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本文中通过常规多模光纤的SBS阈值模型,推导图 1结构中SBS阈值理论模型。耦合环辅助多模光纤的布里渊增益谱服从洛伦兹分布,可以由布里渊频移、线宽和峰值增益系数等参量来描述。
布里渊频移不仅与散射光角度有关,且其值取决于光纤中传播光模式的有效折射率和相互作用的声模式的速度,耦合环的宽度及其折射率的改变都将引起传播光模式的有效折射率变化。在图 1中,耦合环辅助多模光纤中的布里渊频移[14]可以表示为:
$ \nu_{\mathrm{B}}=\frac{2 n_{\mathrm{eff}} v}{\lambda} \sin (\theta / 2) $
(1) 式中: neff是光纤中传输模式的有效折射率; v为光纤中的声速; θ为散射角。νB与散射角的关系为:布里渊频移在θ=π时得到最大值,定义为νB, max; 在θ=0时得到最小值,定义为νB, min。
图 1中耦合环辅助多模光纤中所传输模式的布里渊散射谱线宽[20]为:
$ \Delta \nu_{\mathrm{B}}=\frac{16 {\rm{\mathsf{π}}}^2 n_{\mathrm{eff}}^2 \eta}{\lambda^2 \rho} $
(2) 式中: ΔνB是布里渊散射谱的3 dB带宽,其值与光纤传播模式的有效折射率有关; ρ是光纤材料密度; η是运动粘滞系数。
耦合环辅助多模光纤可以有效减少多模光纤中的模式耦合现象,其布里渊散射增益谱[20]为:
$ \begin{gathered} g_{\mathrm{B}}(\nu)=g_0 \frac{\Delta \nu_{\mathrm{B}} / 2}{\nu_{\mathrm{B}, \max }-\nu_{\mathrm{B}, \min }} \times \\ {\left[\arctan \left(\frac{\nu_{\mathrm{B}, \max }-\nu}{\frac{\Delta \nu_{\mathrm{B}}}{2}}\right)-\arctan \left(\frac{\nu_{\mathrm{B}, \min }-\nu}{\frac{\Delta \nu_{\mathrm{B}}}{2}}\right)\right]} \end{gathered} $
(3) 式中: g0为纯石英布里渊峰值增益。当入射光与Stokes光的频率差ν=νB时,布里渊散射谱的增益系数达到峰值,从式(3)得出其表达式[21]为:
$ g_{\mathrm{B}}=g_{\mathrm{B}}\left(\nu_{\mathrm{B}}\right)=\frac{2 {\rm{\mathsf{π}}}^2 n_{\text {eff }}^7 p_{12}^2}{c \lambda^2 \rho \nu \Delta \nu_{\mathrm{B}}} $
(4) 式中: p12是光纤弹光系数; c是真空中光速。
阈值增益系数不仅与耦合环辅助多模光纤的有效长度有关,还与光纤的有效截面积和耦合环辅助多模光纤纤芯半径等很多因素有关,其表达式[17]为:
$ G \approx \ln \left[\frac{4 A_{\mathrm{eff}} \nu_{\mathrm{B}} \sqrt{G^{\prime 3} {\rm{\mathsf{π}}}}}{g_0 k T \varGamma \nu_0 L_{\mathrm{eff}}}\right] $
(5) 式中: 布里渊阈值增益系数G的临界值G′≈21;k为玻尔兹曼常量; T为绝对温度; Γ为声子衰减速率; ν0为抽运波频率。光纤的有效长度Leff随着光纤长度L的变化与信号衰减系数α的关系表达式[17]为:
$ L_{\mathrm{eff}}=\frac{1-\exp (\alpha L)}{\alpha} $
(6) 图 1中由于耦合环的加入,其纤芯的有效模横截面积Aeff有所变化,表达式为:
$ A_{\text {eff }}=S^2 {\rm{\mathsf{π}}}\left[r_{\mathrm{co}}{ }^2-r_0{ }^2-\left(r_2{ }^2-r_1{ }^2\right)\right] $
(7) 式中: S为图 1中光纤模场面积与纤芯面积之比。多模光纤中SBS阈值的通用表达式[17]为:Pth=GAeff/(gBLeff)。将式(7)代入,可得图 1耦合环辅助多模光纤的SBS阈值表达式为:
$ P_{\mathrm{th}}=\frac{G S^2 {\rm{\mathsf{π}}}\left[r_{\mathrm{co}}{ }^2-r_0{ }^2-\left(r_2{ }^2-r_1{ }^2\right)\right]}{g_{\mathrm{B}} L_{\mathrm{eff}}} $
(8) 式中: r0为中心低折射率耦合环半径;r1为中心耦合环外纤芯的外环半径;r2为高阶折射率耦合环的外环半径;rco为耦合环辅助多模光纤的纤芯半径。
耦合环辅助多模光纤受激布里渊散射阈值分析
Analysis of the stimulated Brillouin scattering threshold of coupling ring-assisted multi-mode fiber
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摘要: 为了降低模间串扰, 提出了一种耦合环辅助多模光纤结构, 通过仿真软件对其结构进行建模, 并建立了其受激布里渊散射增益谱的数学模型; 理论分析了耦合环辅助多模光纤的受激布里渊散射阈值以及光纤参量和模式对受激布里渊散射光谱阈值的影响, 并通过了仿真实验验证。结果表明, 耦合环辅助多模光纤增大关键模式有效折射率差值为原来的1.75倍, 可有效抑制模间串扰; 受激布里渊散射光谱阈值跟随光纤长度的增加, 从急剧下降变至缓慢, 最终在18 km处趋于定值30 dBm; 在其它条件不变的情况下, 阈值随着衰减系数、纤芯的有效截面积和光纤模式阶数的增加呈线性关系增加, 随着纤芯半径的增加呈指数形式增长。该研究为提升光纤通信系统的传输距离和信道容量提供了理论参考。Abstract: In order to reduce inter-mode crosstalk, a coupling ring-assisted multi-mode fiber structure was proposed, and the mathematical model of its excited Brillouin scattering gain spectrum was established. Theoretically, the excited Brillouin scattering threshold of the coupling ring-assisted multi-mode fiber was analyzed, as well as the effects of fiber parameters and modes on the excited Brillouin scattering spectral threshold. The results show that the coupling ring-assisted structure increases the effective refractive index difference to 1.75 times that of the conventional type step-index multi-mode fiber. The threshold of the excited Brillouin scattering spectrum in coupling ring-assisted multi-mode fiber follows the increase of the fiber length from a sharp decrease to a slow one and finally converges to a constant value of 30 dBm at 18 km, which is higher than that of the conventional step refractive index multi-mode fiber. Other things being equal, the threshold increases linearly with the attenuation coefficient, the effective cross-sectional area of the fiber core, and the mode order of the fiber; it increases exponentially with the increase of the core radius. This study provides a theoretical reference for enhancing the transmission distance and channel capacity of fiber optic communication systems.
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