矢量贝塞尔涡旋波束在均匀单轴各向异性介质中的传输

刘佳伟; 李海英; 吴振森; 白璐; 李正军

doi:doi:10.3788/AOS201838.0226001

光学学报, 2018, 38 (2): 0226001, 网络出版: 2018-08-30

矢量贝塞尔涡旋波束在均匀单轴各向异性介质中的传输下载： 707次

Propagation of Vector Bessel Vortex Beams in Homogeneous Uniaxial Anisotropic Media

论文大纲

刘佳伟李海英 ^*吴振森白璐李正军

作者单位

西安电子科技大学物理与光电工程学院, 陕西西安 710071

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摘要

基于电磁场的矢量波函数展开方法和不同坐标系中矢量波函数转化关系,得到了圆对称矢量贝塞尔涡旋波束的圆柱矢量波函数展开系数;结合电磁场边界条件,获得了圆对称矢量贝塞尔涡旋波束以任意方向照射均匀单轴各向异性介质层的反射场、透射场和内部场的展开系数;数值计算了沿波束传播方向横截面上的入射、反射和透射电场强度分布以及xOz平面内传播路径方向的总电场强度分布。结果表明,圆对称贝塞尔涡旋波束入射单轴各向异性介质的反射场基本保持同心圆环结构,但强度分布不再呈圆对称;透射场出现两束交错折射光,总电场强度轮廓显著扭曲。

Abstract

We obtain vector wave function expansion coefficients of the circularly-symmetric vector Bessel vortex beam based on the expansion of electromagnetic field in terms of vector wave function and the transformation of vector wave functions in different coordinate systems. Then, we obtain the expansion coefficients of reflection field, transmission field, and internal field of homogeneous uniaxial anisotropic media, which is illuminated by a circularly-symmetric vector Bessel vortex beam in any direction under the electromagnetic field boundary conditions. At last, we simulate the electric field distributions of reflection field, transmission field, and internal field in the cross sections and the total electric field distribution in xOz plane. The simulated results show that for a circularly-symmetric Bessel vortex beam incidence to uniaxial anisotropic media, the reflection field contour keeps a concentric ring structure, whereas the reflection field intensity distribution is asymmetric; there are two refracted rays in the transmission field, which brings about the distortion of the electric field intensity contour.

1 引言

携带轨道角动量(OAM)的有形波束在光学操控、光学通信和光学探测等领域具有很大的应用潜力。自实验证实拉盖尔-高斯波束具有OAM后^[1],这类有形波束的产生方法、传输特性及其应用研究备受学者们的关注^[2-4]。高阶贝塞尔波束作为一种涡旋波束,其性质也逐渐引起人们的关注。在应用方面,Qian等^[5]实现了高阶贝塞尔波束与有机半导体微型激光器的结合,使激光器的抽运阈值小于100 μJ/cm²;朱清智等^[6]提出轴棱锥聚焦贝塞尔-谢尔模型光束产生空心光束的方法,有望应用于光镊等实际应用。在特性分析方面,Volke-Sepulveda等^[7]研究了高阶贝塞尔波束的OAM;Wang等^[8]给出了圆对称贝塞尔波束的多级展开系数; Porras等^[9]详细阐述了高阶贝塞尔波束的无衍射特性; Mitri等^[10]模拟了高阶贝塞尔波束对磁电瑞利体的牵引力。随着各向异性介质,尤其是单轴各向异性介质^[11]在光学信号处理、光纤设计中得到越来越多的应用^[12-13],有形波束与这类介质的相互作用成为新的研究热点。但是矢量涡旋波束电磁场空间振幅和相位比较复杂,且光在单轴各向异性介质中产生双折射,从而使关于矢量涡旋波束在单轴各向异性介质中传输特性的研究还存在一定的难度。

电磁场的矢量波函数展开方法为研究有形波束与各向异性介质的相互作用提供了新渠道。耿友林等^[14]将球矢量波函数应用到单轴各向异性球体中研究平面波的电磁散射特性,并与传统洛伦兹-米氏理论方法作比较,证实了该方法的可行性。随后,Wu等^[15-16]将这种方法推广到高斯波束、零阶贝塞尔波束,并对单轴各向异性球的散射求解,该方法也同样适用于手征球体和其他各向异性介质体的散射^[17-18]。结合不同坐标系下矢量波函数之间的转换关系,分层介质中的电磁波传播也可以用矢量波函数展开方法求解,如Wang等^[19-20]研究了单轴晶体对高斯波束的反射与透射,Yan等^[21]利用该方法分析了高斯波束在手征介质板中的传播特性。

本文将电磁场的矢量波函数展开方法推广到矢量涡旋波束的传输中,研究了高阶矢量贝塞尔涡旋波束斜入射单轴各向异性介质层的反射与透射特性。通过球矢量波函数与圆柱矢量波函数的转换关系,结合电磁场的边界条件,导出各个区域高阶贝塞尔波束电磁场的圆柱矢量波函数的展开系数,对高阶贝塞尔波束斜入射到均匀单轴各向异性介质层的入射场、反射场和透射场的横截面场强分布,以及xOz平面内传输路径电场强度空间分布进行了仿真。

2 基本原理

考虑圆对称贝塞尔涡旋波束斜入射放置于空气中的均匀单轴各向异性介质层,建立如图1所示Oxyz全局坐标系。x轴与介质层上边界重合,入射方向与z轴夹角为β,介质层厚度为d,介质的光轴沿z轴方向。入射坐标系Ox_Iy_Iz_I、反射坐标系Ox_Ry_Rz_R的原点与全局坐标系的原点重合,各个局部坐标系的z_p(p=I,R,T)轴正方向与波束中心轴方向平行。本文中,时间因子取exp(-iωt)。

图 1. 矢量贝塞尔涡旋波束斜入射单轴各向异性介质层示意图

Fig. 1. Vector Bessel vortex beam oblique incidence to uniaxial anisotropic media

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2.1 正入射圆对称矢量贝塞尔涡旋波束的球矢量波函数展开

根据赫兹矢势可知,在局部坐标系Ox_Iy_Iz_I中,圆对称矢量贝塞尔涡旋波束的电场分量表示为^[22]

\begin{matrix} \begin{matrix} E_{xI}^{(1, i)} = \frac{1}{2} E_{0} {(- i)}^{l} \exp (il φ_{I}) \exp (i k_{zI} z_{I}) [\frac{(k + k_{zI})^{2}}{2} J_{l} (k_{τI} ρ_{I}) + \frac{(x_{I} + i y_{I})^{2}}{2 ρ_{I}^{2}} {k^{2}}_{τI} J_{l + 2} (k_{τI} ρ_{I})], (1) \\ E_{yI}^{(1, i)} = \frac{1}{2} E_{0} {(- i)}^{l} \exp (il φ_{I}) \exp (i k_{zI} z_{I}) [i \frac{(k + k_{zI})^{2}}{2} J_{l} (k_{τI} ρ_{I}) - i \frac{(x_{I} + i y_{I})^{2}}{2 ρ_{I}^{2}} {k^{2}}_{τI} J_{l + 2} (k_{τI} ρ_{I})], (2) \\ E_{zI}^{(1, i)} = \frac{1}{2} E_{0} {(- i)}^{l} \exp (il φ_{I}) e x p (i k_{zI} z_{I}) [\frac{i (x_{I} - y_{I})}{ρ_{I}} k_{τI} (k + k_{zI}) J_{l + 1} (k_{τI} ρ_{I})], (3) \end{matrix} \end{matrix}

式中i为虚数单位,E₀表示电场振幅,k为空气中入射波波数,l为贝塞尔涡旋波束的阶数,(1,i)代表右旋圆极化,J(·)为柱贝塞尔函数。ρ_I= $\begin{matrix} \sqrt[]{x_{I}^{2} + y_{I}^{2}} \end{matrix}$ ,φ_I=tan(y_I/x_I),k_τ_I=ksinα₀,k_z=kcosα₀,其中α₀为波束的半锥角。根据Maxwell方程组易得相应的磁场分量表达式。

利用球矢量波函数,矢量贝塞尔涡旋波束电磁场展开为^[22]

\begin{matrix} \begin{matrix} E = & E_{0} \overset{\infty}{\sum_{n = 1}} \overset{n}{\sum_{m = - n}} [a_{mn} M_{mn}^{(1)} (kr, θ ″, φ ″) + b_{mn} N_{mn}^{(1)} (kr, θ ″, φ ″)], (4) \\ H = & E_{0} \frac{k}{iω μ_{0}} \overset{\infty}{\sum_{n = 1}} \overset{n}{\sum_{m = - n}} [a_{mn} N_{mn}^{(1)} (kr, θ ″, φ ″) + b_{mn} M_{mn}^{(1)} (kr, θ ″, φ ″)], (5) \end{matrix} \end{matrix}

式中r、θ″、φ″为对应球坐标系的半径、天顶角与方位角,下文中的(r,θ,φ)同理; $\begin{matrix} M_{mn}^{(1)} \end{matrix}$ 、 $\begin{matrix} N_{mn}^{(1)} \end{matrix}$ 为平行于直角坐标系的球矢量波函数,上标(1)表示该球矢量波函数里所包含的贝塞尔函数为第一类球贝塞尔函数,下标m、n为电磁场用球矢量波函数展开时用来表示各通项的变量,两者取值范围如(4)式所示;kr为波束与半径的乘积,a_mn、b_mn为贝塞尔涡旋波束的展开系数;ω为波束角频率;μ₀为空气中的磁导率。为描述方便,将两者默认为左旋圆极化圆对称情况下的贝塞尔涡旋波束展开系数。基于x极化、y极化以及右旋圆极化波束之间的关系^[23],a_mn、b_mn可表示为

\begin{matrix} \begin{matrix} a_{mn} = b_{mn} = - i^{n + 1} \frac{\sqrt[]{2} (2 n + 1) (n - m)!}{n (n + 1) (n + m)!} \times \exp (- i k_{z} z_{0}) π \times {i^{l + 1 - m} J_{l + 1 - m} (ρ_{0}) \times \\ [π_{mn} (\cos α_{0}) + τ_{mn} (\cos α_{0})] \times \exp [i (l + 1 - m) φ_{0}]}, (6) \end{matrix} \end{matrix}

式中φ₀=arctan(y₀/x₀),ρ₀= $\begin{matrix} \sqrt[]{x_{0}^{2} + y_{0}^{2}} \end{matrix}$ 。(x₀,y₀,z₀)为波束中心点即入射坐标系的原点在全局坐标系Oxyz中的位置,角函数π_mn与τ_mn的表达式为π_mn(cosα₀)=m $\begin{matrix} P_{n}^{m} \end{matrix}$ (cosα₀)/sinα₀,τ_mn(cosα₀)= $\begin{matrix} d P_{n}^{m} \end{matrix}$ (cosα₀)/dα₀,其中 $\begin{matrix} P_{n}^{m} \end{matrix}$ (·)为连带勒让德多项式。

2.2 斜入射圆对称矢量贝塞尔涡旋波束的球矢量波函数展开

为了研究波束以任意角度斜入射到介质层的电磁场特性,用任意直角坐标系下的球矢量波函数来表示圆对称矢量贝塞尔涡旋波束的电磁场分量。

图2给出了坐标系Ox″y″z″与Oxyz的转换关系,两者可通过欧拉角相互转换^[24]。为简化数值运算,考虑α、γ均为零,β不为零的情况。需要注意的是,图2所示的Oxyz坐标系即为图1中的全局坐标系,为描述方便,入射角β取负值。

图 2. 任意直角坐标系之间的旋转关系示意图

Fig. 2. Rotation relationship between arbitrary rectangular coordinate systems

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不同直角坐标系中球矢量波函数关系式为^[25]

\begin{matrix} \begin{matrix} {(M, N)}_{mn}^{(1)} (kr, θ ″, φ ″) = \overset{n}{\sum_{s = - n}} ψ (m, s, n) {(M, N)}_{sn}^{(1)} (kr, θ, φ), (7) \\ ψ (m, s, n) = {(- 1)}^{s + m} {[\frac{(n + m)! (n - s)!}{(n - m)! (n + s)!}]}^{\frac{1}{2}} ex p (- isβ) u_{sm}^{(n)} (β), (8) \\ u_{sm}^{(n)} (β) = {[\frac{(n + s)! (n - s)!}{(n + m)! (n - m)!}]}^{\frac{1}{2}} \times \\ \sum_{σ} \frac{{(- 1)}^{n - s - σ} (n + m)! (n - m)!}{(n - s - σ)! (m + s + σ)! (n - m - σ)! σ!} {(\cos \frac{β}{2})}^{2 σ + s + m} {(\sin \frac{β}{2})}^{2 n - 2 σ - s - m}, (9) \end{matrix} \end{matrix}

式中s为(7)式中球矢量波函数各通项的变量,σ的取值范围由阶乘函数自变量的非负性来决定。

将(7)式代入(4)式并交换m、s的求和顺序,得到斜入射下圆对称矢量贝塞尔涡旋波束的球矢量波函数展开式为

\begin{matrix} E^{I} = E_{0} \overset{\infty}{\sum_{n = 1}} \overset{n}{\sum_{m = - n}} [a'_{mn} M_{mn}^{(1)} (kr, θ, φ) + b'_{mn} N_{mn}^{(1)} (kr, θ, φ)], (10) \end{matrix}

式中(a'_mn,b'_mn)= $\begin{matrix} \overset{n}{\sum_{s = - n}} \end{matrix}$ ψ(s,m,n)(a_sn,b_sn)。

2.3 斜入射圆对称矢量贝塞尔涡旋波束的圆柱矢量波函数展开

根据球矢量波函数和圆柱矢量波函数的转化关系,可以得到圆对称矢量贝塞尔涡旋波束在任意直角坐标系下圆柱矢量波函数的展开表达式为

\begin{matrix} \begin{matrix} E^{I} = & E_{0} \overset{\infty}{\sum_{m = - \infty}} {\int_{π}}_{0} [A_{m} (ζ) m_{mχ}^{(1)} (h) + B_{m} (ζ) n_{mχ}^{(1)} (h)] \exp (i h z) dζ, (11) \\ H^{I} = & \frac{k}{iω μ_{0}} E_{0} \overset{\infty}{\sum_{m = - \infty}} {\int_{π}}_{0} [A_{m} (ζ) n_{mχ}^{(1)} (h) + B_{m} (ζ) m_{mχ}^{(1)} (h)] \exp (i h z) dζ, (12) \end{matrix} \end{matrix}

式中χ=ksinζ,h=kcosζ; $\begin{matrix} m_{mχ}^{(1)} \end{matrix}$ 、 $\begin{matrix} n_{mχ}^{(1)} \end{matrix}$ 是圆柱矢量波函数,上标(1)表示该函数内的贝塞尔函数为第一类柱贝塞尔函数。k=χ $\begin{matrix} \dot{ρ} \end{matrix}$ +h $\begin{matrix} \dot{z} \end{matrix}$ , $\begin{matrix} \dot{ρ} \end{matrix}$ 与 $\begin{matrix} \dot{z} \end{matrix}$ 为全局坐标系Oxyz所对应圆柱坐标系半径和z轴的基矢。ζ为k与z轴正方向的夹角。展开系数A_m(ζ)和B_m(ζ)分别为

\begin{matrix} \begin{matrix} A_{m} (ζ) = \overset{\infty}{\sum_{n = | m |}} [a'_{mn} c_{mn} (ζ) + b'_{mn} d_{mn} (ζ)], (13) \\ B_{m} (ζ) = \overset{\infty}{\sum_{n = | m |}} [a'_{mn} d_{mn} (ζ) + b'_{mn} c_{mn} (ζ)], (14) \end{matrix} \end{matrix}

式中c_mn= $\begin{matrix} \frac{i^{m - n - 1}}{2 k} \end{matrix}$ τ_mn(cosζ),d_mn= $\begin{matrix} \frac{i^{m - n - 1}}{2 k} \end{matrix}$ π_mn(cosζ)。

2.4 入射场、反射场、透射场及内部场的圆柱矢量波函数展开系数

在自由空间内,(11)式中的入射电场E^I可以分解为两部分,即E^I= $\begin{matrix} {E^{I}}_{1} \end{matrix}$ + $\begin{matrix} {E^{I}}_{2} \end{matrix}$ ,其中 $\begin{matrix} {E^{I}}_{1} \end{matrix}$ 的表达式为

\begin{matrix} \begin{matrix} {E^{I}}_{1} = & E_{0} \overset{\infty}{\sum_{m = - \infty}} \int_{0}^{\frac{π}{2}} A_{m} (ζ) m_{mχ}^{(1)} (h) + B_{m} (ζ) n_{mχ}^{(1)} (h)] \exp (i h z) dζ 。 (15) \end{matrix} \end{matrix}

$\begin{matrix} {E^{I}}_{2} \end{matrix}$ 的表达式与(15)式类似,ζ的积分区间为[π/2,π]。实际入射时,只有 $\begin{matrix} {E^{I}}_{1} \end{matrix}$ 入射在介质上表面。

类似地,反射场、透射场的电场展开式为

\begin{matrix} \begin{matrix} E^{R} = & E_{0} \overset{\infty}{\sum_{m = - \infty}} \int_{0}^{\frac{π}{2}} C_{m} (ζ) m_{mχ}^{(1)} (- h) + D_{m} (ζ) n_{mχ}^{(1)} (- h)] \exp (- i h z) dζ, (16) \\ E^{T} = & E_{0} \overset{\infty}{\sum_{m = - \infty}} \int_{0}^{\frac{π}{2}} G_{m} (ζ) m_{mχ}^{(1)} (h) + I_{m} (ζ) n_{mχ}^{(1)} (h)] \exp (i h z) dζ, (17) \end{matrix} \end{matrix}

式中C_m(ζ)、D_m(ζ)和G_m(ζ)、I_m(ζ)分别为反射场和透射场的展开系数。已知单轴各向异性介质的介电常数张量ε为

\begin{matrix} ε = [\begin{matrix} ε_{t} & 0 & 0 \\ 0 & ε_{t} & 0 \\ 0 & 0 & ε_{z} \end{matrix}], \end{matrix}

介质的主光轴沿着z轴方向,ε_t为介质沿垂直于主光轴方向的介电常数,ε_z为介质沿主光轴方向的介电常数。介质内部区域的电磁场为^[26]

\begin{matrix} \begin{matrix} {E^{W}}_{1} = E_{0} \overset{2}{\sum_{q = 1}} \overset{\infty}{\sum_{m = - \infty}} \int_{0}^{\frac{π}{2}} E_{mq} (ζ) [α_{q} (ζ) m_{mχ}^{(1)} (h_{q}) + β_{q} (ζ) n_{mχ}^{(1)} (h_{q}) + γ_{q} (ζ) l_{mχ}^{(1)} (h_{q})] \exp (i h_{q} z) dζ, (18) \\ {E^{W}}_{2} = E_{0} \overset{2}{\sum_{q = 1}} \overset{\infty}{\sum_{m = - \infty}} \int_{0}^{\frac{π}{2}} F_{mq} (ζ) [α_{q} (ζ) m_{mχ}^{(1)} (- h_{q}) + β_{q} (ζ) n_{mχ}^{(1)} (- h_{q}) - γ_{q} (ζ) l_{mχ}^{(1)} (- h_{q})] \exp (- i h_{q} z) dζ, (19) \end{matrix} \end{matrix}

式中 $\begin{matrix} a_{1}^{2} \end{matrix}$ =ω²ε_tμ₀, $\begin{matrix} a_{2}^{2} \end{matrix}$ =ω²ε_zμ₀;h₁= $\begin{matrix} \sqrt[]{a_{1}^{2} - χ^{2}} \end{matrix}$ ,h₂= $\begin{matrix} \frac{a_{1}}{a_{2}} \end{matrix} \begin{matrix} \sqrt[]{a_{2}^{2} - χ^{2}} \end{matrix}$ ;k₁=a₁,k₂= $\begin{matrix} \frac{1}{a_{2}} \end{matrix} \begin{matrix} \sqrt[]{a_{1}^{2} a_{2}^{2} - (a_{1}^{2} - a_{2}^{2}) χ^{2}} \end{matrix}$ ;α₁(ζ)=β₁(ζ)=γ₁(ζ)=α₂(ζ)=0;β₂(ζ)=-i $\begin{matrix} \frac{a_{2}^{3}}{χ} \end{matrix}$ × $\begin{matrix} \frac{1}{\sqrt[]{a_{1}^{2} a_{2}^{2} - (a_{1}^{2} - a_{2}^{2}) χ^{2}}} \end{matrix}$ ;γ₂(ζ)=- $\begin{matrix} \frac{(a_{1}^{2} - a_{2}^{2}) a_{2}}{a_{1}} \end{matrix}$ × $\begin{matrix} \frac{χ \sqrt[]{a_{2}^{2} - χ^{2}}}{a_{1}^{2} a_{2}^{2} - (a_{1}^{2} - a_{2}^{2}) χ^{2}} \end{matrix}$ 。E_mq、F_mq(q=1,2)分别为内部场 $\begin{matrix} {E^{W}}_{1} \end{matrix}$ 和 $\begin{matrix} {E^{W}}_{2} \end{matrix}$ 的展开系数。

根据电场的边界条件,在介质的上表面有 $\begin{matrix} {E^{I}}_{1 r} \end{matrix}$ + $\begin{matrix} {E^{R}}_{r} \end{matrix}$ = $\begin{matrix} {E^{W}}_{1 r} \end{matrix}$ + $\begin{matrix} {E^{W}}_{2 r} \end{matrix}$ , $\begin{matrix} {E^{I}}_{1 φ} \end{matrix}$ + $\begin{matrix} {E^{R}}_{φ} \end{matrix}$ = $\begin{matrix} {E^{W}}_{1 φ} \end{matrix}$ + $\begin{matrix} {E^{W}}_{2 φ} \end{matrix}$ ,在介质的下表面有 $\begin{matrix} {E^{W}}_{1 r} \end{matrix}$ + $\begin{matrix} {E^{W}}_{2 r} \end{matrix}$ = $\begin{matrix} {E^{T}}_{r} \end{matrix}$ , $\begin{matrix} {E^{W}}_{1 φ} \end{matrix}$ + $\begin{matrix} {E^{W}}_{2 φ} \end{matrix}$ = $\begin{matrix} {E^{T}}_{φ} \end{matrix}$ 。通过电场和磁场以及圆柱矢量波函数之间的转换关系,可以得到对应于电场边界条件的磁场边界条件。

将各场的展开式代入到边界条件中,可以得到展开系数C_m(ζ)、G_m(ζ)、E_m₁(ζ)、F_m₁(ζ)和D_m(ζ),I_m(ζ)、E_m₂(ζ)、F_m₂(ζ)其可用A_m(ζ)、B_m(ζ)表示为

\begin{matrix} \begin{matrix} C_{m} (ζ) = \{2 h \frac{(h_{1} - h) \exp [i (h + h_{1}) d] + (h + h_{1}) \exp [i (h - h_{1}) d]}{(h + h_{1})^{2} \exp [i (h - h_{1}) d] - (h_{1} {- h)}^{2} \exp [i (h + h_{1}) d]} - 1\} A_{m} (ζ), (20) \\ G_{m} (ζ) = \frac{4 h h_{1}}{(h + h_{1})^{2} \exp [i (h - h_{1}) d] - (h_{1} {- h)}^{2} \exp [i (h + h_{1}) d]} A_{m} (ζ), (21) \\ E_{m 1} (ζ) = \frac{2 h (h + h_{1}) \exp [i (h - h_{1}) d]}{(h + h_{1})^{2} \exp [i (h - h_{1}) d] - (h_{1} {- h)}^{2} \exp [i (h + h_{1}) d]} A_{m} (ζ), (22) \\ F_{m 1} (ζ) = \frac{2 h (h_{1} - h) \exp [i (h + h_{1}) d]}{(h + h_{1})^{2} \exp [i (h - h_{1}) d] - (h_{1} {- h)}^{2} \exp [i (h + h_{1}) d]} A_{m} (ζ), (23) \\ \begin{matrix} D_{m} (ζ) = & \{2 Y \frac{(X + Y) \exp [i (h - h_{2}) d] + (X - Y) \exp [i (h + h_{2}) d]}{{(Y + X)}^{2} \exp [i (h - h_{2}) d] - {(X - Y)}^{2} \exp [i (h + h_{2}) d]} - 1\} B_{m} (ζ), (24) \\ I_{m} (ζ) = & \frac{4 XY}{{(Y + X)}^{2} \exp [i (h - h_{2}) d] - {(X - Y)}^{2} \exp [i (h + h_{2}) d]} B_{m} (ζ), (25) \\ E_{m 2} (ζ) = & \frac{2 h}{k_{0}} \frac{(X + Y) \exp [i (h - h_{2}) d]}{{(Y + X)}^{2} \exp [i (h - h_{2}) d] - {(X - Y)}^{2} \exp [i (h + h_{2}) d]} B_{m} (ζ), (26) \\ F_{m 2} (ζ) = & \frac{2 h}{k_{0}} \frac{(X - Y) \exp [i (h + h_{2}) d]}{{(Y + X)}^{2} \exp [i (h - h_{2}) d] - {(X - Y)}^{2} \exp [i (h + h_{2}) d]} B_{m} (ζ), (27) \end{matrix} \end{matrix} \end{matrix}

式中X=β₂(ζ) $\begin{matrix} \frac{h_{2}}{k_{2}} \end{matrix}$ -iγ₂(ζ),Y= $\begin{matrix} \frac{h k_{2}}{k_{0}^{2}} \end{matrix}$ β₂(ζ),将上述展开系数代入相应电场的展开式中可以求出对应区域的电场强度。

3 数值仿真结果分析

3.1 入射场、反射场和透射场的归一化场强分布

利用各个场的圆柱矢量波函数展开式数值计算入射场、反射场和透射场的归一化场强 $\begin{matrix} |E^{I} / E_{0}| \end{matrix}$ 、 $\begin{matrix} |E^{R} / E_{0}| \end{matrix}$ 、 $\begin{matrix} |E^{T} / E_{0}| \end{matrix}$ 分布,入射波束、反射波束在z=0介质上表面,透射波束在z=d介质下表面,三者沿着传播方向横截面上的归一化场强的分布情况如图3所示。其中,矢量贝塞尔涡旋波束的阶数l=4,半锥角α₀=20°,波长 λ=0.6328 μm;单轴各向异性介质层的厚度d=10λ,ε_t=3k²/(ω²μ₀),ε_z=2k²/(ω²μ₀)。入射角β=π/4。各区域中电场的圆柱矢量波函数展开式中m的取值范围为[-60,60]。

图 3. (a)入射场在z=0处、 (b)反射场在z=0处、 (c)透射场在z=d处的横截面的场强分布

Fig. 3. Normalized intensity distributions in cross section of (a) incident field at z=0, (b) reflection field at z=0, and (c) transmission field at z=d

下载图片查看所有图片

由图3(a)可知,圆柱矢量波函数展开方法能够很好地重现圆对称矢量贝塞尔涡旋波束的场强分布特征;图3(b)中反射场的场强分布出现了不对称现象,在x>0的区域尤为明显。这可能是由于入射波束在单轴各向异性介质中多次反射和透射叠加造成的;此外,由于单轴各向异性介质的双折射现象,在图3(c)所示的透射场中,两个光斑产生重合交错,使场强圆形轮廓发生明显扭曲,偏离同心圆环形状。

3.2 传输区域xOz平面内总电场场强分布

为了直观地观察高阶贝塞尔涡旋波束入射单轴各向异性介质的传输情况,数值模拟该波束的反射与透射过程中总电场强度在xOz平面内的分布,图4给出了入射场、反射场、内部场与透射场的归一化场强分布。参数设置与3.1节一致。

图 4. 高阶矢量贝塞尔涡旋波束在单轴各向异性介质中入射场、反射场、内部场与透射场的归一化场强分布

Fig. 4. Normalized intensity distribution of incident, reflection, internal, and transmission fields of high-order vector Bessel vortex beam incidence to uniaxial anisotropic media

下载图片查看所有图片

由图4可知,当主光轴垂直于介质表面的单轴各项异性介质板时,高阶矢量贝塞尔涡旋波束斜入射该介质板之后,折射光主要部分为寻常光,遵循折射定律,且沿着波束传播方向传播,xOz平面上的电场基本呈对称分布,但透射场和反射场强度分布不再呈圆对称分布。

4 结论

利用电磁场的球矢量波函数展开方法以及球矢量波函数与圆柱矢量波函数之间的转换关系,导出了矢量贝塞尔涡旋波束的圆柱矢量波函数展开系数,研究了单轴各向异性介质层中矢量贝塞尔涡旋波束的传输特性,数值计算了高阶贝塞尔波束在入射单轴各向异性介质时入射场、反射场及透射场中波束沿传播方向横截面上的归一化场强分布,并给出了该波束在xOz面内的场强分布。提供了一种研究高阶贝塞尔波束在单轴各向异性介质中传输特性的新思路,该思路同样也适用于手性介质和旋光各向异性等其他介质,这对研究高阶贝塞尔波束在介质传输过程中各类性质诸如能量、极化状态、相位分布以及轨道角动量的变化具有重要的意义,在光学信息、光学探测和利用涡旋波束作为信息载体的通信领域均具有广泛的应用前景。

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刘佳伟, 李海英, 吴振森, 白璐, 李正军. 矢量贝塞尔涡旋波束在均匀单轴各向异性介质中的传输[J]. 光学学报, 2018, 38(2): 0226001. Jiawei Liu, Haiying Li, Zhensen Wu, Lu Bai, Zhengjun Li. Propagation of Vector Bessel Vortex Beams in Homogeneous Uniaxial Anisotropic Media[J]. Acta Optica Sinica, 2018, 38(2): 0226001.

矢量贝塞尔涡旋波束在均匀单轴各向异性介质中的传输下载： 707次

1 引言

2 基本原理

图 1. 矢量贝塞尔涡旋波束斜入射单轴各向异性介质层示意图

Fig. 1. Vector Bessel vortex beam oblique incidence to uniaxial anisotropic media

图 2. 任意直角坐标系之间的旋转关系示意图

Fig. 2. Rotation relationship between arbitrary rectangular coordinate systems

3 数值仿真结果分析

图 3. (a)入射场在z=0处、 (b)反射场在z=0处、 (c)透射场在z=d处的横截面的场强分布

Fig. 3. Normalized intensity distributions in cross section of (a) incident field at z=0, (b) reflection field at z=0, and (c) transmission field at z=d

图 4. 高阶矢量贝塞尔涡旋波束在单轴各向异性介质中入射场、反射场、内部场与透射场的归一化场强分布

Fig. 4. Normalized intensity distribution of incident, reflection, internal, and transmission fields of high-order vector Bessel vortex beam incidence to uniaxial anisotropic media

4 结论

Article Outline

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矢量贝塞尔涡旋波束在均匀单轴各向异性介质中的传输 下载： 707次

1 引言

2 基本原理

图 1. 矢量贝塞尔涡旋波束斜入射单轴各向异性介质层示意图

Fig. 1. Vector Bessel vortex beam oblique incidence to uniaxial anisotropic media

图 2. 任意直角坐标系之间的旋转关系示意图

Fig. 2. Rotation relationship between arbitrary rectangular coordinate systems

3 数值仿真结果分析

图 3. (a)入射场在z=0处、 (b)反射场在z=0处、 (c)透射场在z=d处的横截面的场强分布

Fig. 3. Normalized intensity distributions in cross section of (a) incident field at z=0, (b) reflection field at z=0, and (c) transmission field at z=d

图 4. 高阶矢量贝塞尔涡旋波束在单轴各向异性介质中入射场、反射场、内部场与透射场的归一化场强分布

Fig. 4. Normalized intensity distribution of incident, reflection, internal, and transmission fields of high-order vector Bessel vortex beam incidence to uniaxial anisotropic media

4 结论

Article Outline

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