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长周期光纤光栅,通常指周期为数十或数百微米的光纤光栅,其模式耦合表现为前向传输的纤芯基模与同向传输的一阶各次包层模的耦合,可实现满足相位匹配条件的纤芯模到包层辐射模的耦合,以抑制某些特定频率的光波[1],表现为透射型光栅[2]。相比于光栅周期低于1 μm,耦合模式表现为前向传输的纤芯模与反向传输的纤芯模之间耦合的反射型短周期光栅,长周期光栅因其无后向反射模式而在温度、应力、折射率传感领域和光纤激光器等系统具有重要的应用价值[3-4]。不同的应用对光栅特性有不同的需求,作为传感器件的长周期光纤光栅(LPFGs),为了提高探测灵敏度,要求峰值带宽较窄;而为了光纤放大器系统的增益均衡,则要求光谱带宽较宽和峰值功率较大[5]。因此,为了制作出符合实际需求的LPFGs,光纤光栅的仿真设计显得尤为必要。2002年,徐新华等提出矩形调制型光栅仿真法对光栅透射谱特性进行了研究[6];2007年,吴清海等提出了一种仿真长周期光纤光栅透射谱的新方法,通过将某波长下纤芯基模与各奇次包层模耦合的透射率作为该波长的总透射率,来求出整个光栅的透射谱[7];2017年,高敏采用传输矩阵法对两个不同参数的长周期光纤光栅级联后的传输谱进行了分析,并研究了两个光纤光栅长度、平均有效折射率调制、级联光栅长度等参数对级联光栅传输谱的影响[8]。本文基于耦合模理论和光纤简化三层模型,利用MATLAB模拟软件对长周期光纤光栅的光谱特性与光栅参数(如周期大小,光栅长度,折射率调制参量等)之间的关系进行了模拟仿真,获得了最佳光谱特征和光栅参数之间的定量功率。该研究对长周期光纤光栅刻写过程中透射谱的高精度定位和长周期光纤光栅的应用提供了一定的理论依据。
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在理想条件下的均匀光纤波导结构中,纤芯与包层的模式之间是彼此正交的,并且没有模式耦合和能量交换。长周期光纤光栅的实质是由于引入周期性介电扰动,破坏了光纤光学性质的一致性,各模式之间发生了能量交换,满足耦合条件的光栅透射谱波长处出现损耗峰,表现为透射型光栅[1]。根据耦合模理论,通过合理近似,可以把实验中采用的阶跃折射率单模光纤近似为3层介质圆柱形波导(图1)。忽略材料吸收损耗,其中:n1、n2、n3分别为纤芯、包层和外界空气的折射率,a1、a2分别为纤芯和包层半径[9]。
图 1 阶跃折射率光纤三层模型截面图
Figure 1. Cross section of three-layer structure for the step refractive index fiber
通常光纤光栅的有效折射率neff的变化可表示为[10]:
$$ \Delta {n_{{\rm{eff}}}}({\textit{z}}) = \bar \Delta {n_{{\rm{eff}}}}({\textit{z}})\left[ {1 + s\cos \left(\frac{{2{\text π} }}{\varLambda }{\textit{z}} + \varphi ({\textit{z}})\right)} \right], $$ (1) 式中:s为条纹可见度,Λ为光纤光栅周期,
$ \bar \Delta {n_{{\rm{eff}}}}\left({\textit{z}} \right)$ 为光栅周期的平均折射率变化,φ(z)为光栅周期的啁啾或相移。当光栅未被写入时,各模式之间相互独立,但是,空间相位光栅的形成导致理想波导出现微扰动。在理想状态下,包层模的传播常数为[11]:$$ \beta = \frac{{2{\text π} }}{\varLambda }{n_{{\rm{eff}}}}. $$ (2) 其沿+z和−z方向传播的第m阶模的慢变振幅可表示为[12]:
$${\left\{\begin{aligned} & \frac{{{\rm{d}}{A_m}}}{{{\rm{d}}{\textit{z}}}} = i\displaystyle\sum\limits_q {A_q}(C_{qm}^T \!+\! C_{qm}^L){e^{i({\beta _q} \!-\! {\beta _m}){\textit{z}}}} +i\displaystyle\sum\limits_q {{B_q}(C_{qm}^T \!-\! C_{qm}^L){e^{ - i({\beta _q} + {\beta _m}){\textit{z}}}}}\\ & \frac{{{\rm{d}}{B_m}}}{{{\rm{d}}{\textit{z}}}} =\!-\! i\displaystyle\sum\limits_q {A_q}(C_{qm}^T - C_{qm}^L){e^{i({\beta _q} \!+\! {\beta _m}){\textit{z}}}}\! -\! i\displaystyle\sum\limits_q {{B_q}(C_{qm}^T \!+\! C_{qm}^L){e^{- i({\beta _q}-{\beta _m}){\textit{z}}}}}\end{aligned}\right.,}$$ (3) 式中:Am、Bm分别表示正向传输与反向传输的第m阶模振幅,
${\beta _q}$ 、${\beta _m}$ 阶分别表示第q阶和第m阶包层模的传输常数,$ C_{qm}^T\left({\textit{z}} \right)$ 为第m阶和第q阶模之间的横向耦合系数,在光纤中,纵向耦合系数$ C_{qm}^L \ll C_{qm}^T$ ,可忽略不计。$ C_{qm}^T$ 可表示为[13-14]:$$ C_{qm}^T({\textit{z}}) = \frac{\omega }{4}\iint\limits_\infty {\Delta \varepsilon (x,y,{\textit{z}})e_q^T}(x,y) \cdot e_m^{T * }(x,y){\rm{d}}x{\rm{d}}y, $$ (4) 式中,
$ \Delta {\rm{\varepsilon }}\left( {{{x}},{{y}},{{\textit{z}}}} \right)$ 为介电常数微扰量。根据光纤光栅的基本参数,忽略纵向耦合系数,将式(1)、式(2)带入式(3)中,即可得到长周期光纤光栅的透射谱表达式。
进一步对光栅传输模式进行近似,忽略包层模与模之间的耦合,认为仅存在包层模与纤芯模的耦合,式(3)可简化为[15]:
$$ \left\{ \begin{array}{l} \dfrac{{{\rm d}{A^ + }}}{{{\rm d}{\textit{z}}}} = i{\xi ^ + }{A^ + }({\textit{z}}) + i\kappa {B^ + }({\textit{z}}) \\ \dfrac{{{\rm d}{B^ + }}}{{{\rm d}{\textit{z}}}} = - i{\xi ^ + }{B^ + }({\textit{z}}) + i{\kappa ^ * }{A^ + }({\textit{z}}) \end{array}\right., $$ (5) 式中,
${\rm{\kappa = }}{{\rm{\kappa }}_{{\rm{21}}}}{\rm{ = }}{{\rm{\kappa }}_{{\rm{12}}}}^*$ 是交叉耦合系数。振幅可定义为:$$ \left\{ \begin{array}{l} {A^ + }({\textit{z}}) = {A_1}{{\rm{e}}^{i({\xi _{11}} + {\xi _{22}})\frac{{\textit{z}}}{2}}}{{\rm{e}}^{i{\delta _{\rm{d}}}{\textit{z}} - \textstyle\frac{\varphi }{2}}} \\ {B^ + }({\textit{z}}) = {A_2}{{\rm{e}}^{ - i({\xi _{11}} + {\xi _{22}})\frac{{\textit{z}}}{2}}}{{\rm{e}}^{ - i{\delta _{\rm{d}}}{\textit{z}} + \textstyle \frac{\varphi }{2}}} \end{array} \right., $$ (6) 式中,
${\xi _{11}}$ 和${\xi _{22}}$ 是自耦合系数,一般自耦合系数可定义为:$$ {\xi ^ + } = {\delta _d} + \frac{{{\xi _{11}} - {\xi _{22}}}}{2} - \frac{1}{2} \times \frac{{{\rm d}\varphi }}{{{\rm d}{\textit{z}}}} , $$ (7) 式中,对于LPFGs,
${\xi ^ + }$ 和$\kappa $ 是常数,$\delta _d$ 是模间失谐量,可表示为:$$ {\delta _d} = {\text π} \Delta {n_{{\rm{eff}}}}\left(\frac{1}{\lambda } - \frac{1}{{{\lambda _B}}}\right), $$ (8) 式中
${\lambda _B}$ 是LPFGs的波长,定义${\lambda _B} = \Delta {n_{{\rm{eff}}}}\varLambda = $ $(n_{{\rm{eff}}}^{{\rm{co}}} - n_{{\rm{eff}}}^{{\rm{cl}}})\varLambda $ 。假定入射光仅有一种模式,即
${\textit{z}}= - \infty $ ,即初始条件为$\left[ {{A^ + }(0) = 1,{B^ + }(0) = 0} \right]$ ,则透射光可表示为[16]:$${ \left\{ \begin{aligned} & \dfrac{{{{\left| {{A^ + }({\textit{z}})} \right|}^2}}}{{{{\left| {{A^ + }(0)} \right|}^2}}} = {\cos ^2}({\textit{z}}\sqrt{{\kappa ^2} + \xi ^{+2}}) + \dfrac{{\xi^{+2}}}{{{\kappa ^2} + \xi ^ {+2}}}{\sin ^2}\left({\textit{z}}\sqrt {{\kappa ^2} + \xi^{+2}} \right) \\ & \dfrac{{{{\left| {{B^ + }({\textit{z}})} \right|}^2}}}{{{{\left| {{A^ + }(0)} \right|}^2}}} =\dfrac{{{\kappa ^2}}}{{{\kappa ^2} + \xi^{+2}}}{\sin ^2}({\textit{z}}\sqrt {{\kappa ^2} + {\textit{z}}^{+2}} ) \end{aligned}\right.}, $$ (9) 式中,
$ \dfrac{{{{\left| {{A^ + }\left( Z \right)} \right|}^2}}}{{{{\left| {{A^ + }\left( 0 \right)} \right|}^2}}}$ 表示束缚纤芯模的标准透射响应,$ \dfrac{{{{\left| {{B^ + }\left( Z \right)} \right|}^2}}}{{{{\left| {{A^ + }\left( 0 \right)} \right|}^2}}}$ 代表耦合到包层模的模式与纤芯模LP01的功率比[17]。当
${\xi ^ + } = 0$ 时,包层与纤芯模完全耦合,即:$$ \frac{{{{\left| {{B^ + }({\textit{z}})} \right|}^2}}}{{{{\left| {{A^ + }(0)} \right|}^2}}} = {\sin ^2}(\kappa L). $$ (10) 相关波长近似为:
$$ {\lambda _{\max }} \approx \left(1 + \frac{{\Delta {{\bar n}_{{\rm{eff}}}}}}{{\Delta {n_{{\rm{eff}}}}}}\right){\lambda _B}. $$ (11) 在弱长周期光纤光栅中,归一化带宽可表示为:
$$ \frac{{\Delta {\lambda _0}}}{\lambda } = \frac{{2\lambda }}{{\Delta {n_{{\rm{eff}}}}L}}\sqrt {1 - \left(\frac{{\kappa L}}{{\text π} } \right)}. $$ (12) -
根据耦合模理论,在长周期光纤光栅模式耦合中,纤芯基模与包层模之间进行能量交换。影响其传输特性的参数主要是光纤纤芯和包层的半径a1和a2,光纤纤芯和包层的折射率n1和n2,以及光栅刻写周期Λ、光栅刻写长度L和光栅刻写过程中的折射率调制深度δn。在保持其他条件不变的情况下,改变其中任意一个参数,光栅光谱特性和变化规律将有所不同。
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仿真光纤结构参数均采用康宁公司的普通单模光纤(Corning SMF-28)参数进行模拟,相关参数如下:a1=4.15 μm,a2=62.5 μm,n1=1.45,n2=1.445,n3=1,光栅周期取Λ=400 μm,光栅长度取L=35 mm,条纹可见度s=0.5,折射率调制深度δn=2.5×10−4。在其他条件不变的情况下,首先改变光栅刻写周期,分析光栅刻写周期对光栅光谱传输特性的影响。选取LPFGs刻写周期分别为350、400 、450、500 μm,仿真得到LPFGs归一化透射谱变化情况,见图2。
图 2 (a) LPFGs透射谱随光栅周期变化曲线;(b)谐振波长随光栅周期的变化曲线
Figure 2. (a) Transmission spectrum of LPFGs at different periods; (b) relationship between resonance wavelength and grating period
图2(a)给出了不同光栅刻写周期时LPFGs的光谱特性。由图2(a)可以看出,随着光栅刻写周期的增加,光栅透射谱损耗峰峰值逐渐减少且发生红移。图2(b)进一步对不同谐振波长与光栅周期变化关系进行了线性拟合,发现光栅透射谱的谐振波长与光栅周期的变化呈现线性关系,且高次模谐振波长向长波的移动速度明显高于低阶模,即高次模对光栅周期的依赖性更为敏感。
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保持光栅周期Λ=400 μm不变,研究光栅刻写长度对光栅透射谱的影响。依次选取光栅刻写长度L为2.8、3.2、3.6、4.0、4.4、4.8、5.2 cm,模拟分析LPFGs归一化透射谱的变化情况,如图3所示。图3(a)中的损耗峰从左到右依次为纤芯基模LP01与一阶包层模LP02、LP03、LP04耦合形成的谐振损耗峰。从图3(a)可以看出,谐振峰波长无明显漂移,谐振峰损耗强度随光栅刻写长度的增加而增大,且带宽逐渐变窄。这是由于随着光栅长度的增加,纤芯模与包层模之间的耦合不断增强。同时,当光栅刻写长度为5.2 cm时,光栅谐振损耗峰值达到最大,光栅刻写长度进一步增大,光栅将发生过耦合现象而致使透射谱谐振峰损耗减小。图3(b)进一步给出了光栅透射谱谐振峰损耗随光栅刻写长度的变化,其中由最下面的曲线可以看出,纤芯基模LP01与包层模LP04间出现了过耦合现象。
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保持光栅其他参数不变,仅使折射率调制深度δn由3×10−4逐渐增大至δn=5×10−4,仿真分析光栅折射率调制深度δn对LPFGs归一化透射谱的影响,如图4所示。从图4(a)可以看出,随着折射率调制深度的增加,纤芯基模LP01与一阶包层模LP02、LP03、LP04耦合形成的谐振波长向长波方向呈现轻微的红移,而谐振峰损耗最大值从高次模向低次模逐渐过渡。图4(b)进一步给出了谐振波长随折射率调制深度的变化。可以看出,随着折射率调制深度的增大,谐振波长线性增大,即谐振波长呈现轻微红移,这一结果与采用变区间二分法求解的结果一致[18]。图4(c)则进一步给出了谐振峰损耗随调制深度的变化关系。
图 4 LPFGs透射谱随折射率调制深度δn的变化曲线。 (a) LPFGs透射谱;(b)谐振波长和(c)谐振峰损耗值随折射率调制深度δn的变化
Figure 4. Transmission spectra of LPFGs at different depthes of refractive index modulation. (a) Transmission spectra of LPFGs; relation of the resonance wavelength (b) and transmission loss (c) with depth of refractive index modulation.
可以看出,随着折射率调制深度δn的增大,纤芯基模LP01与一阶包层模LP02、LP03耦合形成的损耗峰呈现增大趋势,而与LP04耦合在折射率调制深度δn<3.5×10−4之前呈现增大趋势,在折射率调制深度δn>3.5×10−4之后呈现减小趋势,且当折射率调制深度为δn=3.5×10−4时,其纤芯基模LP01与包层模LP04耦合,最高次模式损耗峰值达到最大,且透射谱带宽最窄。随后,光栅发生过耦合现象,呈现谐振峰损耗位置由高次模LP04向低次模LP03转移的趋势。
基于以上LPFGs理论设计,对比分析了本课题组在长周期光纤光栅制备的实验结论[19-21],结果发现理论分析与实验结果具有极好的相似性,进一步证明了该理论模拟在LPFGs实验制备上具有指导意义。此外,在农药和有毒物质检测等方面,此类长周期光纤光栅具有潜在应用[22-24]。
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通过对光纤耦合模理论的研究分析,以康宁公司普通单模光纤(Corning SMF-28)参数为基础,仿真研究了普通单模光纤上刻写的LPFGs透射谱谐振波长、谐振峰损耗值、带宽与光栅刻写参数如周期、刻写长度以及折射率调制深度之间的关系。研究结果表明,LPFGs透射谱谐振波长随着光栅刻写周期和折射率调制深度的增大呈现线性增加,即透射谱谐振波长发生了红移现象,且纤芯基模LP01与一阶包层模高次模LP04的耦合模表现更为敏感;透射谱带宽主要取决于光栅刻写长度,随着光栅刻写长度的增加,透射谱带宽逐渐变窄,且当光栅刻写长度大于5.2 cm时,光栅呈现过耦合现象而使谐振峰损耗下降;同时,随着折射率调制深度的增加,光栅依次呈现不完全耦合、完全耦合和过耦合现象,且谐振峰损耗位置逐渐由高次模向低次模转移。该研究为长周期光纤光栅的高精度刻写及其刻写过程中透射谱的精确定位提供一定的参考价值。
Numerical simulation of transmission spectra characterization of long-period fiber grating
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摘要: 基于耦合模理论,利用传输矩阵法求解出长周期光纤光栅(Long Period Fiber Gratings,LPFGs)的透射谱表达式,模拟分析了LPFGs的光谱特性与光栅参数如周期、刻写长度以及折射率调制深度之间的关系。研究结果表明:LPFGs谐振波长随着周期和折射率调制深度的增大向长波方向移动,且高次模谐振波长对光栅周期更为敏感;光谱带宽的变化主要取决于光栅的刻写长度,随着光栅刻写长度的增加,带宽逐渐变窄,且当光栅刻写长度大于5.2 cm时,光栅存在过耦合区域;随着折射率调制深度的增加,光栅存在不完全耦合、完全耦合和过耦合现象,且谐振损耗最大值位置随着折射率调制深度的增加逐渐向低次转移。该研究结论对长周期光纤光栅的理论分析和实际应用中的参数设计具有重要参考价值。Abstract: Based on the coupled-mode theory, the equation for the transmission spectrum of Long Period Fiber Gratings (LPFGs) is solved by using the transmission matrix method and the relationship between the spectral characteristics of LPFGs and grating parameters (such as grating period, writing length and the depth of refractive index modulation) is simulated. The results show that the resonant wavelength of LPFGs is red-shifted with an increase in the grating period and refractive index modulation depth, and that the resonant wavelength with higher-order mode is relatively more sensitive on the grating period. At the same time, the change in the spectral bandwidth mainly depends on the writing length of the grating. The bandwidth narrows gradually with an increase in the length of the grating and over-coupling occurs when the grating length is higher than 5.2 cm. With an increase in the refractive index modulation depth, the grating has the phenomena of incomplete coupling, complete coupling and over-coupling, and the position of the maximum resonance loss will gradually transfer to the lower-order mode. The results of this research have important referential significance for the theoretical research of LPFGs and parameter designs in practical applications.
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图 4 LPFGs透射谱随折射率调制深度δn的变化曲线。 (a) LPFGs透射谱;(b)谐振波长和(c)谐振峰损耗值随折射率调制深度δn的变化
Figure 4. Transmission spectra of LPFGs at different depthes of refractive index modulation. (a) Transmission spectra of LPFGs; relation of the resonance wavelength (b) and transmission loss (c) with depth of refractive index modulation.
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