## Abstract

Fiber figure of merit (FOM), derived from the GN-model theory and validated by several experiments, can predict improvement in OSNR or transmission distance using advanced fibers. We review the FOM theory and present design results of optimal fiber for large capacity long haul transmission, showing variation in design results according to system configuration.

© 2017 Optical Society of America

## 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.

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

*N*

_{s}that is a valid approximation in DWDM systems [12]. It should be pointed out that NLI accumulation is more generally described by

*P*

_{NLI}∝

*N*

_{s}

^{1+ε}, and that even in that case the FOM theory presented below is still accurate in comparing systems having the same distance.

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,

*C*

_{1}= (10/3)⋅log{

*F*/(2k

_{NLC}η)} is a term independent of the fiber characteristics. Note that each of the symbols introduced in Fig. 1 is a number without dimension defined as a ratio of a certain physical quantity to a unit quantity, so that it can be used as argument in log function. Consequently, the maximum OSNR at a given

*N*

_{s}and the maximum

*N*

_{s}at a given OSNR are

*C*

_{2}= (−10/3)⋅log

_{10}{(27/4)

*F*

^{2}k

_{NLC}η} is a term independent of the fiber characteristics. We define FOM in dB scale as

*L*and

*D*

_{T}are defined as dimension-less numbers in Fig. 1 so that they can become arguments of log functions.

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 *C*_{2} 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 *C*_{1} 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 *n*_{2}, 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 *n*_{2} 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 *n*_{2} = 22 μm^{2}/GW and *D* = 21 ps/nm/km.

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 μm^{2} 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.

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.

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 80µm^{2}-*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}.

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}.

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.

## References and links

**1. **R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. **28**(4), 662–701 (2010). [CrossRef]

**2. **A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. **28**(4), 423–433 (2010). [CrossRef]

**3. **E. Ip and J. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. **26**(20), 3416–3425 (2008). [CrossRef]

**4. **R. Dar and P. J. Winzer, “On the limits of digital back-propagation in fully loaded WDM systems,” IEEE Photonics Technol. Lett. **28**(11), 1253–1256 (2016). [CrossRef]

**5. **A. D. Ellis, M. Tan, M. A. Iqbal, M. A. Z. Al-Khateeb, V. Gordienko, G. S. Mondaca, S. Fabbri, M. F. C. Stephens, M. E. McCarthy, A. Perentos, I. D. Phillips, D. Lavery, G. Liga, R. Maher, P. Harper, N. Doran, S. K. Turitsyn, S. Sygletos, and P. Bayvel, “4 Tb/s transmission reach enhancement using 10 × 400 Gb/s super-channels and polarization insensitive dual band optical phase conjugation,” J. Lightwave Technol. **34**(8), 1717–1723 (2016). [CrossRef]

**6. **T. Umeki, T. Kazama, A. Sano, K. Shibahara, K. Suzuki, M. Abe, H. Takenouchi, and Y. Miyamoto, “Simultaneous nonlinearity mitigation in 92 × 180-Gbit/s PDM-16QAM transmission over 3840 km using PPLN-based guard-band-less optical phase conjugation,” Opt. Express **24**(15), 16945–16951 (2016). [CrossRef] [PubMed]

**7. **M. Hirano, T. Haruna, Y. Tamura, T. Kawano, S. Ohnuki, Y. Yamamoto, Y. Koyano, and T. Sasaki, “Record low loss, record high FOM optical fiber with manufacturable process,” OFC 2013, paper PDP5A.7.

**8. **Y. Yamamoto, Y. Kawaguchi, and M. Hirano, “Low-loss and low-nonlinearity pure-silica-core fiber for C- and L-band broadband transmission,” J. Lightwave Technol. **34**(2), 321–326 (2016). [CrossRef]

**9. **S. Makovejs, C. C. Roberts, F. Palacios, H. B. Matthews, D. A. Lewis, D. T. Smith, P. G. Diehl, J. J. Johnson, J. D. Patterson, C. R. Towery and S. Y. Ten, “Record-low (0.1460 dB/km) attenuation ultra-large Aeff optical fiber for submarine applications,” OFC 2015, paper Th5A.2.

**10. **S. Makovejs, J. D. Downie, J. E. Hurley, J. S. Clark, I. Roudas, C. C. Roberts, H. B. Matthews, F. Palacios, D. A. Lewis, D. T. Smith, P. G. Diehl, J. J. Johnson, C. R. Towery, and S. Y. Ten, “Towards superior transmission performance in submarine systems: leveraging ultralow attenuation and large effective area,” J. Lightwave Technol. **34**(1), 114–120 (2016). [CrossRef]

**11. **D. Peckham, A. Klein, P. I. Borel, R. Jensen, O. Levring, K. Carlson, M. Yan, P. Wisk, D. Trevor, R. Lingle, Jr., A. McCurdy, B. Zhu, Y. Zou, R. Norris, B. Palsdottir and D. Vaidya, “Optimization of large area, low loss fiber designs for C+L band transmission,” OFC 2016, paper Tu3G.1.

**12. **P. Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. Lightwave Technol. **30**(24), 3857–3879 (2012). [CrossRef]

**13. **M. Hirano, Y. Yamamoto, V.A.J.M. Sleiffer and T. Sasaki, “Analytical OSNR formulation validated with 100G-WDM experiments and optimal subsea fiber proposal,” OFC2013, paper OTu2B.6.

**14. **Y. Yamamoto, M. Hirano, V.A.J.M. Sleiffer, and T. Sasaki, “Analytical OSNR formulation considering nonlinear compensation,” OECC 2013, paper WR4–3.

**15. **V. Curri, A. Carena, G. Bosco, P. Poggiolini, M. Hirano, Y. Yamamoto and F. Forghieri, “Fiber figure of merit based on maximum reach,” OFC2013, paper OTh3G.2.

**16. **H. Yamaguchi, Y. Yamamoto, T. Hasegawa, T. Kawano, M. Hirano and Y. Koyano, “Ultra-low loss and large Aeff Pure-silica core fiber advances,” SubOptic 2016, paper EC07.

**17. **Sumitomo Electric Industries, Ltd., “Sumitomo Submarine Optical Fibers–Field-proven over 3 decades–“, http://global-sei.com/fttx/images_n/TR-16042SumitomoSumarineFibers-1.pdf.

**18. ** Corning Incorporated, “Corning® Vascade® optical fiber”, https://www.corning.com/media/worldwide/coc/documents/Fiber/PI1445_07_14_English.pdf

**19. **O. F. S. Fitel, LLC, “TeraWave™ ocean optical fiber - SCUBA”, http://fiber-optic-catalog.ofsoptics.com/Asset/TeraWave-Scuba-Ocean-Fibers-fiber-150-web.pdf

**20. **Y. Yamamoto, M. Hirano, K. Kuwahara, and T. Sasaki, “OSNR-enhancing pure-silica-core fiber with large effective area and low attenuation,” OFC 2010, paper OTuI2.

**21. **M. Hirano, Y. Yamamoto, Y. Tamura, T. Haruna, and T. Sasaki, “Aeff enlarged pure-silica-core fiber having ring-core profile,” OFC 2012, paper OTh4I.2.