Conclusions

Chapter 6

Conclusions

In this work we addressed two open problems regarding the estimation of the nonlinear interference arising in modern ber-optic communications, namely the accuracy of the split-step Fourier method (SSFM) and the modeling of the interaction of the polarization dependent loss (PDL) and ber-optic Kerr nonlinearity. Both problems arise from the need in optical communication to precisely estimate the impact of the Kerr nonlinearity on the signal-to-noise ra-tio (SNR) of optical transmissions, which is nowadays one of the main limiting factors of the capacity of an optical channel.

The rst part of the thesis was about the SSFM algorithm, focusing on the scaling of its accuracy to simulate the propagation of optical signals in transmissions aected by Kerr nonlinearity. We have analyzed the accuracy of this technique by treating the numerical error of the simulation as a perturba-tion to the signal, thus worsening the SNR of the transmission. Such a study was motivated by the urge of modern optical communications to increase the transmission bandwidth beyond the current 5 THz of the Erbium-doped ber ampliers available band, where the accuracy of the SSFM has never been investigated.

By exploiting a parallelism between the SSFM and some known numerical quadrature rules we have detailed the reasons of the numerical error, showing

that it is set by an over-estimation of the four-wave mixing (FWM) along each step of the simulation. Following this analysis, we proposed a new parameter to set the rst step of the simulation based on bounding the FWM oscillations on it to grant a constant error on the SNR of the transmission for variations of signal power, bandwidth, and ber dispersion. In particular, we showed that the

rst step of the simulation should be power-independent, and it should shrink quadratically with the signal bandwidth. Such a new rule has been compared with two popular rules to set the step of the simulation, namely the nonlinear phase criterion [8] and the constant local error rule [9,10]. We showed that such rules, which are both power-dependent, do not yield a constant SNR error in decibel. In particular, the nonlinear phase criterion showed a decrement of the simulation accuracy for increasing bandwidth, thus it is not reliable to simulate the propagation of wideband signals, while the constant local error rule, due to its power-dependence, shrinks the step of the simulation excessively, resulting in over-accurate simulations at the expense of computational eort. Contrarily, our proposal showed an almost-constant accuracy for signal bandwidths up to 5 THz, thus it can be an extremely useful plug-and-play universal parameter to set up SSFM simulations for ultra-wide bandwidth transmissions.

Moreover, we compared in terms of computational eort the two dierent step-updating rules, that can be inferred by the nonlinear phase criterion and the constant error rule, both triggered with a rst step chosen by our proposal.

We showed that, for practical accuracies around 0.01 dB of SNR error, the step-updating rule inferred by the constant local error rule shows a lighter computational eort with respect to the one inferred by the nonlinear phase criterion, while the latter results more convenient when the propagation is simulated with a small number of steps, thus it can be preferable in applications where keeping a light computational eort is the main goal, such as in the framework of digital back-propagation, even at the expense of the accuracy.

In the second part of this work, we analyzed the impact of random PDL in nonlinear optical transmissions, showing that its interaction with nonlinearity changes the statistics of the SNR in a dierent way than the interaction

be-6.0. Conclusions 143

tween PDL and the ASE noise. We have shown that this dierent interaction, which is not accounted for by any model to predict the PDL impairments, can signicantly vary the statistics of the SNR at the receiver in nowadays trans-missions where the variance of the nonlinear interference is comparable to the ASE noise one. Such a result should be accounted for in the set up of the SNR margin to limit the outage of the optical system.

Finally, we extended the Gaussian noise (GN) model for the prediction of the NLI variance to include the nonlinear interaction between signal and PDL.

Our extension showed an excellent match both with accurate numerical simu-lation and experiments and resulted in extremely fast computational time. It is thus an ideal tool to perform fast statistical investigations on the randomness of the SNR aected by the PDL.

Appendix A

Nonlinear phase noise investigation

In this thesis we investigated the SNR of optical links by proposing useful rules to i) setup a numerical SSFM simulation (Chapter 3), ii) understand the impact of XPM and XPolM in presence of PDL (Chapter 4), and iii) estimate the outage probability due to PDL in nonlinear regime (.Chapter 5).

Although the SNR is widely adopted as a key paramenter for system design, the designer may be more interested in other metrics, such as the BER. Under the assumption of circular noise, the conversion between BER and SNR is straightforward. For instance, we have:

BER = 1

2erfcpSNR/2 QPSK

BER ' 2

log2M

 1 − 1

√ M

 erfc

s 3

2 (M − 1)SNR

!

M-QAM However, expetially in short links, the NLI may not be circular due to a non-negligible amount of phase noise. Aim of this section is to clarify this aspect and show that after a carrier phase estimation with practical number of taps the circualrity is still a reasonable approximation.

A.1 The problem of nonlinear phase noise

Most of the models of NLI characterization in optical bers share the com-mon assumption that the NLI generated on the link can be approximated by an additive signal with Gaussian statistics [14, 15,1719], and are thus known by the name of Gaussian noise (GN) model. The validity of such an assump-tion has been conrmed by several simulaassump-tions and experiments in dispersion uncompensated (DU) links [15,19,5456]. However, These models result inac-curate in particular scenarios, such as links over short distances or dispersion managed (DM) transmissions. One of the reason of such an inaccuracy can be attributed to the strong nonlinear phase noise in these links [9496], thus the circularity assumption of the GN model can be argued. In particular, in [94]

the authors showed that the amount of phase noise on the NLI in WDM trans-mission strongly depends on channel spacing. As a consequence, while the NLI induced by neighboring channels is a balanced mix of phase and circular noise, the NLI induced by far away channels is mostly phase noise. Such a phase noise could be recovered in the DSP at the receiver, thus not impacting the transmission performance.

The aim of this Appendix is to study more in depth the phase noise nature of the NLI for variable channel spacing by numerically analyze the role of phase noise in the XPM and XPolM contribution to the NLI, and its possible mitigation by using a carrier phase estimation (CPE) at the receiver.