## Efficient algorithm and optimization for broadband Raman amplifiers

Optics Express, Vol. 12, Issue 4, pp. 564-573 (2004)

http://dx.doi.org/10.1364/OPEX.12.000564

Acrobat PDF (208 KB)

### Abstract

A hybrid genetic algorithm (HGA) assisted by stochastic perturbation and the adaptive technique is proposed. Compared with our previous reports, the proposed HGA can exploit better solutions and greatly shorten the amount of run time. An example shows that the design of multipump Raman amplifiers involves the multimodal function optimization problem with multiple variables. With the new HGA, relationships of the optimal signal bandwidth with the span length and the ON-OFF Raman gain are obtained. A movie demonstrates the detailed interaction in pump-to-signal and signal-to-signal procedures. The corresponding optical signal-to-noise ratio of optimal results is obtained.

© 2004 Optical Society of America

## 1. Introduction

1. 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]

3. S. Faralli and E. Di Pasquale, “Impact of double Rayleigh scattering noise in distributed higher order Raman pumping schemes,” IEEE Photon. Technol. Lett. **15**, 804–806 (2003). [CrossRef]

2. X. Zhou and M. Birk, “Performance limitation due to statistical Raman crosstalk in a WDM system with multiple- wavelength bidirectionally pumped Raman amplification,” J. Lightwave Technol. **21**, 2194–2202 (2003). [CrossRef]

5. 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]

4. Y. Hadjar*et al*., “Enhanced double Rayleigh backscattering in second order Raman amplification and system performance implications,” Opt. Commun. **229**, 419–423 (2004). [CrossRef]

6. P. B. Hansen, L. Eskilden, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiBiovanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” IEEE Photon. Technol. Lett. **10**, 159–161 (1998). [CrossRef]

7. P. C. Xiao, Q. J. Zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. **15**, 206–208 (2003). [CrossRef]

8. M. Yan*et al*., “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. **13**, 948–950 (2001). [CrossRef]

9. X. Zhou, C. Lu, P. Shum, and T. H. Cheng, “A simplified model and optimal design of a multiwavelength backward-pumped Raman amplifier,” IEEE Photon. Technol. Lett. **13**, 945–947 (2001). [CrossRef]

10. X. M. Liu and B. Lee, “Optimal design for ultrabroadband amplifier,” J. Lightwave Technol. **21**, 3446–3455 (2003).. [CrossRef]

12. 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]

10. X. M. Liu and B. Lee, “Optimal design for ultrabroadband amplifier,” J. Lightwave Technol. **21**, 3446–3455 (2003).. [CrossRef]

10. X. M. Liu and B. Lee, “Optimal design for ultrabroadband amplifier,” J. Lightwave Technol. **21**, 3446–3455 (2003).. [CrossRef]

12. 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]

## 2. Theoretical model

15. 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]

16. H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. **11**, 530–532 (1999). [CrossRef]

7. P. C. Xiao, Q. J. Zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. **15**, 206–208 (2003). [CrossRef]

12. 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]

18. 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]

18. 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]

**230**, 425–431 (2004). [CrossRef]

16. H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. **11**, 530–532 (1999). [CrossRef]

*k*=1, 2,…,

*n*and

*k*=

*n*+1,…,

*n*+

*m*represent pump and signal waves, respectively.

*P*,

_{k}*v*,

_{k}*γ*, and

_{k}*α*are the power, frequency, Rayleigh scattering, and attenuation coefficient for the kth wave, respectively.

_{k}*P*

_{ASE,K}is the ASE noise power in one mode in the frequency resolution Δ

*v*, and its superscripts ‘+’ and ‘-’ denote forward- and backward-propagating ASE waves, respectively.

*h*,

*k*, and

_{B}*T*are Planck’s constant, Boltzmann’s constant, and temperature, respectively.

*A*is the effective area of the optical fiber. The factor of Γ accounts for polarization randomization effects, whose value lies between 1 and 2.

_{eff}*g*(

_{R}*v*-

_{j}*v*) is the Raman gain coefficient from wave

_{k}*j*to

*k*. The frequency ratio

*v*/

_{k}*v*describes vibrational losses. The minus and plus signs on the left-hand side of Eq. (1) describe the backward-propagating pump waves and forward-propagating signal waves, respectively. The frequencies

_{j}*v*are enumerated in decreasing order of frequency (

_{k}*k*=1, 2,…,

*n*+

*m*).

## 3. Multimodal function for multi-pump Raman amplifiers

*λ*with pump wavelengths

*λ*

_{2}and

*λ*

_{3}, where the wavelengths of two other pumps are specified as

*λ*

_{1}=1434.72 and

*λ*

_{4}=1497.75 nm. Figure 1(b) shows the magnification of the white dashed frame in Fig. 1(a). The color scale in the inset of Fig. 1 illustrates the distribution of Δ

*λ*. In optimizing Fig. 1, we assume that Γ=2,

*L*=40 km; there are 55 signal channels spaced at 200 GHz, and the signal power of each channel is 1 mW; the gain spectrum

*g*(Δ

_{R}*v*) and attenuation spectrum

*α*(

*v*) of the fiber are from Ref. [10

**21**, 3446–3455 (2003).. [CrossRef]

*G*

_{ON-OFF}>

*αL*); and the gain peak-to-peak ripple Δ

*G*is less than 1.1 (i.e., Δ

*G*<1.1).

*λ*

_{1},

*λ*

_{2},

*λ*

_{4}are equal and the difference of

*λ*

_{3}is small (

*λ*

_{3}=1467.25 nm for A and

*λ*

_{3}=1467.81 nm for B), their powers are greatly different (

*P*

_{1, 2, 3, 4}=216.89, 160.11, 88.72, 151.38 mW for A and

*P*

_{1, 2, 3, 4}=240.87, 167.33, 91.22, 151.27 mW for B). If

*λ*

_{4}=1497 nm instead of

*λ*

_{4}=1497.75 nm, the signal bandwidth is Δ

*λ*=61.3 instead of 82.5 nm, although other parameters are the same as the case of A. Therefore, the signal bandwidth is sensitive to the parameter set of pumps, whose values should be optimized in order to obtain the global optima.

## 4. Algorithm

**21**, 3446–3455 (2003).. [CrossRef]

**230**, 425–431 (2004). [CrossRef]

19. X. Liu and Y. Li, “Optimal design of DFG-based wavelength conversion based on hybrid genetic algorithm,” Opt. Express **11**, 1677–1688 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-14-1677. [CrossRef] [PubMed]

**21**, 3446–3455 (2003).. [CrossRef]

**230**, 425–431 (2004). [CrossRef]

*λ*of each pump into fixed range

_{fa}*λ*and variable range

_{f}*λ*, which are shown in Fig. 2. Figure 2 illustrates the distribution of wavelength

_{a}*λ*and power

*P*of four pumps, where the y coordinate represents the pump power and the abscissa denotes the pump wavelength. Then it is a nine-dimensional optimization problem in the case of the four-pump Raman amplifier, including four wavelengths, four powers, and variable range

*λ*in the code of each chromosome. From Fig. 2, we can see that

_{a}*λ*

_{1}∈[

*λ*

_{0},

*λ*

_{0}+

*λ*+

_{f}*λ*],

_{a}*λ*

_{2}∈[

*λ*

_{0}+

*λ*+

_{f}*λ*,

_{a}*λ*

_{0}+2·(

*λ*+

_{f}*λ*)],

_{a}*λ*

_{3}·[

*λ*

_{0}+2·(

*λ*+

_{f}*λ*),

_{a}*λ*

_{0}+3∈(

*λ*+

_{f}*λ*)] and

_{a}*λ*

_{4}∈[

*λ*

_{0}+3·(

*λ*+

_{f}*λ*),

_{a}*λ*

_{0}+4·(

*λ*+

_{f}*λ*)]. To see the detailed procedure of encoding the chromosome, we use Fig. 1 as an example. In simulating Fig. 1, we assume that

_{a}*λ*=14 nm,

_{f}*λ*

_{0}=1430 nm,

*λ*∈[0, 2 nm] (

_{a}*λ*has a different value for each chromosome per generation), and

_{a}*P*

_{1, 2, 3, 4}∈[0, 600 mW]. Here,

*λ*is from the experiential value and

_{f}*λ*

_{0}is calculated from the signal spectra and

*λ*. In fact, the simulated results show that we can obtain the global maximum when

_{f}*λ*∈[14 nm, 19 nm]. Because

_{f}*λ*of the pumps is an adaptive variable in each chromosome and

_{fa}*λ*is less sensitive to its initial value, the adaptive technique makes our proposed HGA greatly robust and flexible. Then, this technique increases the exploring capability of HGA.

_{f}*λ*is perturbed (e.g.,

_{f}*λ*=14.3 nm instead of

_{f}*λ*=14 nm) when the maximum of each cluster is unchangeable. After this perturbation, the new search space is explored and the trapped local maximum is tackled.

_{f}*N*=800, the number of clusters

*N*=7, the normalized niche radius

_{c}*D*=0.2, and the other parameters are the same as in Fig. 1. The parameter set here is also the same as in Figs. 4–6 except for the specified values in those figures. The normalized Euclidean distance

*d*between two individuals

_{ij}*i*and

*j*is calculated as follows:

*d*among the centers of all clusters must be more than the critical distance

_{ij}*D*. After the optimization, all centers of the seven clusters reach the global maxima, and the results are shown in Fig. 1(b). Seven white points represent the centers of the clusters (i.e., maxima), the region in each circle is considered a cluster, and the centers of the other clusters must be out of this region. The optimized parameter set is tabularized in Table 1. The signal spectra of the center of each cluster are demonstrated in Fig. 3. In Fig. 1(b), Fig. 3, and Table 1,

*n*

_{j}(j=1, 2,…,7) denote the centers of seven clusters. The dots in each curve of Fig. 3 represent the channels. From Fig. 3 and Table 1, we can see that ① this case of four-pump Raman amplifiers is a six-variate problem; ② the pump spectra of the seven solutions differ greatly, although the optimal results of Δ

*λ*are the same; ③ to reach the specified bandwidth, there are many combinations of parameter sets. The numerical results also show that ① although the old version of our HGA can obtain the global optima [10

**21**, 3446–3455 (2003).. [CrossRef]

**230**, 425–431 (2004). [CrossRef]

## 5. Optimal results and noise performance

18. 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]

20. 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]

*γ*=10

^{-7}m

^{-1};

*N*,

_{c}*D*,

*N*, Γ,

*g*(Δ

_{R}*v*), α(

*v*) and signal spectra are the same as Fig.3.

### 5.1. Optimized Results for Bandwidth Δλ with L and G_{ON-OFF}

*λ*and the span length

*L*or the ON-OFF (or gross) Raman gain

*G*

_{ON-OFF}in Figs. 4 and 5, respectively. In calculations, we assumed that

*G*

_{ON-OFF}>

*αL*and Δ

*G*<1.1 dB in Fig. 4, and

*L*=50 km and Δ

*G*<1.1 dB in Fig. 5.

*λ*approximately linearly decreases with increasing

*L*and

*G*

_{ON-OFF}, and the relationship is that Δ

*λ*=-0.88322×

*L*+114.59 nm and Δ

*λ*=-7.09127×

*G*

_{ON-OFF}+71.35 nm in our optimizations, respectively. Therefore, extending the fiber span length

*L*and increasing the gross Raman gain

*G*

_{ON-OFF}are done at the cost of decreasing signal bandwidth Δ

*λ*.

*L*=80 km in Fig. 4, we can obtain eight global optima in this HGA, whose parameter set is tabulated in Table 2 (in calculations, we assume that the number of maximum

*N*=8). Compared with our previous reports [10

_{c}**21**, 3446–3455 (2003).. [CrossRef]

**230**, 425–431 (2004). [CrossRef]

*λ*in the experiments, there are several best candidates after optimizing the parameter set of pump spectra based on the new HGA.

### 5.2. A movie for the detailed procedure of FRA

*L*=40 km, Δ

*λ*≥82.5 nm on the conditions of Δ

*G*<1.1 dB and

*G*

_{ON-OFF}>

*αL*, and Δ

*G*of all signal is less than 2.5 dB. Other parameters are the same as in Fig. 4. The red dots in the movie represent the signal channels. The optimal parameters of four pumps are that

*P*

_{1, 2, 3, 4}=207.66, 186.44, 111.28, 139.68 mW, and

*λ*

_{1, 2, 3, 4}=1432.68, 1446.77, 1466.14, 1498.65 nm. The movie shows that ① there are strong interactions of signal-to-signal and pump-to-signal; ② the pump-to-signal interaction can compensate the attenuation of signals and increase signal power when

*z*>~30 km; ③ SRS effects of signal-to-signal make the power of higher-frequency (i.e., shorter wavelength) waves flow into that of lower-frequency (i.e., longer-wavelength) waves.

### 5.3. Optical signal-to-noise ratio

*L*=40 and 50 km in Fig. 4. In calculating Eq. (2), a midpoint shooting algorithm is used, which is from Ref. [17] [In calculating Eqs. (1) and (2), different shooting algorithms are adopted. Equation (1) is simulated by a “pure” shooting algorithm that has faster computational speed, and Eq. (2) is done with a mid-point shooting algorithm that has better stability. Their detailed comparisons are shown in Ref. [17]). From Fig. 7, we find that the OSNR in FRAs decreases with extending span length

*L*, and OSNR of shorter-wavelength signals is less than that of higher-wavelength signals. To equalize OSNR tilt, one can use the bidirectionally pumping scheme [17].

## 6. Discussions

16. H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. **11**, 530–532 (1999). [CrossRef]

21. D. Dahan and G. Eisenstein, “Numerical comparison between distributed and discrete amplification in a point-to-point 40 Gbit/s 40-WDM- based transmission system with three different modulation formats,” J. Lightwave Technol. **20**, 379–388 (2002). [CrossRef]

**11**, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

*et al*. have employed the traditional GA to optimize the gain-flatness and gain-bandwidth performance, and the optimal results are exciting, but they obtain only a single optimal solution in each domain [9

9. X. Zhou, C. Lu, P. Shum, and T. H. Cheng, “A simplified model and optimal design of a multiwavelength backward-pumped Raman amplifier,” IEEE Photon. Technol. Lett. **13**, 945–947 (2001). [CrossRef]

7. P. C. Xiao, Q. J. Zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. **15**, 206–208 (2003). [CrossRef]

8. M. Yan*et al*., “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. **13**, 948–950 (2001). [CrossRef]

*λ*are obtained on the same conditions. Therefore, our algorithm can offer a powerful tool for optimizing the structure of FRAs, and our optimal results have important applications on the practical design of FRAs.

## 7. Conclusions

**21**, 3446–3455 (2003).. [CrossRef]

**230**, 425–431 (2004). [CrossRef]

*λ*with

*L*and

*G*

_{ON-OFF}are obtained, and Δ

*λ*approximately linearly decreases with the increase of

*L*and

*G*

_{ON-OFF}; i.e., Δ

*λ*=-0.88322×

*L*+114.59 nm and Δ

*λ*=-7.09127×

*G*

_{ON-OFF}+71.35 nm in our optimizations, respectively. A movie based on an optimal example demonstrates that there are strong interactions of pump-to-signal and signal-to-signal, and the SRS effect makes the power of higher-frequency waves transfer into that of lower-frequency waves. The corresponding OSNR of optimal results of

*L*=40 and 50 km exhibits that (1) the noise performance deteriorates with increasing the span length

*L*and (2) shorter-wavelength signals have less OSNR than higher-wavelength signals.

## Acknowledgments

## References and links

1. | P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, “Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,” J. Lightwave Technol. |

2. | X. Zhou and M. Birk, “Performance limitation due to statistical Raman crosstalk in a WDM system with multiple- wavelength bidirectionally pumped Raman amplification,” J. Lightwave Technol. |

3. | S. Faralli and E. Di Pasquale, “Impact of double Rayleigh scattering noise in distributed higher order Raman pumping schemes,” IEEE Photon. Technol. Lett. |

4. | Y. Hadjar |

5. | M. Tang, P. Shum, and Y. D. Gong, “Design of double-pass discrete Raman amplifier and the impairments induced by Rayleigh backscattering,” Opt. Express |

6. | P. B. Hansen, L. Eskilden, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiBiovanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” IEEE Photon. Technol. Lett. |

7. | P. C. Xiao, Q. J. Zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. |

8. | M. Yan |

9. | X. Zhou, C. Lu, P. Shum, and T. H. Cheng, “A simplified model and optimal design of a multiwavelength backward-pumped Raman amplifier,” IEEE Photon. Technol. Lett. |

10. | X. M. Liu and B. Lee, “Optimal design for ultrabroadband amplifier,” J. Lightwave Technol. |

11. | X. M. Liu and B. Lee, “Optimal design of fiber Raman amplifier based on hybrid genetic algorithm,” IEEE Photon. Technol. Lett.16, (to be published). |

12. | X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. |

13. | S. W. Mahfoud, “Niching methods for genetic algorithms,” Ph.D. dissertation (University of Illinois, Urbana, Ill., 1995). |

14. | D. E. Goldberg, |

15. | A. A. B. Tio, P. Shum, and Y. D. Gong, “Wide bandwidth flat gain Raman amplifier by using polarization-independent interferometric filter,” Opt. Express |

16. | H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. |

17. | X. M. Liu, “Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers,” Opt. Express. (this issue). |

18. | X. M. Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” Opt. Express |

19. | X. Liu and Y. Li, “Optimal design of DFG-based wavelength conversion based on hybrid genetic algorithm,” Opt. Express |

20. | X. M. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express |

21. | D. Dahan and G. Eisenstein, “Numerical comparison between distributed and discrete amplification in a point-to-point 40 Gbit/s 40-WDM- based transmission system with three different modulation formats,” J. Lightwave Technol. |

**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: January 5, 2004

Revised Manuscript: February 3, 2004

Published: February 23, 2004

**Citation**

Xueming Liu and Yanhe Li, "Efficient algorithm and optimization for broadband Raman amplifiers," Opt. Express **12**, 564-573 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-4-564

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### References

- 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]
- X. Zhou and M. Birk, �??Performance limitation due to statistical Raman crosstalk in a WDM system with multiple- wavelength bidirectionally pumped Raman amplification,�?? J. Lightwave Technol. 21, 2194-2202 (2003). [CrossRef]
- S. Faralli and E. Di Pasquale, �??Impact of double Rayleigh scattering noise in distributed higher order Raman pumping schemes,�?? IEEE Photon. Technol. Lett. 15, 804-806 (2003). [CrossRef]
- Y. Hadjar et al., �??Enhanced double Rayleigh backscattering in second order Raman amplification and system performance implications,�?? Opt. Commun. 229, 419-423 (2004). [CrossRef]
- 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]
- P. B. Hansen L. Eskilden, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiBiovanni, �??Rayleigh scattering limitations in distributed Raman pre-amplifiers,�?? IEEE Photon. Technol. Lett. 10, 159-161 (1998). [CrossRef]
- P. C. Xiao, Q. J. Zeng, J. Huang, and J. M. Liu, �??A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,�?? IEEE Photon. Technol. Lett. 15, 206-208 (2003). [CrossRef]
- M. Yan et al., �??Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,�?? IEEE Photon. Technol. Lett. 13, 948�??950 (2001). [CrossRef]
- X. Zhou, C. Lu, P. Shum, and T. H. Cheng, �??A simplified model and optimal design of a multiwavelength backward-pumped Raman amplifier,�?? IEEE Photon. Technol. Lett. 13, 945�??947 (2001). [CrossRef]
- X. M. Liu and B. Lee, �??Optimal design for ultrabroadband amplifier,�?? J. Lightwave Technol. 21, 3446-3455 (2003). [CrossRef]
- X. M. Liu and B. Lee, �??Optimal design of fiber Raman amplifier based on hybrid genetic algorithm,�?? IEEE Photon. Technol. Lett. 16, (to be published).
- 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]
- S. W. Mahfoud, �??Niching methods for genetic algorithms,�?? Ph.D. dissertation (University of Illinois, Urbana, Ill., 1995).
- D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, New York, 1989).
- 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]
- H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, �??Pump interactions in a 100-nm bandwidth Raman amplifier,�?? IEEE Photon. Technol. Lett. 11, 530-532 (1999). [CrossRef]
- X. M. Liu, �??Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers,�?? Opt. Express. (this issue).
- 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]
- X. Liu and Y. Li, �??Optimal design of DFG-based wavelength conversion based on hybrid genetic algorithm,�?? Opt. Express 11, 1677-1688 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-14-1677">.http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-14-1677</a> [CrossRef] [PubMed]
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