Perturbative approach to the SSFM error

3.1 SSFM error on SNR

The accuracy of a simulation should be referred to the target parameter on which performance is evaluated. Typically, the error of the SSFM is considered an additive contribution on the true solution, i.e.:

e_SSFM, bA (z, t) − A (z, t) (3.1) whereA (z, t)b is the numerical solution of the NLSE and A (z, t) is the unknown true solution. Since such an error is a random process in time t, the accuracy is often translated in a more practical form as [812]:

ke_SSFMk kA (z, t)k =

A (z, t) − A (z, t)b

kA (z, t)k (3.2)

where k.k indicates a signal norm. The normalization in (3.2) relates thus the numerical error to the signal amplitude, which is useful to control such an error over the pulse shape of the signal.

However, in coherent optical communications, the performance of an optical system is typically expressed in terms of the SNR. The SNR can be usually directly converted to bit error rate (BER), for instance under the assumption of additive Gaussian noise. In this framework, the accuracy of the simulation is therefore the accuracy in the SNR estimation. Moreover, since the SNR is generally expressed in decibels, a reliable SSFM simulation should be such that the SNR is estimated with a bounded accuracy error in dB, as sketched in Fig. 3.1 by the constant error bars. The dashed line indicates the SNR considering just the numerical error of the simulation, that should have a xed gap in a dB scale from the true SNR in order to have a xed accuracy. We proceed now to analyze the numerical error under this novel point of view.

3.1. SSFM error on SNR 31

Figure 3.1: Visual representation of an ideal simulation setup on the SNR of the communication. The error bars on the estimated curve should remain constant whatever the transmitted signal power. The dashed curve in b) (SNR_SSFM) represents the SNR due only to the numerical error, that should have a xed gap from the estimated SNR curve.

thus depend on how dierent is the oscillating function e^Lξ inside the step with respect to the xed value e^Lξ0. Although the SSFM does not need a perturbative assumption to solve the NLSE as the RP1 approximation does, it is useful to use some of the assumptions of the RP1 for the NLI to obtain some properties on e_SSFM. This is not a limitation since the error of a good SSFM simulation usually is small. First of all, we consider a perturbative assumption of the Kerr nonlinearity [14,16,17]. In this framework, the error of SSFM e_SSFM, which is generated by a bad" simulation of the NLI, will be a perturbation to the signal too. The channel can be thus considered as an additive noise channel with two dierent noise sources: one the NLI and one the SSFM error, as depicted in Fig. 3.2. By assuming the NLI and the SSFM error uncorrelated for simplicity, a new formulation of the SNR can be dened from (2.33) in the

Figure 3.2: Progressive abstraction of the ber channel rst by the SSFM approximation and then by the perturbative assumption of e_SSFM.

framework of SSFM as¹:

SNR ,[ P

σ²_ASE+ σ²_NLI+ σ_SSFM²

where P is the signal power, while σ_NLI² and σ_SSFM² are respectively the vari-ances of NLI [14] and SSFM error under the perturbative assumptions, with uncorrelation between NLI, SSFM error and the propagating signal. σ²_ASEis the variance of the ASE introduced by EDFAs, which is also uncorrelated with the signal and all the other noises. The true SNR under perturbative assumptions is (2.33):

SNR , P

σ²_ASE+ σ²_NLI which corresponds to [SNR σ_SSFM² = 0

. It is worth noting that in the frame-work of SSFM the obtained SNR estimation is always smaller than the true one due to σ²_SSFM.

1Such an expression of the SNR describe completely the system only in the case of additive circular noises, as in the assumption of the GN model. In some optical links the circularity of the NLI can be argued, since the NLI could manifest partially as phase noise. However, as we demonstrate in Appendix A, the eventual non-circularity of the NLI (and by extension of eSSFM) does not aect its total variance, thus this SNR formula can still be used.

3.1. SSFM error on SNR 33

−600 −50 −40 −30 −20 −10 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

σ²

ASE [dB]

SNR err [dB]

Figure 3.3: Example of SNR error as a function of the ASE noise variance σ_ASE² . In this example σ_NLI² = −20.28dB and σ_SSFM² = −29.1086dB.

By referring now to Fig. 3.1 and remembering that the SNR is expressed in dB, we can dene the error on the SNR as:

SNR^err, SNR

SNR[ , 1 + σ_SSFM²

σ²_ASE+ σ²_NLI. (3.3) Please note that SNR and [SNR in (3.3) are expressed in linear scale, thus in dB the SNR error is the usual

SNR^errdB =SNRdB− [SNRdB

Equation (3.3) is a monotonic decreasing function of ASE noise, as depicted in Fig. 3.3. For increasing values of σ²_ASE, after σ_ASE² > σ_NLI² the SNR error approaches zero dB. The SNR error in dB is thus bounded between:

0 ≤

SNR SNR[



dB

≤



1 +σ_SSFM² σ_NLI²



dB

. (3.4)

Unfortunately, we cannot control the error, hence a good simulation should bound the worst error case, i.e., the upper bound in (3.4). Such a bound is reached in absence of ASE noise.

Moreover, by considering the perturbative assumption we can neglect the SSFM error due to higher order contributions to the nonlinearity. This way σ_SSFM² can be treated as σ_NLI² , hence it is expected to scale with the cube of the power, as we showed in Section 2.1.4 for σ_NLI² . We thus nd convenient to work with normalized variances:

a_NL , σ_NLI²

P³ , a_SSFM , σ_SSFM²

P³ (3.5)

with a_NL and a_SSFM the unit-power variances of the NLI [52] and the SSFM error, respectively. It has been proved that a_NL is constant with the power P as long as the perturbative assumption holds. By the same arguments it can be claimed that even aSSFM is power independent. Although this is just an assumption, we will show later that when the NLI is small, i.e., for typical values of SNR, the perturbative assumption on NLI grants a_SSFMto be constant with the input signal power.

Focusing thus on the SNR error upper bound in (3.4) and considering (3.5), under the perturbative assumption the SNR error can thus be expressed as:

SNR

SNR[ , 1 +a_SSFM

a_NL (3.6)

whatever the power of the transmitted signal. In the limit of absence of nu-merical error, i.e., a_SSFM = 0, the SNR error ^SNR

SNR[ = 1, which means that the solution of the NLSE estimated by the SSFM coincide with the true one.

The main conclusion is thus that the SNR error of the simulation in (3.6), under perturbative assumption of eSSFM, is independent of the signal power P. The power-independent parameter ^aSSFM_aNL is thus the reference parameter to setup a SSFM simulation, and it should remain constant when varying all the other simulation parameters to grant a constant error on the SNR of the simulation. However, typical rules in literature do not follow such a rule [710].