_{0} d ω}{2 \int N_{0} d ω \cdot \int ({| u_{11} (ω) |}^{2} + {| u_{22} (ω) |}^{2}) S_{S} (ω) d ω}

where u_ij(ω) and w_ij(ω) denote the i,j^th entry of U(ω) and W(ω) respectively and S_S(ω) is the power spectral density of the input signals E_in_,_x₍_y₎(ω). In addition to Λ_MPI, another mode-coupling-induced performance penalty Λ_HOM stems from the loss of signals in higher-order modes at the end of each span. This penalty can be approximated by the power remaining in the fundamental mode at the end of each span, i.e.

Λ_{H O M} \approx \lim_{K \to \infty} \prod_{k = 1}^{K} (1 - η_{k}) = e^{- κ L}

Compared with standard fiber systems, the QSM system has a larger effective area, smaller nonlinearity coefficient which in turn result in an OSNR gain

G_{O S N R} = A_{e f f, F M F} / A_{e f f, S t d . f i b e r} .

Assuming the fundamental mode loss coefficient are the same for standard fiber systems and QSM systems and taking into account MPI effects and higher-order mode stripping, the overall performance gain G_overall of QSM relative to standard single-mode fiber transmission systems is then given by

Fig. 5 Q vs. OSNR for 32Gbaud-PM-16-QAM transmissions with and without MPI compensation for a 20-span QSM system with a random realization of the transfer function H(ω). The span length is 100 km, coupling strength κL = 0.1, DMD Δτ = 1.28 ns/km and DML Δα = 0.1 dB/km.

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G_{o v e r a l l} = G_{O S N R} (dB) - Λ_{M P I} (dB) - Λ_{H O M} (dB) .

Figure 6 shows the pdf of overall performance gain G_overall of a 20-span QSM transmission system in presence of mode-coupling-induced penalties Λ_MPI and Λ_HOM and an effective area A_eff,FMF 2.5 times that of a standard single mode fiber for different coupling strengths. Fiber nonlinearity effects are not included in the simulation as we want to highlight the effects of MPI and higher-order mode stripping loss and nonlinearity should induce the same amount of OSNR penalty for both QSM and standard fiber systems at their respective optimal signal launched powers. It can be seen that, when MPI is appropriately compensated, the higher-order mode loss Λ_HOM plays a larger role than residual MPI-induced penalties in limiting the overall performance gain. One can define an appropriate threshold, say 1 dB from optimal performance, to statistically characterize acceptable level of performance and outage probabilities of QSM transmission systems.

Fig. 6 PDF of the overall performance gain G_overall of a 20-span QSM transmission system in presence of mode-coupling-induced penalties Λ_MPI and Λ_HOM and an effective area A_eff_,_FMF 2.5 times that of a standard single mode fiber. The span length is 100 km, DMD Δτ = 1.28 ns/km and the DML Δα = 0.5 dB/km.

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3. 256 Gb/s PM-16-QAM QSM Transmission Experiments over 2600 km with MPI Compensation

Experiments are also conducted to investigate the performance of QSM transmissions with MPI compensation in practical settings. The FMF used for QSM transmission in our experiments is a step-index silica core fiber with core radius around 9 μm and supports 6 polarization modes. The nominal effective area for the fundamental mode is 220 μm². The DMD is measured to be around 1.12 ns/km. The higher order modes suffers a higher loss compared with the fundamental mode. The average measured attenuation of two 100 km spans constructed is about 19.1 dB including splices and connectors. At the beginning and end of each span, the FMF is connected with standard single mode fiber jumpers through a bridge fiber to ensure compatibility with single-mode components such as EDFAs and transmitters/receivers. The total loss on the input side including connector loss, splice from single-mode to bridge fiber, and splice from bridge fiber to FMF is about 0.6 dB. At the output of each span, the same level of connection and splicing loss is expected, in addition to a loss rising from the mode stripping. The average fiber loss is approximately in the range of approximately 17.5 dB and the rest is due to splices and connectors. Since the fibers are drawn from two different blanks, measurements of the accumulated MPI or crosstalk level of each of the two 100 km spans shows significantly different behavior with estimated levels of about −24 dB for one span and −30 dB or less for the second span.

The two 100-km spans of FMF are placed in a re-circulating loop configuration for transmission measurements. The experimental setup is shown in Fig. 7. Ten lasers spaced by 50 GHz are modulated together in the 16-QAM format at a symbol rate of 32 Gbaud, and then polarization-multiplexed to produce 256 Gb/s PM-16-QAM channels. The channel under measurement is encoded on an external cavity laser (ECL) while the other 9 channels are encoded on DFB lasers. The channels are passed through a short piece of positive dispersion fiber to de-correlate adjacent channels by a few symbols before being launched in the loop. EDFAs are used to amplify the signals at the output of each FMF span.

Fig. 7 Experimental setup for QSM transmission and MPI compensation. GEF: gain equalization filter; LSPS: loop synchronous polarization scrambler.

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The receiver DSP algorithms consist of orthogonalization and normalization followed by frequency-domain CD compensation. The received data is then re-sampled to 2 samples per symbol for timing phase recovery [14]. CMA and CMMA on a 21-tap $2 \times 2$ MIMO filters are used as a two-stage algorithm for polarization de-multiplexing and PMD compensation. The signals are then down-sampled to 1 sample per symbol followed by frequency offset estimation and CPE [15,16]. A joint-polarization DD-LMS filters with maximum 501 taps is inserted after CPE for MPI compensation. This is followed by symbol decisions and BER calculation.

Figure 8(a) shows the BER vs. OSNR for linear transmission for one of the FMF spans cut into different lengths without using DD-LMS for MPI compensation. These results are obtained for straight-line transmission through the two fiber lengths shown with ASE noise-loading before the receiver. The presence and distributed nature of MPI is evident from the fact that performance degrades with increasing FMF length. We then conducte QSM transmission over 600 km, or 3 loops, and vary the channel launched power to determine the optimal channel power. The measured Q values (derived from $Q = \sqrt{2} e r f c^{- 1} (2 \cdot B E R)$ ) are shown in Fig. 2(b) along with results over a comparable 600-km standard single mode fiber system. The optimal launched power for the QSM system is about 3-4 dB higher than that of the standard fiber system by virtue of the larger effective area. Without DD-LMS, the FMF system has a maximum Q value about 0.5 dB lower than the standard single mode fiber system at this distance because of significant MPI. However, a 101-tap joint-polarization DD-LMS significantly improves QSM performance and a 501-tap DD-LMS enables QSM to outperform the standard fiber by more than 2 dB in Q. The convergence time for the DD-LMS is in the order of 50000 symbols in most of our data sets.

Fig. 8 (a) Q vs. OSNR for segments of span no. 1 with different lengths in linear transmission. (b) Q vs. channel power of central channel for standard single mode fiber and QSM transmission over 600 km using DD-LMS for MPI compensation.

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It should be noted that the 501-tap DD-LMS does not fully compensate the MPI and further improvement, albeit marginal, can be obtained with increasing number of DD-LMS taps. However, as the delay between x and y polarization in our experimental setup is around 280 symbols, extending the number of taps beyond 560 taps results in double counting the signal and results in an under-estimate of BER. This is clearly evident in Fig. 9 for a single-span 50-km QSM system. Therefore, we choose the maximum number of taps to be 501 in our studies to ensure that our results are free from signal double counting.

Fig. 9 BER vs. number of taps of the DD-LMS filter used to compensate MPI for a single-span 50-km QSM link in which the relative symbol delay between the x- and y-polarization is 280 symbols. When the number of taps exceed 560, the DD-LMS double counts the signal and result in an abrupt BER reduction.

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With the channel power set at the optimum for both FMF and standard single mode fiber systems, we then measured the Q of the central channel as a function of distance. The results are shown in Fig. 10. Without DD-LMS, QSM transmission over the FMF suffers from severe performance degradation due to MPI for all transmission distances. The detrimental effects of MPI offset the benefits of reduced nonlinearity so that the QSM transmission is inferior to standard fiber transmission. However, a joint-polarization 101-tap DD-LMS increases the Q considerably so that it becomes slightly better than that of the standard fiber system. The Q continues to improve with increasing number of taps until around 501 taps when only marginal improvement is obtained. This may imply that most of the MPI has been compensated with 301-taps DD-LMS. Furthermore, the transmission distance for a SD-FEC threshold of 2E-2 BER (or Q of 6.25 dB) has been extended from 1200 km to 2600 km with the help of DD-LMS. It is worth noting that at 2600 km, the calculated total accumulated MPI for the QSM transmission would be greater than −12 dB. In comparison, the standard single mode fiber system has a maximum reach of about 1500 km and the same strength DD-LMS brings negligible performance improvement, thus verifying that the DD-LMS is in fact compensating MPI in QSM systems.

Fig. 10 Q vs. distance at optimal launch power using DD-LMS for MPI compensation.

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To further quantify the effectiveness of DD-LMS in MPI compensation, we show the Q value of the central channel as a function of measured OSNR at the receiver in Fig. 11(a) for both QSM and standard fiber systems at their respective optimal signal launched powers. Note that as the optimal signal launched power is higher for QSM than standard fiber systems due to its reduced fiber nonlinearity, same OSNR actually means a higher signal launched power for QSM system with proportionately more noise, which in turn means the QSM link is longer compared to the corresponding standard fiber system with the same OSNR. The close agreements between QSM with 301- and 501-taps DD-LMS and the standard fiber suggests that the MPI in the FMF is nearly fully compensated. The improvement of the DD-LMS filter for standard fiber systems is minimal, indicating that the transceiver circuitry and components are more or less optimized and there are negligible residual impairments and the DD-LMS are solely compensating MPI effects. Furthermore, Fig. 11(b) shows the Q factor vs. distance at −1 and 3 dBm signal launched power and the DD-LMS provides considerable performance improvements in both cases, illustrating that the DD-LMS is not compensating fiber nonlinearity.

Fig. 11 (a) Q vs. OSNR at optimal signal launched power with DD-LMS for QSM and standard fiber systems and (b) Q vs. distance at −1 and 3 dBm signal launched power with and without DD-LMS filter.

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The DD-LMS taps after convergence allow us to recover/estimate the impulse response of MPI impairment. Alternatively, applying the method described in Ryf et al. [17], the estimated magnitude squared of the self-polarization impulse responses |h_xx(t)|² and |h_yy(t)|² and cross-polarization impulse responses |h_xy(t)|² and |h_yx(t)|² are depicted in Fig. 12 for various transmission distances. The impulse responses shown are obtained by averaging estimates over 5 independent data sets. It is clear that the MPI in QSM systems are distributed in nature as its strength grows with fiber length. Also, self-polarization MPI is more dominant than cross-polarization MPI. It should be noted that the unusual spike at the t < 0 region is merely an artifact of our 16-QAM generation method based on cascading a QPSK signal with its delayed and scaled version. Such configuration has no effect on the overall 16-QAM signal processing and BER performance.

Fig. 12 Estimated magnitude squared of self-polarization and cross polarization impulse responses for (a) 600, (b) 1200 and (c) 2400 km of QSM transmission in the presence of MPI.

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4. Conclusions

In this paper, we analytically investigated polarization-multiplexed long-haul QSM transmission systems using 6-spatial-polarization-mode FMF parameterized by the differential mode delay and differential mode loss between the fundamental mode group and higher-order mode group. In the weak coupling regime, a multi-span QSM channel can be modeled as a Gaussian random process in frequency. Mode coupling in FMF result in MPI and additional signal loss at each span of a QSM system. We derived appropriate filters for MPI compensation and showed that they induce minimal noise enhancement and performance penalty. In addition, we experimentally demonstrated 256 Gb/s PM-16-QAM QSM transmission over a record distance of 2600 km for 100-km spans by using a DD-LMS filter to compensate MPI. We showed that QSM transmissions can be a practical and viable option for transmission reach and capacities upgrade for future long-haul systems. Further studies will include higher-order modulation formats, OFDM as well as super-channel QSM transmissions.

Appendix I: Derivation of the expected impulse response power for QSM transmission systems

We consider the auto-correlation function of H_1-_span(ω) in frequency i.e.

E [H_{1 - s p a n}^{*} (ω') H_{1 - s p a n} (ω' + ω)] = E [[\begin{matrix} 1 & 0 \end{matrix}] \prod_{k = 0}^{K - 1} (R_{K - k}^{*} G^{*} (ω')) [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] \prod_{k = 1}^{K} (G (ω' + ω) R_{k}) [\begin{matrix} 1 \\ 0 \end{matrix}]] .

Since R_m is statistically independent of R_n for

m \neq n

, we have

E [H_{1 - s p a n}^{*} (ω') H_{1 - s p a n} (ω' + ω)] = E [[\begin{matrix} 1 & 0 \end{matrix}] \prod_{k = 0}^{K - 2} R_{K - k}^{*} G^{*} (ω') \cdot E [R_{1}^{*} G^{*} (ω') [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] G (ω' + ω) R_{1}] \cdot \prod_{k = 2}^{K} G (ω' + ω) R_{k} [\begin{matrix} 1 \\ 0 \end{matrix}]],

in which

E [R_{1}^{*} G^{*} (ω') [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] G (ω' + ω) R_{1}] = E [[\begin{matrix} 1 - η_{1} & \sqrt{η_{1} (1 - η_{1})} e^{- j θ_{1}} \\ \sqrt{η_{1} (1 - η_{1})} e^{j θ_{1}} & η_{1} \end{matrix}]] = [\begin{matrix} 1 - κ Δ L & 0 \\ 0 & κ Δ L \end{matrix}] .

Next, if we let ξ = e⁻⁽^jω^Δ^τ ^{+ Δ}^α^)Δ^L,

\begin{matrix} E [R_{2}^{*} G^{*} (ω') R_{1}^{*} G^{*} (ω') [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] G (ω' + ω) R_{1} G (ω' + ω) R_{2}] \\ = E [R_{2}^{*} G^{*} (ω') [\begin{matrix} 1 - κ Δ L & 0 \\ 0 & κ Δ L \end{matrix}] G (ω' + ω) R_{2}] . \\ = [\begin{matrix} {(1 - κ Δ L)}^{2} + κ^{2} Δ L^{2} ξ & 0 \\ 0 & (1 - κ Δ L) κ Δ L (ξ + ξ^{2}) \end{matrix}] \end{matrix}

and

\begin{array}{l} E [R_{3}^{*} G^{*} (ω') R_{2}^{*} G^{*} (ω') R_{1}^{*} G^{*} (ω') [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] G (ω' + ω) R_{1} G (ω' + ω) R_{2} G (ω' + ω) R_{3}] \\ = [\begin{matrix} {(1 - κ Δ L)}^{3} + (1 - κ Δ L) κ^{2} Δ L^{2} (2 ξ + ξ^{2}) & 0 \\ 0 & (1 - κ Δ L) κ Δ L (ξ + ξ^{2} + ξ^{3}) \end{matrix}] \end{array}

and the process can be continued recursively. For a fiber with K sections, we have

\begin{array}{l} E [\prod_{k = 0}^{K - 1} (R_{K - k}^{*} G^{*} (ω')) [\begin{matrix} 1 \\ 0 \end{matrix}] [\begin{matrix} 1 & 0 \end{matrix}] \prod_{k = 1}^{K} (G (ω' + ω) R_{k})] \\ \approx [\begin{matrix} {(1 - κ Δ L)}^{K} + {(1 - κ Δ L)}^{K - 2} κ^{2} Δ L^{2} \cdot \sum_{p = 1}^{K} (K - p) ξ^{p} & 0 \\ 0 & (1 - κ Δ L) κ Δ L \cdot \sum_{p = 1}^{K} ξ^{p} \end{matrix}] \end{array}

in the weak coupling regime and neglect κ^pΔL^p for p ≥ 3. In this case,

E [H_{1 - s p a n}^{*} (ω') H_{1 - s p a n} (ω' + ω)] = {(1 - κ Δ L)}^{K} + {(1 - κ Δ L)}^{K - 2} κ^{2} Δ L^{2} \cdot \sum_{p = 1}^{K} (K - p) ξ^{p}

And its Fourier transform is given by

E [{| h_{1 - s p a n} (t) |}^{2}] = {(1 - κ Δ L)}^{K} (δ (t) + {(\frac{κ Δ L}{1 - κ Δ L})}^{2} \sum_{p = 1}^{K} (K - p) e^{p Δ α Δ L} δ (t - p Δ τ Δ L)) .

Finally, in the limit K→∞, we obtain

lim_{K \to \infty} E [{| h_{1 - s p a n} (t) |}^{2}] = e^{- κ L} (δ (t) + \frac{κ^{2} L}{Δ τ} e^{- \frac{Δ α t}{Δ τ}} (1 - \frac{t}{Δ τ L})) .

Appendix II: Derivation of the transfer function of QSM transmission systems

In the weak coupling regime, we consider the channel output consists of signals that have stayed in the fundamental mode along transmission as well as accumulated MPI from signals that have ‘coupled up’ to higher-order mode and ‘coupled back down’ to fundamental mode only once during transmission. There are K(K-1)/2 such configurations in a span of K sections and hence H_1-_span(ω) can be expressed as

\begin{matrix} H_{1 - s p a n} (ω) = \prod_{k = 1}^{K} \sqrt{1 - η_{k}} + \prod_{k = 1}^{K} \sqrt{1 - η_{k}} \cdot \sum_{m = 1}^{K - 1} \sum_{l = 1}^{K - m} \sqrt{\frac{η_{l} η_{m + l}}{(1 - η_{l}) (1 - η_{m + l})}} e^{- m Δ α Δ L + j (θ_{m + l} - θ_{l} + m Δ τ Δ L ω)} \\ = \prod_{k = 1}^{K} \sqrt{1 - η_{k}} + \prod_{k = 1}^{K} \sqrt{1 - η_{k}} \cdot \sum_{m = 1}^{K - 1} \sum_{l = 1}^{K - m} \sqrt{\frac{η_{l} η_{m + l}}{(1 - η_{l}) (1 - η_{m + l})}} e^{- m Δ α Δ L + j \sum_{q = l}^{m + l - 1} x_{q}}, \end{matrix}

where m denotes the number of consecutive sections the signal ‘stays’ in the higher-order mode and x_q = θ_q₊₁ – θ_q + ΔτΔLω. Note that the x_q’s are i.i.d. random variables with distribution U[0, 2π] and is in fact independent of ω due to the modulo 2π nature of phase. Consequently, the distribution of H_1-_span(ω) is also independent of ω. As K→∞,

\prod_{k = 1}^{K} \sqrt{1 - η_{k}} \to e^{- κ L / 2}

and hence

E [H_{1 - s p a n} (ω)] \to e^{- κ L / 2}

and

V a r (H_{1 - s p a n} (ω)) = \int E [| h_{M P I, 1 - s p a n} (t) |^{2}] d t = \frac{1}{N} P_{M P I} .

For an N-span QSM transmission system, denote

Y_{n} = \sum_{m = 1}^{K - 1} \sum_{l = 1}^{K - m} \sqrt{\frac{η_{l} η_{m + l}}{(1 - η_{l}) (1 - η_{m + l})}} e^{- m Δ α Δ L + j \sum_{q = l}^{m + l - 1} x_{q}}

as the MPI terms incurred in the n^th span. In the weak coupling regime, the transfer function with per-span amplification gain e^κL becomes

H (ω) = \prod_{n = 1}^{N} (1 + Y_{n}) \approx (1 + \sum_{n = 1}^{N} Y_{n}) \overset{N large}{\to} N (1, \frac{1}{2} P_{M P I} I)

by the central limit theorem as N gets large. Together with the derived

E [| h (t) |^{2}]

in the paper, one can show that H(ω) is a stationary complex Gaussian random process in ω. Generalizations to the QSM transfer function matrix H(ω) for polarization-multiplexed transmissions in 6-spatial-polarization mode FMF systems are straight forward and are given by

H_{x x (y y)} (ω) \sim N (1, \frac{1}{8} P_{M P I} I) and H_{x y (y x)} (ω) \sim N (0, \frac{1}{8} P_{M P I} I)

Acknowledgements

The authors would like to acknowledge the financial support of the Hong Kong Government General Research Fund (GRF) under project number PolyU 519211 and the Hong Kong Polytechnic University under project H-ZG1E and H-ZDA9.

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Long-haul quasi-single-mode transmissions using few-mode fiber in presence of multi-path interference

Abstract

1. Introduction

2. Mode-coupling dynamics and charactering the QSM channel

2.1 QSM channel model

2.2 Transmission penalties and DSP mitigation strategies for QSM systems

3. 256 Gb/s PM-16-QAM QSM Transmission Experiments over 2600 km with MPI Compensation

4. Conclusions

Appendix I: Derivation of the expected impulse response power for QSM transmission systems

Appendix II: Derivation of the transfer function of QSM transmission systems

Acknowledgements

References and links

Cited By

Figures (12)

Equations (35)

Optics Express