## Optimizing the incoherent pump spectrum of low-gain-ripple distributed fiber Raman amplifier for a given main pump wavelength

Optics Express, Vol. 15, Issue 1, pp. 45-55 (2007)

http://dx.doi.org/10.1364/OE.15.000045

Acrobat PDF (188 KB)

### Abstract

The method for designing the incoherent pump spectrum for the distributed fiber Raman amplifiers (DFRAs) of low gain ripple is studied, in which the wavelength of the maximum power spectral density can be assigned. The assigned wavelength is called the main pump wavelength. Incoherent pump spectrum is described with the power spectral density function (PSDF) that comprises a set of piece-wise continuous functions. PSDF is optimized for the minimum gain ripple with the least-square minimization method. An extremum pump wavelength condition is applied to PSDF. A proper initial trial PSDF is given so that the optimized PSDF converges to the desired result and the power spectral density at the extremum pump wavelength is the maximum. With this design method, we show the optimized PSDFs for the DFRAs using backward pumping and bidirectional pumping. The gain ripples of considered DFRAs are less than 0.1 dB for 20-dB ON-OFF Raman gain over 70-nm bandwidth. The reduction of average effective noise figure with shorter main pump wavelength is shown and investigated.

© 2007 Optical Society of America

## 1. Introduction

5. E. Lichtman, R. Waarts, and A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fiber,” J. Lightwave. Technol. **7**,171–1174 (1989). [CrossRef]

6. X. Zhou, M. Birk, and S. Woodward, “Pump-noise induced FWM effect and its reduction in a distributed
Raman fiber amplifiers,” IEEE Photon. Technol. Lett. **14**,1686–1688 (2002). [CrossRef]

10. H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, “1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology,” IEEE Photon. Technol. Lett. **12**,903–905 (2000). [CrossRef]

2. T. Zhang, X. Zhang, and G. Zhang, “Distributed fiber Raman amplifiers with incoherent pumping,” IEEE Photon. Technol. Lett. **17**,1175–1177 (2005). [CrossRef]

2. T. Zhang, X. Zhang, and G. Zhang, “Distributed fiber Raman amplifiers with incoherent pumping,” IEEE Photon. Technol. Lett. **17**,1175–1177 (2005). [CrossRef]

## 2. Amplifier model and design method

14. I. Mandelbaum and M. Bolshtyansky, “Raman amplifier model in singlemode optical fiber,” IEEE Photon. Technol. Lett. **15**,1704–1706 (2003). [CrossRef]

15. S. Wen, T.-Y. Wang, and S. Chi, “Self-consistent pump depletion method to design optical transmission
systems amplified by bidirectional Raman pumps,” Int. J. Nonlinear Opt. Phys. **1**,595–608 (1992). [CrossRef]

2. T. Zhang, X. Zhang, and G. Zhang, “Distributed fiber Raman amplifiers with incoherent pumping,” IEEE Photon. Technol. Lett. **17**,1175–1177 (2005). [CrossRef]

*v*. Its spectrum is divided into

*N*sub-bands. A PWCF for the

*i*-th sub-band is defined as [4]

*i*= 1,2,…,

*N*-1, and

*M*is the polynomial order of a sub-band;

*a*’s are the polynomial coefficients;

_{ij}*v*

_{i-1}and

*v*are the boundary optical frequencies of the

_{i}*i*-th sub-band. As there is no negative PSD, the PSDF of the

*i*-th sub-band is taken as

*i*= 1,2,…,

*N*-1; the prime represents the first derivative with respect to

*v*. Note that the pump band lies within the end-point frequencies

*v*and

_{0}*v*. The function and its first derivative are set to be zeros at

_{N}*v*and

_{0}*v*, i.e.

_{N}*λ*

_{ 0}=

*c*/

*v*

_{0}and

*λ*=

_{ N}*c*/

*v*, where

_{N}*c*is the speed of light in vacuum. The spectral range within the end-point pump frequencies or wavelengths is called the pump band. For the assigned wavelength

*λ*that corresponds to a local extremum of PSDF, we have the extremum pump wavelength condition

_{ e}*v*=

_{e}*c*/

*λ*. When the PSD at

_{ e}*λ*is the maximum,

_{ e}*λ*becomes the main pump wavelength of PSDF. From the results shown in Ref. [4], the main pump wavelength is usually close to

_{ e}*λ*. Therefore, we set

_{ N}*λ*to be near

_{ e}*λ*. Multiple extremum wavelengths can be assigned for tailoring a PSDF but that may increase the gain ripple of optimized result. We therefore assign only one extremum wavelength for the DFRAs considered in this paper.

_{ N}*λ*is close to the main pump wavelength of the optimized PSDF that is not constrained by Eq. (5). Otherwise the use of the null IT-PSDF may result in the following undesirable results: (i) there exists the other extremum wavelength that its PSD is comparable to or even larger than the PSD at

_{ e}*λ*, (ii) the PSD at

_{ e}*λ*is a local minimum, and (iii) the gain ripple is large. From Eq. (5), in fact

_{ e}*λ*is only the wavelength corresponding to a local extremum that may be either the local maximum or the local minimum. These undesirable results can be avoided by the use of a proper non-null IT-PSDF. A non-null IT-PSDF can be obtained by fitting the coefficients of a PSDF to the following function

_{ e}*N*(

*M*+1) polynomial coefficients in Eq. (1). Equations (3)–(5) give 2(

*N*-1) +3 conditions. Therefore, the total number of un-constrained polynomial coefficients is

*N*(

*M*-1)-3. These unconstrained coefficients are optimized for the minimum gain ripple, in which the objective function representing gain ripple is defined in Ref. [4]. Modified Levenberg-Marquardt method is used to minimize the objective function [16]. In searching for the minimum gain ripple, the signal gains are obtained by solving the coupled differential equations of DFRA. We will show the numerical results for the DFRAs using backward pumping and bidirectional pumping in the next two sections. It requires two sets of PWCFs for describing the PSDFs of co-pump and counter-pump for the DFRA using bidirectional pumping. The polynomial coefficients of the two PSDFs are optimized simultaneously. We set the same

*N*and

*M*for the two PSDFs for simplicity. Because the characteristics of the DFRA using bidirectional pumping mainly depends on its co-pump, we assign the main pump wavelength to its co-pump and no extremum wavelengths are assigned to its counter-pump for simplifying the analysis of numerical results. The IT-PSDF of the counter-pump is set to be null. For all the considered DFRAs,

*N*= 9 and

*M*= 4 are taken.

*G*) and effective noise figure (ENF) are taken to evaluate the gain and noise performance of DFRAs.

_{on-off}*G*is the ratio of the signal power with pumps ON over the signal power with pumps OFF. The gain variation of the amplifier is usually represented by the gain ripple that is defined as the difference of the maximum gain and the minimum gain of all signal channels. For the DFRAs designed in this paper, we require the gain ripple to be less than 0.1 dB. ENF is defined as

_{on-off}*P*

^{+}

*is the output power of forward ASEN; Δ*

_{ASE}*v*is the bandwidth of the ASEN power;

*hv*is the photon energy at the optical frequency

*v*. The target gains can be chosen to just compensate for fiber loss and they vary with signal wavelength. However, in this paper, we take

*G*= 20 dB for all the signal channels as is usually taken in the literatures. The comparison of ENFs relates to the comparison of the forward ASEN photon-number spectral densities for the same gain. Because ENF varies with signal wavelength, we define the average ENF as the average value of the ENFs of all signal channels for comparing the noise performance of DFRAs. In addition, the ENF variation is defined as the difference of the maximum ENF and the minimum ENF of all signal channels.

_{on-off}## 3. DFRA using backward pumping

*λ*to the short-wavelength pump lobe. We set

_{ e}*λ*

_{ 0}= 1520 nm and

*λ*=

_{ N}*λ*-20nm.

_{ e}*λ*= 1420 nm. In the figures, the cases using the non-null IT-PSDFs with several

_{ e}*p*

_{0}and Δ

*λ*are shown. The gain ripples of all the cases shown in Fig. 1(a) are less than 0.1 dB. From Fig. 1(a), one can see that the optimized pump PSDFs are insensitive to their IT-PSDFs. As is shown in Ref. [4], the main pump wavelength of the optimized PSDF that is not constrained by Eq. (5) is close to 1420 nm, in which the pump band is the same as the cases shown in Fig. 1(a). Therefore the optimized pump PSDFs converge to similar solutions under the requirement of low gain ripple. The total pump power of the case using the null IT-PSDF is 917 mW.

_{ w}*λ*shifts the maximum ASEN away from the signal band and decreases ENF but at the expense of higher pump power [2

_{ e}**17**,1175–1177 (2005). [CrossRef]

*λ*= 1380 nm and using the null IT-PSDF, we show its optimized pump PSDF and ENF in Figs. 2(a) and 2(b), respectively. The gain ripple is 0.068 dB. For this case, there is the pump lobe at 1410 nm in addition to 1380 nm. The power of the pump lobe at 1410 nm is larger than that at 1380 nm. Compared with the DFRA with

_{ e}*λ*= 1420 nm, the ENF of the DFRA with

_{ e}*λ*= 1380 nm and using the null IT-PSDF is not appreciably decreased because its main pump lobe is at 1410 nm.

_{ e}*λ*= 40 nm and several

_{ w}*p*

_{0}. The gain ripples are 0.073 dB, 0.026 dB, and 0.054 dB for the cases with

*p*

_{0}= 10 mW/100GHz, 15 mW/100GHz, and 20 mW/100GHz, respectively. The PSD at 1380 nm does not necessarily increase with

*p*

_{0}because the non-null IT-PSDF just gives a starting point for searching the PSDF that minimizes gain ripple by the minimization routine. Considering ENF, one can see that the case with Δ

*λ*= 40 nm and

_{ w}*p*

_{0}= 10 mW/100GHz is preferred for the cases shown in Fig. 2. The average ENF and total pump power of this preferred case are decreased and increased by 0.49 dB and 216 mW, respectively, compared with the DFRA with

*λ*= 1420 nm and using the null IT-PSDF.

_{ e}*λ*= 1380 nm and Δ

_{ e}*λ*= 30 nm. The gain ripples are 0.12 dB, 0.37 dB, and 0.35 dB for the cases with

_{ w}*p*

_{0}= 10 mW/100GHz, 15 mW/100GHz, and 20 mW/100GHz, respectively. The total pump powers are 1247 mW, 1339 mW, and 1396 mW for the cases with

*p*

_{0}= 10 mW/100GHz, 15 mW/100GHz, and 20 mW/100GHz, respectively. Comparing Fig. 3(b) with Fig. 2(b), one can see that the ENF for the case with Δ

*λ*= 30 nm can be decreased at the expense of larger gain ripple and pump power.

_{ w}## 4. DFRA using bidirectional pumping

*λ*to be the main pump wavelength of the co-pump.

_{ e}*λ*= 1410 nm. In the figures, the cases using the non-null IT-PSDFs with several Δ

_{ e}*λ*and

_{ w}*p*0 are shown. In Fig. 4(a), the pump bands are chosen to be:

*λ*

_{ 0}= 1460 nm and

*λ*= 1400 nm for co-pump; and

_{ N}*λ*

_{ 0}= 1520 nm and

*λ*= 1450 nm for counter-pump as in Ref. [4]. The gain ripples for all the cases shown in Fig. 4(a) are less than 0.1 dB. From Fig. 4(a), the optimized pump PSDFs are insensitive to their IT-PSDFs. As is shown in Ref. [4], the main pump wavelength of the optimized co-pump PSDF that is not constrained by Eq. (5) is close to 1410 nm. Therefore the optimized pump PSDFs converge to similar solutions under the requirement of low gain ripple. The total pump powers of the case using the null IT-PSDF are 650 mW and 491 mW for co-pump and counter-pump, respectively. Note that there is the double-knee structure in the power spectrum near 1427 nm and 1450 nm for the co-pump shown in Fig. 4(a). The short-wavelength signals from 1530 nm to 1560 nm are mainly amplified by the double-knee structure. There is the double-lobe structure for the counter-pump shown in Fig. 4(a). The long-wavelength signals from 1560 nm to 1600 nm are mainly amplified by the counter-pump.

_{ N}*λ*for the DFRA using bidirectional pumping. When

_{ e}*λ*is decreased, we find that the pump band of counter-pump has to be adjusted for reducing gain ripple. For the case with

_{ e}*λ*= 1370 nm, we take the pump bands to be: λ

_{ e}_{ 0}= 1460 nm and

*λ*= 1350 nm for co-pump; and λ

_{ N}_{ 0}= 1520 nm and

*λ*= 1430 nm for counter-pump. Figures 5(a) and 5(b) show the optimized pump PSDF and ENF, respectively, for the case using the null IT-PSDF. The gain ripple is 0.02 dB. From Fig. 5(a), there is no double-knee structure for the co-pump. For this case, the total pump powers of co-pump and counter-pump are 989 mW and 519 mW, respectively. Compared with the case with

_{ N}*λ*= 1410 nm and using the null IT-PSDF, the total pump powers are increased but the average ENF is not decreased. The average ENFs for the cases with

_{ e}*λ*= 1410 nm and 1370 nm are -6.41 dB and -6.3 dB, respectively. The ENF variations are 4.86 dB and 3.26 dB for the cases with

_{ e}*λ*= 1410 nm and 1370 nm, respectively. The ENF variation of the case with

_{ e}*λ*= 1370 nm is decreased but its average ENF is slightly increased.

_{ e}*λ*increases the PSD at 1370 nm and decreases ENF. Figures 5(a) and 5(b) also show the optimized pump PSDFs and ENFs, respectively, for the DFRAs using the non-null IT-PSDFs with Δ

_{ w}*λ*= 25 nm and several

_{ w}*p*

_{0}. Note that the PSD at 1370 nm also does not necessarily increase with

*p*

_{0}as the cases shown in Fig. 2(a). The gain ripples are 0.032 dB, 0.099 dB, and 0.021 dB for the cases with

*p*

_{0}= 10 mW/100GHz, 15 mW/100GHz, and 20 mW/100GHz, respectively. The average ENFs are -6.51 dB, -6.11 dB, and -6.92 dB for the cases with

*p*

_{0}= 10 mW/100GHz, 15 mW/100GHz, and 20 mW/100GHz, respectively. The ENF variations are 2.87 dB, 2.64 dB, and 2.47 dB for the cases with

*p*

_{0}= 10 mW/100GHz, 15 mW/100GHz, and 20 mW/100GHz, respectively. From Fig. 5(b), one can see the significant reduction of the ENF variation for the case with

*p*

_{0}= 20 mW/100GHz, in which the total pump powers of co-pump and counter-pump are 1396 mW and 489 mW, respectively.

*λ*= 1370 nm and

_{ e}*p*

_{0}= 20 mW/100GHz shown in Fig. 5(a). In Fig. 6, the case with

*λ*= 1410 nm and using the null IT-PSDF is also shown for comparison. One can clearly see the low ASEN PSD in signal band for the case with

_{ e}*λ*= 1370 nm. The evolutions of the forward ASEN PSDs at 1530 nm, 1570 nm, and 1600 nm for the cases shown in Fig. 6 are shown in Fig. 7. Figure 8 shows the evolutions of the co-pump PSDs at 1370 and 1430 nm for the case with

_{ e}*λ*= 1370 nm shown in Fig. 6. From Fig. 7, the ASEN at 1530 nm is significantly amplified by the co-pump around 1430 nm near the input end of the transmission fiber. Although the input PSD of the co-pump around 1430 nm is low for the case with

_{ e}*λ*= 1370 nm, it can be amplified by the co-pump PSD around 1370 nm near the input end as is shown in Fig. 8. Because the ASEN at 1530 nm is attenuated in the middle section of the transmission fiber and is less amplified near the output end of the transmission fiber, the ENF at 1530 nm is the smallest in signal band.

_{ e}*λ*= 1370 nm shown in Fig. 6. Note that the frequency difference of 1456 nm and 1370 nm is 12.9 THz which is close to the frequency difference of the maximum Raman gain. From Fig. 8, the short-wavelength pump lobe of counter-pump near 1456 nm is effectively amplified by the co-pump around 1370 nm. Thus, the ENF around 1570 nm decreases with the input power of the short-wavelength lobe of counter-pump. Because the use of the larger co-pump PSD at 1370 nm can decrease the required input power of the short-wavelength lobe of counter-pump, the ENF around 1570 nm decreases as the co-pump PSD at 1370 nm increases.

_{ e}*λ*ranging from 1350 nm to 1390 nm are shown. The corresponding effective noise figures are shown in Fig. 10. The parameters of the initial trial solutions for the cases shown in Fig. 9 are Δ

_{ e}*λ*= 40 nm, 40 nm, 25 nm, 25 nm, and 40 nm for the cases with

_{ w}*λ*= 1350 nm, 1360 nm, 1370 nm, 1380 nm, and 1390 nm, respectively, and

_{ e}*p*

_{0}= 8 mW/100GHz, 8 mW/100GHz, 21 mW/100GHz, 10 mW/100GHz, and 5 mW/100GHz for the cases with

*λ*= 1350 nm, 1360 nm, 1370 nm, 1380 nm, and 1390 nm, respectively, so that their PSDs at

_{ e}*λ*are the maximum and their gain ripples are less than 0.1 dB. Note that the required initial trial bandwidth is larger for the case with

_{ e}*λ*away from 1370 nm. The average ENFs are -5.99 dB, -5.95 dB, -7.16 dB, -6.0 dB, and -5.07 dB for the cases with

_{ e}*λ*= 1350 nm, 1360 nm, 1370 nm, 1380 nm, and 1390 nm, respectively. The ENF variations are 2.63 dB, 2.84 dB, 2.4 dB, 2.28 dB, and 2.23 dB for the cases with

_{ e}*λ*= 1350 nm, 1360 nm, 1370 nm, 1380 nm, and 1390 nm, respectively. One can see that the average ENF is the smallest for the case with

_{ e}*λ*= 1370 nm, because it is able to pump the short-wavelength lobe of counter-pump the most effectively among the cases shown in Fig. 9.

_{ e}## 5. Conclusion

## Acknowledgment

## References and links

1. | D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, L. Knopp, C. Lu, Y. Shen, G. Rodes, S. Vote, P. Wang, and X. Zhu, “Raman amplification using high-power incoherent semiconductor pump sources,” OFC 2003, Paper PD47. |

2. | T. Zhang, X. Zhang, and G. Zhang, “Distributed fiber Raman amplifiers with incoherent pumping,” IEEE Photon. Technol. Lett. |

3. | B. Han, X. Zhang, G. Zhang, Z. Lu, and G. Yang, “Composite broad-band fiber Raman amplifiers using incoherent pumping,” Opt. Express |

4. | S. Wen, “Design of the pump power spectrum for the distributed fiber Raman amplifiers using incoherent pumping ,” Opt. Express |

5. | E. Lichtman, R. Waarts, and A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fiber,” J. Lightwave. Technol. |

6. | X. Zhou, M. Birk, and S. Woodward, “Pump-noise induced FWM effect and its reduction in a distributed
Raman fiber amplifiers,” IEEE Photon. Technol. Lett. |

7. | T. Kung, C. Chang, J. Dung, and S. Chi, “Four-wave mixing between pump and signal in a distributed
Raman amplifier,” J. Lightwave. Technol. |

8. | J. Bouteiller, L. Leng, and C. Headley, “Pump-pump four-wave mixing in distributed Raman amplified
systems,” J. Lightwave. Technol. |

9. | S. Sugliani, G. Sacchi, G. Bolognini, S. Faralli, and F. Pasquale, “Effective suppression of penalties induced by parametric nonlinear interaction in distributed Raman amplifiers based on NZ-DS fibers,” IEEE Photon.
Technol. Lett. |

10. | H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, “1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology,” IEEE Photon. Technol. Lett. |

11. | M. Islam, “Raman amplifiers for telecommunications,” IEEE J. Sel. Tops. Quantum Electron. |

12. | V. Perlin and G. Winful, “On distributed Raman amplification for ultrabroad-band long-haul WDM
systems,” J. Lightwave. Technol. |

13. | J. Bromage, “Raman amplification for fiber communication systems,” J. Lightwave. Technol. |

14. | I. Mandelbaum and M. Bolshtyansky, “Raman amplifier model in singlemode optical fiber,” IEEE Photon. Technol. Lett. |

15. | S. Wen, T.-Y. Wang, and S. Chi, “Self-consistent pump depletion method to design optical transmission
systems amplified by bidirectional Raman pumps,” Int. J. Nonlinear Opt. Phys. |

16. | J. Moré, B. Garbow, and K. Hillstrom, |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(060.2330) Fiber optics and optical communications : Fiber optics communications

(190.5650) Nonlinear optics : Raman effect

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 7, 2006

Revised Manuscript: December 22, 2006

Manuscript Accepted: December 28, 2006

Published: January 8, 2007

**Citation**

Senfar Wen, Chun-Chia Chen, and Jiun-Wei Ou, "Optimizing the incoherent pump spectrum of low-gain-ripple distributed fiber Raman amplifier for a given main pump wavelength," Opt. Express **15**, 45-55 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-1-45

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### References

- D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, L. Knopp, C. Lu, Y. Shen, G. Rodes, S. Vote, P. Wang, and X. Zhu, "Raman amplification using high-power incoherent semiconductor pump sources," OFC 2003, Paper PD47.
- T. Zhang, X. Zhang, and G. Zhang, "Distributed fiber Raman amplifiers with incoherent pumping," IEEE Photon. Technol. Lett. 17, 1175-1177 (2005). [CrossRef]
- B. Han, X. Zhang, G. Zhang, Z. Lu, and G. Yang, "Composite broad-band fiber Raman amplifiers using incoherent pumping," Opt. Express 14, 3752-2762 (2006).
- S. Wen, "Design of the pump power spectrum for the distributed fiber Raman amplifiers using incoherent pumping, " Opt. Express 13, 6023-6032 (2005).
- E. Lichtman, R. Waarts, and A. Friesem, "Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fiber," J. Lightwave. Technol. 7, 171-1174 (1989). [CrossRef]
- X. Zhou, M. Birk, and S. Woodward, "Pump-noise induced FWM effect and its reduction in a distributed Raman fiber amplifiers," IEEE Photon. Technol. Lett. 14, 1686-1688 (2002). [CrossRef]
- T. Kung, C. Chang, J. Dung, and S. Chi, "Four-wave mixing between pump and signal in a distributed Raman amplifier," J. Lightwave. Technol. 21, 1164-1170 (2003). [CrossRef]
- J. Bouteiller, L. Leng, and C. Headley, "Pump-pump four-wave mixing in distributed Raman amplified systems," J. Lightwave. Technol. 22, 723-732 (2004). [CrossRef]
- S. Sugliani, G. Sacchi, G. Bolognini, S. Faralli, and F. Pasquale, "Effective suppression of penalties induced by parametric nonlinear interaction in distributed Raman amplifiers based on NZ-DS fibers," IEEE Photon. Technol. Lett. 16, 81-83 (2004). [CrossRef]
- H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, "1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology," IEEE Photon. Technol. Lett. 12, 903-905 (2000). [CrossRef]
- M. Islam, "Raman amplifiers for telecommunications," IEEE J. Sel. Tops. Quantum Electron. 8, 548-559 (2002). [CrossRef]
- V. Perlin and G. Winful, "On distributed Raman amplification for ultrabroad-band long-haul WDM systems," J. Lightwave. Technol. 20, 409-416 (2002). [CrossRef]
- J. Bromage, "Raman amplification for fiber communication systems," J. Lightwave. Technol. 22, 79-93 (2004). [CrossRef]
- I. Mandelbaum and M. Bolshtyansky, "Raman amplifier model in singlemode optical fiber," IEEE Photon. Technol. Lett. 15, 1704-1706 (2003). [CrossRef]
- S. Wen, T.-Y. Wang, and S. Chi, "Self-consistent pump depletion method to design optical transmission systems amplified by bidirectional Raman pumps," Int. J. Nonlinear Opt. Phys. 1, 595-608 (1992). [CrossRef]
- J. Moré, B. Garbow, and K. Hillstrom, User Guide for MINPACK-1, Argonne National Laboratory Report ANL-80-74, (Argonne National Laboratory, Argonne, Illinois, 1980).

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