## Pump scheme for gain-flattened Raman fiber amplifiers using improved particle swarm optimization and modified shooting algorithm

Optics Express, Vol. 18, Issue 11, pp. 11033-11045 (2010)

http://dx.doi.org/10.1364/OE.18.011033

Acrobat PDF (1836 KB)

### Abstract

An effective pump scheme for the design of broadband and flat gain spectrum Raman fiber amplifiers is proposed. This novel approach uses a new shooting algorithm based on a modified Newton-Raphson method and a contraction factor to solve the two point boundary problems of Raman coupled equations more stably and efficiently. In combination with an improved particle swarm optimization method, which improves the efficiency and convergence rate by introducing a new parameter called velocity acceptability probability, this scheme optimizes the wavelengths and power levels for the pumps quickly and accurately. Several broadband Raman fiber amplifiers in C + L band with optimized pump parameters are designed. An amplifier of 4 pumps is designed to deliver an average on-off gain of 13.3 dB for a bandwidth of 80 nm, with about ± 0.5 dB in band maximum gain ripples.

© 2010 OSA

## 1. Introduction

4. V. E. Perlin and H. G. Winful, “On distributed Raman amplification for ultrabroad-band long-haul WDM systems,” J. Lightwave Technol. **20**(3), 409–416 (2002). [CrossRef]

5. M. Giltrelli and M. Santagiustina, “Semianalytical approach to the gain ripple minimization in multiple pump fiber Raman amplifiers,” IEEE Photon. Technol. Lett. **16**(11), 2454–2456 (2004). [CrossRef]

8. M. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, “New design scheme for multiple-pump Raman fiber amplifiers using a simulated annealing algorithm,” Microw. Opt. Technol. Lett. **30**(6), 434–436 (2001). [CrossRef]

9. H. Jiang, K. Xie, and Y. Wang, “Photonic crystal fiber for use in fiber Raman amplifiers,” Electron. Lett. **44**(13), 796–798 (2008). [CrossRef]

11. H. M. Jiang, K. Xie, and Y. F. Wang, “C band single pump photonic crystal fiber Raman amplifier,” Chin. Sci. Bull. **55**(6), 555–559 (2010). [CrossRef]

7. M. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. **13**(9), 948–950 (2001). [CrossRef]

8. M. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, “New design scheme for multiple-pump Raman fiber amplifiers using a simulated annealing algorithm,” Microw. Opt. Technol. Lett. **30**(6), 434–436 (2001). [CrossRef]

12. B. Neto, A. L. Teixeira, N. Wada, and P. S. André, “Efficient use of hybrid Genetic Algorithms in the gain optimization of distributed Raman amplifiers,” Opt. Express **15**(26), 17520–17528 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-26-17520. [CrossRef] [PubMed]

15. V. E. Perlin and H. G. Winful, “Optimal design of flat-gain wide-band fiber Raman amplifiers,” J. Lightwave Technol. **20**(2), 250–254 (2002). [CrossRef]

16. J. Zhou, J. Chen, X. Li, G. Wu, Y. Wang, and W. Jiang, “Robust, compact, and flexible neural model for a fiber Raman amplifier,” J. Lightwave Technol. **24**(6), 2362–2367 (2006). [CrossRef]

17. P. Xiao, Q. Zeng, J. Huang, and J. Liu, “Pump optimal configuration algorithm for multi-pumped sources of distributed Raman fiber amplifiers,” Proc. SPIE **4906**, 433–441 (2002). [CrossRef]

18. Z. Lalidastjerdi, F. Kroushavi, and M. Rahmani, “An efficient shooting method for fiber amplifiers and lasers,” Opt. Laser Technol. **40**(8), 1041–1046 (2008). [CrossRef]

22. Q. Han, J. Ning, H. Zhang, and Z. Chen, “Novel shooting algorithm for highly efficient analysis of fiber Raman amplifiers,” J. Lightwave Technol. **24**(4), 1946–1952 (2006). [CrossRef]

## 2. Raman coupled equations

23. X. Liu and M. Zhang, “An effective method for two-point boundary value problems in Raman amplifier propagation equations,” Opt. Commun. **235**(1-3), 75–82 (2004). [CrossRef]

4. V. E. Perlin and H. G. Winful, “On distributed Raman amplification for ultrabroad-band long-haul WDM systems,” J. Lightwave Technol. **20**(3), 409–416 (2002). [CrossRef]

15. V. E. Perlin and H. G. Winful, “Optimal design of flat-gain wide-band fiber Raman amplifiers,” J. Lightwave Technol. **20**(2), 250–254 (2002). [CrossRef]

23. X. Liu and M. Zhang, “An effective method for two-point boundary value problems in Raman amplifier propagation equations,” Opt. Commun. **235**(1-3), 75–82 (2004). [CrossRef]

*P*,

_{j}*P*and

_{i}*P*is the optical power level in the

_{k}*j*’th,

*i*’th and

*k*’th channel respectively, and

*N*is the total number of channels including pump and signal waves, “+” is designated to the forwards traveling waves and “-” is designated to the backwards traveling waves along the fiber,

*g*

_{R}is the Raman gain coefficient of the fiber,

*ν*,

_{j}*ν*and

_{i}*ν*is the frequency of the optical signals in the

_{k}*j*’th,

*i*’th and

*k*’th channels respectively, α

_{j}is the absorption coefficient accounting for the fiber loss at the frequency

*ν*,

_{j}*K*

_{eff}≈2 is the polarization factor,

*A*

_{eff}is the effective overlap core area between waves of different channels. Without loss of generality, the channels are so numbered that the frequency is descending from the first channel to the

*N*th channel. Equation (1) indicates that when light wave of frequency

*ν*propagates along the fiber with waves of other frequencies, due to the stimulated Raman scattering, it receives energy from light waves with frequency larger than

_{j}*ν*, in the meantime it losses energy to light waves with frequency smaller than

_{j}*ν*. Equation (1) has no analytical solution in general. It has to be dealt with numerically.

_{j}## 3. Modified shooting algorithm

### 3.1 The basic shooting algorithm for counter pumped Raman coupled equations

*m*signals propagating from Port A to Port B and

*n*pumps propagating from Port B to Port A in an RFA. This physical picture means that the power levels of signals

*P*

_{As}

*,*

_{i}*i*= 1, 2, …,

*m*are known at Port A for every channel and the power levels of pumps

*P*

_{Bp}

*,*

_{j}*j*= 1, 2, ...,

*n*are known at Port B for every channel. The interaction between signals and pumps can be analyzed using Eq. (1) if power levels are known at Port A for all channels including signals and pumps. Since the power levels of pumps are given at Port B instead of at Port A in a counter pumped RFA configuration, an initial guess for the pumps at Port A has to be made. Then for given power levels of signals

*P*_{As}= [

*P*

_{As1},

*P*

_{As2}, ···,

*P*

_{As}

*] and guessed power levels of pumps*

_{m}**’**

*P*_{Ap}= [

*P*’

_{Ap1},

*P*’

_{Ap2}, ···,

*P*’

_{Ap}

*] at Port A, Eq. (1) can be used to build up the evolution picture of the optical channels from Port A to Port B by calculating all power levels step by step along the fiber. The final results obtained at Port B are designated as*

_{n}

*P*_{Bs}= [

*P*

_{Bs1},

*P*

_{Bs2}, ···,

*P*

_{Bs}

*] for the signals and*

_{m}**’**

*P*_{Bp}= [

*P*’

_{Bp1},

*P*’

_{Bp2}, ···,

*P*’

_{Bp}

*] for the pumps. The above calculation process is the so called “shooting” and*

_{n}**’**

*P*_{Bp}is called the “shot values”. The shot values

**’**

*P*_{Bp}is then compared with the given pump power levels

*P*_{Bp}= [

*P*

_{Bp1},

*P*

_{Bp2}, ···,

*P*

_{Bp}

*] at Port B, and their discrepancy is designated as an error vector*

_{n}**= [**

*E**P*’

_{Bp1}-

*P*

_{Bp1},

*P*’

_{Bp2}-

*P*

_{Bp2}, ···,

*P*’

_{Bp}

*-*

_{n}*P*

_{Bp}

*]. The Euclidean norm of*

_{n}**is a measure of the fitness of the initial guess. Obviously, when all the components of the error vector**

*E***reduce to zero, the guessed values**

*E***’**

*P*_{Ap}correctly reflect the true power levels of the pumps that would emerge from Port A. At this point the searching process for solution of Eq. (1) under the boundary conditions

*P*_{As}and

*P*_{Bp}finishes. In order to minimize the error vector

**the Newton-Raphson method is used, with the following formulae to update the guessed values**

*E***’**

*P*_{Ap}. where ∆

**’**

*P*_{Ap}and

**are respectively the change of**

*J***’**

*P*_{Ap}and Jacobian matrix.

### 3.2 The method of initial guess and the contraction factor

*P*_{Bp}for pumps at port B is a good start, because after long distance consuming the output power level should not exceed the input power level. Still a better confine could be derived from the following considerations. Firstly, let the first individual pump wave with power level

*P*

_{Bp1}to propagate on its own along the fiber from Port B to Port A. As it reaches Port A its power level is numerically obtained as

*P*’

_{Ap1max}. Secondly, let both the first and the second pump lights of

*P*

_{Bp1}and

*P*

_{Bp2}to propagate along the fiber from Port B to Port A, and the power level of the second pump at Port A is obtained as

*P*’

_{Ap2max}. This process is repeated till the last pump channel. The obtained vector

**’**

*P*_{Apmax}= (

*P*’

_{Ap1max},

*P*’

_{Ap2max}, ···,

*P*’

_{Ap}

_{n}_{max}) can then be used as a better upper limit for the initial guess for

**’**

*P*_{Ap}. Why this is the case? For the first pump the reason is quite obvious. When propagates with other channels it losses power to the other channels so its power level cannot exceed

*P*’

_{Ap1max}in reaching Port A. For other channels the situation is more complicated, since they can receive power from channels of higher frequencies when propagate with other channels. However, this character has in fact been taken into account in the procedure of obtaining

**’**

*P*_{Apmax}. Taking the second pump for example, the parameter

*P*’

_{Ap2max}is obtained in the case when the second pump propagates along the fiber with the first pump. In this case the power the second pump received is already the maximum it could receive from the first pump. When the second pump propagates along the fiber with all other channels, it would receive less power from the first pump than that in the case of propagating with the first pump alone. Because the first pump now also losses power to other channels so it has less power to deliver to the second pump. In the mean time the second pump also losses power to channels of lower frequencies. For the two reasons the power level of the second pump in reaching Port A should be smaller than

*P*’

_{Ap2max}. Similar arguments apply to all other pump channels. It is therefore established that

**’**

*P*_{Apmax}represents an upper boundary for the possible power levels of pumps at Port A.

*P*

_{Ap}

*, is less than the corresponding component*

_{j}*P*’

_{Ap}

_{j}_{max}of

**’**

*P*_{Apmax}. It falls between zero and

*P*’

_{Ap}

_{j}_{max}. By defining a contraction factor

*d*that is a parameter ranging from 0 to 1, the vector

*d*⋅

**’**

*P*_{Apmax}could be used as a good initial guess for the pump power levels at Port A. The use of contraction factor

*d*in the design practice of broad and flat band RFAs produces a good start point for the shooting process. This scheme is tested in many numerical simulations in this study. It is found that the convergence of the shooting program is guaranteed in all cases.

### 3.3 The adjustment of step sizes

**’**

*P*_{Ap}for the updating. It is found that with this algorithm the shooting process fails to converge from time to time. A further analysis reveals two pitfalls exist for the algorithm. In one occasion the value of

*P*’

_{Ap}

_{j}^{new}is found fall out of the interval (0,

*P*

_{Ap}

_{j}_{max}). Obviously, the case

*P*’

_{Ap}

_{j}^{new}< 0 corresponds to no physical situation, and the case

*P*’

_{Ap}

_{j}^{new}>

*P*’

_{Ap}

_{j}_{max}would lead to a worse error vector

**. In other occasions it is found that even though 0 <**

*E**P*’

_{Ap}

_{j}^{new}<

*P*’

_{Ap}

_{j}_{max}is satisfied, the Euclidean norm of the error vector

**, ||**

*E***||, increased in the succeeding step. In order to obtain a more reasonable amount of adjustment ∆**

*E***’**

*P*_{Ap}, the scheme outlined in the dashed square in Fig. 1 is proposed, where

*l*

_{1}and

*l*

_{2}is the number of Loop 1 and Loop 2 respectively. The criterion for the new adjustment ∆

**’**

*P*_{Ap}is to meet both 0 <

*P*’

_{Ap}

_{j}^{new}<

*P*’

_{Ap}

_{j}_{max}and ||

*E**|| < ||*

_{k}

*E*

_{k}_{-1}||, where

*E*

_{k}_{-1}and

*E**is the error vector for the (*

_{k}*k*-1)’th and

*k*’th shooting respectively. If the two conditions are not met, the adjustment ∆

**’**

*P*_{Ap}would be reduced till the two conditions are satisfied. The flowchart of the proposed modified shooting algorithm for the coupled equations in counter-pumped RFAs is illustrated in Fig. 1. The termination criterion for the shooting process is the maximum number of shooting or the minimum of ||

**||.**

*E*## 4. Improved particle swarm optimization

### 4.1 The standard particle swarm optimization and its social psychology basis

*D*dimensional space there is a swarm consists of

*m*particles. The position of the

*i*’th particle is represented by a

*D*dimensional vector

*x**= (*

_{i}*x*

_{i}_{1},

*x*

_{i}_{2}, ···,

*x*), where

_{iD}*i*= 1, 2, ···,

*m*. The substitution of

*x**into an objective function generates a fitting value and by which the fitness of*

_{i}

*x**is measured. The flying velocity of the*

_{i}*i*’th particle is another

*D*dimensional vector

*v**= (*

_{i}*v*

_{i}_{1},

*v*

_{i}_{2}, ···,

*v*). Suppose the optimal position of the

_{iD}*i*’th particle obtained so far is

*p**= (*

_{i}*p*

_{i}_{1},

*p*

_{i}_{2}, ···,

*p*), and the best particle position found so far in the whole swarm is

_{iD}

*p*_{g}= (

*p*

_{g1},

*p*

_{g2}, ···,

*p*

_{g}

*), then the position and velocity of the*

_{D}*i*’th particle for the next optimization step is calculated as [34] where

*k*is the number of iteration,

*d*= 1, 2, ···,

*D*,

*w*is a non-negative constant or variable called inertia weight,

*c*

_{1,2}is another non-negative constant or variable called learning factor, and

*r*

_{1},

*r*

_{2}∈ [0,1] are random numbers. In practice, if boundary exists the following restrictions also need to be enforced:

*v*∈ [

_{id}*v*

_{d}_{min},

*v*

_{d}_{max}],

*x*∈ [

_{id}*x*

_{d}_{min},

*x*

_{d}_{max}], where

*v*

_{d}_{min},

*v*

_{d}_{max},

*x*

_{d}_{min}, and

*x*

_{d}_{max}are pre-specified constants. When

*v*and (or)

_{id}*x*exceed(s) the boundary, it is reset to the boundary value. In this PSO scheme,

_{id}

*x**is a potential solution of the problem in hand.*

_{i}### 4.2 Velocity acceptability probability and improved particle swarm optimization

## 5. Results and discussions

36. X. Liu and Y. Li, “Efficient algorithm and optimization for broadband Raman amplifiers,” Opt. Express **12**(4), 564–573 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-4-564. [CrossRef] [PubMed]

37. X. Liu and Y. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. **230**(4-6), 425–431 (2004). [CrossRef]

*d*is set to 0.3 and the velocity acceptability probability

*c*

_{3}is set to 0.75. The target of optimization is to find a pump scheme that delivers an average on-off gain of about 13 dB over the whole spectrum from 1530 nm to 1610 nm, with in band ripples of less than ± 0.5 dB.

- 1. Initializing stochastically the positions and velocities of the 20 particles in the pre-specified ranges. Here the positions represent the pumps’ wavelengths and power levels, and velocities represent the increments of these quantities.
- 2. Calculating the on-off gain spectrum of the RFA with the modified shooting algorithm.
- 3. Judging if the target of optimization is met, or if the maximum number of iteration has reached. If the answer is ‘Yes’, terminate the routine and deliver the results, otherwise go to step 4.

**||, which is a measurement of the overall discrepancy between current status and the target status, versus the number of shooting is also listed in Table 2. It can be seen that the value of ||**

*E***|| changes from 259.8 mW for the first shooting to 0.7 mW for the third shooting. The speed of convergence of the shooting is satisfactory. The evolution of the maximum amplitude of gain ripples ∆G**

*E*_{max}for standard PSO and improved PSO via the normalized time is illustrated in Fig. 4 . It can be seen that ∆G

_{max}of both PSO reduces dramatically in the initial stage. In the improved PSO when pitfalls are encountered the particles jump out of the local extremum quickly and the optimization processes to its destination steadily. However, in the standard PSO the progress is trapped into local extremum longer so the algorithm takes a much longer time to converge to the target. The calculation time for the improved PSO to reach the final result is 22.5% of that of the standard PSO, even though its starting point is slightly worse (initial value of ∆G

_{max}is 3.9dB for the improved PSO, while it is 3.8dB for the standard PSO).

31. A. Mowla and N. Granpayeh, “Design of a flat-gain multipumped distributed fiber Raman amplifier by particle swarm optimization,” J. Opt. Soc. Am. A **25**(12), 3059–3066 (2008). [CrossRef]

*c*

_{3}). This is one of the advantages of our method. Our RFA1 is also more cost-effective. It is seen that the number of pumps is reduced by one third while the number of signal channels is increased by more than one third in RFA1. This indicates that the complexity of an amplifier can be reduced significantly by the proposed scheme. This is the second advantage of our method, among others.

32. A. Mowla and N. Granpayeh, “Optimum design of a hybrid erbium-doped fiber amplifier/fiber Raman amplifier using particle swarm optimization,” Appl. Opt. **48**(5), 979–984 (2009). [CrossRef] [PubMed]

32. A. Mowla and N. Granpayeh, “Optimum design of a hybrid erbium-doped fiber amplifier/fiber Raman amplifier using particle swarm optimization,” Appl. Opt. **48**(5), 979–984 (2009). [CrossRef] [PubMed]

_{max}obtainable from an RFA of 6 pumps or 8 pumps is smaller than that of an RFA of 4 pumps. However, from the viewpoint of practical application the use of more pumps than necessary is not cost-effective. The distribution of optimized wavelengths and power levels of 6- and 8-pump RFAs confirms this point. Some wavelengths listed in Table 3 are very close. This fact indicates that some of the multiple pumps could be combined into one pump with almost the same total pump power. The total pump power of 4-, 6- and 8-pump RFA are respectively 716.3, 758.2, and 735.4 mW.

*P*

_{signal},

*N*

_{shooting}respectively represent the power level of signal per channel, the number of shooting when convergence is reached or break-down occurs, T and F stand for True (convergence) and False (divergence).

*P*

_{signal}of the optimized 4-pump RFA while keeping all the other parameters unchanged, and compare the outcomes of the two shooting algorithms. It can be seen from Table 4 that for small

*P*

_{signal}(0.1 - 0.18 mW) both BSA and MSA converge in 3 shootings. When

*P*

_{signal}is increased to moderate power levels (0.20 - 0.8 mW), the BSA becomes unstable. It either needs more shooting cycles to converge, or fails to converge like in the case of 0.2 mW where numerical overflow occurs or in the case of 0.7 mW where oscillatory behavior of the Euclidean norm of

**takes place. The maximum number of shooting allowed is 100 in this example. The approaching processes to the target values of pump power levels and the norm of error vector, ||**

*E***||, for**

*E**P*

_{signal}= 0.8 mW are illustrated in Fig. 5 and Fig. 6 respectively. We can see from these figures, the BSA needs 7 shootings to converge the pump power levels and ||

**|| to their target values. While the MSA needs only 4 shootings to achieve same results. The calculation efficiency is therefore improved by approximate 43.0% in the proposed scheme. When**

*E**P*

_{signal}is increased further (> 0.9 mW) the BSA fails completely. On the contrary, the MSA converges for any level of

*P*

_{signal}. It produces the right solution within a reasonable number of shootings (usually 3 - 4 and in the worst case 7 shootings). It is impressive that in the deep saturated region (

*P*

_{signal}= 0.9 - 2.0 mW, corresponding to total signal power of as high as 72.9 – 162 mW) when the BSA fails completely, the MSA still converges to the solution within a very modest number of shootings. Obviously the MSA is more efficient and stable than the BAS, at least in these examples. We owe the better performances of the MSA to the better initial pump power and the new adjustment mechanics. Details are described in the sections 3.2 and 3.3.

*d*and

*c*

_{3}plays an important role in the design process. With regard to

*d*, smaller values should be used in circumstances of stronger pump or signal, or longer fiber, due to the heavy pump depletion. Generally speaking, if

*d*= (0.3 - 0.8) is used, convergence could usually be achieved in less than 5 shootings. However, if the pump power is very high and/or the fiber is very long, a smaller value than 0.1 is usually needed for

*d*, and at the mean time the number of required shootings may increase. As examples, we calculated the cases of the 4-pump Raman amplifier when the power level of each pump is 500 mW (regarded as higher), 1000 mW (regarded as high enough) respectively for fiber length of 80 km and 100 km (respectively regarded as longer and long enough), as suggested by [23

23. X. Liu and M. Zhang, “An effective method for two-point boundary value problems in Raman amplifier propagation equations,” Opt. Commun. **235**(1-3), 75–82 (2004). [CrossRef]

*d*and the required number of shootings are both list in Table 5 . It is found that the pump power levels have larger impact than the length of fiber on the value of contraction factor and the number of shootings. This is because after some extent of a fiber length the interactions between different optical channels decline and fiber losses dominate at this time. The complexity of the amplification system reduces.

**235**(1-3), 75–82 (2004). [CrossRef]

*c*

_{3}, it could be viewed as an adjustor between the efficiency and the global searching ability for the improved PSO. A bigger

*c*

_{3}could increase the efficiency of PSO, but decrease the global searching ability at the meantime. Through lots of tests on many benchmark functions it is found that if

*c*

_{3}= (0.35 - 0.75) is used, a good balance between the efficiency and the global searching ability for the improved PSO could be achieved.

## 6. Conclusions

*d*, a more practical initial guess can be put forward for the problem of counter pumping. Better stability and higher efficiency is achieved for the program. To obtain the optimal wavelengths and power levels for the pump scheme, we resort to the particle swarm optimization. By introducing a new parameter called velocity acceptability probability (

*c*

_{3}) the standard particle swarm optimization is recast to improve efficiency and convergence rate. With proper choice of

*c*

_{3}, both efficiency and global searching ability are improved over the standard PSO. Combining the modified shooting algorithm with the improved particle swarm optimization, an optimized pump scheme for a Raman fiber amplifier with broadband, flat gain spectrum and small gain ripples in interesting band could be designed efficiently. As an example, a 4-pump Raman fiber amplifier with 80nm bandwidth in C + L band is designed. The average on-off gain is about 13 dB, with about ± 0.5 dB in band ripples. This design procedure provides a good alternative over others for the search of optimal pump scheme for broadband and flat gain spectrum Raman fiber amplifiers.

## Acknowledgments

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20. | J. Hu, B. S. Marks, Q. Zhang, and C. R. Menyuk, “A shooting algorithm to model backward-pumped Raman amplifiers,” In Proc. LEOS, |

21. | J. Ning, Q. Han, Z. Chen, J. Li, and X. Li, “A powerful simple shooting method for designing multi-pumped fibre Raman amplifiers,” Chin. Phys. Lett. |

22. | Q. Han, J. Ning, H. Zhang, and Z. Chen, “Novel shooting algorithm for highly efficient analysis of fiber Raman amplifiers,” J. Lightwave Technol. |

23. | X. Liu and M. Zhang, “An effective method for two-point boundary value problems in Raman amplifier propagation equations,” Opt. Commun. |

24. | S. M. Roberts, and J. S. Shipman, “ |

25. | R. Eberhart, and J. Kennedy, “A new optimizer using particle swarm theory,” In Proc. Int. Symp. Micromechatronics. Hum. Sci., 1995, pp. 39–43. |

26. | J. Kennedy, and R. Eberhart, “Particle swarm optimization,” In Proc. IEEE Int. Conf. Neural Networks, 1995, pp. 1942–1948. |

27. | E. C. Laskari, K. E. Parsopoulos, and M. N. Vrahatis, “Particle swarm optimization for minimax problems,” In: Proc. IEEE Conf. Evol. Comput., 2002, pp. 1576–1581. |

28. | R. Mendes, P. Cortez, M. Rocha, and J. Neves, “Particle swarms for feed forward neural network training,” In Proc. Int. Jt. Conf. Neural Networks, 2002, pp. 1895–1899. |

29. | H. M. Jiang, K. Xie, and Y. F. Wang, “Design of multi-pumped Raman fiber amplifier by particle swarm optimization,” J. Optoelectron., Laser |

30. | H. M. Jiang, K. Xie, and Y. F. Wang, “Novel design method for a multi-wavelength backward-pumped fiber raman amplifier to achieve a flat gain spectrum,” In Proc. Int. Conf. Opt. Commun. Networks, 2005, pp. 30–32. |

31. | A. Mowla and N. Granpayeh, “Design of a flat-gain multipumped distributed fiber Raman amplifier by particle swarm optimization,” J. Opt. Soc. Am. A |

32. | A. Mowla and N. Granpayeh, “Optimum design of a hybrid erbium-doped fiber amplifier/fiber Raman amplifier using particle swarm optimization,” Appl. Opt. |

33. | H. M. Jiang, K. Xie, and Y. F. Wang, “Shooting algorithm and particle swarm optimization based raman fiber amplifiers gain spectra design,” Opt. Commun. (to be published), doi:. [PubMed] |

34. | Y. Shi, and R. Eberhart, “A modified particle swarm optimizer,” In Proc. IEEE Conf. Evol. Comput., 1998, pp. 69–73. |

35. | X. Yao, Y. Liu, and G. Lin, “Evolutionary programming made faster,” IEEE Trans. Evol. Comput. |

36. | X. Liu and Y. Li, “Efficient algorithm and optimization for broadband Raman amplifiers,” Opt. Express |

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

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

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

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

(290.5910) Scattering : Scattering, stimulated Raman

(230.4480) Optical devices : Optical amplifiers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: March 18, 2010

Revised Manuscript: April 25, 2010

Manuscript Accepted: April 26, 2010

Published: May 11, 2010

**Citation**

Hai-ming Jiang, Kang Xie, and Ya-fei Wang, "Pump scheme for gain-flattened Raman fiber amplifiers using improved particle swarm optimization and modified shooting algorithm," Opt. Express **18**, 11033-11045 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-11033

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- J. Kennedy, and R. Eberhart, “Particle swarm optimization,” In Proc. IEEE Int. Conf. Neural Networks, 1995, pp. 1942–1948.
- E. C. Laskari, K. E. Parsopoulos, and M. N. Vrahatis, “Particle swarm optimization for minimax problems,” In: Proc. IEEE Conf. Evol. Comput., 2002, pp. 1576–1581.
- R. Mendes, P. Cortez, M. Rocha, and J. Neves, “Particle swarms for feed forward neural network training,” In Proc. Int. Jt. Conf. Neural Networks, 2002, pp. 1895–1899.
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- H. M. Jiang, K. Xie, and Y. F. Wang, “Shooting algorithm and particle swarm optimization based raman fiber amplifiers gain spectra design,” Opt. Commun. (to be published), doi:. [PubMed]
- Y. Shi, and R. Eberhart, “A modified particle swarm optimizer,” In Proc. IEEE Conf. Evol. Comput., 1998, pp. 69–73.
- X. Yao, Y. Liu, and G. Lin, “Evolutionary programming made faster,” IEEE Trans. Evol. Comput. 3(2), 82–102 (1999). [CrossRef]
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