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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 4 — Feb. 23, 2004
  • pp: 545–550
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Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers

Xueming Liu  »View Author Affiliations


Optics Express, Vol. 12, Issue 4, pp. 545-550 (2004)
http://dx.doi.org/10.1364/OPEX.12.000545


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Abstract

A mid-point shooting algorithm using the Newton–Raphson method is adopted for solving nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers. A series of novel backward-differentiation methods are constructed for the first time to our knowledge. Their combination can form a powerful solution for fiber amplifiers. Numerical results show that the approach can solve Raman amplifier propagation equations on various conditions including co-, counter-, and bidirectionally pumped cases. The computation speed of the present methods is about four times that of the backward-differentiation methods previously adopted.

© 2004 Optical Society of America

1. Introduction

Wavelength-division multiplexed (WDM) transmission using distributed Raman amplification, counterpumped Raman/erbium-doped fiber amplification, and hybrid amplification of semiconductor optical amplifiers (SOAs) and Raman amplifiers has recently been reported [1

1. M. Tang, P. Shum, and Y. D. Gong, “Design of double-pass discrete Raman amplifier and the impairments induced by Rayleigh backscattering,” Opt. Express 11, 1887–1893 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-16-1887. [CrossRef] [PubMed]

6

6. A. A. B. Tio, P. Shum, and Y. D. Gong, “Wide bandwidth flat gain Raman amplifier by using polarization-independent interferometric filter,” Opt. Express 11, 2991–2996 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-2991. [CrossRef] [PubMed]

]. Theoretical analysis and experimental results demonstrate that longer wavelength channels have larger output optical signal-to-noise ratio (OSNR) than shorter wavelength channels in backward-pumped Raman amplifiers [3

3. X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. 230, 425–431 (2004). [CrossRef]

,7

7. C. R. S. Fludger and V. Handerek, “Fundamental noise limits in broadband Raman amplifiers,” in Optical Fiber Communication Conference (OFC 2000), Vol. 37 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D. C., 2000), paper MA-5.

]. However, a bidirectional pumping scheme can equalize OSNR tilt by addition of proper forward pump diodes [8

8. Y. Chen et al., “Bi-directionally pumped broadband Raman Amplifier,” ECOC’2001, Tu.L.3.4.

, 9

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

].

In WDM systems with fiber Raman amplifiers, the analysis of the bidirectional pump/signal propagation in fibers is essential. Therefore, the boundary value problems of model equations must be solved in fiber amplifier systems, and a shooting algorithm is adopted typically [10

10. X. M. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express 11, 1452–1461 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452. [CrossRef] [PubMed]

]. In the co- and counterpumped Raman amplifiers, the gain profile and noise performance can be obtained numerically [3

3. X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. 230, 425–431 (2004). [CrossRef]

, 8

8. Y. Chen et al., “Bi-directionally pumped broadband Raman Amplifier,” ECOC’2001, Tu.L.3.4.

, 11

11. X. M. Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” Opt. Express 11, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

, 12

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

]. However, the traditional algorithms fail to accommodate bidirectional pumping schemes with multiple pump diodes and high power. In this paper, a powerful solution is proposed to overcome the difficulty of bidirectionally multipumped Raman amplifiers, for the first time to our knowledge.

2. Physical model and mathematical algorithm

2.1. Physical model

Wave propagation and noise propagation in fiber Raman amplifiers are characterized by a variety of physical effects, and they can be described by a set of coupled equations in the steady state [11

11. X. M. Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” Opt. Express 11, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

, 12

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

], i.e.,

dPv±dz=αvPv±±ηvPv±Pv±μ>vgμvΓAeff[Pμ++Pμ]
±hviμ>vgμvAeff[Pμ++Pμ][1+(eh(μv)KT1)1]Δv
Pv±μ>vvμgμvΓAeff[Pμ++Pμ]2hviPv±μ>vvμgμvAeff[1+(eh(vμ)KT1)1]Δv,
(1)

where Pv+ and Pv are optical power of forward- and backward-propagating waves within the infinitesimal bandwidth around v, respectively; αv , ηv , h, K, and T are attenuation coefficient, Rayleigh-backscattering coefficient, Planck’s constant, Boltzmann constant, and temperature, respectively; Aeff is effective area of optical fiber; gµv is Raman gain parameter at frequency v resulting from the pump at frequency µ; and the factor of Γ accounts for polarization randomization effects, whose value lies between 1 and 2.

Conveniently, Eq.(1) is rewritten as the standard two-point boundary value problem [15, 16

16. H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Blaisdell, Waltham, Mass., 1968), pp. 192.

], i.e.,

dPi(z)dz=Pi(z)fi(z,P1,P2,PN),(i=1,2,,N).
(2.a)

At z 1 (i.e., z=0) and z 2 (i.e., z=L), the solutions are supposed to satisfy

B1j(z1,P1,P2,PN)=0,(j=1,2,,n1),
(2.b)
B2k(z2,P1,P2,PN)=0,(k=1,2,,n2).
(2.c)

Here, we specify a set of N coupled differential equations of Eq. (1), satisfying n 1 boundary conditions at the starting point z 1, and a remaining set of n 2=N-n 1 boundary conditions at the end point z 2.

2.2. Shooting algorithm

Now we propose a mid-point shooting algorithm, where the solution of Eq. (2) is implemented by integrating from both sides of the interval and trying to match continuity conditions at an intermediate point zf (e.g., zf =L/2). At the same time, the multidimensional Newton–Raphson method is employed in the implementation of the new algorithm. The detailed procedure is as follows:

Step 1 Given an n 2-vector V (1) of starting parameters at z 1, a particular Pi (z 1) is generated from Eq. (2.b), i.e., Pi(z1)=Pi(z1;V(1)1,V(1)2,,V(1)n2). Then, Eq. (2.a) is integrated in the interval [z 1, zf ], and an N-vector F1 is constructed from the solution P=(P 1, P 2,…, P N)T of Eq. (2.a), i.e., F1=F[P(zf , V (1))].

Step 2 Given an n 1-vector V (2) of final parameters at z 2, a particular Pi (z 2) is produced from Eq. (2.c), i.e., Pi(z2)=Pi(z2;V(2)1,V(2)2,,V(1)n1). Then, Eq. (2.a) is integrated in the interval [z 2, zf ], and an N-vector F2 is formed from the solution P=(P 1, P 2,…, P N)T of Eq. (2.a), i.e., F2=F[P(zf , V (2))].

Step 3 Now, we want to find a vector of V=(V (1), V (2)) that zeros the vector value of W (V)=F1-F2, i.e.,

W (V)=F[P(zf , V (1))]-F[P(zf , V (2))]=0.

Based on the Newton–Raphson method, we can solve above nonlinear equations as δV=-[J]-1 W, where

δV=(δV(1)1,δV(1)2,,δV(1)n2,δV(2)1,δV(2)2,,δV(1)n1)T

and Jacobian matrix

[J]=[WV]|z=zf=[WV(1),WV(2)]|z=zf

Step 4 Then, the revised value V(1)new and V(2)new are obtained, i.e.,

V(1)new=V (1)V (1)

and

V(2)new=V (2)V (2).

Step 5 If ‖(F1-F2)/(F1+F2)‖>ε(ε is the specified relative error), go to Step 1. Otherwise, output results.

2.3. Novel backward-differentiation methods

For co- and bipumped Raman amplifiers with high power and multiple diodes, Eq. (1) will be stiff differential equations. The traditional methods such as the Runge–Kutta and Adams methods may fail to solve Eq. (1). And backward-differentiation methods (BDFs) have to be applied to solve it. Although the traditional BDFs can solve stiff equations, their efficiency is poor because of Jacobian matrix calculation and multiple iterations at per step. To improve the computational efficiency, a series of novel BDFs are constructed.

Similar to the operational procedure in Refs. [10

10. X. M. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express 11, 1452–1461 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452. [CrossRef] [PubMed]

, 11

11. X. M. Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” Opt. Express 11, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

, 13

13. X. M. Liu, H. Y. Zhang, and Y. L. Guo, “A novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. 15, 392–394 (2003). [CrossRef]

], Eq. (2.a) can be solved from the traditional BDF formulas, i.e.,

ln(Pt)=j=1kαjln(Ptj)+Δzβf(zt,Pt),
(3.a)

where Pt and P t-j are the value of P at the tth step and (t-j)th step, and the step size Δz=z t+1-zt . Here we use bold scripts only to shorten notation for a set of equations, i.e., P=(P 1, P 2,…, P N) and f=(f 1, f 2,…, f N). Hence, hereafter the division or multiplication of bold scripts does not have any meaning of vector calculus. After the manipulation, Eq. (3.a) is simplified as

Pt=(j=1kPtjαj)·exp(Δzβf(zt,Pt)).
(3.b)

The coefficients of Eq. (3) up to order k=6 are demonstrated in Table 1. For an implicit method such as BDFs, a nonlinear system of equations must be solved at each step [15], namely,

g(Pt)=Pt(j=1kPtjαj)·exp(Δzβf(zt,Pt))=0.
(4)

Table 1. Coefficients of Eq. (3) up to order 6

table-icon
View This Table

By using the Newton-Raphson method, we can obtain the iterative expression of Eq. (3)from Eq. (4), i.e.,

Ptn+1=Ptn(IA·Δzβ·fPtn)1(PtnA)
(5.a)
A=(j=1kPtjαj)·exp(Δzβf(zt,Ptn)),
(5.b)

where the superscript n is the current iteration, and n=0, 1, 2, ….

3. Simulation results

From the proposed shooting algorithm and BDFs, a bidirectionally distributed multipump Raman amplifier is simulated. In the calculations, we assumed that Γ=2, L=100 km, α=0.2, and 0.35 dB/km for signals and pumps, respectively, and ignored ASE, Rayleigh scattering, and other noises. There are 19 signal channels spaced 200 GHz from 188.85 to 192.45 THz. The signal power of each channel is 1 mW. Four backward-propagating and three forward-propagating pumps are used; their powers are all 300 mW, and their wavelengths are 1425, 1445, 1465, 1485, 1435, 1455, and 1475 nm. The estimated values of each signal and each copropagating pump at z=L are 1 mW and 0.1 mW, and each counterpropagating pump at z=0 is estimated as 0.1 mW, which is shown in Fig. 1(a).

Moreover, the novel algorithm is less sensitive to the estimated starting parameters in the calculations so that it is greatly more robust than the traditional algorithms. The assumed value of intermediate point zf affects the efficiency and stability of the proposed shooting algorithm. Usually, it can be specified as zf =L/2. Although the traditional BDFs can also obtain Fig. 1 on the basis of our shooting algorithm, its CPU time is increased about four times in comparison with the novel BDFs under the same conditions.

Fig. 1. Power evolution of pumps and signals along the fiber position, where for (a) 1st iteration (i.e., estimated values), (b) 2nd iteration, (c) 3rd iteration, and (d) 4th iteration.

4. Discussions

When fiber Raman amplifiers work in the saturation region or/and the bidirectional signal transmission, ASE and Rayleigh scattering will affect the gain characteristics. Specially, double Rayleigh scattering is suggested as the major limiting factor for fiber Raman transmission systems, and its impairment grows with the increase of distributed gain [17

17. Y. Zhu et al., “Experimental comparison of all-Raman and Raman/EDFA hybrid amplifications using 40 Gbit/s-based transmissions over 400 km TW-RS fibre,” Electron. Lett. 38, 893–895 (2002). [CrossRef]

, 18

18. J. Ko, S. Kim, J. Lee, S. Won, Y. S. Kim, and J. Jeong, “Estimation of performance degradation of bidirectional WDM transmission systems due to Rayleigh backscattering and ASE noises using numerical and analytical models,” J. Lightwave Technol. 21, 938–946 (2003). [CrossRef]

]. Because of the intrinsic great stability, the proposed BDFs can solve the stiff cases of Eq. (1) if including ASE and Rayleigh scattering terms. However, because the power of backscattering pumps and backscattering signals is lower by ~30 dB and ~20 dB than their original power, and because the power of forward and backward noise is less than that of input signals by ~30 dB [11

11. X. M. Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” Opt. Express 11, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

, 13

13. X. M. Liu, H. Y. Zhang, and Y. L. Guo, “A novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. 15, 392–394 (2003). [CrossRef]

, 14

14. X. M. Liu, “Corrections to: a novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. 15, 1321–1321 (2003). [CrossRef]

], we divide the simulation procedure into two steps. Firstly, ignoring all noise terms, the gain profile is obtained. From the obtained gain profile, second, ASE and Rayleigh scattering can be calculated from Eq. (1). In fact, some researchers have simulated the performance of Rayleigh scattering on the assumption that the pump is nondepleted (i.e., the small signal model) [19

19. P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, “Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,” J. Lightwave Technol. 21, 2224–2228 (2003). [CrossRef]

]. The numerical results show that, compared with the exact values, the error of our two-step method is less than 3%, whereas that of the small signal model is more than 20% in simulating the noise performance of Rayleigh scattering. Therefore, our method can have much less error in comparison with the method in Ref. [19

19. P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, “Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,” J. Lightwave Technol. 21, 2224–2228 (2003). [CrossRef]

].

By comparing with the “pure” shooting algorithm [10

10. X. M. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express 11, 1452–1461 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452. [CrossRef] [PubMed]

, 11

11. X. M. Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” Opt. Express 11, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

], we see that the mid-point shooting algorithm can offer greater stability. For example, when the fiber length in Fig. 1 is reduced to be one third of original length and other parameters remain the same, the “pure” shooting algorithm still fails to solve it. Therefore, the effective span length solved by the mid-point shooting algorithm is over 3 times as long as that solved by the “pure” shooting algorithm. Simulations demonstrate that, on the basis of the parameter set in Ref. [8

8. Y. Chen et al., “Bi-directionally pumped broadband Raman Amplifier,” ECOC’2001, Tu.L.3.4.

], the results obtained by the present methods are consistent with the reports in Ref. [8

8. Y. Chen et al., “Bi-directionally pumped broadband Raman Amplifier,” ECOC’2001, Tu.L.3.4.

].

5. Conclusions

We have constructed a series of new backward-differentiation methods for the first time to our knowledge. An effective shooting algorithm for Raman amplifier propagation equations has been proposed. Their combination can offer a powerful solution for fiber amplifiers. Simulated results demonstrate that the proposed algorithms can be used effectively to solve model equations of Raman amplifiers under various conditions including co-, counter-, and bidirectionally pumped cases. New methods increase the computing speed by about four times in comparison with backward-differentiation methods in our simulations.

Xueming Liu is currently engaged in research at the School of Electrical Engineering, Seoul National University, Korea.

References and links

1.

M. Tang, P. Shum, and Y. D. Gong, “Design of double-pass discrete Raman amplifier and the impairments induced by Rayleigh backscattering,” Opt. Express 11, 1887–1893 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-16-1887. [CrossRef] [PubMed]

2.

H. H. Lee et al., “A gain-clamped-semiconductor-optical-amplifier combined with a distributed Raman-fiber-amplifier: a good candidate as an inline amplifier for WDM networks,” Opt. Commun. 229, 249–252 (2004). [CrossRef]

3.

X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. 230, 425–431 (2004). [CrossRef]

4.

H. Suzuki, N. Takachio, H. Masuda, and K. Iwatsuki, “Super-dense WDM transmission technology in the zero-dispersion region employing distributed Raman amplification,” J. Lightwave Technol. 21, 973–981 (2003). [CrossRef]

5.

A. Pizzinat, M. Santagiustina, and C. Schivo, “Impact of hybrid EDFA-distributed Raman amplification on a 4×40-Gb/s WDM optical communication system,” IEEE Photon. Technol. Lett. 15, 341–343 (2003). [CrossRef]

6.

A. A. B. Tio, P. Shum, and Y. D. Gong, “Wide bandwidth flat gain Raman amplifier by using polarization-independent interferometric filter,” Opt. Express 11, 2991–2996 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-2991. [CrossRef] [PubMed]

7.

C. R. S. Fludger and V. Handerek, “Fundamental noise limits in broadband Raman amplifiers,” in Optical Fiber Communication Conference (OFC 2000), Vol. 37 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D. C., 2000), paper MA-5.

8.

Y. Chen et al., “Bi-directionally pumped broadband Raman Amplifier,” ECOC’2001, Tu.L.3.4.

9.

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

10.

X. M. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express 11, 1452–1461 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452. [CrossRef] [PubMed]

11.

X. M. Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” Opt. Express 11, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

12.

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

13.

X. M. Liu, H. Y. Zhang, and Y. L. Guo, “A novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. 15, 392–394 (2003). [CrossRef]

14.

X. M. Liu, “Corrections to: a novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. 15, 1321–1321 (2003). [CrossRef]

15.

http://www.nr.com.

16.

H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Blaisdell, Waltham, Mass., 1968), pp. 192.

17.

Y. Zhu et al., “Experimental comparison of all-Raman and Raman/EDFA hybrid amplifications using 40 Gbit/s-based transmissions over 400 km TW-RS fibre,” Electron. Lett. 38, 893–895 (2002). [CrossRef]

18.

J. Ko, S. Kim, J. Lee, S. Won, Y. S. Kim, and J. Jeong, “Estimation of performance degradation of bidirectional WDM transmission systems due to Rayleigh backscattering and ASE noises using numerical and analytical models,” J. Lightwave Technol. 21, 938–946 (2003). [CrossRef]

19.

P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, “Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,” J. Lightwave Technol. 21, 2224–2228 (2003). [CrossRef]

OCIS Codes
(000.3860) General : Mathematical methods in physics
(000.4430) General : Numerical approximation and analysis
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

ToC Category:
Research Papers

History
Original Manuscript: December 22, 2003
Revised Manuscript: February 1, 2004
Published: February 23, 2004

Citation
Xueming Liu, "Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers," Opt. Express 12, 545-550 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-4-545


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References

  1. M. Tang, P. Shum, and Y. D. Gong, �??Design of double-pass discrete Raman amplifier and the impairments induced by Rayleigh backscattering,�?? Opt. Express 11, 1887�??1893 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-16-1887">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-16-1887</a> [CrossRef] [PubMed]
  2. H. H. Lee et al., �??A gain-clamped-semiconductor-optical-amplifier combined with a distributed Raman-fiber-amplifier: a good candidate as an inline amplifier for WDM networks,�?? Opt. Commun. 229, 249�??252 (2004). [CrossRef]
  3. X. M. Liu and Y. H. Li, �??Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,�?? Opt. Commun. 230, 425�??431 (2004) [CrossRef]
  4. H. Suzuki, N. Takachio, H. Masuda, and K. Iwatsuki, �??Super-dense WDM transmission technology in the zero-dispersion region employing distributed Raman amplification,�?? J. Lightwave Technol. 21, 973�??981 (2003). [CrossRef]
  5. A. Pizzinat, M. Santagiustina, and C. Schivo, �??Impact of hybrid EDFA-distributed Raman amplification on a 4 x 40-Gb/s WDM optical communication system,�?? IEEE Photon. Technol. Lett. 15, 341�??343 (2003). [CrossRef]
  6. A. A. B. Tio, P. Shum, and Y. D. Gong, �??Wide bandwidth flat gain Raman amplifier by using polarization-independent interferometric filter,�?? Opt. Express 11, 2991�??2996 (2003),<a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-2991"> http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-2991</a> [CrossRef] [PubMed]
  7. C. R. S. Fludger and V. Handerek, �??Fundamental noise limits in broadband Raman amplifiers,�?? in Optical Fiber Communication Conference (OFC 2000), Vol. 37 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D. C., 2000), paper MA-5
  8. Y. Chen et al., �??Bi-directionally pumped broadband Raman Amplifier,�?? ECOC�??2001, Tu.L.3.4.
  9. Z. Tong, H. Wei, and S. S. Jian, �??Theoretical investigation and optimization of bi-directionally pumped broadband fiber Raman amplifiers,�?? Opt. Commun. 217, 401�??413 (2003). [CrossRef]
  10. X. M. Liu and B. Lee, �??Effective shooting algorithm and its application to fiber amplifiers,�?? Opt. Express 11, 1452�??1461 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452</a> [CrossRef] [PubMed]
  11. X. M. Liu and B. Lee, �??A fast and stable method for Raman amplifier propagation equations,�?? Opt. Express 11, 2163�??2176 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163"> http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163</a> [CrossRef] [PubMed]
  12. S. Namiki and Y. Emori, �??Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high power laser diodes,�?? IEEE J. Sel. Top. Quantum Electron. 7, 3�??16 (2001). [CrossRef]
  13. X. M. Liu, H. Y. Zhang, and Y. L. Guo, �??A novel method for Raman amplifier propagation equations,�?? IEEE Photon. Technol. Lett. 15, 392�??394 (2003). [CrossRef]
  14. X. M. Liu, �??Corrections to: a novel method for Raman amplifier propagation equations,�?? IEEE Photon. Technol. Lett. 15, 1321�??1321 (2003). [CrossRef]
  15. <a href= "http://www.nr.com.">http://www.nr.com</a>
  16. H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Blaisdell, Waltham, Mass., 1968) pp. 192.
  17. Y. Zhu et al., �??Experimental comparison of all-Raman and Raman/EDFA hybrid amplifications using 40 Gbit/s-based transmissions over 400 km TW-RS fibre,�?? Electron. Lett. 38, 893�??895 (2002). [CrossRef]
  18. J. Ko, S. Kim, J. Lee, S. Won, Y. S. Kim, and J. Jeong, �??Estimation of performance degradation of bidirectional WDM transmission systems due to Rayleigh backscattering and ASE noises using numerical and analytical models,�?? J. Lightwave Technol. 21, 938�??946 (2003). [CrossRef]
  19. P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, �??Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,�?? J. Lightwave Technol. 21, 2224�??2228 (2003). [CrossRef]

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