1 引言
波导光栅在波分复用[1]、信号处理[2-3]、激光[4-5]、传感[6-7]、光波的滤波及耦合[8-9]等方面得到了广泛研究及应用,这些研究和应用主要基于波导光栅的功率谱。对于波导光栅的相位谱而言,其具有较复杂的非线性,且对结构变化极其敏感,同时相位谱的获取也相对较难,这些因素束缚了相位谱的研究和应用。时延谱与相位谱、色散谱是密切相关的,可相互表征和转换,存在相近的信息等效性。因此,时延谱携带了丰富的相位、时延及色散信息。基于时延应用的波导光栅通常是非均匀的波导布拉格光栅(WBG)。随着非均匀光栅制作[10]及其应用技术的发展,更多情况下人们需要分析WBG的时延或色散特性,如在超快激光及射频光子滤波[11-13]、慢光效应及高灵敏传感[14-16]、波分复用及色散补偿[17-19]等应用领域,人们需分析WBG的时延或相位谱、色散量。目前,分析WBG时延谱的方法主要有耦合模理论、传输矩阵法、Riccati方程数值解法和分层法等[8,20-23]。耦合模理论主要用于分析均匀波导光栅,难以直接分析非均匀光栅的时延特性。传输矩阵法是基于耦合模理论的数值分析法[21],通过分段均匀化来建立非均匀光栅的传输矩阵,其分析效率和精度都较低。Riccati方程数值解法通过数值求解基于折射率微扰的Riccati微分方程而得到时延谱。分层法的分层数和计算量远大于传输矩阵法,计算效率最低。上述方法可分析比较规范的非均匀光栅,但没有封闭形式的时延谱通解,不能准确分析非规范的、难以用函数表征的、有随机性折变分布的波导光栅。时延谱的通解或解析解具有通用、直观和准确的特点。然而,WBG结构的多样性和复杂性以及求解非常系数耦合方程的困难度导致近半个世纪来一直没有时延谱的通解。为此,本文利用傅里叶模式耦合(FMC)理论[24]求解波导光栅的相位谱,进而求解出简单、精确、通用、适合各种WBG的时延谱半解析通解。通过仿真和实测的相位及时延谱,验证了该时延谱模型的正确性和复杂度,为分析、设计和应用各种WBG建立了精确统一的基础理论。
2 通解
2.1 相位谱通解
设WBG是在波导横截面上均匀的、在光传播方向上周期或准周期变化的折射率微扰分布。它可将波导内的传导模式反向耦合到同一波导内,其模式耦合情况如图1所示。
图 1. 波导布拉格光栅中的模式耦合示意图
Fig. 1. Schematic of mode coupling in WBG
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当WBG无过耦合时,在仅考虑原传输模和耦合模两个模式的耦合的情况下,根据FMC理论,传输模和耦合模本征电场的振幅系数Am和Bs同时满足反向耦合方程和流守恒定律[24-25],即
式中:z为光波的传输方向;L为光栅长度;下标m和s分别为原传输模和耦合模;δn(z)为折射率微扰分布;v=(nm+ns)/λ为与波长有关的类频率量,λ为波长;nm和ns分别为传输模和耦合模的有效折射率;k=0.5ωε0n0EmdD为耦合常数,Em和Es分别为传输模和耦合模在横截面D(面积)上的归一化本征电场分布,*为取共轭,n0为波导折射率,ω为光波角频率,ε0为真空介电常数。实际上,在(1)式右侧的积分是在波导光栅内对δn(z)的傅里叶变换。设该傅里叶变换的实部和虚部分别为γ和η,则有
WBG的相位响应是其耦合光相对于入射光的相位变化量。光波的耦合条件是相位匹配,当折射率微扰谐波峰值点的各反射光之间的相位差是2π的整数倍时,各反射光相干叠加。相干叠加后的幅值受谐波分量幅值的调制,与折射率微扰谐波分量的初相位无关。但该谐波分量的初相位决定了耦合光的初始反射点,对WBG的相位响应有影响。根据反向耦合的边界条件Bs(L)=0,求解(1)~(3)式的耦合方程可得
进而可求得在z=0处耦合模与传输模的振幅系数比Bs(0)/Am(0)。该振幅系数比包含了WBG的相位响应。WBG中的反向耦合光总是滞后于入射光,其相位响应只能取负值且在[-2π, 0]内变化。如果用振幅系数比的虚部和实部之比的反正切函数来计算相位,其相位值的范围为[-π/2, π/2]。这将使相位谱的计算值与实际值的值域不匹配,需要根据实部和虚部的符号把相位平移到[-2π, 0]内,同时需要保证实部在0附近时的连续性。因此,由WBG的振幅系数比可得其相位响应φ(v)为
类频率v是波长λ的函数,所以相位ϕ(v)既是类频率的函数,也是波长的函数。对于均匀、相移、超结构、摩尔、高斯和余弦类变迹等WBG而言,它们的折射率微扰具有解析型的傅里叶变换结果,这类WBG可称为A类光栅(A-WBG)。而对于啁啾光栅、难以用数学函数表征微扰的光栅、有随机性相变或折变量的光栅、任意复杂折射率微扰的光栅等,它们的折射率微扰没有解析型的傅里叶变换结果,但都有离散傅里叶变换(DFT)的数值结果,这类WBG可称为B类光栅(B-WBG)。A类WBG的γ和η是解析型结果,对应的φ(v)是解析型相位谱,采用γη符号的判定条件。B类WBG的γ和η是DFT的数值结果,其ϕ(v)是半解析型相位谱,采用η符号的判定条件。
2.2 时延谱的通解
WBG的时延τ是其反向耦合光相对于输入光的时间延迟,也是相位谱ϕ对角频率ω的导数,即τ=dϕ/dω。对于A类光栅,可由相位随波长变化的相位解析解ϕ(λ),用相位谱对波长的导数计算出时延谱。根据角频率与波长的关系ω=2πc/λ(其中c是真空中的光速),时延谱τ=-
·
。在±π/2及其附近相位的正切函数是不连续的且趋于无穷大,则在η=0处计算的相位是不连续和奇异的,相位的导数也不存在。在计算机中很容易通过差分来计算微分值。为此采用微分值来近似计算相位的导数,即用相位的微分值Δϕ(λ)和波长间隔Δλ直接计算时延谱τ=-·
。对于B类光栅,只能通过DFT得到其半解析型的相位谱。DFT的自变量是类频率v,与角频率的关系为ω=2πvc/(nm+ns)。根据(5)式可直接计算类频率v处的相位值,采用相位对类频率的微分能更好地计算时延谱。另外,在用计算机实现DFT时,折射率微扰将被平移到DFT的中心位置。设该位置平移量为L0,对应会产生相位移动量πvL0,则根据DFT结果计算得到的相位量ϕd(v)=ϕ(v)+πvL0,用相位微分值Δϕd和DFT中类频率间隔值Δv可计算得到的时延谱τ(v)=(Δϕd/Δv)(nm+ns)/(2πc)。因此,WBG时延谱τ的半解析型通解为
类频率v是波长的函数,所以由(6)式计算的时延响应都是波长的函数。用(5)式计算出随波长λ或类频率v变化的相位谱,并计算出相邻波长或类频率间的相位微分值Δϕ后,可按(6)式直接计算出WBG的时延谱。光波传输时的相位变化是可以连续的,但用反正切函数计算相位时,通解在±π/2值域附近存在不连续的奇异点,即在η=0的邻域内相位值会存在伪突变,故需要去除该伪突变量。事实上,目前所有的相位分析方法如耦合模理论、传输矩阵法和Riccati方程数值计算法等,都存在该相位伪突变的问题。
傅里叶变换(3)式没有限定折射率微扰的分布形式,故所得的通解适合于所有类型的WBG或任意的折射率微扰分布。无论是A-WBG还是B-WBG,其折射率微扰的解析型或离散型傅里叶变换总是存在的,可直接代入(5)式得到其相位谱,进而可按(6)式计算得到其时延谱。这说明该时延谱的解适合所有类型的任意WBG,是WBG时延谱的通解。同时,在得到解析或数值化的傅里叶变换结果后,基于该结果的相位谱解和时延谱解,即(5)和(6)式,都是封闭形式的解析型表达式。因此,所得的时延谱解,即(6)式,是适用于各种任意WBG的、半解析型的时延谱通解。
3 通解验证
验证时延谱通解的过程是:首先,按(3)式计算WBG折射率微扰的傅里叶变换;其次,根据傅里叶变换结果按(5)式计算相位谱及其相位微分值Δϕ;最后,按(6)式用相位微分值Δϕ计算WBG的时延谱。A类WBG的折射率微扰存在解析型傅里叶变换,其相位谱是全解析的且可直接计算出不同波长处的相位,此时采用γ和η为解析值时的时延谱公式。B类WBG只有数值化的DFT结果,其相位谱是半解析的且可直接计算出不同类频率的相位,此时采用γ和η为DFT值时的时延谱公式。下面以均匀光纤光栅和线性啁啾光纤光栅为例,分别验证A类和B类WBG时延谱的分析方法的正确性。
3.1 A类WBG的时延谱
以单模光纤中的均匀光纤Bragg光栅(FBG)为例,说明基于解析傅里叶变换的时延谱分析方法的正确性。均匀FBG的折射率n(z)及其微扰δn(z)可分别表示为n(z)=n0+δn(z)和δn(z)=δ[1-cos(2πz/Λ)],式中:δ为折射率微扰的幅值;Λ为光栅周期。在单模光纤中,传输模和耦合模都是纤芯的基模,则nm=ns和v=2nm/λ。根据(3)式对均匀FBG的δn(z)进行傅里叶变换,可得其解析型的实部γ和虚部η为
式中:σ=πL(2nm/λ-1/Λ)为失谐量;sincx=
,又称辛格函数。将(7)式代入(5)式可得到均匀FBG的相位谱为
图 2. 计算所得的均匀FBG相位谱。(a) 解析解; (b) 耦合模理论
Fig. 2. Calculated phase spectra of uniform FBG. (a) Analytic solution; (b) coupled-mode theory
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根据FBG的长度L、微扰幅值δ、周期Λ、有效折射率nm等,可直接按(8)式计算各波长处的相位,用该相位谱即可计算出相位微分值,再根据(6)式可计算出时延谱。设单模光纤FBG的结构参数为:L=4 mm、Λ=0.534 μm、nm=1.45131、δ=0.0001。在λ=1550 nm附近计算的基模耦合常数k=2530.62π N/s,波长计算范围为1548~1552 nm,波长分辨率为1 pm,用相位解析解(8)式计算所得的相位谱如图2(a)所示,用传统耦合模理论计算所得的相位谱如图2(b)所示。由图2可知,用相位解析解和传统耦合模理论计算的相位谱结果是一致的,可推断采用相位谱通解分析均匀WBG的相位是准确的。在方正S360R、Vista操作系统和Matlab7.1环境下,基于(8)式计算相位谱约需要0.5 s,具有线性复杂度和最小计算量,与用耦合模理论计算相位的时间相近。
下面计算并比较均匀FBG的时延谱。时延谱是用相位谱的微分值计算的,所以比较结果可同时验证相位谱和时延谱通解的正确性。FBG耦合带以外的光波只有极小或没有反射,用反正切函数计算相位将产生耦合带外时延谱的奇异点,而人们关注的只是耦合带以内的谱特性,故只需比较反射带以内的时延谱。设在单模光纤中均匀FBG的结构参数为:L=10 mm、Λ=0.525 μm、nm=1.4823、δ=4×10-5。在波长1556 nm附近计算的耦合常数为k=2520.54π N/s,波长计算范围为1554~1558 nm,波长间隔为2 pm,计算所得的反射率如图3(a)所示,耦合中心波长约为1556.4 nm;用相位解析解(8)式和时延通解(6)式计算所得的相位谱和时延谱分别如图3(b)和(c)所示。与图3(a)有相近幅值谱、长度为10 mm的均匀FBG的时延谱测量结果如图4所示。在FBG所在位置与测量点之间的一段光纤导致了约2 ns的时延,所以FBG的时延谱是时延测量值相对于-2 ns的部分。比较图3(c)和图4可知,相对于时延-2 ns而言,在1556.33~1556.5 nm反射带内的时延测量值接近于0且是慢变的,而时延的计算值也基本上接近于0且是慢变的。因此,反射带内时延谱的计算值与其实际测量结果是一致的。
图 3. 用解析解计算均匀FBG的谱特性。(a)反射率; (b)相位谱; (c)时延谱
Fig. 3. Calculated spectral properties of uniform FBG with analytic solutions. (a) Reflectivity; (b) phase spectrum; (c) delay spectrum
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图 4. 均匀FBG相对时延谱的测量值
Fig. 4. Measured relative delay spectrum of uniform FBG
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3.2 B类WBG的时延谱
B类WBG有半解析型的时延谱解,其时延谱的验证过程与A类WBG的过程相似,不同点在于:1)用DFT计算δn(z)的傅里叶变换;2)以类频率v为变量计算相位谱及时延谱τ(v),再将τ(v)转换为基于波长的时延谱τ(λ)。线性啁啾FBG(CFBG)具有微扰周期随位置线性变化的特点,其时延随波长线性变化,CFBG的时延谱特性最有代表性,在光纤色散补偿中应用较多。下面以CFBG为例,验证B类光栅时延谱的分析方法的正确性。CFBG的折射率微扰可表示为δn(z)=δ,式中F为光栅周期的啁啾量。设在单模光纤的nm=1.4774,CFBG的参数为:折射率微扰量5×10-5、初始周期Λ=0.5267 μm、啁啾量2.45×10-4、长度108 mm。在波长1556 nm附近计算的耦合常数为2520.86π N/s,离散傅里叶变换的计算点数为222,离散化δn(z)的采样间隔是初始周期的1/9,波长分辨率为3.34 pm,计算复杂度为O(Nlb N)(N为计算点数)。该CFBG的反射率和利用(6)式计算所得的时延谱分别如图5(a)和(b)所示,在反射带内的时延差为1.065 ns,与其理论时延差1.0637 ns基本一致。在方正S360R、Vista操作系统和Matlab7.1环境下,时延谱的平均计算时间约为4.17 s,小于传输矩阵法和Riccati方程数值法的时间(约12 s)。长度为108 mm、具有相近反射率分布的CFBG的时延谱的测量结果如图5(c)所示,反射带内的时延差约为1.06 ns。图4和图5(c)所示的时延谱都是用射频调制方法测量得到的[20]。从图5(b)和(c)的时延谱可知,用半解析通解计算所得的时延谱与实测的时延谱在时延差、带宽、有效区域、时延随波长的变化率及趋势等方面都是精确一致的。这证明了用时延谱的半解析通解分析B类光栅的结果也是正确的。
图 5. CFBG的谱特性。(a) 反射率; (b) 计算时延谱; (c) 测量时延谱
Fig. 5. Spectral properties of CFBG. (a) Reflectivity; (b) calculated delay spectrum; (c) measured delay spectrum
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4 分析讨论
现有的相位或时延计算方法都采用了反正切函数,本文的时延谱通解也采用了反正切函数。这将使计算的相位在±π/2附近产生不连续的突变,导致时延谱计算值存在奇异点。分析相位谱及时延谱的通解和仿真数据可知,产生该现象的原因是:当折射率微扰经傅里叶变换后的虚部η从+0变到-0或从-0变到+0时,计算所得的相位值将从π/2变为-π/2或从-π/2变为π/2,产生了π相位突变量,故由相位微分值计算的时延谱将出现脉冲式突变的奇异值。该相位突变量和时延奇异值是由反正切函数引起的虚假量值,需要被消除以保证时延谱的计算值与实测值是一致的。在图3(c)和图4所示的均匀光纤光栅时延谱中,其计算值与实测值在反射带以内是一致的,但在1556~1556.32 nm的反射带以外存在一些差异。产生该差异的原因有:1)基于时延谱通解的时延值在反射带内外都是可计算的,在反射带以外的傅里叶变换虚部η为0或为在0附近波动的极小值。在0附近存在大量的正负符号变化,从而产生大量的相位突变和时延奇异值。2)在测量时延的射频调制法中,可调谐单色光经射频信号调制和WBG反射后,再与射频信号比相从而测得时延量。比相也会产生相位突变和时延奇异值,而且WBG在反射带以外将无反射或仅有极其微弱的反射,产生近似于噪声的极弱信号而非有效的比相信号。3)反射带以外的光波经比相所得的时延值将存在很大的比相(测量)误差和伪奇异值,该测量误差和伪奇异值可使不同波长段的光波的时延有差异或有不同的奇异值,时延谱的计算值和实测值也有不同的误差源及奇异源。上述因素共同导致了反射带外的时延谱计算值与测量值之间的差异。这也表明,在反射带以外的WBG时延谱具有较低的实用价值。另外,计算和测量的相位谱对计算误差及测量噪声是极其敏感的,很小的波长间隔在微分相位时也会放大误差和噪声。因此,在实际分析计算时,需要选择合适的波长分辨率并减小误差及噪声。
根据折射率微扰的傅里叶变换是否具有全解析型结果,把WBG分为了A类和B类光栅。根据相位及时延谱的通解可知,A类WBG具有全解析型的傅里叶变换和相位谱解。基于全解析型相位解的相位及时延谱计算具有简单、直观、快速、线性复杂度(最小计算量)低等优点,其计算精度、分辨率及效率与傅里叶变换无关。B类光栅没有解析型的傅里叶变换结果,只有基于DFT的相位谱解。用该相位谱解分析计算相位和时延谱时,其分析特性由DFT的精度、分辨率和复杂度决定,其计算量至少是A类光栅计算量的lb N倍。DFT中的N值一般都较大,分析B类光栅的计算量会远大于A类光栅的计算量,且不够简单直观。A类和B类光栅可有不同形式的相位谱及时延谱解,可采用不同的分析方法,具有不同的分析性能。当然,DFT可计算任意折射率微扰的傅里叶变换,适用于所有类型的WBG。所以,B类光栅的时延谱解和分析方法是通用的,也适用于A类光栅。尽管如此,我们仍然把A类光栅分类出来,并求解出其独立形式的时延谱解。这样可以充分体现全解析与半解析解的差异性,有助于认识和利用A类光栅全解析解的优越性,更好地利用时延谱解去分析和设计A类光栅。另外,WBG时延谱对角频率的变化率决定了其色散特性,所建立的相位或时延谱通解既可用于分析相位和时延谱,又可用于分析色散特性。
5 结论
用FMC理论求解WBG的相位谱,基于该相位谱建立了WBG时延响应的半解析型通解模型。该半解析型时延谱通解具有很强的通用性,适用于各种类型的WBG或任意复杂度的折射率微扰分布。对于均匀、相移、超结构、摩尔、高斯和余弦类变迹等A类WBG,可得到全解析型的相位谱解,用该解析型相位谱解可简单、快速地计算其时延谱,所得的时延谱的计算具有线性复杂度(最小计算量)。对于B类光栅(包括啁啾光栅、复杂折射率微扰光栅、难以用数学函数表示微扰的光栅、有随机性相变或折变的光栅等),虽然其没有解析型的相位谱解,但可以有基于折射率微扰DFT的相位谱解。利用该相位谱也能容易地计算其时延谱,所得的时延谱的计算具有O(Nlb N)的复杂度。为了验证半解析型时延(或相位)谱通解的正确性和分析效率,用该时延谱通解计算了均匀FBG的相位和时延谱,并与基于耦合模理论的相位谱和实际测量的时延谱进行了对比分析。同时,计算了最有代表性的CFBG的反射率和时延谱,并和具有相近反射率分布的CFBG的实测时延谱进行了比较分析,同时对比了二者计算所需时间。分析结果表明,所建立的时延谱通解能快速、准确地分析各类WBG或任意折射率微扰的时延谱,间接表明了分析相位谱的正确性;对A类和B类光栅的分析复杂度分别是O(N)和O(Nlb N),分析效率均优于现有的分析方法,而且A类光栅的分辨率及效率与傅里叶变换无关。所建立的半解析型时延谱通解更新了对波导光栅响应谱解的传统认知,为分析、设计和应用WBG提供了基础理论和分析方法,在基于WBG时延特性的更高性能应用中可发挥重要作用。
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