## 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

Because of its flexible gain band, broad gain bandwidth, and low noise figure, Raman fiber amplifiers (RFAs) have recently attracted much attention in the wavelength division multiplexing (WDM) systems for optical fiber communications [1–4]. In order to achieve a broad and flat gain bandwidth multiple pumps are usually used [5–8] besides the use of media with inherent flat Raman gain efficiency [9–11]. In this circumstance, the wavelengths and power levels for these pumps must be chosen carefully so a comprehensive design procedure has to be adopted to ensure a good performance. Several optimization algorithms have been developed to deal with such problems, including simulated annealing [7,8] generic algorithm [12–15], neural network training [16,17] etc.

In this work the optimal wavelengths and power levels of pumps are searched for by an improved particle swarm optimization (PSO). In the searching process it is necessary to evaluate the fitness of every combination of wavelengths and power levels of the pumps. Therefore a set of Raman coupled equations governing the fiber amplifiers have to be solved. For backward-pumped Raman coupled equations, the shooting method [18–22] is one of the widely used algorithms, but it is a very time consuming task and may fail to get the solution in many occasions. In order to overcome these difficulties, three improvements are proposed in this study. Firstly, a parameter called contraction factor is introduced to obtain good values of initial guess. This method excludes the use of certain unreasonable combinations of pump parameters by consulting with the physical picture of stimulated Raman scattering, so that program breakdown caused by bad initial guess could be prevented. Secondly, the Newton-Raphson method is modified by the proposed adjustable step size. Thus a new shooting algorithm with more efficiency and stability is resulted. Thirdly, the standard PSO is improved by introducing a new parameter called velocity acceptability probability, which improves the efficiency and convergence rate. By combining the three improvements, a powerful approach to the pump scheme for flat gain spectrum and broadband Raman fiber amplifiers is developed. A 4-pump Raman fiber amplifier with an average gain of 13.3 dB for a bandwidth of 80 nm and with in-band maximum ripple levels of ± 0.5 dB is designed successfully.

## 2. Raman coupled equations

The basic operation mechanism of RFAs is the stimulated Raman scattering (SRS). When pumps and signals propagate along the fiber SRS occurs between pumps and pumps, between pumps and signals, and between signals and signals at different wavelengths. Due to the fact that power levels of various noise sources, such as spontaneous Raman scattering, Rayleigh scattering, thermal noise, etc., are far below that of the pumps and signals, they can be ignored in the calculation of the on-off gain of signals [23]. Therefore in steady state, the behaviors of Raman fiber amplifiers are governed by the following coupled set of equations [4,15,23]:

*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

When an RFA is counter pumped the signals are launched from one end (hereinafter, Port A) of the fiber and the pumps are launched from the other end (hereinafter, Port B) of the fiber. Finding the solution of Eq. (1) in this situation becomes a two point boundary value problem in mathematics. Generally, the solution can be found with shooting algorithm. A typical shooting algorithm of counter pumped Raman coupled equations using Newton-Raphson [24] method is summarized below.

Suppose there are *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}.

**’**

*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

In the practical shooting process, there are two important aspects need to be taken care of. One is the finding of “good” initially guessed values for the shooting; the other is the development of a “suitable” mechanism to approach the true values from the guessed values. Among them the finding of a good initial guess is very important as the shooting method is sensitive to the initial values used. The accuracy of the initial guess plays a crucial rule in the convergence of the shooting method. In order to increase the quality of initial guess, the range of possible pump power levels needs to be determined. The range can provide a guideline for the initial guess so that to avoid unreasonable guesses that not only waste computational resources but also increase the risk of break down for the shooting process. In this work, the physical picture of stimulated Raman scattering is analyzed, a smaller range of possible pump power levels and more accurate initial guess than before is derived.

Obviously the values of pump power levels could not be negative, so zero is the lower limit for the initial guess. For the upper limit, the given power levels *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.

When all the pumps and signals propagate together, from the above analysis, the resulting output power level of each pump at Port A, *P*
_{Ap}
* _{j}*, is less than the corresponding component

*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

As mentioned before, the mechanism to approach the true values from the initially guessed values in the shooting algorithm is also important. In the basic shooting algorithm Eq. (3) is used to get ∆** 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

The particle swarm optimization is a modern heuristic algorithms based on the social metaphor. It was first proposed as an evolutionary computation methodology [25,26]. It has ties to bird flocking, fish schooling, and swarming theory. This method has now been widely used in function optimization [27], neural network training [28], etc. To the best of our knowledge, the PSO has not found application in the design of RFAs until our works [29,30]. We are the first group that introduced the PSO into the area of RFAs and made progress. Afterwards, the PSO was adopted in more works [31–33] for the design of RFAs. The PSO used in the literatures so far is the standard PSO. In this work an improvement on the standard PSO is proposed. By introducing a new parameter called velocity acceptability probability into the flying velocity formulae, the efficiency and convergence rate of PSO is improved.

The principle of the PSO is as follows. In a *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]

*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}Analyzed from the social psychology view point, the first term on the right hand side of Eq. (5) is the weighted velocity of the particle from the previous iteration, which can be regarded as the particle’s movement inertia. So the value of *w* represents the impact of movement inertia on the particle. The second term is self cognition, which can be regarded as the particle’s ability of thinking according to its historical experiences. Thus, the value of *c*
_{1} describes the impact of historical experiences on individual. The third term reflects particle’s sociality. It can be regarded as the particle’s decision according to historical experiences of all the other particles. Thereby, the value of *c*
_{2} reflects the impact of swarm on individuals. These psychological associations of PSO have deep theoretical basis. During the evolution of social communities, individuals always act according to the experiences of their own and the experiences of the community they belong to.

#### 4.2 Velocity acceptability probability and improved particle swarm optimization

The standard PSO is simple and elegant. Its core code can be implemented by a few lines of computer language. Its required computer resources in term of memory and CPU are modest. However, the operation of the standard PSO is sometime prematurely trapped in a local extremum, in which case the particles stop moving so that the optimization stops from progressing to better values. In order to overcome such difficulty a new parameter is introduced, along with a mechanism for the particles to jump out of the local extremum in such circumstance. The inspiration is originated from the social psychology discussion above. The status of a particle is dependent on three mechanisms, namely, the inertia, self cognition and sociality. From the viewpoint of social psychology, a decision made by an individual according to its behavior inertia, cognitive capability and social impact may sometimes be not good enough in a social system. There is always misjudgment or implicit obedience that causes a wrong decision to be made. Similarly, the velocities of particles derived from Eq. (5) may not always be the best or completely feasible. The particle should be allowed with certain probability to jump out of the given position, just like an individual in a social system is sometime given a chance to correct his misjudgment. This can be incorporated into the PSO by a modification on Eq. (5). The modified formula reads:

*r*

_{3}, like

*r*

_{1}and

*r*

_{2}, is a random number sampled from (0, 1) before each

*v*is generated,

_{id}*c*

_{3}is a predefined constant ranging between 0 and 1. Obviously, when

*c*

_{3}= 1, the improved PSO degenerates into the standard PSO. While if

*c*

_{3}= 0, no particle will move. So the parameter

*c*

_{3}is a measure of possibility if the velocity would be accepted or not. Hereinafter it is called the velocity acceptability probability. The modified PSO is applied to several benchmark functions including the Six-hump camel-back, Goldstein-Price, Schaffer f6, etc., and excellent agreement with the existing theoretical results [35] are obtained. It is found that by the introduction of

*c*

_{3}the PSO is improved remarkably both in convergence rate and efficiency.

## 5. Results and discussions

As we known, the optimal gain bandwidth of a Raman amplifier inversely changes with fiber length and the on-off gain [36,37]. The purpose of the current work is to effectively find the optimal or quasi-optimal wavelengths and pump power levels that lead to the desired on-off gain and gain ripples in the specified signal bandwidth for given parameters. Using the modified shooting algorithm and improved PSO,a pump scheme for a C + L broadband flat gain spectrum 4-pump RFA is optimized with the following parameters. The gain media is a 50 km conventional fiber. The wavelengths and power levels to be optimized for the 4 pumps range from 1400 nm – 1500 nm and 50 mW – 300 mW respectively. The wavelengths of signals range from 1530 nm to 1610 nm in 1 nm steps, and the power level of each signal is 0.1 mW. In optimization, the contraction factor *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.

The key steps of the optimization are listed below:

- 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.

The overall optimization procedure is summarized in Fig. 2 .

The wavelengths and power levels for the optimized 4-pump scheme are listed in Table 1
. The gain spectrum of the corresponding RFA is shown in Fig. 3
. It is seen that the resulting gain spectrum of the RFA has a bandwidth of 80 nm, an average gain of about 13 dB, and in band ripples of less than ± 0.5 dB. The target of optimization is fulfilled. The target values and intermediate values of pump power levels in shooting iterations for the four-pump RFA are listed in Table 2
. For the first pump, its power level for the first shot is 217.1 mW higher than its target value. The discrepancy of the first shot power level and the target one for pump 2 is about 33.7 mW. It is seen from Table 2 that after 3 shootings, the discrepancies between the shot values and the target values reduce to less than 0.5 mW for all four pumps. The value of ||** E**||, 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 ||

**|| 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).

Comparing our RFA (denoted as RFA1 thereinafter) with that of [31] (counter pumped by six pumps with optimized wavelengths and powers, denoted as RFA2 thereinafter), it is found that (1) The number of signal channels of RFA1 is 81 while that of RFA2 is 57; (2) The number of pumps of RFA1 is 4 while that of RFA2 is 6; (3) The in-band gain ripples of RFA1 is ± 0.5 dB while that of RFA2 is ± 0.4 dB; (4) The gain bandwidth of RFA1 is 80 nm while that of RFA2 is 92.1 nm. These data show that RFA1 uses fewer pumps while achieves a similar level of in-band gain ripples and gain bandwidth to that of RFA2.

It is worth noting that the number of signal channels of RFA1 is 24 larger than that of RFA2. The total number of equations via Eq. (1) dealt with in RFA1 is therefore 22 larger than that of RFA2. The complexity of the set of nonlinear Eq. (1) increases quickly with the number of equations so the numerical task dealt with in RFA1 is far more complicated than that of RFA2. It is demonstrated that the improved PSO can find results in a more complicated nonlinear system accurately and efficiently. This improvement owes to the introduction of the velocity acceptability probability parameter (*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.

By adding up the gain spectra of an EDFA and an RFA, the advantage of a hybrid erbium-doped fiber amplifier/Raman fiber amplifier (EDFA/RFA) is the large bandwidth achievable by this approach [32]. However, it is not as cost-effective as a pure Raman fiber amplifier. Its level of in-band gain ripples is difficult to control. Its design is also more involved. In [32], a hybrid EDFA/RFA with a bandwidth of 97.9 nm was put forward. This hybrid EDFA/RFA includes an EDFA and a 6- or a 10-pump RFA. The level of in-band gain ripples is 2.91 dB (2.03 dB) for the hybrid EDFA/RFA of the 6-pump (10-pump) RAF. For our pure RFA1, the level of in-band gain ripples is ± 0.5 dB and the gain bandwidth is 80 nm. Although the bandwidth of our pure RFA1 is slightly narrower than that of the hybrid EDFA/RFAs, we have used fewer pumps. The level of in-band gain ripples is also reduced greatly in our scheme. Small in-band gain ripple is essential in many applications. On the structure side the shortcoming of a hybrid erbium-doped fiber amplifier/fiber Raman amplifier is the demand of an additional EDFA, which is constructed on a section of erbium-doped fiber as gain media. A pure fiber Raman amplifier uses the transmission fiber itself as gain media, therefore not only costs much less but also is easy to construct

For further verification, the proposed method is applied to the design of RFAs of 6 pumps and 8 pumps. The results are listed in Table 3
. Generally speaking, the ripple size ∆G_{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.

The comparison between basic shooting algorithm (BSA) and modified shooting algorithm (MSA) is made in terms of efficiency and convergence characteristic. The results are listed in Table 4
, and illustrated by Fig. 5
- Fig. 6
, where *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).

In order to test the performance of the shooting algorithms in both unsaturated and saturated regime, we increase the signal power *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 ** E** 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, ||

**||, 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.

Our modified shooting algorithm is found robust and works for all complex conditions, include the case of high enough pump power and long enough amplifiers. The selection of suitable values for parameters *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]. The other parameters are kept unchanged. The suitable values of *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.

It is worth pointing out that although the basic Newton-Raphson method could be used for the case of very long fiber and/or very strong pump power levels, by splitting the fiber into 2 segments as suggested in [23], the calculation efficiency decreases dramatically. Because in this circumstance, not only the backward pump powers at Port A, but also the powers of forward pumps and signals are unknown and need to be guessed initially. According to Eq. (2) – Eq. (4), a large proportion of calculation effort will be spent on the calculation of Jacobian matrix, especially when the total number of pumps and signals are large. In this sense our proposed scheme is much superior.

In regarding *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

In pursuing the optimal pump scheme for broadband and flat gain spectrum Raman fiber amplifiers, the improved particle swarm optimization method is introduced into this area for the first time. To find the solution of the counter-pumped Raman coupled equations more efficiently, a new shooting algorithm is proposed in this work. The modified Newton-Raphson method that reflects the physical meaning of the solution more closely is used to improve the convergence of the process. The range of initially guessed values for the pump power is reduced effectively to better reflect the physical picture of stimulated Raman scattering, so that unreasonable initial guess is avoided. With the use of the contraction factor *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

The authors would like to acknowledge the financial support by NSFC (Project Nos. 60607005, 60588502, and 60877033), the Science and Technology Bureau of Sichuan Province (Project No. 2006z02-010-3) and the Youth Science and Technology Foundation of UESTC (Project No. JX0628).

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