Abstract

We study long-haul Quasi-Single-mode (QSM) systems in which signals are transmitted in the fundamental modes of a few-mode fiber (FMF) while keeping other system components such as amplifiers and receivers are kept single-moded. The large-effective-area nature of the FMF fundamental modes improves system nonlinear tolerance in the expense of mode coupling along FMF transmissions which induces multi-path interference (MPI) and needs to be compensated. We analytically investigate 6-spatial-polarization mode QSM transmission systems in presence of MPI and show that in the weak coupling regime, the QSM channel is a Gaussian random process in frequency. MPI compensation filters are derived and performance penalties due to MPI and signal loss from higher-order modes are characterized. We also experimentally demonstrate 256 Gb/s polarization multiplexed (PM)-16-QAM QSM transmissions over a record distance of 2600 km with 100-km span using decision directed least mean square (DD-LMS) algorithm for MPI compensation.

© 2015 Optical Society of America

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 Ein(ω) is the transmitted signal launched into the fundamental mode with unit power and Eout(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.

 figure: Fig. 1

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

Eout(ω)=[10]k=1K(G(ω)Rk)[Ein(ω)0]
where
G(ω)=[100e(jωΔτ+12Δα)ΔL]
takes into account the effects of DMD and DML. The matrix
Rk=[1ηkηkejθkηkejθk1ηk]
models random mode coupling at the kth section through independent identically distributed (i.i.d.) uniform random variables ηk and θk 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

H1span(ω)=[10]k=1K(G(ω)Rk)[10]
and let h1-span(t) denote the corresponding impulse response. We are interested in the statistics of h1-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[|h1span(t)|2]FE[H1span*(ω')H1span(ω'+ω)].
Appendix I shows that in the weak coupling regime where κL is small,
E[|h1span(t)|2]={eκLδ(t)+eκLκ2LΔτeΔαtΔτ(1tΔτL)E[|hMPI,1span(t)|2],0tΔτL0otherwise,
where E[·] denotes expectation and E[|hMPI,1span(t)|2] are the MPI component of h1-span(t). For a multi-span QSM system with per-span amplification, let denotes convolution and hMPI,n(t) be the MPI component of the nth span of fiber. In this case, the overall impulse response h(t) is given by
h(t)=(δ(t)+hMPI,1(t))(δ(t)+hMPI,2(t))...(δ(t)+hMPI,N(t))(δ(t)+n=1NhMPI,n(t))
in the weak coupling regime where hMPI,n(t) is small. Now, since hMPI,n(t) are statistically independent for different spans,
E[|h(t)|2]δ(t)+n=1NE[|hMPI,n(t)|2]={δ(t)+Nκ2LΔτeΔαtΔτ(1tΔτL)E[|hMPI(t)|2]0tΔτL0otherwise
and the corresponding MPI power is
PMPI=0ΔτLE[|hMPI(t)|2]dt=Nκ2ΔαL+eΔαL1Δα2.
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

[Eout,x(ω)Eout,y(ω)]=H(ω)Ein(ω)+V(ω)=[Hxx(ω)Hxy(ω)Hyx(ω)Hxx(ω)][Ein,x(ω)Ein,y(ω)]+[Vx(ω)Vy(ω)],
where Vx(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 ηk 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[|hxx(t)|2]=E[|hyy(t)|2]=δ(t)+14E[|hMPI(t)|2]
and

E[|hxy(t)|2]=E[|hyx(t)|2]=14E[|hMPI(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

Hxx(yy)(ω)N(1,18PMPII)andHxy(yx)(ω)N(0,18PMPII),
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[|hxx(yy)(t)|2] and E[|hxy(yx)(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[|hxx(yy)(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.

 figure: Fig. 2

Fig. 2 Theoretical and simulation results for (a) Expected impulse response power E[|hxx(yy)(t)|2] and (b) pdf of transfer function Hxx(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 τMPI as the length of time that contains 99% of the MPI power, we have

0τMPI|hMPI(t)|2dt=0τMPI(1tΔτL)eΔαtΔτdt=0.990ΔτL(1tΔτL)eΔαtΔτdt
and the solution is given by
τMPI=ΔτΔα(ΔαLW((0.01(ΔαL1)0.99eΔαL)eΔαL1)1)ΔαLlargeΔτΔαln100
where W(∙) is the Lambert W function [13]. Figure 3 shows the MPI length τMPI for various DMD and DML values for a 32Gbaud QSM transmission system.

 figure: Fig. 3

Fig. 3 The MPI length τMPI 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 Hxx(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)SS(ω)+N0I)1H(ω)SS(ω),
where H(ω) is the conjugate transpose of H(ω). SS(ω) is the power spectral density of the input signal. N0 is the power spectral density of the amplifier noise Vx(y)(ω) and the filter output is given by

 figure: Fig. 4

Fig. 4 (a) A sample realization of Hxx/yy(ω) and (b) pdf of the amplitude |Hxx/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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[Rout,x(ω)Rout,y(ω)]=W(ω)H(ω)Ein(ω)+W(ω)V(ω)=U(ω)Ein(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=2erfc1(2BER). 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 Hxx(yy)(ω). If we also consider the residual ISI as an additional source of noise, this MPI-induced penalty ΛMPI will be given by

ΛMPI=(|u12(ω)|2+|u21(ω)|2)SS(ω)dω+(|w11(ω)|2+|w12(ω)|2+|w21(ω)|2+|w22(ω)|2)N0dω2N0dω(|u11(ω)|2+|u22(ω)|2)SS(ω)dω
where uij(ω) and wij(ω) denote the i,jth entry of U(ω) and W(ω) respectively and SS(ω) is the power spectral density of the input signals Ein,x(y)(ω). In addition to ΛMPI, another mode-coupling-induced performance penalty ΛHOM 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.
ΛHOMlimKk=1K(1ηk)=eκL
Compared with standard fiber systems, the QSM system has a larger effective area, smaller nonlinearity coefficient which in turn result in an OSNR gain
GOSNR=Aeff,FMF/Aeff,Std.fiber.
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 Goverall of QSM relative to standard single-mode fiber transmission systems is then given by

 figure: Fig. 5

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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Goverall=GOSNR(dB)ΛMPI(dB)ΛHOM(dB).

Figure 6 shows the pdf of overall performance gain Goverall of a 20-span QSM transmission system in presence of mode-coupling-induced penalties ΛMPI and ΛHOM and an effective area Aeff,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 ΛHOM 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.

 figure: Fig. 6

Fig. 6 PDF of the overall performance gain Goverall of a 20-span QSM transmission system in presence of mode-coupling-induced penalties ΛMPI and ΛHOM and an effective area Aeff,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.

 figure: Fig. 7

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 2×2MIMO 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=2erfc1(2BER)) 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.

 figure: Fig. 8

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.

 figure: Fig. 9

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.

 figure: Fig. 10

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.

 figure: Fig. 11

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 |hxx(t)|2 and |hyy(t)|2 and cross-polarization impulse responses |hxy(t)|2 and |hyx(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.

 figure: Fig. 12

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 H1-span(ω) in frequency i.e.

E[H1span*(ω')H1span(ω'+ω)]=E[[10]k=0K1(RKk*G*(ω'))[1000]k=1K(G(ω'+ω)Rk)[10]].
Since Rm is statistically independent of Rn for mn, we have
E[H1span*(ω')H1span(ω'+ω)]=E[[10]k=0K2RKk*G*(ω')E[R1*G*(ω')[1000]G(ω'+ω)R1]k=2KG(ω'+ω)Rk[10]],
in which
E[R1*G*(ω')[1000]G(ω'+ω)R1]=E[[1η1η1(1η1)ejθ1η1(1η1)ejθ1η1]]=[1κΔL00κΔL].
Next, if we let ξ = e−(Δτ + ΔαL,
E[R2*G*(ω')R1*G*(ω')[1000]G(ω'+ω)R1G(ω'+ω)R2]=E[R2*G*(ω')[1κΔL00κΔL]G(ω'+ω)R2].=[(1κΔL)2+κ2ΔL2ξ00(1κΔL)κΔL(ξ+ξ2)]
and
E[R3*G*(ω')R2*G*(ω')R1*G*(ω')[1000]G(ω'+ω)R1G(ω'+ω)R2G(ω'+ω)R3]=[(1κΔL)3+(1κΔL)κ2ΔL2(2ξ+ξ2)00(1κΔL)κΔL(ξ+ξ2+ξ3)]
and the process can be continued recursively. For a fiber with K sections, we have
E[k=0K1(RKk*G*(ω'))[10][10]k=1K(G(ω'+ω)Rk)][(1κΔL)K+(1κΔL)K2κ2ΔL2p=1K(Kp)ξp00(1κΔL)κΔLp=1Kξp]
in the weak coupling regime and neglect κpΔLp for p ≥ 3. In this case,
E[H1span*(ω')H1span(ω'+ω)]=(1κΔL)K+(1κΔL)K2κ2ΔL2p=1K(Kp)ξp
And its Fourier transform is given by
E[|h1span(t)|2]=(1κΔL)K(δ(t)+(κΔL1κΔL)2p=1K(Kp)epΔαΔLδ(tpΔτΔL)).
Finally, in the limit K→∞, we obtain

limKE[|h1span(t)|2]=eκL(δ(t)+κ2LΔτeΔαtΔτ(1tΔτL)).

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 H1-span(ω) can be expressed as

H1span(ω)=k=1K1ηk+k=1K1ηkm=1K1l=1Kmηlηm+l(1ηl)(1ηm+l)emΔαΔL+j(θm+lθl+mΔτΔLω)=k=1K1ηk+k=1K1ηkm=1K1l=1Kmηlηm+l(1ηl)(1ηm+l)emΔαΔL+jq=lm+l1xq,
where m denotes the number of consecutive sections the signal ‘stays’ in the higher-order mode and xq = θq+1θq + ΔτΔ. Note that the xq’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 H1-span(ω) is also independent of ω. As K→∞, k=1K1ηkeκL/2 and hence E[H1span(ω)]eκL/2 and
Var(H1span(ω))=E[|hMPI,1span(t)|2]dt=1NPMPI.
For an N-span QSM transmission system, denote
Yn=m=1K1l=1Kmηlηm+l(1ηl)(1ηm+l)emΔαΔL+jq=lm+l1xq
as the MPI terms incurred in the nth span. In the weak coupling regime, the transfer function with per-span amplification gain eκL becomes
H(ω)=n=1N(1+Yn)(1+n=1NYn)NlargeN(1,12PMPII)
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

Hxx(yy)(ω)N(1,18PMPII)andHxy(yx)(ω)N(0,18PMPII)

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.

References and links

1. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]  

2. S. Randel, “Space Division Multiplexed Transmission,” in Proc. Opt. Fiber Commun. Conf. (OFC) 2013, paper OW4F.

3. F. Yaman, N. Bai, B. Zhu, T. Wang, and G. Li, “Long distance transmission in few-mode fibers,” Opt. Express 18(12), 13250–13257 (2010). [CrossRef]   [PubMed]  

4. F. Yaman, N. Bai, Y. K. Huang, M. F. Huang, B. Zhu, T. Wang, and G. Li, “10 x 112Gb/s PDM-QPSK transmission over 5032 km in few-mode fibers,” Opt. Express 18(20), 21342–21349 (2010). [CrossRef]   [PubMed]  

5. J. D. Downie, J. E. Hurley, D. V. Kuksenkov, C. M. Lynn, A. E. Korolev, and V. N. Nazarov, “Transmission of 112 Gb/s PM-QPSK signals over up to 635 km of multimode optical fiber,” Opt. Express 19(26), B363–B369 (2011). [CrossRef]   [PubMed]  

6. N. Bai, C. Xia, and G. Li, “Adaptive frequency-domain equalization for the transmission of the fundamental mode in a few-mode fiber,” Opt. Express 20(21), 24010–24017 (2012). [CrossRef]   [PubMed]  

7. Q. Sui, H. Zhang, J. D. Downie, W. A. Wood, J. Hurley, S. Mishra, A. P. T. Lau, C. Lu, H.-Y. Tam, and P. K. A. Wai, “256 Gb/s PM-16-QAM Quasi-Single-Mode Transmission over 2600 km using Few-Mode Fiber with Multi-Path Interference Compensation,” in Proc. Opt. Fiber Commun. Conf. (OFC)2014, paper M3C.5.

8. A. P. T. Lau, Q. Sui, H. Y. Tam, C. Lu, P. K. A. Wai, J. D. Downie, W. A. Wood, J. Hurley, and S. Mishra, “Long-haul Quasi-Single-Mode Transmission using Few-Mode Fiber with Multi-Path Interference Compensation” in Proc. of International Conference on Optical Internet (COIN)2014, paper FB3–1. [CrossRef]  

9. F. Yaman, E. Mateo, and T. Wang, “Impact of Modal Crosstalk and Multi-Path Interference on Few-Mode Fiber Transmission,” in Proc. Opt. Fiber Commun. Conf. (OFC) 2012, paper OTu1D.2. [CrossRef]  

10. J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems,” Proceedings of ECOC 2011, Paper Tu.5.B.2 (2011). [CrossRef]  

11. D. Marcuse, C. R. Manyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15(9), 1735–1746 (1997). [CrossRef]  

12. P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996). [CrossRef]  

13. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5(1), 329–359 (1996). [CrossRef]  

14. F. M. Gardner, “A BPSK/QPSK Timing-Error Detector for Sampled Receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986). [CrossRef]  

15. A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007). [CrossRef]  

16. X. Zhou, “An improved feed-forward carrier recovery algorithm for coherent receivers with M-QAM modulation format,” IEEE Photon. Technol. Lett. 22(14), 1051–1053 (2010). [CrossRef]  

17. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30(4), 521–531 (2012). [CrossRef]  

References

  • View by:

  1. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
    [Crossref]
  2. S. Randel, “Space Division Multiplexed Transmission,” in Proc. Opt. Fiber Commun. Conf. (OFC) 2013, paper OW4F.
  3. F. Yaman, N. Bai, B. Zhu, T. Wang, and G. Li, “Long distance transmission in few-mode fibers,” Opt. Express 18(12), 13250–13257 (2010).
    [Crossref] [PubMed]
  4. F. Yaman, N. Bai, Y. K. Huang, M. F. Huang, B. Zhu, T. Wang, and G. Li, “10 x 112Gb/s PDM-QPSK transmission over 5032 km in few-mode fibers,” Opt. Express 18(20), 21342–21349 (2010).
    [Crossref] [PubMed]
  5. J. D. Downie, J. E. Hurley, D. V. Kuksenkov, C. M. Lynn, A. E. Korolev, and V. N. Nazarov, “Transmission of 112 Gb/s PM-QPSK signals over up to 635 km of multimode optical fiber,” Opt. Express 19(26), B363–B369 (2011).
    [Crossref] [PubMed]
  6. N. Bai, C. Xia, and G. Li, “Adaptive frequency-domain equalization for the transmission of the fundamental mode in a few-mode fiber,” Opt. Express 20(21), 24010–24017 (2012).
    [Crossref] [PubMed]
  7. Q. Sui, H. Zhang, J. D. Downie, W. A. Wood, J. Hurley, S. Mishra, A. P. T. Lau, C. Lu, H.-Y. Tam, and P. K. A. Wai, “256 Gb/s PM-16-QAM Quasi-Single-Mode Transmission over 2600 km using Few-Mode Fiber with Multi-Path Interference Compensation,” in Proc. Opt. Fiber Commun. Conf. (OFC)2014, paper M3C.5.
  8. A. P. T. Lau, Q. Sui, H. Y. Tam, C. Lu, P. K. A. Wai, J. D. Downie, W. A. Wood, J. Hurley, and S. Mishra, “Long-haul Quasi-Single-Mode Transmission using Few-Mode Fiber with Multi-Path Interference Compensation” in Proc. of International Conference on Optical Internet (COIN)2014, paper FB3–1.
    [Crossref]
  9. F. Yaman, E. Mateo, and T. Wang, “Impact of Modal Crosstalk and Multi-Path Interference on Few-Mode Fiber Transmission,” in Proc. Opt. Fiber Commun. Conf. (OFC) 2012, paper OTu1D.2.
    [Crossref]
  10. J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems,” Proceedings of ECOC 2011, Paper Tu.5.B.2 (2011).
    [Crossref]
  11. D. Marcuse, C. R. Manyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15(9), 1735–1746 (1997).
    [Crossref]
  12. P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
    [Crossref]
  13. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5(1), 329–359 (1996).
    [Crossref]
  14. F. M. Gardner, “A BPSK/QPSK Timing-Error Detector for Sampled Receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986).
    [Crossref]
  15. A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
    [Crossref]
  16. X. Zhou, “An improved feed-forward carrier recovery algorithm for coherent receivers with M-QAM modulation format,” IEEE Photon. Technol. Lett. 22(14), 1051–1053 (2010).
    [Crossref]
  17. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30(4), 521–531 (2012).
    [Crossref]

2012 (2)

2011 (1)

2010 (3)

2008 (1)

2007 (1)

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
[Crossref]

1997 (1)

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15(9), 1735–1746 (1997).
[Crossref]

1996 (2)

P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

1986 (1)

F. M. Gardner, “A BPSK/QPSK Timing-Error Detector for Sampled Receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986).
[Crossref]

Bai, N.

Bolle, C.

Burrows, E. C.

Chen, Y.-K.

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
[Crossref]

Corless, R. M.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

Downie, J. D.

Esmaeelpour, M.

Essiambre, R.-J.

Gardner, F. M.

F. M. Gardner, “A BPSK/QPSK Timing-Error Detector for Sampled Receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986).
[Crossref]

Gnauck, A. H.

Gonnet, G. H.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

Hare, D. E. G.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

Huang, M. F.

Huang, Y. K.

Hurley, J. E.

Ip, E.

Jeffrey, D. J.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

Kahn, J. M.

Kaneda, N.

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
[Crossref]

Knuth, D. E.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

Koc, U.-V.

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
[Crossref]

Korolev, A. E.

Kuksenkov, D. V.

Leven, A.

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
[Crossref]

Li, G.

Lingle, R.

Lynn, C. M.

Manyuk, C. R.

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15(9), 1735–1746 (1997).
[Crossref]

Marcuse, D.

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15(9), 1735–1746 (1997).
[Crossref]

McCurdy, A. H.

Menyak, C. R.

P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

Mumtaz, S.

Nazarov, V. N.

Peckham, D. W.

Randel, S.

Ryf, R.

Sierra, A.

Wai, P. K. A.

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15(9), 1735–1746 (1997).
[Crossref]

P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

Wang, T.

Winzer, P. J.

Xia, C.

Yaman, F.

Zhou, X.

X. Zhou, “An improved feed-forward carrier recovery algorithm for coherent receivers with M-QAM modulation format,” IEEE Photon. Technol. Lett. 22(14), 1051–1053 (2010).
[Crossref]

Zhu, B.

Adv. Comput. Math. (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5(1), 329–359 (1996).
[Crossref]

IEEE Photon. Technol. Lett. (2)

A. Leven, N. Kaneda, U.-V. Koc, and Y.-K. Chen, “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett. 19(6), 366–368 (2007).
[Crossref]

X. Zhou, “An improved feed-forward carrier recovery algorithm for coherent receivers with M-QAM modulation format,” IEEE Photon. Technol. Lett. 22(14), 1051–1053 (2010).
[Crossref]

IEEE Trans. Commun. (1)

F. M. Gardner, “A BPSK/QPSK Timing-Error Detector for Sampled Receivers,” IEEE Trans. Commun. 34(5), 423–429 (1986).
[Crossref]

J. Lightwave Technol. (4)

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30(4), 521–531 (2012).
[Crossref]

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
[Crossref]

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15(9), 1735–1746 (1997).
[Crossref]

P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

Opt. Express (4)

Other (5)

Q. Sui, H. Zhang, J. D. Downie, W. A. Wood, J. Hurley, S. Mishra, A. P. T. Lau, C. Lu, H.-Y. Tam, and P. K. A. Wai, “256 Gb/s PM-16-QAM Quasi-Single-Mode Transmission over 2600 km using Few-Mode Fiber with Multi-Path Interference Compensation,” in Proc. Opt. Fiber Commun. Conf. (OFC)2014, paper M3C.5.

A. P. T. Lau, Q. Sui, H. Y. Tam, C. Lu, P. K. A. Wai, J. D. Downie, W. A. Wood, J. Hurley, and S. Mishra, “Long-haul Quasi-Single-Mode Transmission using Few-Mode Fiber with Multi-Path Interference Compensation” in Proc. of International Conference on Optical Internet (COIN)2014, paper FB3–1.
[Crossref]

F. Yaman, E. Mateo, and T. Wang, “Impact of Modal Crosstalk and Multi-Path Interference on Few-Mode Fiber Transmission,” in Proc. Opt. Fiber Commun. Conf. (OFC) 2012, paper OTu1D.2.
[Crossref]

J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems,” Proceedings of ECOC 2011, Paper Tu.5.B.2 (2011).
[Crossref]

S. Randel, “Space Division Multiplexed Transmission,” in Proc. Opt. Fiber Commun. Conf. (OFC) 2013, paper OW4F.

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Figures (12)

Fig. 1
Fig. 1 The 2-mode coupling model for one span of QSM transmission.
Fig. 2
Fig. 2 Theoretical and simulation results for (a) Expected impulse response power E[ | h xx(yy) (t) | 2 ] and (b) pdf of transfer function Hxx(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.
Fig. 3
Fig. 3 The MPI length τMPI for various DMD Δτ and DML Δα for a 32 Gbaud polarization-multiplexed QSM transmission system.
Fig. 4
Fig. 4 (a) A sample realization of Hxx/yy(ω) and (b) pdf of the amplitude |Hxx/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.
Fig. 5
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.
Fig. 6
Fig. 6 PDF of the overall performance gain Goverall of a 20-span QSM transmission system in presence of mode-coupling-induced penalties ΛMPI and ΛHOM and an effective area Aeff,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.
Fig. 7
Fig. 7 Experimental setup for QSM transmission and MPI compensation. GEF: gain equalization filter; LSPS: loop synchronous polarization scrambler.
Fig. 8
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.
Fig. 9
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.
Fig. 10
Fig. 10 Q vs. distance at optimal launch power using DD-LMS for MPI compensation.
Fig. 11
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.
Fig. 12
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.

Equations (35)

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E out (ω)=[ 1 0 ] k=1 K ( G(ω)R k )[ E in (ω) 0 ]
G(ω)=[ 1 0 0 e (jωΔτ+ 1 2 Δα)ΔL ]
R k =[ 1 η k η k e j θ k η k e j θ k 1 η k ]
H 1span (ω)=[ 1 0 ] k=1 K ( G(ω)R k )[ 1 0 ]
E[ | h 1span (t) | 2 ] F E[ H 1span * (ω') H 1span (ω'+ω) ].
E[ | h 1span (t) | 2 ]={ e κL δ(t)+ e κL κ 2 L Δτ e Δαt Δτ ( 1 t ΔτL ) E[ | h MPI,1span (t) | 2 ] , 0tΔτL 0 otherwise ,
h(t)=( δ(t)+ h MPI,1 (t) )( δ(t)+ h MPI,2 (t) )...( δ(t)+ h MPI,N (t) ) ( δ(t)+ n=1 N h MPI,n (t) )
E[ | h(t) | 2 ]δ(t)+ n=1 N E[ | h MPI,n (t) | 2 ] ={ δ(t)+ N κ 2 L Δτ e Δαt Δτ ( 1 t ΔτL ) E[ | h MPI (t) | 2 ] 0tΔτL 0 otherwise
P MPI = 0 ΔτL E[ | h MPI (t) | 2 ]dt =N κ 2 ΔαL+ e ΔαL 1 Δ α 2 .
[ E out,x (ω) E out,y (ω) ]=H(ω) E in (ω)+V(ω)=[ H xx (ω) H xy (ω) H yx (ω) H xx (ω) ][ E in,x (ω) E in,y (ω) ]+[ V x (ω) V y (ω) ],
E[ | h xx (t) | 2 ]=E[ | h yy (t) | 2 ]=δ(t)+ 1 4 E[ | h MPI (t) | 2 ]
E[ | h xy (t) | 2 ]=E[ | h yx (t) | 2 ]= 1 4 E[ | h MPI (t) | 2 ].
H xx(yy) ( ω )N( 1, 1 8 P MPI I ) and H xy(yx) ( ω )N( 0, 1 8 P MPI I ),
0 τ MPI | h MPI (t) | 2 dt= 0 τ MPI ( 1 t ΔτL ) e Δαt Δτ dt =0.99 0 ΔτL ( 1 t ΔτL ) e Δαt Δτ dt
τ MPI = Δτ Δα ( ΔαLW( ( 0.01( ΔαL1 )0.99 e ΔαL ) e ΔαL1 )1 ) ΔαL large Δτ Δα ln100
W( ω )= ( H ( ω )H( ω ) S S ( ω )+ N 0 I ) 1 H ( ω ) S S ( ω ),
[ R out,x (ω) R out,y (ω) ]=W(ω)H(ω) E in (ω)+W(ω)V(ω)=U(ω) E in (ω)+W(ω)V(ω)
Λ MPI = ( | u 12 ( ω ) | 2 + | u 21 ( ω ) | 2 ) S S ( ω )dω + ( | w 11 ( ω ) | 2 + | w 12 ( ω ) | 2 + | w 21 ( ω ) | 2 + | w 22 ( ω ) | 2 ) N 0 dω 2 N 0 dω ( | u 11 ( ω ) | 2 + | u 22 ( ω ) | 2 ) S S ( ω )dω
Λ HOM lim K k=1 K ( 1 η k )= e κL
G OSNR = A eff,FMF / A eff,Std.fiber .
G overall = G OSNR ( dB ) Λ MPI ( dB ) Λ HOM ( dB ).
E[ H 1span * (ω') H 1span (ω'+ω) ]=E[ [ 1 0 ] k=0 K1 ( R Kk * G * (ω') ) [ 1 0 0 0 ] k=1 K ( G(ω'+ω) R k ) [ 1 0 ] ].
E[ H 1span * (ω') H 1span (ω'+ω) ]=E[ [ 1 0 ] k=0 K2 R Kk * G * (ω') E[ R 1 * G * (ω')[ 1 0 0 0 ]G(ω'+ω) R 1 ] k=2 K G(ω'+ω) R k [ 1 0 ] ],
E[ R 1 * G * (ω')[ 1 0 0 0 ]G(ω'+ω) R 1 ]=E[ [ 1 η 1 η 1 (1 η 1 ) e j θ 1 η 1 (1 η 1 ) e j θ 1 η 1 ] ]=[ 1κΔL 0 0 κΔL ].
E[ R 2 * G * (ω') R 1 * G * (ω')[ 1 0 0 0 ]G(ω'+ω) R 1 G(ω'+ω) R 2 ] =E[ R 2 * G * (ω')[ 1κΔL 0 0 κΔL ]G(ω'+ω) R 2 ] . =[ (1κΔL) 2 + κ 2 Δ L 2 ξ 0 0 (1κΔL)κΔL( ξ+ ξ 2 ) ]
E[ R 3 * G * (ω') R 2 * G * (ω') R 1 * G * (ω')[ 1 0 0 0 ]G(ω'+ω) R 1 G(ω'+ω) R 2 G(ω'+ω) R 3 ] =[ (1κΔL) 3 +(1κΔL) κ 2 Δ L 2 (2ξ+ ξ 2 ) 0 0 (1κΔL)κΔL( ξ+ ξ 2 + ξ 3 ) ]
E[ k=0 K1 ( R Kk * G * (ω') ) [ 1 0 ][ 1 0 ] k=1 K ( G(ω'+ω) R k ) ] [ (1κΔL) K + (1κΔL) K2 κ 2 Δ L 2 p=1 K (Kp) ξ p 0 0 (1κΔL)κΔL p=1 K ξ p ]
E[ H 1span * (ω') H 1span (ω'+ω)]= (1κΔL) K + (1κΔL) K2 κ 2 Δ L 2 p=1 K (Kp) ξ p
E[ | h 1span (t) | 2 ]= (1κΔL) K ( δ(t)+ ( κΔL 1κΔL ) 2 p=1 K (Kp) e pΔαΔL δ(tpΔτΔL) ).
lim K E[ | h 1span (t) | 2 ]= e κL ( δ(t)+ κ 2 L Δτ e Δαt Δτ ( 1 t ΔτL ) ).
H 1span (ω)= k=1 K 1 η k + k=1 K 1 η k m=1 K1 l=1 Km η l η m+l (1 η l )(1 η m+l ) e mΔαΔL+j( θ m+l θ l +mΔτΔLω) = k=1 K 1 η k + k=1 K 1 η k m=1 K1 l=1 Km η l η m+l (1 η l )(1 η m+l ) e mΔαΔL+j q=l m+l1 x q ,
Var( H 1span (ω) )= E[ | h MPI,1span (t) | 2 ]dt = 1 N P MPI .
Y n = m=1 K1 l=1 Km η l η m+l (1 η l )(1 η m+l ) e mΔαΔL+j q=l m+l1 x q
H(ω)= n=1 N ( 1+ Y n ) ( 1+ n=1 N Y n ) N large N( 1, 1 2 P MPI I )
H xx(yy) ( ω )N( 1, 1 8 P MPI I ) and H xy(yx) ( ω )N( 0, 1 8 P MPI I )

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