## 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

With their intrinsic merits, fiber Raman amplifiers (FRAs) have rapidly become a critical technology for 40-Gbit/s long-haul transmission and 10-Gbit/s ultra-long-haul transmission [1–3]. Because the performance of the entire transmission system is strongly influenced by the spectral gain and spectral noise performance [2–5], detailed knowledge about the spectral characteristics of the amplifier is key to the advanced design of WDM systems. Amplified spontaneous emission (ASE) and the Rayleigh backscattering effect are two main aspects of FRAs noise performance [4]. Single Rayleigh backscattering of ASE and double Rayleigh backscattering of optical signals in the transmission fiber grow with increasing distributed gain [6], and their impairments impose limits on the maximum allowable distributed gain in a system.

Since WDM signals propagating in a fiber experience gain tilts resulting from the energy transfer to longer wavelengths, optimizing the power and wavelength of pumps is used to control gain tilts. So, the design of multipump flat-gain Raman amplifiers presents a grand challenge to numerical simulation. Moreover, optimizing algorithms play a critical role in the design, analysis, and control of WDM transmission systems and are in widespread use in the research of FRAs. Recently, several different approaches to the design of wideband FRAs have been reported, such as the neural network method [7], the simulated annealing algorithm [8], the traditional genetic algorithm (GA) [9], and the hybrid GA (HGA) [10–12].

GA, a multivariate stochastic optimization algorithm based on the mechanics of natural selection and natural genetics, is generally able to find good solutions in a reasonable amount of run time [13]. Owing to a large amount of implicit parallelism, GAs perform searches in complex, large, and multimodal landscapes, and they provide optimal (or near-optimal) solutions for objective or fitness functions of an optimization problem. However, the traditional GA tends to converge toward a single solution and is even trapped in local optima of the search space because of selection pressure, selection noise, and operator disruption [10, 13, 14]. For successfully solving multiple optima in the domains in the required run time, a key task of the GA is to combat premature convergence and to strike a trade-off between exploration and exploitation.

The optimal design of multipump flat-gain Raman amplifiers is a multivariate optimizing problem with multiple maximal values. For instance, it is an eight-dimensional optimizing problem for four pump spectra (including both wavelength and power), and it has multiple global maxima and many local maxima (see Fig. 1). Although the HGA in our previous reports can offer multiple global maxima in the design of FRAs [10–12], it may get only part of all global maxima in the required amount of time. To explore more area in the search space covered, to exploit the better solutions, and to shorten the amount of run time, a HGA assisted by stochastic perturbation and the adaptive technique is proposed in this paper. The simulation results show that the novel algorithm can obtain more global maxima and greatly shorten the run time. From the new HGA, we obtain the optimal signal bandwidth and the corresponding noise performance.

## 2. Theoretical model

Wave and noise propagation in FRAs are characterized by a variety of physical effects [15, 16], the major influences of which, for the gain profile design of backward-propagating multipump Raman amplifiers, are pump-to-pump, signal-to-signal, and pump-to-signal stimulated Raman scattering (SRS), as well as the fiber loss experienced by both pump and signal waves [7–12, 17]. In the steady state, the coupled equation can be described as

Because backscattering powers of pumps and signals are usually approximately 30 and 20 dB lower than their original powers, respectively [18], the backscattering pumps and backscattering signals are ignored in simulating ASE waves. Furthermore, forward and backward noise powers are also less than input signal powers by ~30 dB [18]. Then the model equations for ASE waves include such physical effects as attenuation, SRS, spontaneous Raman scattering, Rayleigh scattering, thermal noise, and so on; namely [12, 16, 17],

$$+{P}_{\mathrm{ASE},k}^{\pm}\sum _{j=1}^{k-1}\frac{{g}_{R}({v}_{j}-{v}_{k})}{\Gamma {A}_{\mathit{eff}}}{P}_{j}^{\pm}\left[1+\frac{2h{v}_{k}}{{P}_{\mathrm{ASE},k}^{\pm}}\left({1+({e}^{\frac{h({v}_{j}-{v}_{k})}{{k}_{B}T}}-1)}^{-1}\right)\Delta v\right],$$

$$-{P}_{\mathrm{ASE},k}^{\pm}\sum _{j=k+1}^{n+m}\frac{{v}_{k}}{{v}_{j}}\frac{{g}_{R}({v}_{k}-{v}_{j})}{\Gamma {A}_{\mathit{eff}}}\left[{P}_{j}^{\pm}+4h{v}_{k}\left({1+({e}^{\frac{h({v}_{k}-{v}_{j})}{{k}_{B}T}}-1)}^{-1}\right)\Delta v\right]$$

In Eqs. (1) and (2), the indexes *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

Model equations of multipump Raman amplifiers are multivaritate problems; e.g., the design of the four-pump Raman amplifier has eight variables (including four wavelengths and four powers). Then their design problems comprise the multimodal function optimizations, and Fig. 1 demonstrates an example with four pumps. Figure 1 shows the contour of optimal signal bandwidth Δ*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 _{R}*(Δ

*v*) and attenuation spectrum

*α*(

*v*) of the fiber are from Ref. [10]; the gross Raman gain can compensate the loss of signals (i.e.,

*G*

_{ON-OFF}>

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

*G*is less than 1.1 (i.e., Δ

*G*<1.1).

From Fig. 1, we can see that there are multiple global maxima and many local maxima in the four-pump Raman amplifier. To reach the global maxima, the parameters of the pump spectra have to carefully accounted for. For example, in the cases of A and B global maxima in Fig. 1(b), although their wavelengths *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.

Owing to the intrinsic weakness of traditional GA, it offers only a single optimal solution in the search space and may even be trapped in a local optimum. To obtain global optima of much as possible, a powerful algorithm should be formed.

## 4. Algorithm

To be successful, GAs must strike a balance between exploration of the search space and exploitat ion of knowledge of the problem. Too much explorat ion of a problem tends to be slow and inefficient, and too much exploitation tends to yield lower-quality answers. From the traditional GA, in our previous reports [10–12, 19], the HGA is proposed with such advanced techniques as clustering, sharing, crowding, elitist replacement, fitness scaling, and adaptive genetic operators. Although this HGA can obtain the global optima, its optimized results often include some local optima [10–12], and the amount of run-time is more.

To obtain a global optima of as much as possible and shorten the amount of run time in optimizing the design of multipump flat-gain Raman amplifiers, from the old version of our HGA, an new HGA assisted by the adaptive technique and stochastic perturbation is proposed. In the code of pumps, we divide the wavelength range *λ _{fa}* of each pump into fixed range

*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}On the other hand, when the procedure of HGA inherits some generations, the maximum of each cluster approaches to a certain value. But part of these values may be the local maxima instead of the global maxima. To escape from being trapped to the local maximum, the technique of stochastic perturbation is adopted. After some generations, the value of *λ _{f}* is perturbed (e.g.,

*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}From the new HGA, a simulated example is tested. In the simulation, we assume that the population size *N*=800, the number of clusters *N _{c}*=7, the normalized niche radius

*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:

During the HGA procedure, the distance *d _{ij}* among the centers of all clusters must be more than the critical distance

*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 [10–12], the optimized solutions include some local optima instead of all global optima; ② to obtain the same number of global optima, the old HGA costs twice the run-time as compared with the new HGA.

## 5. Optimal results and noise performance

In the following simulations, we employ the predictor-corrector method and the “pure” shooting algorithm to solve Eqs. (1) [18, 20] and assume that: *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}

To reveal the influence of some major parameters on the FRA bandwidth, we plot the figures that show the relationships between the optimal signal bandwidth Δ*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.

The red solid curves in Figs. 4 and 5 are the fitted lines based on all circles (each circle corresponds to an optimized global maximum), respectively. We can see that, from Figs. 4 and 5, the optimal signal bandwidth Δ*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*.

The simulated results show that the new HGA can offer more than seven global optima in parallel. For example, when *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 _{c}*=8). Compared with our previous reports [10–12], the number of global optima is greatly increased. At the same time, the amount of run time is decreased by more than half on the condition of finding the same global optima (e.g., three global optima) by comparison of the new HGA with our old HGA. Table 2 illustrates that, to realize the fixed Δ

*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

To give a clearer understanding of the relationship of pump-to-signal and signal-to-signal, we present a movie that demonstrates the evolution of all channels transmitting along the fiber. In simulations, we assumed that *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

Figure 7 exhibits the corresponding optical signal-to-noise ratio (OSNR) of *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

In Ref. [16], eight pumps are used to design the gain spectrum of 100 nm with 1.1-dB peak-to-peak ripple for a 45-km fiber length. Six pumps are required for obtaining the signal bandwidth of 33 nm for contenting 18-dB gain, 0.8-dB peak-to-peak ripple, and 75-km span length in Ref. [21]. To realize an 80-nm bandwidth with a gain ripple of less than 1 dB, eight pumps are employed for the span length of 80 km [18]. Zhou *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], and even their methods may be trapped in local optima of the search space. In Ref. [7] the neural network method is used to optimize the Raman-gain spectrum; however, it only partly obtains the optimization in pump power excluding wavelengths. Although the simulated annealing algorithm has been adopted to give automatic pump configuration [8], the optimal bandwidth is ~50 nm with a gain ripple of 2.6 dB at five pumps and a fiber length of 25 km. Obviously, their results are far from the global optimum. Comparing the above examples with our optimal results, we can see that not only is the bandwidth greatly broadened after the optimization of HGA, but also a number of global maxima Δ*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

A HGA assisted by stochastic perturbation and the adaptive technique has been constructed in this paper. In comparison with the old version of our HGA [10–12], the new HGA can exploit the better solutions (e.g., it can in parallel offer more than seven global maxima that are shown in Tables 1 and 2, but the old HGA offers only part of the global maxima accompanied with some local maxima) and shortens the amount of run time by more than half in our simulations. An example of a four-pump FRA shows that the design of multipump Raman amplifiers involves the multimodal function optimization problems with multiple variables. From the new HGA, relationships of the optimal signal bandwidth Δ*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

This research has been supported in part by the National Natural Science Foundation of China (60132020). The authors thank Xin-Jun Liu and Xinjie Yu, Tsinghua University, respectively, for helping to make three movies and for fruitful discussions on the GA.

Xueming Liu is currently engaged in research at the School of Electrical Engineering, Seoul National University, Korea.

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