## Abstract

We derive analytic formulas for the improvement in effective optical signal-to-noise ratio brought by a digital nonlinear compensator for dispersion uncompensated links. By assuming Gaussian distributed nonlinear noise, we are able to take both nonlinear signal-to-signal and nonlinear signal-to-noise interactions into account. In the limit of weak nonlinear signal-to-noise interactions, we derive an upper boundary of the OSNR improvement. This upper boundary only depends on fiber parameters as well as on the total bandwidth of the considered wavelength-division multiplexing (WDM) signal and the bandwidth available for back propagation. We discuss the dependency of the upper boundary on different fiber types and also the OSNR improvement in practical system conditions. Furthermore, the analytical formulas are validated by numerical simulations.

©2012 Optical Society of America

## 1. Introduction

Digital equalization for fiber nonlinearity is an indispensable technology to extend the transmission reach of high spectral efficiency (SE) transmission systems, which combine coherently detected higher order modulation formats with wavelength division multiplexing (WDM). In particular, digital signal processing (DSP) using back propagation nonlinear compensation (NLC) has been applied [1–4] to compensate for nonlinearity due to the Kerr effect except for limitations due to inter-channel nonlinearity, amplified spontaneous emission (ASE) noise [5,6], and polarization-mode dispersion (PMD) [7,8]. The performance improvements by NLC combined with single- and/or multi-channel coherent reception have been demonstrated with both experiments and numerical simulations [1–4] and discussed [5–8]. However, so far no simple analytical formula has been derived, which is able to describe the OSNR improvement brought by NLC in WDM links. The main issue is that the NLC performance depends not only on the considered NLC scheme itself but also on transmission system parameters, e.g. the fiber launch power, the fiber type and the dispersion map. Although numerical simulations based on the well-known Manakov equation can be used for evaluation of the NLC performance, the approach of numerical simulation would be time-consuming and the results strongly depend on certain system setup.

On the other hand, the Gaussian distributed nonlinear noise model, which is proposed in [9] and applied to digital coherent transmission combined with dispersion uncompensated links in [10–13], enables us to utilize a simple formulation of nonlinear impairments caused by the transmission instead of using extensive numerical simulations. In first results which have been presented in [14], we have derived a simple formula to estimate the effective OSNR improvement brought by a digital nonlinear compensator for dispersion uncompensated links, assuming Gaussian distributed nonlinear noise caused by signal-to-signal interference.

In this paper, we further extend this analysis and theoretically derive a simple formula of effective OSNR improvement brought by NLC taking both signal-to-signal and signal-to-noise interference into account. The theoretical predictions are verified with numerical simulations and show good agreement with the simulation results if the fiber launch power is so high that the system requires NLC. In the limit of weak signal-to-noise interference, we end up with a simple formula for the OSNR improvement brought by NLC as an upper boundary [14]. It should be noted that this simple formula is independent of fiber span length, number of spans, symbol rate, and channel spacing of the WDM channels, and the OSNR improvement therefore can be calculated with the help of the fiber parameters, the bandwidth used for NLC, and the total optical bandwidth occupied by all WDM channels. We discuss the dependency of the upper boundary on different fiber types and also the OSNR improvement in practical system conditions.

## 2. Modeling of a back propagation NLC with transmission links

Figure 1 shows a block diagram of the system configuration under study. The aim of this section is to derive an analytical model of a back propagation NLC. First we start to formulate the performance of a system without NLC. Assuming a Gaussian distribution of the nonlinear noise in a dispersion uncompensated link, the system performance after multi-span transmission with erbium doped fiber amplifiers (EDFAs) depends on an effective OSNR as follows [9–13]:

where*P*is the optical power per channel,

_{Tx,ch}*P*is the amplified spontaneous emission (ASE) power,

_{ASE}*P*is the equivalent noise power generated by the nonlinear Kerr effect over the transmission fibers,

_{NL}*N*is the number of transmission spans,

_{s}*F*is the EDFA noise figure (NF),

*h*is Plank's constant,

*υ*is the center frequency of channel under study,

*A*is the span loss, and

_{span}*B*is bandwidth for OSNR.

_{n}*P*consists of

_{NL}*P*and

_{NL, sig-sig}*P*which correspond to the nonlinear noise power generated by nonlinear signal-to-signal and signal-to-ASE interactions, respectively:Assuming quasi Nyquist WDM systems, meaning near rectangular, and near symbol rate spaced spectra, we can describe the nonlinear noise power generated by signal-to-signal interactions falling within the bandwidth

_{NL, sig-ASE}*B*after a

_{n}*N*-span transmission link [10]:

_{s}*γ*is the fiber nonlinearity coefficient,

*α*is the fiber loss coefficient,

*β*is the fiber dispersion,

_{2}*R*is the symbol rate of the signal,

_{s}*Δf*is the WDM channel spacing,

*B*=

_{WDM}*N*×

_{ch}*Δf*is the total optical bandwidth occupied by the

*N*WDM channels,

_{ch}*B*=

_{o}*α*/ (2π

^{2}|

*β*|

_{2}*B*) is the parameter of the walk-off bandwidth. Equation (4) shows that the nonlinear noise power by signal-to-signal interactions scales cubically with

_{WDM}*P*through a system depending constant.

_{Tx,ch}The nonlinear noise power caused by signal-to-ASE interference in the *m-th* span is proportional to${G}_{ASE}(m)\text{\hspace{0.17em}}{G}_{Tx,ch}^{2}=\left(m{P}_{ASE}/\left({B}_{n}{N}_{s}\right)\right){\left({P}_{Tx,ch}/\left({R}_{s}\Delta f\right)\right)}^{2}$, where *G _{ASE}*(

*m*) is the power spectral density of the ASE noise at the

*m-th*span and

*G*is the power spectral density of the signal [5,6]. In the summation of the nonlinear noise power of each span, we assume the span contributions to be uncorrelated random variables. Then the nonlinear noise power due to signal-to-ASE interference after

_{Tx,ch}*N*-spans transmission is given by [10]

_{s}*P*[6]. Note that these formulas so far do not take into account that nonlinear impairments are mitigated by using back propagation NLC.

_{Tx,ch}Considering NLC with a limited bandwidth *B _{NLC}*, the nonlinear noise generated by signal-to-signal interactions within this bandwidth can be removed, while the nonlinear noise power generated from outside this bandwidth and the noise power generated by signal-to-ASE interactions remain as residual noise. Substituting the integration range of

*f*in Eq. (4) to be (

*B*/2,

_{NLC}*B*/2) with the condition

_{WDM}*B*>

_{WDM}*B*>

_{NLC}*B*, the residual nonlinear noise power after NLC can be expressed as:

_{o}*B*is the optical bandwidth used for NLC, typically corresponding to the electrical bandwidth of a single coherent receiver. The OSNR improvement generated by a back propagation NLC is defined as the effective OSNR-difference between the achievable maximum effective OSNR with and without the use of NLC, respectively.

_{NLC}In the limit of weak signal-to-ASE interference (${P}_{NL,sig-ASE}\to 0$), a maximum value of effective OSNR without nonlinear compensation is given by

*P*. From Eqs. (7) and (8), the effective OSNR improvement in [dB] by using NLC without signal-to-ASE interference can be derived as [14]:

_{Tx,ch}## 3. System setup for numerical simulation

The proposed theoretical NLC model was evaluated in numerical simulations of 112 Gb/s dual-polarization quadrature phase-shift keying (DP-QPSK), 37.5 GHz-spaced WDM transmission. The system under consideration is depicted in Fig. 2(a)
. The measured channel at the central wavelength of 1552.52 nm and the 10 surrounding channels were independently modulated at a symbol rate of 28 Gbaud. The transmission line consisted of 120 spans of 80 km standard single-mode fiber (SMF) (*α* = 0.2 dB/km, *n _{2}* = 2.7 × 10

^{−20}m

^{2}/W,

*A*= 80 μm

_{eff}^{2}, and

*D*= 17 ps/nm/km), resulting in a transmission link of 9,600 km length. The input power to each fiber span was varied in a range between −3 dBm to + 2 dBm per channel. The EDFAs (NF = 5dB) at each span compensated for the optical power loss due to transmission. After the transmission, digital back propagation with limited bandwidth

*B*was carried out to compensate for both chromatic dispersion and nonlinear impairments. Figure 2(b) shows the channel assignment and bandwidth used for NLC. The NLC parameters including number of steps were optimized for each case. After the back propagation, the measured channel was selected by using a 5th order Bessel filter with 35 GHz 3-dB bandwidth. After CMA-based adaptive equalization with an 11-taps butterfly structured FIR filter, frequency offset compensation, Viterbi and Viterbi carrier recovery, and symbol decision, the effective Q-factors were calculated by measuring the bit error ratio.

_{NLC}To compare the results of the numerical simulations with the theoretical results, a back-to-back model [15] was used. The effective OSNR by theory can be converted to a corresponding Q-factor. Assuming DP-QPSK modulation, the Q-factor is given by Q = *k* OSNR *B _{n}*/

*R*where

_{s}*k*is a parameter in the back-to-back model [15] (

*η*in [15] was set to zero.) The

_{XT}*k*, which introduces a rigid translation of the BER vs. OSNR curve, can be obtained from back-to-back simulations. In this setup, the k-parameter in the back-to-back model was evaluated to be 0.88 and this value was used in following analysis. The net Q-gain due to NLC is defined as the Q-difference between the achievable maximum Q-factors with and without the use of NLC, respectively.

## 4. Results and discussion

First, we compared the agreement between Q-factors obtained by theory ('signal-signal model' and 'signal-signal + signal-ASE model') and numerical simulation with a single channel NLC (*B _{NLC}* = 35 GHz) after transmission over 120 spans SMF. Figure 3
shows Q-factors after transmission for theory and simulation. We observed, at least in the linear regime, a good agreement between the analytical 'signal-signal model' and the simulation, where the Q-factor increases initially with launched power, reaches a peak value, and then decreases with a further increase in power because of the onset of the nonlinear effects. However, the difference between the results for the analytical 'signal-signal model' and the simulation is larger for higher launch powers. This is due to the nonlinear interaction between signal and ASE, which are not included in the analytical formula. With modification of the model to include signal-ASE interference, we observed an improved agreement between Q-factors derived by theory and simulation. The differences between Q-factor predicted by 'signal-signal + signal-ASE' model and numerical simulation were within 0.5 dB regardless of NLC. These results support that the derived equations can be used to evaluate the performance of NLC.

Next, we investigated the dependency of the improvement caused by the NLC on the bandwidth used for the nonlinear compensation. Figure 4
shows the simulated and theoretical Q-gains as a function of bandwidth used for NLC *B _{NLC}* after transmission over 120 spans SMF. The total bandwidth is

*B*= 11 × 37.5 GHz. The net Q-gain brought by the NLC shows consistent improvement over the entire range of

_{WDM}*B*from 20 to 200 GHz for both simulation and theory. It can be seen that the difference between the predicted Q-gain by the 'signal-signal model' and the 'signal-signal + signal-ASE model' was increasing with bandwidth used for NLC. This is due to the residual signal-to-ASE interference, which is not included in the 'signal-signal model'. The differences between Q-gain predicted by 'signal-signal + signal-ASE model' and numerical simulation were within only 0.15 dB. The optimum fiber input power with NLC

_{NLC}*P*to achieve maximum values of OSNR monotonically increases with the bandwidth used for NLC. In particular, employing 'signal-to-signal model' ('signal-to-signal + signal-to-ASE model)', the values of

_{TX,ch,opt,w/NLC}*P*are calculated as 0 (−0.3), 0.6 (0.2), 1.1 (0.5), and 1.5 (0.9) dBm per channel at the bandwidth used for NLC of 50, 100, 150, and 200 GHz, respectively. Note that the optimum powers are reduced when also accounting for nonlinear signal-to-ASE interactions.

_{TX,ch,opt,w/NLC}According to Eq. (9), the upper boundaries for the benefit caused by the NLC can be described with only *B _{WDM}*,

*B*and fiber parameters in

_{NLC}*B*. We evaluate the upper boundaries for transmission links consisting of various fibers types. To obtain the upper limit of the benefit, we assumed a system in which all channels were propagated through the same fiber link, e.g. corresponding to a point-to-point system and/or super-channel transmission system. Figure 5(a) shows the upper boundary for the OSNR improvement as a function of

_{o}*B*for transmission over SMF, non-zero dispersion-shifted fiber (NZ-DSF) and ultra-large–effective-area fiber (ULAF). The whole C-band of 4 THz was taken into account for

_{NLC}*B*. The fiber parameters of each fiber are described in the caption of Fig. 5(a). The system using NZ-DSF shows less benefit from the NLC than the systems with SMF and ULAF, which show almost the same improvement. The fiber-type dependency of the improvement is relatively small and the difference among the three fiber types is within 0.3 dB. The OSNR improvement itself monotonically increases with the bandwidth used for NLC.

_{WDM}Finally, we consider the OSNR improvement by using the NLC model including nonlinear signal-to-ASE interactions. In this case, the OSNR improvement by NLC depends not only on *B _{NLC}*,

*B*, and

_{WDM}*B*but also on other system parameters. To highlight the dependence on system parameters, we swept the number of transmission spans

_{o}*N*from 20 to 150. Figure 5(b) illustrates the influence of the number of spans (

_{s}*N*) on the OSNR improvement by NLC. To maximize values of the OSNR,

_{s}*P*were optimized at each condition of

_{Tx,ch}*N*and

_{s}*B*in Fig. 5(b). The other simulation parameters were fixed at the values described in the caption of Fig. 5(b). We observed that increasing the number of spans degrades the achievable OSNR improvement by NLC through nonlinear signal-to-ASE interactions. After 20-spans transmission, the optimum fiber launch power was increased from −1.4 dBm to 0.5 dBm per channel for increasing bandwidth

_{NLC}*B*from 20 GHz to 1 THz even in the presence of nonlinear signal-to-ASE interactions. The increase of optimum fiber launch power enabled by a higher bandwidth

_{NLC}*B*allows achieving a higher Q-factor. On the other hand, the optimum launch power is reduced with increasing number of spans

_{NLC}*N*through nonlinear signal-to-ASE interactions. For example, the optimum fiber launch power was decreased from 0.5 dBm to 0 dBm per channel when increasing the number of spans of

_{s}*N*from 20 to 150 spans and keeping the bandwidth

_{s}*B*constant at 1 THz.

_{NLC}## 5. Conclusions

We have derived analytical formulas for the OSNR improvement brought by a digital back propagation nonlinear compensator for dispersion uncompensated transmission links. These formulas are based on an analytical model, which assumes Gaussian distributed nonlinear noise, and is able to take both nonlinear signal-to-signal and signal-to-ASE interactions into account. The obtained analytical results were verified by numerical simulations of DP-QPSK WDM transmission. In the limit of weak nonlinear signal-to-ASE interactions, we derived an upper boundary for the improvement brought by NLC that only depends on the fiber parameters, the overall WDM bandwidth and the bandwidth available for back-propagation. For a particular fixed transmission system, the improvement monotonically increases with the bandwidth used for NLC.

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