1. Introduction

Since the beginning of optical communications, fiber manufacturer have tried to give their contribution to system performance improvement by designing new fiber types with parameters tailored to reduce the impact of propagation effects. After the development of the standard single-mode fiber (SSMF), fibers with low dispersion were introduced to ease in-line optical dispersion compensation. Then, fibers with large effective area to reduce the impact of nonlinear effects were developed, and recently new fibers with low loss and even larger effective area have been introduced in the market.

In order to compare fiber types, different figure of merits (FoM) have been proposed encompassing the effects of both nonlinearity and loss [1,2]. With the introduction of coherent detection and electronic dispersion compensation [3] for 40 and 100 Gbps systems, a new paradigm for system design has arisen: it appears that uncompensated links outperform dispersion managed solutions and that larger dispersion fibers outperform lower dispersion ones [4,5]. In order to deal with this scenario, in this paper we propose a FoM definition taking into account the effects of chromatic dispersion on system performance, extending results presented in [6] to channel spacing larger than the symbol rate. The presented closed-form of FoM is based on a model for the impact of non-linear propagation proposed in [7].

First, in Sec. 2, we give the FoM definition based on the system margin. Then, in Sec. 3, we recall the model of nonlinear interference [6] to obtain a closed-form expression of FoM at the Nyquist limit, i.e., for channel spacing (Δf) equal to the symbol rate (R_S). In Sec. 4, we compare the proposed FoM with the ones earlier presented in the literature [1,2] showing how it is fundamental to include the role of chromatic dispersion in order to obtain a fair comparison between different fiber types. In order to validate the proposed FoM we simulated a 10 channels Nyquist WDM PM-QPSK link and, in Sec. 5, displayed how simulative estimates of FoM based on system margin are in excellent agreement with the proposed closed-form expression. In Sec. 6, we tested by simulation the applicability of the proposed FoM to channel spacings larger than the Nyquist limit, showing that the presented FoM works properly also in such scenarios. Finally, in Sec. 7, we validate experimentally the performance prediction obtained through the new FoM for different fiber types and in conclusion (Sec. 8) we discuss the obtained results.

2. The figure of merit based on system margin

We consider a multi-span periodic link based on optical amplification operating at a target bit error-rate BER_target. In such a scenario we can define the following parameters.

Using these two parameters we can then define the system margin as the tolerable excess span loss with respect to fiber loss:

In traditional links based on dispersion management, the system margin strongly depends on the dispersion map, modulation format and data-rate, therefore it cannot be used as universal parameter to weight the performance of a fiber type. If we focus our analysis on transmission based on coherently-received modulation formats, we know that the optimal links are uncompensated [4,5]. Therefore, we can suppose to use the system margin as quality parameter for a specified fiber type. In this paper, we propose to define as fiber figure of merit (FoM) for uncompensated links transmitting coherently-received modulation formats, the system margin, i.e.:

In general, the FoM is difficult to be interpreted as an absolute quality parameter, therefore, it can be useful to refer to a differential FoM (ΔFoM) defined with respect to a reference fiber as

3. Closed-form expression for channel spacing equal to the symbol rate

The FoM definition proposed in Eq. (2) has a general validity, provided that it is applied to uncompensated links transmitting coherently-received modulation formats. In general, the FoM can be estimated by simulation or measurements. In this section, we demonstrate that a closed-form expression for the FoM can be derived for multichannel transmission with channel spacing equal to the symbol rate R_s.

We focus our analysis on multi-span systems based on EDFA lumped amplification completely recovering span loss. According to [7], the performance of modulation formats coherently-received over uncompensated links can be inferred from an effective optical signal-to-noise ratio OSNR_NL that accounts for nonlinear interference (NLI) as additional Gaussian noise with equivalent noise power P_NLI, so that:

The amount of ASE noise at the receiver (Rx) is

Substituting Eqs. (5) and (6) into the expression of OSNR_NL, the span loss A resulting for a given target OSNR_target can be evaluated as a function of all system parameters. Maximizing A with respect to the transmitted power P_Tx,ch, the following expression for the maximum span budget, valid for practical scenarios with A_max >> 1, is obtained:

According to definition reported in Eq. (2), using Eq. (8) we can express the FoM in the following closed-form:

Note that various system parameters (F, N_s, OSNR_target and the span length L_s) giving the first factor of Eq. (8) do not appear in Eq. (9) because they can be neglected as they exactly cancel out when calculating $\Delta FoM$. Analyzing in detail Eq. (9), we observe that for a large occupied bandwidth B_WDM = N_ch·R_s (e.g., C-band and R_s = 30 Gbaud, means N_ch≈160), log(πN_ch²)>>log(|β₂|L_effR_s²). Hence, the term log(|β₂|L_effR_s²) becomes negligible, and the term R_s²/log(πN_ch²) is fiber-independent and cancels out when calculating $\Delta FoM$. As a result, for practical scenarios using several channels, we can re-write the FoM expression in more compact form:

4. Earlier FoM definitions and fiber comparison

After introducing the original closed-form expression for the FoM (Eqs. (8) and (9)), we compared it with two earlier FoMs proposed in the literature [1,2]:

Note that their original expressions used the effective area (A_eff) rather than the nonlinear parameter γ = 2π·n₂/(λ·A_eff). We have replaced A_eff with γ to ease the comparison with the one we propose. FoM₁ takes into account only the scaling in the nonlinear coefficient together with the difference in span loss. FoM₂ is an upgrade of FoM₁ considering also the interplay between nonlinearity and loss through the effective length. The FoM we propose in Eq. (10) is therefore a further upgrade that takes into account the simultaneous effect of nonlinearity and fiber dispersion, together with fiber loss.

We compared the three FoMs considering different fiber types: a standard single mode fiber (SSMF), a pure silica core fiber (PSCF) and a non-zero dispersion-shifted fiber (NZDSF). Fiber parameters are displayed in Table 1 together with ΔFoM values, where SSMF has been selected as the reference fiber. Chromatic dispersion of the NZDSF is very low, therefore it can be considered as a worst-case.

Table 1. Fiber parameters and ΔFoM comparison with L_s = 100 km.

View Table

Note that for the FoM we propose, results are shown either using the simplified expression given by Eq. (10), or using the complete expression (Eq. (9)) with only 10 channels. It can be observed that for large dispersion (PSCF) there are no differences, while reducing dispersion a limited difference appears: −7 dB against −6.2 for NZDSF.

Analyzing Table 1, for the PSCF it can be observed that the three ΔFoMs present small differences limited within 0.7 dB. While, for the NZDSF, whose dispersion value is much smaller than the reference fiber (SSMF) one, ΔFoM₁ and ΔFoM₂ display a mismatch larger than 4 dB with respect to the ΔFoM we propose. Such a large difference confirms the need for the inclusion of the dispersion parameters in a FoM definition that allows fair comparison between fibers presenting large differences in transmission parameters.

In order to highlight the dependence of ΔFoM on fiber dispersion, in Fig. 1 we plotted it with respect to β₂ for three different pairs of (α, γ) values, again taking as reference the SSMF of Table 1. It can be clearly observed that on the dispersion range of practical transmission fibers the FoM varies over a range of 6 dB independently of the value of nonlinear and loss coefficients. This plot gives a further confirmation of the need to properly include dispersion in a quality parameter allowing comparison between fiber types.

 

Fig. 1 ΔFoM as a function of dispersion, for span length L_s = 100 km, with fiber loss and non-linearity as parameters. Labeled dots represent the fibers listed in Table 1

Download Full Size | PDF

5. Simulative validations

In order to validate the proposed FoM definition, we carried out an analysis aimed at calculating by simulation the FoM for different fiber types and comparing such results with the closed-form expressions of Eqs. (8) and (9). For simulations we used the same set-up described in [8] with Δf = R_s in order to have a reference system scenario used also for the experimental validation that is presented in Sec. 7. The channel comb was made of 10 optically-shaped Nyquist-WDM PM-QPSK channels at R_S = 30 Gbaud (120 Gbps) spaced Δf = R_S. The link was a 8 spans uncompensated link with each span made of L_S = 100 km of fiber, a variable optical attenuator (VOA) and an EDFA completely recovering the span loss.

We considered the three fiber types (SSMF as reference, PSCF and NZDSF) whose parameters are reported in Table 1. The target BER was established to be BER_target = 3·10⁻³ corresponding to OSNR_target = 16.3 dB that includes realistic Tx impairments and crosstalk by fitting to the back-to-back performance of the experimental set-up.

For each considered fiber type we swept the transmitted power per channel P_tx,ch from −5 up to + 7 dBm and varied the attenuation of the VOA in order to evaluate the maximum span loss, the span budget A_max, ensuring the transmission operating below BER_target after 8 spans. Results are plotted in Fig. 2 for the three fiber types. The plotted curves display the expected qualitative parabolic behavior characterized by an optimal power representing the best trade-off between advantages of power enlargement ad detrimental effects of nonlinearities. Also the hierarchy between fibers is the expected one showing performance advantages with the increasing of fiber dispersion due to beneficial effects of chromatic dispersion in mitigating nonlinearities.

 

Fig. 2 Simulative results of maximum span budget vs. the power per channel for the fibers whose parameters are listed in Table 1 used in the considered 8 spans, PM-QPSK, R_S = 30 Gbaud,10 channel system with Δf = R_s. Evaluations of FoM according to Eq. (2) are reported on the graph.

Download Full Size | PDF

From a quantitative evaluation of maxima for each curve, and with the knowledge of the fiber loss parameter and span length, the FoM can be easily evaluated using Eq. (2). Calculations are reported on the graph showing FoM = 12.1 dB for the PSCF, FoM = 8.0 dB for the SSMF and FoM = 1.7 dB for the NZDSF. Considering the SSMF as a reference, we can evaluate ΔFoM, obtaining ΔFoM = 4.1 dB for the PSCF and ΔFoM = −6.3 dB for the NZDSF: results in excellent agreement with the ones based on the closed-form expressions presented in Table 1. Such an agreement validates the proposed FoM for Δf = R_s and confirms the need of inclusion of dispersion in giving a proper hierarchy to fiber types used to transmit coherently-received modulation formats on uncompensated links.

6. Extension of FoM definition to Δf > R_s

The proposed closed-form expression of FoM is so far proposed and validated only for links operated at the Nyquist limit, i.e., multi-channel systems based on Δf = R_S. The reason is that for such scenarios the NLI presents an easy-to-be-handled closed-form expression [7].

For larger channel spacings there indeed exists an expression for the NLI that still is a noise-like Gaussian process [9], but it does not have a closed-form, therefore it is difficult to infer the general trends with respect to the system parameters.

Therefore, we followed a heuristic approach and tested by simulation the applicability of the proposed FoM definition to channel spacings larger than the symbol rate. To pursue such an objective, we considered the same system scenario used for the simulative validation presented in Sec. 5 and redid simulations enlarging the channel spacing from 30 GHz (Δf = R_S) up to 50 GHz (Δf = 5/3·R_S). For each considered scenario, we swept the transmitted power per channel and varied the VOA level in order to estimate by simulation the maximum span budget at the link receiver for BER_target = 3·10⁻³. Then, from the collected results we were able to evaluate the FoM according to Eq. (2) for each scenario, and considering the SSMF as a reference we derived ΔFoM vs. Δf for PSCF and NZDSF.

Results of this analysis are pictorially presented in Fig. 3 as ΔFoM vs. Δf for the considered fibers. It can be clearly observed that the behavior of the curves is practically flat with respect to Δf and the constant level is in excellent agreement with the values predicted by Eqs. (8) and (9) reported in Table 1, last column. This result confirms the applicability of the proposed FoM definition also to system scenarios based on channel spacing larger than the Nyquist limit. And from a physical interpretation point of view, it says that the scaling of performance hierarchy between fibers with respect to loss, dispersion and nonlinearity is independent of the channel spacing.

 

Fig. 3 Simulative evaluations of ΔFoM for the fibers whose parameters are listed in Table 1 used in the considered 8 spans, PM-QPSK, R_S = 30 Gbaud,10 channels system with Δf = 1·R_s up to Δf = 5/3·R_s

Download Full Size | PDF

7. Experimental validation

As a further validation, we carried out experiments aiming at measuring the maximum achievable span budget and consequently evaluate the FoM. The experimental set-up was the same as described in [8] with ten 30 Gbaud (120 Gbps) PM-QPSK optically-shaped channels.

We tested experimentally SSMF and NZDSF with parameters specified in Table 1. Span lengths were L_s = 102 km (SSMF) and L_s = 100 km (NZDSF). In order to reduce the linear crosstalk between channels, we did not consider the Nyquist limit case and we used a channel spacing Δf = 1.1·R_s, i.e., 33 GHz.

Results in terms of maximum span budget vs. P_Tx for BER = 3·10⁻³, after 8 spans propagation, are shown in Fig. 4b together with the analytical prediction based on the NLI model [9]. The measured maximum span budgets were 31.2 dB and 25.7 dB, for SSMF and NZDSF, respectively. In Fig. 4a simulative results are presented for the same scenario, i.e., for Δf = 33 GHz, for three fiber types. The excellent agreement between simulations, experiments and theory can be clearly observed giving a further cross-validation of both theoretical predictions and simulative algorithms. Calculating from the experimental results the system margin difference according to Eq. (1), we obtained a $\Delta FoM$ of −6.1 dB for NZDSF vs. SSMF. Comparing this measurement with the analytical result of Table 1 (ΔFoM = −6.2 dB) and the simulative calculation (ΔFoM = −6.3 dB), we can observe that differences are of the order of fractions of dBs, i.e., they are within the inaccuracy of both experiments and simulations. Hence, we can conclude that also the experiments confirm the validity of the proposed FoM within a scenario of channel spacing larger than the Nyquist limit. Note that, instead, the ΔFoMs calculated according to FoM₁ and FoM₂ definitions are −3.0 dB and −2.8 dB, respectively, as reported in Table 1. Clearly, they greatly underestimate propagation penalty confirming the need of inclusion of dispersion effect in FoM definition in order to fairly compare fiber types over a wide range of chromatic dispersion values.

 

Fig. 4 Simulative (a) and experimental (b) evaluations of maximum span budget for the fibers whose parameters are listed in Table 1 used in the considered 8 spans link, PM-QPSK, R_S = 30 Gbaud,10 channels system with Δf = 1.1·R_s (33 GHz). Together with experimental results plotted as points (b), NLI model [9] results are plotted as continuous lines.

Download Full Size | PDF

7. Conclusions

We formalized a novel definition of fiber figure of merit based on the system margin of multi-span uncompensated links transmitting coherently received modulation formats extending results presented in [6]. On the base of the model of NLI presented in [7], we introduced a closed-form expression for the FoM. We validated it by simulation and experimentally and we proved it can be applied also to system scenarios based on channel spacings larger than the Nyquist limit.

We compared the proposed FoM to earlier definitions proposed in the technical literature showing how it is fundamental to include the role of fiber dispersion within a figure of merit allowing a fair comparison between fiber types over a wide range of chromatic dispersion.