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自2010年德国Jena大学[7]首次观测到TMI现象以来,科研人员对其展开了大量的研究探索,已经逐步发展出多种不同的物理模型来描述TMI[6, 8-12]。目前,研究人员普遍认为高功率光纤激光中TMI发生的根源在于量子亏损引起的热效应以及大模场光纤纤芯支持多个传输模式。当信号光注入大模场光纤时,虽然插入了模场适配器来过渡,仍然会不可避免地激发少量的高阶模式[13]。基模和高阶模沿着光纤的轴向进行传输,纵向的模间干涉效应会诱发增益光纤中不同位置的反转粒子数呈周期性分布。上能级反转粒子数的分布与光纤放大器中的增益系数相对应,并直接影响量子损耗导致的热沉积,通过热光效应,调制光纤折射率分布呈周期性变化。该周期满足光栅的相位匹配条件,因而可以实现基模和高阶模之间的能量耦合[14]。
文中基于丹麦科技大学Hansen等[8, 15]建立的半解析理论模型,分析了大模场光纤中各高阶模成分与基模LP01之间的非线性耦合作用,从而简要评估出光纤激光系统在不同弯曲半径下的TMI阈值大小。输出激光中的高阶模成分是评价激光模式特性的重要指标之一。因此,引入高阶模成分占比
$\xi $ 来量化评价TMI现象,$\xi (L)$ 可以定义为:$$\xi (L) = \dfrac{{{P_2}(L)}}{{{P_1}(L) + {P_2}(L)}}$$ (1) 式中:
${P_2}$ 为高阶模输出功率;${P_1}$ 为基模输出功率。文中的理论模型主要研究量子噪声和注入信号光的强度噪声等[16]诱发的TMI效应。当初始入射的信号光中只有基模而无任何高阶模成分时,TMI现象的发生主要由量子噪声(
$\hbar {\omega _0}$ )引起,此时高阶模的成分占比$\xi (L)$ 可表示为:$$\xi (L) \approx \dfrac{{\hbar {\omega _0}\sqrt {\dfrac{{2\pi {\varGamma _1}}}{{\left| {\chi ''({\Omega _p})} \right|}}} \dfrac{{{P_1}{{(L)}^{({\varGamma _2}/{\varGamma _1} - 3/2)}}}}{{{P_{0,1}}^{{\varGamma _2}/{\varGamma _1}}}}\exp \Bigg[\dfrac{{\chi ({\Omega _p})}}{{{\varGamma _1}}}({P_1}(L) - {P_{0,1}})\Bigg]{\rm{exp( - }}{\alpha _{{\rm{coil}}}}{L_{{\rm{coil}}}}{\rm{)}}}}{{1{\rm{ + }}\hbar {\omega _0}\sqrt {\dfrac{{2\pi {\varGamma _1}}}{{\left| {\chi ''({\Omega _p})} \right|}}} \dfrac{{{P_1}{{(L)}^{({\varGamma _2}/{\varGamma _1} - 3/2)}}}}{{{P_{0,1}}^{{\varGamma _2}/{\varGamma _1}}}}\exp \Bigg[\dfrac{{\chi ({\Omega _p})}}{{{\varGamma _1}}}({P_1}(L) - {P_{0,1}})\Bigg]{\rm{exp( - }}{\alpha _{{\rm{coil}}}}{L_{{\rm{coil}}}}{\rm{)}}}}$$ (2) 而当初始入射的信号光中原本就有高阶模(
$\xi (0)$ )存在时,输入信号光的强度噪声也会诱发TMI效应,此时高阶模的成分占比$\xi (L)$ 可表示为:$$\xi (L) \approx \dfrac{{\xi (0){{\left[ {\dfrac{{{P_{0,1}}}}{{{P_1}(L)}}} \right]}^{1 - \dfrac{{{\varGamma _2}}}{{{\varGamma _1}}}}}\left\{ {1 + \dfrac{1}{4}{R_N}({\Omega _p})\sqrt {\dfrac{{2\pi {\varGamma _1}}}{{{P_1}(L)\left| {\chi ''({\Omega _p})} \right|}}} } \right.\left. {\exp \Bigg[\dfrac{{\chi ({\Omega _p})}}{{{\varGamma _1}}}({P_1}(L) - {P_{0,1}})\Bigg]} \right\}{\rm{exp( - }}{\alpha _{{\rm{coil}}}}{L_{{\rm{coil}}}}{\rm{)}}}}{{1{\rm{ + }}\xi (0){{\left[ {\dfrac{{{P_{0,1}}}}{{{P_1}(L)}}} \right]}^{1 - \dfrac{{{\varGamma _2}}}{{{\varGamma _1}}}}}\left\{ {1 + \dfrac{1}{4}{R_N}({\Omega _p})\sqrt {\dfrac{{2\pi {\varGamma _1}}}{{{P_1}(L)\left| {\chi ''({\Omega _p})} \right|}}} } \right.\left. {\exp \Bigg[\dfrac{{\chi ({\Omega _p})}}{{{\varGamma _1}}}({P_1}(L) - {P_{0,1}})\Bigg]} \right\}{\rm{exp( - }}{\alpha _{{\rm{coil}}}}{L_{{\rm{coil}}}}{\rm{)}}}}$$ (3) 其中,
$$\chi (\Omega ) = 2\pi \dfrac{{\eta ({\lambda _s} - {\lambda _p})}}{{\kappa {n_c}{\lambda _s}{\lambda _p}}}{\rm{Im}} [{G_{2121}}(\Omega )]$$ (4) $$\begin{aligned} {G_{2121}}(\Omega ) = & \pi \displaystyle\int_0^b {{R_1}(r)} {R_2}(r)r\int_0^a {{R_1}(r')} \times {\rm{ }}{R_2}(r'){{\rm{g}}_1}(r,r',\Omega )r'{\rm{d}}r'{\rm{d}}r \\ \end{aligned} $$ (5) $$ {{{g}}_1}(r,r',\Omega ){\rm{ = }}\left\{ {\begin{array}{*{20}{c}} {{I_1}(\sqrt q r)[\dfrac{{{K_2}(\sqrt q b) + {K_0}(\sqrt q b) - A{K_1}(\sqrt q b)}}{{{I_2}(\sqrt q b) + {I_0}(\sqrt q b) + A{I_1}(\sqrt q b)}}{I_1}(\sqrt q r') + {K_1}(\sqrt q r')]\;\;\;{\rm{ 0}} \leqslant r < r'}\\ {{I_1}(\sqrt q r')[\dfrac{{{K_2}(\sqrt q b) + {K_0}(\sqrt q b) - A{K_1}(\sqrt q b)}}{{{I_2}(\sqrt q b) + {I_0}(\sqrt q b) + A{I_1}(\sqrt q b)}}{I_1}(\sqrt q r) + {K_1}(\sqrt q r)]\;\;\;r' \leqslant r < b} \end{array}} \right. $$ (6) 式中:
$\chi (\Omega )$ 是传输模式间的非线性耦合系数;${\Omega _p}$ 是$\chi (\Omega )$ 最大值所对应的频率;${R_N}({\Omega _p})$ 是相对强度噪声;变量q和A的指代形式分别为$i\rho C\Omega /\kappa $ 和$2{h_q}/\sqrt q \kappa $ ;函数In和Kn (n=0, 1, 2)则分别是第一类和第二类修正的贝塞尔函数;a和b是光纤的纤芯半径和外轮廓半径;${P_{0,\;1}}$ 为初始注入的信号功率;${\varGamma _1}$ 和${\varGamma _2}$ 为对应耦合模式成分的重叠因子;${\alpha _{{\rm{coil}}}}$ 是高阶模的损耗系数;${L_{{\rm{coil}}}}$ 是弯曲光纤的长度。根据Marcuse光纤弯曲的损耗理论[17],可分别计算出各传输模式在不同弯曲半径下的损耗系数,弯曲损耗的表达式如下:$${\alpha _{\rm{R}}} = \dfrac{{\sqrt \pi {U^2}\exp \left( - \dfrac{{2{W^3}}}{{3{a^3}{\beta ^2}}}R\right)}}{{2{e_\nu }{W^{3/2}}\sqrt {aR} {V^2}{K_{\nu - 1}}(W){K_{\nu + 1}}(W)}}$$ (7) 其中,
$${e_\nu } = \left\{ {\begin{array}{*{20}{c}} {2\;\;\;\;\;\;\nu {\rm{ = 0}}} \\ {1\;\;\;\;\;\;\nu \ne 0} \end{array}} \right.$$ (8) $${V^2} = {U^2}{\rm{ + }}{W^2}$$ (9) 式中:R是光纤的弯曲半径;
$\beta $ 是纵向传播常数;V、U和W分别是归一化工作频率、横向相位参数和衰减参数。利用大模场光纤25/400 μm的基本参数:NA=0.06,V=4.43@1064 nm,可分别计算出各传输模式在不同弯曲半径下的损耗系数,如图1所示。对比高阶模的弯曲损耗,图中基模LP01的损耗系数较小,两者之间存在着数量级的差别。因此,在理论评估光纤激光系统TMI阈值大小的时候,仅考虑增加了高阶模的弯曲损耗,而忽略了基模能量的损耗衰减,这也与参考文献[18]中的简化方式相一致。图 1 25/400 μm光纤中各模式的损耗系数与弯曲半径的变化关系
Figure 1. The relationship between the loss coefficient and bend radius for the modes of a 25/400 μm fiber
根据表1所示的光纤激光的系统参数,并通过利用公式(4)~(6),可模拟仿真出大模场光纤25/400 μm中各高阶模成分与基模LP01之间的非线性耦合系数,数值计算结果如图2所示。从图中可以明显看出,相较于截止频率(
${V_c} = 3.8$ )较高的模式LP02和LP21,次高阶模LP11(${V_c} = 2.4$ )与基模LP01之间的耦合强度最高,所以高功率光纤激光系统中TMI效应的发生总是最先建立在这两个模式间的能量耦合和转换。表 1 光纤激光系统的参数设置
Table 1. Parameters of fiber laser system
Parameter Value Parameter Value ${n_c}$ 1.45 $b$ 300 μm $\kappa $ 1.4 W/(m·K) $\eta $ 3.5×10−5 K−1 ${h_q}$ 1000 W/(m2·K) $\rho C$ 1.67×106 J/(m3·K) ${\lambda _p}$ 976 nm ${\lambda _s}$ 1064 nm 图 2 25/400 μm光纤中基模LP01与不同高阶模之间的非线性耦合系数
Figure 2. Nonlinear coupling coefficient between fundamental mode LP01 and different high order modes in 25/400 μm fiber
基于LP01模和次高阶模LP11两者之间的非线性耦合作用,代入计算公式(2)和(3),可分别有效评估出不同弯曲半径下高阶模成分占比的变化趋势。计算过程中,当
$\xi (L) = 0.05$ 时,即认为发生了横模不稳定现象,此时${P_1}$ 对应的功率量级即为TMI出现的阈值功率[8, 15]。当信号光中无高阶模成分存在时,TMI由光纤中的量子噪声引起,初始注入的信号功率
${P_{0,1}} = 17\;{\rm{ W}}$ ,不同弯曲半径条件下光纤激光器TMI阈值功率的变化趋势如图3(a)所示。而图3(b)描述的则是信号光中强度噪声引起的TMI效应,初始高阶模的成分占比为1%,即$\xi (0) = 0.01$ ,强度噪声的大小取为${R_N} = $ $ {10^{{\rm{ - 1}}3}}{\rm{H}}{{\rm{z}}^{-1}}$ 。对比两种不同诱因导致的TMI效应,不管是同一弯曲半径下TMI阈值的量级,还是不同弯曲半径对TMI阈值大小的影响,数值模拟的计算结果都基本类似。由图3可知,当大模场光纤25/400 μm的弯曲半径从6.5 cm缩小至4.5 cm的过程中,表1所示光纤激光系统的TMI阈值范围近似可从300 W量级提升到4000 W量级,直观说明了弯曲限模对TMI现象的有效抑制,得到不同弯曲半径条件下光纤激光系统的TMI阈值大小。
Influence of bending on transverse mode instability of large mode area fiber
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摘要: 横模不稳定效应已经逐渐成为引起高功率光纤激光光束质量急剧恶化并限制其输出功率进一步提升的首要瓶颈问题。基于全光纤化正向泵浦的窄线宽高功率放大平台,对大模场光纤激光器中的横模不稳定效应进行了一系列的探索研究。根据耦合模方程的计算结果,所用大模场光纤25/400 μm中LP01、LP11模之间的非线性耦合强度最大,这也直接诱导了横模不稳定效应的发生。为了抑制LP11模在主放大级的产生和放大,通过弯曲限模这种可操作性强的模式滤波技术,将主放增益光纤的弯曲半径从6 cm缩小至5 cm的过程中,高功率光纤激光系统的横模不稳定阈值从1000 W量级提高到了1600 W量级,而且激光器的其他输出性能几乎没有受到影响。这为构建实际的窄线宽高功率全光纤化的激光系统提供了强有力的实验参照。Abstract: The effect of transverse mode instability has gradually become the primary problem that causes laser beam quality degradation and limits power scaling of high-power fiber lasers. This paper conducts a series of study on the transverse mode instability (TMI) in large mode area (LMA) fiber based on a co-pumped all-fiberized narrow linewidth high power amplification platform. According to the calculation results of the coupled mode equations, the nonlinear coupling strength between the LP01 and LP11 modes of the LMA fiber 25/400 μm is the largest, which directly induces the TMI. In order to suppress the generation and amplification of the LP11 mode at the main amplifier, the fiber coiling method was used as an operational mode filtering technique to achieve mode control. The threshold of TMI increased from 1000 W to 1600 W while reducing the bending radius of the main amplifier gain fiber from 6 cm to 5 cm, and the other output performance of the fiber laser was hardly affected. This provides a powerful experimental reference for us to build an actual narrow linewidth, high power, all-fiberized laser system.
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Key words:
- fiber laser /
- transverse mode instability /
- bending mode /
- nonlinear coupling coefficient
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表 1 光纤激光系统的参数设置
Table 1. Parameters of fiber laser system
Parameter Value Parameter Value ${n_c}$ 1.45 $b$ 300 μm $\kappa $ 1.4 W/(m·K) $\eta $ 3.5×10−5 K−1 ${h_q}$ 1000 W/(m2·K) $\rho C$ 1.67×106 J/(m3·K) ${\lambda _p}$ 976 nm ${\lambda _s}$ 1064 nm -
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