Digital non-linear compensation for high-speed long-haul transmission

Marco Forzati, Jonas Mårtensson, Hou-Man Chin, Marco Mussolin, Danish Rafique1 Department for Networking and Transmission, Acreo AB, 164 40 Kista, Sweden

1Photonic Systems Group, Tyndall National Institute and Department of EE, UCC, Cork, Ireland Tel: +46 8 632 7753, Fax: +46 8 750 5430, e-mail: marco.forzati@acreo.se

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 so devise methods to increase the non-linear tolerance of optical signals by means of digital signal processing. In this 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: fiber 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, H(ω), describes the effects of the fibre CD and PMD, and of filtering at the transmitter and receiver. Therefore an of the transmitted signal can be obtained by applying a linear equaliser H−1(ω) to the received signal. This approach allows removing in principle all linear distortions, so that the received and processed signal, A(w, L)⋅H(ω), is in principle a copy of the transmitted signal A0(ω) plus accumulated noise, filtered by the linear equaliser, N⋅H(ω). In order to reduce the relative intensity of noise, and hence reduce the bit error ratio, the optical signal to noise ratio (OSNR) can be simply increased by increasing the power launched in the transmission line 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 effects of polarisation on the nonlinear interactions takes the following form:

where only the Kerr non-linearity is taken into account. In the equation above, 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] for the Ax polarisation component:

and its coupled equation for Ay. This equation does not have a closed form solution, unless γ = 0, or β2 = 0. In standard linear DSP, it is assumed γ = 0. In the following sections we will describe methods in which an approximation is made to Eq. 2 in an effort to compensate transmission impairments, including non-linear effects.

2. STANDARD DIGITAL BACK PROPAGATION (DBP) As mentioned in the previous section, Eq. 2 does not have a closed form solution, unless γ = 0, or β2 = 0. In standard linear DSP, it is assumed γ = 0. In the former case the system is linear and the NLSE can be written in the frequency domain and an inverse linear transfer function can be calculated easily, as discussed above. 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]. These FIR+NLC core steps can then be repeated for each span, until an estimate of the signal at the transmitter is obtained (see [4] 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 (WDBP) [5] 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 WDBP 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 identical since if the Manakov model holds [2]. Further details on optimization of these parameters can be found in [5]. It also is worth mentioning that the complexity reduction via WDBP 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. 2 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. 2), and disregarding cross-polarisation effects for the sake of simplicity, yields [6]:

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) [6]:

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 [7], [8]. 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 signal at the A(ω, L) [6].

5. TESTING NON-LINEAR COMPENSATION We have tested the techniques described above numerically and experimentally, here we give an overview of the latest results. An example of the efficiency of standard DBP was given in where a PolMux QPSK signal at 100 Gb/s was generated in the laboratory and transmitted up to 2000 km, in a loop experiment where DPB gave a small but non-negligible improvement at 2000 km of uncompensated fibre [4]. We were later able to extend the reach of transmission, and to show that a considerable improvement can be obtained for longer transmission distances, both in terms of reach and margin (see Fig. 1).

Figure 1 - Reach achievable using linear compensation only, and using standard DBP non-linear compensation In a more recent work, we have been investigating the effectiveness of WDBP on a 112 Gb/s PM-QPSK signal after being propagated over a 1600 km uncompensated transmission link (2x80km). We tested WDBP 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 Fig. 2). For further details, please refer to the upcoming Optics Express article [5]. We are currently investigating the effectiveness of Volterra-based non-linear compensation on different simulation and experimental set-ups.

Figure 2 - Nonlinear threshold for no linear compensation only (square), standard DBP (triangles), WDBP (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.)

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