## Design of the pump power spectrum for the distributed fiber Raman amplifiers using incoherent pumping

Optics Express, Vol. 14, Issue 9, pp. 3752-3762 (2006)

http://dx.doi.org/10.1364/OE.14.003752

Acrobat PDF (205 KB)

### Abstract

The method to design the incoherent pump power spectrum described with a set of piece-wise continuous functions (PWCFs) for the distributed fiber Raman amplifier (DFRA) is presented. The pump power spectrum is divided into a number of sub-bands, in which each sub-band is described with a polynomial. The power spectral density function (PSDF) is the absolute value of the set of PWCFs, in which the polynomial coefficients are optimized with the least-square minimization method for reducing the signal gain ripple. Two 100-km TW-Reach DFRAs using backward pumping and bidirectional pumping respectively are taken as examples. The numerical results show that the gain ripple of less than 0.02 dB over 70-nm bandwidth can be achieved. The spectral characteristics of the optimized PSDF for the ultra-low gain ripple are investigated. The optimized PSDF can be synthesized with multiple incoherent pumps. The synthesis examples using the multiple Gaussian incoherent pumps are shown, in which the gain ripples are increased to 0.3 dB due to the discrepancy between the optimized PSDF and the synthesized PSDF. The gain ripples can be reduced to 0.05 dB by further optimizing the parameters of the multiple Gaussian incoherent pumps.

© 2006 Optical Society of America

## 1. Introduction

1. A. Yariv, “Signal-to-noise considerations in fiber links with periodic or distributed optical amplification,”
Opt. Lett. **15**, 1064–1066 (1990). [CrossRef] [PubMed]

9. 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]

10. 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]

2. 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]

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

8. B. Han, X. Zhang, G. Zhang, Z. Lu, and G. Yang, “Composite broad-band fiber Raman amplifiers using incoherent pumping,” Opt. Express **13**, 6023–6032 (2005). [CrossRef] [PubMed]

7. 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 optimization 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]

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

8. B. Han, X. Zhang, G. Zhang, Z. Lu, and G. Yang, “Composite broad-band fiber Raman amplifiers using incoherent pumping,” Opt. Express **13**, 6023–6032 (2005). [CrossRef] [PubMed]

*v*. The spectrum is divided into N sub-bands. A PWCF for the i-th sub-band is defined as

*i*= 1, 2, …, N-1, and

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

*a*

_{ij}’s are the polynomial coefficients;

*v*

_{i}-1 and

*v*

_{i}are the boundary optical frequencies of the

*i*-th sub-band. The entire pump spectrum lies within the frequencies

*v*

_{0}and

*v*

_{N}which are called the end-point pump frequencies. The corresponding end-point pump wavelengths λ

_{0}=

*c*/

*v*

_{0}and λ

_{N}=

*c*/

*v*

_{N}, where

*c*is the speed of light in vacuum. The spectral range within the end-point pump frequencies or wavelengths is called the pump band. Piece-wise continuous boundary condition is applied to the neighboring sub-bands, which can be written as

*i*= 1, 2,…,N -1; the prime represents the first derivative with respect to v. We take the function and its first derivative to be zeros at the end-point frequencies, i.e.,

*N*(

*M*+1) polynomial coefficients. Eqs. (2) and (3) give 2(

*N*-1)+4 conditions. Therefore, the total number of the polynomial coefficients that are not constrained by the Eqs. (1b) and (3) is

*N*

_{A}-dimension vector

*A*. For the case of unidirectional pumping,

*N*

_{A}=

*U*. The optimization can be carried out with the least-square minimization methods. The objective function to be minimized is defined as

*is the vector to be optimized;*

**A***N*

_{S}is the number of signal channels which is also the number of gain requirements for the minimization problem;

*G*

_{k}(

*) is the signal gain of the*

**A***k*-th channel using the pump PSDF specified by

*;*

**A***G*

_{kT}is the target signal gain of the

*k*-th channel.

*conventional least-square minimization routine starts with a trial solution of*

**A***. In searching for the minimum objective function,*

**A***is changed according to the minimization algorithm. The modified Levenberg-Marquardt method is used to minimize the objective function [16]. In every searching step, the signal gains are obtained by solving the coupled differential equations of the DFRA.*

**A***. We use the method with the simplest formulas relating the components of*

**A***and the coefficients constrained by the Eqs. (2) and (3). The components of*

**A***are the coefficients*

**A***a*

_{i2},

*a*

_{i3}, …,

*a*

_{iM}of the sub-bands labeled with

*i*= 1, 2, ‥, N-2 in the Eq. (1a), and the coefficients

*a*

_{i3}, …,

*a*

_{iM}of the sub-bands labeled with

*i*=

*N*-1 and

*N*in the Eq. (1a). That is

*a*

_{10}=

*a*

_{11}= 0 from the Eqs. (3a) and (3c); for the

*N*-th sub-band,

*a*

_{N0}=

*a*

_{N1}= 0 from the Eqs. (3b) and (3d); the coefficients

*a*

_{i0}and

*a*

_{i1}of the i-th sub-band (

*i*= 2, 3,‥,

*N*-1), and the coefficients

*a*

_{(N-1)2}and

*a*

_{N2}can be easily solved from the Eqs. (2a) and (2b) for a given

*. For the*

**A***i*-th sub-band (

*i*= 2, 3,‥,

*N*-1),

*v*

_{i-1}=

*v*

_{i-1}-

*v*

_{i-2}. For the (

*N*-1)-th and

*N*-th sub-bands,

*i*=

*N*-1,

*k*=

*N*; for

*i*=

*N*,

*k*=

*N*-1; Δ

*v*

_{N-1}=

*v*

_{N-1}-

*v*

_{N-2}and Δ

*v*

_{N}=

*v*

_{N-1}-

*v*

_{N}. From the Eqs. (7)-(10), it requires

*M*≥ 3. It is found that the optimized results slightly depend on

*M*for

*M*≥ 4. Therefore we take

*M*= 4 for all the examples shown in the next section.

*are the unconstrained coefficients of the two PSDFs. We set the same*

**A***N*and

*M*for the two PSDFs for simplicity. In this case, the dimension of

**is 2**

*A**U*, where

*U*is given by the Eq. (5).

*U*= 25 for

*N*= 9 and

*M*= 4. Therefore, the optimized pump PSDF depends on the initial trial solution because the minimization process only freezes to a local minimum of the objective function, Eq. (6). A proper non-null initial trial solution can be applied for the minimization routine. In this paper, we take a null initial trial solution so that the optimized pump PSDF is not pre-biased by the initial trial solution, i.e., the components of the initial trial solution of

*are all taken to be zeros.*

**A***N*may not result in lower gain ripple because of the less effective optimization resulting from too large dimension of the vector

*. Several design examples of low gain ripple using the different numbers of sub-bands are shown in the next section. From the results, though the spectral shapes of the optimized pump PSDFs change with the number of sub-bands, their characteristics are similar.*

**A**## 3. Numerical results and discussion

*G*

_{on-off}) and effective noise figure (ENF) are taken to evaluate the gain and noise performance of the DFRAs.

*G*

_{on-off}is the ratio of the signal power with pumps ON over the signal power with pumps OFF. ENF is defined as

*v*is the bandwidth of the ASEN power;

*hv*is the photon energy at the optical frequency

*v*. The power spectrum of either the forward or backward ASEN is sliced into a number of beams. The bandwidth of every ASEN beam is 100 GHz. The target gains can be chosen to just compensate for the fiber loss and they vary with the signal wavelength. However, in this paper, we take

*G*

_{on-off}= 20 dB for all the signal channels, i.e.,

*G*

_{kT}= 20 dB for

*k*= 1, 2, …,

*N*

_{S}in the Eq. (6) 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.

_{N}= 1400 nm and λ

_{0}= 1500 nm, it is found that the minimum gain ripple is as high as 0.6 dB because the long-wavelength signal gains near 1600 nm are not enough. Therefore we take λ

_{0}= 1520 nm to provide higher gains for the long-wavelength signals. Figure 1(a) shows the optimized pump PSDFs of the pump band λ

_{N}= 1400 nm and λ

_{0}= 1520 nm for the cases of

*N*= 6, 9, and 12. The corresponding gains, effective noise figures, and output forward ASEN PSDFs for the three cases are shown in Figs. 1(b)–1(d), respectively. The gain ripples are 0.042 dB, 0.051 dB, and 0.014 dB for the cases of

*N*= 6, 9, and 12, respectively. The total pump powers are 910 mW, 912 mW, and 930 mW for the cases of

*N*= 6, 9, and 12, respectively. One can see that the spectral characteristics of the three cases are similar. There are a high-power lobe and a low-power lobe in the pump spectrum near 1420 nm and 1490 nm respectively. The low-power lobe is used to amplify the long-wavelength signals. The high-power lobe is used to amplify the short-wavelength signals and the low-power lobe. The result agrees to the optimized PSDF with two counter-incoherent 20-nm bandwidth pumps in [7

**17**, 1175–1177 (2005). [CrossRef]

*N*= 6, 9, and 12 are about the same. The case of the largest difference is that the effective noise figure of the case of

*N*= 6 is 0.15-dB lower than the case of

*N*= 9 at the shortest signal wavelength. Comparing the Figs. 1(a) and 1(d), the peak ASEN near 1500 nm comes from the Rayleigh back-scattering of the low-power pump lobe and the peak ASEN near 1515 nm results from the amplification of the ASEN by the high-power pump lobe. For reducing the effective noise figure, the wavelength of the high-power pump lobe should be shorter so that the wavelength of the peak ASEN amplified by the high-power pump lobe can be shifted away from the signal band. For the high-power pump lobe of the case of

*N*= 6, its wavelength is shorter than the case of

*N*= 9 and its peak power is less than the case of

*N*= 12. Such characteristics of the high-power pump lobe results in the lowest effective noise figures for the case of

*N*= 6.

_{N}= 1400 nm and λ

_{0}= 1460 nm for the co-pump; λ

_{N}= 1450 nm and λ

_{0}= 1520 nm for the counter-pump. The pump band of the co-pump is chosen to be in shorter wavelength region so that the amplification of the ASEN near the shortest signal wavelength by the co-pump can be attenuated in the middle section of the transmission fiber and the effective noise figures can be reduced. Figure 2(a) shows the optimized pump PSDFs for the cases of

*N*= 6, 9, and 12. The corresponding gains, effective noise figures, and output forward ASEN PSDFs for the three cases are shown in Figs. 2(b)– 2(d), respectively. The gain ripples are 0.0033 dB, 0.019 dB, and 0.011 dB for the cases of

*N*= 6, 9, and 12, respectively. The pump powers of co-pumps are 607 mW, 652 mW, and 598 mW for the cases of

*N*= 6, 9, and 12, respectively. The pump powers of counter-pumps are 505 mW, 501 mW, and 511 mW for the cases of

*N*= 6, 9, and 12, respectively. One can see that the spectral characteristics of the three cases are also similar. For the co-pumps, their wavelengths of the maximum PSDF are near 1410 nm. In addition, there are the double-knee structures in their power spectra near 1427 nm and 1450 nm. For the counter-pumps, there are the double-lobe structures in their power spectra near 1465 nm and 1500 nm. These results indicate that, to achieve the ultra-low gain ripple, there requires three and two incoherent pumps for the co-pump and counter pump respectively, in which the bandwidth of the individual incoherent pump is about 10 nm.

*N*= 6, 9, and 12 are about the same except in the long signal wavelength region. The case of the largest difference is that the effective noise figure of the case of

*N*= 9 is 0.29-dB lower than the case of

*N*= 12 at 1591.51 nm. The wavelengths of their maximum effective noise figures are near 1573 nm. From the Figs. 2(a) and 2(d), the peak ASEN near 1500 nm comes from the Rayleigh back-scattering of the long-wavelength pump lobe of the counter-pump and the peak ASEN near 1573 nm is due to the amplification of the ASEN by the short-wavelength pump lobe of the counter-pump. Figure 3 shows the evolutions of the forward ASEN PSDs at 1530 nm and 1600 nm for the three cases of

*N*= 6, 9, and 12 shown in Fig. 2(a). One can clearly observe that, the amplification of the ASEN at 1530 nm is more near the input end by co-pump but is less near the output end by counter-pump. This results in the low effective noise figures for the short-wavelength signals. For the ASEN at 1600 nm, it is mainly amplified by the counter-pump because the wavelength difference between the co-pump and the ASEN is far beyond the 13.2-THz Raman gain bandwidth. Although the counter-pump power of the case of

*N*= 9 is lower than the other two cases, it is higher than the other two cases near the input end of the transmission fiber because it is amplified near the input end by the co-pump of the largest power among the three cases. From the Fig. 3, one can see that the amplification of the 1600- nm ASEN for the case of

*N*= 9 is more than the other two cases near the input end and is less than the other two cases near the output end. This results in the lower effective noise figures for the case of

*N*= 9.

## 4. Synthesis of pump PSDF

*P*

_{p},

*v*

_{c}and Δ

*v*

_{W}are the amplitude, central optical frequency and spectral width (FHWM) of the PSDF. The spectral width in wavelength is defined as Δλ

_{W}=

*c*Δ

*v*

_{W}/

*N*= 9 shown in the Fig. 1(a), in which the DFRA is backwardly pumped. The optimized pump PSDF is well synthesized except near 1400 nm. The gain ripple is increased to 0.3 dB with this synthesized PSDF. We further optimize the parameters of the four Gaussian incoherent pumps for reducing the gain ripple. The optimized results are shown in the Table 1. The total spectrum of the four optimized Gaussian incoherent pumps is also shown in the Fig. 4, in which it slightly differs from the original synthesized pump PSDF but the corresponding gain ripple is reduced to 0.047 dB. The gains, effective noise figures, and output forward ASEN PSDFs are shown in the Figs. 1(b)–1(c), respectively, in which the effective noise figures are about the same as the original case of

*N*= 6. We have also used three Gaussian incoherent pumps to synthesize the pump PSDF. The original PSDF is poorly synthesized. However, we further optimize the parameters of the three Gaussian incoherent pumps. It is found that the gain ripple after the optimization is about 0.3 dB.

*N*= 6 shown in the Fig. 2(a) for the co-pump and counter-pump, in which the DFRA is bidirectional pumped. One can see the optimized pump PSDFs are not well synthesized especially for the counter-pump. The gain ripple is increased to 0.28 dB using the two synthesized PSDFs. We also further optimize the parameters of the five Gaussian incoherent pumps for reducing the gain ripple. The optimized results are shown in the Table 2. The total spectrum of the three optimized Gaussian incoherent co-pumps and the total spectrum of the two optimized Gaussian incoherent counter-pumps are also shown in the Fig. 5, in which they are blue-shifted with respect to the original synthesized pump PSDFs. Using the pump PSDFs of the five optimized Gaussian incoherent pumps, the gain ripple is reduced to 0.015 dB. The gains, effective noise figures, and output forward ASEN PSDFs are shown in the Figs. 2(b)–2(d), respectively. One can see the effective noise figures are larger than the original case of

*N*= 6 shown in the Fig. 2(c).

## 5. Conclusion

## Acknowledgments

## References and links

1. | A. Yariv, “Signal-to-noise considerations in fiber links with periodic or distributed optical amplification,”
Opt. Lett. |

2. | 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. |

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

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

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

6. | 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. |

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

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

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

10. | 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. |

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

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

13. | 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. |

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: March 7, 2006

Manuscript Accepted: April 8, 2006

Published: May 1, 2006

**Citation**

Senfar Wen, "Design of the pump power spectrum for the distributed fiber Raman amplifiers using incoherent pumping," Opt. Express **14**, 3752-3762 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-9-3752

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### References

- A. Yariv, "Signal-to-noise considerations in fiber links with periodic or distributed optical amplification," Opt. Lett. 15, 1064-1066 (1990). [CrossRef] [PubMed]
- 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. Top. 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]
- 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 13, 6023-6032 (2005). [CrossRef] [PubMed]
- 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]
- 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, Illinois, 1980.

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