In this paper, we propose an efficient and accurate method that combines the Genetic Algorithm (GA) with the Nelder-Mead method in order to obtain the gain optimization of distributed Raman amplifiers. By using these two methods together, the advantages of both are combined: the convergence of the GA and the high accuracy of the Nelder-Mead. To enhance the convergence of the GA, several features were examined and correlated with fitting errors. It is also shown that when the right moment to switch between methods is chosen, the computation time can be reduced by a factor of two.
©2007 Optical Society of America
Raman fiber amplifiers are assuredly one of the most seriously renewed research subjects in the field of optical fiber communication systems. The major benefits of such amplifiers are the upgradeability of the existing systems, the noise improvements in repeatered systems, the bandwidth/flatness, and the theoretical possibility of obtaining gain at all wavelengths. A wide bandwidth (encompassing the S, C and the L transmission bands) and a flat spectral gain profile are achievable thanks to the combination of several pumping lasers operating at specific powers and wavelengths. The composite amplification is determined by the mutual interactions among the pump and the signal wavelengths [1–3]. Those interactions are described through a set of coupled nonlinear differential equations. The composite amplification spectrum may, thus, be modified by selecting the appropriate pump wavelength and power, being this selection the reason for the study of many numerical optimization methods and algorithms. The simulation of the Raman effect is often based on the direct integration of the Raman nonlinear set of coupled differential equations. The optimization procedure includes two major tasks: the integration of Raman differential equations and the gain optimization based on the minimization of the ripple function that computes the deviation of the obtained gain in relation to a target value.
The Raman integration task is on its own a tough task. Although, some methodologies based on the shooting method have proved to be effective [4, 5], in this study we chose to use the average power analysis developed by Min et al .
The optimization of the gain spectrum has so far been widely performed by making use of several global search methods, such as neural networks , simulated annealing  and genetic algorithms (GA) , but the high simulation time remains a matter of concern. To enhance the efficiency of the GA, without any expense of accuracy, some hybrid approaches have been proposed. A combination of GA and simulated annealing is employed in  to optimize the power levels of 20 equally spaced pumps. GA is implemented and a simulated annealing step is inserted after mutation to renew the population of solutions and avoid premature convergence. Despite the flattening of the spectral gain was achieved over a 100 nm bandwidth, the obtained ripple remains relatively high (0.74 dB). Hybrid GA based upon clustering, sharing, crowding and adaptative probability is used in . This methodology explores the possibility of finding multiple optima instead of only one by forcing the GA to maintain a diverse population throughout its search. Therefore, several pumps schemes were tested (with 2, 3, 4, 5, 6 and seven pumps) having been demonstrated that the improvements in flattening the gain decrease the signal bandwidth.
This paper is also related to the efficient use of hybrid GA in the gain optimization of Raman amplifiers. However, our approach, although based on the GA, differs from the latter because two different methods are performed independently in the optimization: GA and Nelder-Mead simplex method. The Nelder-Mead method is a “direct” method that requires no derivatives. The objective function is evaluated at the vertices of a simplex, and the movement is away from the poorest value. The process is adaptive, causing the simplexes to be continually revised to best conform to the nature of the response surface .
As far as we know, it is the first time that a combination of GA and Nelder-Mead method is applied to solve the complex problem of optimizing the gain of multipump Raman amplifiers. The GA is used to find the vicinity of the global minimum rather than the minimum itself. The remaining search is performed by the Nelder-Mead method. Thus, using the two methods together, the advantages of both are combined: the convergence of GA and the high accuracy of Nelder-Mead. If we only applied the GA, the optimization would reveal a problem of low accuracy. If we only used the Nelder-Mead method, the optimization would be possibly trapped in a local minimum. The question is to decide the right moment to switch from GA to Nelder-Mead. To improve the GA search, it is necessary to dimension properly the population size, the initial range and the number of generations that are necessary to accomplish which allows for starting of the Nelder-Mead simplex method. It must be noted that if the search space is not large, it can be searched exhaustively and the best possible solution will be probably found. Hence, we considered some empirical knowledge about the pumps powers and wavelengths in lack of further information about the function behaviour. Our hybrid method also takes into account the problem of premature convergence by using the appropriate selection. Indeed, the selection operator plays an important role in pressing the system to evolve. Strong selection means that suboptimal highly fitted solutions will take over the population reducing the necessary diversity for further change. Weak selection implies very slow evolution. The Stochastic Uniform Selection, that we tried to prove to be the more efficient selection method, mimics the statistical results expected for population with a high number of individuals, without expense of efficiency [13, 14]. The other operators, crossover and mutation were also chosen according to their efficiency in the scope of this problem.
The pith of our work is the efficient optimization of Raman gain using a hybrid GA. Hence, the implementation of GA must be as efficient as possible. However, the algorithm becomes really efficient when the right moment to switch to the Nelder-Mead method is properly chosen. Using this methodology, we demonstrate that the simulation time can be reduced by a factor of two.
2. Raman propagation equations integration
Wave propagation in the backward-multipump Raman amplifier is characterized by a large number of effects. In steady state, the system (signals and pumps, ASE and Rayleigh scattering) is described by the following set of coupled nonlinear differential equations, which has properly accounted for the interactions among the pumps, signals, forward and backward ASE and co-and counterpropagating Rayleigh scattering components .
The ± signs relate to the forward and backward propagating waves, respectively. Pi, υi and αi are the optical power, the optical frequency and attenuation coefficient of the ith wave, respectively. Aeff is the optical fiber effective area, the factor Γ is a dimensionless quantity comprised between 1 and 2 that accounts the polarization random effects and gR(υj-υi) is the Raman gain coefficient from wave j to wave i . The frequency values υi are sequenced in decreasing order (i=1,2,…, m), noticing that the terms from j=1 to j=i-1 cause amplification and the terms from j=i+1 to j=m cause depletion. h and k are the Planck and Boltzmann constants, respectively, T is the fiber temperature and Δυ is the spectrum resolution bandwidth.
In the analytical average power analysis developed by Min et al., the fiber is divided into a series of concatenated elemental sections. In each section, the varying optical powers are replaced by average values.
It must be noted that the numerical problem of integrating a set of nonlinear coupled differential equations is reduced into a problem where the input power values are computed into an average power, being the output values in each section obtained directly from the input values of the this section. This procedure assumes that all the input parameters are defined at one end of the fiber and the output values are obtained iteratively up to the other end. The reason for the computational time reduction is explained by considering the fact that the intensive numerical integrations are now replaced by simpler algebraic calculations.
In the backward pumping situation, the inputs are located at both sides of the fiber, disabling the possibility of a one directional integration. So, the solution must be found out in another way. Since the core of the problem relies on the pump direction of propagation, one approach is to assume that the pumps are located at the other end and multiply the pumps equations by (-1), and to integrate all the equations from z=0 to z=L . The pumps powers at z=0 are unknown parameters that the optimization algorithm is going to compute. Therefore, we only have to bear in mind that despite the optimization algorithm determines the pumps inputs at z=0, by integrating the equations using the optimized inputs at z=0 we can determine the effective pump power inputs at z=L. This approach can work, without lack of efficiency, for all types of regimes, such as strong pump depletion, considering also the effect of the signal tilt.
3. Pump optimization procedure
The purpose of this work is to find the optimum Raman pump parameters (wavelengths and powers) that produce a gain profile as close as possible to the target one (value and ripple). The gain ripple will be used as the fitness for the GA. Given a sample input vector, we can find the error using the following steps: i) we first calculate the spectral gain for the input parameters ii) using a predefined error function, we then estimate the deviation of the calculated resonate wavelengths with respect to the target value. This function indicates to the GA how to ascertain the weight of the different parts of the data. The fitness will hopefully converge to a minimal value of the defined ripple function. To avoid the exploration of some search domain areas that are irrelevant to our problem context, we stipulated a high arbitrary error value. Thus, it is expectable that the eligible “individuals” remain in the domain of interest. When the GA attains the maximum number of generations, the search algorithm is switched to the Nelder-Mead.
To enlighten the conclusions provided by our numerical approach, we present a practical case. A scheme of the implemented system is displayed in Fig. 1. It is composed by 5 backward pumps injected into the fiber by a multiplexer coupler and 16×400 GHz spaced C band probe signals. Pumps and probe signals are then combined and injected into 25 km of SMF fiber. The signals were analysed recurring to an optical spectrum analyser (OSA).
The gain optimization simulation with this hybrid GA algorithm provided a mean ripple below 0.17 dB. The optimized values for the pumps powers and wavelengths are listed on table 1.
Although the hybrid GA is always able to perform the gain optimization with a ripple below 0.5 dB, some steps can be taken to improve the optimization in order to reduce the number of function evaluations. The initial population where the GA is achieved, as well as the GA operators, can be properly dimensioned so that the algorithm is brought nearer the optimum in a faster way. Another important feature is the number of generations needed to perform the GA before switching to the direct search method. It is expectable that the increase in the number of generations provides more chances to produce good fitted individuals enhancing in this way the process of finding the minima. But the use of the hybrid configuration, i. e, the use of the GA followed by a direct search method can avoid this necessity whenever the minima reached by the GA are small enough.
3.1 Initial population range
The initial population range is the limit established for pump powers and wavelengths at one end of the fiber. In the backward pumping scheme, the input power values are not the real ones because the equation is integrated reversely. So, it is advisable to try previously some values and to look to the output power values at the other end of the fiber. Since the higher frequency pumps transfer more power to the other waves it is expectable that at the end of the fiber length their powers are lower. Therefore, it is advisable to try lower power values for the higher frequency pumps and then increase them for the lower frequency pumps. In regard to the pumps wavelengths, theoretical assumptions about Raman gain spectrum state that in order to obtain a maximum gain, the signal beam is downshifted from the pump frequency by 13.2 THz (about 100 nm in the 1500 nm region) [1, 3]. So, it is advisable to divide our spectral range into the number of pumps and then downshift those values by 13.2 THz.
A good choice for the initial population range, i.e., a population range that allows an exhaustive search from the algorithm is very important, because the next generations are framed upon the best fitted members of the previous generation. Based on the above mentioned statements and taking our five backward pumps problem, the initial population for the powers are comprised between 1 mW and 10 mW for the first three higher frequency pumps. The powers for the other pumps are settled between 10mW and 100 mW. The initial population for the wavelengths are [1430; 1440] nm, [1440; 1450] nm, [1450; 1460] nm; [1460; 1470] nm and [1470; 1480] nm, respectively.
In general it can be stated that the higher the number of individuals, the better the fitness. Therefore, a better minimum will be found at the end of the run of the GA. The inconvenient is that the running time can be too long because the simulation time is proportional to the population size. In our scenario, we decided to perform the GA 10 times (since the random nature of GA will produce a different result at each run) and to record the optimized ripples at the end of the first generation. Populations with 25, 50, 75 and 100 individuals were tested. For each population, the mean of the optimized ripples and the standard deviation were listed in Table 2. Although the listed values suggest that higher sized populations present better minima, the improvement is not worthwhile because of the increase in the simulation time. It should be noted that exceptions to that rule sometimes occur on account of the above mentioned random nature of this algorithm.
4.2 Best selection, crossover and mutation
The selection is the way to choose from the individuals of a population of possible solutions the ones who will create offsprings for the next generations, and how many offsprings each of them will create. The purpose of selection is to distinguish between the fitter individuals in the population in hope that their offsprings will in turn have even better fitness [13, 14]. The process of selection holds two important issues: the maintenance of diversity within the population and the selective pressure. These factors are strongly related to each other being important to strike a balance between them .
The idea of crossover consists of the exchange of string fragments between two selected members of the population. There are many ways of implementing the crossover, for example having a single crossover point or multiple points which are randomly selected. A priori, it is not possible to identify the most adequate crossover because its success and failure depends on the particular fitness function, on the encoding and on other details.  The selection combined with the crossover gives the GA the bulk of its processing power. The mutation is a random change in an element of a string within the given population. As a matter of fact the mutation operator plays a secondary role in the GA, being considered as a secondary mechanism of its adaptation.
The available selection functions are: stochastic uniform, roulette wheel, tournament and remainder. For crossover we can try different functions as heuristic, single point, two points, intermediate, and scattered. The mutation can be performed in two ways: uniform or gaussian. Several computational experiments for the five backward pumps problem were performed taking different kinds of selection methods: stochastic uniform, roulette wheel, tournament and remainder. For each selection method, ten simulations were carried out and the optimized ripples at the end of the GA were registered. Those values are shown in table 3. From the listed results (mean and standard deviation) it is possible to conclude that the stochastic uniform method is the most appropriate selection method for our purposes, being the tournament method the one that presents the worse results.
In order to find the best crossover function for our purposes, a GA with a population size of 25 individuals was run along with 10 generations for the five backward pumps problem. Among the available crossover functions, we concluded that best one is the scattered
Lastly, the same procedure was implemented for the mutation operator, being the optimized ripples (mean and standard deviation) listed in table 5. The best mutation operator is uniform.
In conclusion the best GA operators are stochastic uniform for selection, scattered for crossover and uniform for mutation.
4.3 Hybrid Genetic Algorithm
After checking the most adequate GA operators for the defined 5 backward pumps problem, it is time to implement the hybrid algorithm (the GA followed by the Nelder-Mead method). We said previously that this methodology was efficient and accurate. To prove that the hybrid GA is more accurate then the simple GA, we have implement it and registered the optimized ripples of the simple GA and hybrid GA with the number of generations. A population of 50 individuals was considered and the adequate operators were used. The Nelder-Mead algorithm runs until the average change of the objective function (best attained ripple) is less than 1×10-6.
In Fig. 2, the GA and the hybrid GA minima are plotted for different number of generations. It is clear that even when the algorithm is switched at a higher minimum (the 5 generations situation) the Nelder-Mead method is able to reach a desirable minimum. Hence, the hybrid method is assuredly more accurate then the simple GA. It is also noticeable that the GA minima present exponential decay behaviour (r2=0.99549) with the number of generations and consequently present an asymptotic behaviour as the number of generations increase unlimitedly. Thus, we can conclude that it is not worthwhile the unlimited increase the number of generations in the hope of improving the method’s accuracy.
To improve the efficiency of the hybrid GA, we have to identify which situation is the fastest or, alternatively, to identify the right moment to switch from one method to another. By examining Fig.2, we realize that the ripple variation performed by the Nelder-Mead method is higher for GA with smaller number of generations (bellow 15), so it is expectable that more function evaluations are needed and consequently more computational time.
Considering the 5 generations situation as a reference, we decided to plot the relative computational time and the ripple variation attained by the Nelder-Mead method against the number of generations and display the results in Fig.3. Since both results decay when the number of generations is increased, the Nelder Mead time is primarily dependent of the ripple value where it starts the search.
In Fig. 4, we plotted the GA, the Nelder-Mead and the total relative simulation times against the number of generations. Here, we took the 40 generations situation as a reference. We noticed that the computation time of the simple GA increases linearly with the number of generations and that the tendency of the Nelder-Mead simulation time is to decay. An optimal situation is attained when both the GA and the Nelder-Mead achieve their computation taking a similar amount of time; thus, we realize that this situation is verified when 17 generations are used in the GA.
In conclusion, an efficient and accurate hybrid GA was presented for a 5 backward pumps distributed Raman amplifier. This method combines the GA and the Nelder-Mead method by making the most of them: the convergence of GA and accuracy of Nelder-Mead. In our study, the GA was dimensioned so that its search became more efficient. We began by enhancing the search domain of the GA with some empirical knowledge about the pumps powers and wavelengths. The proper choice of the GA operators elected stochastic uniform for selection, scattered for crossover and uniform for mutation. Furthermore, we demonstrated that the hybrid GA is more accurate than the simple GA and also that its efficiency can be improved by a factor of two when the right moment to switch methods is properly chosen. Our approach, that we demonstrated to be accurate and efficient, was able to optimize 5 backward pumps to obtain a spectral gain ripple below 0.17 dB within a bandwidth of 50 nm.
This work was supported by the POSC program, by the European Union FEDER fund and by the Portuguese scientific program. The authors also greatly acknowledge the project ARPA POSI/EEA-CPS/55781/2004) and the project TECLAR (POCI/A072/2005). B. Neto gratefully acknowledges the Ph. D. grant financed by FCT (SFRH/BD/28904/2006).
References and links
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