1. Introduction

As digital coherent transmission technologies advance and enhance the transmission capacity, nonlinear interference (NLI) in transmission fiber gains increasing importance because it obstructs capacity improvement to approach the fundamental linear Shannon limit [1,2]. For NLI in already deployed fibers, nonlinearity compensation (NLC) methods such as digital back propagation (DBP) [3,4] and optical phase conjugation (OPC) [5,6] are effective to mitigate NLI. On the other hand, for minimizing NLI in newly deployed systems, it should be beneficial to employ advanced fibers having lower nonlinearity in addition to lower loss [7–11]. The merits of such advanced fibers can be analyzed based on the Gaussian noise model (GN model) theory [12] that formulates the power spectral density of NLI as a function of fiber characteristics. By treating NLI as an additional noise, the theory derives generalized optical signal to noise ratio (OSNR), and predicts the optimal signal power that maximizes OSNR. Since the GN model theory predicts that NLI in a DWDM system accumulates almost linearly to the number of amplified spans, improvement in OSNR becomes equivalent to improvement in transmission distance. Consequently the relative improvement in OSNR or transmission distance is quantified as fiber figure of merit (FOM), which is validated by an excellent agreement with the experimental results [7,13–15]. The FOM theory is now utilized in a number of works on development of advanced fibers [7–9], and should also be useful for system designers in choosing fibers. In this paper, we review our theoretical works on FOM formulation and fiber design based on the FOM theory for various cases of system configuration.

2. Fiber figure of merit (FOM)

For large capacity long haul transmission, digital coherent transmission over a dispersion uncompensated link provides great benefits such as high spectral efficiency of the signal with optimal modulation format and low nonlinearity of the transmission link. Another important benefit is that NLI can be predicted by the GN model theory [12] and improvement in OSNR and transmission distance can be predicted by the fiber figure of merit (FOM) theory [7,13–15] shown in what follows.

We assume an uncompensated DWDM transmission link shown in Fig. 1 that is multiple spans composed of a transmission fiber and an EDFA without distributed Raman amplification, which is typical for the state-of-the-art submarine transmission systems. The loss of each span is caused by fiber loss and splice losses at both the ends of the transmission fiber, and is assumed to be compensated by the amplifier gain exactly. Figure 1 also summarizes the definitions of the symbols representing the physical parameters used in the FOM theory.

 

Fig. 1 Schematic model of a transmission link and definitions of the symbols.

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According to the GN model, NLI is treated as an additive noise so that OSNR is generalized as OSNR = P_ch / (P_ASE + P_NLI), where P_ASE and P_NLI is respectively the power of amplifier ASE and NLI at the launching end of the link, given by

Assuming a linear accumulation of NLI, OSNR also has linear dependence on N_s, so that OSNR and N_s are maximized at the same optimal signal power P_opt(dB), which is expressed in dB scale as,

In practice, a signal power of P_opt is sometimes not feasible because of limitation in the output power of the EDFA. In such cases with an arbitrary signal power P_ch = R⋅P_opt, FOM in Eq. (6) and OSNR_max(dB) in Eq. (7) are replaced by

3. Validation of FOM

The theory of FOM was validated based on the experimental results [13–15]. In ref [13], the Q factors in a 100G DP-QPSK transmission experiment over a standard single mode fiber (SSMF) with distances from 2,400 km to 7,800 km, and a large area pure silica core fiber (PSCF) with distances from 3,280 km to 9,840 km were reproduced from OSNR_R(dB) by Eq. (9) assuming the Q factor proportional to the OSNR_R(dB) in dB scale, a fiber-independent fitting parameter C₂ of 38.4 dB, and the splice loss estimated from the actual span loss. The results showed an excellent agreement with errors less than 0.3 dB. In addition, a fiber-independent fitting parameter C₁ was derived to be −6.6 dBm/ch from the measured Q factors as a function of P_ch.

It is further verified that even in the case with NLC by DBP, the Q factors in the experiment agreed well with the theoretical prediction by Eq. (9) using an NLI mitigation coefficient k_NLC = 0.6 [14].

In ref [15], the maximum distances in a 100G Nyquist WDM PM-QPSK transmission and two 100G standard WDM PM-QPSK transmission experiments over SSMF, PSCF and non-zero dispersion shifted fibers (NZDSF) were reproduced by the relative FOM formula equivalent to Eq. (6). Again, the results showed an excellent agreement with errors less than 0.4 dB. Consequently, we can say that a number of experimental results show the validity of the FOM theory.

4. Fiber design to maximize FOM

For maximizing the capacity at a given transmission distance, FOM_R should be maximized under constraints on the span length L and the upper limit in the signal power P_ch due to repeater performance and cost. As suggested by Eq. (8), a transmission fiber is better to have a low nonlinear index n₂, a large effective area A_eff, a large absolute dispersion |D|, a low fiber loss α, and a low splice loss α_sp. In most cases an advanced transmission fiber has a pure silica core with similar n₂ and loose confinement resulting in a similar D dominated by material, so that significant difference would be made by A_eff, α_sp and α. Since α_sp mostly correlates to A_eff, it is reasonable to map FOM_R as contour lines in a plane defined by A_eff and α, as shown in Fig. 2(a). In the calculation of Fig. 2 and the following figures, we assume pure silica core fibers with constant n₂ = 22 μm²/GW and D = 21 ps/nm/km.

 

Fig. 2 (a): Contour lines of FOM_R with values shown in the parentheses. The calculation conditions are: D_T = 10000km, L = 80km, P_ch ≤ −2dBm/ch and the relationship between α_sp and A_eff shown in (b), where a base splice loss α_sp0 = 0.02dB and an Aeff factor k = 1.1 are assumed to fit experimental results. The points corresponds to the experimental values in the literature.

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We also assume a relationship between α_sp and A_eff shown in Fig. 2(b), unless otherwise noted. Figure 2(a) also shows the performances of the commercial submarine fibers by plots. Although detailed comparison depends on the assumption on system configuration as shown later in this section, we can see that most of the advanced fibers having high FOM_R’s are categorized in a range of A_eff ≥ 130 μm² and α ≤ 0.158 dB/km.

The effects of increasing A_eff and decreasing α on FOM_R are shown in Fig. 3. We consider four typical structures having (A_eff, α) of A (150, 0.150), B (150, 0.160), C (110, 0.150), and D (110, 0.160). We further assume the transmission system conditions same as in Fig. 2(a) and the Q-value being proportional to the OSNR, and calculate expected Q-values as a function of P_ch, as shown in Fig. 3(b). Comparing A with B, or C with D, a decrease in α from 0.160 to 0.150 results in an almost constant improvement in Q for the low P_ch region where ASE is dominant and Q linearly increases with P_ch. This should be attributed to reduction in the span loss that results in reduction in the amplifier gain and P_ASE. In a high P_ch region where NLI is dominant, the difference caused by α tends to diminish.

 

Fig. 3 (a): Four example structures to illustrate the effects of A_eff and loss on FOM_R contour map same as Fig. 2(a). (b): Q value vs P_ch for the 4 structures shown in (a). The plots shows the maximum Q value at the optimal or the upper limit of signal power. The solid lines show valid Q values with P_ch below the upper limit of −2dBm/ch. The crosses show the conditions corresponding to P_opt.

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On the other hand, comparing A with C, or B with D, an increase in A_eff from 110 to 150 results in increases in P_opt and maximum Q values. This should be attributed to reduction in P_NLI that allows higher P_ch. In practice, P_ch is also limited by the upper limit due to system configuration, and merits by larger A_eff can be limited in such cases.

In order to improve FOM_R by increasing A_eff, it is most important to suppress increase in micro-bending loss, caused by coupling to higher order leaky modes due to random micro bending of fiber glass. Since micro-bending is caused by random lateral forces acting on coated optical fibers, mechanical characteristics of coatings are important. Figure 4 shows that micro bending loss increases with larger A_eff and it can be reduced using an improved coating having a lower Young’s modulus in the primary layer [8,20]. In order to enlarge A_eff further without increasing micro-bending loss, improvement in mechanical characteristics in coatings are important.

 

Fig. 4 Increase in micro-bending loss with larger A_eff for PSCF-110 with a conventional coating and PSCF-130 with an improved coating with a lower Young’s modulus primary layer. The typical schematic index profile of the PSCFs are also shown.

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In addition to micro-bending loss, macro-bending loss and dissimilar fiber splice loss can also limit the upper limit in A_eff. Regarding macro-bending loss, it is known that a depressed cladding profile shown in Fig. 4 and a shifted cutoff wavelength is effective to suppress it within a similar level with SSMF [8]. Since the cable cutoff wavelength typically has to be kept below the upper limit of 1530 nm set by ITU-T G.654 recommendation, larger A_eff is generally in trade-off against tighter manufacturing tolerance. Regarding dissimilar fiber splice loss α_sp to a standard 80µm²-A_eff fiber in repeaters, larger A_eff generally causes a higher splice loss as shown in Fig. 2(b). In order to reduce dissimilar fiber splice loss, it is effective to employ a ring core structure that reduces the mode field diameter while maintaining large A_eff [21], or to apply elaborated splicing techniques such as a bridge fiber [9] or tapering [10].

It is also worth noting that although FOM is an effective tool to compare fibers in terms of transmission performance, the results of comparison vary depending on assumed system configurations. Therefore, an advancement in system technology will affect the choice of optimal fiber. For example, increase in the upper limit of P_ch favors larger A_eff. As shown in Fig. 5(a), if we remove the upper limit of P_ch, the slopes of FOM_R contour lines in the large A_eff region get less steep so that a large A_eff contributes more to high FOM than in the case of limited P_ch.

 

Fig. 5 (a): The influence of the upper limit of P_ch on FOM. The solid lines show FOM_R at P_ch ≤ −2dBm/ch shown in Fig. 2 (a) and the dashed line shows FOM with unlimited P_ch. (b): The influence of the span length L on FOM. Note that the absolute FOM_R values are not always same between 80km and 90km at the crossing points of the contours.

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On the other hand, a longer span length favors lower loss. As shown in Fig. 5(b), if the span length is increased from 80km to 90km, the slopes of the FOM_R contour lines get steeper showing that a lower loss improves FOM rather than a larger A_eff.

An advance in splice technique will also affect the optimal choice of fiber. As shown in Fig. 6(a), if it could become practical to reduce dissimilar fiber splice loss between a large core fiber and a standard A_eff fiber in repeaters, for example by half in dB scale from the current losses shown in Fig. 2(b), the slopes of FOM_R contour lines in the large A_eff region get less steep similarly to the case in an increased P_ch shown in Fig. 5(a). Therefore, reduction in dissimilar fiber splice loss also results in an increased merit of large A_eff.

 

Fig. 6 (a): The effect of reduction in dissimilar fiber splice loss. The solid lines show FOM_R for the reference case with the splice loss shown in Fig. 2(a) and the dashed lines show that for the splice loss reduced to 50% of the reference case in dB scale. (b): The effect of application of NLI mitigation. The solid lines show FOM_R without NLI mitigation shown in Fig. 2(a) and the dashed lines show that with NLI mitigation of a 0.5dB SNR gain.

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Finally, if NLI mitigation technique such as DBP and OPC is employed and NLI power P_NLI is reduced, a lower loss would gain increased importance. As shown in Fig. 6(b), we assume that the peak SNR gain of 0.5 dB as a best case for single channel DBP [4], and also assume that the SNR gain is independent of the fiber characteristics. As a result, the slopes of the FOM_R contour lines get steeper similarly to the longer span cases shown in Fig. 5(b). Therefore, application of NLI mitigation results in an increased merit of lower loss.

5. Conclusions

In digital coherent transmission over uncompensated transmission links, we can apply the FOM theory to predict improvement in OSNR and transmission distance. Several preceding works showed the validity of the FOM theory by excellent agreements with the results of the transmission experiments. Using the FOM theory, one can quantitatively predict merits of advanced fibers with larger effective areas and lower losses. It should also be noted that such merits predicted by the FOM theory also depends on assumed system configuration. For example, higher channel power and lower splice loss will favor larger effective area, and longer span length and adoption of NLI mitigation will favor lower loss.