折射率调制矩阵的Bragg光栅光谱分布特性研究

Research on Bragg Spectral Distribution Based on Refractive Index Modulation Matrix

为了实现对光纤光栅回波光谱分布的控制,利用传输矩阵法构建了分段调制折射率的数学模型。通过在各个分段中以不同形式的折射率调制组合实现对回波光谱分布的控制,研究了基于不同折射率分布条件下的谱形特性,为实现获取任意形态的Bragg光谱分布提供了理论支撑。系统结合耦合模理论与矩阵传输算法,当分段后子光纤光栅尺寸符合边界条件时,即仍可应用耦合模理论计算,同时又可以将多段的耦合方程以正、反向模式的形式通过矩阵函数进行表达。由此可知,虽然任意折射率调制组合构成的整个光纤光栅不具备通解形式,但分段后的子光纤光栅具有可解析的特性,同时再利用矩阵传输算法可以将m段子光纤光栅的正反向模式进行计算,就能将任意形式光纤光栅的折射率调制函数转化为传输矩阵组,再对其反射率分布场进行解析。最终,可以得到整个光纤光栅的等效正、反模式,即实现回波光谱分布的控制。由理论部分可知,回波光谱分布特性主要受正反向导模耦合系数、纤芯位置、分段数决定,可由^σ(z)和k(z)表示。通过MATLAB仿真分析可知,两参数在(0,1)范围内对反射率函数具有明显的调制作用。随着控制参数阶数的增大,反射率调制斜率也会增大;当k(0. 38,0. 48),^σ(0. 52,0. 58)时,反射率调制符合单调特性。从而得到了不同控制参数条件下反射率函数的分布变化,讨论了耦合系数对回波谱形控制的量化效果。实验利用AVESTA公司的Ti:Sapphire飞秒激光器完成了四种不同结构光纤光栅的制作,采用了四种折射率分段调制的FBG结构,分别是:(1)在m段中Λ1和Λ2交替均匀分布;(2)在m/2段中Λ1和Λ2交替均匀分布,其余段随机分布;(3)在m/2段中Λ1和Λ2随机分布,其余段也随机分布;(4)在整个光纤光栅段折射率随机分布。对以上四种FBG的回波光谱分布进行测试与比较,可知采用分段折射率调制对Bragg光谱特性的作用效果。实验结果显示:当以矩阵组形式的FBG若在m段分布时,则与传统串联型均匀FBG测试效果一致,具有两个明显的Bragg特征峰;而矩阵组在m/2段中分布时,测试光谱仍能明显获取折射率调制特征信息,即存在两个Bragg特征峰,但峰峰值减小,噪声谱增大,半宽变窄。同时,相比交替模式而言,随机分布形式此种变化趋势更为明显。由此可见,通过控制矩阵组分布对回波谱形中特征峰值、半宽及功率谱进行调制。该方法在预先设计折射率调制矩阵的条件下,可以对Bragg光谱分布进行精确控制,实现目标回波谱形的获取。

According to the form of its engraved grid,FBG sensor can be divided into uniform type,chirp type and so on.The spectrum dist ributions of FBG are different for different grid forms.At present,it has been reported that existing structural parameters were mainly analyzed by the litera ture.The functional model was studied for obtaining the spectral distribution,which can be from any kind of FBG,and its parameter design was realized.In ord er to realize the control of the echo spectrum distribution by FBG,a mathematic al model of the segmented modulation index was established by using transfer mat rix method.The spectrum distribution of the echo was controlled by the combinat ion of different refractive index modulations in each segment,and the spectral characteristics under different refractive index distributions were studied.It provided theoretical support for obtaining the Bragg spectrum distribution in an y form.In the system,coupled-mode theory and matrix transmission algorithms w ere used in combination.Compared with the traditional uniform FBG,σand k were no longer constants,but ratherσ(z)and k(z)as a function of form,so for any FBG structure did not have an analytic solution.However,if the F BG was divided into m sections,m sub-FBGs could be obtained from the con creteσ(z)and k(z)functions on each small section,so that the overa ll effects of the FBG could be obtained by the matrix transmission method.The F BG was divided into m small sections in the z-axis direction,and m su b-FBGs could be obtained from the specificσ(z)and k(z)functions in e ac h small section,so that the overall effects of the FBG can be obtained by the matrix transmission method.When the size of sub-FBG segmented meets the b oundary conditions,the coupling mode theory can still be applied.At the same time,it can express the multi-section coupled equations through the matrix f unction in the form of positive and negative modes.It can be seen from this th a t the entire FBG composed of arbitrary refractive index modulation does not have the generalized form,but it can be resolvable for the segmented sub-FBGs.A nd the matrix transmission algorithm can be used to calculate the positive and n egative modes of the m-segment sub-FBGs.So the refractive index modulation function of any type of FBG can be transformed into a transmission matrix group.The reflectivity distribution field can also be analyzed.Finally,the equi v alent positive and negative modes of the whole FBG can be obtained,so as to re alize the control of the echo spectrum distribution.As can be seen from the th eoretical part,the spectrum distribution characteristics of echo are mainly de termined by the coupling coefficients of the forward and reverse guided modes,the position of the core and the number of segments.They can be represented byσ(z)and k(z).Through MATLAB simulation analysis showed that the two parameters have significant modulation effects on the reflectivity function in the range of(0,1).As the order of control parameters increases,the slope of r eflectivity modulation will also increase.In case of k(0.38,0.48),σ(0.52,0.58),it is monotonically tuned for reflectivity modulation.The distribution of the reflectivity function under different control parameters was obtained.The quantitative effects of the coupling coefficient on the control of the echo spectrum were discussed.Taking two specific pitchΛ1 andΛ2 as an example,after splitting the whole FBG into m sub FBGs,Λ1 andΛ2 were placed on different sub-sections.The spectrum distribution patterns of FBG were analyzed according to different grid layouts.If the spect rum characteristics of FBG change by the parameters and distribution forms ofΛ1 andΛ2,and they are resolvable,the Bragg spectral characteris tics can be considered as controllability.Through the parameter control any sp ectrum distribution can be achieved.In the experiment,AVESTA’s Ti:Sapphir e femtosecond laser(Its center wavelength 800 nm,frequency 1 kHz,peak pulse energy 800 nJ.)was used to fabricate four different structured fiber gratings.Four kinds of refractive index modulation FBG segment structure were employed.Respectively:(1)Λ1 andΛ2 were evenly distributed alternately in the m section;(2)Λ1 andΛ2 were evenly distributed alte r nately in the m/2 section,and the rest of section were randomly distribute d;(3)Λ1 andΛ2 were randomly distributed in the m/2 section,a nd th e rest of section were randomly distributed,too;(4)The refractive index of the entire fiber grating segment was randomly distributed.Echo spectrum distri b ution of the above four FBGs was tested and compared,so the spectrum propertie s of Bragg was studied by the segmented refractive index modulation.The experi mental results showed that when the FBGs in the form of matrix group are distrib ut ed in the m section,they are consistent with the traditional series homogeneou s FBG test and have two obvious Bragg characteristic peaks,and they are locate d at 1 551.485 and 1 563.572 nm,respectively,and have a high signal-to-n oise ratio.It is consistent with the test results of two series-connected FBG s with fixed pitch and is also a special solution to the piecewise modulated fun ction.In case 2,its characteristic peak positions are 1 551.499 and 1 563.551 nm,its absolute error is better than 0.030 nm.Its half-width is better than that of case 1,but its noise power increases greatly and the signal-to-noise ratio decreases.In case 3,the absolute error of the characteristic p eak position is better than 0.050 nm,and the sharpness of the characteristic peak is further increased,and the noise power is further increased,and the sig nal-to-noise ratio is the worst.When the matrix group is distributed in the m/2 segment,the refractive index modulation characteristic information can still be obtained obviously in the test spectrum,that is,there are two Bragg c haracteristic peaks,but the peak-peak value decreases,and the noise spectrum increases,and the half-width narrows.At the same time,the trend of stochasti c distribution is more obvious than that of alternation.Thus,the characteristi c peak,half-width and power spectrum in the echo spectrum can be modulated by controlling the matrix group distribution.The method can accurately control the Bragg spectrum distribution under the pre-designed refractive index modulation matrix to obtain the target echo spectrum.