Use of a genetic algorithm to optimize multistage erbium-doped fiber-amplifier systems with complex structures

We propose optimizing multifunctional multistage erbiumdoped fiber amplifiers (EDFAs) with complex structures by use of a genetic algorithm. With this method, we investigated optimum configurations of Cand L-band gain-flattened multistage EDFAs containing gain-flattening filters and high-loss interstage elements for dense wavelength-division multiplexing systems in detail and compared the amplifiers with various kinds of configurations under different design criteria. With the guidance of optimization results, the roles of all the factors such as pumping schemes, pump-power allocation, component position, and insertion loss in the optimization of EDFAs have been studied, and useful guidelines for optimizations have been provided. © 2004 Optical Society of America OCIS codes:(060.2320) Fiber optics amplifiers and oscillators; (060.2410) Fibers, erbium; (060.2360) Fiber optics links and subsystems. References and links 1. K. Wundke, “Advanced amplifier design: physics and systems limitations,” in Optical Fiber Communication Conference (OFC 2003) , Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003). 2. J. Lee, U.-C. Ryu, S. J. Ahn, and N. Park, “Enhancement of power conversion efficiency for an L-band EDFA with a secondary pumping effect in the unpumped EDF section,” IEEE Photon. Technol. Lett. 11,42-44 (1999). 3. A. Yeniay and R. Gao, “Single stage high power L-band EDFA with multiple C-band seeds,” in Opt cal Fiber Communication Conference (OFC 2002) , Vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002). 4. Z. Michalewicz,Genetic Algorithms + Data Structures = Evolution Programs (Springer-Verlag, New York, 1992). 5. C. R. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 271-283 (1991). 6. T. G. Hodgkinson, “Improved average power analysis technique for erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett.4, 1273-1275 (1992). 7. R. Lebref, B. Landousies, T. Georges, and E. Delevaque, “Theoretical study of the gain equalization of a stabilized gain EDFA for WDM applications,” J. Lightwave Technol. 15,766-770 (1997). 8. Z. Tong, H. Wei, T. Li, and S. Jian, “Optimal design of L-band EDFAs with high-loss interstage elements,” Opt. Commun.224,63-72 (2003). 9. R. D. Muro, P. N. Kean, S. J. Wilson, and J. Mun, “Dependence of L-band amplifier efficiency on pump wavelength and amplifier design,” in Optical Fiber Communication Conference (OFC 2000) , Vol. 37 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000). (C) 2004 OSA 23 February 2004 / Vol. 12, No. 4 / OPTICS EXPRESS 531 #3539 $15.00 US Received 16 December 2003; revised 28 January 2004; accepted 2 February 2004


Introduction
The rapid development of optical fiber communications along with maturing large-capacity broadband transmission systems has led to the growth of erbium-doped fiber amplifiers (ED-FAs) from simple gain block designs to complicated systems consisting of multiple functional elements.Many additional features such as the amplified spontaneous noise (ASE) filter, the gain-flattening filter (GFF), and the dispersion compensator are incorporated in the midsection of amplifiers [1].Because the parameters of an amplifier's structure have a great effect on the amplifier's performance, an effective optimization method for designing EDFAs with a complex structure is necessary.In previous literature, most papers about EDFA design have focused on amplifiers with new structures [2,3]; few papers touch on optimization methods.
In this paper we describe the optimizations of multistage EDFAs by the genetic algorithm (GA) method.The paper is organized as follows.In Section 2, a procedure on how to realize the optimization is introduced in detail.There are two subsections in Section 3: the first presents optimizations for amplifiers with various configurations by use of the GA method.In the second, we systematically analyze the optimization results.Some useful guidelines for designing a multistage EDFA with high pump efficiency to satisfy the system's requirements are also discussed in this subsection.

Theory
As amplifiers' structures become more complex, additional parameters must be concurrently optimized.Furthermore, the accurate analytic expression between the parameters of an EDFA's structure and performance is often not available.Compared with conventional methods, the genetic algorithm (GA) has no requirement such as connectivity or convexity on searching space.It can efficiently search large and poorly understood searching space where expert knowledge is limited or inaccessible.And it has less probability of being trapped in a local optimum solution [4].
In the GA, an "individual" is a feasible solution that is described by a coded datum called a "chromosome" (in amplifier optimization, the coded datum contains amplifier information such as the length of the erbium-doped fiber (EDF), the pump power, and the position of optical components and pumps).After generating an initial population that contains a certain number of individuals, we perform a series of processes (selection, crossover, and mutation) on the population to generate the next generation circularly.The fittest individuals of any population tend to reproduce and survive into the next generation, thus improving successive generations.
In amplifier optimization we should give a numerical value to represent the degree of satisfaction with the amplifier performance.Using the average inversion ratio iteration method or the average power analysis (APA) method based on the Giles model [5][6][7], we can simulate the amplifier in a static state.Because there are many optical components inserted into the EDFA, the numerical model should take them into consideration.Assuming that a component is located at z j , the optical power at the two ports of the component are Pv L (z j ) and Pv R (z j ).The relations between them are F ± (ν) is used to denote the component's property (ν is the frequency of the light, and +,-denote forward and backward direction, respectively; e.g., for an ideal optical isolator (OI) Then the fitness value can be derived from the amplifier's simulation result.For various optimization purposes, the evaluation function is different.Commonly, the optimized amplifier should have high gain, flattened gain profile, and low noise figure (NF), so the evaluation func-tion can be calculated as follows: where α 1 , α 2 , α 3 are weight factors; G ave is the average value of the amplifier's gain; n f is the NF; and u f is the flatness of the amplifier's gain profile.The weight factors are selected according to the requirements of amplifier performance.Simulated results have been compared with measurement results to validate the accuracy of the numerical model.Figure 1  The effectiveness of the GA method has also been verified.Figure 3 gives the relationship between average gain and fitness value to L 1 , L 2 of an L-band multistage EDFA (Fig. 2) [we assume that the pump power and the position of the dispersion-compensation module (DCM) are fixed to make the optimization a comparatively simple problem that is easy to express with three-dimensional graphics).From Fig. 3 we can see that the optimization problem is a multiple-peaked problem with discontinuous properties.These properties seriously limited the application of conventional optimization methods.Although we can use the enumeration algorithm to find the global optimum solution by computing all the feasible solutions, the large amount of calculation makes the enumeration algorithm inefficient.It takes 326 min.for us with a Pentium IV 2-GHz computer to find the optimum solution in this simple example that has only two parameters to be optimized.When the searching space is large, it is impractical to compute all the feasible solutions.With the genetic method, setting the population to be 40, after 50 generations of evolution, we get the optimum solution that is the same as we derived with the enumeration algorithm.Figure 4 shows the trace of the optimization.It indicates that 20 generations of evolution are sufficient for obtaining the optimum solution, with a computing time of 39 min.

Generation
Fitness value Fig .4 The trace of the optimization

Applications for EDFA optimization
We derive the optimal parameters for EDFAs with different structures by the GA method and then compare the optimized EDFAs to find the one with best performance under certain conditions.
Optimizations for both C-and L-band EDFA have been performed.First, we optimized a C-band EDFA pumped with two laser diodes.The amplifier is composed of EDF (Table 1, conventional EDF for C-band), pump units [laser diode (LD) and wavelength-division multiplexing (WDM) coupler], a gain-flattening filter (GFF) and a DCM (with 12-dB loss for pump and signal power).We use the GFF [made of fiber Bragg grating (FBG) and OI] to keep the flatness of the gain for all the channels less than ±0.4 dB.The DCM is inserted into the EDFA to derive better system performance.There are 40 input wavelength channels on the International (C) 2004 OSA 23 February 2004 / Vol. 12, No. 4 / OPTICS EXPRESS 534 Telecommunication Union (ITU) grid with 100-GHz spacing, between 1528.77 and 1559.79 nm.The signal power is -17 dBm/channel.The sum power of the amplifier's two pumps is fixed at 380 mW, whereas the power ratio of two pumps can be selected.The goal of optimization is assembling the components to form an EDFA with optimized structure (Fig. 5).
The optimized amplifier should have the highest gain, with a gain flatness lower than ±0.4 dB, and at the same time a NF lower than a certain request value.The parameters used in the optimizations are shown in Tables 1 and 2. Extra insertion loss of GFF@1550 1.0dB A series of optimizations for different kinds of EDFA under various requests on NF (NF < 5.8-10 dB) have been carried out.Six kinds of pumping schemes have been taken into consideration: the forward-backward (FB) pumping and dual-forward (FF) pumping-direction schemes; and for each pumping-direction scheme there are three kinds of pump wavelength configurations: 980 nm + 1480 nm, 980 nm + 980 nm, and 1480 nm + 1480 nm.We compared the optimized configuration for each structure (Fig. 6) to find which pumping scheme has the best performance.
The L-band EDFAs with six kinds of pumping schemes have been optimized under different allowed NF requests by the GA method in the same manner as we used for the C-band EDFAs.The structures for L-band EDFAs [Figs.5, 6(c), and 6(d)] are not the same as those of C-band.To derive a higher pump-utilization ratio, an OI unit with pump path (composed of two pumpsignal WDM couplers) is used to restrain the backward ASE, and the pump power (include backward pumping) is coupled into the high-concentration EDF (Table 1) at a selected position to utilize the enhancement of power conversion efficiency with the secondary pumping effect [2,8].The GFF used in the C-band is not needed in the L-band EDFA because it has a flat gain profile.In the L-band EDFA optimizations, the sum power of the two-pump LDs is 440 mW, and there are 40 input channels with 100-GHz spacing (-17 dBm/channel, 1571.24-1602.74nm).The optimized amplifiers should have the highest possible gain.The gain flatness of the amplifiers should be lower than ±1.5 dB for all 40 channels and less than ±0.8 dB for the channels between 1574 and 1598 nm; at the same time, the NF of the optimized EDFAs should be lower than requirements (NF < 5.8-10dB).
The optimization results such as the output characteristics of some optimized EDFAs, gain performance for amplifiers with various configurations under different allowed NF, and amplifier structures are given in Figs. 7, 8, and 9, respectively.
Figure 7 shows the output optical spectra, gain, and NF of the optimized EDFAs.The results indicate that the optimized EDFAs can surely satisfy the requirements, so the optimizations are confirmed to be effective.
The performance of the optimized EDFAs is shown in Fig. 8   band).The FF and FB pumping schemes have almost the same performance, whereas the pump wavelength has great effects on the EDFA.For C-band EDFA, the 980 nm + 1480 nm pumping scheme can provide higher gain than other pumping schemes in a wide range of allowed NF.
When the allowed NF is very low, the 980 nm + 980 nm pumping scheme is a better choice.
In the practically allowed NF region, EDFAs with a 1480 nm + 1480 nm pumping scheme have the lowest gain among these C-band amplifiers with different kinds of pumping schemes.However, this situation is quite different from that of the C-band EDFAs; the dual 1480-nmpumped L-band EDFAs can have better performance in a wide range of allowed maximal NF.
The derived optimized structures of C-and L-band EDFAs for different allowed NF conditions are presented in Fig. 9, in which the position of components is expressed by the ratio of the EDF's length between the input end and the component to the whole length of the EDF, and pump-power allocation is expressed by the ratio of the front stage pump's power to the sum power of the two pumps.The optimization results reflect the compromise between noise performance and output power; when the requirement for noise is less strict (allowed maximal NF is higher), we can adjust the structure of the EDFA to get higher gain.Figures 9(a) and 9(b) give the parameters of the optimized C-band EDFA structure with FF and FB pumping schemes (980 nm +1480 nm) for different allowed NF conditions.We find that the order of the components along the EDF has not changed when the allowed NF becomes higher, but the ratio of the pump's power at the former stage should be decreased, and the components should be moved to the input end.The total length of the EDF should be slightly longer (from 25 to 38 m for the FF pumping scheme and 28 to 41 m for the FB pumping scheme for the allowed NF < 5.8 to NF < 10).The derived optimized structures of L-band EDFAs (dual-1480-nm pumping scheme) are given in Figs.9(c) and 9(d).The relationship between the components' position and the allowed NF are similar to with C-band EDFA: When the allowed NF becomes higher, (C) 2004 OSA 23 February 2004 / Vol. 12, No. 4 / OPTICS EXPRESS 538 the ratio of the pump's power at the former stage should be decreased, and the components should be moved to the input end to derive higher gain; only the OI remains at a relatively fixed position.

Discussion
The pumping scheme, component position, and pump-power allocation play different roles in amplifier performance.Under the guidance of optimization results, we discuss all these factors in detail.

Pumping scheme
From the optimization results (Fig. 8) we can see that the pumping wavelength is the most critically important factor; selecting the right pumping wavelength to satisfy the requirement should be done first.

Inversion ratio
Fiber length (m) Fig. 10 Inversion ratio for different kinds of pumping schemes In the Giles model, the EDF's absorption parameter is high and the emission parameter is zero at 980 nm, whereas at 1480 nm, the absorption parameter is relatively lower and the emission parameter is not zero.So the 980-nm pump power can be absorbed more rapidly and derive a higher inversion ratio than with the 1480-nm pumping scheme (Fig. 10).The high inversion ratio is beneficial for making the C-band EDFA have better noise performance (lower NF).However, as the wavelength of the 1480-nm pump is much longer than that of the 980-nm pump, with the same pump power, the 1480-nm pump can provide more pump photons than the 980-nm pump does, so the 1480-nm pumping scheme can provide higher power conversion efficiency (PCE).Therefore, there is also a compromise between noise performance and output power in selecting the pumping wavelength.
The sharp contrast between the performance of the dual-1480-nm pumping scheme for the C-and L-bands can be ascribed to the different gain property for C-and L-band EDFA.Two reasons make the dual-1480-nm pumping scheme have inferior performance in this kind of Cband amplification.One is the low inversion ratio that makes the noise performance far from what is desired; what is more, the insertion loss of DCM deteriorates the noise performance and makes the problem more serious.The other factor is the gain profile.Because the EDFA's inversion ratio is dissimilar under different wavelength pumping, the gain profile and the GFF's profile of it are also different (Fig. 11).The GFFs for the 1480-nm pumping scheme have (C) 2004 OSA 23 February 2004 / Vol. 12, No. 4 / OPTICS EXPRESS 539 higher loss than 980-nm pumping schemes in a wider wavelength range (Fig. 12); thus it can have unfavorable effects on amplifier performance.Wavelength (nm) Loss of GFF (dB) Fig. 12 GFF for 980+1480nm and dual-1480nm pumping For the multistage C-band EDFA, the NF is mainly determined by the former stage.And the 1480-nm pump at rear stage can provide higher PCE.So the 980 nm + 1480 nm pumping scheme can have better performance than other pumping schemes do.
To get a flat gain profile, the average inversion ratio for L-band EDFA is low and the EDF is longer than that of C-band EDFA.Then the backward C-band ASE plays an important role.It accumulated and amplified in EDFA, and near the input end, it becomes very high.This ASE power can cause pump-power depletion, decrease the inversion ratio, and then affect amplifier noise performance.It was shown that 1480-nm pumping leads to more-uniform gain distribution along the fiber length (Fig. 10).Thereby, it is beneficial to prevent short-wavelength ASE from being built up.The improvement in L-band EDFA efficiency resulting from pump wavelength tuning that has been reported [9] is also based on these reasons.
In contrast to pumping wavelength, the pumping direction of multistage EDFA has a small effect on amplifier performance for both C-and L-band EDFAs.The optimized FF-and FBpumped EDFAs with the same pumping wavelength have almost the same performance (Fig. 7).The FF-pumped EDFA has better performance at a low-NF region, whereas the FB-pumped (C) 2004 OSA 23 February 2004 / Vol. 12, No. 4 / OPTICS EXPRESS 540 EDFA can provide higher gain when the allowed NF is higher.But the difference between these two kinds of pumping scheme is very small.

Component position, pump-power allocation
In all the optimizations we have done, the components of the EDFA can be located at any position of the EDF, and then the optimized structure is obtained by the GA method.Analyzing the optimized results, we can get some useful guidance.
The rule for the optimized EDFA's structure under different allowed NF is that generally for all the amplifiers we have derived when the sum pump power (P) of the amplifier is fixed, increasing the power ratio for the pump at former stage (P1/P) and locating the components at the position farther from the input end can make the amplifier have better noise performance with lower NF and relatively low gain; in contrast, increasing the pump power ratio of rear stage and locating the components at the position nearer to the input end lead to an increase in gain and degradation in noise performance.Compromise between the gain and NF performance is the key to amplifier optimization, and pump-power allocation is the determining factor for the optimized amplifier's gain and NF.With a fixed pump ratio, the optimization of EDFA is limited drastically; improving performance at (C) 2004 OSA 23 February 2004 / Vol. 12, No. 4 / OPTICS EXPRESS 541 one aspect (higher gain or lower NF) can be realized only with the high expense of performance degradation in the other aspect (Fig. 13).For L-band EDFA this phenomenon is more obvious [Fig.13(b)].
Besides the general rules, we can also find some subtle relationships between the components' position and the amplifier's performance that should be paid attention to.
For the L-band EDFA, utilization of the C-band ASE as the second pump is a very effective method to improve the amplifier's performance.A section of unpumped EDF can be used for this purpose [2,8].But from the results of optimizations we find that this method should be utilized at proper conditions.Using a section of unpumped EDF after a backward pump is an ideal structure that can not only increase pump utilization but also improve the noise performance by suppression the ASE at shorter wavelength (see Fig. 14).However, when using unpumped EDF before a forward pump, we must beware of the degradation of noise property.In fact, this structure cannot be used at the front stage because its low conversion ratio can degrade the noise property.Only when the requirement of noise performance is not very high can we use this method to improve the gain of the EDFA.In Fig. 9(c), we can see that when we allow NF < 6.5dB, the DCM and the forward pump are located at the same position; only when we allow NF > 6.5dB can a section of the EDF be put between the DCM and the forward pump for the rear section of the amplifier.
For all the optimized C-band EDFA, the components (e.g., DCM) with high insertion loss should be located just between the two stages.(The DCMs in this kind of FF-pumped amplifiers are all adjacent to the forward pumps at rear stage.)But as far as the GFF is concerned, the optimized GFF's position is different for different pumping schemes.The GFF for 980 nm + 980 nm and 980 nm + 1480 nm pumped C-band EDFA is located at the rare stage [Figs.6(a) and 6(b)], whereas the GFF for 1480 nm + 1480 nm pumped EDFA is located at the former stage.The main reason is that the peak of gain spectrum of 980-nm-pumped EDFA is much higher than that of the 1480-nm-pumped EDFA, then the GFF must have higher loss at ∼1530 nm.When the GFF is nearer to the input end the NF profile of the EDFA is determined by the GFF's loss profile.The high loss of the GFF at 1530 nm can cause NF increased greatly at that wavelength.

Insertion loss
Because the multistage amplifier has a complex structure and so many components, the insertion loss of components and splicing loss of standard single-mode fiber (SMF) and EDF do greatly affect the amplifier's property, especially the NF and PCE (Fig. 15 presents the comparison between the results for the optimizations that do and do not take the loss into consideration).Through calculation we know that if there is no insertion loss, the optimized L-band EDFA can have very high PEC (>60%) for the allowed NF < 5.5 dB, but when we take the loss into account, the PCE decreases to 42% even for the allowed NF of 6.5 dB.
The configurations for the EDFAs with different DCM loss are also analyzed (Figs.16 and  17).When the loss of DCM becomes higher, the NF of the amplifier increases, and it cannot satisfy the requirement; then the structure of the amplifier needs to be adjusted (P 1 /P should be larger, and components should be moved toward output end) to have better noise performance at the expense of decrease in gain.

Summary
Selecting proper pumping wavelength is the most important issue in designing an EDFA, and it should be done as the first step.In the C band, the 980 nm + 1480 nm pumping scheme has advantages over other pumping schemes for a wide range of allowed NF, whereas the dual-1480-nm pumping scheme is quite suitable for L-band amplification.Compared with pumping wavelength, the pumping direction of the amplifier's rear stage has a lesser effect.Although the dual-forward pumping scheme is slightly more suitable for the situation of low allowed NF, and the forward-backward pumped EDFA can provide higher gain when the allowed NF is high, the difference between these two kinds of pumping schemes is very small.
A compromise between the noise and gain performance is the main issue in EDFA optimization; choosing the pump-power allocation and component position for multistage EDFA is key to the compromise.The components, especially the DCM, GFF, and setup for use of unpumped EDF, should be placed properly according to the allowed noise and gain performance under the guidance that we have introduced.
Insertion loss of components and splicing loss of different kinds of fibers can degrade the performance of multistage EDFAs greatly.Therefore, minimizing the insertion loss is very important to get EDFA with good performance.

Conclusions
For multistage EDFAs with complex structures, many factors such as the pumping schemes, pump-power allocation, components position, and insertion loss determine the performance of EDFA jointly.GA is an effective method for optimizing EDFAs, taking all those factors into consideration.Using the GA method, we have done a series of optimizations; then under the guidance of optimization results, a systematic account of the subtle relations between amplifiers' performance and all those factors have been derived.

Fig . 1 Fig. 1 .Fig . 2 Fig. 2 .
Fig .1 Comparison of simulated results with measurement results for C (a) and L-band (b) EDFA Fig. 1.Comparison of simulated results with measurement results for C-(a) and L-band (b) EDFA.

Fig . 3 Fig. 3 .
Fig .3The relationship between average gain (a) and fitness value (b) to L1, L2 for an L-band multi stage EDFA Fig. 3. Relationship between average gain (a) and fitness value (b) to L 1 , L 2 for an L-band multistage EDFA.

Fig. 5 .
Fig.5Use GA method to assemble the components to form EDFAs with optimized structure (IO module is for L-band EDFA and GFF is for C-band EDFA only ) Fig.5.Use of the GA method to assemble the components to form EDFAs with optimized structure (IO module is for L-band EDFA, and GFF is for C-band EDFA only).

Fig. 6 Fig. 6 .
Fig.6 Some structures for C-band (a,b) and L-band (c,d) EDFA (input and output OI have not been included in the figure) Fig. 6.Some structures for C-band (a, b) and L-band (c, d) EDFA (input and output OI have not been included in the figure).

Table 1 .
Parameters of the EDF used in this paper.

Table 2 .
Other parameters used in the optimizations.