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多芯超模掺铒光纤结构如图1(a)所示。该光纤中含有19个纤芯,分别为中心纤芯(1)和第一层纤芯(2~7)以及第二层纤芯(8~19),19个纤芯呈正六边形均匀分布。纤芯的半径为$ r $,相邻两个纤芯间距为$ d $,包层半径为$ R $,纤芯相对包层的折射率差为$ \Delta n $。其中,$ r ={5 \; \text{μ m}} $,$ d ={12 \; \text{μ m}} $,$ R = {{62.5}}{ \; \text{μ m}} $,$ \Delta n = 0.17 \text{%} $。多个纤芯分布在同一包层中,共同形成等效纤芯区域,其模场被限制在等效纤芯区域内,进而实现模场调控。
图 1 19芯超模光纤的折射率分布(1~19为纤芯编号)以及模场分布。(a)折射率分布;(b)模场分布
Figure 1. Refractive index distribution (cores numbered 1-19) and mode field distribution of the 19-core supermode fiber. (a) Refractive index distribution; (b) Mode field distribution
所提多芯超模光纤支持10模式($ {{\rm{LP}}_{01}} $,$ {{\rm{LP}}_{11{\text{a}}}} $,$ {{\rm{LP}}_{11 {\rm{b}}}} $,$ {{\rm{LP}}_{21 {\text{a}}}} $,$ {{\rm{LP}}_{21 {\text{b}}}} $,$ {{\rm{LP}}_{02}} $,$ {{\rm{LP}}_{31 {\text{a}}}} $,$ {{\rm{LP}}_{31 {\text{b}}}} $,$ {{\rm{LP}}_{12 {\text{a}}}} $和$ {{\rm{LP}}_{12 {\text{b}}}} $)同时传输。在1 550 nm信号波长处10模式沿光纤截面的归一化光强分布如图1(b)所示。可以看出,MCSMF相较于普通单芯少模光纤,信号光10种超模分布于光纤的等效纤芯区域中,拥有超大的有效模场面积$ {A_{eff}} $,使得非线性效应的影响显著降低。模场面积$ {A_{eff}} $的表达式为:
$$ {A_{eff}} = \frac{{{{\left| {\displaystyle\iint {I\left( {x,y,z} \right){\rm{d}}x{\rm{d}}y}} \right|}^2}}}{{\displaystyle\iint {{I^2}\left( {x,y,z} \right){\rm{d}}x{\rm{d}}y}}} $$ (1) 式中:$ I\left( {x,y,z} \right) $为模式光强分布。
多芯超模光纤的模式特性如图2所示。MCSMF的有效模场面积$ {A_{eff}} $和有效折射率差$ \Delta {N_{{\text{e}}ff}} $随$ d/r $的变化而变化,通过调整选取合适的$ d/r $,可以得到较大的有效模场面积和较小的模式串扰;此外,不同超模的差分群时延(Different Group Delay,DGD)较小,使得MCSMF的串扰性能优于等效的少模光纤。模式的功率限制因子Γ描述了光纤对模式的约束能力,由图2(c)可知,选取合适的$ d/r $可以得到较大的Γ,即光纤对光场的约束能力较强,但同时高阶模因其模场分布较为分散,其Γ与低阶模的相比较小,从而导致模式损耗增加。
图 2 (a) MCSMF的$ {A_{eff}} $和$ \Delta {N_{{\text{e}}ff}} $随$ d/r $的变化;(b) MCSMF的群时延随信号波长的变化;(c) MCSMF的Γ随$ d/r $的变化
Figure 2. (a) Variation of $ {A_{eff}} $ and $ \Delta {N_{{\text{e}}ff}} $ of MCSMF with $ d/r $; (b) Variation of group delay of the MCSMF with signal wavelength; (c) Variation of Γ of the MCSMF with $ d/r $
多芯超模光纤中铒离子在纤芯内单层均匀掺杂,掺杂浓度(体积分数,下同)按照纤芯的编号分别记为 $ {N}_{1} $,$ {N}_{2} $,···$ {N}_{19} $。
在各纤芯均匀掺杂的条件下,即设置$ {N}_{1} $,$ {N}_{2} $,···$ {N}_{19} $均为$ 4 \times {10^{24}}\;{{\text{m}}^{ - 3}} $,将上述多芯超模掺铒光纤作为增益光纤,对MC-SM-EDFA增益性能进行研究。设置各信号模式输入功率均为−10 dBm,信号波长为1 550 nm,泵浦波长为1 480 nm。
首先研究MC-SM-EDFA 10模式的增益和DMG随泵浦功率的变化,如图3(a)所示。设定光纤长度为8 m,仿真结果表明,当泵浦功率低于3 W时,随着泵浦功率的增大,各信号模式增益迅速增大。当泵浦功率达到3 W时,10模式的平均增益高达21.82 dB,DMG约为1.33 dB。当泵浦功率超过3 W时,各信号模式的增益逐步增大并趋于饱和。同时,DMG随着泵浦功率的变化逐步增加并趋于平稳,最终稳定在1.58 dB以下。
图 3 MC-SM-EDFA的增益和DMG随泵浦功率和光纤长度的变化。(a)泵浦功率;(b)光纤长度
Figure 3. Variation of gain and DMG of MC-SM-EDFA with pump power and fiber length. (a) Pump power; (b) Fiber length
MC-SM-EDFA中光纤长度对各信号模式的影响如图3(b)所示。设定泵浦功率为3 W。仿真结果表明,MC-SM-EDFA各信号模式的增益均随着光纤长度的增大而增大。当光纤长度为8 m时,10模式的平均增益为21.82 dB。当光纤长度超过8 m时,各信号模式增益逐步增大并趋于饱和。在改变光纤长度的过程中,$ {\mathrm{L}\mathrm{P}}_{01} $模式的增益最大,$ {\mathrm{L}\mathrm{P}}_{12} $模式的增益最小。10模式DMG的大小主要决定于$ {\mathrm{L}\mathrm{P}}_{01} $和$ {\mathrm{L}\mathrm{P}}_{12} $模的增益,并随光纤长度增大而增大,但DMG总是小于1.70 dB。
后续通过粒子群优化算法优化各纤芯内掺铒浓度来降低不同超模的交叠积分因子,进一步减小DMG,从而实现各信号模式的增益均衡。
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多芯超模掺铒光纤是MC-SM-EDFA的核心,其信号的增益情况可根据速率方程计算得出,各信号模式的增益情况与信号光和泵浦光的归一化光强分布以及纤芯内铒离子分布有关,将光纤横截面上信号光、泵浦光以及铒离子分布的交叠程度定义为交叠积分因子[19],其表达式为:
$$ {\eta _k} = \iint {{\varGamma _{s,k}}\left( {x,y,z} \right){\varGamma _p}\left( {x,y,z} \right)}{N_0}\left( {x,y,z} \right){\rm{d}}x{\rm{d}}y $$ (2) 式中:$ {N_0}\left( {x,y,z} \right) $为纤芯内铒离子分布;$ {\varGamma _{s,k}}\left( {x,y,z} \right) $和$ {\varGamma _p}\left( {x,y,z} \right) $分别为信号光和泵浦光的归一化光强分布。
由速率方程和公式(2)可知,MC-SM-EDFA内,各信号模式间的增益情况与多芯超模掺铒光纤的交叠积分因子密切相关。通过粒子群算法优化降低交叠积分因子的差异,可实现MC-SM-EDFA中不同超模间的增益均衡。
PSO将个体鸟儿抽象成“粒子”,鸟群所寻找的食物则为“最优解”,是一种可在目标范围内搜索多变量优化问题最优解的智能算法。对于图1(a)中的光纤结构,需要同时优化多个纤芯的铒离子掺杂浓度,故采用粒子群算法进行优化计算。PSO对于光纤掺杂结构的优化问题而言:粒子群的规模$ M $为光纤结构的总数;种群搜索的目标范围为掺铒浓度的取值范围;种群飞行的空间维度$ D $为MCSMF的纤芯个数;个体的适应值$ fit $用于评估每个光纤掺杂结构的增益性能。取任一粒子对其优化过程进行描述,设该粒子当前的掺杂结构为$ {x_i} $,粒子此时的飞行速度为$ {v_i} $,则粒子在飞行过程中所经过的最优掺杂结构$ {P_{best}} $的表达式为:
$$ {P_{best}}_{_{i + 1}}\left( {t + 1} \right) = \left\{ {\begin{array}{*{20}{c}} {{P_{best}}_{_i}\left( t \right),f\left( {{x_i}\left( {t + 1} \right)} \right) \leqslant f\left( {{P_{best}}_{_i}\left( t \right)} \right)} \\ {{x_i}\left( {t + 1} \right),f\left( {{x_i}\left( {t + 1} \right)} \right) > f\left( {{P_{best}}_{_i}\left( t \right)} \right)} \end{array}} \right. $$ (3) 种群中当前的最优掺杂结构$ {g_{best}} $的更新表达式为:
$$ {g_{best}}\left( t \right) = \max \left\{ {f\left( {{P_{best}}_{_1}\left( t \right)} \right),f\left( {{P_{best}}_{_2}\left( t \right)} \right), \cdots \cdots ,f\left( {{P_{best}}_{_M}\left( t \right)} \right)} \right\} $$ (4) 粒子的飞行速度$ {v_i} $的更新表达式为:
$$ \begin{split} v_{id}^{t + 1} =& w \times v_{id}^t + {c_1} \times {r_1} \times \left( {{P_{id}} - x_{id}^t} \right) +\\ & {c_2} \times {r_2} \times \left( {{P_{gd}} - x_{id}^t} \right) \\ \end{split} $$ (5) 式中:$ w $为惯性权重;$ {c_1} $、 $ {c_2} $为学习因子;$ {r_1} $、$ {r_2} $为随机数;各项的上下标表示粒子$ i $在第$ t $次迭代中的$ d $维方向上的向量,如$ {v_{id}} $表示为速度分量、$ {x_{id}} $为位移分量、$ {P_{id}} $与$ {P_{gd}} $为粒子当前掺杂结构与种群最优掺杂结构的分量。光纤掺杂结构$ {x_{id}} $的更新表达式为:
$$ x_{id}^{t + 1} = x_{id}^t + v_{id}^{t + 1} $$ (6) 使用粒子群优化算法对MC-SM-EDFA的优化过程如图4所示。
其中利用适应值函数$ fit $来评估多芯超模光纤掺杂结构的增益性能,并将其定义为:
$$ fit = \frac{{\overline G }}{{{G_{{max} }} - {G_{{min} }}}} $$ (7) 式中:$ \overline G $为各信号模式的平均增益;${G_{{{max}} }} - {G_{{{min}} }}$为DMG。表示各信号模式的平均增益较大和DMG较小的光纤掺杂结构较易保留,因此能较快获得结构最优多芯超模掺铒光纤。适应值函数对PSO的优化结果起到决定性的作用。
Design of gain equalization for multi-core supermode fiber amplifier (inside back cover paper)
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摘要: 多芯超模光纤(MCSMF)的芯间距较小,多个芯子共同形成芯区支持多个超模传输,与普通单芯少模光纤相比,其具有较大的有效模场面积和较小的模式串扰,备受关注。MCSMF用于长距离传输时,与其相匹配的新型增益均衡放大器是实现信号中继并保持信号稳定传输的必要器件。文中提出了一种基于粒子群优化算法的19芯超模光纤增益均衡放大器,该光纤支持10个超模共同传输。通过粒子群算法分别优化各纤芯内掺铒浓度来降低不同超模的交叠积分因子,从而减小模式增益差(DMG)。结果表明,在包层泵浦条件下,最大DMG从1.33 dB (各纤芯均匀掺杂)降低至0.20 dB,在1 550 nm信号波长处10模式的平均增益为27.79 dB,且该放大器在整个C波段的增益平坦度低于1 dB。Abstract:
Objective Multi-core supermode fiber (MCSMF) with small inter-core spacing enable multiple cores to form a core region that supports the transmission of multiple supermodes. In comparison to conventional single-core few modes fiber, MCSMF has a larger effective mode field area and lower mode crosstalk, making it highly attractive. When used for long-distance transmission, novel gain equalization amplifier that is compatible with MCSMF is a necessary device to achieve signal relaying and maintain stable signal transmission. Current research on MCSMF mainly focuses on increasing the number of spatially multiplexed channels, optimizing pumping methods, and adjusting the length of the erbium-doped fiber (EDF) to expand communication capacity and reduce differential mode gain (DMG). However, there are few reports on the structural design of MCSMF. Therefore, it is of great significance to optimize the fiber's structural parameters and erbium ion distribution to further reduce DMG. In this study, the particle swarm optimization algorithm was employed to flexibly control the erbium doping concentration in each fiber core, determining the optimal doping structure of the EDF. This approach reduces the overlap integral factors of different supermodes and achieves gain equalization for $ {{\rm{LP}}_{01}} $, $ {{\rm{LP}}_{11{\text{a}}}} $, $ {{\rm{LP}}_{11{\text{b}}}} $, $ {{\rm{LP}}_{21{\text{a}}}} $, $ {{\rm{LP}}_{21{\text{b}}}} $, $ {{\rm{LP}}_{02}} $, $ {{\rm{LP}}_{31{\text{a}}}} $, $ {{\rm{LP}}_{31{\text{b}}}} $, $ {{\rm{LP}}_{12{\text{a}}}} $ and $ {{\rm{LP}}_{12{\text{b}}}} $ in the MC-SM-EDFA. Methods The designed MCSMF in this study consists of 19 fiber cores (Fig.1), including a central core (1), the first layer of cores (2-7), and the second layer of cores (8-19). These 19 cores are uniformly distributed in a hexagonal pattern. In the MCSMF, erbium ions are uniformly doped within a single layer of each fiber core. The doping concentration (volume fraction) in each core is denoted as $ {N}_{1} $, $ {N}_{2} $, ···$ {N}_{19} $ according to the core numbering. The particle swarm optimization algorithm is utilized to optimize the erbium doping concentration in each fiber core, aiming to reduce the overlap integral factors of different supermodes and further minimize DMG. This optimization process enables the achievement of gain equalization for various signal modes. Results and Discussions After optimization, the MC-SM-EDFA achieved an average gain of 27.79 dB, DMG of only 0.20 dB, and NF below 3.79 dB at a signal wavelength of 1 550 nm. Furthermore, the MC-SM-EDFA exhibited gains higher than 25 dB and gain flatness below 1 dB for different signal wavelengths in the C-band (Fig.7). The noise figure ranged from 3.4 dB to 4.4 dB, and the DMG showed minimal variation with signal wavelength. Additionally, using the Monte Carlo method, this study conducted simulations to analyze the impact of erbium ion doping concentration deviations on the balancing performance of the MC-SM-EDFA. The results demonstrated that the proposed MC-SM-EDFA structure exhibits good robustness (Fig.8). Conclusions The proposed MC-SM-EDFA in this study supports simultaneous amplification and gain equalization of 10 modes. Simulation results demonstrate that when erbium ions are flexibly doped at different concentrations in each fiber core of the MC-SM-EDFA, the DMG at a signal wavelength of 1 550 nm for the 10 modes is 0.20 dB. In the C-band (1 530-1 565 nm), all signal modes achieve gains exceeding 26.99 dB, with DMG below 0.26 dB and NF below 4.37 dB. Additionally, the gain flatness in the C-band is below 1 dB. Furthermore, the tolerance analysis of DMG to fiber manufacturing deviations indicates stable gain performance of the proposed MC-SM-EDFA. Moreover, the MC-SM-EDFA achieves gain equalization by uniformly doping erbium ions in a single layer within each fiber core, eliminating the need for a layered doping design, as required in few-mode erbium-doped fiber amplifier (FM-EDFA). Therefore, the MC-SM-EDFA offers certain advantages in terms of design and manufacturing. -
图 2 (a) MCSMF的$ {A_{eff}} $和$ \Delta {N_{{\text{e}}ff}} $随$ d/r $的变化;(b) MCSMF的群时延随信号波长的变化;(c) MCSMF的Γ随$ d/r $的变化
Figure 2. (a) Variation of $ {A_{eff}} $ and $ \Delta {N_{{\text{e}}ff}} $ of MCSMF with $ d/r $; (b) Variation of group delay of the MCSMF with signal wavelength; (c) Variation of Γ of the MCSMF with $ d/r $
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