Non-linear compensation techniques for coherent fibre transmission Marco Forzatia*, Jonas Mårtenssona, Hou-Man China, Marco Mussolina, Danish Rafiqueb, Fernando Guiomarc

aAcreo AB, 164 40 Kista, Sweden bPhotonic Systems Group, Tyndall National Institute and Department of EE, UCC, Cork, Ireland

cInstituto de Telecomunicações, 3810-193 Aveiro, Portugal

ABSTRACT Thanks to coherent detection and digital signal processing (DSP), linear distortions such as chromatic dispersion (CD) and polarisation mode dispersion (PMD) can in principle be completely compensated for in high-speed optical transmission. And indeed, effective algorithms have been devised and extensively investigated that allow CD- and PMD-resilient transmission of high-speed signals over long distances, leaving optical noise accumulation and non-linear impairments as the factors ultimately limiting reach. Considerable research has been dedicated in the last couple of years to devise methods to increase the non-linear tolerance of optical signals by means of digital signal processing. In this review paper, we present an overview of the most promising techniques, show some examples of their application and outline the status of research on this important topic.

Keywords: fibre optics, high-speed transmission, DSP, non-linear effects, non-linear compensation

1. INTRODUCTION The propagation of a light signal through an optical-fibre communication link is at a first-order approximation described by a linear system, where the transfer function, Hc(ω), describes the effects of the fibre chromatic dispersion (CD) and polarisation-mode dispersion (PMD), and of filtering at the transmitter and receiver. Therefore a copy of the transmitted signal can be obtained by filtering the received signal through a linear equaliser with transfer function He(ω) – or, equivalently, with an impulse response he(t) – such that He(ω) = Hc

−1(ω). This approach allows removing in principle all linear distortions, so that the received and processed signal, is in principle a copy of the transmitted signal. However, as signal propagates through the fibre, noise is also accumulated. Hence, if A(t, z) is the complex envelope [1] of the linearly-distorted optical field at position z in the fibre, the signal at the receiver is ARx(t) = A(t, L) + n(t), where L is the fibre transmission length, and n(t) is the noise accumulated during transmission, mainly due to amplified spontaneous emission (ASE) in the amplifiers [1]. Therefore the resulting copy produced by an ideal equaliser (for which is He(ω) = Hc

−1(ω)) is:

which is the sum of the transmitted signal A0(t) = A(t, 0), and the noise accumulated during transmission, filtered by the equaliser (the symbol represents linear convolution). In order to reduce the relative intensity of noise, and hence reduce the bit error ratio, the signal power (and hence the Optical Signal-to-Noise Ratio, OSNR) can be increased. Optical fibres, however, are only approximately linear. Propagation of a light signal in an optical fibre is described by the nonlinear Schrödinger equation (NLSE) [1] which, taking into account the nonlinear polarisation interactions, takes the following form:

* marco.forzati@acreo.se, phone: +46 8 632 7753, fax: +46 8 750 5430, acreo.se

where only the Kerr non-linearity is taken into account. In Eq. 2, Ax and Ay are the complex envelope of two orthogonal polarisation components of the electric field, α is the attenuation coefficient, β2 is the dispersion parameter, γ is the nonlinear coefficient, z and t are the propagation direction and time, respectively (higher order dispersion is neglected here for the sake of clarity, i.e. β3=0, but the same principles would hold throughout the paper if it were included). The small birefringence exhibited by common optical fibres, however, is sufficient to randomly scatter the polarisation of the electric field over a much shorter length (typically around 100 m [1]) than the nonlinear interaction length (typically above 10 km) so that the resulting nonlinearity is averaged over the entire Poincaré sphere. In that situation the expected non-linear interaction is described by the Manakov equation [2]:

This equation does not have a closed form solution, unless γ = 0, or β2 = 0. In the following sections we will describe three methods in which an approximation is made to Eq. 3 in an effort to compensate transmission impairments, including non-linear effects. Finally, we present recent results that we have obtained using the three methods.

2. STANDARD DIGITAL BACK PROPAGATION (DBP) If γ = 0, the system is linear and the NLSE, Eq. 3, can be written in the frequency domain and an inverse linear transfer function can be calculated easily. On the other hand, if chromatic dispersion is neglected (β2 = 0) the equation can be easily solved in the time domain instead, to give:

where φx

NL(t,z) is the nonlinear phase shift for the Ax component:

Now, although both CD and non-linear phase shift are present at the same time during transmission and the equation above does not hold strictly speaking, one can use the knowledge of the transmission line to find an approximate solution to the NLSE and hence equalise for both linear and non-linear impairments. In long-haul transmission systems the signal is periodically amplified throughout transmission to compensate for fibre loss. The optical link is therefore composed by a number of spans composed of an EDFA, followed by an optical fibre (and possibly a section of dispersion-compensating fibre). The signal power is highest at the fibre input and decreases to half typically after 10-20 km, during which the intensity profile of the signal is modified moderately by the accumulated CD (typically 150-300 ps/nm). We can then roughly approximate that nonlinearities mostly take place in the initial section of the fibre, where the power is higher, whereas propagation during the rest of the fibre span takes place in a linear regime. Indeed, we can simplify things further by assuming that the nonlinear phase shift takes place instantaneously at the fibre input, and that CD alone is present between the fibre input and the fibre output. This method is based on the same split-step approximation to

solve the non-linear Schrödinger equation (NLSE) when modelling fibre transmission [1], albeit with a much coarser step size [3].

Following this approach, we can then estimate the signal at the latest amplifier stage by backward- propagating the received signal: first by removing the CD accumulated during the span (by means of a FIR filter) and then by removing the nonlinear phase shift generated at the fibre input:

where a is the intra-polarisation nonlinearity parameter (SPM) and b is the cross-polarisation nonlinearity parameter (XPolM) and in principle have to be optimised (if the Manakov model is valid, we will obtain a = b). This can be implemented by a butterfly structure referred to as Non-Linear Compensator (NLC) core [4], [5]. These FIR+NLC core steps can then be repeated for each span, until an estimate of the signal at the transmitter is obtained (see [5] for further details).

3. WEIGHTED DBP While standard DBP has been proven to work effectively in reducing SPM, its complexity is a source of concern. One way to reduce complexity is to reduce the number of non-linear compensation stages to below the number of amplifier stages. This obviously invalidates the assumption that transmission takes place linearly between each NLC. Weighted DBP (W-DBP) [6] is a digital back-propagation algorithm, in which the nonlinear shift at a specific symbol location is correlated with the power of various consecutive symbols, rather than itself only, thus taking into account the fact that the non-linear phase shift is not instantaneous but accumulates over transmission, during which dispersion induces power “spilling” between neighbouring symbols. Specifically, the nonlinear phase shift on a given symbol is a weighted average of phase shifts arising from a number of symbols:

(7a)

(7b)

where Ax,yout and Ax,y

in are the electric fields for orthogonal polarization states before and after W-DBP for x and y polarization states, a and b represent intra-polarization and inter-polarization parameters [8], N represents the number of symbols (or filter length) to be considered for a nonlinear phase shift, ck is the weighing vector, k is the delay order, and Ts is the symbol period. Again, the values for a and b are identical if the Manakov model holds [2]. Further details on optimization of these parameters can be found in [6]. It also is worth mentioning that the complexity reduction via W-DBP is primarily achieved from the reduction in required FFTs rather than the reduced number of steps for NLC calculations.

4. VOLTERRA-BASED NON-LINEAR COMPENSATION In the previous sections, Split-Step Fourier methods have been shown, in which Eq. 3 was solved for fibre sections during which propagation was assumed to take place with either γ = 0, or β2 = 0. A different approach can be followed. Taking the Fourier transform of the NLSE (Eq. 3), and disregarding cross-polarisation effects for the sake of simplicity, yields [8]:

This equation can be integrated over z, from the transmitter to the transmission length L to give the Volterra series transfer function between the launched signal A(ω) and the signal at the receiver A(ω, L) [5]:

where H1 = exp{(–α – jβ2ω

2)z/2} is the linear kernel, and H3 is the third order kernel:

The approximation symbol in Eq. 9 is due to the fact that chromatic dispersion causes higher order kernels to appear when integrating the Fourier-domain equation. However a good approximation of the solution is general accepted considering H3 only [9], [10], [11]. Also, it should be noted that even-order kernels are not included since optical fibre does not have even order nonlinearities [1]. In order to compensate for the fibre linear and non-linear impairments, it is enough to inverse H1 and H3 and cascading them with the received signal at the ARx(ω) = A(ω, L) + n(ω), so that the transmitted field is digitally calculated as:

where ω, ωi and ωj all represent discrete frequencies, K1 = H1–1

and K3 = H1–1

H3 H1–1

are the inverse Volterra operators. The VSTF includes only odd order terms up to the third order as third order truncation has been shown to be provide sufficient accuracy compared to Split Step Fourier (SSF) [8], [12].

Figure 1 – 3rd order Volterra Inverse system for compensation of a span of optical fibre

5. TESTING NON-LINEAR COMPENSATION An example of the efficiency of standard DBP is shown in Fig. 1, showing the result of a laboratory experiment in which a PolMux QPSK signal at 100 Gb/s was generated in the laboratory and transmitted up to 3800 km, in a loop experiment (see [4] for a detailed description of the lab set-up) where DPB lead to a considerable improvement both in terms of transmission reach and power margin.

In a recent work, we have investigated the effectiveness of W-DBP on a 112 Gb/s PM-QPSK signal after being propagated over a 1600 km uncompensated transmission link (2x80km). We tested W-DBP with optimised filter taps (the ck coefficients in Eq. 7) and filter shape, and we found that up to 80% reduction in number of non-linear compensation steps could be achieved with no performance degradation with respect to standard DBP (see Figure 3).

We also tested the algorithm on a 224 Gb/s 16 QAM transmission experiment, in single-channel and WDM (eight channels) transmission over a 250 km straight-line fibre link consisting of ultra-large area fibre (ULAF), see. We found a performance enhancement enabled by weighted digital back propagation (W-DBP) method, using only one back-propagation step for the entire link, of up to a 3 dB improvement in power tolerance with respect to linear compensation only. This is more or less the same improvement that can be obtained with the standard, computationally heavy, non-

weighted digital back propagation (NW-DBP) employing one step per span. On the other hand, we found that self-phase modulation compensation is inefficient in WDM transmission, because cross-phase phase shift (XPM) is dominant here. However, for higher baud-rates, intra-channel effects become more dominant [13] so the improvements provided by DBP may become more interesting for higher baud-rate transmission systems.

Figure 2 – Reach achievable using linear compensation only, and using standard DBP non-linear compensation.

Figure 3 – Nonlinear threshold for no linear compensation only (square), standard DBP (triangles), W-DBP (circles);

for different number of steps utilised to equalise the 20 x 80 km link. (The nonlinear threshold is defined as the launch power at which a 3 dB penalty in required OSNR for BER=10-3 is observed, with respect to back-to-back.)

Figure 4 – Experimental setup for 224 Gb/s PM-16QAM transmission system with 3 total spans.

Figure 5 – a) Q versus OSNR and, b) versus launch power, for single-channel 28 Gbaud 16QAM transmission over 250

km ULAF with various digital compensation scenarios; different values of OSNR in the Fig. 2a were obtained by varying the launch power. Linear compensation (squares), weighted digital back-propagation (1 step per link, triangles), non-weighted digital back-propagation (3 steps per link, stars), non-weighted digital back-propagation (60 steps per link, circles).

Figure 5 depicts the Q-factor of the transmitted 28 Gbaud PM-16QAM signal (single-channel) as a function of received OSNR, for 250 km transmission, and as a function of launch power. The different curves refer to different electronic signal processing used at the receiver: with linear compensation (LC) only, with W-DBP (1 step for the whole link), and with NW-DBP (1 and 20 steps per span). At lower launch powers, where performance is limited by noise, Q increases with increasing launch power, and eventually reaches a maximum at an optimum launch power, above which it starts increasing again due to the accumulation of fibre nonlinearities. It can be seen that the Q curves for LC and DBP overlap in the noise-limited regime. However, the Q of the LC system reaches a maximum of ~11.5 at 3 dBm launch power (giving an OSNR of 31.7 dB), and then rapidly degrades due to intra-channel nonlinear effects. Such effects are alleviated by DBP techniques. Specifically, it can be seen that both NW-DBP with one step per span and W-DBP with one step for the whole link allows for a 3 dB increase of power tolerance (defined as the point at which Q reaches again the LC optimum of ~11.5 dB). This confirms that the computational burden of DBP can be greatly reduced using weighing, since only one extra FFT/IFFT stage is required in W-DBP with one step for the whole link (though an extra filtering stage with respect to only LC is needed, as well as one non-linear phase estimation stage), without compromising significantly performance improvement.

Finally, we have tested the Volterra series nonlinear equalizer (VSNE) by means of numerical simulations. A single-channel and single-polarization 20 Gbaud QPSK signal is transmitted over a link of 20 uncompensated spans (for further details about the transmission link and the receiver front end, please refer to [14]. In the DSP block, the VSNE is implemented in a span-by-span basis using the output modification proposed in [9] to correct for the energy divergence problem that occurs at high launch powers. Two different propagation scenarios were analysed: a highly dispersive link, composed by uncompensated standard single-mode fibre (SSMF) and a highly nonlinear link, composed by non-zero dispersion shifted fibre (NZDSF). Furthermore, the impact of the sampling rate on the equalizer performance has also been studied, changing the number of samples per symbol (Nsp) that are fed to digital equalization. The results are shown in Figure 7, in which we use the error vector magnitude (EVM) between the equalized signal and the ideal constellation as a figure of merit for equalization performance. A direct comparison with standard DBP (increasingly number of steps per span) and chromatic dispersion equalization (CDE) is also provided, providing a benchmark for the accuracy of nonlinear equalization. We denote DBP with Nsteps steps per span as DBPNsteps

.

(a) (b)

(d) (e)

Figure 7 – Performance assessment of the Volterra series nonlinear equalizer in terms of EVM against input power in the link. a) 20 ×

80 km of SSMF, Nsp=3; b) 20 × 80 km of SSMF, Nsp=2; c) 20 × 80 km of NZDSF, Nsp=2; a) 20 × 80 km of NZDSF, Nsp=3.

As can be seen from Figure 7 the equalization performance is primarily limited by temporal resolution. Above this limit increasing the spatial resolution in DBP, i.e. the number of steps per span, becomes useless. In fact, for the SSFM link we have found that the maximum DBP performance is obtained at 8 steps per span. In contrast with DBP, the VSNE performance has no spatial dependence and therefore its maximum accuracy is solely set by the receiver sampling rate. This explains why a VSNE based on a third-order approximation is able to perform as good as a highly iterative DBP at 3 samples per symbol (see Figures 7 a) and c)): both are being limited by temporal resolution. Reducing the sampling rate to the Nyquist limit of 2 samples per symbol (see Figures 7 b) and d)), we may observe a severe penalty on DBP, which is due to the generation of aliasing components arising from time-domain implementation of the nonlinear operator and subsequent evaluation of the linear operator in frequency-domain. In the opposite side, the fully frequency-domain VSNE avoids this aliasing generation phenomena, and thus it is now able to surpass the maximum performance of DBP. A particularly interesting case is shown in Figure 7 e): the highly nonlinear propagation regime associate with low temporal resolution sets the DBP limit to 1 step per span. However, despite of the extreme conditions, VSNE maintains a high accuracy, largely surpassing standard DBP, as we can see by the constellations taken at 6 dBm input power. Further simulation results and numerical complexity assessment can be found in [12], whereas we are currently testing the VSTF method on the same 16-QAM experimental setup of Figure 1: we aim at presenting the results at an upcoming conference.

6. CONCLUDING CONSIDERATIONS The recent advent and rapid uptake of DSP-based coherent communication systems has made it possible to compensate most linear impairments arising in long-haul high-speed fibre optic transmission. As more advanced and fast DSP become available, the transmission length is becoming more and more noise-limited, and high signal power becomes more and more desirable in order to increase transmission distance: non-linear impairments become then the obvious limiting factor. In this paper we have given an overview of non-linear Kerr effect (the major non-linear effect in practical

systems today), described two different approaches to model and compensate for that, and presented three DSP-based techniques together with some examples of their application.

The complexity of these techniques is currently rather high, but several steps have been made towards complexity reduction, and we have shown examples of simplified methods yielding similar performance with considerable complexity reduction, notably using weighted digital back-propagation (W-DBP). We believe that progress can continue, and that DSP will continue to grow more powerful, so that real-time implementations can become feasible in the near future. On the other hand, the problem of non-linear cross talk between neighbouring channels in a WDM system remains the major limitation of currently proposed methods, and need to be tackled if DSP-based non-linear compensation is to become of practical use.

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