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RFA将信号光放大是通过受激喇曼散射效应实现的。当具有高功率的抽运光与信号光同时注入光纤时,会产生受激喇曼散射效应。在实际应用中,主要考虑抽运光对信号光的受激喇曼散射效应,忽略其它非线性效应,通过求解N信道单向受激喇曼散射耦合波方程[11]:
$ \left\{ {\begin{array}{*{20}{c}} {\frac{{{\rm{d}}{n_i}(z)}}{{{\rm{d}}z}} = -{\alpha _i}{n_i}\left( z \right) + \sum\limits_{j = 1}^N {{r_{ij}}{n_j}\left( z \right){n_i}\left( z \right), } }\\ {\left( {i = 1, \ldots , N} \right)}\\ {{n_i}\left( z \right)\left| {_{_{z = 0}}} \right. = {n_i}\left( 0 \right), (i = 1, \ldots , N)} \end{array}} \right. $
(1) 式中,i=1,2, …, N指的是一共有1到N个信道;ni(z)和nj(z)分别代表z处第i, j个信道中前向传输的光子通量;ni(0)是各信道z在0处入射的初始光子通量,它是恒定值;αi表示第i个信道中信号光的线性衰减系数;rij=gR/Ae为第i信道与第j信道之间前向传输光子通量的喇曼增益效率,Ae表示光纤的有效截面积[12],gR是抽运光与其它信道的信号光之间的喇曼增益系数。
经过(1)式,令1,2信道为抽运光信道,可以得到级联RFA的信号光喇曼增益G表达式为:
$ \begin{array}{l} G = 10{\rm{lg}}\frac{{{\mathit{P}_\mathit{i}}\left( \mathit{Z} \right)\left| {_{_{_{Z = {L_1} + {L_2}}}}} \right.}}{{{P_i}(0)}} = \\ 10{\rm{lg}}{\left\{ {\frac{{{{\rm{e}}^{-{\alpha _1}{L_1}}}\left[ {\frac{{{{\bar v}_1}}}{{{\nu _{11}}}}{P_{11}}\left( 0 \right) + \frac{{{{\bar v}_1}}}{{{\nu _{12}}}}{P_{12}}\left( 0 \right)} \right]}}{{\frac{{{{\bar v}_1}}}{{{\nu _{11}}}}{P_{11}}\left( 0 \right){{\rm{e}}^{{G_{11i}}}} + \frac{{{{\bar v}_1}}}{{{\nu _{12}}}}{P_{12}}(0){{\rm{e}}^{{G_{12i}}}}}}} \right._1} \times \\ \frac{{{{\rm{e}}^{-{\alpha _2}{L_2}}}[\frac{{{{\bar v}_2}}}{{{\nu _{21}}}}{P_{21}}\left( 0 \right) + \frac{{{{\bar v}_2}}}{{{\nu _{22}}}}{P_{22}}\left( 0 \right) + \frac{{{{\bar v}_2}}}{{{\nu _{23}}}}{P_{23}}\left( 0 \right)]}}{{\frac{{{{\bar v}_2}}}{{{\nu _{21}}}}{P_{21}}\left( 0 \right){{\rm{e}}^{{G_{21i}}}} + \frac{{{{\bar v}_2}}}{{{\nu _{22}}}}{P_{22}}\left( 0 \right){{\rm{e}}^{{G_{22i}}}} + \frac{{{{\bar v}_2}}}{{{\nu _{23}}}}{P_{23}}\left( 0 \right){{\rm{e}}^{{G_{23i}}}}}} \end{array} $
(2) $ \begin{array}{*{20}{c}} {{\rm{ }}{G_{11i}} = \frac{{{g_{i11}}}}{{M{A_{\rm{e}}}}}{P_{11}}\left( 0 \right){L_{{\rm{e}}, 1}}, {\rm{ }}{G_{12i}} = \frac{{{g_{i12}}}}{{M{A_{\rm{e}}}}}{P_{12}}\left( 0 \right){L_{{\rm{e}}, 1}}, }\\ {{\rm{ }}{G_{21i}} = \frac{{{g_{i21}}}}{{M{A_{\rm{e}}}}}{P_{21}}\left( 0 \right){L_{{\rm{e}}, 2}}, {\rm{ }}{G_{22i}} = \frac{{{g_{i22}}}}{{M{A_{\rm{e}}}}}{P_{22}}\left( 0 \right){L_{{\rm{e}}, 2}}, }\\ {{G_{23i}} = \frac{{{g_{i23}}}}{{M{A_{\rm{e}}}}}{P_{23}}\left( 0 \right){L_{{\rm{e}}, 2}}} \end{array} $
(3) 式中, v1与v2分别为每段光纤中各抽运光νij的均值; L1和L2分别为两段光纤的长度; Pi为第i信道的光功率;假设每段光纤上的衰减系数相同,α1和α2分别为As-S光纤和碲基光纤的衰减系数;P11, P12分别表示第1段光纤中抽运光1, 2的功率,P21, P22, P23分别表示第2段光纤中抽运光1, 2和3的功率;M是光纤的保偏系数; gi11, gi12为第1段光纤中第i信道与第1、2信道之间的喇曼增益系数; gi21, gi22, gi23为第2段光纤中第i信道与3个抽运光之间的喇曼增益系数;Le, 1, Le, 2分别是第1段与第2段光纤的有效作用距离。
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粒子群优化算法是一种较为先进的算法[13],相比于遗传算法它不需要变异和交叉这两个步骤,使算法具有简单和搜索速度快的特点[14]。粒子群优化算法可以动态地追踪当前的搜索情况并调整搜索范围,十分适用于求解非线性微分方程[15-16]。
对于喇曼光纤放大器的适应度函数为增益平坦度GΔ,函数表示为:
$ {G_\Delta } = {\rm{max}}(G)-{\rm{min}}(G) $
(4) 第i个粒子的极值被记作pbest,即每个粒子的最优位置表示为:
$ {p_{\rm best}} = ({p_{i1}}, {p_{i2}}, \ldots , {p_{iD}}), \left( {i = 1, 2, \ldots ,M} \right) $
(5) 式中,D为搜索维度,pi1, pi2, …, piD分别被记作1~M个粒子每一个维度下的最优位置。整个粒子群的最优解值记作gbest,即全局最优位置可以表示为:
$ {g_{best}} = ({g_1}, {g_2}, \ldots , {g_D}) $
(6) 式中,g1, g2, …, gD被记作每一个维度下的全局最优位置。
根据图 1中的算法流程将每个抽运光的波长和功率看作是两个维度[17],并带入算法中计算,最终可以得到满足最大输出增益和最优增益平坦度条件下的最优的抽运光参量配置。
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利用5个抽运光和As-S光纤与碲基光纤级联的RFA结构图如图 2所示。图中,OF为光滤波器(optical filter),OC为光耦合器(optical coupler)。在第1段光纤加入N个信号光与两个抽运光,通过光复用单元(optical multiplexer unit, OMU)进入长度为L1的As-S光纤放大,将两抽运光滤除。在第2段光纤处注入3个抽运光,与放大后的信号光一同进入长度为L2的碲基光纤实现补偿作用[18],最后经过光解复用单元(optical demultiplexer unit, ODU)输出信号光。
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图 3分别为As-S光纤与碲基光纤的喇曼增益谱。相比于常见的石英光纤增益谱增益大大提高。可以清晰地看到, As-S光纤喇曼频移[19]范围在[260cm-1, 350cm-1]时,喇曼增益谱处于上升阶段用作第1段放大光纤;碲基光纤喇曼频移[20]范围在[400cm-1, 510cm-1]时,喇曼增益谱处于下降阶段可以对第1段光纤实现增益补偿从而达到平坦。
利用As-S光纤喇曼频移范围在[260cm-1, 350cm-1]与碲基光纤喇曼频移范围在[400cm-1, 510cm-1]处的喇曼增益谱进行前增益后补偿实现增益平坦。按上述范围进行拟合直线得:gR, 1(Δν)=k1Δν+b1,频移Δν∈[260cm-1, 350cm-1],gR, 2(Δν)=k2, Δν+b2Δν∈[400cm-1, 510cm-1], 其中,斜率k1=1.391×10-13m·cm/W,截距b1=-3.731×10-11m/W;斜率k2=-1.220× 10-14m·cm/W,截距b2=7.45×10-12m/W。仿真参量配置如下:两段光纤有效截面积分别是Ae, 1=2.67×10-11m2, Ae, 2=5.5×10-11m2;As-S光纤的衰减系数α1=550dB/km, 碲基光纤的衰减系数α2=26dB/km,保偏系数均为M=2, 两段光纤长度分别为L1=0.02km, L2=0.402km;凭借工程经验,第1段As-S光纤设置2个抽运光,抽运光功率分别为P11=2W, P12=2.5W, 抽运光的波长分别为λ11=1480.0nm, λ12=1475.5nm;第2段碲基光纤设置3个抽运光,抽运光功率分别为P21=2W, P22=2.1W, P23=2.6W,抽运光的波长分别为λ21=1488.2nm, λ22=1467.6nm, λ23=1476.8nm。
如图 4所示, 96路信号光经过第1段0.02km的放大光纤,使信号光功率得到明显的放大,再经过第2段0.402km的补偿光纤,使获得放大后的信号光随距离增加, 共同收敛于0.7W左右。由图 5可以看出,通过级联光纤和多抽运的作用方式下,其最终获得的放大器的输出增益平均值达到45.55dB,平坦度为1.2dB,相比于普通的光纤放大器增益提高了许多,但平坦度不是十分理想,后期可以通过使用优化算法来使其达到最优值。
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在优化放大器功率之前,需要对一些必要的参量进行设置。如表 1所示,共采用5个抽运激光器对信号光放大,每一个抽运光由两个方面决定,分别为抽运光功率和抽运光波长,所以每一个粒子的维度就设置为10;抽运光的功率由Pmax和Pmin两个值限制功率搜索范围,抽运光波长由λmax和λmin来限制波长搜索范围,也就是确定两个边界条件。
Table 1. Basic parameter setting
parameter name values number of iterations T 1000 total number of particles M 50 dimension of each particle D 10 inertia weight w 0.8 learning factor c1, c2 1, 1 number of pumps n 5 effective sectional area of Ae, 1, Ae, 2 26.7μm, 55μm wavelength upper limit λmax 1410nm lower wavelength limit λmin 1490nm upper limit of power Pmax 3.5W lower power limit Pmin 2W signal optical power P 0.01mW 根据表 1中的参量设置并结合粒子群优化算法,每次运行后都能分别得到5个抽运光的波长及其对应的功率, 以及级联光纤放大器优化后的输出增益和增益平坦度。对其进行多次优化,并对获得的结果对比分析,就可以得到最佳的抽运光设置值。本文中共进行3次优化,如表 2所示(分别对应表中的1,2,3)。其中P11, P12, P21, P22和P23分别表示5个抽运光的功率值,λ11, λ12, λ21, λ22, λ23分别表示抽运光的波长值;G表示放大器最后的输出增益的平均值;GΔ表示放大器的增益平坦度。
Table 2. The optimization results
parameter name values 1 2 3 P11/W 2.7901 2.5200 2.8001 P12/W 2.6500 2.4201 2.7500 P21/W 3.2205 2.5205 3.2310 P22/W 3.1711 2.6712 3.1691 P23/W 2.9511 2.9531 2.9601 λ11/nm 1451.4 1460.8 1486.4 λ12/nm 1481.9 1459.7 1489.8 λ21/nm 1454.6 1451.1 1444.2 λ22/nm 1440.8 1440.7 1449.8 λ23/nm 1440.8 1440.7 1449.8 G/dB 49.20 53.25 44.80 GΔ/dB 0.39 0.30 0.40 通过表 2优化后的数值仿真后,得到图 6所示的输出增益图。粒子群优化算法主要对抽运光的波长和功率进行优化,采用2组优化的数值仿真,利用短波长高功率的抽运光对长波长低功率的信号光进行放大的原理,得到3组不同增益和不同平坦度的增益谱。
结合表 2与图 6可知, 第1组的平均增益为49.20dB,增益平坦度为0.39dB;第2组的平均增益为53.25dB,增益平坦度是0.30dB;第3组的平均增益为44.80dB,增益平坦度为0.40dB;而由图 6可以清晰地看出, 第2组的优化结果在获得较大的增益的同时增益平坦度最小符合放大器的要求。表 2中第2组优化结果抽运光的整体功率相比于1组和3组都相对比较小,但却获得了3组中最大的输出增益。这是由于抽运光功率逐渐增加,使抽运光之间的非线性效应也随之增强,能量转换就越来越强,抽运光对于信号光的放大作用就会逐渐减弱。因此可以得出,抽运光功率并不是越大越好,而是有一定的变化范围。另外,对于级联光纤放大器两段的抽运功率设置都在相同范围,由表 2可以看出,第2段碲基光纤的抽运光功率, 每一组优化结果都相对于第1段硫系光纤的优化后抽运光功率要大,这是由于硫系光纤的增益谱要高于碲基光纤,只有将碲基光纤的抽运光功率加大,才能够使其最后输出的增益平坦。
基于粒子群优化算法的级联喇曼光纤放大器
Cascaded Raman fiber amplifier based on particle swarm optimization
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摘要: 为了在较高的净增益条件下实现最小化喇曼增益平坦度,采用粒子群优化算法对As-S光纤与碲基光纤级联的光纤放大器各抽运光参量优化配置的方法,进行了理论分析和实验验证。结果表明,应用上述方法对得到的3组优化结果进行对比,在带宽为40nm的级联喇曼光纤放大器上,达到了平均增益为53.25dB、增益平坦度为0.30dB的较高性能。与传统光纤放大器和现有的级联光纤放大器相比,使用粒子群优化算法对各抽运光的功率和波长优化后,会使级联光纤放大器性能明显提高,这在未来的光纤通信中具有一定实用价值。Abstract: In order to achieve the purpose of minimizing the flatness of Raman gain under the condition of high net gain, particle swarm optimization (PSO) algorithm was used to optimize the pump parameters of As-S fiber and tellurium based fiber amplifier. Theoretical analysis and experimental verification were carried out. Using the above method to compare the obtained three sets of optimization results, it is found that the average gain is 53.25dB and the corresponding gain flatness is 0.30dB on the cascaded Raman fiber amplifier with a bandwidth of 40nm. The results show that, compared with the traditional fiber amplifiers, the performance of the cascaded fiber amplifier will be significantly improved after the optimization of the power and wavelength of each pump light by using particle swarm optimization algorithm, which has a certain practical value in the future fiber communication.
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Table 1. Basic parameter setting
parameter name values number of iterations T 1000 total number of particles M 50 dimension of each particle D 10 inertia weight w 0.8 learning factor c1, c2 1, 1 number of pumps n 5 effective sectional area of Ae, 1, Ae, 2 26.7μm, 55μm wavelength upper limit λmax 1410nm lower wavelength limit λmin 1490nm upper limit of power Pmax 3.5W lower power limit Pmin 2W signal optical power P 0.01mW Table 2. The optimization results
parameter name values 1 2 3 P11/W 2.7901 2.5200 2.8001 P12/W 2.6500 2.4201 2.7500 P21/W 3.2205 2.5205 3.2310 P22/W 3.1711 2.6712 3.1691 P23/W 2.9511 2.9531 2.9601 λ11/nm 1451.4 1460.8 1486.4 λ12/nm 1481.9 1459.7 1489.8 λ21/nm 1454.6 1451.1 1444.2 λ22/nm 1440.8 1440.7 1449.8 λ23/nm 1440.8 1440.7 1449.8 G/dB 49.20 53.25 44.80 GΔ/dB 0.39 0.30 0.40 -
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