## Abstract

A theoretical investigation of bidirectionally dual-order pumped distributed Raman amplifiers is presented in detail, and comparisons with other Raman amplification schemes, i.e., bidirectional first-order pumping and Raman-plus-erbium-doped fiber hybrid amplification, are carried out, for the first time to the authors’ knowledge, at identical nonlinear phase shifts. The results show that symmetric bidirectional dual-order pumping can achieve the best optical signal-to-noise ratio performance by appropriate choice of the second-order pump wavelength and second-to-first-order pump power ratio for both short- and long-span conditions, which will be helpful for designing long-haul transmission systems.

©2004 Optical Society of America

## 1. Introduction

Distributed fiber Raman amplifiers have become key devices in broadband, long-haul optical communication systems and are attracting increasing attention because of their flat and seamless bandwidth, outstanding noise performance, and flexible applications [1]. According to the conclusions of Perlin and Winful, a more nearly uniform power distribution will lead to better optical signal-to-noise ratio (OSNR) performance at constant nonlinear phase shift, which shows the advantage of utilizing distributed Raman amplifiers (DRAs) [2]. Generally, backward first-order pumping is adopted to suppress the pump-to-signal relative intensity noise (RIN) transfer [3]. However, several new pumping schemes, such as bidirectional pumping and high-order pumping [4–6], have been put to use recently because of improved performance of Raman pump diodes. These configurations can achieve flatter power distributions than backward pumping, and thus they can produce lower amplified spontaneous emission (ASE) output at identical nonlinearity. Obviously, the uniformity of signal power distributions as well as the ASE performance will be further improved when bidirectional, dual-order Raman pumping (BiDP) is used. In the research reported in Ref. 7, a quasi-constant power distribution was achieved in an 80-km transmission by use of a BiDP structure, and the output ASE noise was proved to be less than in other pumping schemes. And recently an efficient method for suppressing RIN transfer in forward dual-order Raman pumping schemes was proposed that makes BiDP more practical and reliable [8].

Moreover, by using proper Raman pumping methods, one can enlarge the transmission span length drastically to reduce system costs without significantly sacrificing the quality of the OSNR. Long-haul transmission systems with span lengths as long as 160 and 200 km have been reported when Raman plus erbium-doped fiber amplifier (Raman–EDFA) hybrid amplification and bidirectional first-order pumping (BiFP), respectively, were used [9, 10]. Unlike in short-span conditions, double Rayleigh backscattering (DRB) of the signal channel cannot be ignored in long-span conditions. DRB noise will increase rapidly with span length and with on–off Raman gain and may degrade receiver sensitivity considerably, which makes conventional backward pumping or single-directional high-order pumping unsuitable. In Ref. 11 we showed that a balanced BiFP scheme has a better OSNR performance than hybrid amplification for most long-span conditions at identical nonlinearity. However, a performance comparison of BiFP and BiDP for both short-span and long-span applications has still not been reported.

In this paper a theoretical investigation of BiDP and of its optimization is presented in detail, and comparisons among BiDP, BiFP, and hybrid amplification for short- and long-span conditions at constant fiber nonlinearity are reported. The results show that a symmetric BiDP structure can achieve the best OSNR performance for most conditions by appropriate choice of the second-order pump wavelength and second-to-first-order pump power ratio. In addition, some useful guidelines are proposed through extensive numerical simulations that we believe will be helpful for designing bidirectionally dual-order pumped Raman amplifiers as well as long-haul transmission systems.

## 2. Configuration and theory model

Three amplification structures (designated types 1, 2, and 3, which are BiDP, BiFP, and backward-pumped Raman–EDFA hybrid schemes, respectively, are shown in Fig. 1. The effects of dispersion compensation modules on noise performance are not considered. For type 1, the input power of the first-order pump is defined as *P*
_{1} , whereas *P*
_{2} denotes the second-order pump power. Subscripts *f* and *b* represent forward and backward pumps, respectively (as they do for type 2). The theoretical model of DRAs is based on the coupled Raman power equations that govern the effects of Raman interactions, the interdependence of ASE and temperature, and amplified multi-Rayleigh scattering .The basic equation can be written as [12]

$$\pm \mathit{hv}\sum _{\xi >v}\frac{{g}_{r}(v-\xi )}{{A}_{\mathit{eff}}(\xi )}[{P}^{\pm}(z,\xi )+{P}^{\mp}(z,\xi )]\xb7[1+\frac{1}{{e}^{\frac{h(\xi -v)}{kT}}-1}]\Delta v$$

$$\mp {P}^{\pm}(z,v)\xb7\sum _{\xi >v}\frac{v}{\xi}\xb7\frac{{g}_{r}(v-\xi )}{{A}_{\mathit{eff}}{(v)K}_{\mathit{eff}}}\xb7[{P}^{\pm}(z,\xi )+{P}^{\mp}(z,\xi )]$$

$$\mp 2\mathit{hv}{P}^{\pm}(z,v)\xb7\sum _{\xi <v}\frac{{g}_{r}(v-\xi )}{{A}_{\mathit{eff}}(v)}\xb7[1+\frac{1}{{e}^{\frac{h(v-\zeta )}{jkT}}-1}]\Delta v,$$

where *v* and *ξ* denote optical frequencies, *P*
^{+}(*z*, *v*) and *P*
^{-} (*z*, *v*) are forward- and backward-propagating optical power, respectively, within infinitesimal bandwidths about *v*, and *α*(*v*) and *R*_{s}
(*v*) are the fiber attenuation and the backward Rayleigh scattering coefficient, respectively, at frequency *v*. *h*, *K*, and *T* are Planck’s constant, Boltzmann’s constant, and temperature in degrees Kelvin, respectively. 12.5 GHz (0.1 nm) is the effective spontaneous-emission bandwidth. In addition, *K*_{eff}
is the polarization factor between the pump and the Stokes signals, and fully scrambled polarization states are assumed in the transmission fiber; i.e., *K*_{eff}
= 2. Finally, *g*_{r}
(*v* - *ξ*) is the Raman gain coefficient from a higher pump frequency *v* to a lower Stokes frequency *ξ*, which depends significantly on pump frequency *v* and frequency shift *v* - *ξ* ; *A*_{eff}
(*v*) represents the effective area of optical fiber at frequency *v*, which can be calculated approximately by use of a Gaussian mode-field distribution in single-mode fiber. On the right-hand side of Eq. (1), the second term describes the influence of Rayleigh scattering, the fourth term is the growth of spontaneous Raman emission and thermal noise, and the last two terms are related to the depletion of pumps and signals. An average-power analysis is used to simplify the ordinary differential equations given above [13], and we adopt Broyden’s method [14] to find the roots of the boundary-value problems, which can ensure efficient and global convergence, especially for long-span conditions.

Here DRB is assumed to be an incoherent, ASE-like noise within a receiver bandwidth, in accordance with Ref. 15. Hence the output OSNR including DRB noise for types 1 and 2 can easily be written as

where *P*_{s}
(0) represents the input signal power, *G*_{Raman}
is Raman on–off gain, and *L* is the span length. *T* = *α*_{s}*L* denotes the total loss of the fiber span. *P*_{ASE}
(*L*) represents output ASE power within 0.1-nm bandwidth, and *P*_{DRB}
(*L*) is the output DRB power. For type 3, the expression for the OSNR should be changed to

where *G*_{EDFA}
denotes the EDFA output gain and *n*_{sp}
is the spontaneous-emission factor. In Eq. (3) the DRB noise generated by the EDFA is ignored because *G*_{EDFA}
is not too large. Furthermore, nonlinear effects in DRAs are evaluated by a total nonlinear phase shift along the span length, which is defined as *K*_{NL}
= *γ*${\int}_{0}^{L}$
*P*_{s}
(*z*)*dz* [4], where *γ* is the fiber nonlinearity coefficient. This parameter has proved to be simple and effective for assessment of the nonlinear performance that corresponds to the optimal dispersion management [16]. Therefore the output OSNRs of the three configurations should be optimized and compared at identical values of *K*_{NL}
, i.e., the same span integral of signal power rather than constant input signal power. Here, one can keep *K*_{NL}
unchanged by altering the value of the input power for a given span length. In this paper, a reference *K*_{NL}
value is defined as *K*_{NL-ref}
, and the target values of *K*_{NL}
for different span lengths can be determined by *K*_{NL-ref}
/*L*
_{0} to ensure the same average nonlinearity, where *L*
_{0}= 100 km. In addition, the span’s net gain is fixed at 0 dB to produce transparent transmission, and we obtain the required gain by modifying Raman pump power iteratively. It should be pointed out that the OSNR performance of BiDP as well as of the BiFP will be reduced as the net gain diverges from 0 dB owing to the increased nonuniformity of the signal distribution. However, for general applications, a net gain of approximately 0 dB can always be satisfied.

## 3. Optimization and numerical results

To start with, each amplification type should be optimized individually to approach the best output OSNR at constant *K*_{NL}
for a given span length. For simplicity single-channel amplification is assumed in our calculations, and the first- and second-order pump wavelengths are defined as *λ*
_{P1} and *λ*
_{P2}, respectively, the signal wavelength is 1550 nm, and *λ*
_{P1} is fixed at 1450 nm for maximum Raman efficiency. For type 2, both the minimum output ASE and DRB are achieved when 50% forward pumping is used, and, for type 3, one can derive the best noise characteristics simply by changing the Raman gain/total gain ratio [11]. However, it is more difficult to optimize BiDP scheme than other types of pumping because BiDP has four degrees of freedom to modify, i.e., the wavelength of the second-order pump (*λ*
_{P2} ), the forward and backward second-to-first-order pump ratios (defined as *r*_{f}
= *P*
_{2f} / *P*
_{1f} and *r*_{b}
= *P*
_{2b} / *P*
_{1b}, respectively), and the forward-to-total-first-order pump ratio [defined as *r*
_{1} = *P*
_{1f} /(*P*
_{1f} + *P*
_{1b})]. In the following subsections, we optimize type 1 and compare it in short- as well as long-span conditions.

#### 3.1 Short-span applications

When span length is relatively short (≤ 80 km), DRB noise is insignificant compared with ASE power and can be ignored; as a result, only ASE noise is considered here. To find the optimal design of type 1, we calculated contour maps of output OSNR versus *r*_{b}
and *r*
_{1} (Fig. 2; *r*_{f}
is set at 13 and 20 dB and *λ*
_{P2} is fixed at 1365 nm), where the fiber loss is 0.192 dB/km at 550 nm, 0.24 dB/km at 1450 nm, and 0.32 dB/km at 365 nm, the peak Raman coefficient is 0.62 W^{-1} km^{-1}, *γ* = 1.6 W^{-1} km, *L* = 80 km, and *K*_{NL-ref}
is set at 0.09 rad (which equals the nonlinear phase shift value of a 100-km passive transmission with 4-dBm input power). It can be observed that the optimum OSNR is obtained when *r*_{b}
= *r*_{f}
as well as *r*
_{1} = 0.5 , which means that a symmetric BiDP structure (*P*
_{1f} = *P*
_{1b}, *P*
_{2f} = *P*
_{2b} , and *P*
_{1f} = *P*
_{2f}) can lead to the best output OSNR. In Fig. 3 the ratio of output OSNR to *r*_{f}
has been calculated, which shows that an approximately 19-dB (80:1) second-to-first-order pump ratio will cause the optimal noise performance and that the output OSNR will decrease when larger *r*_{f}
is applied, because the excursion of the signal distribution tends to increase. The influence of different values of *λ*
_{P2} on optimal OSNR can be neglected in short-span conditions through our computations, however, and thus *λ*
_{P2} is commonly chosen at 1365 nm to produce the maximum Raman efficiency, as reported in several published papers [7, 17]. In addition, when DRB noise is not considered, only all-Raman amplification can reach the optimal performance for type 3; i.e., *G*_{EDFA}
=0 dB.

The optimized OSNRs of the three types relative to *K*_{NL}
and fiber loss are shown in Fig. 4 for an 80-km span length. It is noted that BiDP is superior to the other two types, especially when fiber loss becomes larger (which causes more-severe power nonuniformity along the fiber span). The reason can be found in Fig. 5, which gives the power distributions of the three types, and it is obvious that type 1 has the most consistent power curve, as expected (there is less than a 0.8-dB power ripple).

#### 3.2 Long-span applications

With the increase of the span length and the Raman on–off gain, DRB cannot be ignored and will become a major component of the output noise power. In long-span (>120-km) conditions, four degrees of freedom of the BiDP scheme should be altered carefully. Similarly, the contour maps of OSNR versus *r*_{f}
and *r*
_{1} are calculated in Fig. 6 (*r*_{b}
is set at 13 and 20 dB, whereas *λ*
_{P2} is fixed at 1365 nm), where *L* = 160 km and *R*_{s}
= 8 × 10^{-5} km^{-1} , and other parameters are the same as those used in Fig. 2. In Fig. 5, the conclusion that symmetric BiDP (*P*
_{1f} = *P*
_{1b} , *P*
_{2f} = *P*
_{2b} , and *P*
_{1f} = *P*
_{2f}) results in optimal performance still holds, even including DRB noise, mainly because balanced power evolutions can also approach the minimum DRB output [18]. And then the other two parameters, *λ*
_{P2} and *r* = *r*_{f}
= *r*_{b}
, should be adjusted properly to further improve the OSNR performance through appropriate
changes in the power distributions.

In Fig. 7 the output OSNR is compared with *r* and *λ*
_{P2} (*L* = 200 km), and it is straightforward to find that different values of *λ*
_{P2} have relatively large influence on output OSNR. As discussed above, for short-span applications for which DRB noise can be neglected, the second-order pump wavelength is generally set at 1365 nm to achieve the maximum Raman gain and pump conversion efficiency (for 1450-nm first-order pump wavelength) because various values of *λ*
_{P2} have only a small influence on the optimal OSNR. However, for long-span situations the best OSNR performance cannot be derived at identical *K*_{NL}
when *λ*
_{P2} =1365nm, since the DRB noise is relatively large in this case. Although the power excursion is reduced and ASE performance is improved, the output OSNR is limited by DRB noise, as shown in Fig. 7(b). As *λ*
_{P2} gradually increases, better DRB and OSNR performance can be derived owing to the changes in the signal power distribution caused by the second-order pumping. But, when *λ*
_{P2} is too large, the effects of dual-order pumping will be diminished significantly because of the small gain experienced by the first-order pump, which will degrade the OSNR performance instead.

Our computations show that the optimal *λ*
_{P2} is approximately 1395 nm, and, when the DRB power increases, the optimal *λ*
_{P2} will shift to a slightly longer wavelength, as shown in Fig. 8. The result described above is also valid for other span lengths and fiber types. Moreover, the *P*
_{2} /*P*
_{1} ratio should be large enough to produce a satisfying performance compared with that achieved in short-span conditions. For a 200-km span length a satisfactory OSNR can be achieved when *r*_{f}
= *r*
_{b1} is 36 dB or so (larger ratios leads to only small improvements and larger cost), which corresponds to *P*
^{1} = 0.47 mW and *P*
_{2} = 1.78 W in this situation, as shown in Fig. 9. The optimized OSNR curves of the three types are plotted in Fig. 10 as functions of span length (here *n*_{sp}
= 1.4 for type 3, which corresponds to a 4.5-dB EDFA noise figure, and the effects of various *n*_{sp}
values on optimal OSNR can also be neglected because the noise performance of a cascaded amplifier is determined mainly by the first stage), which clearly shows that BiDP is superior to the other two commonly used schemes, as expected for typical parameters, and that the OSNR enhancement will increase with span length. From Fig. 9, a greater than 2-dB total OSNR as well as a 3-dB improvement in ASE performance can be observed compared with values for the BiFP scheme when the length is 20km.

For more-general conclusions, the curves for OSNR versus *K*_{NL}
, fiber loss, and the Rayleigh scattering factor are shown in Figs. 10(a), 10(b), and 10(c), respectively. As is shown in Fig. 10(a), type 1 has better performance than the other types, especially when *L* or the fiber loss becomes larger, because type 1 is more tolerant of power-nonuniformity-induced ASE penalties. Increasing *K*_{NL}
and the fiber Rayleigh backscattering factor, however, will enlarge the ratio of the DRB power to the total noise power, which will cancel the advantage of type 1 to some extent or even make type 3 become the best choice [11]. In fact, for most transmission applications, a low Rayleigh backscattering factor, small *K*_{NL}
, moderate fiber loss, and a span net gain of approximately 0 dB can always be satisfied; therefore BiDP is always the best choice for long-span systems.

Finally, it should be pointed out that the conclusions given above are also valid for multichannel conditions when the signal gain band is not very wide (20–30 nm or so), or the advantages of high-order pumping will be canceled at the band edges (in fact, the gain bandwidth of second-order pumping schemes is always less than 40 nm in the experiments described here). Moreover, further investigations of power transmission of BiDP schemes are required for signal channel add–drop situations.

## 4. Conclusions

To summarize, a bidirectional dual-order Raman pumping scheme has been numerically optimized and compared with other widely used amplification configurations for both short-and long-span transmission systems. The results show that symmetric BiDP exhibits the optimal OSNR performance at identical nonlinearity for most conditions. For short-span conditions, a 1365-nm second-order pump plus a relatively small second-to-first-order pump power ratio can achieve a satisfactory output OSNR. However, the best noise performance requires a second-order pump wavelength of approximately 1395 nm and a much larger
*P*
_{2}/*P*
_{1} ratio. We believe that these conclusions will be helpful in the design of Raman-assisted long-haul transmission systems.

## Acknowledgments

This research is supported by the National 863 High Technology Project (China; grant 2001AA122012) and the paper foundation of Beijing Jiaotong University.

## References and Links

**1. **M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Sel. Top. Quantum Electron. **8**, 548–559 (2002). [CrossRef]

**2. **V. E. Perlin and H. G. Winful, “Optimizing the noise performance of broadband WDM systems with distributed Raman amplification,” IEEE Photon. Technol. Lett. **14**, 1199–1201 (2002). [CrossRef]

**3. **J. S. Wei, D. L. Butler, M. F. V. Leeuwen, L. G. Joneckis, and J. Goldhar, “Crosstalk bandwidth in backward pumped fiber Raman amplifiers,” IEEE Photon. Technol. Lett. **11**, 1417–1419 (1999). [CrossRef]

**4. **Z. Tong, H. Wei, and S. Jian, “Theoretical investigation and optimization of bidirectionally pumped broadband fiber Raman amplifiers,” Opt. Commun. **217**, 401–413 (2003). [CrossRef]

**5. **K. Rottwitt, A. Stentz, T. Nielson, P. Hansen, K. Feder, and K. Walker, “Transparent 80km bi-directionally pumped distributed Raman amplifier with second order pumping,” in *Proc. European Conference on Optical Communication (ECOC’99*, Institute of Electrical and Electronics Engineers, Nice, France), p. II-144 (1999).

**6. **J. C. Bouteiller, K. Brar, J. Bromage, S. Radic, and C. Headley, “Dual-order Raman pump,” IEEE Photon. Technol. Lett. **15**, 212–214 (2003). [CrossRef]

**7. **J. C. Bouteiller, K. Brar, and C. Headley, “Quasi-constant signal power transmission,” in *Proc. European Conference on Optical Communication (ECOC’02*, Institute of Electrical and Electronics Engineers, Copenhagen, Denmark), symposium 3.04 (2002).

**8. **M. D. Mermelstein, K. Brar, and C. Headley, “RIN transfer suppression technique for dual-order Raman pumping schemes,” IEEE Photon. Technol. Lett. **15**, 1354–1356 (2003). [CrossRef]

**9. **Y. Zhu, W. S. Lee, C. Scahill, C. Fludger, D. Watley, M. Jones, J. Homan, B. Shaw, and A. Hadjifotiou, “1.28Tbit/s (32×40Gbit/s) transmission over 1000km NDSF employing distributed Raman amplification and active gain flattening,” Electron. Lett. **37**, 43–45 (2001). [CrossRef]

**10. **J. Bromage, J.-C. Bouteiller, H. J. Thiele, K. Brar, L. E. Nelson, S. Stulz, C. Headley, J. Kim, A. Klein, G. Baynham, L. V. Jørgensen, L. Grüner-Nielsen, R. L. Lingle Jr., and D. J. DiGiovanni, “High co-directional Raman gain for 200-km spans, enabling 40^{*}10.66Gb/s transmission over 2400km,” in *Optical Fiber Communication Conference (OFC)*, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D. C., 2003), paper PD24-1 (2003).

**11. **Z. Tong, H. Wei, and S. Jian, “Comparison of different Raman amplification schemes in long-span fiber transmission systems with double Rayleigh backscattering,” IEEE Photon. Technol. Lett. **15**, 1782–1784 (2003). [CrossRef]

**12. **S. Namiki and Y. Emori, “Ultrabroad-band Raman amplifiers pumped and gain-equaliazed by wavelength-division-multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. **7**, 3–16 (2001). [CrossRef]

**13. **B. Min, W. J. Lee, and N. Park, “Efficient formulation of Raman amplifier propagation equations with average power analysis,” IEEE Photon. Technol. Lett. **12**, 1486–1488 (2000). [CrossRef]

**14. **R. L. Burden and J. D. Faires, *Numerical Analysis* (7th edition, Brooks-Cole, 2001), Chap. 10.

**15. **R. Hainberger, T. Hoshida, T. Terahara, and H. Onaka, “Comparison of span configurations of Raman-amplified dispersion-managed fibers,” IEEE Photon. Technol. Lett. **14**, 471–473 (2002). [CrossRef]

**16. **Z. Tong, H. Wei, and S. Jian, “Impacts of SPM/XPM on multi-span transmission systems using distributed Raman amplification at identical nonlinear phase shift,” IEEE Photon. Technol. Lett. **16**, 933–935 (2004). [CrossRef]

**17. **C. Martinelli, D. Mongardien, J. C. Antona, C. Simnneau, and D. Bayart, “Analysis of bi-directional and second-order pumping in long-haul systems with distributed Raman amplification,” in *Proc. European Conference on Optical Communication (ECOC’02*, Institute of Electrical and Electronics Engineers, Copenhagen, Denmark*)*, p. 3.30 (2002).

**18. **M. Nissov, K. Rottwitt, H. D. Kidorf, and M. X. Ma, “Rayleigh crosstalk in long cascades of distributed unsaturated Raman amplifiers,” Electron. Lett. **35**, 997–998 (1999). [CrossRef]