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We employ a Genetic Algorithm for the purpose of minimization of the maximum differential modal gain (DMG) over all the supported signal modes (at the same wavelength) of cladding-pumped four-mode and six-mode-group EDFAs. The optimal EDFA designs found through the algorithm provide less than 1 dB DMG across the C-band (1530-1565 nm) whilst achieving more than 20 dB gain per mode. We then analyze the sensitivity of the DMG to small variations from the optimal value of the erbium doping concentration and the structural parameters, and estimate the fabrication tolerance for reliable amplifier performance.
© 2014 Optical Society of America
1. Introduction
Space Division Multiplexing (SDM) has attracted considerable attention in high-capacity fiber-optic communication systems as a radical approach to increase the capacity-per-fiber by employing multiple distinguishable spatial information channels through the same fiber [1–3]. Such an approach is now needed since all other dimensions i.e. wavelength, polarization, phase and amplitude are already close to fully exploited in Single-Mode Fiber (SMF) based systems. One form of SDM uses Few Mode Fibers (FMFs) which guide a restricted number of modes that are used to define the independent spatial channels. So far, both 2 and 4-mode group systems have been reported, with both transmission fibers and a compatible associated inline Few-mode Erbium Doped Fiber Amplifier (FM-EDFA) developed [4,5]. The majority of the FM-EDFAs demonstrated so far have been core-pumped with either one or two single-mode pump diodes [4–6]. The gain equalization for both core-pumped 4M-EDFA and 6M-EDFA at a single wavelength (e.g. 1550nm) was theoretically investigated using a combination of selected pump modes and sophisticated erbium doping profiles [7]. In several recent works on designing core pumped FM-EDFAs, numerical optimization methods (e.g. Gradient Descent Optimization) were applied to optimize the modal composition of the pump beam and erbium doping profile [8,9]. Obviously, as the number of modes supported by a single fiber is increased, the total signal output power (and hence the required pump power needed for high gain, low noise amplification) increases significantly, dictating the use of multiple, expensive single mode pumps. Reduction in the cost per transmitted bit is essential for SDM to be seriously considered for commercial deployment and cladding-pumping using high-power multimode pump diodes (which has already been demonstrated for multi-core EDFAs [10]) provides a more practical and potentially much cheaper way in terms of $/W to generate and deliver the pump radiation. The first demonstration of a cladding-pumped EDFA supporting four mode groups (4M-EDFA) reported a DMG (i.e. the maximum gain difference among all the signal modes at the same wavelength) of around 4 dB [11]. When developing FM EDFAs minimizing DMG is of paramount importance in ensuring the best overall system performance and the DMG fundamentally arises from the difference in the overlap between pump, signal and rare earth dopants between modes. In the cladding-pumping scheme, where a double-clad fiber (DCF) is used, the heavily multimode pump light is guided in the inner-cladding, which is intentionally made asymmetric (e.g. D-shaped) to promote mode mixing for efficient pump absorption [12]. So in the modeling of cladding-pumped amplifiers, a simple and established way to represent the pump beam, is to assume that the pump intensity profile is uniform across the doped core [13–15]. Careful tailoring of the erbium doping profile in an active multimode fiber is therefore central to minimizing the DMG in cladding-pumped FM-EDFAs.
Very recently, we reported theoretical designs of cladding pumped FM-EDFA supporting four and six mode groups [16]. In this previous work a Genetic Algorithm (GA) was employed to help optimize the rare earth doping profile of the cladding pumped FM-EDFAs. In this paper we report a significant advance on this earlier work with a much more rigorous fitness function applied in the GA and optimisation of the fiber design for WDM operation. Using this approach we identify optimum fiber designs providing more than 20 dB gain and less than 1 dB DMG across the full C-band (1530-1565nm). In the last section of this work, the sensitivity of the DMG to variations in the erbium doping concentration and the structural parameters is also presented in order to investigate the design tolerance.
2. Simulation
The mode dependent gain is determined by a combination of the signal and pump intensity profiles and their overlap with the erbium dopant distribution. The FM-EDFA was simulated using the simulation model described in our previous work [17] with the absorption and emission cross sections taken from [18]. As mentioned previously we assume that the pump light is uniformly distributed across the inner-cladding of the fiber along the full fiber length. For the noise calculation, we split the wavelength band from 1500 nm to 1600 nm into 40 equal-width (2.5 nm) wavelength slots. In order to manipulate the erbium ion distribution, we chose to divide the erbium-doped core into several layers whilst keeping the Fiber Refractive Index Profile (FRIP) unchanged, so that the optimal dopant concentration in each core layer can be numerically investigated for minimization of the DMG. For the purpose of reducing fabrication complexity, the number of erbium-doped layers should be as few as possible. Through initial trials, we found that at least three-layer and four-layer doping structures (shown in Fig. 1) are required in order to achieve a DMG of less than 1 dB in the C- band (1530-1565 nm) for the 4M-EDFA and 6M-EDFA respectively. Thus for the 4M-EDFA, there are two structural parameters (i.e. a1, a2 shown in Fig. 1(a)) determining the dimensions of the three-core layers and three doping concentration parameters (i.e. ρ1, ρ2, ρ3) describing the erbium ion concentration of each core layer to be optimized. Similarly, there are three structural parameters (i.e. a1, a2, a3 shown in Fig. 1(b)) and four doping concentration parameters (i.e. ρ1, ρ2, ρ3, ρ4) to be optimized for the 6M-EDFA. In addition, the fiber length is another key parameter that can be used as a free parameter to optimize the performance of the amplifier. Due to the involvement of more than 5 free parameters, we implemented a GA to establish the optimum choice of the dopant concentration and the dimension of each core layer for the minimization of the DMG. The GA approach is a generic optimization method for optimizing multiple parameters simultaneously based on a natural selection process that mimics biological evolution. The evolution starts from a population of randomly generated ‘genes’ (constrained by the initial conditions pre-defined) which are used as “parents” to produce the “children” for the next generation. In each generation, the “fitness” of every ‘child’ (and associated set of “genes”) in the population is evaluated; where the “fitness” is defined by the value of the objective function in the optimization problem being solved. The “fittest children” are then selected to become “parents” and over successive generations, the population “evolves” toward an optimal solution. The GA approach has been successfully applied to design several fiber types, see for example Refs [9,19–21]. In the GA implemented here, the fitness function, “F”, that is used to evaluate the ‘quality’ of a given structure and minimized by the algorithm, is defined as:
where is the differential modal gain calculated at wavelength and the is the average gain over the distinct special modes at wavelength . The sum of the is performed over 15 WDM signals (uniformly spaced in the C-band from 1530nm to 1565nm). The free parameters (or the ‘genes’ of the algorithm) are the structural parameters ai (i.e. a1, a2, …), the doping concentration parameters ρi (i.e. ρ1, ρ2, …) of the i-th core layer and the fiber length of the amplifier. It is to be noted that other fitness functions can easily be chosen and incorporated within the GA depending on the end design objectives. Our view is that our current choice, as shown in Eq. (1), is a reasonable one for minimizing the overall DMG and maximizing the average gain over the C-band. For all modeling work in this paper the WDM signals are assumed to have an input signal power of −20 dBm per-spatial-mode per-wavelength and to be co-propagating with a multimode pump at 980 nm with an input power of 2.5 W. Note that for the non-LP0m mode groups (e.g. LP11), an equal power of −20 dBm is assigned to the two degenerate modes (e.g. LP11a and LP11b modes), so that the sum of the two LP modes has no azimuthal dependence. The gains for the two degenerate modes (e.g. LP11a and LP11b modes) of the non-LP0m mode group are identical under the weakly guiding approximation and the uniform pump intensity approximation in the case of cladding pumping. This pump power level was chosen based on past experience with the goal of ensuring more than 20 dB gain over the C band for a WDM signal power of −20 dBm per channel. The tolerance threshold of the fitness function “F” is set to be 1 × 10−5 dB and the maximum iteration number is set to be 200. The algorithm terminates either when there is no further minimization of the fitness function “F” after 20 successive generations or the maximum iteration number is reached. In general, it requires 50 to 70 generations for the GA to converge. Due to the nature of the GA techniques for a given set of starting conditions the algorithm might only converge to a local rather than globally optimum solution. To mitigate this risk we performed an initial coarse survey of starting conditions to map out the parameter space and then constrained the starting conditions to focus on the most promising fitness solution. Then with these constraints in place we ran the optimization tool several times (typically 5) to robustly identify the “fittest” solution. As we shall show we then investigated the robustness of the design to variation in each design parameter for the “optimal” fiber design in each instance.
Fig. 1 The fiber refractive index profile, the signal mode intensity distributions, and the doping profile of (a) 4M-EDFA, denoted as ‘F1’ and (b) 6M- EDFA, denoted as ‘F2’, to be optimized through the GA; where ρi (m−3) is the doping concentration of the i-th core layer.
3. Optimized fiber
3.1 Four-mode-group EDFA
On the basis of our amplifier and fiber fabrication experience, we chose to study a double-clad, step-index erbium doped fiber (EDF) with a core-to-inner clad NA of 0.15 and a core diameter of 17μm. We assume that the fiber is weakly guiding and supports the lowest four transverse mode groups over the range 1530 nm to 1565 nm, whose modal intensity profiles are shown in Fig. 1(a). The fiber diameter was chosen such that the LP02 mode is strongly guided whilst the LP31 mode is extremely leaky when the fiber is bent down to a radius of 10cm. The bend losses of the guided modes were calculated by using the imaginary part of the effective mode indices (computed using the COMSOL Multiphysics® software). The diameter of the inner-cladding was chosen to be 70 μm, compatible with our preferred choice of pigtailed pump diode, which creates a uniform pump intensity of 6.5 × 108 W/m2 at the input end of the fiber. The optimum fiber length selected by the GA was 6.2 m. The best fiber design (a1 = 1.3 µm, a2 = 6.0 µm, ρ1 = 1.8e + 25 m−3, ρ2 = 8.0e + 24 m−3, ρ3 = 1.2e + 25 m−3) found through the GA is shown in Fig. 1(a), here denoted as ‘F1’. This ‘W’ shaped erbium dopant distribution provides similar amounts of overlap between the dopants and the signal modes. The gain and noise characteristics of the best fiber design are plotted in Fig. 2(a). As can be seen from the red dotted curve in Fig. 2(a), the DMG is controlled to better than 0.6 dB from 1530 nm to 1565 nm, with a minimum of 0.3 dB at 1542.5 nm and maximum of 0.6 dB at 1530 nm. The modal gains are found to be more than 20 dB with a average gain of 24.8 dB, for the input signal/pump power conditions described in section 2. The NFs shown in Fig. 2(a) are found to be between 3.2 and 5.3 dB and are comparable to those of the cladding-pumped multi-core EDFA proposed in [22].
Fig. 2 The modal gain (continuous line), noise figure (dashed line) and DMG (red dotted line) characteristics of the best (a) 4-mode-group EDFA and (b) 6-mode-group EDFA calculated by the GA.
From more extensive simulations, we have also found that changes (up to ± 20%) in length or pump power of the amplifiers based on fiber F1 only result in a very small variation in the DMGs (DMG still less than 1.0 dB over the C band).
3.2 Six-mode-group EDFA
In this instance the core diameter of the EDF was scaled up to 22 μm so as to support six-mode-groups while keeping the NA exactly the same as in the previous case. The fiber diameter is chosen such that the LP12 mode is well-guided and the LP41 mode becomes very leaky when the fiber is bent down to a radius of 10 cm. The FRIP and signal intensity profiles of the 6M-EDFA (denoted as ‘F2’) and its optimal erbium dopant profile (a1 = 1.9 µm, a2 = 6.8 µm, a3 = 9.1 µm, ρ1 = 6.3e + 24 m−3, ρ2 = 9.6e + 24 m−3, ρ3 = 6.1e + 24 m−3, ρ4 = 1.5e + 25 m−3) obtained from the GA are shown in Fig. 1(b). The corresponding optimum fiber length found by the GA was 5.9 m. The gain and NF characteristic of the optimium 6M-EDFA are shown in Fig. 2(b). As shown in Fig. 2(b), the DMG (the red dotted line) is found to be below 1 dB with a minimum value of 0.5 dB at 1565 nm and a highest value of 0.9 dB at 1530 nm. The gain and noise characteristics of F2 are similar to those of F1. The modal gains are well above 20 dB with an average gain of 24.0 dB. The NFs shown in Fig. 2(b) are found to be between 3.4 and 5.2 dB.
Again, we have also found that changes (up to ± 20%) in length or pump power of the amplifiers based on fiber F2 only result in a small variation in the DMGs (DMG still less than 1.5 dB over the C band).
4. Fabrication tolerance discussion
Having demonstrated that very low DMG in both 4M and 6M cladding-pumped EDFAs can be theoretically realized, we went on to investigate how the inevitable imperfections introduced during the EDF fabrication process affect the EDFA performance. Firstly, a tolerance check on the doping concentration of each erbium-doped layer is studied in this section to assess the fabrication challenge in achieving the predicted levels of amplifier performance. We have run a set of simulations on both F1 and F2, in which we modified the doping concentration of each core layer (i.e. ρ1, ρ2 …) by ± 5% and ± 10% from the optimum value. Results for F1 and F2 are presented in Fig. 3 and Fig. 4 respectively. Note that in the tolerance studies, the fiber lengths for both F1 and F2 were fixed at the optimum lengths selected by the GA .
Fig. 3 Variation of the DMG versus signal wavelength as the doping concentration of (a) the 1st, (b) the 2nd and (c) the 3rd core layer is changed for fiber F1. Continuous lines represent a “+” variation, while dashed lines represent a “-” variation.
Fig. 4 Variation of the DMG versus signal wavelength as the doping concentration of (a) the 1st, (b) the 2nd, (c) the 3rd and (d) the 4th core layer is changed for the fiber F2. Continuous lines represent a “+” variation, while dashed lines represent a “-” variation.
From Fig. 3(a), it is evident that less than 1 dB DMG can be achieved over the full C-band with up to ± 10% variation in doping concentration of the 1st layer of F1. Decreasing the doping concentration of the 2nd layer of F1 by 5% brings about a rise in the DMG at shorter wavelengths whilst the DMG values at longer wavelengths are even smaller than for the optimum fiber. On the contrary, increasing the doping concentration of the 2nd layer of F1, results in a large increase in the DMG over the entire C-band. For the 3rd layer of F1, increasing the doping concentration ρ3 up to 10% results in an increase in DMG at shorter wavelengths, but the DMG is still below 1.2 dB over the entire C-band. In particular, when ρ3 is increased by 5% (shown by the red solid line in Fig. 3(c)), the DMG of the design is about 0.2 dB higher than that of the F1 at 1530 nm, but at 1565nm the DMG is about 0.1 dB lower than the optimum design. This means that the “ρ3+5%” is as good as the “F1”. But F1 is still the best solution providing the lowest DMGs over the whole C-band. Decreasing ρ3 results in an increase in gain for the LP01 mode relative to the other signal modes at longer wavelengths, thus increasing the DMG as shown by the dashed lines in Fig. 3(c). In general, the performance of F1 is robust for up to −5% changes in erbium concentration of the 2nd core layer and up to + 10% for the 3rd core layer.
Figure 4(a) and 4(b) show that the performance of F2 is acceptable (with DMG lower than 1.5 dB over the C band) for up to ± 10% change in the doping concentration of the first layer (ρ1), and for up to + 5% change in ρ2. However, even a small decrease in the doping concentration of the 2nd layer (ρ2) from its optimum value results in the DMG increasing to nearly 2dB at the short wavelength edge of the C-band. For the doping concentration of the 3rd layer of F2, as indicated in Fig. 4(c), decreasing ρ3 by 10% results in an increase in DMG but still below 1.5 dB over the C-band. The 4th layer presents a slightly more complicated story, as shown in Fig. 4(d). Down to −5% change of ρ4 is acceptable. At wavelengths longer than 1535 nm, the DMG is lower than 1.5 dB when ρ4 changes by up to + 10%. However, the DMGs are increased considerably at shorter wavelengths when ρ3 is increased by more than + 5%. For example, if ρ3 increases by + 10% from its optimum position whilst the simulation conditions and other parameters of F2 are kept unchanged, the gain of the LP01 mode slightly decreases and the gain of the LP31 mode increases. This results in a larger DMG (i.e. 2 dB).
A tolerance check of the structural parameters (i.e. a1, a2 ...) of both F1 and F2 was also undertaken. Once again we ran a set of simulations on both F1 and F2, in which we modified the structural parameters (i.e. a1, a2 ...) by ± 5% and ± 10% from the optimum value. Results for F1 and F2 are presented in Fig. 5 and Fig. 6 respectively. Again note that in the tolerance studies, the fiber lengths for both F1 and F2 were fixed at the optimum lengths selected by the GA .
Fig. 5 Variation of the DMG against signal wavelength as the structural parameter (a) a1, (b) a2 is changed for the fiber F1. Continuous lines represent a “+” variation, while dashed lines represent a “-” variation.
Fig. 6 Variation of the DMG versus signal wavelength as the structural parameter (a) a1, (b) a2, (c) a3 is changed for the fiber F2. Continuous lines represent a “+” variation, while dashed lines represent a “-” variation.
From Fig. 5(a), it can be seen that increasing the a1 of F1 to + 10% or decreasing the a1 by 10%, results in a large rise (above 1.5dB) in the DMG across the C-band (1545 - 1565nm). If a performance of DMG< 1.5 dB for the full C band is required, the variation of a1 in F1 should be kept to ± 5%. However, from Fig. 5(b), it is evident that the DMG values for the full C-band are below the upper limit (1.5 dB) with up to ± 10% variation in a2 for F1.
Figure 6(a) and 6(b) show that the performance of F2 is acceptable for changes from −5% to + 10% in the a1, and −5% to + 5% variations in a2 if a DMG <1.5 dB across the C-band can be tolerated. a3 determines the thickness of the 3rd and the 4th core layers of F2, so that any changes in a3 mostly impact the overlap between the erbium dopants and the LP31 and LP12 modes. Simulations show that the gains for the LP12 and LP01 modes become the largest amongst all the modes when a3 is increased; while the gain for the LP31 mode becomes highest when a3 is decreased. Overall the results in Fig. 6 show that the parameter a3 is the most critical. Only small variations in a3 are tolerable and particular care must be applied to try and match the optimum value to maintain low DMG.
Such complex fibers (e.g. F1 and F2) with multiple-layer-cores are likely to be challenging to realize using conventional MCVD/solution doping preform manufacturing techniques (although they are by no means impossible). However, alternative approaches offering improved spatial dopant control exist (e.g. using microstructured optical fiber [23], or chelate-based vapour phase technology [24]), and may be preferable in practice.
5. Conclusion
We have applied a Genetic Algorithm to minimize the differential modal gain in cladding-pumped EDFAs supporting four and six-mode-groups. The optimum 4-mode-group EDFA and 6-mode-group EDFA designs, exhibiting three-layer and four-layer core structures (denoed as ‘F1’ and ‘F2’) respectively, provide less than 1 dB DMG across the C-band. Over 20 dB gain across the C-band is obtained for both F1 and F2 using a forward pump power of 2.5W and EDF lengths of 6.2 m and 5.9 m respectively. We have also studied the impact of variation in doping concentration and physical dimension of each core layer of F1 and F2 on DMG. For both F1 and F2, the sensitivity to the changes in concentration is the least (up to ± 10% of its optimal value) for the inner most layer. The dimension of the inner-most layer (i.e. a1) of both F1 and F2 are tolerant to changes from −5% to + 5% variation of its optimal value. Simulations show that the a3 of F2 is the most critical, and the optimum value should be targeted if a low DMG profile is required.
Acknowledgments
This work was supported by the European Communities 7th Framework Program under grant agreement 258033 (MODE-GAP).
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